# LIQUID CRYSTALS

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					This is a new and greatly revised edition of Professor Chandrasekhar's
classic book Liquid crystals, first published in 1977. In this second edition
the author has brought the subject completely up to date.

The subject of liquid crystals has now grown to become an exciting
interdisciplinary field of research with important practical applications.
This book presents a systematic and self-contained treatment of the
physics of the different types of thermotropic liquid crystals - the three
classical types, nematic, cholesteric and smectic, composed of rod-shaped
molecules, and the newly discovered discotic type composed of disc-shaped
molecules. The coverage includes a description of the structures of these
four main types and their polymorphic modifications, their thermo-
dynamical, optical and mechanical properties and their behaviour under
external fields. The basic principles underlying the major applications of
liquid crystals in display technology (for example, the twisted and
supertwisted nematic devices, the surface stabilized ferroelectric device,
etc.) and in thermography are also discussed.

This book will be of great value to advanced students and research workers
in condensed matter physics, chemical physics, materials science and
technology with an interest in the physics, chemistry and applications of
liquid crystals.
LIQUID CRYSTALS
LIQUID CRYSTALS
Second edition
S. CHANDRASEKHAR, F.R.S.
Centre for Liquid Crystal Research, Bangalore

CAMBRIDGE
UNIVERSITY PRESS
The Pitt Building, Trumpington Street, Cambridge CB2 1RP
40 West 20th Street, New York, NY 10011-4211, USA
10 Stamford Road, Oakleigh, Victoria 3166, Australia

© Cambridge University Press 1977, 1992

First published 1977
Second edition 1992

A catalogue record for this book is available from the British Library

Library of Congress cataloguing in publication data
Chandrasekhar, S. (Sivaramakrishna), 1930—
Liquid crystals/S. Chandrasekhar. - 2nd ed.
p. cm.
Includes bibliographical references (p. ).
ISBN 0 521 41747 3 (hardback). - ISBN 0 521 42741 X (pbk.)
1. Liquid crystals. I. Title.
QD923.C46 1992
530.4'29 - dc20     91 38434 CIP

ISBN 0 521 41747 3 hardback
ISBN 0 521 42741 X paperback

Transferred to digital printing 2004

UP
Contents

Preface to the first edition                                     page   xiii
Preface to the second edition                                            xv
Introduction                                                      1
.1     Thermotropic liquid crystals: structure and classification
of the mesophases                                                1
.1.1  Liquid crystals of rod-like molecules                            1
.1.2 Liquid crystals of disc-like molecules                            8
.1.3 Polymer liquid crystals                                          10
.2    Lyotropic liquid crystals                                       12
.3    Polymorphism in thermotropic liquid crystals                    14
2       Statistical theories of nematic order                           17
2.1     Melting of molecular crystals: the Pople-Karasz model           17
2.1.1   Order-disorder in positions and orientations                    17
2.1.2 Plastic crystals and liquid crystals                              22
2.1.3 Pressure-induced mesomorphism                                     27
2.2     Phase transition in a fluid of hard rods                        29
2.3     The Maier-Saupe theory and its applications                     38
2.3.1 Definition of long-range orientational order                      38
2.3.2 The mean field approximation                                      41
2.3.3 Evaluation of the order parameter                                 43
2.3.4 Theory of dielectric anisotropy                                   51
2.3.5   Relationship between elasticity and orientational order         57
2.4     Hybrid models: hard rods with a superposed attractive
potential                                                        60
2.5     Short-range order effects in the isotropic phase                 61
2.5.1 The Landau-de Gennes model                                         61
2.5.2 Magnetic and electric birefringence                                63
2.5.3 Light scattering                                                   66

vii
viii                            Contents

2.5.4   Flow birefringence                                        69
2.5.5   Comparison with the Maier-Saupe theory                    70
2.6     Near-neighbour correlations: Bethe's method               71
2.6.1   The Krieger-James approximation                           71
2.6.2   Antiferroelectric short-range order                       75
2.7     The nematic liquid crystal free surface                   80
3       Continuum theory of the nematic state                     85
3.1     The Ericksen-Leslie theory                                85
3.1.1   Conservation laws and the entropy inequality              86
3.1.2   Constitutive equations                                    88
3.1.3   Coefficients of viscosity                                 91
3.1.4   Parodi's relation                                         93
3.2     Curvature elasticity: the Oseen-Zocher-Frank equations    94
3.3     Summary of equations of the continuum theory              97
3.4     Distortions due to magnetic and electric fields: static
theory                                                     98
3.4.1   The Freedericksz effect                                    98
3.4.2   The twisted nematic device                                106
3.4.3   The Freedericksz effect in highly anisotropic nematics:
periodic distortions                                      113
3.4.4   The constant k13                                          115
3.5     Disclinations                                             117
3.5.1   Schlieren textures                                        117
3.5.2   Interaction between disclinations                         120
3.5.3   Non-singular structures (s = ± 1): escape in the third
dimension                                                 123
3.5.4   Twist disclinations                                       126
3.5.5   Singular points                                           129
3.5.6   Interaction between disclinations and surfaces            132
3.5.7   Defects in the presence of an external field              135
3.5.8   Some consequences of elastic anisotropy                   139
3.5.9   The core structure                                        143
3.6     Flow properties                                           144
3.6.1   Miesowicz's experiment                                    144
3.6.2   Tsvetkov's experiment                                     145
3.6.3   Poiseuille                        flow                    148
3.6.4   Shear                           flow                      152
3.6.5   Transverse pressure and secondary              flow       157
3.7     Reflexion of shear waves                                  159
3.8     Dynamics of the Freedericksz effect                       161
Contents                              ix

3.8.1    Twist deformation                                           161
3.8.2    Homeotropic to planar transition: backflow and kickback
effects                                                     162
3.9      Light scattering                                            167
3.9.1    Orientational                     fluctuations              167
3.9.2    Intensity and angular dependence of the scattering          170
3.9.3    Eigenmodes and the frequency spectrum of the scattered
light                                                       173
3.10     Electrohydrodynamics                                        177
3.10.1   The experimental situation                                  177
3.10.2   Helfrich's theory                                           183
3.10.3   DC excitation                                               186
3.10.4   Square wave excitation                                      187
3.10.5   Sinusoidal excitation                                       192
3.11     Hydrodynamic instabilities                                  19 5
3.11.1   Homogeneous instability in shear                 flow       195
3.11.2   Roll instability in shear                   flow            197
3.12     Thermal instability: stationary convection                  202
3.13     Flexoelectricity                                            205
3.13.1   Determination of the flexoelectric coefficients             205
3.13.2   Influence of flexoelectricity on electrohy drody namic
instability                                                 210
3.13.3   Order electricity                                           211
4        Cholesteric liquid crystals                                 213
4.1      Optical properties                                          213
4.1.1    Propagation along the optic axis for wavelengths ^ pitch    213
4.1.2    Propagation along the optic axis for wavelengths ~pitch:
analogy with Darwin's dynamical theory of X-ray
diffraction                                                 222
4.1.3    Exact solution of the wave equation for propagation along
the optic axis: the Mauguin-Oseen-de Vries model            237
4.1.4    Equivalence of the continuum and the dynamical theories     241
4.1.5    Oblique incidence                                           245
4.1.6    Propagation normal to the optic axis                        247
4.2      Defects                                                     248
4.2.1    /-disclinations                                             248
4.2.2    Lattice disclinations                                       252
4.2.3    Dislocations                                                254
4.3      Leslie's theory of thermomechanical coupling                258
4.4      The Lehmann rotation phenomenon                             262
x                                Contents

4.5     Flow properties                                          267
4.5.1   Flow along the helical axis                              270
4.5.2   Flow normal to the helical axis                          274
4.6     Distortions of the structure by external fields          277
4.6.1   Magnetic field normal to the helical axis: the
cholesteric-nematic transition                           277
4.6.2   Magnetic field along the helical axis: the square grid
pattern                                                  281
4.6.3   Electric field along the helical axis                    286
4.7     Anomalous optical rotation in the isotropic phase        289
4.8     The blue phases                                          292
4.9     Some factors influencing the pitch                       296
4.10    Molecular models                                         298
5       Smectic liquid crystals                                  300
5.1     Classification of the smectic phases                     300
5.2     Extension of the Maier-Saupe theory to smectic A:
McMillan's model                                         302
5.3     Continuum theory of smectic A                            310
5.3.1   The basic equations                                      310
5.3.2   The Peierls-Landau instability                           311
5.3.3   The Helfrich deformation                                 313
5.3.4   Fluctuations and Rayleigh scattering                     317
5.3.5   Damping rate of the undulation mode                      320
5.3.6   Ultrasonic propagation and Brillouin scattering          323
5.3.7   Breakdown of conventional hydrodynamics                  325
5.4     Defects in smectic A                                     327
5.4.1   Focal conic textures                                     327
5.4.2   Edge dislocations                                        333
5.4.3   Screw dislocations                                       338
5.4.4   Disclinations                                            338
5.5     The smectic A-nematic transition                         340
5.5.1   Phenomenological theory of the smectic A-nematic
transition                                               340
5.5.2   Pretransition effects in the nematic phase               341
5.6     Smectic A polymorphism                                   350
5.6.1   Smectic A phases of strongly polar molecules             350
5.6.2   The phenomenon of re-entrance                            355
5.6.3   Smectic A* or the twist grain boundary phase             358
5.7     The hexatic phase                                        360
5.8     Smectic C                                                362
Contents                           xi
5.8.1  Description of the structure                             362
5.8.2  Continuum theory of smectic C                            365
5.8.3  Defects in smectic C                                     369
5.8.4  The smectic C-smectic A transition                       370
5.9    The nematic-smectic A-smectic C multicritical point      374
5.10   Ferroelectric liquid crystals                            378
5.10.1 The properties of smectic C*                             378
5.10.2 The smectic C*-smectic A transition                      383
5.10.3 Applications of ferroelectric liquid crystals            386
6       Discotic liquid crystals                                388
6.1    Description of the liquid crystalline structures         388
6.2    Extension of McMillan's model of smectic A to discotic
liquid crystals                                          394
6.3    The columnar liquid crystal: applications of the
continuum theory                                         398
6.3.1 Fluctuations                                              398
6.3.2 X-ray scattering                                          399
6.3.3 Light scattering                                          400
6.3.4 Mechanical instabilities                                  400
6.3.5 Acoustic wave propagation                                 401
6.4    Defects in the columnar liquid crystal                   403
6.4.1 Dislocations                                              404
6.4.2 Disclinations                                             409
6.4.3 Developable domains                                       410
6.5    The discotic nematic phase                               411
6.6    The biaxial nematic phase                                414
References                                                      418
Index                                                           451
Preface to the first edition

' I am at a loss to give a distinct idea of the nature of this liquid, and
cannot do so without many words.'
The Narrative of Arthur Gordon Pym of Nantucket,
Edgar Allan Poe

The foundations of the physics of liquid crystals were laid in the 1920s but,
surprisingly, interest in these substances died down almost completely
during the next three decades. The situation was summarized by F. C.
Frank in his opening remarks at a Discussion of the Faraday Society in
1958: 'After the Society's successful Discussion on liquid crystals in 1933,
too many people, perhaps, drew the conclusion that the major puzzles were
eliminated, and too few the equally valid conclusion that quantitative
experimental work on liquid crystals offers powerfully direct information
about molecular interactions in condensed phases.' In the last few years
there has been a resurgence of activity in this field, owing partly to the
realization that liquid crystals have important uses in display technology.
An exposition of the physics of liquid crystals involves many disciplines:
continuum mechanics, optics of anisotropic media, statistical physics,
crystallography etc. In covering such a wide field it is difficult to define
to present as far as possible a self-contained treatment of each of these
different aspects of the subject. Naturally, discussion of some topics has
had to be curtailed for reasons of space. For example, we have not dealt
with lyotropic systems, whose complex structures are only just beginning
to be elucidated; or the special applications of magnetic resonance
techniques, as these have been adequately reviewed elsewhere; or the very
recent results of neutron scattering experiments. The primary aim of this
monograph is to provide an insight into the variety of new phenomena
exhibited by these intermediate states of matter.

Xlll
xiv                      Preface to the first edition

This book would probably never have been completed without the
unstinting cooperation of my young colleagues in Bangalore. I am
particularly indebted to N. V. Madhusudana, G. S. Ranganath, R.
Shashidhar, U. D. Kini, R. Nityananda and T. G. Ramesh, whose notes
and critical comments were of inestimable value at every stage of the
writing, and to K. A. Suresh and B. R. Ratna for their assistance in
preparing the diagrams and the list of references. Finally, it is a pleasure to
express my thanks to Michael Woolfson for his encouraging interest and

Raman Research Institute                                   S. Chandrasekhar
August 1975
Preface to the second edition

Since the publication of the first edition 15 years ago, the subject of liquid
crystals has grown enormously to become a fascinating interdisciplinary
field of study. A variety of new thermotropic phases have been discovered,
including over a dozen different smectic modifications, discotics, biaxial
nematics, etc., which have opened up a veritable treasure-house for the
theoretical condensed matter physicist. On the technological side, the
advances have been no less spectacular: portable computers and hand-held
TV sets using liquid crystal display devices are being sold in large numbers,
and high-definition LCD-TV would seem to be just round the corner. The
aim of the present edition is to bring the coverage up to date. The chapters
dealing with the classical nematic, cholesteric and smectic types of liquid
crystals have been revised substantially and a new chapter has been
included on discotics. However, mainly for reasons of space, special topics
like the applications of magnetic resonance techniques, non-linear optical
properties, etc., have not been discussed here as these have been
comprehensively reviewed elsewhere.
It is my privilege to express my thanks to my young colleagues who kept
me alive to the subject: in particular to V. N. Raja, S. Krishna Prasad,
D. S. Shankar Rao, S. M. Khened and Geetha Nair for their invaluable
help throughout the preparation of this book, and to Sriram Ramaswamy
and U. D. Kini for their advice on certain theoretical points. I am indebted
to the Council of Scientific and Industrial Research, New Delhi, for a
Bhatnagar Fellowship which made it easier for me to undertake this task.

Centre for Liquid Crystal Research                         S. Chandrasekhar
March 1992

xv
1
Introduction

The term liquid crystal signifies a state of aggregation that is intermediate
between the crystalline solid and the amorphous liquid. As a rule, a
substance in this state is strongly anisotropic in some of its properties and
yet exhibits a certain degree of fluidity, which in some cases may be
comparable to that of an ordinary liquid. The first observations of liquid
crystalline or mesomorphic behaviour were made towards the end of the
last century by Reinitzer(1) and Lehmann.(2) Several thousands of organic
compounds are known now to form liquid crystals.(3) An essential
requirement for mesomorphism to occur is that the molecule must be
highly geometrically anisotropic in shape, like a rod or a disc. Depending
on the detailed molecular structure, the system may pass through one or
more mesophases before it is transformed into the isotropic liquid.
Transitions to these intermediate states may be brought about by purely
thermal processes {thermotropic mesomorphism) or by the influence of
solvents (lyotropic mesomorphism).

1.1 Thermotropic liquid crystals: structure and classification of the
mesophases
1.1 A Liquid crystals of rod-like molecules
The vast majority of thermotropic liquid crystals are composed of rod-like
molecules. Following the nomenclature proposed originally by Friedel,(4)
they are classified broadly into three types: nematic, cholesteric and
smectic.
The nematic liquid crystal has a high degree of long-range orientational
order of the molecules, but no long-range translational order (fig. 1.1.1).
Thus it differs from the isotropic liquid in that the molecules are

1
1. Introduction

Crystal

Nematic

Fig. 1.1.1. Schematic representation of molecular order in the crystalline, nematic
and isotropic phases.

spontaneously oriented with their long axes approximately parallel. The
preferred direction usually varies from point to point in the medium, but
a uniformly aligned specimen is optically uniaxial, positive and strongly
birefringent. The mesophase owes its fluidity to the ease with which the
molecules slide past one another while still retaining their parallelism.
X-ray studies (56) indicate that some nematics possess a lamellar type of
short-range order, i.e., they consist of clusters of molecules - called
cybotactic groups(7) - the molecular centres in each cluster arranged in
layers. In ordinary nematics the cybotactic groups, if they do exist, are
smaller than can be detected by X-ray methods (see figs. 1.1.2 and 1.1.3).
1.1 Thermotropic liquid crystals

(a)

(b)

Fig. 1.1.2. X-ray diffraction patterns from an 'ordinary' nematic liquid crystal:
4-n-propyloxy-benzylidene-4/-n-propylaniline: (a) an unaligned (though not quite
randomly oriented) sample; (b) a well aligned sample. (De Vries.(6))

A biaxial modification of the nematic has been discovered recently (see
§6.6).
The cholesteric mesophase is also a nematic type of liquid crystal except
1. Introduction

(a)

(b)

Fig. 1.1.3. X-ray diffraction patterns from a 'cybotactic' nematic: bis-(4'-n-
octyloxybenzal)-2-chloro-l,4-phenylenediamine: (a) an unaligned (though not
quite randomly oriented) sample; (b) a well aligned sample. (De Vries.(6))

that it is composed of optically active molecules. As a consequence the
structure acquires a spontaneous twist about an axis normal to the
preferred molecular directions (fig. 1.1.4). The twist may be right-handed
or left-handed depending on the molecular conformation. Optically
inactive molecules or racemic mixtures result in a helix of infinite pitch
which corresponds to the true nematic. The energy of twist forms only a
minute part (~10~ 5 ) of the total energy associated with the parallel
alignment of the molecules,(8) so much so that when a small quantity of a
1.1 Thermotropic liquid crystals

Fig. 1.1.4. The cholesteric liquid crystal: schematic representation of the helical
structure.

cholesteric substance,(4) or even a non-mesomorphic optically active
substance,(9) is added to a nematic the mixture adopts a helical con-
figuration. The spiral arrangement of the molecules in the cholesteric is
responsible for its unique optical properties, viz., selective reflexion of
circularly polarized light and a rotatory power about a thousand times
greater than that of an ordinary optically active substance. Cholesterics of
low pitch, less than about 5000 A, exhibit what are known as blue phases.
1. Introduction

(a)

Illllllllllllllllll
Illlllllllllllllll
Illllllllllllllllll
Fig. 1.1.5. Schematic representation of the molecular arrangement in (a) smectic A
and (b) smectic C.

These phases exist over a small temperature range (~ 1 °C) between the
liquid crystal phase and the isotropic liquid. Their structures will be
discussed in §4.8.
Smectic liquid crystals have stratified structures but a variety of
molecular arrangements are possible within each stratification. In smectic
A the molecules are upright in each layer with their centres irregularly
spaced in a 'liquid-like' fashion (fig. I A.5 (a)). The interlayer attractions
are weak as compared with the lateral forces between molecules and in
consequence the layers are able to slide over one another relatively easily.
Hence this mesophase has fluid properties, though it is very much more
viscous than the nematic. Smectic C is a tilted form of smectic A, i.e., the
molecules are inclined with respect to the layer normal (fig. 1.1.5(6)).
Several polymorphic forms of smectics A and C exist, as we shall see later.
Over a dozen other distinct smectic modifications have been identified/ 10 ' U)
Some of them (e.g., SB, S E , SG, SH, ST and SK) have three-dimensional long-
range positional order as in a crystal, though with weak interlayer forces
(and hence energetically weak interlayer ordering), while some others,
referred to as hexatic phases, have three-dimensional long-range 'bond-
orientational' order, but without any long-range positional order. (12) The
1.1 Thermotropic liquid crystals

Fig. 1.1.6. (a) Threads in a nematic liquid crystal. Crossed polarizers. Film
thickness ~ 100 jum. (b) Schlieren texture in a nematic film of thickness ~ 10 jum.
Crossed polarizers. (Sackmann and Demus. (14))
8                              1. Introduction

D phase has a cubic structure(13) and would appear to be an exception to
the rule that smectics have layered structures.
The energy required to deform a liquid crystal is so small that even the
slightest perturbation, caused say by a dust particle or a surface
inhomogeneity, can distort the structure quite profoundly. Thus when a
liquid crystal is taken between glass plates and examined under a polarizing
microscope one rarely sees the familiar interference figures expected from
the equilibrium structures shown in figs. 1.1.1,1.1.4 and 1.1.5. Instead, one
usually obtains a rather complex optical pattern. For example, a nematic
film shows a characteristic threaded texture from which this mesophase
derives its name (fig. 1.1.6) and a smectic A film a focal conic texture (fig.
1.1.7). These textures are useful in the optical identification of the
mesophases (1516) and their nature is generally well understood, as we shall
see in later chapters. 'Single crystal' films with the molecules aligned
perpendicular to the plates (homeotropic structure) or parallel to them
(homogeneous or planar structure) can be prepared by suitable prior
treatment of the glass surfaces.
From purely geometrical arguments, Herrmann (17) concluded that there
should be 18 distinct mesomorphic groups between the perfectly ordered
crystalline arrangement and the truly amorphous one. Examples of some
of these groups have been found in plant virus preparations (18) and in
surfactant-water compositions,(19) but it is not yet clear whether all of
them can give rise to energetically feasible configurations. Thus Friedel's
nomenclature offers a convenient basis for the classification of thermo-
tropic liquid crystals and is universally adopted at present. The term
calamitic has come into use in recent years to describe liquid crystals
composed of rod-like molecules and to distinguish them from discotic
systems, which will be discussed in the next section.

1.1.2 Liquid crystals of disc-like molecules
The first liquid crystals of disc-shaped molecules, now generally referred to
as discotic liquid crystals, were prepared and identified in 1977.(20) Since
then a large number of discotic compounds have been synthesized and a
variety of mesophases discovered.<21) Structurally, most of them fall into
two distinct categories, the columnar and the nematic. The columnar phase
in its simplest form consists of discs stacked one on top of the other
aperiodically to form liquid-like columns, the different columns con-
stituting a two-dimensional lattice (fig. 1.1.8 (a)). The structure is somewhat
similar to that of the hexagonal phase of soap-water and other lyotropic
1.1 Thermo tropic liquid crystals

Fig. 1.1.7. Focal conic textures in smectic A. (a) The polygonal texture. Crossed
polarizers. (Friedel.(4)) (b) Simple fan-shaped texture. Crossed polarizers. (Sack-
mann and Demus.(14))

systems (see fig. 1.2.2), but a number of variants of this structure have been
identified: hexagonal, rectangular, tilted, etc. The nematic phase has an
orientationally ordered arrangement of the discs without any long-range
translational order (fig. 1.1.8(6)). Unlike the classical nematic of rod-like
molecules, this phase is optically negative. A cholesteric (or twisted
nematic) phase has also been identified. A smectic-like lamellar phase has
been reported(22) but the disposition of the molecules in the layers has not
yet been resolved.
10                               1. Introduction

(a)

Fig. 1.1.8. The structure of (a) the columnar phase and (b) the nematic phase of
disc-like molecules.

(b)

Fig. 1.1.9. Polymers that exhibit liquid crystalline phases. The basic monomer units
are low molar mass rod-like or disc-like mesogens which may form part of the main
chain (a) or attached as side groups (b).

1.1.3 Polymer liquid crystals
The structures of polymers that form liquid crystals are illustrated
schematically in fig. 1.1.9. The basic monomer units are low molar mass
mesogens, rod-like or disc-like, which are attached to the polymer
backbone in the main chain itself (fig. 1.1.9 (a)), or as side groups (fig.
1.1.9(6)). The nature of the mesophase depends rather sensitively on the
1.1 Thermotropic liquid crystals                        11

(a)

to

Fig. 1.1.10. Mesophases of discotic polymers: (a) the hexagonal columnar phase.
The diagram illustrates intercolumnar binding as well as intracolumnar back-
folding of the main chain (Herrmann-Schonherr, Wendorff and Ringsdorf (24) );
(b) the' sanidic' nematic phase composed of boards stacked parallel to one another
(Herrmann-Schonherr et al.{25)); (c) the columnar nematic phase (Ringsdorf
et al (26) ).

backbone, the mesogenic unit and the spacers. With rod-shaped repeating
units, mesophases similar to the nematic, cholesteric and smectic types of
rod-like molecules are observed(23) whereas with disc-shaped repeating
units, some new kinds of mesophase structures have been found. For
12                             1. Introduction
example, a polyester with (disc-shaped) triphenylene as the repeating unit
in the main chain separated by flexible spacers forms a hexagonal columnar
structure (24) (fig. 1.1.10 (a)). On the other hand, rigid aromatic polyamides
and polyesters with disc-shaped units in the main chain exhibit a different
type of mesophase which has been described as a ' sanidic' (or board-like)
nematic,(25) the boards stacked parallel to one another as shown in fig.
1.1.10(6). The addition of electron acceptor molecules to discotic polymers
results in the formation of charge transfer complexes which stabilize - or in
certain non-mesomorphic materials, induce-mesophases. (26) A new type
of induced mesophase identified in such a system is the' columnar nematic'
illustrated in fig. 1.1.10(c).

1.2 Lyotropic liquid crystals
Lyotropic liquid crystals are made up of two or more components. (27)
Generally, one of the components is an amphiphile (containing a polar
head group attached to one or more long hydrocarbon chains) and another
is water. A familiar example of such a system is soap (sodium dodecyl
sulphate) in water. As the water content is increased several mesophases
are obtained. The types of molecular packing in these mesophases are
represented schematically in figs. 1.2.1 and 1.2.2, but several modifications
of these structures exist.(28)
In the lamellar or neat phase, water is sandwiched between the polar
or in a liquid-like configuration, are in a non-polar environment (fig. 1.2.1).
In the cubic or viscous isotropic phase the layers are bent to form spherical
units, the polar heads lying on the surface of the sphere and the
hydrocarbon chains filling up the inside. The spherical units form a body-
centred cubic arrangement, (29) water taking up the space between the units.
In the hexagonal or middle phase, the layers are rolled up into cylinders.
The cylindrical units of indefinite length are arranged parallel to one
another in a hexagonal array (fig. 1.2.2) rather like the columnar phase of
disco tics. A nematic type of ordering has also been observed in some soap
systems.(30) It is believed that there exists a cylindrical superstructure
similar to the hexagonal phase, but this does not appear to have been
proved conclusively.(31)
In hydrophobe-dominated compositions, such as aerosol OT-water
systems,(32) inverted middle and inverted viscous isotropic phases can occur
in which the tails point outward towards the hydrophobic medium while
water is trapped inside.(33)
1.2 Lyotropic liquid crystals                    13

Fig. 1.2.1. The lamellar or neat phase of soaps.

Fig. 1.2.2. The hexagonal or middle phase of soaps.

Two mesophases may coexist over small ranges of composition and
temperature. Phase diagrams have been constructed for a large number of
binary systems. Transitions may be brought about from any one of the
mesophases directly to the isotropic solution at appropriate temperatures.
Ternary and higher-component systems exhibit essentially the same types
of structures, the phase diagrams being of course much more complicated.
Cholesteric liquid crystals are formed by solutions of synthetic poly-
pep tides, e.g., poly-y-benzyl-L-glutamate, in organic liquids when the
concentration exceeds a certain critical value. (34)
14                              1. Introduction
Lyotropic liquid crystals occur abundantly in nature, being ubiquitous
in living systems.(35) Their structures are quite complex and are only just
beginning to be elucidated. However, in this monograph we shall be
confining our attention mainly to the physics of low molecular weight
thermotropic liquid crystals and do not propose to discuss polymer and
lyotropic systems in any further detail. In chapters 2-5, we deal with the
nematic, cholesteric and smectic mesophases of rod-like molecules and in
chapter 6 discotic systems.

1.3 Polymorphism in thermotropic liquid crystals
Extensive tables are now available listing practically all known mesogenic
compounds, their transition temperatures and latent heats. (336) We
therefore give just a few typical examples of rod-like mesogens to illustrate
the rich polymorphism exhibited by some of these thermotropic materials.
The latent heats (in kilojoules per mole) are also presented for those cases
for which data are available. Three nematogenic compounds (1.3.1)—(1.3.3)
are included in the list for a special reason: the first has a permanent electric
dipole moment inclined at a large angle (~ 60°) with respect to the long
molecular axis, the second a strong longitudinal dipole moment, while the
third is non-polar. This brings out the fact emphasized earlier that it is the
molecular shape that plays the major role in determining mesomorphic
behaviour. All the transitions listed here, except one, are enantiotropic, i.e.,
they take place reversibly on heating and cooling, though the reversal to
the solid phase is usually accompanied by supercooling. The one exception
is cholesteryl nonanoate (1.3.5) which shows a monotropic transition - the
smectic A phase occurs only on cooling. Examples of disc-like mesogens
will be given in chapter 6.
A considerable body of experimental data has been accumulated over
the years in regard to the relationship between mesomorphism and
chemical constitution, especially for rod-like molecules, but we shall not be
discussing this aspect of the problem. For a concise review of current
trends in the chemistry of mesogenic compounds, with emphasis on the
special requirements for technological applications, reference may be
made to a recent book edited by Gray.(37)
1.3 Polymorphism in thermotropic liquid crystals                                                15

4, 4'-Di-methoxyazoxy benzene (/?-azoxyanisole/ ^

CH.O                                                                               OCH,               (1.3.1)

...     118.2 °C                .   135.3 °C  .
solid -*           •         nematic -*        • isotropic
29.57 kJ                      0.57 kJ

4'-n-Pentyl-4-cyanobiphenyl'I (39)

(1.3.2)
111
\     /       /       \               / /

22.5 °C                      35 °r
solid   ^          •        nematic -*                •       isotropic
17.2 kJ                      o.8 kJ

7?-Quinquephenylv                }

401 °C                    445 °C                                                       (1.3.3)
solid -*          •         nematic -*       •                isotropic

4'-n-Octyl-4-cyanobiphenyl (39)

\
n       C8H17-                                                                                     (1.3.4)
t\
24 °C             +.  A     34 °C                      .    42.6 °C
solid -*              •     smectic A -*        •                nematic -*       •    isotropic
25.3 kJ                     0.13 kJ                           0.97 kJ
16                                           1. Introduction
Cholesteryl nonanoate (monotropic smectic A) (41>42)

O

CH 3 (CH 2 ) 7       C       O

78.6 °C              91 °C                           91.3 ° C
solid -•     • cholesteric -*            •    blue phase I -*          • blue phase II
0.017 kJ                        0.017 kJ

75.5 °C 0.25 kJ                                     91.45 °C 0.53 kJ

smectic A                                          isotropic         (1.3.5)

N-(4-n- Pentyloxybenzylidene)-4'- n- hexylaniline,(43)

C5HnO-                              =
-CH —                              -C6H13

36 °C                    38 4 °C               42.4 °C
solid -*       •       smectic G -*—:     »• smectic F -*       •        smectic B
15.2 kJ                    0.39 kJ              0.15 kJ

50 °C 25 kJ

isotropic ^ 7 2 - 8 c »    nematic               smectic A ~m——• smectic C                 (1.3.6)
0.66 kJ                 0.42 kJ                very
small
2
Statistical theories of nematic order

2.1 Melting of molecular crystals: the Pople-Karasz model
X-ray analyses of the crystal structures of typical nematogenic com-
pounds(13) have established that the long narrow molecules are more or
less parallel and interleave one another to form what was described by
Bernal and Crowfoot(1) as an 'imbricated' arrangement (fig. 2.1.1). The
transformation from the solid to the nematic phase is characterized by the
breakdown of the positional order of the molecules but not of the
orientational order. The mesophase is fluid and at the same time
anisotropic because of the facility with which the molecules slide over one
another while still preserving their parallelism. The degree of orientational
order in the liquid crystal drops gradually on heating, whereas certain
other thermodynamic properties, such as the specific heat, thermal
expansion and isothermal compressibility, increase rapidly as the tem-
perature approaches the nematic-isotropic point T^r At Tm a 'weak' first
order (discontinuous) phase transition takes place, accompanied by the
complete breakdown of the long-range orientational order. The changes of
entropy and volume attending this transition are typically only a few per
cent of the corresponding values for the solid-nematic transition.
In this section, we shall consider an elementary model(4"6) which
illustrates the principal mechanisms responsible for this unusual type of
melting phenomenon.

2.1.1 Order-disorder in positions and orientations
The simplest treatment of the melting of inert gas crystals is that due to
Lennard-Jones and Devonshire(7) (LJD) who regarded the mechanism of
fusion as a positional order-disorder phenomenon. They postulated that

17
18                 2. Statistical theories ofnematic order
cJ2

Fig. 2.1.1. Molecular arrangement in the crystalline phase of anisaldazine: (a)
projection of the structure along [010]; [b) projection along [001]. Crystals are
monoclinic with a= 17.46 A, c = 8.45 A, /? = 113° 48'. Space group Cc with
four molecules per unit cell. (After Galigne and Falgueirettes.(3))

the atoms may occupy sites on one of two equivalent interpenetrating
lattices, which we shall refer to as ^4-sites and 2?-sites. The lowest energy
configuration (corresponding to the state of perfect order) is that in which
all atoms occupy the same kind of sites, say the ^4-sites. With increasing
temperature, the number of atoms in the interstitial 2?-sites increases till a
critical stage is reached when there is complete collapse of the long-range
order and both kinds of sites become equally populated. The system in
which there is an equal number of occupied and unoccupied ^4-sites
corresponds to a liquid, for under such circumstances migration from one
site to another becomes easily possible. By assuming an appropriate
volume dependence for the energy of interaction between atoms in A and
2.1 Melting of molecular crystals                      19

Table 2.1.1. Melting entropy of molecular crystals(8) (in entropy units,
AS/R)

Inert gas crystals
Ne           3.26         Kr        3.36
A           3.35         Xe        3.40
Crystals in which the rotational transition precedes the melting transition
O2          2.0         PH 3       1.92
SiH4          1.8         CD 4       2.42
CF 4         2.0          CH 4       2.47
Crystals in which the two transitions coalesce
CO 2         9.25         SiCl4      9.06
N2O          8.58        HCN         7.73
C2H4          7.70         CS 2      6.51
SO2         8.95        (CN) 2      7.90

2?-sites, Lennard-Jones and Devonshire showed that the melting par-
ameters predicted by the theory agree well with the data for a number of
spherical or nearly spherical molecules.
However, the theory fails for anisotropic molecules where the effects of
orientational disorder become important. The thermodynamic data on
melting suggest that there are two classes of molecular crystals: those
which undergo phase transitions associated with rotational motions at
temperatures below the melting point and those in which the rotational
and melting transitions coalesce. The former have entropies of fusion
lower than the inert gas crystals, while the latter have much higher
entropies of fusion (table 2.1.1).
Pople and Karasz ( 4 ) proposed a simple extension of the L J D model
which at once provides an interpretation of these variations in the melting
properties. They assumed that the molecule can take u p one of two
orientations on any site, so that it now has four possibilities, Ax, A2, B1 and
B2. The state of perfect order (or the solid at zero temperature) may then
be regarded as one in which all molecules occupy sites and orientations of
the same type, say A19 and the state of complete disorder (or the liquid
phase) as one in which all four configurations are equally populated.
Clearly, there can also be states with positional order and no orientational
order and vice versa.
When a molecule is turned into an unfavourable orientation, local
strains are set up and consequently there is an increased tendency for the
molecules in the vicinity to move to interstitial sites. T o allow for this
20                 2. Statistical theories ofnematic order

coupling between orientational and positional ordering, Pople and Karasz
made the simple assumption that the orientational component of the AB
interactions is negligible compared with that of the A A or BB interactions,
so that the B-sites near a misorientation in an ^4-lattice are favoured.
Accordingly the energy required for a molecule to diffuse to an interstitial
5-site in an ^4-lattice is determined only by the AB interactions regardless
of orientation, and that required for a molecule to assume an A2-
orientation in an ^-lattice (or a i?2-orientation in a ^-lattice) is
determined by the AXA2 (or BXB2) interactions only.
Let W be the energy of an AB interaction and W the orientational
energy of an AXA2 or B1B2 interaction. We neglect interactions between
more distant neighbours. If for any configuration of the system of N
molecules, NAB is the number of neighbouring AB pairs, NA A the number
of relative misorientations on neighbouring ^4-sites and NB B is similarly
defined, the partition function for the whole assembly is
N
z=f n,
(AU W+NAiA% W + NBxB% W')/kBTl
where the sum is over all configurations taking into account all possible
arrangements in positions and orientations;/is the partition function per
molecule in a state of perfect order which is treated as a function of volume
per molecule and temperature only, and kB is the Boltzmann constant.

Approximate solution of the cooperative order-disorder problem
We define the degree of positional order as q = 2J — 1, where J is the
fraction of molecules in the ^4-sites, and the degree of orientational order
as s = 2£f'— 1, where Sf is the fraction of molecules in 1-orientations. If TV
is the total number of molecules in the system, there are evidently
\N(\ -Vq)(\ +s) molecules in Ax positions,
\N{\ +q)(\ —s) molecules in A2 positions,
\N(\ — q){\ +s) molecules in B1 positions,
\N(\ —q)(l—s) molecules in B2 positions.
Let there be z 5-sites adjacent to each ^4-site and z^-sites closest to each
^4-site. This implies, of course, that there are also z ^4-sites adjacent to each
B-site and z'5-sites closest to each i?-site.
Applying the Bragg-Williams (or zeroth order) approximation,(9)
the configurational partition function
2.1 Melting of molecular crystals                    21

where

r(q,s) = \       m

f
la
1UJV(1 - ? ) ( 1 +^)]![17V(1 -

Using Stirling's approximation

n+s

W(\-q
2 )   KT\
B J    4

K1AA)
kBr{
To evaluate the contribution of the disorder to the various thermodynamic
quantities it is necessary to specify the dependence of PFand W on volume.
It is found(5)6) that the empirical trends in the properties, especially for
large orientational barriers which is the main region of interest in the
present discussion, are reproduced well by the model when W —
W0(V0/vy and W = WO(VO/V)Z where the suffix zero denotes the value
corresponding to the equilibrium intermolecular separation as defined by
the Lennard-Jones potential. When W = 0, the theory reduces identically
to the LJD model for spherical molecules. Applying the equilibrium
conditions,
8 In Q 8 In Q
=       = 0,
dq       ds
we obtain

<2U)
-(H-FWr)*
where v = z'W'JzWQ is a measure of the relative barriers for the rotation
of a molecule and for its diffusion to an interstitial site.
22                2. Statistical theories ofnematic order

The Helmholtz free energy may be conveniently expressed as F =
F' + F'\ where F' = —NkBT\nfis the contribution due to the perfectly
ordered system and F" = — kBT\nQ is that due to disorder. The pressure
due to disorder/' = -@F"/dV)T, so that from (2.1.1),

7w    vx             v                 l
NkBT      kBT

where q and s are now the equilibrium values determined by (2.1.2)
and (2.1.3). If/?' is the ordered part of the pressure, the total pressure
p = pf +//'. Extensive calculations of p' V'/'NkB Tand other thermodynamic
parameters have been carried out as functions of V/ Vo and kB T/s for a
face-centred cubic lattice in terms of the Lennard-Jones 6—12 potential, e
being the minimum energy of interaction of a pair of particles.(10) Using
these values and (2.1.4) the complete isotherm can be obtained. In the
actual calculations the ratio WJe was adjusted empirically to give the
correct melting temperature of argon. The parameters used were z = 6 for
an interpenetrating face-centred cubic lattice, and W^/e = 0.977. The
melting properties of a range of materials can then be investigated as the
non-dimensional parameter v increases from zero to a high value.

2.1.2 Plastic crystals and liquid crystals
Hereafter q and s will be understood to refer to those values that satisfy
(2.1.2) and (2.1.3). For low zW/kB T, q = s = 0 is the only solution that
minimizes the free energy for any given kB T/e. The behaviour for higher
zW/kB T depends critically on the strength of the orientational barrier. The
variation of the equilibrium values of q and s with zW/kBT for three
typical values of v and the corresponding isotherms are shown in figs.
2.1.2-2.1.4. The sigmoid portions of the isotherms represent phase
transitions, i.e., the two phases will be in equilibrium at a given pressure
when the areas enclosed above and below the pressure line are equal. For
small v (less than about 0.3), the theory predicts two transitions, a solid
state rotational transition followed by a melting transition. The in-
termediate phase corresponds to the orientationally disordered crystal,
sometimes referred to as the plastic crystal For v between 0.3 and 0.975,
the two transitions coalesce and there occurs a single transition with a
much greater entropy of fusion. For v > 0.975 there are again two
transitions; the positional melting now precedes the rotational melting,
and there occurs an intermediate phase similar to the nematic liquid crystal
2.1 Melting of molecular crystals                         23

-0.2

-0.4

15
zW/kBT
Fig. 2.1.2. (a) Variation of the equilibrium value of the positional order parameter
q and the orientational order parameter s with zW/kB rfor v = 0.3. (b) Theoretical
isotherm for v = 0.3 showing rotational transition in the solid state followed by the
melting transition. (The isotherm is drawn for the solid rotational transition
temperature kB T/e = 0.593.)

L (b)

Fig. 2.1.3. (a) Variation of the equilibrium values of q and s with zW/kB T for
v = 0.8. (b) Theoretical isotherm for v = 0.8 showing a single transition in which
both positional and orientational orders collapse simultaneously (k B T/e = 0.625).

with orientational order but no translational order. (Over a small range of
v, 0.975 < v < 1.047, the theory predicts a second order (continuous)
nematic-isotropic transition, whereas experimentally it is found that this
transition is always discontinuous, though weakly so. This limitation of the
model probably arises from the restriction that the molecule can take up
only two orientations. An extension of the treatment to more than two
discrete orientations has been given by Amzel and Becka (11) for low v.)
24                2. Statistical theories ofnematic order

1.0

q, s

Fig. 2.1.4. (a) Variation of the equilibrium values of q and s with zW/kB T for
v = 1.3. (b) Theoretical isotherm for v = 1.3 showing solid-nematic and nematic-
isotropic transitions. (The isotherm is drawn for the solid-nematic transition
temperature kB T/e = 0.678.) (After reference 5.)

0.6

•0.4

0.2

0.4                0.8               1.2

Fig. 2.1.5. Reduced transition temperature versus v. S-S, solid-solid; S-I,
solid-isotropic liquid; S-N, solid-nematic; N-I, nematic-isotropic. (After
reference 6.)
2.1 Melting of molecular crystals                   25

a-1

3 --

\S-N

2 -
S-I          '

1 -

s-s

0 -
1

0
1

0.2
1     i

0.4
i

0.6
i

0.8
i

1.0
N-I/

i

1.2

Fig. 2.1.6. Entropy of transition versus v (see legend of fig. 2.1.5).

The entropy S, specific heat cv and isothermal compressibility p can be
evaluated from the thermodynamic relations:

S" = -
ZT);

\     + 2 T ( \ ]
)/2T{ aril*

Figs. 2.1.5-2.1.7 give the reduced temperatures of transition and the
corresponding entropy and volume changes as functions of v. For a certain
range of v, AS/R and AV/Vfor the nematic-isotropic transition are only
very small fractions of the values for the solid-nematic transition. This is
indeed a distinctive feature of such transitions in general (table 2.1.2).
Curves for the order parameter derived from the theory are presented in
fig. 2.1.8(a) and as can be seen they are closely similar to the experimental
results (fig. 2.1.8(6)). The theoretical curves(5) for the thermal expansion,
specific heat and isothermal compressibility are also found to reproduce
quite well the observed trends in the properties of the nematic phase/ 14 ' 15 ' 20)
26                    2. Statistical theories ofnematic order

Table 2.1.2. Latent heat and volume change data for some nematogenic
compounds

Solid-nematic             Nematic-isotropic
transition                 transition

Latent        Volume         Latent         Volume
heat        change           heat         change
Compound                    (kJ mole-1)    AV/V(%)       (kJ mole-1)    AV/V(%)
4-(4-n-Propylmercapto
benzalamino)-
azobenzene(12)            31.30           5.07         0.368          0.09
/?-Azoxyanisole              29.57(14)      11.03 (13)    0.574(14)      0.36(15)
/7-Azoxyphenetole            26.87 (14)      84d6)        1.366(14)      0.60(16)
4- Methoxy benzy lidene-
4/-butylaniline           18.06 (17)    —              0.582(17)      0.40(18)

S-N
0.2                                               ^—

0.1

s-s/                          N-I /
n                                             ^
1                1               i             i
0.4            0.8          1.2

Fig. 2.1.7. Relative change of volume on transition versus v (see legend offig.2.1.5).

Thus while a simple model of this type cannot be expected to be applicable
in detail to any particular substance, it does serve to bring out the effects
of orientational disordering on the thermodynamics of the melting process.
However, as near-neighbour correlations are neglected completely in the
Bragg-Williams approximation, the theory is unable to account for the
specific heat and other anomalies in the isotropic phase. An attempt has
been made to extend the theory by using the quasi-chemical or first
2.1 Melting of molecular crystals                                    27

Fig. 2.1.8. Orientational order parameters versus temperature: (a) theoretical
curves for v = 1.15,1.18 and 1.20; (b) experimental curves of s denned by (2.3.1) for
/7-azoxyanisole (PAA), anisaldazine and /?-azoxyphenetole (PAP).(19) The dashed
portions of the curves represent supercooled regions. (After reference 5.)

(b)
0.2 -                                  / iNemat
(«)
0.2
Solid /
/ Isotropic
pVo         Solid /Nematic/Isotropic       T~
"^                                   / /
0.1 -
0.1
/
/
/             i              i

0.67   0.68    0.69   0.70            0.6                      0.7            0.8
kBT/e                                                kBT/e

Fig. 2.1.9. Theoretical variation of the transition temperatures with pressure:
(a)v= 1.15; (b)v = 0.95.

approximation.(21) This results in a very weak anomaly in the specific heat
above 7^I? but not in the thermal expansion or isothermal compressibility.
It is clear that the model is far too elementary for describing these and
other short-range order effects in the isotropic phase (see §2.5).

2.1.3 Pressure-induced mesomorphism
Theoretical phase diagrams for two representative values of v are illustrated
in fig. 2.1.9. In fig. 2.\.9{a) the orientational barrier is large enough for the
nematic phase to occur at zero pressure. As the pressure is raised, both the
solid-nematic and the nematic-isotropic transition temperatures increase,
28                     2. Statistical theories ofnematic order

500

400

Solid         Nematic        /   Isotropic
0/
A   300

200

100   -

390           400    410                430
T(K)
Fig. 2.1.10. Experimental phase diagram for PAA. (After reference 22.)

the slope AT/dp for the latter being greater in accordance with the
experimental data (fig. 2.1.10). Of course, such a behaviour is also to be
expected from the Clausius-Clapeyron equation.
Fig. 2.1.9(6) represents a more interesting case. Here v is just below the
critical value for the nematic phase to occur at zero pressure. As the
pressure is raised, there is initially a single transition, viz, the solid-liquid
melting transition, but at higher pressures branching takes place and
there are two transitions. The branching point represents the solid-
nematic-isotropic triple point.
Experiments were done on the first two members of the p-n-
alkoxybenzoic acid series to verify this prediction. (22) These two com-
pounds, methoxy- and ethoxybenzoic acids, do not form liquid crystals
at atmospheric pressure, whereas propoxybenzoic acid and the higher
homologues show at least one liquid crystalline phase. As the pressure is
raised, both compounds exhibit mesophases, initially a nematic phase and
2.2 Phase transition in a fluid of hard rods                    29

2
e
200 -

Solid
1/
1
' Smectic / Nematic

/                      /
/isotropic

7
100 -           /                        /
Triple point tS

k
^            7

/
Triple point
/                                 i            i
i
205                 215          225
Temperature ( C)

Fig. 2.1.11. Experimental phase diagram for /?-ethoxybenzoic acid showing
the solid-nematic-isotropic and solid-smectic-nematic triple points. (After
reference 22.)

then, at higher pressures, a smectic phase as well (fig. 2.1.11). (The smectic
phase is not included in the framework of the theory in its present form. It
would require the introduction of another order parameter, viz, a lamellar
order parameter (see chapter 5).) The transitions were detected by
differential thermal analysis and the mesophases identified by microscopic
observations using a high pressure optical cell (fig. 2.1.12). These studies
also established for the first time the existence of the solid-nematic-
isotropic and the solid-smectic-nematic triple points in single com-
ponent systems.

2.2 Phase transition in a fluid of hard rods
We have emphasized that the asymmetry of the molecular shape is an
important factor in determining whether or not a substance exhibits the
liquid crystalline phase. Consider a fluid of long thin rods without any
forces between them other than the one preventing their interpenetration.
30                2. Statistical theories ofnematic order

Fig. 2.1.12. Pressure-induced mesophases in /?-ethoxybenzoic acid: (a) nematic
schlieren textures; (b) focal conies and batonnets in the smectic phase.
(Reference 22.)
2.2 Phase transition in a fluid of hard rods               31
At sufficiently low densities the rods can assume all possible orientations
and the fluid will be isotropic. As the density increases, it becomes
increasingly difficult for the rods to point in random directions and
intuitively one may expect the fluid to undergo a transition to a more
ordered anisotropic phase having uniaxial symmetry. That this does indeed
happen was first proved by Onsager.(23) Other hard rod theories have since
been proposed, which we shall refer to briefly at the end of this section, but
Onsager's approach, based on an exact density expansion for the free
energy, is still probably the most satisfying. However, the mathematical
analysis involved in the theory is rather complex - even in the lowest order
it leads to a non-linear integral equation. Truncation of the series after the
linear term, as was done by Onsager, would appear to be satisfactory only
for very long rods and not for shorter ones, but attempts to evaluate the
coefficients in the expansion beyond that of the linear term have so far been
unsuccessful because of computational difficulties.
Zwanzig(24) proposed a simplified version of Onsager's theory which
enables the virial coefficients to be evaluated to a much higher order. The
simplification consists in the following assumptions: (a) the rods can take
up only discrete orientations along three mutually perpendicular axes,
whereas in the original theory they can point in any direction, and (b) the
length to breadth ratio of the rods is very large (tending to infinity).
Though these assumptions may be somewhat too drastic as far as real
liquid crystal systems are concerned(25) the model does serve to illustrate
the essential principles of Onsager's method and may prove to be useful as
a starting point for possible extensions and applications of the general
theory.
In Zwanzig's model the rods are rectangular parallelopipeds of length /
and square cross-sectional area d2. Assuming that the potential energy of
V
interaction of T such hard rods is a sum of pair potentials, we may write

UN = \tuij9                               (2.2.1)
where Utj is oo if the molecules / and j intersect and zero if they do not. The
intersection may occur either in a parallel or perpendicular configuration
as shown in fig. 2.2.1. Let the position vector of they'th rod be represented
by Rj and its orientation by up and let the set of all positions be abbreviated
by R and the set of all orientations by u. The configurational integral for
V
the system consisting of T rods in the volume V can then be expressed as

QN(T, V) = ~ -L £ JdR exp ( - UN/kB T),
32                   2. Statistical theories of nematic order

where TV! allows for the indistinguishability of the particles and 3^ for the
total number of orientational states. The potential energy of the system
depends on the sets R and u. Since the configurational space available for
each molecule is V,

I dR exp ( - UN/kB T) = VN exp [ - <pN(u)/kB T],

where the new function <pN depends only on the orientations. Thus

tT}.

Now for any given configuration of the system, there will be N(\) molecules
pointing in the direction u(l), N(2) in the direction u(2) and N(3) in the
direction u(3). Therefore q>N becomes a function of the occupation numbers
in the three allowed directions, i.e.,
<pN =        <pN[N(l),N(2),N(3)].
The indistinguishability of the molecules requires that for any given set of
occupation numbers N(l), N(2) and N(3) there are N\/N(l)]N(2)\NQ)\
equivalent configurations, so that

=   J^_       »      »         »       N\exp(-<pN/kBT)
V*       N\lN   N {?)=a N£)=o   N   ^   N(\)\N{2)\N{?)\   '

where N(\) + N(2) + NQ) = N. Putting

) Nil) M3)l             ^
we have
Q»= £              £       £ t[N(l),N(2),N(3)].             (2.2.2)
iV(l)=0 A^(2)=0 iV(3)=0

To evaluate this configurational integral, we make use of the ' maximum
term method' of statistical mechanics,(26) i.e., in the limit N^ oo, V^ oo,
but N/V = p = constant, QN can be approximated by the largest term fmax
in the sum. The problem therefore reduces to one of maximizing t with
respect to the occupation numbers. Using Stirling's approximation and
introducing the mole fraction x a = N(tx)/N of the various components,

N-1]nt=l-]np-]n3-txttlnxa-(<pN/NkBT),                            (2.2.3)
2.2 Phase transition in a fluid of hard rods                    33

Fig. 2.2.1. Intersection of rods in the parallel and perpendicular configurations.

subject to the condition

To proceed further, we need to know the dependence of the function cpN on
density, which can be obtained by resorting to the virial expansion. Now,
the molecules having different orientations may be regarded as belonging
to different species, so that in effect we have a multicomponent system. The
virial expansion of the free energy of a gaseous mixture in ascending
powers of the density is well known :(27) for our present purpose we use the
form
N(3)]
N(2) JV(3)

(2.2.4)

where the virial coefficient B[N(l), N(2), N(3)] is defined as

/ i s the Mayer function given by
y;, = e x p ( - ^ . / / : B r ) - i ,
{ —1 if / and y intersect,
0 otherwise.
34                 2. Statistical theories ofnematic order

is t n e standard abbreviation in the cluster theory of imperfect gases
which signifies the integral over all positions of a sum of products of Mayer
/functions taken over all irreducible clusters (or graphs) involving N{\)
molecules of species 1, N(2) molecules of species 2 and N(3) molecules of
species 3.
The maximum value of t is obtained from the condition

^(N-1lnt)    = 0,    a =1,2,3.

The configurational free energy of the system is given by

F=-kBTlnQN
and the pressure by

Thus by determining fmax, the pressure can be derived as a function of the
density.

Calculation of the virial coefficients
The simplification introduced by restricting the rods to three discrete
orientations will now be apparent. The rods must intersect only in those
three directions, so that the cluster integrals in three dimensions can be
factorized into products of cluster integrals in one dimension. Consider for
example the second virial coefficient for parallel rods. By definition

°' 0) =
The first volume integral is evaluated directly so that

We observe that the Mayer function/ 12 is zero everywhere except in the
range — / to / in the x direction and — d to d in the other two directions. In
the range where/is non-zero, its value is — 1. Hence

5(2,0,0) = -4W 2 .
2.2 Phase transition in a fluid of hard rods             35

Similarly, the second virial coefficient for two rods in an orthogonal
configuration

B { 1 l 0 )

ni+d)/2     r(i+d)/2 ra
d3R2 = -
J-(l+d)/2   J-(l+d)/2    J -d

When the rods are very long and very thin, I P d,

2?(l,l,0)^-2/ 2 41+2(<///)],
£(2,0,0)        ~-4l2d(d/l).

Thus, the virial coefficient for parallel configuration is smaller than that for
orthogonal configuration by an order d/l. Higher order virial coefficients
exhibit the same behaviour. To simplify the calculation Zwanzig neglected
all terms of order d/l, i.e., he confined himself to the limit /-> oo, d-^0,
l2d = constant, thereby reducing considerably the number of coef-
ficients to be considered. It is readily verified that only coefficients of the
type B(m,n,0), B(0,m,n) and B(m,0,n) survive, and that by symmetry
B(m, 0, n) = B(n, 0,ra)= B(0,ra,n) = B(0, n, ra).
Let us now choose the z axis (/ = 3) as a preferred direction. By
symmetry the numbers of molecules in the x and y directions are equal, so
that the mole fractions along the three directions can be expressed in terms
of a single order parameter x:
v —v
x2 = x,
x3 = 1— 2x.

Therefore, (2.2.4) reduces to

VN    _ sr ^r       D/,^
NkBT
(2.2.6)
The mole fraction x is then determined by

— (AT1 In 0 = 0,
ax
i.e.,
x \             d /      cpN
2 In
\-2x)             dx\      NkBT)'
36                 2. Statistical theories ofnematic order

For example, in the second virial approximation this yields

The solution x = | corresponding to the isotropic distribution is trivially
satisfied at all densities and in any approximation. Below a certain critical
density, (2.2.7) has only the isotropic root. Above this density two new
roots appear; the one which corresponds to low x gives / max and the others
are discarded. Using (2.2.5), the pressure can then be evaluated as a
function of density or volume. The transition predicted by Onsager's
theory in the second virial approximation is confirmed by this model even
when all virial coefficients up to the seventh are included but the properties
of the anisotropic phase depend rather sensitively on the order of the
approximation. Fig. 2.2.2 gives the isotherm obtained in the sixth
approximation, and it can be seen that the shape of the curve is
characteristic of a first order transition.
A Pade analysis of Zwanzig's model by Runnels and Colvin (28) has
shown that the character of the transition is stable against increase in the
order of the approximation. However, the model does not converge to the
Onsager limit with increasing number of discrete orientations and
Straley<25) has concluded that it does not adequately represent real liquid
crystal systems.

Other hard rod theories
Most statistical treatments of the hard rod system, including Onsager's
theory and Flory's well known lattice model, (29) are valid only for very long
systems, and are therefore not applicable to low molecular weight nematics
for which l/b is usually between 3 and 5.(30) However, a formulation of the
'scaled particle theory' due to Cotter (31) has proved to be convenient for
evaluating the excess free energy for shorter rods at high densities. This
theory provides a means of deriving approximate expressions for the
chemical potential and pressure of the fluid by considering the reversible
work necessary to insert a scaled particle (i.e., a 'solute' particle which is
a scaled replica of the ' solvent' particles) at an arbitrary point in the fluid.
The method was introduced by Reiss, Frisch and Lebowitz (32) for hard
spheres, first applied to hard rods by Cotter and Martire (33) assuming
restricted orientations (as in Zwanzig's model) and extended by Lasher (34)
for continuous orientations. The theory was later presented in a more
rigorous fashion by Cotter. (31) The transition parameters and the properties
2.2 Phase transition in a fluid of hard rods              37

3.0

2.0

0.2              0.4             0.6          0.8
Volume
Fig. 2.2.2. Theoretical isotherm evaluated from Zwanzig's model in the
sixth approximation showing a first order nematic-i so tropic transition. (After
Zwanzig.(24))

of the anisotropic phase predicted by this model for a system of hard
spherocylinders, i.e., cylinders capped at either end by a hemisphere, are in
reasonable agreement with those of a real system, viz, PAA, for l/b =
2 5 (31,35) Another theory which leads to comparable results, is a method
proposed by Andrews(36) for hard spheres, extended to spherocylinders.(35)
It is based on the idea that the reciprocal of the thermodynamic ' activity'
is merely the probability of being able to insert a particle into the system
without overlapping with other particles. The method enables an extrapo-
lation of the 'exact' results of Monte Carlo and molecular dynamic
simulations in the isotropic phase carried out by Vieillard-Baron(37) and
others(38) for l/b = 2 and 3, to the region of the N - I transition. Other
treatments include the y-variable expansion(39) (where y = P/{\ — V0P))9
the application of the density functional theory,(40) the functional scaling
approach of Lee,(41) Monte Carlo simulation studies on prolate and oblate
spheroids,(42) etc. Hard rod models have also been used to study the effect
of molecular flexibility on the density change at the transition,(43) the
contribution of the alkyl end-chain,(44) the properties of mixtures(45) and
the influence of molecular biaxiality.(46)
However, there is a basic difficulty with all hard particle models. Their
properties are essentially 'athermaP and depend entirely on the density.
Thus if one defines a quantity

= (31nr/ain/>).                     (2.2.8)
38                2. Statistical theories of nematic order
which is a measure of the relative sensitivity of the order parameter s to
changes in density versus changes in temperature, it follows that y = oo for
a system of hard particles, whereas experimentally y = 4 for PAA (see
(2.3.18)). Clearly, therefore, attractive interactions must also be included
to give a proper description of the nematic phase.

2.3 The Maier-Saupe theory and its applications
An approach that has proved to be extremely useful in developing a theory
of spontaneous long-range orientational order and the related properties
of the nematic phase is the molecular field method, closely analogous to
that introduced by Weiss in ferromagnetism. Each molecule is assumed to
be in an average orienting field due to its environment, but otherwise
uncorrelated with its neighbours. The first molecular field theory of the
nematic state was proposed in 1916 by Born (47) who treated the medium as
an assembly of permanent electric dipoles and demonstrated the possibility
of a transition from an isotropic phase to an anisotropic one as the
temperature is lowered. Though this result is important qualitatively we
need not discuss this particular theory because it is now well established
that permanent dipole moments are not necessary for the occurrence of the
liquid crystalline phase (see (1.3.3)). Moreover, the theory predicts that the
aligned phase should be ferroelectric, which does not appear to be the case
even when the molecules are polar. The most widely used treatment based
on the molecular field approximation is that due to Maier and Saupe. (48)

2.3.1 Definition of long-range orientational order
We begin by defining the long-range orientational order parameter in the
nematic phase. Suppose that the liquid crystal is composed of rod-like
molecules in which (i) the distribution function is cylindrically symmetric
about the axis of preferred orientation n and (ii) the directions n and — n
are fully equivalent, i.e., the preferred axis is non-polar. Subject to these
two symmetry properties, and assuming the rods to be cylindrically
symmetric, the simplest way of defining the degree of alignment is by the
parameter s, first introduced by Tsvetkov,(49)
.s = K 3 c o s 2 0 - l > ,                (2.3.1)
where 6 is the angle which the long molecular axis makes with n, and the
angular brackets denote a statistical average. For perfectly parallel
alignment s = \9 while for random orientations s = 0. In the nematic
2.3 The Maier-Saupe theory and its applications                39
phase, s has an intermediate value which is strongly temperature
dependent.
The order parameter can be directly related to certain experimentally
determinable quantities - say, for example, the diamagnetic anisotropy of
the liquid crystal.(50) Let us choose a space-fixed cartesian coordinate
system xyz with z parallel to n. If rj1 and rj2 are the principal diamagnetic
susceptibilities of the molecule referred to its own principal axes, the
average z component of the susceptibility per unit volume in the nematic
phase is evidently

(2.3.2)

where fj = \{n1 + 2n2) and n the number of molecules per unit volume.
Similarly

Therefore
n2).                    (2.3.4)
r/1 and n2 can be obtained from the susceptibilities of the solid crystal if the
structure is known. Thus a measurement of xz>Xz o r t n e anisotropy xz~Xx
in the nematic phase gives the absolute value of s. In a similar manner,
s can be related to the optical anisotropy (birefringence (51) and linear
dichroism(50)) though, as is well known, there is a difficulty in this case as
there is no rigorous way of correcting for the effects of the polarization field
in the medium. Approximate methods have been proposed (50>52) which
appear to yield satisfactory results/ 53 ' 54)
Another important tool for investigating molecular order is nuclear
magnetic resonance (NMR) spectroscopy. (55) The principle of this method
is that if there is a pair of protons in a molecule, each spin is coupled to the
external magnetic field H as well as to the dipole field created by its
neighbour. The latter gives rise to a small perturbation of the eigenstates of
the Zeeman Hamiltonian so that the signal appears as a doublet. The
doublet separation is easily seen to be proportional to (3cos 2 # — l)/r 3 .
where 9 is the angle which the interproton vector rtj makes with H. In the
isotropic phase xtj can assume all possible orientations with respect to H
and <3cos 2 # —1> vanishes, but in the nematic phase there is a splitting
which is a measure of the order parameter s. The analysis is, of course, not
so straightforward for complex molecules as it requires precise knowledge
of the interproton distances and angles. The quadrupole splitting in the
magnetic resonance spectrum of 14N(56) or of deuterium(57) has also been
employed for studying molecular order. Other methods involve the use of
40                   2. Statistical theories of nematic order

0.7 r

0.6

a 0.5
O
6
0.4

I
0.3
-40         -30           -20            -10
r-rNI(°Q
Fig. 2.3.1. The temperature variation of the long-range orientational order
parameter s in PAA. Open circles from NMR measurements (McColl and Shih(66)),
filled circles from diamagnetic anisotropy (Gasparoux, Regaya and Prost(67));
triangles from refractive indices.(65)

NMR (58) or electron spin resonance spectroscopy (59) of geometrically
anisotropic molecules aligned in nematic solvents. We shall not discuss any
of these techniques here as a number of excellent reviews on the subject are
available. (605) In fig. 2.3.1 we present experimental values of the order
parameter of PAA determined by three different methods.
NMR and dielectric measurements (see §2.3.5) indicate that there is a
fair degree of rotational freedom of the molecules about their long axes.
Nevertheless the assumption that the molecules are cylindrically symmetric
is not valid in general and can sometimes lead to errors in the determination
of s.{6S) For a molecule of arbitrary shape, (2.3.1) may be written in the
generalized form(69)

where a,/? = x9y9z refer to the space-fixed axes, ij = x\y\z' refer to the
principal axes of the molecule, /a, jp denote the projection of the unit vectors
i, j along a,/?, and 3a/3, S(j are Kronecker deltas. $"/is symmetric in /, j and in a,/?. It is also a traceless tensor with respect to either pair, i.e., where repeated tensor indices imply the usual summation convention. The generalization of (2.3.2) and (2.3.3) is 2.3 The Maier-Saupe theory and its applications 41 where ntj is the molecular susceptibility tensor which can, in principle, be determined from the crystal data. s$ is a 3 x 3 matrix. When diagonalized the matrix has the form
slx     0            0

0      0   -(Sll      + s2i.
In the uniaxial nematic case, s n = s22.
If the molecule is flexible more parameters are required to define the
degree of alignment of its different parts. (64) However, in a large part of the
discussion that follows on the theory of the nematic state we shall ignore
all these details and assume the molecules to be cylindrically symmetric
rods.
2.3.2 The mean field approximation
We assume that each molecule is subject to an average internal field which
is independent of any local variations or short-range ordering. Consistent
with the symmetry of the structure, viz, the cylindrical distribution about
the preferred axis and the absence of polarity, we may postulate that the
orientational energy of a molecule
ut oc |(3 cos2 6i—\)s,
where 0i is the angle which the long molecular axis makes with the
preferred axis and s is given by (2.3.1).
The exact nature of the intermolecular forces need not be specified for
the development of the theory. However, in their original presentation
Maier and Saupe(48) assumed that the stability of the nematic phase arises
from the dipole-dipole part of the anisotropic dispersion forces. The
second order perturbed energy of the Coulomb interaction between a pair
of molecules 1 and 2 is given by

^oo = i      L                      3
r>2
K
Y ( 1 ) jr < 2 )

(2.3.6)

where x$,x$       etc. are the components of the transition moments defined
as
f
_J
4? = £ f^2)*42)42Vv2)dr(2), etc.,
k J
42                 2. Statistical theories ofnematic order
y/^\ y/[2) and EM,EV are respectively the eigenfunctions and eigenvalues of
the unperturbed molecules corresponding to states ju and v, EMV = EM + Ev9
^ 1) ,xj 1) ^ 1) zj 1) are the charges and their coordinates in the first molecule
measured in the space-fixed coordinate system with the origin at the centre
of mass of molecule 1, R = (X, Y,Z) is the position of the centre of mass
of molecule 2, e(*\ x^y^z™ are the charges and their coordinates in
molecule 2 measured in the coordinate system obtained by translating the
one defined above by XYZ. The second summation in (2.3.6) is over terms
with x9 y and z changed cyclically.
We assume that the rotational motions of the molecules are independent
of their translational motions. Let ^rja)C(1) and f<2y2)f<2) be the two
molecular coordinate systems with their origins at the centres of mass, £
coinciding with the long axis of each molecule. We define the Eulerian
angles as follows: 6 = the angle between z and £, 0 = angle between £ and
the normal to the z£ plane and 0 ' = angle between x and the normal to the
z£ plane. The components of the transition moments are

where (p is the eigenfunction in the £r/C coordinate system. Suppose now
that the nematic medium has cylindrical symmetry about z so that all
values of 0 ' are equally probable, and further that the molecules rotate
freely about £ so that all values of 0 are equally probable. Setting
<cos#(1)cos#(2)> = 0, i.e., the molecules do not distinguish between 9 = 0
and n, and assuming a spherical distribution of the intermolecular radius
vectors, the orientation dependent part of the potential energy of the /th
molecule can be written in the mean field approximation as

(237)

where M is a function of the intermolecular vector Rik only and

is a measure of the anisotropy of the 0-// transition. Maier and Saupe
expressed (2.3.7) in the simple form
A       /3cos2^-l\
s                                 (    }
[ j
where V is the molar volume and A is taken to be a constant independent
of pressure, volume and temperature.
2.3 The Maier-Saupe theory and its applications                   43
Cotter(70) has examined the postulates underlying the mean field
approximation in the light of Widom's analysis of this general problem and
has concluded that thermodynamic consistency requires that ut should be
proportional to V'1 regardless of the nature of the intermolecular pair
potential. However, in what follows we have assumed a V~2 dependence as
in the original formulation of the theory by Maier and Saupe.

2.3.3 Evaluation of the order parameter
The excess thermodynamic properties of the ordered system relative to
those of the disordered one can now readily be derived on the basis of
(2.3.8). The internal energy per mole

I K<exp(-w</fcBr)d(cos0<)
- ^                          = -\NkBTBs\                   (2.3.9)
exp(-ui/kBT)d(cos0J
Jo
where B = A/kB TV2 and N is the Avogadro number. The partition
function for a single molecule

ft = f exp ( - ut/kB T) d(cos 0,),                 (2.3.10)
Jo
so that the entropy

k
= -NkBkBs(2s+I)-In                    rexp(ffecos20,)d(cos0jj. (2.3.11)

The Helmholtz free energy
F=     U-TS

(2.3.12)

The condition for equilibrium is

IT)
\   U l )
-o-
/V,T
or
3<cos^)   +   l=0.
OS

This equation is satisfied when
<cos20,> = <cos20>,                         (2.3.13)
which may be called the consistency relation.
44                    2. Statistical theories ofnematic order

0.05 -                                  T>TNJ

yL.
0.0

\
-0.05

-0.1

i           1           1          1
0.2        0.4         0.6          0.8
s

Fig. 2.3.2. Variation of the free energy with the order parameter calculated from the
Maier-Saupe theory for different values of A/kB TV2. The minima in the curves
occur at values of 5 which fulfil the consistency relation (2.3.13).(71)

The plot of F versus s at different temperatures evaluated from (2.3.12)
is shown in fig. 2.3.2. The minimum of the free energy occurs at that value
of s which satisfies the consistency relation (2.3.13). When T < Tm, there is
only one minimum which corresponds to the stable ordered phase. When
Tis slightly greater than 7 ^ there are two minima, but the one at s = 0, i.e.,
the isotropic phase, represents the absolute minimum. At T = Tm, there
are again two minima, one at s = 0 and the other at s = sc, but the two
states now have equal free energies; at this temperature therefore a
discontinuous transition takes place with no change of volume but an
abrupt change in the order parameter. Putting F = 0 at the transition yields

Vl = 4.541,                         (2.3.14)
0.4292,                             (2.3.15)

where Vc is the molar volume of the nematic phase at TN1. As an example,
for PAA Tm = 408 K, Vc = 225 cm3, and A = 13.0 erg cm6.
Strictly speaking it is the Gibbs free energy G=U-TS+PV       which
2.3 The Maier-Saupe theory and its applications                 45

0-7                       (b)

2 0.6

"2
O
0.5

0.4
-40          -30        -20        -10
T-Tm C
Fig. 2.3.3. Order parameter s versus temperature predicted by the Maier-Saupe
theory, (a) ut oc V~2; (b) ut oc V\

must be equated to zero at the transition. This leads to the following
expression for the volume change at Tmm'72)

or
2 In    exp (|fts0 cos2 0) d(cos 6) - Bsc(sc + 1) , (2.3.16)
V       Bs*X
from which sc can be determined if A V is known from experiment.
However, AV is usually so small (see table 2.1.2) that sc derived by this
method differs hardly by 1-2 per cent from that given by (2.3.15). In effect,
therefore, the theory predicts a universal value of sc « 0.44 for all nematic
materials. While this value is in fair agreement with the observed data for
a number of compounds, there are systematic deviations/ 53 ' 60) The
introduction of higher order terms in the Maier-Saupe potential function
to allow for contributions other than the purely dipole-dipole part of the
dispersion energy has been suggested, (71"3) which enables calculations to be
made for specific cases. Further, the long wavelength orientational
fluctuations in the medium (see §3.9) will diminish the effective order
parameter and this has to be taken into account in the theory. (62) The
assumption that the molecule is a rigid rod is also an oversimplification.
Realistic calculations of the role of the flexible end-chain in the ordering
process have been made by Marcelja (74) and by Luckhurst,(75) whose results
will be discussed presently.
Empirical data suggest that the volume dependence of ut may be
46                       2. Statistical theories of nematic order

V (cm3 mole- 1 )
224       222        220    218

6.05   -

6.00   -

5.38

Fig. 2.3.4. Lines of constant order parameter s versus In Kand In T for PAA. The
slope (aIn T/ein V) = -4.0. (After McColl and Shih.(66))

different from that assumed in (2.3.8). In general, if ut ccVm, where m is a
number, then (2.3.14) gets modified to
TmV™ = 4.541                  (2.3.17)
but sc still has the same value. The rate of variation of s with temperature
does not depend too critically on the exponent m, as the isobaric change of
volume with temperature over the entire nematic range is usually only of
the order of 1-2 per cent (fig. 2.3.3). However, its precise value becomes
rather important in accounting for the effect of pressure on the order
parameter/ 66 ' 76>77) Pressure studies (6676) have established the following
results for PAA:

(i) (8In r / d l n V)s = -4 (fig. 2.3.4).
= -y as defined in (2.2.8).             (2.3.18)
(ii) The thermal range of the nematic phase at constant volume is about
2.5 times that at constant pressure.
(iii) The order parameter sc at the transition point is very nearly
independent of the pressure.
Now, it is clear from (2.3.12) and (2.3.13) that B should have a constant
value if s is to be invariant. If B is to be kept constant while T and V are
varied, as is implied in (2.3.18), TVm should be constant, or

(d\nT/d\nV)s = -                    (2.3.19)

Empirically, therefore, m = 4 for PAA. It turns out that with this value of
rn, result (ii) follows at once from the theory; on the other hand with m = 2,
2.3 The Maier-Saupe theory and its applications                47
the thermal range at constant volume is only 0.74 times the experimental
value.(78) The pressure invariance of sc is, of course, a direct consequence of
(2.3.14) or (2.3.17). Thus, the experimental data can be fitted satisfactorily
with m = 4. However, it is doubtful if this has any theoretical significance,
in view of Cotter's result,(70) referred to earlier, that m should be 1
irrespective of the nature of the intermolecular pair potential.
A drawback of the theory becomes apparent when we calculate the
latent heat of transition from the nematic to the isotropic phase. The heat
of transition is easily shown to be given by(48>72)
H = Txl[(a/P)AV-S(Ve,      Tm)],               (2.3.20)
where a and p are respectively the coefficients of thermal expansion and
isothermal compressibility of the isotropic liquid at Tm. (Although AFis
usually small, its contribution to H is not negligible since H itself is rather
small.) However, the theoretical values of//are found to be much too high,
usually by a factor of about 2 or 3.<72) Large discrepancies are also found in
the specific heat Cv and the isothermal compressibility /? in the nematic
phase. The failure of the theory in this respect is evidently because the
molecular field method neglects completely the effect of short-range order.
A naive way of accounting for this discrepancy is to assume that the
molecules form small clusters of 2-3 molecules each, and that the single
particle potential (2.3.8) applies to each cluster rather than to the individual
molecules/ 48 ' 72) The theory is then able to give reasonable values for //, Cv
and /?.(72) However, such a procedure cannot account for the remarkable
effects of short-range order in the isotropic phase (see §2.5).
As observed earlier, the assumption that the molecule is cylindrically
symmetric is clearly not valid for real systems, and consequently the use of
a single order parameter is not adequate. Most molecules are lath-shaped
and have a biaxial character. Therefore two order parameters are required
to describe the uniaxial nematic phase composed of biaxial molecules. (68) If
£, rj, C are the principal axes of the molecule (£ defining the molecular long
axis), it is necessary to introduce an additional order parameter
/) = f<sin 2 /9cos2^>,                     (2.3.21)

where y/ is the Eulerean angle made by £ with the line of intersection of the
£n plane and the xy plane (i.e., the plane normal to the director). A finite
value of D indicates that there is a difference in the tendency of the two
transverse molecular axes to project on the z axis. Theories have been
developed(79) by incorporating an additional term proportional to D in the
Maier-Saupe potential function. It emerges that molecular biaxiality
48                2. Statistical theories ofnematic order
decreases the value of s at the transition, and leads to better agreement with
experiment. Further, since different properties have, in general, different
biaxial anisotropies, the order parameters determined by different tech-
niques (optical anisotropy, diamagnetic anisotropy, etc.) need not
necessarily agree with one another if the biaxial order parameter (2.3.21) is
included in the calculations.
Until the 1970s only s — (P2(cos9)} (where P2 is the second order
Legendre polynomial) was accessible experimentally, but a technique is
now available for measuring both </>2) and <i^> (where P4 is the fourth
order Legendre polynomial). It involves polarized Raman scattering
measurements on aligned samples. Jen et al.m) who developed this
technique, made use of the C = N stretch vibration of a cyano-compound
dissolved in the nematic phase as a probe for the Raman measurements. A
number of pure nematogenic cyano-compounds have since been inves-
tigated by this method/81'82) It turns out that in every case studied so far,
<P4> is negative for at least part of the nematic range. While the observed
<P2> can be fitted quantitatively by including higher order terms in the
Maier-Saupe potential, the negative <P4> cannot be accounted for
theoretically. Using the experimental value of <P2> and <P4> one can
calculate an orientational distribution function for the molecule :(80)

f(cos6) = £            —j— <i>(cos0)>i>cos0         (2.3.22)
I (even)     ^

truncated after the third term 1 = 4. The results show that the molecule has
a strong tendency to be tipped away from the nematic axis - very much
more than predicted by the Maier-Saupe theory (fig. 2.3.5). In the three
pure cyanobiphenyl compounds studied/81'82) it is found that the greater
the chain length, the greater is the magnitude of the negative <,P4> (fig.
2.3.6). This may possibly provide a clue to the origin of the observed
angular distribution function. However, further studies need to be carried
out before drawing any definite conclusions.

The odd-even effect
From chemical studies it has long been known that the end-chains play a
significant part in the stability of the mesophase. The transition tem-
perature r m (83) and a number of other properties (e.g., the order parameter,
the excess specific heat, the transition entropy, etc.) show a pronounced
alternation as the homologous series is ascended, i.e., as the number of
carbon atoms in the end-chain is increased. This is often referred to as the
23 The Maier-Saupe theory and its applications                      49

3.0                                  T-T=2°C

2.4

0.6

-0.3            I    i    I                  I     °,OQ 6
1.0   0.6       0.8           0.4   0.2       0.0
cos 6
Fig. 2.3.5. Orientational distribution function /(cos 6) as defined by (2.3.22) in the
nematic phase of MBBA. Circles represent values from Raman measurements and
the line gives the distribution function derived from a two-term Maier-Saupe
potential in which the parameters are adjusted to give a good fit with the observed
<P2(cos#)>. The negative values of/(cos 0) at high angles arise from truncation
errors. (After Jen et al.m))

odd-even effect. Qualitatively, the origin of this effect can be understood
from a consideration of the molecular structure (fig. 2.3.7). In the even
members of this series, the disposition of the end group is such as to
enhance the molecular anisotropy and hence the molecular order, whereas
in the odd members it has the opposite effect. As the chains become longer
their flexibility increases and the odd-even effect becomes less pronounced,
until for long chains it becomes negligible. A quantitative calculation of the
contribution of the end-chain to the ordering has been made by
Marcelja.(74)
The possible conformations of the alkyl chain and the corresponding
internal energies can be worked out accurately, as was shown by Flory. (84)
The valence angle between each successive bond is constant, while the
energy as a function of the azimuthal angle has three minima defining three
possible states for each successive bond, viz, the extended trans state and
two symmetrical gauche states. Marcelja incorporated the configurational
50                 2. Statistical theories ofnematic order

-o.i   -

-0.2   -

Fig. 2.3.6. Orientational order parameter <P 4(cos#)> in three nematic liquid
crystals from Raman measurements: (a) 4 /-n-pentyl-4-cyanobiophenyl (5CB); (81)
(b) 4/-n-heptyl-4-cyanobiphenyl (7CB);(82) (c) 4/-n-octyloxy-4-cyanobiphenyl
(80CB).(82) (After reference 82.)

Fig. 2.3.7. Structure of 4,4/-di-n-alkoxyazoxybenzenes. The addition of an even-
numbered carbon atom in the preferred trans conformation is along the major
molecular axis. This is not the case for an odd-numbered carbon atom.
2.3 The Maier-Saupe theory and its applications              51
statistics of the end-chains in the theory of the nematic phase. In addition
to the conformational energy, each C—C bond is subject to a mean field
which depends on the orientational order of the rigid central part of the
molecule as well as that of the end-chain. The detailed calculations yield
results in good accord with the observed trends. An interesting conse-
quence of the theory is that depending on the strength of the interaction
between the rigid part of the molecule and the end-chain, there is a rising
trend in T^ versus n for some homologous series of compounds, and a
decreasing trend for others. This theory has been presented in a more
refined form by Luckhurst, (75) who applied it successfully to molecules
composed of two cyanobiphenyl moieties linked by flexible spacers in
which the odd-even alternation in T^ is as high as 100 °C for the first few
members of the homologous series. Some of Luckhurst's results are
presented in fig. 2.3.8.

Binary nematic mixtures
The Maier-Saupe theory has been extended by Humphries, James and
Luckhurst,(85) and more recently by Palffy-Muhoray et al.,m) to investigate
the properties of binary mixtures. The latter treatment (86) has led to some
significant conclusions:
(i) For certain values of the parameters, it is possible to have a
nematic-nematic coexistence region. Coexisting nematic phases have
in fact been observed in mixtures of polymeric and low molecular
weight nematogens,(87) as also in mixtures of rod-shaped and disc-
shaped nematogens. (88)
(ii) The order parameter of the mixture and its variation with temperature
agree very closely with the Maier-Saupe universal curve. However, the
order parameters of the individual components of the mixture may
differ appreciably. (62) The fact that the two components can have
different order parameters is of practical importance in the design of
dye displays. The dye molecules are chosen to have a high anisotropy,
and hence a high order parameter which, in turn, improves the contrast
ratio.

2.3.4 Theory of dielectric anisotropy
As an example of an application of the mean field method we shall consider
the theory of the dielectric anisotropy of the nematic phase. (89>90) The low
frequency dielectric anisotropy of a molecule is determined by two factors:
(i) the polarizability anisotropy aa which for the elongated molecules of
nematogenic compounds always makes a positive contribution (i.e., a
52                    2. Statistical theories of nematic order

260 r

220

200

160

120
12

Fig. 2.3.8. The variation of the nematic-isotropic transition 7 ^ with the number of
methylene groups in the flexible spacer for a,o>bis(4,4'-cyanobiphenyloxy)alkanes,
the molecular structure of which is shown at the top of the diagram. The dashed
lines represent the theoretically calculated values. (After Luckhurst.(75))

larger contribution for the measuring field parallel to the long molecular
axis) and (ii) the dipole orientation effect. The sign of the latter contribution
is positive if the net permanent dipole moment of the molecule makes only
a small angle with its long axis and is negative if the angle is large. In the
last case the sign of the net anisotropy depends on the relative magnitudes
of the two contributions. Thus different nematic materials can exhibit
widely different dielectric properties (fig. 2.3.9).
In contrast to diamagnetism where the magnetic interactions between
molecules can be neglected, the polarization field in the medium becomes
important when discussing dielectric anisotropy (see, e.g., Bottcher(92)).
Maier and Meier(89) took this into account by applying the Onsager
theory.(93) The effective induced dipole moments per molecule along and
2.3 The Maier-Saupe theory and its applications                      53
5.9 ,- (a)

tn   5.8

I
I 5.7
s
5.6

120                  130                      140
7TQ

30 —

20

10

1
100                  120                       140
7TQ
Fig. 2.3.9. Principal dielectric constants of (a) PAA (after Maier and Meier(90)) and
(b) n(4/-ethoxybenzylidene) 4-aminobenzonitrile (after Schadt (91)). s1 and e2 refer to
the values along and perpendicular to the optic axis of the nematic medium and £is
is the isotropic value.

perpendicular to the unique axis of the nematic liquid crystal are then given
by expressions essentially similar to (2.3.2) and (2.3.3) but with appropriate
correction factors for the polarization field:
(2.3.23)
(2.3.24)
where h = 3e/(2s+ 1) is the cavity field factor, e the mean dielectric
54                 2. Statistical theories ofnematic order
constant, F = 1/(1 — af) the reaction field factor, a the mean polarizability,
aa the polarizability anisotropy (the direction of maximum polarizability is
assumed to be the long molecular axis),/= 4nNp(2e — 2)/3M(2s+ 1), p the
density, M the molecular weight, s the order parameter defined by (2.3.1),
and E the applied field. Strictly speaking F and h have to be corrected for
the anisotropy of the dielectric constant but we shall ignore these
corrections.
To calculate the effective permanent dipole moment, we choose XYZ as
the space-fixed coordinate system, Z being parallel to the unique axis of the
medium, and £rj( as the molecule-fixed coordinate system, the £ axis
coinciding with the long axis of the molecule. Let v be the Eulerian angle
between the £ axis and the line of intersection of the XYand £rj planes, and
V the angle between this line and X. Suppose that the permanent dipole
moment ju is inclined at an angle ft with respect to the long molecular axis.
In dielectric measurements, low fields are usually employed so that the
potential energy of the dipoles due to the external field is small. We can
therefore write the effective dipole moments along the field directions as

,T)]juzy/(0)sin0d0dvdv/

[l+(M,hE1/kBT)]yf(ff)sin9d0dvdv'
o Jo Jo
hE2/kB T)] jux y/{0) sin 0 d6 dv dv'
> (2.3.26)
[l+(MxhE2/kBT)]¥(S)sm8dvdv'
Jo Jo Jo
Jo Jo Jo
where

fiz = ///[cos P cos 6 + sinfi sin v sin #],
jux = Fju[cos P sin V sin 6 + sin /?(cos v cos V — sin v sin V cos 9)]

and y/(0) d6 is the probability of a molecule having an orientation be-
tween 6 and 6 + d6. If y/(0) is given by the Maier-Saupe distribution,
exp( — ut/kB T) where ut is defined by (2.3.8), the integrals reduce to

^          1           ,                        (2.3.27)

ft = ^T"^[l+i(l-3cos a i8)5]AF 2 £ 2 .                (2.3.28)
2.3 The Maier-Saupe theory and its applications                  55
Since

^f[\-(I-3       cos2 p)s

(2.3.29)
and similarly

s2= i + 4 ^ ^ ^ | a - i a ^ + ^;[l+Kl-3cos 2 y 9)^J. (2.3.30)

Therefore

£a = e1-e2 = ^^hFW~^-f(l               -3cos 2 £)L.         (2.3.31)

Clearly if /? is small, the two terms in the square brackets of (2.3.31) add to
give rise to a strong positive ea, whereas if /? is sufficiently large £a may be
negative. In PAA, for example, ju and /? are estimated to be 2.2 debyes and
62.5° respectively from Kerr constant measurements in dilute solutions
and dielectric measurements in the isotropic phase. Substituting for the
other known parameters for this compound and using s from theory, it
turns out that ea should be weakly negative. The absolute value of ea as well
as its temperature variation are in fair agreement with the experimental
data.(90)
The dielectric constants are, of course, frequency dependent.(94) The
dipole orientation part of the polarization parallel to the preferred
direction (1-direction) may be expected to be characterized by a relatively
long relaxation time. This arises because of the strong hindering of the
rotation of the longitudinal component of the dipole moment about a
transverse axis. On the other hand the orientational polarization along the
2-direction will have a much faster relaxation, comparable to the Debye
relaxation in normal liquids, as this involves rotation about the long axis
of the molecule. If there are additional dipoles in parts of the molecule,
with their own internal degrees of rotation, the corresponding relaxation
times will again be similar to that in a liquid. The expected form of the
dispersion curves for a compound like /7-azoxyanisole is illustrated in fig.
2.3.10. The trends in the curves have been confirmed experimentally.(95>96)
Meier and Saupe(97) have discussed the mechanism of dipole orientation in
PAA and have shown that the relaxation time for polarization parallel to
56                  2. Statistical theories of nematic order

e 3

10 12

Fig. 2.3.10. Expected form of the dispersion of the principal dielectric constants of
4,4/-di-n-alkoxyazoxybenzenes. The suffix 0 refers to the static values and the suffix
oo to the optical values. ex shows the low frequency relaxation and both £x and e2
show the normal Debye high frequency relaxation. (After Maier and Meier.(95))

A \\ \l/7 ///

II.   I
Splay
^

Twist

Bend

Fig. 2.3.11. The three principal types of deformation in a nematic liquid crystal.
2.3 The Maier-Saupe theory and its applications                57

the 1-direction should increase from its value in the isotropic phase by a
'retardation factor' which may amount to several orders of magnitude
depending on the strength of the nematic potential. However, the
Maier-Saupe potential yields a retardation factor smaller than the
experimental value. This is not surprising since short-range order may be
expected to play a dominant part in the relaxation process and the mean
field theory neglects this completely. No theory of the relaxation
mechanism has yet been proposed taking into account near-neighbour
correlations.

2.3.5 Relationship between elasticity and orientational order
As remarked in chapter 1, a uniformly oriented film of nematic liquid
crystal may be prepared by prior treatment of the surfaces with which it is
in contact. If the preferred orientation imposed by the surfaces is perturbed,
let us say by a magnetic field, a curvature strain will be introduced in the
medium. The theory of such a deformation will be discussed at length in
§3.2; for the present it will suffice to state some of the important results.
The free energy per unit volume of the deformed medium relative to the
state of uniform orientation is

where n is a unit vector called the director representing the preferred
molecular direction, dnx/dx, dnx/dy and dnx/dz are respectively the splay,
twist and bend components of curvature at any point, the curvature being
defined with respect to a right-handed cartesian coordinate system XYZ
with Z parallel to the preferred direction at the origin. The three principal
types of deformations are illustrated schematically in fig. 2.3.11.
At the molecular level, it is obvious that curvature elasticity is a
consequence of the orientational order in the liquid crystal. A quantitative
relationship between them was established by Saupe (98) using the mean
field theory. From (2.3.6) is it seen that the dipole-dipole part of the
dispersion energy of interaction of a molecule / in the average field due to
its neighbours k is given by

k
1
*Kik
^
juv* 00   J
1
K       Y2     \
o f j f c i \ ay (fc)
J
D2      l
}XOtiXOv
+ 3—^
58                 2. Statistical theories ofnematic order
The internal energy per mole due to orientational order in the mean field
approximation is then
U = \NAV~2s\
where V is the molar volume. Now let us suppose that the director at the
site of the fcth molecule is turned through an angle ak with respect to that
at /. If we define a new system of axes X Y'Z' at the site of the /:th molecule,
Z ' making an angle (xk with respect to the Z axis,
C 0 S                  i
*oT = ^OM               <** ~   ^O
v'(fc) _ v(k)

Since we are dealing with small deformations, we assume that the director
continues to have cylindrical symmetry about the Z ' axis at the site of the
fcth molecule and that the order parameter s is unchanged in magnitude.
Substituting for x(etc. in (2.3.33) in terms of x'0{*\ defining the Eulerian angles between the molecular coordinate system £<fcy*>£<*> and the X'Y'Z' system and taking appropriate averages, the extra internal energy per mole of the system in the deformed state turns out to be 1 r/ z 2 \2 z 2 Y21 JLsin,flJ 3 ^ _ ! - 9 ^ 4 (2.3.34) ^ik L\ *^ik / ^ik J Now for a pure bend distortion da/dX = da/d Y = 0 and we may set sin ock « ak « 7^ifc 3a/6Z. Assuming a spherically symmetric distribution, Xik/Rik etc. may be taken to be constant (independent of temperature), and therefore the second sum in (2.3.34) is proportional to F~4/3. The free energy of deformation per unit volume can then be expressed as SF where C is a constant. But by definition so that k3S = CV~7/3s2. (2.3.35) Similar expressions are obtained for the splay and twist elastic constants. The temperature dependence of the elastic constants of' simple' nematics is represented well by (2.3.35).(99'100) To calculate the absolute values of the elastic constants, one has to evaluate the second summation in the right-hand side of (2.3.34). Saupe(98) evaluated the mean value of this sum for an isotropic distribution of 2.3 The Maier-Saupe theory and its applications 59 molecules and found fcn:fc22:fc33 to be 17: —7:11, whereas for PAA at 120 °C it is 1.6:1:3.2. Saupe and Nehring(101) have attributed this discrepancy partly to the neglect of k13 in the expression for the free energy of deformation (see (3.2.8)). When this coefficient is included, k'^'.k^-.k'^ turns out to be 5:11:5, where k'xl = kxl — k13 and k'ss = kss — kls. Of course, these calculations neglect the anisotropy of molecular shape. Attempts have been made to evaluate the elastic constants on the basis of hard rod models(102~5) (strictly valid only for very long and thin rods) and hybrid models incorporating both hard rod features and attractive forces.(106"9) A corollary to (2.3.35) is that though the order parameter as well as the elastic constants decrease rapidly with rise of temperature, the intensity of light scattering should be practically constant throughout the nematic range. Light scattering from nematic liquid crystals is governed by the long-range orientational fluctuations as described by the continuum theory. It is shown in §3.9 that the differential scattering cross-section may be written approximately as da [ns»\2kBT where Q is the irradiated volume of the liquid crystal, £a the dielectric anisotropy at optical frequencies, X the vacuum wavelength, ken an effective average elastic constant and q the magnitude of the scattering vector. Now the variation of kB r o v e r the nematic range can be neglected. Since £a varies approximately as s (see (2.3.4)) and kett approximately as s2, a should be nearly temperature independent. Experimentally, this result was first established by Chatelain(110) in PAA and PAP. Some measurements on 4-methoxybenzylidene-4/-butylaniline (MBBA) are presented in fig. 2.3.12. An important source of error in these calculations is the neglect of short- range order. In particular, the theory fails for the bend and twist elastic constants when smectic-like short-range order is present in the nematic liquid crystal. Under such circumstances these two constants exhibit a critical divergence as the temperature approaches the smectic-nematic point and the light scattering also shows a marked temperature de- pendence. The present treatment is then inadequate and more elaborate models have been proposed/ 111 ' 112) The phenomenological theory of this aspect of the problem will be discussed in chapter 5. 60 2. Statistical theories of nematic order CD 3 1.2 D 'S Q o % io A 0.9 0.8 290 300 310 320 T(K) Fig. 2.3.12. The temperature dependence of the intensity (/) of light scattering from 4-methoxybenzylidene-4/-butylaniline (MBBA) in the nematic phase. The values (plotted as kB T/I) are normalized to unity at 286 K. The squares, triangles and circles correspond to three different experimental configurations. For one of the configurations (squares) the data were not fully corrected for the effect of the temperature dependence of the refractive indices on the variation of one of the wave-vector components, which probably explains the slight increase near the transition temperatures in this case. (After Haller and Litster.(100)) 2.4 Hybrid models: hard rods with a superposed attractive potential A realistic theory of nematics should, of course, incorporate the attractive potential between the molecules as well as their hard rod features.(113) There have been several attempts to develop such hybrid models. Equations of state have been derived based on the Percus-Yevick and BBGKY approximations for spherical molecules subject to an attractive Maier-Saupe potential/114'115) However, a drawback with these models is that they lead to y = 1 (see (2.3.18)). Cotter(31> has extended her scaled particle theory to include an attractive potential of the form ut = —vQp — v2psP2(cos0i). (2.4.1) The resulting distribution function is similar to that in the Maier- Saupe theory, except that the coefficient of the potential has the form [(v2p/kBT) + A(p)], i.e., a temperature dependent attractive part and an 'athermal' part as given by the scaled particle theory. A similar result can be obtained using the Andrews model as well.(35) These last two approaches appear to be promising; for example, calculations show that y « 4 for l/b ~ 2 without violating Cotter's thermodynamic consistency condition that the mean field potential should be proportional to /?. Further the transition parameters and the properties of the nematic phase are in reasonably good agreement with the experimental values for PAA. Gen- 2.5 Short-range order effects in the isotropic phase 61 100 r- 80 ^ 60 g 40 20 120 TNI 140 160 180 Temperature (°C) Fig. 2.5.1. Magnetic birefringence in the isotropic phase of PAA: the horizontal dashed line gives the value for nitrobenzene at 22.5 °C. (After Zadoc-Kahn.(119)) eralized van der Waals models have also been developed(116 18) which lead to results that are essentially the same as those predicted by the scaled particle theory with a superposed attractive potential. 2.5 Short-range order effects in the isotropic phase 2.5.1 The Landau-de Gennes model We have noted in §2.1.2 that though the long-range order vanishes abruptly at 7^I? certain anomalous effects in the isotropic phase reveal that an appreciable degree of nematic-like short-range order persists above the transition point. The most direct evidence of this is the very high value of the magnetic birefringence, which in the neighbourhood of Tm may be as much as 100 times greater than in ordinary organic liquids (119120) (fig. 2.5.1). Similar anomalies are seen in the flow birefringence,(121) Kerr effect(122) and nuclear spin lattice relaxation,(123) confirming the existence of strong orientational correlations between the molecules. Foex(124) observed many years ago that the magnetic birefringence exhibits a (T— T*)1 dependence and drew attention to the fact that the behaviour is closely analogous to that of a ferromagnet above the Curie temperature. More 62 2. Statistical theories ofnematic order recently, de Gennes(125) has proposed a phenomenological description of these pretransition effects on the basis of the Landau theory of phase transitions.(126) Consider an expansion of the excess free energy of any ordered system in powers of a scalar order parameter s in the following form: F=|^J2-|& 8 + JCJ 4 + ..., (2.5.1) where B > 0 and C > 0. We observe that such an expression predicts a discontinuous transition, for putting F = 0 and dF/ds = 0 at the transition point Tc we get sc = 2B/3C (2.5.2) neglecting higher order terms. A plot of F versus s results in a family of curves similar to that shown in fig. 2.3.2. If B = 0, the transition is continuous and A vanishes at the transition point.(126) This is because in the disordered phase s = 0 corresponds to a = minimum of F only if A > 0, while in the ordered phase s # 0 corresponds to a stable minimum only if A < 0. Thus, since A is positive on one side of the transition point and negative on the other, it must vanish at the transition point itself. In the vicinity of the transition, we may therefore write A = a(T-T*) (2.5.3) where T* is the second order transition temperature. If B > 0, T* lies below Tc and (Tc - T*) = 2B2/9aC. (2.5.4) For a' weak' first order transition, B is small and 2B2/9aC may be expected to be a very small quantity. In principle, a free energy expansion of this type should be valid for nematic liquid crystals, with s denoting the usual orientational order parameter defined by (2.3.1). The term of order ss is not precluded by symmetry, for the states s and — s represent two entirely different kinds of molecular arrangement which are not symmetry related and do not have equal free energies. In the former case, the molecules are more nearly parallel to the unique axis, while in the latter they are more nearly perpendicular to it. However, in the nematic phase s is usually quite large (greater than about 0.4) so that very many more terms have to be included in the expansion in order to draw any valid conclusions. Consequently, the 2.5 Short-range order effects in the isotropic phase 63 model is conveniently applied only to the weakly ordered isotropic phase. We shall discuss some of these applications. 2.5.2 Magnetic and electric birefringence The free energy per mole of the isotropic phase in the presence of an external field (magnetic or electric) may be written as F=\a(T-T*)s2-\Bsz + \Cs\.. + NW(s\ (2.5.5) where W(s) is the average orientational potential energy of a molecule due V to the external field and T the Avogadro number. If the external field is magnetic W{s) = -^H\ (2.5.6) where / a = X\\—%± *s the anisotropy of diamagnetic susceptibility of the molecule. The magnetically-induced order is extremely weak ( ~ 10~5) so that we may neglect cubes and higher powers of s. The condition dF/ds = 0 then leads to the result (2.5.7) and the magnetic birefringence (2.5.8) where, assuming a Lorenz-Lorentz type of relationship for the refractive index n in the absence of a field, C - 2nN2x^n^H\n2 + 2Y/21Vn, (2.5.9) rja is the anisotropy of optical polarizability of the molecule, and V the molar volume.(127) The magnetic birefringence varies essentially as (T— T*)'1 since the dependence of C on temperature is relatively small. Experimentally(128) this is found to be the case, with r N I - r * ^ l K (fig. 2.5.2). As we shall see later a (T— r * ) " 1 law implies a classical mean field. The behaviour is slightly more complicated in the case of electric birefringence because, as explained in §2.3.4, the orientational energy in an electric field E arises from the anisotropy of low frequency polarizability aa and also from the net permanent dipole moment ju. An interesting example is that of PAA in which the Kerr constant actually changes sign at about rn + 5K (122) (fig. 2.5.3). The sign reversal is easily understood.(129) The 64 2. Statistical theories of nematic order o I 4 0 40 44 48 52 56 60 Temperature (°C) Fig. 2.5.2. Reciprocal of the Cotton-Mouton coefficient in the isotropic phase of MBBA for two samples, one slightly impure and having a lower transition point Tm. Both yield the same value of Tm- T* = 1 °C. (After Stinson and Litster.(128)) average orientational energy of the molecule due to the induced dipole moment is (2.5.10) and that due to the permanent dipole moment is W2(s) = -(F2h2ju2E2/6kB T)(3cos2p- \)s, (2.5.11) where the symbols have the same meanings as in (2.3.31). In obtaining W2(s), the distribution function is supposed to involve only even powers of cos 9. This is clearly valid in the present case in view of the assumed form of the free energy expression. Putting W{s) =WX+W2, F=±a(T-T*)s2-±NFh2E2[(xgL-(Fju2/2kBT)(l-3cos2l3)]s. (2.5.12) Proceeding as before NFh2E2[ota-(Fju2/2kB T)(\-3cos2p)] s = • (2.5.13) 3a(T- T*) and (2.5.14) 21nVa(T-T*) A«E can be positive or negative depending on the sign of the quantity in the square brackets of (2.5.14). The polarizability anisotropy aa is always 2.5 Short-range order effects in the isotropic phase 65 20 5 - 15 © X 10 <l -10 -15 r 0 405 Ni 415 425 435 445 Temperature (K) Fig. 2.5.3. The electric birefringence (Tsvetkov and Ryumtsev,(122) open circles) and the reciprocal of the magnetic birefringence (Zadoc-Kahn,(119) full circles) in the isotropic phase of PA A versus temperature. The lines represent the theoretical variations.(129) positive for the long molecules under consideration, but the sign of the dipole contribution depends on the angle /?. If /? is small, A«E will be strongly positive, whereas if /3 is sufficiently large AnK may be negative. Moreover, the second term in the square brackets of (2.5.14) is pro- portional to T~x, so that, in principle, there can occur a reversal of sign of A«E with temperature. Using the values of a and T* derived from the magnetic birefringence of PAA and substituting for //,/?, etc. (see §2.3.4) it is found(129) that there is in fact a reversal of sign of A«E, though it occurs at I NI + 9 K (fig. 2.5.3). Since there is a competition between the polarizability and the permanent dipole contributions, even a small error in any of the parameters will cause an appreciable shift in the temperature at which A«E = 0. Taking this into consideration, the agreement may be regarded as satisfactory. If p = 0, A«E given by (2.5.14) varies essentially as (T— 77*)"1. In such materials, the electric and magnetic birefringence may be expected to exhibit the same type of behaviour over a wide temperature range. Measurements(130) on pure samples of hexylcyanobiphenyl, a nematogen 66 2. Statistical theories ofnematic order 3.2 - - 3.2 2.4 2.4 o X Magnetic s 1.6 <l 0.8 0.8 T* Tm 30 35 40 45 50 55 Temperature (°C) Fig. 2.5.4. Reciprocals of the magnetic and electric birefringence in the isotropic phase of 4-hexyl-4/-cyanobiphenyl versus temperature. Both give the same value of T*(T* = 28 °C, r N I - T* = 1.1 °C). (After reference 130.) of strong positive dielectric anisotropy with the dipole (—C=N) pointing almost exactly along the major molecular axis, confirm this prediction (fig. 2.5.4). 2.5.3 Light scattering Although in the absence of an externally applied field the equilibrium value of s in the isotropic phase is zero, there can occur fluctuations in the order parameter about the zero value.(125) This gives rise to an anomalous scattering of light. We write the free energy expression (2.5.1) in a more general form in terms of the tensor order parameter (2.3.5), with / =j = 3 corresponding to the long molecular axis: F— (2.5.15) = F0 + \a(T- (2.5.16) neglecting higher terms. For a given scattering wave vector q, the differential scattering cross-section per unit volume is given by (see §3.9) (2.5.17) where (2.5.18) e is the mean dielectric constant and As is the dielectric anisotropy when the 2.5 Short-range order effects in the isotropic phase 67 ^ 16 t 14 I I 2 1 •| 10 i I 44 48 52 56 60 Temperature (°C) Fig. 2.5.5. Reciprocal of the intensity of light scattering in the isotropic phase of MBBA versus temperature. (After Stinson and Litster. (128)) molecules are all exactly parallel to one another (see (2.3.2)). Both e and Ae refer, of course, to optical frequencies. If the incident light is linearly polarized along z and the scattered light polarized along x (2.5.19) From the equipartition theorem so that An2 (Ae)2kBT (2.5.20) dQ If the incident and scattered radiations are both polarized along z, (where we have made use of the condition sxx 4- syy + szz = 0) and da^ _ 8TT2 (Ae)2kBT (2.5.21) dQ 68 2. Statistical theories ofnematic order Thus the intensity of scattering should vary essentially as (T— T*)' 1 (fig. 2.5.5) and the ratio of the scattered light polarized along z to that polarized along x should be 4/3. Both these predictions have been verified quantitatively for MBBA. (131) If the order parameter varies gradually from point to point, the free energy expression should include gradient terms as well, which can be written in the form \Li^Py^s^ + \L^as^pspr (2.5.22) where 8a = 6/8xa, repeated indices being subject to the usual summation convention, and L 1? L 2 are constants. To elucidate the consequences of these additional terms let us suppose that szz = s,sxx = syy = —\s,s xy = s yz = szx = 0- Also, let s be a function of z only. The free energy is then p (£J] (2.5.23) where { } a(T-T*)\ may be called the coherence length. The spatial correlation function has the form <>(0) s(R)} = const. kB Tcxp ( - R/Q (R > Q, and the scattering cross-section (2.5.21) for a given scattering wavevector q is modified to (125) (2-5.25) Experimentally it is found that the angular variation of the intensity of scattering is rather small (132133) proving that the coherence length is much smaller than the wavelength of light (q£ < 0.1), but by careful measure- ments Stinson and Litster(133) have established that f oc (T- T*)~1/2, in accordance with (2.5.24). 2.5 Short-range order effects in the isotropic phase 69 2.5.4 Flow birefringence To discuss flow birefringence, we have to make use of some results of the continuum theory developed in chapter 3. In analogy with (3.1.38) we write for the isotropic phase(125) (2.5.26) where t^^cp^ are treated as 'forces' and as 'fluxes'; t^ is the viscous stress tensor; dF ds, 'a/? = -AsxP from (2.5.15); For an incompressible fluid daa = 0. All four tensors are symmetric and traceless. Further from Onsager's relations, ju = ju'. Now consider shear flow along x with a velocity gradient d*;/dz. The flow induces a birefringence proportional to the velocity gradient with the principal axes of the index ellipsoid inclined at 45° to the x, z axes. In the steady state R^ = 0, cpxz = \pi dv/dz. Therefore ju dv Sxz = (2.5.27) ~2a(T-T*)~dz' No other components of s exist, and Transforming to the x\ z' axes which are inclined at 45° to x, z, sxz 0 0 0 0 0 (2.5.28) 0 0 -sv 70 2. Statistical theories of nematic order Therefore x'yz represent the principal axes of the order parameter tensor. The difference between the dielectric constants (at optical frequencies) for polarizations along the x' and z' axes is where, as in (2.5.18), As is the dielectric anisotropy when all the molecules are exactly parallel to one another. Putting Ss = 2ndn and As = 2nAn, the flow birefringence where An is the birefringence when the molecules are all perfectly parallel. The flow birefringence may therefore be expected to show an anomalous increase as the temperature approaches the transition. This was observed by Tolstoi and Fedotov(121) many years ago in PAA. The experiments of Martinoty, Candau and Debeauvais(134) on MBBA have confirmed the temperature dependence predicted by de Gennes's equation (2.5.29) (see fig. 2.5.6). 2.5.5 Comparison with the Maier-Saupe theory It is of interest to examine the relationship between the phenomenological model that we have just discussed and the molecular statistical theory of Maier and Saupe. The free energy of the weakly ordered isotropic phase in the presence of an external magnetic field is, according to the mean field theory, 2 V2 -kBT\n I e x p | ^ r ^ ( 7 7 ^ + ^ a // 2 )cos 2 0|sin0d<9^. (2.5.30) Expanding and integrating jo2 (T-T*)- 0.0762- where T* = A/5kB V\. This expression is identical in form to the free 2.6 Near-neighbour correlations: Bethe's method 71 5 n -i 5 3 ~ 2 45 7\ NI 47 49 51 53 55 57 Temperature (°C) Fig. 2.5.6. Flow birefringence in the isotropic phase of MBBA. Crosses represent the birefringence (3n) per unit shear rate (G) and circles the reciprocal of this quantity. (After Martinoty, Candau and Debeauvais.(134)) energy expansion (2.5.1) of the Landau model. However, it does not yield a satisfactory value of 7*. Since (A/kBTNI V2C) = 4.54, T*/Tm = 0.908. For PAA, TNI = 408K so that 7NI - T* ~ 40 K, whereas empirically r NI -r* - 1 K. Clearly near-neighbour correlations have to be allowed for in the molecular statistical approach to give a better description of the pre- transition effects. A step in this direction was taken in 1973(135~7) which we shall proceed to consider in the next section. 2.6 Near-neighbour correlations: Bethe's method 2.6.1 The Krieger-James approximation The theory is based on a method developed originally by Bethe (138) for treating order-disorder effects in binary alloys. We suppose that every molecule is surrounded by z neighbours (z ^ 3) and that no two of the z neighbours are nearest neighbours of each other. (This implies that in writing down the interactions between the central molecule and its z 72 2. Statistical theories ofnematic order neighbours, we neglect the interaction between any two of the z neighbours.) Let the pair potential between the central molecule 0 and one of its neighbours j be E(9Qj), where 6Qj is a function of the usual spherical coordinates #0, q>0,6p cpp and let every outer shell molecule y be coupled with the remaining (external) molecules of the uniaxial medium by an interaction potential V(9j). The relative weight for a given configuration of a cluster of z + 1 molecules is then flf(00j)g(0^ (2.6.1) 3=1 where The relative probability of the central molecule assuming a certain orientation 80, cp0 is J •f• Jfl/( J- J (2.6.2) while that for an outer shell molecule, say 1, to assume an orientation 015 ^i is JJ/(#oi) S(0i) d(cos 0o) d^ o J... J f l / ( ^ ) g(03) d(cos 0,) d^. (2.6.3) If we postulate that these two probabilities are identical when 0 and 1 have the same orientation, we obtain the following consistency relation due to Chang :(139) eoj) g{6d) d(cos 0,) d J d(cos 90) d<p0 (2.6A) which has to be satisfied for all values of 6, (p. This condition was expressed in a slightly different form by Krieger and James.(140) The relative probability that the central molecule 0 and one of its neighbours, say 1, are oriented along #0, q>0 and 019 cpx respectively is ei)ll V(009tp0;Ol9q>1)=fL001)g(ei)ll ff f f/(0 o ,)^,)d(cos0,)d^,. (2.6.5) 3=2 J J J 2.6 Near-neighbour correlations: Bethe's method 73 Krieger and James postulated that this probability should be the same irrespective of which molecule is regarded as the central one, i.e., ; 0 1 5 cp^ = ^ ( 0 1 ? q>x; 00, (p0), so that g(00) TZI = p (constant), (2.6.6) which again has to be satisfied for all values of 6, q>. We shall suppose that , (2.6.7) V(6}) = - £ Bin P2n(cos 6,). (2.6.8) n The assumption here is that the energy is independent of cp, i.e., that the distribution is cylindrically symmetric. We also ignore the volume dependence of the potential function. For a given value of z, the values of B2, i?4, etc., can be derived in terms of B* at every temperature such that the consistency condition (and hence the thermodynamic equilibrium of the system) is satisfied accurately. All the properties of the system can therefore be deduced in terms of a single parameter B*. By retaining terms up to P4(cos0) in the mean field potential (2.6.8), it is found (137141) that the maximum error in fulfilling Chang's relation (2.6.4) is less than 0.08 per cent for z = 8. Further it has been verified that when terms up to P12(cos 8) are included, the error becomes negligibly small (~ 10"9). The long-range order parameter s is given by ... P 2 (cos 0O) y/(90, (pQ; 0j9 ^ ) d(cos 0O) dy0 d(cos 0 ; ) d ^ = r r • (2-6-9) ... y(009 (p0; 9P (p3) d(cos 0O) d(p0 d(cos 6d) dtp. The internal energy of the system is U = -\NzB*(P2(cos eoj)}, (2.6.10) where </>2(cos0o,)> ... P2(cos 60j) y/(60, <pQ; 6p ^ ) d(cos 0O) dcp0 d(cos 0^) d ^ . (2.6.11) ... y/(0o, <p0; 9p I*,) d(cos 0O) d(p0 d(cos 0§) &q>} J J 74 2. Statistical theories ofnematic order B*/kBT 0.60 0.62 0.64 0.66 'i i 1 i 1 —s I __ i 0.15 \ r phase " Disordered 2nd order / transition S 0.20 - - 1st order transition 0.25 - Ordered phase < 0.30 Fig. 2.6.1. Plot of <P2(cos 0OJ)> (= -2U/NzB*) versus B*/kB Tfor z = 8.(137) At the first order transition temperature the shaded areas are equal so that the Helmholtz free energy of the ordered and disordered phases are the same. At the second order transition temperature <P2(cos6Oj)} = c2 = l/(z—1). The plot of U versus \/T (at constant volume) shows the characteristic sigmoid shape (fig. 2.6.1). The curves for the ordered and disordered phases meet at the apparent second order transition point T*, which is also the temperature at which the short-range order parameter (P 2(cos60j)) in the isotropic phase = l/(z—1). The first order transition point 7 ^ is the temperature at which the shaded areas are equal, i.e., when the Helmholtz free energies of the ordered and disordered phases are the same. The calculation gives (Tm-T*)/TNI = 0.062 for z = 8, 0.04 for z = 4 and 0.03 for z = 3. This is an improvement over the Maier-Saupe value of 0.092, though still much higher than the experimental value of 0.003. The curves for s, <P4(cos#)> and (P2(cos60j)} as functions of tem- perature are shown in fig. 2.6.2. All three parameters change discon- tinuously at Tm; the long-range order drops to zero at the transition, while the short-range order persists even in the isotropic phase. At the transition, s = 0.400 (for z = 8) which is slightly lower than the Maier-Saupe value of 0.4292. Methods involving less stringent assumptions, which have the advantage that the relevant equations are somewhat easier to solve, have since been put forward.(142"4) These are based on approximations used earlier in theories of magnetism. (145146) However, when higher order terms (up to P4(cos0) or higher) are used in (2.6.8), all these methods yield results that are practically identical to those given by the Krieger-James approxi- correlations: Bethe's method 2.6 Near-neighbour — - - - Nematic Isotropic 0.6 0.5 •=-— Order parameter <P2 (cos 60j)) 0.4 —^ 0.3 0.2 <P2 (cos 60j) 0.1 - 1 1 0.90 0.95 1.0 T/Tm Fig. 2.6.2. Short-range order parameter <P2(cos #0;.)> and the long-range order parameters s and <P4(cos#)> versus T/T^ calculated in the Krieger-James approximation.(137) The long-range order disappears at the transition but the short-range order persists even in the isotropic phase. mation. (137141) Lekkerkerker et al.a*7) and Van der Haegen et al.am have used three-particle and four-particle cluster approximations in analogy with a method followed in magnetism. (149) This leads to a slight improvement in (TNI— r*)/7^ I ? the precise value depending on the type of lattice considered. A Monte Carlo simulation study/ 150 ' 151) of a lattice version of the Maier-Saupe theory gives (Tm-T*)/Tm « 0.01. 2.6.2 Antiferroelectric short-range order The vast majority of nematogens are polar compounds but the absence of ferroelectricity in the nematic phase shows that there is equal probability of the dipoles pointing in either direction. Because of this it is generally assumed that the permanent dipolar contribution to the orientational order is negligibly small. However, a simple calculation shows that the interaction between neighbouring dipoles is by no means trivial compared with dispersion forces, particularly in strongly polar materials. We shall now consider a model which takes into account the influence of permanent dipoles and is at the same time consistent with the non-polar character of the medium. (136137) 76 2. Statistical theories of nematic order o Fig. 2.6.3. Preferred orientation of neighbouring dipoles in the end-on and broadside-on positions. However, because of the anisotropic shape of the molecules, situation I is much more important than II in the nematic structure and there results a net antiparallel correlation between neighbouring dipoles. Nematic Isotropic 0.6 - " • . s — — ^ ^ _ 0.4 0.2 (P2 (cos 0O,)> - 0 0 <p 1 (cos eoj): </\ (cos 60j)y ' -0.2 1 1 0.90 0.95 1.0 Fig. 2.6.4. Short-range order parameters (P^cos^.)), <P2(cos^0j)> and the long- range order parameter s versus T/Tm. The negative value of (P^cos^.)) signifies antiparallel ordering. The curves are calculated for A*/B* = 0.5.(137) Suppose as before that the molecules are cylindrically symmetric so that the dipole moment is along the major molecular axis. Now if a dipole is fixed at O as shown in fig. 2.6.3,1 and II represent situations of minimum energy for a neighbouring dipole in the broadside-on and end-on positions 2.6 Near-neighbour correlations: Bethe's method 11 respectively. Evidently, by virtue of the anisotropic shape of the molecule, situation I will be much more important, i.e., there will be a greater tendency for the nearest neighbours to assume an antiparallel orientation. However, the absence of long-range translational order in the nematic fluid precludes the possibility of antiferroelectric long-range order. To express this in a mathematically tractable form we shall resort again to the Bethe approximation. We modify the pair potential (2.6.7) to E(0oj) = y4*P1(cos 00j) - 5*P2(cos 00j), (2.6.12) which favours an antiparallel arrangement of the permanent dipoles, but let the interaction between j and the rest of the medium continue to be (2.6.8) as before. Here P1 is the Legendre polynomial of the first order. Fig. 2.6.4 gives the curves for the long-range parameter s and the short-range parameters <P1(cos0o^)> and <P2(cos#0j)> calculated on the basis of the modified potential, taking A*/B* = 0.5. </>1(cos 0Oj)} is negative signifying antiparallel order. (Tm-T*)/Tm is now 0.059 for z = 8. The first important question that needs to be considered is whether such an antiparallel correlation leads to the right sign and magnitude of the dielectric anisotropy. Proceeding in the usual manner, the principal dielectric constants turn out to be a + | a a ^ + - ^ ( 2 ^ + l ) + ^-<cos6> 0 cos(9 ; .> (2.6.13) J/C B 1 KB 1 J and 1 4nNphF\ x Fju2 , . _ o , M [ -j—— <cos (p0 cos q>i sin 0O sin 0^ , (2.6.14) where the symbols F9 h, etc., have the same meanings as in §2.3.5. Using the theoretically derived s of fig. 2.6.4, e1, e2 as well as s = l(e± + 2e2) calculated from these equations are presented in fig. 2.6.5 for a strongly polar (nitrile) compound of the type studied by Schadt (91) (see fig. 2.3.9(b)). The parameters used in the calculations are: ju = 5 debyes along the major molecular axis, a = 28 x 10~24 cm3 and aa = 15 x 10~24 cm 3 (evaluated approximately by assuming addivity of bond polarizabilities extrapolated 78 2. Statistical theories of nematic order 25 - Nematic Isotropic 20 - I 15 - —-^^^_ 10 - 5 - 1 1 0.85 0.90 0.95 1.0 Fig. 2.6.5. Theoretical variation of the dielectric constants e19 s2 and e = |(e1 2 with T/Tm.a31) The theory predicts that £ in the nematic phase should be slightly lower than the extrapolated value of els, the dielectric constant in the isotropic phase. This is found to be the case experimentally (see fig.2.6.6). to low frequency). Since thermal expansion is ignored, the rate of variation with temperature is reduced especially near the transition, but apart from that it is clear from a comparison with fig. 2.3.9(b) that the dielectric anisotropy £a is of the right order of magnitude. An interesting consequence of the theory is that the mean dielectric constant e should increase by a few per cent on going from the nematic to the isotropic phase because of the diminution in (P^cos 00j)}. This is found to be the case experimentally in a number of strongly positive ma- terials^ 1 ' 130152) (fig. 2.6.6). (A similar increase is seen in some negatively anisotropic materials also, e.g., PAA (90) (see fig. 2.3.9 (a)); this can probably be explained as due to an antiparallel correlation between the longitudinal components of the dipole moments. As far as the transverse components are concerned there will not be on the average any orientational correlation for position I of figure 2.6.3 because of the cylindrically symmetric distribution about the optic axis, but there will be an antiparallel correlation for position II which is, however, likely to be so weak that it can probably be neglected.) Direct X-ray evidence for such antiparallel local ordering in the nematic and isotropic phases of 4/-n-pentyl- and 4/-n-heptyl-4-cyanobiphenyls 2.6 Near-neighbour correlations: Bethe's method 79 18 - 16 J4 I •a I 12 5 10 25 30 35 Temperature (°C) Fig. 2.6.6. Principal dielectric constants of 4/-n-pentyl-4-cyanobiphenyl (5CB); s = i(£i-f2e2) is calculated from the measured values of e1 and e2. The dashed line denotes the extrapolated value of eis. (After reference 130.) (5CB and 7CB), both of which are strongly polar compounds, has been reported by Leadbetter, Richardson and Colling.(153) They have found that the meridional reflexions correspond to a repeat distance along the preferred axis of about 1.4 times the molecular length, which they have interpreted as due to an overlapping head-to-tail arrangement of the neighbouring molecules (fig. 2.6.7). In MBBA, on the other hand, the repeat distance is approximately equal to the molecular length. These studies appear to lend strong support to the model of antiparallel 80 2. Statistical theories ofnematic order 25.7 A Fig. 2.6.7. Schematic diagram of antiparallel local structure in 5CB resulting in a repeat distance along the nematic axis of about 1.4 times the molecular length. (Proposed by Leadbetter, Richardson and Colling.(153)) correlation that we have just discussed. As we shall see in chapter 5 antiparallel ordering has important implications for the phenomena of reentrance and SA polymorphism in polar materials. 2.7 The nematic liquid crystal free surface We now turn our attention very briefly to the nematic liquid crystal surface. A variety of experimental studies have established conclusively that orientationally ordered states, and in certain materials even density modulations, develop in the vicinity of the free surface. We describe below the salient features of these observations. 2.7 The nematic liquid crystal free surface 81 37 35 120 130 140 T(°C) 38r 36 34 160 170 180 190 r(°c) 29 28 -16 -12 -S 0 4 12 16 20 r-r NI (°c) Fig. 2.7.1. Experimental variation of the surface tension with temperature for three nematic liquid crystals, (a) PAA,(156) (b) /?-anisaldazine(156) and (c) 5CB. (158159) Open circles represent measurements using the pendant drop method. Filled circles in (c) are values determined independently by Gannon and Faber(159) using the Wilhelmy plate method. 82 2. Statistical theories ofnematic order Fig. 2.7.2. Schematic form of the density profile p(z) and its gradient \dp{z)/dz\ across the nematic liquid crystal-vapour transition zone. It is a well known result that the gradient of the surface tension (y) versus temperature is directly related to the surface excess entropy per unit area as follows: %- = -(So-SX (2.7.1) where the suffixes o and /? refer to the surface and bulk states. Thus if there is surface ordering SG may be less than Sfi, and y may actually show a positive slope.(154) Such a trend was discovered by Ferguson and Ken- nedy(155) in PAA and some other compounds many years ago and has been confirmed by precise measurements on a number of liquid crystals using different techniques. (1569) Some experimental curves are shown in fig. 2.7.1. The suggestion has been made (154) that the orientational order near the surface is determined by two competing effects: (1) the disordering effect of the spatial delocalization across the liquid-vapour transition zone, and (2) the ordering effect of a surface torque field. The former may be assumed to be proportional to the density profile across the interface and the latter to the gradient of this profile (fig. 2.7.2). The different trends observed in the different materials (fig. 2.7.1) can then be interpreted as arising from the relative strengths of these two opposing effects and their variations with temperature. (160) Light scattering and optical reflectivity studies (161) again reveal the existence of orientational ordering at the surface. For MBBA it is found 2.7 The nematic liquid crystal free surface 83 io 4 - 101 - io1 1.0 l.i QJQo Fig. 2.7.3. High resolution specular reflectivity data for 80CB near the peak due to the formation of smectic-like layers near the free surface of the nematic phase. The open circles refer to the scale on the left and the filled circles to the scale on the right. The temperatures T- TNA are (a) 0.10 °C, (b) 0.21 °C, (c) 0.40 °C and (d) 1.80 °C. It is seen that the peak becomes significantly sharper as the temperature decreases, showing that the number of surface induced layers increases on approaching the nematic-smectic A transition point 7^A. (After Pershan et al.a62)) that the molecular long axis is inclined at about 75° with respect to the free surface, the angle being temperature dependent, while for PAA the angle is zero and temperature independent. Smectic and cybotactic nematic systems may be expected to show an oscillatory density profile near the surface. That this is the case has been strikingly demonstrated by the very fine X-ray reflectivity measurements of Pershan et a/.(162163) (figs. 2.7.3 and 2.7.4). Surface-induced smectic layering is observed in the nematic or isotropic phase that extends into the bulk, the number of layers increasing as the temperature approaches the transition to the smectic phase. 84 2. Statistical theories ofnematic order 10" 10" = (T-T1A)/TlA Fig. 2.7.4. The measured X-ray reflectivity from the free surface of the isotropic phase of 12CB at temperatures very close to the isotropic-smectic A transition point. The step-like form of the intensity curve reveals the quantized nature of the layer growth at the surface. (After Ocko et «/.(163)) Not much progress has yet been made in giving a quantitative molecular statistical description of these remarkable surface phenomena. 3 Continuum theory of the nematic state 3.1 The Ericksen-Leslie theory In this chapter we shall discuss the continuum theory of nematic liquid crystals and some of its applications. Many of the most important physical phenomena exhibited by the nematic phase, such as its unusual flow properties or its response to electric and magneticfields,can be studied by regarding the liquid crystal as a continuous medium. The foundations of the continuum model were laid in the late 1920s by Oseen(1) and Z6cher(2) who developed a static theory which proved to be quite successful. The subject lay dormant for nearly thirty years afterwards until Frank(3) reexamined Oseen's treatment and presented it as a theory of curvature elasticity. Dynamical theories were put forward by Anzelius(4) and Oseen,(1) but the formulation of general conservation laws and constitutive equations describing the mechanical behaviour of the nematic state is due to Ericksen(5>6) and Leslie(7). Other continuum theories have been pro- posed,(8) but it turns out that the Ericksen-Leslie approach is the one that is most widely used in discussing the nematic state. The nematic liquid crystal differs from a normal liquid in that it is composed of rod-like molecules with the long axes of neighbouring molecules aligned approximately parallel to one another. To allow for this anisotropic structure, we introduce a vector n to represent the direction of preferred orientation of the molecules in the neighbourhood of any point. This vector is called the director. Its orientation can change continuously and in a systematic manner from point to point in the medium (except at singularities). Thus external forces and fields acting on the liquid crystal can result in a translational motion of the fluid as also in an orientational motion of the director. 85 86 3. Continuum theory of the nematic state 3.1.1 Conservation laws and the entropy inequality We begin by writing down the conservation or balance laws (Ericksen(6)). We shall employ the cartesian tensor notation, repeated tensor indices being subject to the usual summation convention. The comma denotes partial differentiation with respect to spatial coordinates and the super- posed dot a material time derivative. For example, Tti = dT/dXt, vitj = dvJdXj and T=dT/dt. We shall consider the medium to be incompressible (vt i = 0, where vt is the linear velocity) and at constant temperature (t = T t = 0). We shall assume further that the director is of constant magnitude. This implies that the external forces and fields responsible for elastic deformation, viscous flow, etc., are very much weaker than the molecular interactions giving rise to the spontaneous alignment of the neighbouring molecules. It is indeed a valid assumption in all the static and dynamic phenomena discussed in this chapter. We may therefore conveniently choose n to be a dimensionless unit vector (ntnt = 1 ) . Let the material volume be Vbounded by a surface A. The conservation laws take the following form: Conservation of mass — /?dF=0, (3.1.1) where p is the density. Conservation of linear momentum £ f pv( d V = f f( d V+ f tn dA,, (3.1.2) ai Jv Jv JA where ft is the body force per unit volume and tn the stress tensor. Conservation of energy d fi i - _ f • f 2 2 X l % l dt)v ** Jv JA (3.1.3) where px is a material constant having the dimensions of moment of inertia per unit volume (ML'1), U the internal energy per unit volume, G{ the 3.1 The Ericksen-Leslie theory 87 external director body force (which has the dimensions of torque per unit volume since nt has been chosen to be dimensionless), ti = tn Vj the surface force per unit area acting across the plane whose unit normal is vp and st = nn Vj the director surface force (which has the dimensions of torque per unit area). We assume here that there are no heat sources or sinks. Conservation of angular momentum jA = {^mxJjc + e^n^dV+X {e^ , (3.1.4) JV JA or in vector notation, — [p(rxy)+Pl(nxn)]dV dtjy = I [(rxf) + (nxG)]dF+ | [(rxt) + (nxs)]<U. JV JA Finally, we have Oseerts equation: f Pl nt d V = f (G( +gt) d V+ f w,f d^, (3.1.5) Jv Jv JA where gt is the intrinsic director body force, which has the dimensions of torque per unit volume and whose existence is independent of Gt. Converting surface integrals into volume integrals and simplifying, (3-1.1)—(3.1.5) lead to the following differential equations: p = 0, (3.1.6) PVi=fi + hi,P (3.1-7) U = tn dtj + nn Ntj—giNt, (3.1.8) Pifi^Gi + gt + njij, (3.1.9) where hi - nkj "i, k + gj ni = hi - Xjci "j, ic + gi nP (3.1.10) Nt may be interpreted as the angular velocity of the director relative to that 88 3. Continuum theory of the nematic state of the fluid. It should be emphasized that the stress tensor tn is asymmetric. When ni = 0, (3.1.6)—(3.1.10) reduce to the familiar equations of hydro- dynamics for a normal fluid. In conjunction with the balance equations we make use of the inequality i I sdv^o, where S is the entropy per unit volume. Defining the Helmholtz free energy function per unit volume F=U- TS, we obtain for a system in isothermal equilibrium hi dij + nn Na-gi Nt-F^0. (3.1.11) 3.1.2 Constitutive equations In order to develop the theory, it is necessary to set up constitutive equations for the quantities F, tjt, nn and gi (Leslie(7)). We assume that these quantities are single-valued functions of n0 niJ9 nt and vu. (3.1.12) We now invoke the fundamental principle of classical physics that material properties are indifferent to the frame of reference or the observer.(9) Hence the constitutive equations should be invariant under proper orthogonal transformations. It is seen that ni and vtJ do not transform as tensors. The parameters (3.1.12) must therefore be replaced by ni9 n i p Ni9 a n d dtj. Thus F may be expanded as dF dF d dF • dF d l l 3 dnt dnt dt ' 87Vt- x ddt- dt ij But nt = Ni + Wyii^ Nt-wjtnj9 d — (nt,) = Ntj - wki nk j - (dkj + wkj) nt k and dF d , x dF „ dF dF , dF 3.1 The Ericksen-Leslie theory Therefore . dF Ar dF dF „ dF . dF . dF r V V y dv i ^ 7\rl " tj i> k W ~ A« *^ ^ i V ^ ^ « ^ Pi AT dd 3 tj ^ dnt k J* dF Hence (3.1.11) becomes ) > a (3 ^ -U3) In view of the constitutive assumptions, it is clear that wji9 Nip Nt and dtj can be varied arbitrarily and independently of all other quantities and hence their coefficients must vanish, i.e., or F=F(ni,niJ); (3.1.14) dF dF dF\ ( dF dF Wy«—hn, *.- hw*. y - — «,-—h«, fr- Yn 3 h dn^ ' 8^>Jfc ' dnk J \ dn ^M (3.1.15) nji-^L = 0; (3.1.16) and (3.1.13) reduces to F)F \ / ?\F\ r ,^0. (3.1.17) Let us write the stress and the intrinsic director body force as where the superscript 0 denotes the isothermal static deformation value and the prime the hydrodynamic part. Equations (3.1.14) and (3.1.16) prove that nn does not depend on dtj or Nt so that nn = nQn. Substituting in (3.1.17) 90 3. Continuum theory of the nematic state Since dtj and Nt can be chosen arbitrarily and independently of the static parts fn and gf, tf—g, (3.1.20) t'pdv-g'tN^O. (3.1.21) Further, using (3.1.10) and (3.1.15) ';<+*;«< = '«+& / *r (3-1.22) The incompressibility condition implies that the stress is indeterminate to an arbitrary pressure. Similarly there is a certain degree of indeterminacy in g° and nn for if we replace rfbyy^-jffj/i^ + s? and ni} by p^ + n^ (3.1.8) continues to be satisfied because n{nt = niNi = n.N^ + n.^N, = 0. Thus (3.1.19), (3.1.20) and (3.1.16) become (3.1.23) n k,j t j - — , (3.1.24) nn=P}nt + ^ - , (3.1.25) where p, y and fit are arbitrary constants, while the hydrodynamic components fulfil the inequality (3.1.21). For an isothermal static deformation, (3.1.9) becomes an equation of equilibrium: (3.1.26) Substituting for g* and nn from (3.1.24) and (3.1.25), dF\ dF — - — + ^ + ^ = 0. (3.1.27) 3.1 The Ericksen-Leslie theory 91 3.1.3 Coefficients of viscosity We next consider the nature of t'n and g[, the hydrodynamic components of the stress tensor and the intrinsic director body force. We assume that they are linear functions of ni9 Nt and dtj and omit higher order terms :(5>7) l A2 XT \ ri jikiy k~ ' jik /o i AT, where A and B are functions of ni (at constant T and p). They can therefore be expanded as nt + a4 Sik n, + a5 /if ^ /i^, 2 Sn 3 km B lk = 73 8^ nk + y4 J4Jfc /i^ + yb Sjk nt + y6«, /i^ wfc. Substituting in (3.1.28) and remembering that N^ = du = 0, t'n = («o + a e 4m nk nm) 5n + (ax + a14rffcOT»fc /i m )«, »^ + a13 ^ + a15 ^ /i4 ^ + a16 dki n, nk + a3 nt N, + a4 ^ ^ (3.1.29) where a15 = a9 + a10, a16 = a7 + a n , and ^ = (y0 + 7e dkj nk n,) n{ + y9 dik nk + yx Nt (3.1.30) where But in view of (3.1.22) y9 = a 1 6 -a 1 5 , y1 = a 4 - a 3 . (3.1.31) In addition, t'n and g[ must satisfy the entropy inequality (3.1.21). Substituting (3.1.29) and (3.1.30) in (3.1.21) we get [(a0 + a6 dkm nk nj S{j] dtj + [ax nt n^ dtj - (y0 + ye dkj nk nd) nt d n n km k m 92 3. Continuum theory of the nematic state i.e., ax nt nj dtj + (quadratic in dip Nt) ^ 0. As di:j can be chosen arbitrarily, ax = 0. Also the coefficient of Sjf(= —p say) in tn and that of nt ( = y say) in g[ are arbitrary since du = 0 and ntNt = 0. Putting: a Ml = i4 M* = «13 //2 = a 4 /z5 = a 1 6 we obtain t'a = Mi nk nm dkm nt H, + // 2 w, A^, + // 3 /i4 ^ + // 4rfi4+ ju5 n, nk dki + n% nt nk dkp (3.1.32) g't-^Nt + ^dfl, (3.1.33) where, making use of (3.1.31), ^=^2-^3, /l2 = // 5 -// 6 . (3.1.34) We have omitted the terms of the form/7^, in (3.1.32) and 7^ in (3.1.33) as they do not contribute to the hydrodynamic effects and can be combined with the corresponding terms in (3.1.23) and (3.1.24) respectively. Substituting for t'jt and g{ it is found that the left-hand side of (3.1.21) is a positive and definite quantity if the following inequalities are satisfied :(7) Finally, from (3.1.23), (3.1.24), (3.1.32) and (3.1.33), n PSji ^ K i + Mink nm dkm nt it, On k,j + ju2 ^ Nt + ju3 nt N, + // 4 dn + M5njnkdki + /i6ninkdkj9 (3.1.36) goi+gt yn.-p^-^ + X^ +^ n ^ . (3.1.37) 3.1 The Ericksen-Leslie theory 93 The jus represent the six coefficients of viscosity of a nematic liquid crystal. However, the number of independent coefficients reduces to five if we assume Onsager's reciprocal relations. 3.1.4 ParodVs relation From (3.1.21) we observe that the rate of entropy production per unit volume TS=t'ndij-g'iNi where t'n is given by (3.1.32) and g{ by (3.1.33). Since t'n is an asymmetric tensor it can be resolved into a symmetric component Ytj and an antisymmetric component Zip where Y ij = Mi dkp nk nv nt n, + // 4 dtj + \{pi2 + // 3 ) (nt N, + Nt ft,) "* *, + dkj nk nt), As Zn = —Z^ and dtj = djt, it follows that ^ 4 = 0, and therefore TS-Yfidv-g'tNt. Thus the entropy production can be separated into two parts, one due to the linear motion of the fluid and the other due to the orientational motion of the director. Now h = H x n, where O is the angular velocity of the director. It is then easily shown that where Jn = n^ — n^ is the torque exerted by the director. Consequently Y^Jy may be regarded as 'fluxes' and dip (Qtf —w y) as 'forces'. The relation between the fluxes and forces is (3.1.38) where " K/^5 + Me) (ir Hs Uj + <>js Ur ni)>

%rs = UM2 - Ms) {"i KS Srj + nr nj      S
is)-
94               3. Continuum theory of the nematic state

From Onsager's reciprocal relations in irreversible processes(10) it follows
that [D] is symmetric, i.e.,

or
// 6 -// 5 .                  (3.1.39)
(11)
This is referred to as Parodfs relation.         The number of independent
viscosity coefficients is therefore reduced from six to five.
It is well known that Truesdell(12) is of the view that Onsager's relations
do not apply to phenomena like heat conduction, viscosity and diffusion
since there is no unambiguous way of selecting the fluxes and forces. It
would appear therefore that there may be some doubts as to the validity of
Parodi's relation. Available data indicate that (3.1.39) is satisfied within
experimental limits,(13) but, in any case, the relation has been tacitly
assumed to be true in most discussions.

3.2 Curvature elasticity: the Oseen-Zocher-Frank equations
We have shown that for an incompressible fluid and an isothermal
deformation (see (3.1.14))

We may therefore expand the free energy per unit volume of a deformed
liquid crystal relative to that in the state of uniform orientation as

We neglect higher order terms in the expansion as we are concerned only
with infinitesimal deformations. Since these deformations relate to changes
in the orientation of the director, nitj may appropriately be called the
curvature strain tensor (Frank (3) ). In order to define the components of this
tensor, let us choose a local right-handed system of cartesian coordinates
with n parallel to z at the origin. The components of strain are then

onx     9w
v

Twist:
aw,
6x'

Bend:
9z'     dz
3.2 Curvature elasticity: the Oseen-Zocher-Frank equations                                  95
We ignore dnjdx, dnjdy and dnjdz. The three types of deformation are
shown in fig. 2.3.12. The curvature strain tensor may therefore be written
as

dx           dy    dz
dny         driy dn^                                    (say).
dx          dy    dz                  0       0
0             0         0

Now any of the three types of deformation destroys the centre of
symmetry of the liquid crystal. The strain tensor is therefore an axial
second rank tensor which vanishes identically under a centro-symmetric
operation. Since the free energy is a scalar, the components of k tj also form
an axial second rank tensor:
Vk n
AC               k
K12       k
AC13 1
k —\k                k         k \
^ij   ~      ^21     ^22       ^ 2 3 I •>

l_/c 31   /c 3 2    A:33J

or using an abbreviated one-index notation
Vk      k

K K                                   (3.2.1)

In general kt has nine components, but the presence of symmetry in the
liquid crystal reduces this number.(14) The distribution of molecules around
any point is cylindrically symmetric, so that the choice of the x axis is
arbitrary apart from the requirement that it should be normal to the
director axis z. Therefore

K      0
-kt        K      0
. 0           0    0
Additional symmetry operations reduce the number of moduli even
further:
Enantiomorphic and     polar         k± + 0,                          k2 + 0
Enantiomorphic and     non-polar      x = 0,                             =|
k2 = 0
Non-enantiomorphic     and polar                                      k2 = 0
Non-enantiomorphic     and non-polar kx = 0,                          k2 = 0.
The tensor kijlm (or ktj in the abbreviated notation of (3.2.1)) has 81
96                  3. Continuum theory of the nematic state
components in general, but as a79 a8 and a9 are zero there remain only 36.
The presence of cylindrical symmetry reduces this number to 18 with only
5 independent constants:

kl2     0      — k12                k15            0
k22     0       ^24                 k12            0
0     /Cqq
CO
0                0             0
^24     0       k22                ~^12            0
k12     0      -k12                                0
Lo         0      0        0                   0           ^33-

where k15 = kxl — k22 — k2±. In the absence of polarity or enantiomorphy,
2 = o.
*„ = o.
The free energy of deformation may therefore be written in the form

+ k12(a± + a5) (a2 + aA) - (k22 + £24) (a± ah + a2 aj.                             (3.2.2)
(15)
As the free energy must be positive definite, it can be shown                                     by standard
algebraical methods (16) that

^22

In tensor notation (3.2.2) reads as
F = \klx{s + nit ,)2 + \k22{q + n{ em nk ;.)
+ kM)[nijnji-(niif],                        (3.2.3)
where s = kjkxi is the permanent splay and q = kjk22 the permanent
twist. There is no physical polarity along the direction n in any known
nematic or cholesteric substance. The molecules themselves may be polar
but the absence of ferroelectricity confirms that there is equal probability
of their pointing in either direction. We shall therefore set kY = k12 = 0.
Substituting for F i n (3.1.27) we then obtain
e
~       ijk nj\np   e
pqr Ur, q), fcJ

= 0.     (3-2.4)
It is seen that &24 plays no role in (3.2.4) and can be omitted as far as
equilibrium situations are concerned (Ericksen (17)). When external body
torques are absent {Gi •=0), the solutions of (3.2.4) are

n2 = sin (qz+y/\)
3.3 Summary of equations of the continuum theory               97

where

^ + ? S = 0.                            (3.2.6)
dx* dy*
Equation (3.2.5) describes a cholesteric structure with a twist per unit
length of kjk22. In the absence of enantiomorphy, k2 = 0 and the structure
is nematic. The solutions of (3.2.6) describe the configurations around line
singularities in the structure which we shall consider in some detail in a
later section (§3.5.1).
In vector notation the free energy density may be written in the more
compact form

F = k2(n • x n) + §&n(V • 2 + \k22(n • x n)2 + ^ 33 (n x V x n)2, (3.2.7)
V              n)           V

with k2 = 0 in the nematic case.

3.3 Summary of equations of the continuum theory
In the following sections of this chapter we shall apply the continuum
theory to study the behaviour of the nematic phase in various physical
situations. For convenience we set out below the most important equations
of the theory which we shall be referring to constantly:

pvt=ft + tjU9                          (3.3.1)
px h'i — Gi + gi + nn p                    (3.3.2)
where p is the density of the fluid (assumed to be incompressible and at
constant temperature), px a material constant having the dimensions of
moment of inertia per unit volume, nt a dimensionless unit vector called the
director, vt the linear velocity, f the body force per unit volume, tn the
stress tensor, Gt the external director body force, gt the intrinsic director
body force and nn the director surface stress.
The stress tensor tn may be separated into a static (or elastic) part and a
hydrodynamic (or viscous) part:

tjt = qt + t'ji9                      (3.3.3)
where

$i = -P*ii-^»t.i> (3-3.4) n, nj + JU2 ^ Nt + jus nt N, + //4 dn ^^^^d^n^n^d^ (3.3.5) 98 3. Continuum theory of the nematic state F is the free energy per unit volume given by ,-K2)ninjnhinhp (3.3.6) 2 3 3 (nxVxn) , (3.3.7) N^rit-w^ (3.3.8) 2dtj = vu + v^ (3.3.9) IWt^Vij-v^ (3.3.10) p is an arbitrary (indeterminate) constant, // x ... ju6 are the coefficients of viscosity (generally referred to as Leslie coefficients), and k119 k22 and k33 the elastic constants (or Frank constants). Similarly the intrinsic director body force gt may be written in two parts gi = g°i+g't, (3.3.11) where g°i=yni-PjniJ-dF/dnt9 (3.3.12) g'^KNt + ^dw (3.3.13) y and Pj are arbitrary (indeterminate) constants, and (3.3.14) Also, according to the Onsager-Parodi relation JU2+JU3 = JU6-JU5. (3.3.15) The director surface stress Uji = ^jni + dF/dnij. (3.3.16) For static deformations, = 0. (3.3.17) 3.4 Distortions due to magnetic and electric fields: static theory 3.4.1 The Freedericksz effect The simplest method of measuring the three elastic constants of a nematic liquid crystal is by studying the deformations due to an external magnetic field (Freedericksz and Tsvetkov,(18) Z6cher (2)). The geometry has to be so chosen that the orienting effect of the field conflicts with the orientations imposed by the surfaces with which the liquid crystal is in contact. To develop a static theory of such deformations we apply the equation of 3.4 Distortions due to magnetic and electric fields 99 equilibrium (3.3.17) where Gt is the external director body force due to the magnetic field H. If/y and/ ± are the principal diamagnetic susceptibilities per unit volume along and perpendicular to the director axis respectively, G,=/a//,«,#„ (3.4.1) where / a = / , , - / ± . Let us consider first the case of a nematic film in which the initial undisturbed orientation of the director is throughout parallel to the glass plates. The magnetic field H is now applied perpendicular to the director and to the plates (fig. 3.4.1 (a)). For this geometry, n = (cos #,0, sin 6), H = (0,0,//) and G = (O,O,/ a // 2 sin0). The free energy of elastic deformation (3.3.6) reduces in this case to By straightforward substitution in (3.3.17) and simplification, we obtain the equilibrium condition As is to be expected, the deformation involves the splay and bend moduli, klx and £33 respectively, and not the twist modulus k22. Because of the orienting influence of the glass surfaces, 0 = 0 at z = 0 and d, where dis the thickness of the film. Therefore 9 attains a maximum value #max at z = d/2 and from symmetry considerations 0(z) = 6{d—z). Since d8/dz = 0 at z = d/2, we get o Jo Transforming to a new variable X given by sin X = sin 9/sin 0 r [ M l - sin2 #max sin2 X) +fr33sin2 flmax sin2 Xf l-sin 2 0 max sin 2 A cu. Taking the limit #max = 0 gives H (3A2) '~AtV In other words, deformation occurs only above a certain critical field Hc. This is referred to as the Freedericksz effect. The threshold condition can be used for a direct determination of the splay modulus klv 100 3. Continuum theory of the nematic state z H<H H<Hn — ZZZ H>HC 1 1 ) 1 ) 1 } / / / / (c) H<HC »>Hc/ / / / / I I I I I I II Fig. 3.4.1. The experimental Freedericksz geometries for the determination of the (a) splay, (b) twist and (c) bend elastic constants of a nematic liquid crystal. For H > Hc, the deformation at any arbitrary point can be computed from the expressions*19'20* ..., (3A3) #„ «J 0 Isin 2 ^_-sin 2 0 = I arc sin -, (3.4.4) where q = (k^-k^/k^. Two other important geometries are illustrated in fig. 3.4.1. For n = (cos 0, sin 6,0), H = (0, H, 0) (fig. 3.4.1 (b)), (3.4.5) 3.4 Distortions due to magnetic and electric fields 101 and for n = (sin 0,0, cos 0), H = (//, 0,0) (fig. 3.4.1 (c)), /XJ- (3A6) (21) De Gennes introduced a parameter which he called the magnetic coherence length to define the thickness of the transition layer near the boundary. Consider, for example, a nematic liquid crystal occupying the half space z > 0. Let the wall, the xy plane at z = 0, impose an orientation along x and let the magnetic field be applied along y (analogous to the geometry of fig. 3.4.1 (&)). If cp ( = \n — 6) is the angle made by the director with the field, the equilibrium condition is easily shown to be where £ = (k2Jx$H~x. Integrating, subject to the boundary condition
that when z -> oo, cp -> 0 and dcp/dz -> 0,

£ is the magnetic coherence length. It is usually of the order of a micron for
a field of 104 G and increases with diminishing field. If the sample thickness
d is very much greater than £, most of the material will be aligned in the
field direction.
The experiment for the determination of klx or A;33 consists of measuring
the variation of birefringence for light incident normal to the film. With
linearly polarized light incident and a suitable analyser (e.g., a combination
of a A/4 plate and a linear polarizer), the transmitted intensity shows a
sudden change when the field attains the threshold value. A measurement
of Hc in the geometries (a) and (c) of fig. 3.4.1 therefore gives the elastic
constant kxl or ksz directly. As the field is gradually increased further, the
intensity exhibits oscillations because of the change in phase retardation
(fig. 3.4.2). The observed variation in the retardation is found to be in
conformity with that expected from (3.4.3) and (3.4.4).(20) In principle,
measurements beyond Hc using the geometry (a) of fig. 3.4.1 should yield
both klx and A:33.
However, the threshold for a twist deformation cannot be detected
optically when viewed along the twist axis. This is because of the large
birefringence (Sn) of the medium for this direction of propagation (the case
/? <^ y in §4.1.1). Thus with the experimental geometry of fig. 3.4.1 (ft) in
which the director is anchored parallel to the walls at either end and light
is incident normal to the film, the state of polarization of the emergent
beam is indistinguishable from that of the beam emerging from the
102               3. Continuum theory of the nematic state

~liAAAA/\
123.2 °C

121.3 °Cu

IWIAA/W
119.6 °C
117.5 °C
—JlMAAA/\
-jmmw
115.5 °C

,,-jmm/v
106 °C
101.3 °C

0          1
Magnetic field (kG)
Fig. 3.4.2. Raw recorder traces of interference oscillations due to the change in the
sample birefringence with deformation for hexyloxyazoxybenzene at various
temperatures. Polarizer and crossed analyser are inclined at 45° to the principal
axes of the specimen. The sudden onset of oscillations occurs at the threshold field.
The increase in the threshold for the successive traces illustrates the rapid
temperature variation of the elastic constant. Sample thickness 45 /zm. 7^ =
128.5 °C. (After Gruler, Scheffer and Meier.(20))

untwisted nematic. For this reason Freedericksz and Tsvetkov(18) used a
total internal reflexion technique by letting the light beam fall at an
appropriate angle on the specimen contained between a convex lens and a
prism. A simple and more direct method has been proposed.(22) If the
ellipsoid of refractive index is viewed obliquely, say at 5 or 10° to the
director (fig. 3.4.3), the effective Sn is reduced to a low value and the
deformed medium can be shown to be optically equivalent to a rotator and
3.4 Distortions due to magnetic and electric fields               103

If !•:•• 1
f
— - • i!
,:i I

IS
Ill'll1
!•>:•• I
•••

'iliii!1              (a)

07/

Fig. 3.4.3. (a) The usual experimental configuration for the optical observation of
the Freedericksz effect. Light is incident normal to the film. However, for reasons
discussed in the text, this arrangement is unsuitable for observing a twist
deformation, (b)l Oblique' configuration which enables the optical detection of a
twist deformation.(22) The magnetic field is perpendicular to the plane of the paper
in both cases.

a retarder (the case /? ~ y in §4.1.1). Hence a twist deformation produces a
change in the state of polarization of the emergent beam which can be
detected by the usual optical methods. In fig. 3.4.4 we present k22 for two
compounds determined by this technique.
When the medium is twisted, the principal axes of the equivalent
retarder will evidently be rotated with respect to those of the undistorted
one. The angle of rotation can be measured by observing the concoscopic
interference figures in a convergent beam. Here again it is essential that the
rays make a sufficiently large angle with the twist axis to reduce the
effective Sn. This method has been used by Cladis(23) to determine k22.
It should be emphasized that if the magnetic field is not strictly
perpendicular to the initial undisturbed orientation of the director, the
distortion does not set in abruptly (fig. 3.4.5) and the experimental
104               3. Continuum theory of the nematic state

-30              -20                 -10
r-r NI (°c)
Fig. 3.4.4. Twist elastic constant (k22) versus temperature for PAA and PAP.
Squares, circles and triangles represent independent measurements on different
samples. (Karat. (22))

determination of the elastic constants becomes somewhat unreliable.
When the field is applied exactly at right angles, there is equal probability
of the director turning through an angle cp or — cp with respect to H.
Consequently a number of disclination walls dividing the domains having
different preferred orientations are formed in the specimen (see §3.5.2).
This serves as a useful criterion for checking the alignment of the field.

Orientation at the glass surfaces
Errors may also be introduced by weak anchoring at the glass surfaces. The
usual procedure for obtaining a planar or homogeneous structure is to rub
the glass (which may be coated with a very thin layer of polymer) with dry
lens tissue or cloth along a fixed direction. (24) Berreman has shown that
geometrical factors play an important role in producing such alignment,
for rubbing tends to corrugate the surface.(24) Assuming that both ends of
the molecule have equal affinity for the surface material so that they lie flat
against the surface, which may not always be true, it is obvious that more
elastic energy is required for the molecules to lie across the rubbed
3.4 Distortions due to magnetic and electric fields               105

0.7

0.6

^   0.5

J
0.4

0.3

0.2

0.1

0
i
Magnetic field (kG)
Fig. 3.4.5. Theoretical curves illustrating the relaxation of the Freedericksz
threshold as the magnetic field is tilted away from the normal to the initial
orientation of the director. The calculations have been made for PAA in the
twist geometry for a sample of thickness 12.7 jum. (a) Field normal to the director,
<p = 90°, (b) q> = 89°, (c) (p = 85° and (d) tp = 80°. 0max is the maximum deformation
in the midplane of the sample. (Kini and Ranganath, unpublished.)

(a)                                   (b)
Fig. 3.4.6. The orienting effect of grooves. Extra elastic energy is required for the
nematic director to lie across the grooves on the solid surface as in (a) rather than
to lie parallel to them as in (b).

direction than for them to lie parallel to it (fig. 3.4.6). A simple calculation
shows that this energy difference for the material near the surface is quite
appreciable and almost impossible to overcome by means of a magnetic
field. The anchoring is therefore firm. Deposition of silicon monoxide and
certain other materials on the glass surface at oblique incidence has been
shown to have essentially the same effect as rubbing.(25)
106              3. Continuum theory of the nematic state

If the surface is equally rough in both directions (as can happen if it is
etched and cleaned thoroughly) there would be a tendency for the
molecules to stand upright. Perpendicular or homeotropic alignment may
also be achieved by coating the surface with surfactants in which case the
detailed intermolecular forces probably play a significant part. There is
evidence that homeotropic alignment is not always rigid. The consequences
of weak anchoring on the Freedericksz transition have been discussed by
Rapini and Papoular.(25)

Electric fields
Measurements can also be made using an electric field,(20) or even an
optical field from a laser.(26) There is complete analogy between electric and
magnetic fields as far as the threshold is concerned (except when the
dielectric anisotropy is very large(191)). Above the threshold the analogy
fails in general because the distortion gives rise to a non-uniformity of the
electric field in the medium, a problem which, for all practical purposes,
does not arise in the magnetic field case. An important precaution to be
taken in electric field measurements is that the sample has to be pure to
avoid conduction-induced instabilities (see §3.10).

Capacitance measurements
The deformation at the threshold field in geometries (a) and (c) of fig. 3.4.1
may be detected conveniently by measuring the change in the capaci-
tance/ 2 ^ A slight disadvantage with this technique is that relatively
large areas have to be uniformly oriented which is not easily achieved in
practice. Non-uniform areas and edge effects tend to destroy the sharpness
of the threshold. For optical observations, on the other hand, a well
oriented area of less than 1 mm2 is adequate.

3.4.2 The twisted nematic device
Another configuration of much practical interest is the twisted nematic
film.(2829) The liquid crystal is sandwiched between two glass plates with
the director aligned homogeneously (parallel to the walls). A twist is now
imposed on the liquid crystal by turning one of the plates in its own plane
about an axis normal to the film. A magnetic field above a critical strength
applied along the twist axis results in a deformation as shown schematically
in fig. 3.4.7.
When H = 0, we have n = {cos MX)], sin[#?(z)], 0}, where (p(O) = —(po
and cp{d) = #?0, d being the film thickness. When H = (0,0, H), n =
3.4 Distortions due to magnetic and electric fields                  107

t
H < Hc

t   Hc

Fig. 3.4.7. The twisted nematic cell.

{cos [9(z)] cos [<pO)L C O S [#( Z )1 si n [<P(z)], sin [#(z)]} where # is the angle made
by n with the xy plane. From (3.3.17) we get

and

(3.4.8)

where
f(9) = kxl co
g(9) = (k22 cos2 9 + £33 sin2 9) cos2 9.
After integration, (3.4.7) and (3.4.8) yield

(3.4.9)

and

(3.4.10)

where A and B are constants. Equation (3.4.9) may be rewritten as

(3.4.11)
108               3. Continuum theory of the nematic state

Because of strong anchoring at the walls, 9 = 0 at z = 0 and d, while # max,
the maximum value of 9, occurs at the midplane z = d/2. Since d9/dz = 0
at z = rf/2,
A =           [B2/g(9maK)]+XaH*sm*9max.

Substituting in (3.4.11)

or

Aw)

i//, say.

Similarly from (3.4.8)

7r

Therefore
=2                                           (3.4.12)
J 0
J0
and

(3.4.13)

Transforming to a new variable given by sin X = sin ^/sin 0 max, (3.4.12)
and (3.4.13) become

,=2 r          r__j
and                   Jo L ^ -
Jo
where

1            T 1        1
M{¥)
(si
= [fc33 - 2k22 - (kS3 - k22) (sin 2 W + sin 2 emj]/g(6)   g(0m!lx).
3.4 Distortions due to magnetic and electric fields                109

6 h

Fig. 3.4.8. Computed relative capacitance change AC/C 0 and optical transmission
T between parallel polarizers (both parallel to the director at one of the boundaries)
of a twisted nematic film as functions of H/Hc. The total twist angle is n/2. Film
thicknesses 13 and 54 jum. The threshold for optical transmission increases with the
thickness of the film. (After Van Doorn.(27))

Taking the limit 0 m a x ^ ' ', we have                   ll9   g(0)^k22, dcp/dz
2
2<po/d,B^2k22<po/d,M-        •(k3-2k22)/k'
3             22   and

=2n                                         I
Jo \x.H*- (4tpl/d*){k^-2k22)
or
(3.4.14)

which is the critical field for the deformation to occur. (28) Thus a
measurement of Hc in this geometry gives k22, if kxl and kZ3 are known.
However, for reasons discussed in §4.1.1, the deformation cannot be
detected optically by observation in polarized light at normal incidence
until the field is well above the threshold value Hc. A more convenient way
of detecting Hc would appear to be by measuring the change in the
capacitance (27) (fig. 3.4.8).
A noteworthy feature of the twisted nematic (TN) is that the intensity of
110              3. Continuum theory of the nematic state
the transmitted light (through a pair of polarizers) as a function of field
shows a 'bilevel' behaviour (fig. 3.4.8). It can therefore act as an electric-
ally controllable optical shutter, as was first demonstrated by Schadt
and Helfrich.(29>30) This principle is now finding extensive application
for making liquid crystal display devices (LCDs) for watches, pocket
calculators, automobile dashboards, miniature TVs, etc. A thin nematic
film of positive dielectric anisotropy is sandwiched between two glass
plates, the inside surfaces of which are coated with transparent conducting
material. By surface treatment, the director is aligned parallel to both
plates except that a 90° twist is imposed on the liquid crystal (fig. 3.4.7,
top). Linearly polarized light incident normal to the plates, say with the
vibration direction parallel to the director on the entrance side of the film,
will emerge with its vibration direction rotated through 90° (see (4.1.15)).
The emergent beam will be transmitted by a second polarizer set at the
crossed position with respect to the first one. If now an electric field is
applied, the molecules in the bulk of the sample tend to align themselves
normal to the electrodes when the voltage exceeds a threshold value,
which, by analogy with (3.4.14), is given by

for 2q>0 = n/2, where ea is the dielectric anisotropy. For voltages of the
order of 2 Vc the twist is lost over most of the sample (fig. 3.4.7, bottom), the
polarization is no longer rotated by 90° and the transmitted light is
extinguished by the second polarizer. With parallel polarizers, one can get
the opposite effect, namely bright field in the ON state and extinction in the
OFF state. LCDs are often operated in the ' reflexion' mode: this is done
by having a diffuse reflector on the rear side of the cell. Unlike the dynamic
scattering mode (see §3.10.1) in which conductivity plays a crucial role, the
TN LCD is a field effect device. Therefore, it is advantageous to have high
purity (low conductivity) materials. A major advance in the materials
development was the discovery by Gray and others(31) of highly stable
mesogens of strong positive dielectric anisotropy, like 5CB, 7CB and
related compounds. A wide nematic range, from about —10 °C to 70 °C,
that is necessary for most practical devices, is obtained by making suitable
mixtures of these compounds. With a 90° twist there is equal probability of
the medium acquiring a right-handed or left-handed twist, which results in
the formation of disclination walls. To avoid this, a small quantity of a
cholesteric dopant is added to the nematic mixture in order to favour one
sense of twist throughout the sample. Typically, with such materials, the
threshold voltage is about 1 V, the switching time (for a cell of thickness
3.4 Distortions due to magnetic and electric       fields    111
8 //m) a few tens of milliseconds and the power consumption a //W cm" 2 of
display area. A disadvantage with the TN cell is that the viewing angle is
limited to about ± 45° from the normal, but the advantages far outweigh
this disadvantage. By patterning the electrodes appropriately, it is possible
to display the required information, whether it is digits or letters or any
other symbols.
For higher information content it is convenient to use the dot matrix
configuration, with electrodes as horizontal rows on one glass plate and as
vertical columns on the other. The intersection of a row and a column is a
picture element (pixel). Thus a matrix display with T rows and M columns
V
V
has T x M pixels but only N+M connections. When the number of pixels
becomes large one resorts to an electronic addressing technique known as
multiplexing: each row in the matrix is selected sequentially, while
appropriate data waveforms coded with the information are applied to the
columns.(32) Because of the slow response times of LCDs (~50 ms), each
pixel responds only to the rms of the resulting waveforms. As the number
V
of lines T in the matrix increases, the fraction (I/TV) of the total time that
the selected pixels see the full select pulse decreases, thereby reducing the
ratio J^ ms (sel)/^ ms (unsel). Alt and Pleshko(33) showed that

For T = 100, this ratio is just about 1.1. If the electrooptic response of the
V
display were very steep (like a step function), a small change in the voltage
would produce a large change in the director orientation, and it would be
possible to activate a given pixel without altering the state of the other
V
pixels, but since the response is rather broad (fig. 3.4.8), T is restricted to
about 100 to get a reasonable contrast ratio. To overcome this limitation,
in the pocket TV, for example, an active addressing technique is used in
which each pixel is backed by a thin film transitor (TFT) which enables one
to apply any desired voltage to the ON pixels and zero voltage to the OFF
pixels. With each column replaced by three which are coated with different
pigments, one can get full colour display. Further, the broad electrooptic
response of the material is actually helpful for producing grey scales.
An advance that has extended the application of LCDs to full page
computer terminals and other high information content displays without
having to resort to TFTs is the supertwisted nematic device/ 34 ' 35) It makes
use of the fact that the electrooptic response of the nematic gets
progressively steeper as the twist angle ^ is increased, until at a certain
112              3. Continuum theory of the nematic state

1.0          1.5             2.0         2.5        3.0
Reduced voltage (V)
Fig. 3.4.9. Computed values of the tilt angle 0max in the midplane of the sample
versus voltage for various twist angles <fi for a standard TN mixture. Cell
thickness/pitch = (j>/2n in all cases and the director tilt at the surface 6S= 1°.
(After Scheffer.(32))

value of ^ it becomes infinitely steep. This is illustrated in fig. 3.4.9, which
gives plots of the computed values of the maximum director tilt angle #max
in the midplane of the sample versus voltage for various values of <j> for a
standard TN mixture. In all cases the cell thickness/pitch ratio d/P =
<f>/2n, and the director tilt at the surface 6S= 1°, which is the value
generally obtained with standard nematic materials on rubbed polymer
coatings on glass. As remarked earlier, the steeper the response the higher
the multiplexability, and the infinite slope for <j> = 270° represents the best
condition for rms multiplexing. For higher twist angles the slope is
negative, and in practice the device becomes bistable.(36) The precise value
of </> at which the slope becomes infinite depends on combinations of the
relevant material and device parameters,(37) but for most known nematic
materials it lies in the range 180° < <f> < 360°. The high twist is stabilized in
the cell by the addition of the appropriate quantity of a cholesteric dopant.
Modified forms of these displays have been developed - e.g., the 'dye'
display which has a pleochroic dye dissolved in the nematic material and
requires the use of just one polarizer - but we shall not be discussing them
here. Analytical expressions have been derived(37) which simplify the
computational effort involved in optimizing the material and device
parameters, but one has to rely on numerical modelling to give a complete
description of the dynamical characteristics of these devices.(38) Certain
unusual dynamical effects observed in the TN device, e.g., the 'reverse-
3.4 Distortions due to magnetic and electric fields          113

twist' and 'optical bounce' effects, will be explained very briefly in a later
section while discussing the dynamics of the Freedericksz transition (see
§3.8).

3.4.3 The Freedericksz effect in highly anisotropic nematics: periodic
distortions
The Freedericksz transition discussed in §3.4.1 may be called a 'homo-
geneous' transition since the distortion occurring above the threshold is
uniform in the plane of the sample. In low-molecular-weight nematics,
which as a rule have relatively small elastic anisotropy (klx ~ k33;k119
k33 ~ 2k22), it is the homogeneous transition that is generally observed.
Some polymer nematics, however, are known to exhibit high elastic
anisotropy - an example is a racemic mixture of poly-y-benzyl-glutamate
(PBG) which has klx/k22 =11.4 and k33/k22 = 13.0(39) - and in such cases
more complex types of field-induced deformations are possible.(40)
Let us consider the geometry of fig. 3.4.1 (a): the initial unperturbed
orientation of the director is along x and the magnetic field is applied along
z. Lonberg and Meyer,(41) who investigated PBG in this geometry, found
that above a well defined threshold there appears a periodic distortion with
the wavevector along y (fig. 3.4.10). It emerged from their theoretical
analysis that when klx/k22 is greater than about 3.3, the periodic distortion,
which involves both splay and twist, has a lower threshold than the usual
homogeneous splay distortion, and moreover, close to the threshold the
deformation angles for splay and twist are out of phase with each other
along the direction of periodicity.
The theory of periodic distortions has been discussed thoroughly. (415)
Because of thermal fluctuations, the director orientation is perturbed and
one may therefore write n = (1, ^, #), where 9 and (/> are, respectively, the
splay and twist angles, which are assumed to be small. Bearing in mind the
experimental result, one may assume that 9 and <f> are functions of y and z.
Retaining terms to the first order in 9, (/> and their derivatives, one obtains
the following equations:

where rj = 2z/d, d being the sample thickness, and the subscripts denote
partial derivatives, e.g.,

etc.           (3.4.16)
114               3. Continuum theory of the nematic state

Fig. 3.4.10. Periodic distortion in PBG. Initial unperturbed director is parallel to
the stripes and the magneticfieldperpendicular to the plane of the sample. Sample
thickness ~ 37 //m and distance between two dark bands 32.5 //m. (Lonberg and
Meyer.(41))

The boundary conditions are 6 = <j> = 0 at n = ± 1. Seeking solutions of the
form

(3.4.15) and (3.4.16) lead to the compatibility condition (41)

1
— ^ tanh q 2 — tan qx = 0,
(3.4.17)

where

Qy is the dimensionless wavevector of the periodicity. For given k119 k22, d
and / a , (3.4.17) is satisfied by a certain Hz(Qy) for each value of Qy. In the
limit Qy->0, Hz(Qy)-*Hz(0) which is the splay Freedericksz threshold.
Calculations show that when k1:L/k22 is large enough, Hz{Qy) decreases
3.4 Distortions due to magnetic and electric fields              115

-   0.8   RH

Fig. 3.4.11. Threshold plots of RH (the ratio of the periodic distortion threshold
and the normal (homogeneous) Freedericksz threshold) and of Qyc (the di-
mensionless wavevector of periodicity at threshold) versus k1±/k22. Periodic
distortion is possible only when kxjk22 ^ 3.3; for kxjk22 ^ 3.3 the normal
Freedericksz deformation is favourable. (After reference 42.)

from Hz(0) as Qy increases, reaching a minimum Hz(Qyc) at Qy = Qyc. On
increasing Qy beyond Qyc, Hz(Qy) increases again. Hz(Qyc) is regarded as
the threshold for periodic distortion, and Qyc (and kyc = nd/Qyc) the
wavevector (and wavelength) of periodicity at threshold.
Plots of the ratio RH = Hz(Qyc)/Hz(0) and Qyc as functions of r =
fcn/fc22 are shown in fig. 3.4.11. It is seen that as r decreases RH increases
and Qye decreases. When r tends to a lower limiting value rc ~ 3.3, RH^\,
Qyc-+0, showing that for r < rc, the periodic distortion is no longer
favourable. The transition to the periodically distorted state can be treated
as a second order phase transition with Qy as the order parameter. <43"5)

3.4.4 The constant kls
(46)
Nehring and Saupe put forward the view that linear terms of the second
derivatives of n in the nematic free energy expansion can, in principle,
make contributions of the same order as the quadratic terms of the first
116              3. Continuum theory of the nematic state
derivative, and proposed the following more general expression instead of
(3.3.7):
F = ^ ( V - n ) 2 + y n - V x n ) 2 + ^ ( n x y x n ) 2 + 2yv-(V-n)n]   (3.4.18)
where k'xl = klx-2klz     and k'33 = kS3 + 2kls.
As far as simple splay and bend deformations are concerned, kn and kss
have, in effect, been rescaled and therefore do not yield an estimate of the
absolute value of k13. The term A:13[V • (V • n) n] satisfies the Euler-Lagrange
equation identically, i.e., it does not contribute to the bulk equilibrium
configuration of the director. Thus kls cannot be measured by the usual
techniques. Its influence on the director configuration can be detected only
if the anchoring at the surfaces is weak. (4748) However, Oldano and
Barbero(49) have pointed out that there is a mathematical difficulty in
estimating its value quantitatively. Assuming that 9, the director tilt, and
d8/dz are independent variables at the boundaries, they have shown that
the variational problem does not have a solution corresponding to the k13
term. This leads to a discontinuous variation of 9 at the boundaries, in
contradiction to the basic principles of the continuum theory. Inclusion of
higher order elasticity and surface terms may perhaps restore a continuous
variation of #?(50'51) but little is known at present about these additional
constants. On the other hand, Hinov(52) has argued that a continuous
solution is possible if 9, dO/dz and their variations are treated as dependent
functions.
A determination of kls is of some interest for it has been suggested that
this constant may play a role in certain special situations, as e.g., in the
measurement of the flexoelectric coefficient of a nematic(53) (see §3.13).
Attempts have been made to estimate k13 experimentally using samples
that are weakly anchored at one or both boundaries.(54"6> As an illustration
of the principle involved, we describe one such experiment(56) employing a
' hybrid' aligned cell with weak homeotropic anchoring on one glass plate
and strong homogeneous anchoring on the other. In such a cell the director
makes a small angle 9 with respect to the normal at the weakly anchored
surface. A magnetic field H is then applied perpendicular to the plates to
change the director profile. The tilt angle 9 is measured by optical methods
as a function of //, and the value of k13 extracted from the surface torque
balance equations (assuming the Euler-Lagrange solution to be valid
right up to the surface of the sample). The value was found to be
( 9 ± 4 ) x 10~7 dyn for /?-cyano-//-heptyl-phenyl-cyclohexane at 25 °C. In
view of the difficulties referred to earlier, this can only be taken to be an
approximate effective value which may have contributions from other
3.5 Disclinations                            117

terms influencing the director configuration at the surface. The possible
existence of a spontaneously splayed state in a free standing film when kls
is sufficiently large and the film thickness sufficiently small has also been
investigated/ 57 ' 58)
3.5 Disclinations
5.5.7 Schlieren textures
As remarked in chapter 1, the nematic state is named for the threads that
can be seen within the fluid under a microscope (fig. 1.1.6(0)). In thin films
sandwiched between glass plates these threads can be seen end on. A
typical example of the texture in a plane film of thickness about 10 jum
between crossed polarizers - the structures a noyaux or schlieren textures
- is given in fig. 1.1.6(b). The black brushes originating from the points are
due to 'line singularities' perpendicular to the layer. In analogy with
dislocations in crystals, Frank<3) proposed the term 'disinclinations',
which has since been modified to disclinations in current usage.
The brushes are regions where the director (or the local optical axis) is
either parallel or perpendicular to the plane of polarization of the incident
light. The polarization is unchanged by the material in these regions and is
therefore extinguished by the crossed analyser. Some points have four
black brushes while others have only two. The positions of the points
remain unchanged on rotating the crossed polarizers but the brushes
themselves rotate continuously showing that the orientation of the director
changes continuously about the disclinations. The sense of rotation may be
either the same as that of the polarizers (positive disclinations), or opposite
(negative disclinations). The rate of rotation is about equal to that of the
polarizers when the disclination has four brushes and is twice as fast when
it has only two.
The strength of a disclination is defined as s = |(number of brushes).
Only disclinations of strengths s =+%, —£, + 1 and —1 are generally
observed. Neighbouring disclinations connected by brushes are of opposite
signs and the sum of the strengths of all disclinations in a sample tends to
be zero. At temperatures close to 7^I9 disclinations of opposite signs are
seen to attract each other and coalesce. They may then disappear altogether
*
(^ + 52 = 0) or form a new singularity (s1 + s2 = s').
The significance of these textures was understood by Lehmann(59) and
Friedel,(60) but a mathematical description of the actual configuration
around disclinations was given by Oseen(1) and Frank (3) . The subject has
since been treated in greater detail by Nehring and Saupe,(61) and reviewed
comprehensively in a number of articles.(62~5)
118                3. Continuum theory of the nematic state

0                                        x
Fig. 3.5.1. Director orientation (indicated by arrows) along a polar line making an
angle a. Incident light that is linearly polarized at angle ^ or (j>±n/2 will be
extinguished by a crossed analyser and will give rise to a dark brush.

Consider a planar structure in which the director is confined to the xy
plane (the z axis being normal to the film). Taking the components of the
director to be nx = cos•(/>, ny = sin (/>, nz = 0, and making the simplifying as-
sumption that the medium is elastically isotropic, i.e., k1± = k22 = £33 = k,
(3.3.7) and (3.3.17) reduce to
(3.5.1)
= 0                         (3.5.2)
respectively. We seek simple solutions that are independent of r =
(x2+y2)K The solutions of (3.5.2) are <j> = 0, which is of no interest, and
(3.5.3)

where a = tan" 1 (y/x) and c is a constant. This equation describes the
director configuration around the disclination; the singular line is along
the z axis and the director orientation changes by 2ns on going round the
line. If the orientational order is apolar, it is clear that a rotation of mn
(where m is an integer) in the director orientation </> should correspond to
a rotation of 2n in the polar angle a (fig. 3.5.1). On the other hand, for a
polar medium a change of 2mn in (f> should correspond to a change of 2n
in a. More generally
s = + | , + 1, + § . . . ,    with 0 < c < n   (apolar)
s = ±l, ± 2 , ± 3 , . . . ,   with 0 < c < 2n (polar)
s is called the strength of the disclination.
In terms of the Volterra process<66) one can visualize the topological
features of these disclinations in the following way. Cut the material by a
plane that is parallel to the director. The limit of this cut is a line L called
5.5 Disclinations                            119

* = • *

5=1, C= 0             s=\,c=n/4                S=l,C=7T/2

5=3/2                       5=2
Fig. 3.5.2. Molecular orientation in the neighbourhood of a disclination. (After
Frank.(3))

the disclination line. Rotate one face of the cut with respect to the other by
a relative angle 2ns about an axis perpendicular to the director. Remove
material from the overlapping regions or add material to fill in the voids,
and allow the system to relax. If the axis of rotation is parallel to the line
L, as is true for the case under consideration, the disclinations may be
referred to as wedge disclinations.(67) This distinguishes them from twist
disclinations, which will be considered in §3.5.4.
We shall now show that s = ^number of brushes). If light is incident
normal to the film and linearly polarized at an angle (p with respect to the
120              3. Continuum theory of the nematic state

x axis, it is seen from fig. 3.5.1 that the polarization will be unchanged at
all points on the polar line a and hence will not be transmitted by the
analyser. This will result in a black brush at an angle a. A similar situation
will arise when <j> changes by n/2. The angle between two successive dark
brushes is therefore Aa = A<p/s = n/2s. Thus the number of dark brushes
per singularity is 2n/\Aa\ = 4|s|. Also if the polarizers are turned through
angle co the brushes rotate by an angle co/s. The rate of rotation of the
brushes of the two-brush disclination (s = ± §) is therefore twice as fast as
that of the four-brush variety (s = ± 1). If the polarizers are kept fixed and
the microscope stage is rotated by co the brushes turn by co{s— \)/s.
Observations in polarized light therefore enable one to determine s both in
sign and magnitude. The existence of |.s| = \ in nematic liquid crystals
establishes the absence of polarity in this phase.
If the structure is one in which the director is not normal but inclined
with respect to the z axis (as in the case of smectic C, see §5.8), half integral
values of s are not possible even if the molecular order is apolar.
The molecular orientation in the neighbourhood of a disclination as
given by (3.5.3) is shown in fig. 3.5.2 for a few values of s. For s + 1, a
change Ac in the constant c merely causes a rotation of the figure by
Ac/(1 — s), while for s = 1, the pattern itself is changed.
When the strengths of two neighbouring disclinations are equal and
opposite, the brushes connecting them are circular. By superposition of
solutions of the type (3.5.3)

<f> = </>i + 4>2 = ^i <h + s2 a 2 + c o n s t .
Putting sx = — s2 = s,
(j> = sfi + const.,

where /? = ax —a 2. Thus curves of constant (/> will be arcs of circles passing
through the two disclinations (fig. 3.5.3). For s = \, </> changes by n/2 on
going from one side of the chord to the other (since ft = n—/? or
' = ns — <t>) while for s = 1 it is unchanged. An example of such circular brushes is shown in fig. 3.5.4. 3.5.2 Interaction between disclinations The energy of deformation of an isolated disclination in a circular layer of radius R and of unit thickness is(61>68) W= \FAxAy. (3.5.4) 3.5 Disclinations 121 Fig. 3.5.3. Curves of equal alignment around a pair of singularities of equal and opposite strengths. The orientations marked on the circles refer to the case 5 = 1 , The disclination is supposed to have a core whose energy is not known. To allow for this, we postulate a cut-off radius rc around the disclination and integrate for distances greater than rc to obtain W= Wc + nks2\n(R/rc), (3.5.5) where Wc is the energy of the central region. As R^co, W-^cc logarithmically, i.e., an isolated disclination in an infinitely extended layer has infinite energy, but such a situation does not arise in practice because of the presence of disclination pairs of opposite signs. The energy of a single defect being proportional to s2 according to the planar model that we have just considered, defects of strength \s\ > \ should be unstable and should dissociate into |.s| = \ defects. However, as is evident from fig. 1.1.6(6), stable defects of strength \s\ = 1 (with four brushes) occur very frequently. The reason for this will be discussed in §3.5.3. The interaction between disclinations may be calculated by superposing solutions of the form (3.5.3), t tan" + const. (3.5.6) Proceeding as before, we obtain for a pair of disclinations separated by a distance r12(69) W = nk{s1 + s2f In (R/rc) - 2nksx s2 In (r 12 /2r c ). (3.5.7) The assumption here is that rc <^ r12 <^ R. If s1 = —s 2, E becomes 122 3. Continuum theory of the nematic state {a) Fig. 3.5.4. Brushes connecting a pair of disclinations of equal and opposite strengths, s = 1 and — 1, in nematic MBBA. Crossed polarizers rotated clockwise by 22.5° in each successive photograph. In (d) the directions of extinction are parallel to the edges of the picture. (Nehring and Saupe.(61)) independent of R. (This is also true if there are many defects in the layer and YJ si — O.(7O)) The interaction energy is given by the second term on the right-hand side of (3.5.7). The force between two singularities is therefore 2nks1s2/r12. Accordingly disclinations of like signs repel and those of opposite signs attract, the force being inversely proportional to the distance. The properties of disclinations in nematics bear some striking similarities with screw disclinations in crystals (7172) and vortex filaments in super- fluids/ 73 ' 74) but at the same time there are important differences that cannot be overlooked while drawing detailed analogies. (64) 3.5 Disclinations 123 (a) (b) SO/mi Fig. 3.5.5. (a) s = ± 1 disclinations and (b) s = \ disclination in a nematic film; (left) between crossed linear polarizers; (right) between crossed circular polarizers. (Meyer.(76)) 5.5.5 Non-singular structures (s = ± 1): escape in the third dimension Cladis and Kleman (75) and Meyer(76) showed that the singularity at the origin of the |^| = 1 defect as given by the planar model can be avoided by a non-singular continuous structure of lower energy. The director orientation now 'escapes' in the z direction. Optical observations confirm that this does indeed happen close to the origin of \s\ = 1 defects (fig. 3.5.5). Structures of this type are also formed in thin capillaries (fig. 3.5.6). Let us consider the nematic in a capillary of radius r0 with the director homeotropically aligned at the wall (i.e., with the director normal to the 124 Continuum theory of the nematic state I \ ! \ (a) Fig. 3.5.6. (a) Director 'escape' at the centre of a disclination of strength s = 1 in a thin capillary: the wall alignment is homeotropic and changes by 90° from wall to axis. (Williams, Pieranski and Cladis(77).) (b) Projection of the structure on a plane normal to the capillary axis. Nails signify that the director is tilted with respect to the plane of the paper. Solutions with positive and negative tilts are equally probable. surface). With the planar solution this would lead to the structure s = + 1, c = 0 (see fig. 3.5.2). However, if we allow for the possibility of a director tilt towards the capillary axis (z axis), we may assume in the region 0 < r < r0, nx = sin <j> sin 0, ny = cos sin 0 and nz = cos 0. The energy of
deformation per unit length is then
r
W = \k\ [(V0)2 + sin2 0(V</>)2 + 2 sin 0cos 0{V<p x V0}] dr.            (3.5.8)

The equations of equilibrium (3.3.17) become

(3.5.9)
= 0.
We look for a solution with (j> = ^(a) and 9 = 6{r). Therefore, (3.5.9)
reduce to
- l - f r ? \ - sin 0 cos 9(V</>)2 = 0,    (3.5.10)

8V = 0.                             (3.5.11)
The solution of (3.5.11) is

while 0 = 0(r) can be determined from

1 8 / 80>
= 0.
5.5 Disclinations                                      125

^           S*   T     \           ^
-   ** **        '          ^ ^    -
(   _^ _\^     ^_   f

\ \ I//
(a)                                           (b)
Fig. 3.5.7. Escaped configurations of (a) s = I, c = n/2 and (b) s = — 1 dis-
clinations. Nails signify that the director is tilted with respect to the plane of the paper.

Assuming the boundary condition

0   at r = 0,
= n/2 at r = rQ,
we get
(3.5.12)
This is an escape involving bend and splay (fig. 3.5.6).
In (3.5.8), the first two terms in the square brackets represent volume
integrations, which together give 27^1^1, while the third term is a surface
integration whose value is nks. Thus the total energy

_ (3nk for s = + l,
\nk      for s = — 1.
Interestingly, the energy is independent of r0. On the other hand, the planar
structure gives
W=nk\n(rjrc)+Wc
for s = ± 1, where rc is the core radius and Ec the core energy. As rc is
expected to be of molecular dimensions, the planar solution has much
higher energy than the continuous structure if r0 is large enough for optical
observations. On the other hand if the capillary radius r0 is extremely small,
or the elastic constant very large (as can happen in the vicinity of a
nematic-smectic A transition) the planar solution may be more favourable
energetically.
The escaped configurations for s = 1, c = n/2 (involving bend and
twist), and for s = — 1 (involving twist, bend and splay) are shown in fig.
3.5.7. Structures with the nails pointing in the reverse direction (i.e., with
nz being replaced by —n z) are equally probable.
126                3. Continuum theory of the nematic state

z>0                                                                               z>0
K

z=0

-\//'--—-
z<0                                                                               z<0

= 1                          \ ^--*                          z>0
z>0                                               \ -v - ^    ^

\ w \\\                                                                                                   z=0
\\\ \\ \

z<0
z<0

z>0                                       \ •«.-•//                               z>0
\
1
'   \

1 1
1 1
\           2
z=0
1 1

z<0
/   /   •   *   •   -   •

z<0
•       \

(a)                                                   (ft)
Fig. 3.5.8. Twist disclinations: the director patterns for (<z) s = J, c = 0, n/4 and
7r/2; and (Z?) s = 1, c = 0, rc/4 and n/2. In each case, the disclination line is shown
as a full line in the middle of the z = 0 plane. The patterns for (s, c) and ( —s, —c)
are mirror images of each other.

3.5.4 Twist disclinations
Equation (3.5.2) admits of another type of singular solution (Friedel and
de Gennes (78) ): the director is parallel to the xy plane as before, but <j> is now
a function of x and z. The solution is

(f> = s t2Ln~\z/x) + c

where s is the strength of the disclination and c a constant. The axis of
rotation (z axis) is now at right angles to the singular line (y axis). Thus this
is referred to as the twist disclination. The director fields for a few values of
s and c are illustrated in fig. 3.5.8. Negative strengths are not shown in the
figure, because the structures of (s, c) and ( — s,—c) are mirror images of
5.5 Disclinations                           127

each other. It is seen that all three types of elastic distortions, splay, twist
and bend, are now present whereas planar solutions for the wedge
disclination involve only splay and bend.
In the elastically isotropic medium, the expressions for the energies and
interactions derived for wedge disclinations are exactly applicable to the
present case. Similarly, the non-singular solution (3.5.11) for s = 1 is valid
in this case, except that the energy for the escaped configuration turns out
to be 2nk\s\ per unit length.(76) The structure of the escaped configuration
is, of course, rather more complicated than that for the wedge disclination.
From the nature of the director patterns it is clear that dark brushes of
the schlieren type will not be seen under the polarizing microscope for light
propagating normal to the film (see §4.1.1). Twist disclinations may
therefore be expected to be less conspicuous than wedge disclinations, and
few observations have been reported of their existence in ordinary
nematics. They do, however, reveal themselves under favourable cir-
cumstances in twisted nematics, often as loops separating regions of
different twist.(79)
The Volterra process for creating a loop, i.e., a closed disclination line,
in a nematic is as follows. Let £ be the surface enclosed by the loop L. Call
the two sides of the surface S + and S~. Rotate the molecules in contact with
I + by an angle sn and those in contact with Z~ by —sn about an axis
normal to the unperturbed orientation of the director, where s = ± |, ± 1,
etc., is the strength of the disclination line. At finite distances from S the
director will adjust itself and the resulting configuration will be continuous
everywhere except on I .
We now consider a twist disclination loop in a twisted nematic. The
nematic is supposed to have a planar structure with the director parallel to
the xy plane and an imposed twist of q per unit length about the z axis, and
the disclination loop of radius R is supposed to be in the xy plane. The
director distortions are planar, nx = cos^, ny = sin^, nz = 0. On going
once round the disclination line at any point on the loop, the director
orientation <f> changes by 2ns, the sign of which may be either the same as
that of q or opposite.
The net energy of such a structure is(78)

W = n2ks[2sR In (R/rc) - qR2].

When s and q are of the same sign, W has a maximum value when
128               3. Continuum theory of the nematic state

Fig. 3.5.9. Shrinking of twist disclination loops: (a) thin thread |s| = |, (b) thick
thread \s\ = 1 with an escaped (coreless) structure. Each thread was photographed
twice several seconds apart. (Nehring.(80))

The energy decreases for higher and lower values of R. Thus large loops
with R> Ro may be expected to occur. Smaller loops shrink (fig. 3.5.9) and
disappear. When s and q are of opposite signs, loops may not be expected
to occur at all.
A stability analysis(81) has shown that twist disclinations are less
favourable than wedge disclinations in elastically anisotropic media. This
may explain why the former are so rarely seen in ordinary nematics.
3.5 Disclinations                              129

(a)

Fig. 3.5.10. Singular points in droplets: (a) spherically symmetric radial (hedgehog)
configuration with the director normal to the surface; (b) bipolar structure with the
director tangential to the surface; (c) singular points in a capillary.

3.5.5 Singular points
Point singularities of strength s = ± 1 occur in droplets and can also be
seen in thin capillaries (fig. 3.5.10). To obtain their solutions (68) let us set
nx — cos (/> sin 9, ny = sin <j> sin 9, nz = cos 9, and use spherical coordinates
x = p sin S cos a, y = /?sin^sina, z = pcosS. If we assume that <j> = ^(a)
and 9 = 0(8)9 we find that

while 9 can be obtained from the differential equation

c)29
—:                  = 0.
oo               do
Selecting solutions with 9 -> S for p -> 0,

tan
D-M9I
For \s\ = 1,9 = S. On the other hand, if 9 -> n — S as p -> 0, then c becomes
c + n.
Fig. 3.5.11 gives the director configurations for some typical cases. The
sections through the (x, y), and the (x,z) or (y,z) planes are identical with
the patterns for the +1 and — 1 wedge disclinations in two dimensions.
Any pattern on the left-hand side of fig. 3.5.11 may be combined with any
one on the right to give a possible point singularity.
130               3. Continuum theory of the nematic state
y

s = — 1,    c = n
Fig. 3.5.11. Director field around singular points. Any pattern on the left may be
combined with any one on the right to give the field around a possible point
singularity. (Saupe.(68))

The total energy for a spherically symmetric radial configuration is
E = %nkR.
where R is the radius of the sphere.(82) This configuration (sometimes
referred to as the ' hedgehog' point defect) is realized in droplets with the
director normal to the surface (fig. 3.5.10(a)).
If the boundary condition is tangential, point defects are formed at the
two poles (fig. 3.5.10(b)). The director components in cylindrical polars
may be taken as nr = — sin 6, na = 0, nz = cos 0, and assuming

0 = tan" 1        rz
R2-r
it turns out that for this bipolar structure(82)
E « 5nkR.
Elastic anisotropy modifies the idealized configurations shown in fig.
3.5.10.(83>84) More complex structures with an oblique orientation of the
director at the surface have also been reported/ 85 ' 86)
Point singularities of equal and opposite strengths attract one another
and are annihilated(68'87) (see fig. 3.5.12). As the total energy of elastic
deformation around a point defect increases linearly with the radius of the
3.5 Disclinations                          131

Fig. 3.5.12. A sequence of photographs demonstrating the attraction and
annihilation of singular points of opposite strengths. (Saupe.(68))
132               3. Continuum theory of the nematic state

volume enclosed, it has been suggested that the interaction energy of two
defects grows linearly with separation, analogous to quarks interacting
through a gluon field.(65)

3.5.6 Interaction between disclinations and surfaces
Interaction with a plane surface
From the superposition principle (3.5.6) we know that the director pattern
around a pair of like disclinations located at x = d and — d is given by

X —l

Hence, at all points (0, y) on the midplane z = 0, the director orientation
<p = sn + 0o = 0W. We may therefore conclude that a wall located at the
midplane with the director firmly anchored on it at an angle 0W can be
replaced by an image of the defect. Consequently the wall repels the defect
with a force
/ = nks2/d.

Interaction with an air bubble: formation of a dipole pair
It has been observed that an air bubble in a nematic sample interacts with
a point defect.(87) A (—1) point is attracted to the bubble and settles at
some equilibrium distance from it to form a stable dipole pair (fig. 3.5.13).
We shall consider the analogous, though simpler, problem of the
interaction between a line defect and a cylindrical cavity, the line of
singularity being parallel to the axis of the cylinder(88) (fig. 3.5.14). We shall
assume uniform boundary conditions at the surface of the cylinder, i.e.,
that the director is everywhere inclined at the same angle with respect to the
radius vector. Let the disclination of strength s be at A at a distance D from
the centre of the cylinder of radius R (i.e., r = D and a = 0). Now, let us
place a disclination of strength (l—s) at the centre of the cylinder and
another of strength s at A' at a distance D' such that DD' = R2. The net
orientation at any point P(x, y) on the cylinder is then

y ,| + ,stan(
-DJ               \x-D
which implies a uniform alignment of the director on the surface of the
cylinder. Thus in the presence of a disclination the cylindrical cavity can be
replaced mathematically by two disclinations, one of strength (1 — s) at its
centre and another of strength s at the conjugate point.
3.5 Disclinations                              133

(a)

(b)

Fig. 3.5.13. Stable dipole pairs formed between —1 point disclinations and air
bubbles in a nematic between (a) crossed linear polarizers and (b) crossed circular
polarizers. (Meyer.(87))
134                3. Continuum theory of the nematic state

y

<x,y)

Fig. 3.5.14. A line singularity of strength s is located at A at a distance D from the
centre 0 of a cylindrical cavity of radius R.

The net force on the disclination at A is

s2
D       D—D
J

r0 being a unit vector directed away from the centre towards the singularity.
At large distances from the cylinder (D > R) Consequently, at far off points the cavity behaves as a + 1 disclination located at its centre. A negative disclination (s < 0) at a large distance away will therefore be attracted by the cylinder. In the neighbourhood of the cylinder (D « R), and the disclination is repelled by the cavity. At a certain intermediate distance given by the force / = 0. This represents a position of equilibrium and the disclination and the cavity form a dipole pair, similar to the ones observed by Meyer (fig. 3.5.13). 5.5 Disclinations 135 H H (a) (b) (c) Fig. 3.5.15. Helfrich walls: (a) a twist wall parallel to thefield,(b) a bend-splay wall parallel to the field, and (c) a splay-bend wall perpendicular to the field. On the other hand, a positive disclination will be repelled by the cavity at all distances. 3.5.7 Defects in the presence of an external field Helfrich walls When a nematic of positive diamagnetic anisotropy (ja > 0) is placed in a magnetic field, the director aligns itself parallel or, equivalently, anti- parallel with respect to the field direction. A region of parallel alignment and one of antiparallel alignment can be separated by a wall, inside which the director turns through an angle n. There are three possibilities. (89) (a) A twist wall: The field is along the z axis and the director is confined to the yz plane and twists about x (fig. 3.5.15(#)). The wall, which is parallel to the field direction, is analogous to the Bloch wall in a ferromagnet. (b) A bend-splay wall: The field is along z and the director is confined to the zx plane (fig. 3.5.15 (b)). The transition from + n to — n takes place mainly through a bend deformation, though some splay is also present. The wall is parallel to H and may be compared with the Neel wall in ferromagnetic systems. (c) A splay-bend wall: In this case the transition from + n to — n is predominantly through splay but there is some bend as well (fig. 3.5.15(c)). It is an 'inversion' wall perpendicular to H. The thickness of the Helfrich wall is of the order of 2£, where £ is the magnetic coherence length defined as H'^k/xJ* (see §3.4.1), and the wall energy per unit length 136 3. Continuum theory of the nematic state H<HC H>He H> Hc (a) (b) (c) Fig. 3.5.16. Brochard-Leger walls: (a) initial unperturbed orientation of the director when thefieldHis less than the threshold value Hc; (b) H > Hc: twist wall parallel to thefield,full line for y > 0 and dashed line for y < 0; (c) H > He: splay wall perpendicular to the field. Brochard-Leger walls Such walls are associated with the Freedericksz deformation. (90) With the homeotropic geometry of Fig. 3.4.1 (c), the possible distortions for H > Hc are illustrated in fig. 3.5.16. Since the director tilt has a degeneracy in sign with respect to H, there can arise twist walls parallel to the field (fig. 3.5.16(&)) or splay walls perpendicular to the field (fig. 3.5.16(c)). Similarly with the homogeneous geometry, there can arise bend walls. In this case the wall thickness t ~ £/a, where a2 = 31 1 — d2 t diverges as H-+Hc. The wall energy per unit length W---1 Umbilics Consider a nematic film of negative dielectric anisotropy (ea < 0) aligned homeotropically between glass plates. If an electric field is applied along the director axis (z axis) a distortion will set in when the field exceeds the critical Freedericksz value given by Ec = (n/d)(k3S/Ej. 5.5 Disclinations 137 I I I I I lllll 11111 T T T T T lllll TT TTT iilll T T T T T lllll l l l l l [1 T 1 N \ Fig. 3.5.17. The structure of umbilics of strength 5 = 1 . Nails signify that the director is inclined with respect to the plane of the paper. (The experiment can also be performed with a magnetic field and a negative diamagnetic anisotropy material like a discotic nematic, see §6.5.) Two possible types of distortions are depicted in fig. 3.5.17. In the distorted state we have a n ± component which is degenerate in the xy plane. Therefore, there can be defects in the n ± field, and because of the symmetry in the xy plane, only defects of integral strength can occur. Such defects have been observed*87>68) (fig. 3.5.18) and are called umbilics.m) They are somewhat similar to the s = + 1 of the schlieren texture but differ in detail. Over a distance from the centre of the defect, the director gradually tilts towards the z axis, the tilt angle becoming exactly zero at r = 0. In this sense, umbilics have a collapsed core, but the structure is not exactly the same as that of the s = ± 1 defects discussed in §3.5.3. Planar and linear solitons When a magnetic field is applied normal to a half-integral disclination line in a nematic, there results a domain wall terminating in a singular line. The wall thickness is of the order of £. Figs. 3.5.19(#) and (b) illustrate the director patterns for s = \ and — \. Such walls have been referred to as planar solitons.m) On the other hand, a magnetic field acting on a point defect with a radial configuration will give rise to a cylindrical domain 138 3. Continuum theory of the nematic state Fig. 3.5.18. Umbilics induced by an electric field in a nematic film of negative dielectric anisotropy: crossed polarizers. (Saupe. (68)) (a) Fig. 3.5.19. Planar soliton produced by a field acting on (a) s = \ and (b) s = — \ disclination lines. 3.5 Disclinations 139 ccccc ccccc Fig. 3.5.20. Linear soliton produced by afieldacting on a point defect of strength ^= 1. ending in a singular point (fig. 3.5.20). This has been called a linear soliton. The occurrence of such solitons in the phases of superfluid 3 He, antiferromagnetics, etc. has been discussed by Mineev(93). Other interesting cases have been investigated by Sunil Kumar and Ranganath. (94) 3.5.8 Some consequences of elastic anisotropy Real nematics are, of course, elastically anisotropic. In certain situations, as for example at temperatures close to the nematic-smectic transition, the anisotropy becomes very large and certainly cannot be ignored. We shall now investigate some of the consequences of elastic anisotropy on the properties of disclinations. Solutions for isolated disclinations As before, we shall begin by considering a planar sample in which the director is confined to the xy plane. In such a case, a wedge disclination involves only splay and bend distortions and we need to take into account = j only the splay-bend anisotropy (fcn = fc33). The free energy density (3.3.7) may be written as F=^[l+£cos2(^-a)]/r2, (3.5.13) where (px = d</>/doc, a = tan~ x (y/x), k = ±(k11 + k33) and e = (k11-k33)/ (&n + £33), and the equation of equilibrium is -^)sin2(^-a)], (3.5.14) 2 2 where ^ aa = d <p/da . For any general value of e, (3.5.14) can be solved numerically and energies can be evaluated. (95) However, for small e one can employ a perturbation technique to obtain analytical solutions. (61) For = s # 1, <f> may be expressed as e2y/(2) +.... 140 3. Continuum theory of the nematic state (a) (b) Fig. 3.5.21. The structure of s = \ in the extreme limits of (a) e = (fcn-fcssV^u + A^) = 1 and (6) e = - 1 . Substituting in (3.5.14) and integrating one obtains to a first order in e (3.5.15) where c and c' are constants. The energy of the singularity (for s 4= 1) is (3.5.16) which may be compared with (3.5.5) for the isotropic case. It may be noted that s and — s have different energies because of elastic anisotropy. For s = 1, one obtains the elastically isotropic solution with c = 0 or n/2 to be a solution of (3.5.14) depending upon whether e is negative or positive, and for s = 2 one gets the isotropic solution irrespective of e. In the limit of e = 1, analytical solutions can be obtained for s = — \ and -l : ( 96 ' 97 > (3.5.17) with C = | for s = — \ and C = 2 for s = — 1. The corresponding energy is given by (3.5.18) For s = |, e = 1, one gets a solution with </> = 0 for a between — n/2 and TT/2,and ^ = a —{n/2) for a between n/2 and 3TT/2. In this solution, sketched in fig. 3.5.21 (a), splay is avoided as it requires infinitely greater 3.5 Disclinations 141 0.2 - 1.0 Fig. 3.5.22. Dependence of the energy on e = (k1±—k zz)/(kxl + A;33) for 5 = \ and — | disclinations. Squares are from the Nehring-Saupe approximation and circles from the perturbation calculation in the neighbourhood of s = 1. (After reference 97.) energy than bend when e = 1. The solution for s = |, e = — 1 is also shown in fig. 3.5.21(6). For a general value of e, the corrections can be evaluated by numerical methods. Fig. 3.5.22 gives the energies of s = + | and s = — \ defects in units of k\n(R/rc). The results of the Nehring-Saupe formula are plotted as squares, and those calculated from a perturbation expansion in the neighbourhood of s = 1 are marked as circles. Interaction between disclinations The radial force of interaction between disclination will, of course, be modified because of anisotropy. (97) In addition there will now be an angular component of the force.(98) The physical basis for the angular force can be understood by referring to fig. 3.5.23, which shows the director patterns for two pairs of unlike defects, ( + f, — |) and ( + 1 , - 1 ) , each in two different situations. It is seen that there are significant differences in the 142 3. Continuum theory of the nematic state (-L+!) (-1,+ c= 0 Fig. 3.5.23. The director patterns for (§, — §) and (1, —1) defect pairs in two situations, c = 0 and c = n/2. The double-headed arrow at the centre indicates the director orientation far away from the defect pairs. (After reference 98.) patterns depending on whether the director at large distances is parallel or perpendicular to the line joining the defects. For the ( + |, — |) pair, the central region is predominantly bend for one configuration and pre- dominantly splay for the other. In the case of the ( + 1, — 1) pair, a structure that is mainly splay becomes one that is mainly bend and, moreover, the + 1 defect itself changes from a radial to a circular pattern. Thus depending on the sign of the elastic anisotropy, one or the other configuration is energetically favoured. In other words, given a boundary condition, an angular force comes into play. This force can be computed to a first order in s. It is given by where = \[(O2x-Ol with = s tan" 1 ±tan"1| (3.5.19) x-d 6X = — and 0 y = —. ox dy The positive sign in (3.5.19) is taken for like singularities and the negative sign for unlike singularities. 3.5 Disclinations 143 Fig. 3.5.24. Possible helical configurations of disclination pairs in twisted nematics or cholesterics of large pitch (see figs. 4.2.2 and 4.2.3). The above arguments do not hold for Yjisii = + 1 o r a n v si = + *• This is N because when e = 0, the structure of s = + 1 can be either radial or circular but cannot have the intermediate logarithmic spiral pattern, as was, in fact, first pointed out by Frank. (3) Thus c has to be 0 or n/2 depending on the sign of £. In summary, the presence of elastic anisotropy favours one particular value of the constant c for a disclination pair. This may have a bearing on the structure of disclination pairs in long-pitched cholesterics in which each layer may be regarded as nematic-like. Since each layer is allowed only one value of c, and the layers themselves are rotated about the twist axis, disclination pairs may be expected to adopt helical configurations as illustrated schematically in fig. 3.5.24. As we shall see in §4.2.1, this conclusion seems to be in agreement with experimental observations. 3.5.9 The core structure The nature of the core still remains an interesting unsolved problem. We have seen in §3.1.1 that director distortions have stresses associated with them as given by (3.3.4). In the case of a single disclination the stress is a tension which can be expressed as in the one-constant approximation. Ericksen(99) postulated that since the tension increases on approaching the centre of the defect, below a certain critical radius rc it should be large enough to transform the material from the nematic to the isotropic phase. Thus the core should consist of a 144 3. Continuum theory of the nematic state cylinder of isotropic material. The nematic-isotropic transition being of first order there will be a physical interface between the two regions. Ericksen showed that the surface-director couple vanishes on this interface. By the same argument, it is reasonable to expect a spherical isotropic droplet at the centre of the hedgehog (all radial) singular point (fig. 3.5.10(a)) and isotropic regions near the surface in the bipolar structure (fig. 3.5.10(b)). However, an important parameter that has been ignored in this approach is the surface tension at the interface. The interfacial tension T can be taken into account in an elementary way as is generally done for crystal screw dislocations.(100) The total energy of the disclination in the one-constant approximation, including the energy at the core surface, is E=2nrT+nks2 In (R/r) the minimization of which gives the radius of the core rc = ks2/2T. Typically, rc ~ 10~6 cm. Of course, the complete analysis has to include the surface anchoring energy, latent heat, etc. 3.6 Flow properties 3.6.1 Miesowicz's experiment We shall now discuss the application of the Ericksen-Leslie theory to some practical problems in viscometry.(101) Probably the first precise deter- mination of the anisotropic viscosity of a nematic liquid crystal was by Miesowicz.(102) He oriented the sample by applying a strong magnetic field and measured the viscosity coefficients in the following three geometries using an oscillating plate viscometer: (i) n parallel to the flow; (ii) n parallel to the velocity gradient; (iii) n perpendicular to the flow and to the velocity gradient. In the presence of a strong field, the magnetic coherence length is quite small and one may without sensible error neglect boundary effects and director gradients. The observational data for the three geometries can then be readily interpreted on the basis of the equations given in §3.3. The apparent viscosity for any geometry shear stress Lt rj = velocity gradient 2dtj' 3.6 Flow properties 145 Table 3.6.1. Miesowicz's viscosity coefficients (measurements in 10~2 P) Molecules Molecules Molecules perpendicular parallel parallel to to flow direction to flow velocity and to velocity direction gradient gradient /?-Azoxyanisole 122 °C 2.4 ±0.05 9.2 ±0.4 3.4±0.3 /7-Azoxyphenetole 144.4 °C 1.3 ±0.05 8.3±0.4 2.5±0.3 If we take the flow to be along x and the velocity gradient along y, and xy yx 2 x, y Since director gradients are neglected, the elastic part of the stress tensor t% = 0. Thus, for n = (1,0,0) we have from (3.3.5) or Similarly = X- Miesowicz's results for two compounds are presented in table 3.6.1. 3.6.2 Tsvetkov's experiment In this experiment, a tube containing a nematic liquid crystal is suspended in a uniform magnetic field acting in a horizontal plane and is spun at a constant angular velocity Q about a vertical axis. If the axis of rotation is along z and the magnetic field along y, the components of the bulk velocity of the fluid are vx — — Cly, vy = Qx and vz = 0. Since the director lies in the xy plane n = (cos cp, sin (p, 0). If the diameter of 146 3. Continuum theory of the nematic state the tube is large enough, wall effects and director gradients may be neglected. Therefore (3.3.2) may be written as where g\ is given by (3.3.13) and G{ by (3.4.1). Therefore (3.6.1) where Xx = JU2—JUZ. Below a critical angular velocity Q c , (3.6.1) has the simple solution sin2<p = Q/ft c , (3.6.2) where (3.6.3) Thus when Q < Q c the director makes a constant angle cp with the field. This has been accurately verified by Leslie, Luckhurst and Smith(103) from a study of the electron spin resonance spectrum of a paramagnetic probe dissolved in a nematic liquid crystal which is spun in a magnetic field. The spacing between the hyperfine lines was found to be in quantitative accord with (3.6.2). When Q > Q c (3.6.1) does not have a steady state solution. Assuming director inertia to be negligible O. (3.6.4) For Q > Q c , 2 ^j) ( ^ ^ | Q c ) ^ - g , (3.6.5) where assuming the initial condition that cp = 0 when t = O.(103) Hence the director rotates with a mean angular velocity co = (Q 2 -Q c 2 )i (3.6.6) An alternative method of performing the experiment is to have a stationary sample in a rotating magnetic field. In point of fact Tsvetkov<104) (and later Gasparoux and Prost(105)) used this method and measured the torque exerted by the fluid on the cylinder as a function of Q. With increasing Q, the torque at first increases linearly, reaches a maximum and then starts to decrease. From (3.3.5), the stress tensor 3.6 Flow properties 147 The total torque on a cylinder of length L and radius R is therefore = — VX1 Q c sin 2(p from (3.6.4), where V = nR2L is the volume of the cylinder. When Q < Q,c, T = -Vkxa. (3.6.7) The torque increases linearly with the angular velocity and offers a direct method of determining Ar When Q = Qc, T = \VX*H\ (3.6.8) When Q > Qc (3.6.5) yields the relation 2 tan w sin 2(p = l+tan> 2{QC/Q + (1 - 1 + {Qc/Q + (1 - Q2/Q2)^ tan [(Q2 - Q2)i/ -10]}2' The mean value of the torque is therefore > When Q ^ Qc, f=-^Q2. (3.6.10) Above the critical angular velocity, the torque decreases with increasing Q. The predictions are generally in agreement with observations(105) (fig. 3.6.1). However, the shape of the experimental curve at higher angular velocities appears to be rather sensitive to the nature of the solid surface in contact with the liquid crystal, showing that a complete theory has to take into account boundary effects and the production and migration of disclination walls at the surface.(106) Another method of determining A1 is by studying the damped torsional oscillations of the nematic suspended in a static magnetic field.(107) For i 1, where D is the torsion constant of the wire and V the 148 3. Continuum theory of the nematic state 1.0 0.5 Fig. 3.6.1. Variation of the torque as a function of the angular velocity of the rotating magnetic field. Open circles: theoretical values; filled squares: exper- imental values for PAA, T= 112°C, / / = 2 9 0 0 G (Tsvetkov(104)); crosses: ex- perimental values for MBBA, T=24°C, 77 = 2230 G; triangles: experimental values for the same compound, T = 22 °C, H = 2230 G, but using a solid teflon cylinder immersed in the fluid to measure the torque. (After Gasparoux and Prost.(105)) volume of the sample, the decay in the angular amplitude of the oscillations can be expressed as where K = Xx V/D, neglecting wall effects, backflow, etc. The method is particularly useful for materials with large kv 3.6.3 Poiseuille flow We shall next consider the rigorous theory of Poiseuille flow, i.e., the steady laminar flow, caused by a pressure gradient, of an incompressible fluid through a tube of circular cross-section.(108) We shall suppose that the tube is of infinite length so that end effects can be ignored. Let us choose a cylindrical polar coordinate system rcpz with z along the axis of the tube. In the steady state the only component of the velocity gradient is vZt r = dv/dr. It is natural to expect that the director is everywhere in the rz plane 3.6 Flow properties 149 making an angle 6(r) with z. Thus we seek for the components of the director and velocity fields the solutions nr = sin 0(r), n9 = 0, nz = cos 9{r). vr = 0, v(p = 0, vz = v{r). If a magnetic field with components (Hr9 0, Hz) is applied, the external body force Gz = x^Hr sin 6 + Hz cos 6) Hz, where / a is the diamagnetic anisotropy per unit volume. Substituting in (3.3.1) and (3.3.2) we have in the steady state 2/d0 k HzsinO) = 0, (3.6.11) ar b\ „ , ^ where (3.6.13) (3.6.14) a = -dp/dz, and ft is a constant. Equations (3.6.11) and (3.6.12) are also applicable to flow through the annular space between two coaxial cylinders. For flow through a capillary we put ft = 0 to avoid the singularity in (3.6.12) at r = 0. The two equations can then be used to obtain the orientation and velocity profiles. At high flow rates, the contributions of the elastic terms tend to become small and Xx + X2 cos 2 0 ^ 0 in the absence of a magnetic field. The director orientation then approaches an asymptotic value given by V ^ (3.6.15) or, assuming Parodi's relation (3.3.15), (3.6.16) In ordinary nematics //2 and //3 are both negative and the flow alignment angle 0O is usually small. This equilibrium orientation of the director is 150 3. Continuum theory of the nematic state «> 4.0 i o (c) 3.0 g a, 2.0 0.001 0.01 0.1 1.0 AQInR (cm2 s1) Fig. 3.6.2. Apparent viscosity rj for Poiseuille flow of PAA at 122 °C (homeotropic wall orientation) plotted against the ratio of the flow rate to the radius of the tube. Open circles are experimental data of Fischer and Fredrickson,(110) (a) values obtained from table I, (b) values obtained from table II with //1 = 0, (c) values obtained from table II with ju± = —0.038. (After Tseng, Silver and Finlayson.(111)) Table I Table II ^ = 0.043 (g cm- 1 s-1) ju1 = 0or -0.038 (g cm"1 s"1) H2 = -0.069 H% = -0.068 jus = -0.002 //3 = 0.000 ju, = 0.068 /i4 = 0.068 AiB = 0.047 Ai5 = 0.048 /z6 = -0.023 //6 = - 0 . 0 2 0 ^ - ^ = ^ = -0.067 £/ o -#i, = A, = - 0 . 0 6 8 H - ^ = ; = 0.0705 attained in practice in relatively thick samples at high flow rates so that the aligning effect of the walls has negligible influence. The amount of fluid flowing per second v(r)rdr Jo and the apparent viscosity n = By scaling the radius and the time as r' = hr and f = kt respectively, with k = h2, it is easily shown(109) that in the absence of a magnetic field Q/R is a unique function of aR3. Consequently n plotted versus Q/R should be a universal curve for all tube radii and flow rates. This has been confirmed 3.6 Flow properties 151 0.001 0.01 AQInR (cm2 s ') Fig. 3.6.3. Apparent viscosity rj for Poiseuille flow of PAA versus AQ/nR computed for a tube of radius R = 55.5 jum. The values of/ a // 2 and 0(R), the orientation at the wall, are respectively (a) 0, - T T / 2 , (6) 1.23, -n/2, (c) 0, - T T / 4 , (</) 24.2, - T T / 2 , (e) oo, - 7 i / 2 or 0,0. (After reference 112.) experimentally(110) (fig. 3.6.2). The apparent viscosity increases slightly at lower pressure gradients. Equations (3.6.11) and (3.6.12) with H = 0 have been solved numerically by Tseng, Silver and Finlayson(111) assuming the boundary conditions v(R) = 0, 0(R) = - 7z/2 and 6(0) = 0. The computed apparent viscosity versus shear rate is shown in fig. 3.6.2. The agreement with the experimental data can be seen to be quite good. Calculations have also been made of the effect of an axial magnetic field.(112) The apparent viscosity decreases appreciably in the presence of the field but an experimental study of this effect has not yet been reported. Curves for rj, the orientation and velocity profiles as functions of shear rate and magnetic field are presented in figs. 3.6.3, 3.6.4 and 3.6.5. It is seen from fig. 3.6.3 that the apparent viscosity is very sensitive to 152 3. Continuum theory of the nematic state 0 -0.2 -0.4 -0.6 -0.8 (a) -1.0 -1.2 -1.4 -a/2 0 -0.2 -0.4 -0.6 -0.8 -1.0 -1.2 -1.4 -ir/2 0.4 0.5 0.6 0.7 0.8 0.9 1.0 r/R Fig. 3.6.4. Orientation profile for Poiseuille flow for different shear rates. Wall alignment homeotropic. The values of z a // 2 are 24.2 for (a) and 1.23 for (b). The values of AQ/nR in (a) are (i) 0.003045, (ii) 0.03354, (iii) 0.1345, and in (b) they are (i) 0.001245, (ii) 0.004167, (iii) 0.032438, and (iv) 0.1372. (After reference 112.) 9(R), the director orientation at the boundary. Thus imperfect alignment or weak anchoring can be a serious source of experimental error in the determination of rjam. 3.6.4 Shear flow Consider now the steady laminar flow of a nematic fluid between two parallel plates. If the flow is along x and the velocity gradient along y the components of the velocity and the director are v V x = V y = z = nx = cos 0(y), ny = sin 0(y\ nz = 0, 3.6 Flow properties 153 l.O 0.9 0.8 0.7 g 0.5 0.4 0.3. 0.2 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 y/R Fig. 3.6.5. Velocity profiles for Poiseuille flow; the curves (a) and (b) are for different values oiAQ/nR but for the samefield,while for curves (b) and (c) 4 Q/nR is nearly the same but the fields are different. The values of / a H2 and 4Q/nR are respectively (a) 1.23, 0.001245, (b) 1.23, 0.004167, (c) 24.2, 0.003045 and (d) xaH2 = oo or 4Q/nR = oo (truly parabolic). (After reference 112.) where 6 is the angle made by the director with x. Proceeding as before, the differential equations for the velocity and the director orientation in the presence of a magnetic field (Hx9 Hy, 0) turn out to be(7) ^1 + A2cos2g -n+Tn\*7.) +(ay + W) + 2xa(Hx cos 6 + Hy sin 0) (Hy cos 0-Hx sin 0) = 0, (3.6.17) (3.6.18) where f[6) and g(9) are defined by (3.6.13) and (3.6.14) respectively and a = dp/dx is the pressure gradient and c the constant shear stress applied = | to the fluid. If both plates are stationary a = 0, and taking y = 0 half-way between the plates, c = 0. On the other hand, if there is no pressure 154 3. Continuum theory of the nematic state gradient, and the flow is caused by one of the plates moving at a uniform velocity in its own plane, a = 0 and c + 0. Equations (3.6.17) and (3.6.18) can be solved to yield the apparent viscosity, velocity and orientation profiles under different boundary conditions.(113) If the plate separation is large enough, boundary effects and elastic terms can justifiably be neglected in (3.6.17). For large shear rates and zero magnetic field the director orientation approaches the value 0O, defined by (3.6.15). But if a magnetic field of moderate strength is applied, the orientation profile is modified slightly. Gahwiller(114) has studied this behaviour by measuring the change in birefringence. He used capillaries (5 cm long) of rectangular cross-section (4 mm x 0.3 mm) and measured the rate offlowdue to a pressure gradient. If H is along the flow direction and shear rates are large, we obtain from (3.6.17) and (3.6.18) (/I1 + /l 2 cos2fl)^ = z a // 2 sin2fl. (3.6.19) Gahwiller assumed that the velocity profile may be approximated by the usual parabolic dependence v(y) = vo[l-(4y2/d% (3.6.20) where v0 is the velocity half-way between the plates and d the plate separation (though, as emphasized by O'Neill(115), this assumption may not be strictly valid in practical situations). From (3.6.19) and (3.6.20) tan0*8^oj;//a//2</2. (3.6.21) The phase difference between the two perpendicularly polarized com- ponents when light is incident normal to the plates is then fd/2 \ [n(0)-no]dy, -d/2 where 1 /n2 = (sin2 6/nl) + (cos2 #/«2), where n0 and ne are the ordinary and extraordinary refractive indices. If the magnetic field is applied along the velocity gradient, 2 2 d (3.6.22) and S± can similarly be calculated. In the absence of a magnetic field, an( Thus the three measurements So, <5y and d± yield JU3//I2, JJLJXB. ^ 3.6 Flow properties 155 1.5 1.0 0.7 IT a •s 0.4 0.2 0.1 20 30 40 50 60 Temperature (°C) Fig. 3.6.6. The viscosity coefficients rj19 rj2 and rjs of MBBA as functions of temperature. The temperature scale is linear in T~x. (After Gahwiller.(114)) At high magnetic fields the experiment reduces in effect to Miesowicz's method except that Gahwiller extended it to arbitrary orientations of the magnetic field. If 9 is the angle between the director and the flow direction and cp that between the projection of the director on the yz plane and the velocity gradient, then, neglecting secondary flow (see §3.6.5), one may write approximately rj(9, (p) = jux sin2 9 cos2 9 cos2 cp —1// 2 sin2 9 cos2 cp + IJUS COS2 6 +1// 4 4-1//5 sin2 6 cos2 cp +1// 6 cos2 6 = rjx cos2 6 + (rj2 + jux cos2 9) sin2 8 cos2 cp By choosing 9 and (p appropriately, one can determine rj19 rj2, rjs and juv Using these two sets of data, Gahwiller was able to determine all five independent viscosity coefficients as well as / a . Some of his results are presented in figs. 3.6.6 and 3.6.7. 156 3. Continuum theory of the nematic state 0.1 - 0.06 - 0.03 60 80 100 120 Temperature (°C) Fig. 3.6.7. The viscosity coefficients rjx, rj2 and //3 of /?-n-hexyloxybenzylidene-//- aminobenzonitrile (HBAB) as functions of temperature. The temperature scale is linear in T~\ (After Gahwiller. (114)) +3 - 40 60 80 100 Temperature (°C) Fig. 3.6.8. The ratio // 3 /// 2 versus temperature for MBBA (crosses), HBAB (open circles) and 1:1:1 molar mixture of HBAB, /?-n-butoxybenzylidene-//-amino- benzonitrile and ^-n-octanoyloxybenzylidene-^'-aminobenzonitrile (filled circles). (After Gahwiller.(114)) 3.6 Flow properties 157 Gahwiller discovered that nematics that undergo a transformation to the smectic phase at lower temperatures exhibit an unusual type of instability as the temperature approaches the transition point. The limiting value of the flow alignment angle 0O defined as (see (3.6.16)) decreases rapidly and becomes zero at a certain temperature, below which the steady laminar flow breaks up into many irregular domains. He interpreted this as a reversal in the sign of ju3 below the critical temperature (fig. 3.6.8). Under these circumstances there is no equilibrium value of 90, and in the absence of an orienting effect due to the walls or a strong external field the flow becomes unstable/ 116 ' 117) The effect is associated with the formation of cybotactic (smectic-like) clusters in the nematic phase in the vicinity of the transition to the smectic phase. Consequently, such materials have two distinct flow regimes which have been the subject of several detailed investigations. (155) When I^/AJ > 1, (3.6.15) is evidently not valid. Steady state solutions now exist in which the director is rotated by many turns on going from one plate to the other.(118) However, such a cycloidal configuration may be expected to be unstable. 3.6.5 Transverse pressure and secondary flow We now examine in greater detail the case of oblique orientation of the director in the Miesowicz experiment. Let the flow be along y (in a plane Poiseuille geometry) and the velocity gradient along z, but the director is oriented by a strong magnetic field in the xy plane at an angle <f> with respect to x. (The velocity is assumed to be small enough not to give rise to any distortion in the director orientation.) Under such circumstances theory predicts that there should develop a transverse pressure gradient and secondary flow along x.{119) Let the sample thickness (measured along z) be d, and its lateral width (along x) be /. Strictly speaking, the finite lateral width / of the sample and the appropriate boundary conditions have to be taken into account in treating this problem exactly, but since / is small we shall make the simplifying assumption that there is no secondary flow vx but that a pressure gradient p x builds up. We take n = (cos <j>, sin <j>, 0) throughout the sample, and thus ignore director gradients. In the steady state, the x component of the momentum transport equation (3.3.1) yields giVy,z=P,*z (3.6.23) 158 3. Continuum theory of the nematic state 90° Fig. 3.6.9. (a) Ratio of transverse to longitudinal pressure gradient versus director orientation <j> in plane Poiseuille flow of nematic MBBA. Sample thickness d = 200 jum. Length of cell L = 4 cm and lateral width / = 4 cm. (b) Deflection y/ of flow lines with respect to the y axis versus <f> in wide cells (L/l ~ ^ ) . Full line represents theoretical variation. (From Pieranski and Guyon.(120)) and the y component yields (3.6.24) where = £l Oil ~ Vz) S m ^ C 0 S ^ Imposing the boundary conditions vy{±\d) = 0, we have from (3.6.24) vy=pjx*-±d*)/2g2 and from (3.6.23) the transverse pressure gradient The transverse pressure difference was directly demonstrated by Pieranski and Guyon(120) by measuring the liquid level difference in tubes connected to holes facing each other across the width / of the cell. Its dependence on (j) predicted by (3.6.25) was verified (fig. 3.6.9(a)). It vanishes for <p = 0 and n/2. Further, it was confirmed that it changes sign when the flow is reversed or when the field is rotated so that </> becomes — <f>. For wide cells (1 |> L, the length of the cell measured along y) it was found by observing the motion of dust particles that the flow lines are not along y (in other words, the assumption that vx = 0 is no longer valid). In 3.7 Reflexion of shear waves 159 Transmitted wave (damped) Nematic Fig. 3.7.1. Experimental arrangement for studying the reflexion of ultrasonic shear waves at a solid-nematic interface. the central region of the cell the flow lines were deflected by an angle y/(<f>) with respect to y, which again showed the expected dependence on cp (fig. 3.6.9(6)). Though this experiment deals with a particularly simple situation, the basic principle that it illustrates is relevant to the discussions that follow on hydrodynamic instability (§3.11). 3.7 Reflexion of shear waves The viscosity coefficients may also be determined by studying the reflexion of ultrasonic shear waves at a solid-nematic interface. The technique was developed by Martinoty and Candau. (121) A thin film of a nematic liquid crystal is taken on the surface of a fused quartz rod with obliquely cut ends (fig. 3.7.1). A quartz crystal bonded to one of the ends generates a transverse wave. At the solid-nematic interface there is a transmitted wave, which is rapidly attenuated, and a reflected wave which is received at the other end by a second quartz crystal. The reflexion coefficient, obtained by measuring the amplitudes of reflexion with and without the nematic sample, directly yields the effective coefficient of viscosity. To explain the principle of the method we shall consider the simpler case of normal incidence. Let a shear wave be incident along z with its vibration direction along x, and let the nematic director be anchored firmly at the interface (z = 0) along y. The only non-vanishing component of the velocity of the fluid is vx = v, and the velocity gradient is along z. From (3.3.5) the stress across the interface is (neglecting director gradients) (3.7.1) 160 3. Continuum theory of the nematic state and from (3.3.1) (3.7.2) Now the velocities associated with the incident, reflected and transmitted waves may be written as vt = Ai exp (— F s z) exp (icot), vr = Ar exp (F s z) exp (icot), vt = At exp ( - Tn z) exp (icot), respectively, where F may be complex and the subscripts s and n stand for solid and nematic. Therefore, in the nematic medium tzx = -ZnAt exp (-Tnz) exp (icot), where Z n = |// 4 F n is the mechanical impedance, and pvt = X(8 2iV8z 2). We thus have or F n = (1 -\-i)(pco/ju^ (3.7.3) and Z n = |(1 + i) (pcoju^y. (3.7.4) The complex reflexion coefficient is given by A y _ z r _ 111 _ ^ s ^n Ax Z s + Z n' where Z s is the mechanical impedance of the solid which may be assumed to be a real quantity. If r= |r|exp(-i0), l-Hexp(ifl) n s l + |r|exp(i0) = ^s i . .12 , OI«I /3- (3.7.5) Since according to (3.7.4) the real and imaginary parts of Z n are of equal magnitude, it follows that -^t (3.7.6) 3.8 Dynamics of the Freedericksz effect 161 Thus a measurement of \r\ at once gives Z n , which, in turn, yields //4. Similarly, for the director oriented along x at the interface and for the director along z at the interface The theory can be generalized to the case of oblique incidence.(122) Martinoty and Candau found that the viscosity coefficients determined by the ultrasonic technique compare fairly well with those derived from capillary flow. 3.8 Dynamics of the Freedericksz effect 3.8.1 Twist deformation We shall now extend the theory of the Freedericksz effect to study the dynamical behaviour when the magnetic field is switched on or off suddenly.(123) The analysis is particularly simple for a twist deformation (fig. 3.4.1 (b)) because the torsion exerted on the director does not result in a translational motion of the centres of gravity of the molecules. Neglecting director inertia in (3.3.2) we obtain the following equation of motion for this geometry: d28 2 . W 22 8 ? *a Sm C0S x dt ~ ' where kx is the twist viscosity defined by (3.3.14). If 8 is small, where The most general solution satisfying the boundary conditions 8 = 0 at z = • Neglecting higher harmonics and remembering that 8 has a maximum value #max at z = 0, we take 8 = 8m^(t) cos (nz/d). 162 3. Continuum theory of the nematic state Equation (3.8.1) then gives or - 1] exp {- (2 Thus 0max(O attains the value 0(oo) with a time constant T(H) given by H2). (3.8.2) If the field is now reduced to a value less than i/ c , the decay rate is still given by the same expression only with a negative sign. If the field is switched off from H > Hc to zero, and the decay rate T-\0) = (kJX1)(n*/d*\ The twist viscosity can be determined from a measurement of T. (123) Typically, T(0) for a film of 25 jum is about 10"1 s. This gives an idea of the order of magnitude of the relaxation time for most nematic liquid crystal devices. 3.8.2 Homeotropic to planar transition: backflow and kickback effects The other two geometries used in the Freedericksz experiment are more interesting as they result in a new effect, namely, hydrodynamic flow induced by orientational deformation. This is the inverse of the more familiar property of flow alignment that has been discussed at length in previous sections. Let us consider the homeotropic to planar transition (fig. 3.4.1 (c)). For this geometry, n = (sin 0,0, cos 0), 0 = 0(z), v = vx(z), and vz(z) = 0. Setting klx = fc33 = k, cos 0 « 1 and sin 0 « 0, we get from (3.3.2) ^ ^ ^ = 0 (3.8.3) dz dt dz where and r = (l2- 3.8 Dynamics of the Freedericksz effect 163 Neglecting inertial effects, (3.3.1) reduces in the present case to where, using (3.3.3), ,dvT d6 hx = K/^ + ^ - ^ - ^ + ^ Q p neglecting squares and higher powers of 9. Therefore where a = f(//4 + //5 —fii2) a n d b = ju2. The boundary conditions for n and v are 9 = 0 and vx = 0 at z = ±d/2. The solutions are of the form 9 = 0o[cos qz-cos (qd/2)] exp (t/x\ (3.8.5) vx = i;0[sin qz-(2z/d) sin (qd/2)] exp (t/z). (3.8.6) Substituting in (3.8.3) and (3.8.4) we obtain the following relations, i=° (3-8-7) and / H\2 Ay/2 (iff/A) — tan y/ ,. o o , = (18 8) (HJ le p-tan, ' - where A = k'b/a and y/ = qd/2. From numerical calculations, Pieranski, Brochard and Guyon(123) have shown that the relaxation rate can be expressed in the form T~\H) = (xJ^)(H2-Hl\ (3.8.9) where the apparent viscosity A* is now strongly dependent on H. The translational velocity (3.8.6) has two components in the simplest case: one linear in z and the other oscillatory, the wavelength of the latter diminishing with increasing value of the final field H. The transient velocity profile is illustrated schematically infig.3.8.1. The effect of this backflow is to relax the constraints, i.e., to reduce the apparent viscosity. In the planar to homeotropic transition (fig. 3.4.1 (a)) backflow effects are not usually so pronounced near the threshold. In this geometry, the torque exerted by the director on an elementary volume of the fluid is 164 3. Continuum theory of the nematic state Fig. 3.8.1. Velocity profile in the homeotropic to planar transition (seefig.3.4.1 (c)). The velocity has two components, one linear in z and the other oscillatory, the wavelength A of the latter diminishing with increasing value of H. Fig. 3.8.2. Torques acting on an elementary volume of the fluid when the molecules are rotating with angular velocity D: (a) homeotropic to planar transition, F a = |(2 1 —A 2)Q; (b) planar to homeotropic transition, Tb = ^(<X1 + X2)Q. As a rule, |FJ > | r j . and where Q is the angular velocity (fig. 3.8.2). In many nematic liquids, Ax and X2 are of opposite signs and of comparable magnitude (see, e.g., legend of 3.8 Dynamics of the Freedericksz effect 165 30 25 20 r(0) r(h) 15 10 0 a 10 15 20 Fig. 3.8.3. Theoretical and experimental normalized relaxation rates as functions of h2 = H2/H\. Open circles and triangles are respectively the experimental and theoretical values for the homeotropic to planar transition. Closed circles are the experimental values for the planar to homeotropic transition. The line represents the variation expected from (3.8.9). The departure from this line for the homeotropic to planar transition is a consequence of backflow. Material: MBBA. (After Pieranski et al.a23)) fig. 3.6.2) so that F is small. In other words, A is small and the solutions of (3.8.7) and (3.8.8) reduce to y/ = n/2 and as for the twist geometry. The marked difference in the relaxation rates for the two geometries at higher fields confirms the existence of backflow as predicted by the Leslie equations (fig. 3.8.3). Backflow has also been studied by direct observation of the motion of disclination walls separating two regions of opposite tilt in a film which is subject to a magnetic field.(124) It is instructive to consider the geometry of fig. 3.4.1 (a) and to examine the relaxation when the field is switched off from H 5> Hc to zero. Clark and Leslie(125) have analysed this problem theoretically and have presented the equations in a form that reduces considerably the computational effort required for making detailed predictions in any practical situation. Using 166 3. Continuum theory of the nematic state Fig. 3.8.4. (a) Velocity profiles at (i) 1.09 s, (ii) 1.744 s, (iii) 2.397 s, (iv) 2.943 s, (v) 4.469 s, (vi) 4.905 s and (vii) 8.61 s after the magnetic field is switched off from H ^> Hc to zero in the Freedericksz experiment of fig. 3.4.1 (a). The values are computed for a sample of MBBA of thickness 200 jum using the equations of Clark and Leslie.(125) (b) Director orientation profiles at (i) 0 s, (ii) 0.109 s, (iii) 0.654 s, (iv) 3.161 s, (v) 6.648 s after switch off. (U. D. Kini, unpublished.) their equations the velocity and orientation profiles at different instants of time after the field is switched off have been calculated and are shown in fig. 3.8.4(a) and (b). The velocity is zero at the boundaries (z = ±d/2) and at the middle of the sample (z = 0), and the profile is antisymmetrical about the midplane. At intermediate regions (z ~ ± d/4) the fluid first moves one way and then with passage of time reverses direction before coming to rest. The director orientation in the middle of the sample initially tips over to an angle greater than n/2 and then gradually relaxes to 6 = 0. These have been termed as backflow and kickback effects respectively. The physical explanation of these effects will be clear from fig. 3.8.5. (126) The initial director orientation profile is shown on the left (fig. 3.8.5(a)). 3.9 Light scattering 167 d/2 J V x -d/2 (a) (b) Fig. 3.8.5. Interpretation of backflow and kickback effects (see text(126)). The elastic torque will be greatest around z = ± d/4 where the curvature is greatest, and this is balanced by the magnetic torque. When the field is switched off the unbalanced elastic torque causes a clockwise rotation of the director in this region (as indicated in fig. 3.8.5(b)). Because of the coupling between director rotation and hydrodynamic motion, fluid flow is induced as shown by the long arrows. The fluid motion, in turn, results in a counterclockwise torque on the director in the middle region (z ~ 0). This overcomes the elastic torque, which is weak in this region, and gives rise to a counterclockwise rotation of the director. The director tilts over to an angle greater than n/2; when it relaxes back to 6 = 0, it produces a small amount of fluid motion in the reverse direction. Finally at large values of t the system settles down to the undistorted equilibrium configuration. This transient effect manifests itself in a direct way in the behaviour of a twisted nematic cell (see §3.4.2). When the external field (assumed to be sufficiently strong) is switched off, the light transmission shows an ' optical bounce effect', i.e., it does not decrease monotonically but rises again to a peak before decaying to its 'off' value. Calculations have confirmed that the peak in transmission corresponds approximately to a perpendicular alignment of the director in the central portion of the cell. This is caused by fluid motion, which also gives rise to a reverse-twist. (124127) 3.9 Light scattering 3.9.1 Orientational fluctuations One of the most striking features of the nematic liquid crystal is its turbidity. From systematic observations of the Rayleigh scattering from oriented samples, Chatelain (128) showed that the scattered intensity is strongly depolarized and exhibits a marked angular variation. An early model put forward to explain this phenomenon assumed the medium to be composed of swarms, about 1 jum in diameter, of aligned molecules, the 168 3. Continuum theory of the nematic state orientations of the different swarms being uncorrelated. However, it is now well established(129"31) that the light scattering can be interpreted rigorously in terms of the small amplitude orientational fluctuations as described by the continuum theory. The intensity of this scattering turns out to be very much larger than that arising from the density fluctuations in the fluid, so much so that the latter contribution can be neglected altogether. Let co0, k0 and i be respectively the angular frequency, wavevector and unit polarization vector of the incident beam and co^ kx and f the corresponding quantities for the scattered beam. The scattering process is associated with an angular frequency change co = co0 — co1 and a wavevector change q = ko-k1. We define the differential scattering cross-section per unit volume of the scatterer, per unit solid angle (Q), per unit angular frequency change as to dfidcu where X = 2nc/co0 is the vacuum wavelength and is the mean square fluctuation of the dielectric constant at a given point r and time t. For the uniaxial nematic medium, the dielectric tensor at any point r can be written as (see §2.3.1) eif = s + e&(ntnf-lX (3.9.2) where e = (e||+2e 1 )/3 is the mean dielectric constant (at optical fre- _ quencies), £a = £||— eL is the dielectric anisotropy assumed to be <^£, n{: = n • i and nf = nf. An electric vector polarized along i induces a displacement Df = eifEt along f. The director at r n(r) = n0 + dn, 5 where, to a first approximation, < n • n0 = 0, since we assume the fluctuations to be of small amplitude. Thus, neglecting density fluctuations (i.e., assuming s and £a to be constant) we have from (3.9.2) 3.9 Light scattering 169 (c) Fig. 3.9.1. (a) The two uncoupled modes Snx and Sn2; (b) components of the deformation in the SnY mode, bend and splay; (c) components of the deformation in the 3n2 mode, bend and twist. where i0 = n0 • i and / 0 = n0 • f. Therefore .(r,/)>. (3.9.3) The fluctuations can be analysed into Fourier components. For a given Fourier component of wavevector q, we may conveniently resolve Sn into two components 5nx and Sn2, the former in the q, n0 plane and the latter perpendicular to it (fig. 3.9.1). Let us therefore introduce two unit vectors ex = e2 x n0, where qL is the component of q perpendicular to n0. Defining ia = e a i and /a = e a f ( a = l , 2 ) , 170 3. Continuum theory of the nematic state Hence <Se2} = el [ £ (ijo +/aio)\6nJL09 0) SnJLr, t))]. (3.9.4) a The scattering cross-section is then dV/dQdco = n*lr*el £ (i a / 0 + /0/o)24(q,<*>), (3.9.5) a where Too aq,w j ^ na q, and <S«a(q, /) = (snJLr, t) exp (iq • r) dr. (3.9.7) Here (iaf0 + i0fa)2 is a purely geometric factor while / a is a correlation function which describes the power spectrum of the fluctuations. For a = 1, the director vibrates in the n o ,q plane and the mode is a superposition of bend and splay. For a = 2, the vibration is normal to the n0, q plane and the mode is a superposition of bend and twist. The two modes are shown schematically in fig. 3.9.1. 3.9.2 Intensity and angular dependence of the scattering It is of interest to consider first the intensity of the scattered light integrated over time (or frequency). The differential scattering cross-section is then — = 7l2A~*el YJ 0'a/o + *o/a)2<<^a(q)> (3.9.8) a where Writing <5n(r, t) = <5nexp [i(q • r — cot)] and substituting in (3.3.6) we obtain the free energy of elastic deformation of the system where 3.9 Light scattering 171 Optic axis Q Fig. 3.9.2. A typical experimental configuration used by Chatelain in his light- scattering studies: k and i are the wavevector and unit polarization vector of the incident light, k' and f the corresponding quantities for the scattered light, the suffixes e and o denote the extraordinary and ordinary polarizations with respect to the optic axis of the nematic medium and qx is the wavevector change on scattering. From the equipartition theorem (which is certainly valid in the present problem) where kB is the Boltzmann constant. To get an idea of the order of magnitude of the scattering cross-section let us suppose that k±1 « k22 « kS3 = k\ then dcr ^ (nea\2 ^B T (3.9.10) dQ~VlV ~W' On the other hand, the cross-section due to density fluctuation is given by the well known formula (3.9.11) where /? is the isothermal compressibility. Taking p(ds/dp) ~ ea, da'/da ~ 0kq2. Typically P ~ 10"11 cm2 dyn"1, k ~ 10"6 dyn, q ~ 104 cm"1, so that da'/da ~ 10"8. 172 3. Continuum theory of the nematic state 150 r- 100 I 50 1 0 ^ 50 100 150 200 cot 2 (<p/2) Fig. 3.9.3. Angular dependence of the intensity of scattering for PAA. Circles give the experimental values of Chatelain and the line represents the theoretical variation. (After de Gennes.(129)) Thus the director fluctuations make the predominant contribution to the light scattering, as was first pointed out by de Gennes.(129) A comparison of the polarization factors in (3.9.5) and (3.9.11) at once explains why the light scattering from a nematic liquid crystal is strongly depolarized. The angular dependence of the light scattering is also accounted for in a straightforward manner. Let us, for example, consider one of the geometries used by Chatelain (fig. 3.9.2). The incident and scattered beams are both normal to z; the incident beam is linearly polarized in the plane of scattering while the scattered beam is polarized along z, the optic axis of the medium. If the angle of scattering is <p9 : k0 sin (p/2), qz = 0, ix = cos (<p/2), i , = / 1 = / 2 = 0, fz=L Therefore dcr dQ (3.9.12) 3.9 Light scattering 173 Fig. 3.9.3 compares this relation with the experimental data of Chatelain and as can be seen the agreement is good. In principle a measurement of the intensity of the scattering and its angular variation offers a method of determining the elastic constants. 3.9.3 Eigenmodes and the frequency spectrum of the scattered light For a given mode of vibration of angular frequency co and wavevector q, we can write <5n(r, t) = Sn exp [i(q • r — cot)], Substituting in the basic equations of motion (3.3.1) and (3.3.2) we obtain icopvk = &k-qk(qj^/q2\ (3.9.13) 0. (3.9.14) We ignore inertial effects as well as terms quadratic in Sn; J%. = i where t'ik is the viscous part of the stress tensor defined by (3.3.5), and F is the elastic energy density. As the fluid is supposed to be incompressible v k, k = 0 o r Qk vk = 0. lfv19 v2, vz are the components of v in the frame e19 e2, z, we have dzz = kz vz, dxl = iq± v1 = - \qz vz, where q2 = q\ + q\. Substituting in (3.9.13) and (3.9.14) iQ(q) vz + [Llx co + ^(q)] Sn, = 0,1 { } iCa(q)i;2 + [UlG> +fca(q)](J/i2= 0 j ' ' Sn, = 0,1 K } v2\pco-iP2(q)]-icoQ2(q)Sn2 = 0j ' where K) q\+(K - K 174 3. Continuum theory of the nematic state \+vm q\ ql)/q\ Is = For compatibility of (3.9.15) and (3.9.16) we have the vanishing of the determinant, and therefore [pa - iPa(q)] [xcol, + kJLq)] - C a (q) Qa(q) = 0 or ) = 0, (3.9.17) 6 3 which has two roots. Typically, k ~ 10~ dyn, p ~ 1 g cm" ,77 ~ // ^ O.li5, 2 , C(q) Consequently /?A:a(q) is negligible compared to ^ ^(q) and Ca(q)Qa(q), and therefore the two roots of (3.9.17) are P M ( 3 9 1 8 ) (3919) The subscripts s and f denote 'slow' and 'fast', for cos - ikq2/rj, cot - ir/q2/p, and cojcot ~ pk/rj2 ^ 1. We observe that both modes are purely dissipative (non-propagating); cos involves the elastic constants while cof does not. The slow mode therefore represents the relaxation of the orientational motion of the director, while the fast mode may be looked upon as the diffusion of a vorticity but one in which there is no torque on the molecules. The light scattering, being dependent primarily on the orientational fluctuations, is controlled entirely by the slow mode, as we shall proceed to show. We have resolved the director fluctuations into dnx and Sn2 which from the symmetry of the problem can be seen to be uncoupled. If Gx and G2 are 3.9 Light scattering 175 the forces responsible for these tilts in the director, then in a first order theory dn1=%1G19 ^ « 2 = / 2 G 2 , where / 1 ? / 2 are susceptibilities; more generally We have seen that due to the thermal agitation, there are spontaneous fluctuations in n whose mean square value is defined by 7a(q, co) = (SnJt - q, 0) Sna(q, co)}. According to the fluctuation-dissipation theorem,(132) the relation between and h X, is /a(q, co) = ^ J l m (xJLq, co)), (3.9.20) where Im stands for the imaginary part. We can derive an expression for 7a(q, co) from the equation of motion for the director in the presence of an external field: Accordingly (3.9.15) becomes iC^v^dnJico^ + k^q)] = G19 iC2(q)v2-\-Sn2[icol1-\-k2(q)] = G2. Along with (3.9.16), this can be simplified to obtain AaVM ' ' [pco-iPM][ttiO)+K(<i)]-c«(q)QMv or 7a(q, co) = —y^ —Y—T~T\ 2°\ > (3.9.21) where um = -ico s o c a n d uta = -icof(X. (3.9.22) Thus 7a is a superposition of two Lorentzians. However, as wfa > wsa, we may ignore the second term in the square brackets of (3.9.21) and rewrite the power spectrum as 176 3. Continuum theory of the nematic state 200 150 100 50 0 10 20 30 40 Fig. 3.9.4. Angular dependence of the width of the Lorentzian spectral density for mode 2 in PAA at 125 °C. Open circles denote experimental values in the [ke,k^] configuration and open squares the values in the [ko, k^] case. The curve is obtained by a least squares fit with the theory. (After the Orsay Liquid Crystals Group.(131)) The light scattering is therefore determined entirely by the slow mode. The integrated intensity in agreement with (3.9.10). Also from (3.9.18), (3.9.19) and (3.9.22) (3.9.24) 3.10 Electrohydrodynamics 177 (3.9.25) The integrated intensity gives the elastic constants ku while the half width yields uS(X. It is therefore possible to measure the viscosity coefficients from an analysis of the scattered light using appropriate geometries. As an example, we present in fig. 3.9.4 a convenient geometry for isolating mode 2. The director is aligned parallel to the walls of the glass plates. The incident beam is polarized parallel to the director and the scattered beam perpendicular to it. If ne and no are the extraordinary and ordinary indices of the liquid crystal, and cp the scattering angle, ke = Innjk, k'o = Innjk, qz = k'o sin q>9 q± = ke- k'o cos q>. For small angles of scattering, and from (3.9.25) which enables Xx to be determined. By going to higher angles, it is possible to obtain juJ2fil and rjv/2fi\. A typical curve for the angular dependence of the width of the Lorentzian spectral density is shown in fig. 3.9.4. The experiments are rather difficult because of the large amount of stray radiation, especially in the forward direction, arising from defects in the alignment of the specimen. However, by very careful techniques using a laser light beat spectrometer, and employing various geometries Leger- Quercy(131) has been able to determine four viscosity coefficients of PAA, //2-//5. The values of//1? rj2 and rj3 calculated from these coefficients are in reasonably good agreement with those determined by Miesowicz (see table 3.6.1). 3.10 Electrohydrodynamics 3.10.1 The experimental situation From dielectric studies in the radio frequency region,(133) it has long been known that PAA is negatively anisotropic, i.e., £a = el{ — e± < 0. However, in a number of early investigations on the effect of an external DC electric field it was noticed that the PAA molecules align themselves parallel to the field, rather than perpendicular to it as would be expected of a material of negative dielectric anisotropy. This observation gave rise to some contro- 178 3. Continuum theory of the nematic state Transparent conductive coating Glass LIQUID CR VYSTAL ^> * Spacers Glass t Light beam Fig. 3.10.1. Experimental arrangement for observing Williams domains. versy in the 1930s but it has since been confirmed by the systematic experiments of Carr, (134) who proved that the anomalous alignment is due to the anisotropy of the electrical conductivity of the liquid crystal. His studies showed that there is a critical frequency of the applied field below which the alignment of PAA is anomalous, and that this frequency increases with the conductivity of the material, ranging from 2 to 100 kHz in the samples examined by him. Macroscopic motion of the fluid induced by electric fields was ob- served many years ago by Freedericksz and Zolina, (18) Tsvetkov and Mikhailov,(135) Bjornstahl(136) and Naggiar.(137) Tsvetkov also noted that the flow decreases with increasing frequency of the applied field, and probably recognized the fact that the phenomenon may be connected in some way with the electrical conductivity. More recently, Williams (138) discovered that a thin layer of a nematic material of negative dielectric anisotropy between conducting glass plates forms regular striations when a DC voltage of sufficient magnitude is applied. At higher voltages, the regular pattern gives way to turbulence accompanied by intense scattering of light, which has come to be known as dynamic scattering and has found practical applications in display devices. (139) Similar observations have been reported by other authors. (140) The experimental arrangement for observing the Williams domains is shown in fig. 3.10.1. The nematic film of negative dielectric anisotropy (e.g., PAA or MBBA) is aligned with the director parallel to the glass Fig. 3.10.2. Electrohydrodynamic alignment patterns in nematic liquid crystals, (a) Williams domains in a 38 jum thick sample of /?-azoxyanisole. 7.8 V, 100 Hz. (Penz.(141)) (b) Chevron pattern of oscillating domains in MBBA. Sample thickness ~ 100 //m. Distance between bright lines ~ 5 jum. 260 V, 120 Hz. (Orsay Liquid Crystals Group.(142)) 3.10 Electrohydrodynamics 179 Fig. 3.10.2. For legend see facing page. 180 3. Continuum theory of the nematic state . Fig. 3.10.3. (a) Flow and (b) orientation patterns of Williams domains. The periodic orientation pattern and the consequent refractive index variation has a focussing action for light polarized in the plane of the paper. This gives rise to the bright domain lines as indicated by the stars above and below the sample. (After Penz.(143)) surfaces which are coated with a transparent conducting material. When a DC or low frequency AC field is applied between the transparent electrodes, there appears above a threshold voltage a regular set of parallel striations perpendicular to the initial unperturbed orientation of the director (fig. 3A0.2(a); see, however, §3.13.2). Dust particles are seen to undergo periodic motion in the field of view proving that the domains are due to hydrodynamic motion. The distortion of the director orientation caused by this motion results in a focussing action for light polarized parallel to the director(143) (fig. 3.10.3). This is responsible for the appearance of a set of bright lines with a spacing approximately equal to the film thickness when the microscope is focussed at the top surface. The lines are shifted by about half the spacing when the focal plane is moved down to the bottom surface. The pattern disappears when the light is polarized perpendicular to the director. The threshold voltage is usually a few volts and is practically independent of the sample thickness. It is, however, strongly dependent on the frequency(142) (fig. 3.10.4). There is a cut-off frequency coc above which the domains do not appear, the value of coc increasing with the conductivity of 3.10 Electrohydrodynamics 181 400 - 2 "o 100 - 100 (Hz) 400 600 Frequency (Hz) Fig. 3.10.4. Threshold voltage of the AC instabilities versus frequency for MBBA. Sample thickness 100 jum. Region I: conducting regime (stationary Williams domains); region II: dielectric regime ('chevrons')- Full line is the theoretical curve. The cut-off frequency / c = 89 Hz. (After the Orsay Liquid Crystals Group.(142)) the sample. Below coc, i.e., in the so-called conduction regime, the regular Williams pattern becomes unstable at about twice the threshold voltage and the medium goes over to the dynamic scattering mode. Above coc, in the dielectric regime, another type of domain pattern is observed. Parallel striations, again perpendicular to the initial orientation of the director but with a much shorter spacing (a few microns), are formed in the midplane of the sample. When the field is increased very slightly above the threshold, the striations bend and move to form a chevron pattern (fig. 3.10.2(b)). In this regime, the threshold is determined by a critical field strength rather than a critical voltage. Both the threshold field strength and the spatial 182 3. Continuum theory of the nematic state 150 r Oscillating domains 100 B -a vi Region of stability 50 _—# 50 100 150 200 Frequency (Hz) Fig. 3.10.5. Threshold voltage versus frequency for MBBA. Sample thickness 50/urn. Open circles: sinusoidal excitation; triangles: square wave excitation. (After the Orsay Liquid Crystals Group. (142)) Approaching - optic axis 1 field 1 25 No domains 20 o o "o o " o Optic axis_L field Domains / 10 --_ /C7H15 No domains 5 H 15 C 7 0 1 10 100 1000 10000 Frequency (Hz) Fig. 3.10.6. Frequency dependence of the threshold voltage in /?,//-di-n-hept- oxyazobenzene, a nematic of positive dielectric anisotropy. (After Gruler and Meier.(144)) 3.10 Electrohydrodynamics 183 periodicity of the pattern are frequency dependent - the former increasing with frequency (as OP) and the latter diminishing with it. The relaxation time of the oscillating chevron pattern is a few milliseconds while that of the stationary Williams pattern is typically about 0.1 s for a thickness of 25 jum. The oscillating domain regime is therefore sometimes called the fast turn-off mode. The chevron pattern also gives way to turbulence at about twice the threshold field. An applied magnetic field parallel to the initial orientation of the director increases the threshold voltage in the conduction regime, but has no effect in the dielectric regime except to increase the spacing between the striations. The threshold curve has a pronounced sigmoid shape with square wave excitation (fig. 3.10.5) indicating that at high electric fields there is a quenching of the conductive instability even when co < coc. DC and very low frequency AC voltages produce electrohydrodynamic instabilities in the isotropic phase also (T> Tm), the threshold being comparable to that in the nematic phase. It has been suggested that this is due to charge injection at the electrodes. A frequency of about 10 Hz is usually enough to suppress this effect showing that charge injection is not the primary mechanism for the AC field instabilities in the nematic phase. If the initial alignment of the director is homeotropic, domains as well as turbulence can be produced. The threshold voltage for the domains is somewhat higher than in the case of parallel alignment, but the patterns persist even when the voltage is reduced to a lower value. We have so far discussed only materials of negative dielectric anisotropy. Electrohydrodynamic distortions are observed even in weakly positive materials,(144) but only when the initial orientation of the director is perpendicular to the applied field. Striations appear above a threshold voltage but vanish at still higher voltages and there is no dynamic scattering. The frequency dependence of the threshold voltage is shown in fig. 3.10.6. 3.10.2 Helfrich's theory The basic mechanism for the electric-field-induced instabilities is now quite well understood. The current carriers in the nematic phase are ions whose mobility is greater along the preferred axis of the molecules than perpendicular to it. The ratio of the conductivities o^oL is usually about |. Because of this anisotropy, space charge can be formed by ion segregation in the liquid crystal itself, as was first pointed out by Carr.(134) The manner in which the space charge can build up due to a bend fluctuation is shown 184 3. Continuum theory of the nematic state Fig. 3.10.7. Charge segregation in an appliedfieldEz caused by a bend fluctuation in a nematic of positive conductivity anisotropy. The resulting transversefieldis Ex. schematically in fig. 3.10.7. The applied field acts on the charges to give rise to material flow in alternating directions which, in turn, exerts a torque on the molecules. This is reinforced by the dielectric torque due to the transverse field created by the space charge distribution. Under appropriate conditions, these torques may offset the normal elastic and dielectric torques and the system may become unstable. The resulting cellular flow pattern and director orientations are sketched in fig. 3.10.3. Even a conductivity of the order 10"9 Q" 1 cm"1 is enough to produce this type of fluid motion. Indeed unless very special precautions are taken, the impurity conductivity is usually greater than this value. With a DC field, there may be injection of charge carriers at the solid-liquid interface but its role in the electrohydrodynamics of the nematic phase is not yet fully understood. However, as remarked earlier, a frequency of about 10 Hz is enough to suppress charge injection. We shall therefore neglect it in the present discussion. We shall now outline the theory of electrohydrodynamic instabilities proposed by Helfrich(145) and extended by Dubois-Violette, de Gennes and Parodi(146) and Smith et fl/.(147) We consider a nematic film of thickness d lying in the xy plane and subjected to an electric field Ez along z. Let the initial unperturbed orientation of the director be along JC, and let there also be a stabilizing magnetic field along the same direction. We consider a bend 3.10 Electrohydrodynamics 185 fluctuation in which the director is in the xz plane and makes an angle cp with x. We ignore wall effects and assume that the deflexion cp is a function of x only. Due to the anisotropy of conductivity, space charges will develop as indicated in fig. 3.10.7 till the transverse electric field stops the transverse current. The local transverse field in the steady state is easily seen to be al{ cos2 cp + <7± sin2 cp where cra = o^ — oL > 0, a^ and aL being the principal electrical con- ductivities along and perpendicular to the local director axis. Also, as soon as the electric field is applied there will be a transverse field EEX due to the dielectric anisotropy. Since the transverse displacement is zero, Eex = — [ea cos (p sin ^/(e|( cos2 cp + eL sin2 cp)] Ez. The space charge per unit area produced on any plane normal to the x axis is ± On cos 2 (p + e ± sin 2 (p) (Ez - EEX)/4n, the positive and negative signs standing for the two sides of the plane under consideration. The applied field Ez acting on these charges causes a flow along the z axis; the resulting shear stress is evidently 4 = (e^cos2 (p + eLsm2 cp)(Ex- EEX) EJ An. (3.10.1) From the Ericksen-Leslie theory, we know that the viscous torque is rvisc = n x g ' , (3.10.2) where g' is given by (3.3.13). This is the frictional torque exerted by the molecules on the hydrodynamic motion. Clearly in the present geometry (3.10.2) reduces to ^^t) (3.10.3) where Ax = H2—H3 and X2 = fii~/j,6. Also, in the present geometry, the viscous stress tensor t'n given by (3.3.5) can be simplified to where, making use of Parodi's relation (3.3.15), 186 3. Continuum theory of the nematic state Setting Ty = r elastfy + r dlelfI/ + r m a g f y -r v l 8 C f y = -Aq>9 where A is a force constant, the condition for instability may be written as(148) A = -Ty/<p = 0. (3.10.5) The elastic, dielectric and magnetic torques can be evaluated from the functional derivative where JV and Fis given by (3.3.6). We then have r elast, y = ~ (^33 C O s 2 <P + Kl sir — (fc n — k33) sin cp cos q>(d(p/dx)2, r diei, y = - (47r)- 1 K cos <p sin (p(E2z - El) - ea(sin2 q> - cos2 ^) ^ Ez], F mag y = 0fa cos ^ sin cp) H2. Since we are interested in the threshold conditions, we retain only first order terms in (p, i.e., G 6 \\ \\ Also assuming a spatially periodic fluctuation of the form cp = (p0coskxx, we have JEZ) 3.10.3 DC excitation In the DC case, we may set <p = 0 and the analysis becomes simple. Using (3.10.1), (3.10.3) and (3.10.4), 3.10 Electrohydrodynamics 187 where rj^ = rj'[2X1/(X1 — X2)]2. Applying (3.10.5), we obtain the threshold field (3.10.7) where El = -^(xaH2 + k3Sk2x) (3.10.8) and C is a dimensionless quantity, called the Helfrich parameter, given by (3.10.9) Typically £2 is a small number; e.g., for MBBA it is about 3. For DC (and low frequency AC), kx « n/d. (A numerical solution (149) of the two- dimensional problem with appropriate boundary conditions has confirmed that at the threshold kx is indeed n/d.) Thus when H = 0, we have a voltage threshold independent of film thickness given by \% (3.10.10) where 888 °" ' We have treated the distortion as a pure bend but this is not exactly true. Since the thickness of the sample is of the same order as the periodicity of the distortion, the orientation of the director will vary in the z direction also. There will therefore be a non-negligible splay component. To allow for this, it has been suggested(147) that the bend elastic constant k33 should be replaced by where the wavevector in the z direction kz ~ n/d. With this correction, the DC threshold given by (3.10.10) is in good agreement with the experimental value. For MBBA it is about 8 V. 3.10.4 Square wave excitation In the AC case, the time dependence of cp cannot be neglected. Using (3.10.3), (3.10.5) and (3.10.6) F \/p l visc,y— A \\ H7l b L\ 188 3. Continuum theory of the nematic state In addition we have to take into consideration the conservation of charge. The charge balance equation reads q + dJx/dx = 0, (3.10.12) where q is the excess charge per unit volume and Jx the electric current given by Jx = <7|| Ex + o^Ez(p retaining only the first order terms in cp and Ex. We suppose that diffusion currents make a negligible contribution. From the relation where Dx is the x component of the displacement, we obtain where y/ = d<p/Qx is the local curvature. Therefore + oaE^ = 0, (3.10.14) where T is the dielectric relaxation time given by and 1 1 \ fin Neglecting the inertial term, (3.3.1) may be written as ',<,,+/< = 0, (3.10.15) where/, is the body force per unit volume, which in this case is equal to qEz. We are interested only in the z component of this equation. Since the director orientation q> is assumed to be a function of x only, txz vanishes, and (3.10.15) reduces to or, from (3.10.4), At the threshold, we have the condition o. (3.10.17) 3.10 Electrohydrodynamics 189 From (3.10.3) and (3.10.16) ar v l s c , y = Krj, I l ix1 \ From (3.10.6) and (3.10.13), - — — E2 Therefore, from (3.10.17) we obtain the following equation for the curvature y + yy/ + ?-Ez = 0, (3.10.18) where (3.10.19) T is the decay time for curvature, and rj is an effective viscosity coefficient given by n « Equations (3.10.14) and (3.10.18) are two coupled equations which cannot be solved analytically for any general value oiEz(i) since y itself depends on Ez and t. However for a square wave J + £, 0<t<n/co A ) \-E, n/(D<t<2n/oj. In this case y remains constant in any half-period and the solutions are simpler and enable a physical interpretation of many of the observed phenomena. If the solutions are taken in the form (147) y/ = Cw exp (At/z\ q = Cq exp (At/x), the general solutions become /T)9 (3.10.20) where 190 3. Continuum theory of the nematic state These solutions are valid as long as E is constant. We observe that as q= -<?Jiy/Ex/(\+A1), a change of sign of E implies that q or y/ changes sign but not both, i.e., if (q, y/) is a solution for a half-period, it will be (q, — y/) or (— q, y/) for the other half-period. Subject to these conditions we obtain two sets of solutions given by (3.10.22) and (3.10.23) below: -expH; ± 1—exp (3.10.22) A2t -l-Ao exp Eo^x sx — -exp 2vx exp -\-Ax where v = co/2n is the frequency of the voltage. Here q changes sign with E, but y/ does not, which is the situation in the conduction regime. + expl ^ 2 - ^ 1 \ 2VT (3.10.23) sx — Ax 1+exp 2vx 1 ^T. n A2-Ax • ^ / - + exp 2vx exp In this case y/ changes sign with E, but q does not. This corresponds to the dielectric regime in which the director oscillates. In the above equations, s is a real number which determines whether the system is stable or not, and which takes the value 0 at the threshold. Setting s = 0 and defining 3JO Electrohydrodynamics 191 \ t \ \ j1 \ i / i (a) \ / t \ I ^ q Fig. 3.10.8. Time dependence of the charge q and the curvature y/ over one period of the square wave excitation, (a) Conduction regime (coz <4 1, T> z). The charges > oscillate but the domains are stationary, (b) Dielectric regime (coz ^ 1, T < r). The charges are stationary and the domains oscillate. (After Smith et «/.(147)) it can be easily shown that in the conduction regime <3l0 24) - and in the dielectric regime TTA ±+i = U-l sinh (3.10.25) '2COT' These equations show clearly that the problem falls naturally into two distinct parts. For T > z (3.10.25) has no solution and consequently there is no dielectric regime, while for T < z (3.10.24) has no solution and there is no conduction regime. For T = T, neither equation has a solution (except when co = 0). The theoretical variation of q and y/ over a full period of the exciting wave is illustrated in fig. 3.10.8 for two values of co, one at low frequency in the conduction regime (coz < 1, T > z) and the other at high frequency 192 3. Continuum theory of the nematic state in the dielectric regime (coz > \,T 4, z). In the former case one obtains essentially stationary domains and oscillating charges and in the latter case vice versa. The transition from the conduction to the dielectric regime occurs at a critical frequency coc such that coc z ~ 1. Certain other interesting conclusions can be drawn from the above equations. For example, even at frequencies less than coc, there can be quenching of the conductive instability at high fields. This is because T = (AEl + AQ)" 1 decreases with increasing Ez9 and when it becomes equal to T there can be restabilization. At higher fields, T < z and the system goes over to the dielectric regime. This gives a physical insight into the origin of the sigmoid shape of the experimental threshold curve. Indeed, theoretically the threshold field as a function of frequency for conductive instability may be expected to form a closed loop. Another result that follows from the theory is that the conduction regime can be suppressed altogether by using very thin samples. We have seen that at low frequencies, kx ~ n/d. Now a decrease in sample thickness increases Ao, where from (3.10.19) This, in turn, reduces the curvature relaxation T and when T < z the conductivity instability is eliminated and only a dielectric instability is possible. A similar quenching can be achieved by applying a stabilizing magnetic field. Greubel and Wolff(150) and Vistin(151) observed that for thin and pure samples the spatial periodicity of the pattern in DC excitation decreases with increasing voltage above the threshold. Smith et <z/.(147) have suggested that this variable grating mode(150) is due to non-linear (saturation) effects at higher voltages in samples that are so thin and pure that the conductivity instability has been quenched. 3.10.5 Sinusoidal excitation The behaviour is qualitatively similar when the exciting field is sinusoidal. Putting Ez = EM cos cot, the coupled equations for the charge and the curvature become(146) q + --\-aIly/EMcoscot = 0, (3.10.26) — coscot = 0, (3.10.27) 3.10 Electrohydrodynamics 193 where We have seen in the previous section that in the conduction regime, q changes sign with the field but y/ does not. Let us make the simplifying assumption that y/ is sensibly constant over a period. Since co ~ 1/T and T > T, we may replace cos2 cot by its average value f. Also expanding q{t) as we have Therefore from (3.10.27), Integration of (3.10.26) yields = # —i^— M 2 2 (cos ^ +C0T s m w 0J so that Using (3.10.17) the threshold field is given by = (E\t)} = \El = p^JT{- (3.10.29) Thus the threshold field increases with co. Though the analysis is not strictly valid when coT ~ 1, calculations show that (3.10.29) holds good quite well over the range 0 to coc where COC = (C2-\)*/T, (3.10.30) which represents a critical 'cut-off' frequency beyond which the system goes over to the dielectric regime (fig. 3.10.4). To discuss the effect of the magnetic field, we write (3.10.8) as F2 — — 194 3. Continuum theory of the nematic state where £ is the magnetic coherence length (see §3.4.1). For low magnetic fields, £ > d, we get a voltage threshold independent of the thickness of the sample: __v_w_fv (3.10.31) where 47T£II j V 33 ' For higher magnetic fields, HV> O.10.32) where // c = (n/d)(kzz/xd*- T n e threshold voltage therefore increases with H in agreement with observations. For very high magnetic fields, H ^> Hc, theory predicts a field threshold independent of thickness but proportional to if. When COT P \,q may be taken as constant over a period. Expanding y/(t) as a Fourier series 00 y/(f) = ^ ( ^ cos «co/ + \i/"n sin «co/) n=0 and replacing y/(t) cos cot by its average value \y/[, Equation (3.10.27) can now be integrated but solutions can only be obtained by numerical techniques.(146) Calculations show that in the dielectric regime (a) the threshold field Eth is independent of the wavevector kx, which results in a,fieldthreshold (as distinct from a voltage threshold as in the conduction regime), (b) E*h varies linearly as co, (c) for a given //, k\ varies linearly as co, and (d) for a given co, %a H2 + k3S k\ is a constant. These predictions have been confirmed experimentally. Other aspects of the problem, e.g., the dependence of the instabilities on the magnitude and sign of the dielectric anisotropy and on the conductivity, the behaviour in the vicinity of the threshold on either side, the effect of external stabilizing fields, etc., have been discussed extensively in a number 3.11 Hydrodynamic instabilities 195 of articles. (1525) Some results on the influence of flexoelectricity will be described in §3.13.2. 3.11 Hydrodynamic instabilities 3.11.1 Homogeneous instability in shear flow The anisotropic properties of nematics give rise to certain novel instability mechanisms that are not encountered in the classical problem of hydro- dynamic instability in ordinary liquids. Theoretical work on electro- hydrodynamic instability stimulated systematic studies on two other types of convective processes, viz, thermal and hydrodynamic instabilities, and it was soon established that the basic mechanisms involved in all three cases are closely similar.(156~9) In this section we shall examine the problem of hydrodynamic instabilities in nematics. We shall discuss first what is called homogeneous instability (HI), which is the hydrodynamic analogue of the Freedericksz transition/158'159) Consider simple shear flow between two plane parallel plates (caused by moving the plates parallel to each other at a constant relative velocity v). Let the director be initially along x, the flow along y, and the velocity gradient along z (fig. 3.11.1). At low values of vy the sample retains its planar configuration, but above a critical velocity vc - or a critical shear rate Sc = vc/d, where d is the gap width - the director tilts towards the yz plane, Sc varying inversely as the square of the sample thickness. In the presence of a strong stabilizing magnetic field Hx along x, Sc increases as//;. The physical interpretation of this distortion is as follows. (We shall assume throughout this discussion that the material is an ordinary nematic like MBBA with jus < 0.) Let us suppose that there is a fluctuation 0 > 0 in the xy plane that rotates the director from its initial orientation (1,0,0). The flow now exerts a viscous torque, which is given by Fz = — /z2 SO (fig. 3.11.1). Since ju2 < 0, Tz > 0 and gives rise to a small twist <j> > 0. Now, a deflection ^ results in a viscous torque Ty = //3 S(p, and since //3 < 0, the sign of Ty is such as to increase 6 further. Thus the viscous torques have a destabilizing effect, and above a critical shear rate they overcome the elastic and magnetic torques and the system becomes unstable. There is, in fact, another torque that comes into effect. As we have seen in §3.6.5, a deflection of <j> in the director orientation gives rise to a secondary flow vx. The secondary velocity gradient vXt z causes a torque Ty that makes an additional destabilizing contribution (except close to the boundaries). 196 3. Continuum theory of the nematic state rv<o (a) Fig. 3.11.1. The mechanism for homogeneous instability: the flow is along y and the velocity gradient along z. (a) An angular fluctuation 6 = nz > 0 results in a viscous torque Yz > 0 such that a small twist (/> = ny > 0 is produced, (b) a deformation <j> > 0 results in a torque Yy < 0 such that the initial deformation 0 is enhanced. The rigorous theory of HI threshold has been developed by Manneville and Dubois-Violette(160) and by Leslie(161), but for the present purpose it is enough to discuss the approximate treatment given by Pieranski and Guyon(158) neglecting secondary flow. Taking n = [l,^(z),0(z)] 5 v = (0, Sz, 0), and balancing the viscous torque against the elastic and magnetic torques, one obtains the coupled equations (3.11.1) (3.11.2) Putting 6 = 61 cos (nz/d) and <j> = <t>x cos (nz/d), we get from the condition for the compatibility of (3.11.1) and (3.11.2) the threshold shear rate for H=0: 5 =— (3.11.3) and in the presence of the magnetic field (3.11.4) Hcl and H are the Freedericksz threshold fields for the splay and twist geometries. It is seen that Scccd~2, and for large Hx, ScccH2x; these predictions are in agreement with observations. For a sample thickness d of 200 //m, the zero field critical threshold velocity v c= 11.5 jum s"1 for MBBA.(159) 3.11 Hydrodynamic instabilities 197 100 80 60 40 20 1 2 Frequency (Hz) Fig. 3.11.2. Experimental roll instability threshold curves for MBBA as functions of the applied voltage and frequency of shear for different values of the effective plate velocity veU; d = 240 //m, Hx = 3200 G. (From Pieranski and Guyon.(159)) It is possible to define a dimensionless quantity called the Ericksen number Er = (Sd*/4) (fi2 jus / * u k22)i which from (3.11.3) is seen to be n2/4 at the threshold.(162) The more rigorous calculation, taking into account secondary flow, yields a critical value of Er = 2.309 for MBBA at the threshold.(160) It follows from the exact solutions that ET is not a universal number but varies from one substance to another. 3.11.2 Roll instability in shear flow We have seen that under steady (DC) shear, the HI threshold increases with increasing strength of the stabilizing magnetic field Hx. However, if the field becomes large enough, the instability does not appear as a homogeneous distortion but as a regular series of bright and dark lines parallel to y, the direction of primary flow.(159) This is referred to as the roll instability (RI). The spacing between the lines is of the order of the sample thickness and the pattern is reminiscent of the Williams domains (see fig. 3.10.2 (a)). The lines arise from a cellular motion of the fluid in the xz plane superposed on the primary flow along y. A laser beam produces a diffraction pattern, and as in the case of the Williams domains, the 198 3. Continuum theory of the nematic state i 100 ^ ^ ^ t?eff = 0.148 c m s " 1 v'3 Y v3 > \ Z | Stable X / , TR s .' Y 0.6 0.8 Frequency (Hz) Fig. 3.11.3. Roll instability threshold curve for vett = 0.148 cm s 1 from fig. 3.11.2 showing the regimes Y,Z and TR (see text). (From Pieranski and Guyon.(159)) diffraction pattern vanishes when the light is polarized perpendicular to the long axes of the molecules. Another way of producing this pattern is to apply an AC shear. In this case the RI appears even in the absence of a magnetic field (except, of course, at very low frequencies, in which case it reduces, in effect, to DC shear and HI is regained). Two distinct regimes can be identified by optical observations. These are referred to as the Fand Z regimes. In the Y regime the y component of the director ny changes sign at each half-period of the AC shear but the z component nz does not, and vice versa in the Z regime. Nematic MBBA was used in the experiment with a stabilizing magnetic field Hx as well as a high frequency stabilizing electric field Ez (the latter is stabilizing because £a < 0 for MBBA). Sinusoidal shears of up to about 2 Hz were applied. The threshold curves for different values of the effective velocity vett of the upper plate ( = total plate displacement per half-period) as functions of the frequency of shear and of the applied voltage Vz are shown in fig. 3.11.2. To the left of each curve is the domain of existence of the RI. At larger frequencies rolls do not develop. 3.11 Hydrodynamic instabilities 199 Fig. 3.11.4. Afluctuation^ = ny that is spatially periodic in x results in a secondary velocity vz that is also spatially periodic. This can have a destabilizing effect and generate convective rolls. To understand these curves let us examine the curve for a given vm (fig. 3.11.3). Let us suppose that the frequency of shear is kept constant at 0.8 Hz and the voltage Vz is increased. At V = 0, the instability is of the Y type; at Vx the instability disappears; at V2 > Vx the instability reappears but is now of the Z type; and finally at V3 > V2 it disappears again. The spatial periodicity of the rolls remains of the order of the sample thickness, though it is somewhat larger in the branch just above the cusp. At a frequency slightly less than that at the cusp, 0.6 Hz say, the instability is of the Y type at V = 0. At much larger voltages it is of the Z type, but for V = V's the rolls disappear. The changeover from Y to Z takes place in a transition regime, TR, that is defined by extrapolating the two branches to lower frequencies. In TR the rolls do not extend over large regions but seem to correspond to an interchange between the Y and Z types along a given roll. The laser diffraction pattern now has satellite spots in the y direction and also shows that the spacing between the rolls has doubled. The principal mechanism for the onset of RI is shown schematically in fig. 3.11.4. A spatially periodic <j>fluctuation(or ny fluctuation) of the form cos qx x results in a secondary velocity vz that is also periodic in x. This effect has been referred to as hydrodynamic focussing. Under appropriate conditions, vz can have a destabilizing effect and generate convective rolls. 200 3. Continuum theory of the nematic state c = 45.70 o)Ty/2n Fig. 3.11.5. Calculated roll instability curves fitted to the results of fig. 3.11.3 by adjusting the values at zero frequency, at the cusp and at a point P to define the frequency scale. (From Pieranski and Guyon.(159)) For a given <p there is the torque ju3 S</> due to the primary velocity (as in HI) as well as a torque due to the secondary velocity gradient vZt x and together the total viscous torque Yy brings about a distortion 6. In turn, a 6 distortion gives rise to a torque Tz that increases <f>. (We have already seen in the case of HI that Tz = — ju2 S6, but this value is now modified slightly by a torque due to vy z.) The complete calculation of the threshold for RI involves a solution of the form exp \{qx x + qz z) with boundary conditions for 6, </>, vx9 vy, and ^.<159'160'163> We present only the salient results of the theory. Assuming fcn = k22 = fc33 = k, and ignoring material derivatives in (3.3.1), 3.11 Hydrodynamic instabilities 201 the behaviour under A C shear c a n be expressed in terms of two coupled equations with ny( = <p) a n d nz( = 0) as the variables; 09 (3.11.5) y 0, (3.11.6) where the relaxation rates are yy and yz are effective viscosity coefficients, A and B are functions of the viscosity coefficients and of the wave vectors qx and qz, and q2 = q2x + q\. The value of xz is influenced by the external electric field whereas xy is not; xz > xy gives the Y regime and xz < xy the Z regime. It is seen at once that (3.11.5) and (3.11.6) bear a close similarity to (3.10.14) and (3.10.18) for electrohydrodynamic instability; ny, nz and S correspond to the charge, curvature, and electric field, and the Y and Z regimes correspond to the conduction and dielectric regimes, respectively. Let the shear rate S = S0coscot (co = 2n/T). At steady state ny and nz are periodic functions of time. Suppose that xz > ry, xz > T, and that nz is constant in time. Then from (3.11.6) (3.11.7) ;o Using (3.11.7) in (3.11.5) and integrating AS0 fl,[(cos cot)/ty + co sin cot] x\ n n f i , Yly n*j - —- . \J.Y Y.O) Compatibility between (3.11.7) and (3.11.8) leads to ABS2cxzxy/2(\+co2x2y)=l (3.11.9) which is the threshold condition for the Y regime. On the other hand, with xy > TZ, Xy> T one gets in a similar fashion ABS2cxyxz/2(l+co2x2) = 1 (3.11.10) as the threshold condition for the Z regime. Equations (3.11.9) and (3.11.10) can be cast in a parametric form by defining x = coxy, y = (xy/xz)\ c = ABS2cx2y/2.Then 2 y = c/(l+x2), xpy2, y>\ (3.11.11) describes the Y regime, while x 2 + y * - c y 2 = 0, x p l , y < \ (3.11.12) 202 3. Continuum theory of the nematic state describes the Z regime. The theoretical diagram for threshold is shown in fig. 3.11.5. These expressions do not, of course, describe the complex behaviour in the transition region between the two regimes. 3.12 Thermal instability: stationary convection Let us first recall briefly the classical Benard-Rayleigh problem of thermal convection in an isotropic liquid.(164) When a horizontal layer of isotropic liquid bounded between two plane parallel plates spaced d apart is heated from below, a steady convective flow is observed when the temperature difference between the plates exceeds a critical value ATc. The flow has a stationary cellular character with a spatial periodicity of about Id. The mechanism for the onset of convection may be looked upon as follows. A fluctuation T' in temperature creates warmer and cooler regions, and due to buoyancy effects the former tends to move upwards and the latter downwards. When AT < ATC, the fluctuation dies out in time because of viscous effects and heat loss due to conductivity. At the threshold the energy loss is balanced exactly and beyond it instability develops. Assuming a one-dimensional model in which T' and the velocity vz (normal to the layer) vary as exp (iqyy) with qy « n/d, the threshold is given by the dimensionless Rayleigh number Rc = ATcd*pag/Kri = 7i\ (3.12.1) where p is the density, a the coefficient of thermal expansion, g the acceleration due to gravity, K = K/pC the thermal diffusivity, K the thermal conductivity, C the specific heat and n the viscosity coefficient. Rigorous calculations show that for rigid conducting boundaries 7r4 of (3.12.1) should be replaced by 1704. For a liquid having material parameters comparable with the average values of MBBA (K ~ 10~3, rj ~ 1 and a - 10"3 cgs) the modified form of (3.12.1) gives ATC « 2 °C for d = 1 cm and ATcx2x 103 °C for d = 1 mm. The situation is altered profoundly in the case of a nematic because of its anisotropic transport properties. Dubois-Violette(156) was the first to give an approximate theoretical treatment of thermal convection in a planar (homogeneously aligned) nematic and to show by consideration of torques that such a system will be unstable against cellular flow when the film is heated from below if Kn > 0, or when it is heated from above if Ka < 0, where K^ = K^ — K± is the anisotropy of thermal conductivity (which is positive for all known nematics(165)). Dubois-Violette also showed that the critical temperature gradient fte( = ATc/d) should be much less than that 3.12 Thermal instability: stationary convection 203 (a) T+AT (b) Fig. 3.12.1. (a) Thermal instability in a nematic heated from below. Initial orientation of the director is horizontal (along y). A bend fluctuation causes 'heat focussing' because of the anisotropy of thermal conductivity (K^ > K±) and gives rise to warmer ( + ) and colder ( —) regions. The warmer regions move up and the colder regions down due to buoyancy effects and this, in turn, results in a velocity vy (the fluid being assumed to be incompressible). The transverse velocity gradient vz y induces a major destabilizing viscous torque, while the vertical gradient vy z induces only a very weak stabilizing torque. The resulting torques, shown by the curved arrows, are destabilizing. Here the long straight arrows represent the translational velocities vz and vy9 the short straight arrows the heat fluxes, (b) The same geometry as in (a) but with the top plate at a higher temperature. The system is stable. for an isotropic liquid. Experimental observations of convective rolls in a homogeneously aligned film of MBBA heated from below were reported by Guyon and Pieranski. (157166) As expected, the threshold value of the thermal gradient fic was about 10~3 times that for an isotropic liquid of comparable average physical properties. Optical observations confirmed that the rolls are essentially as depicted in fig. 3.10.2 (a) for the Williams domains. When the bottom plate was heated to a temperature greater than the nematic-isotropic transition point the convection disappeared, show- ing that it is the anisotropy that is responsible for the low threshold. Figures 3.12.1 and 3.12.2 illustrate the destabilizing mechanisms involved. Because of the anisotropy of thermal conductivity, a thermal fluctuation of the director along y creates warmer ( + ) and cooler (—) 204 3. Continuum theory of the nematic state \ T+AT (a) T+AT Fig. 3.12.2. (a) The initial orientation of the director is along z. A splay fluctuation gives rise to warmer and colder regions. In this case, vyz causes a dominant stabilizing viscous torque while vz y causes only a very weak destabilizing torque. The system is therefore stable against stationary convection, (b) The same geometry as in (a) but with the top plate at a higher temperature. The system is unstable (see legend of fig. 3.12.1 regions. This is referred to as heat focussing. Because of the buoyancy effect, the warmer regions move up and the cooler regions move down creating a velocity fluctuation vz9 which in turn gives rise to a torque Tx that stabilizes or destabilizes the orientation depending on the sign of Ka. We seek solutions of the form n = [0, l,0(y,i)]9 \ = [0,0,vz(y,t)], T=-pz+T'(y,t), and using (3.1.6)—(3.1.9) obtain for a variation Qxp(iqyy) of the fluctuation(152) exp (iqyy) [vz0 + (vjzv) - ocgT'o] - (jujp) y/ = 0, (3.12.2) f = 0, (3.12.3) = 0, (3.12.4) where y/ = d6/dy is the director curvature and 019 vz0, T'o are amplitudes of fluctuations. In (3.12.4) T¥ = — ^x/kzzq^ is the relaxation time for the director in the absence of any coupling. Ignoring vz0 in (3.12.2) and 3.13 Flexoelectricity 205 eliminating vz0 in (3.12.3) and (3.12.4) one finds the condition for threshold to be where r a = (K^q\XJii^f1. In an isotropic liquid Ka = 0,r a = oo, and we recover (3.12.1). For a nematic film of thickness d ~ 1 mm, r¥ ~ 103 s, i a ^ 1 s, and ATe reduces to 10~3 times the isotropic value. It is the large ratio T /r a that leads to a very low threshold. A magnetic field Hx (stabilizing) or Hz (destabilizing) can be used to decrease or increase T^, or, equivalently, increase or decrease A Tc; the critical temperature difference varies linearly as H\ (or # z 2 ). (166) Pieranski et al.ae7) then showed that when a homeotropically aligned film of MBBA is heated from above, the orientation becomes unstable above a critical value of AT of the same order as that for planar orientation. They also verified that the film is stable when heated from below. A model of the destabilizing mechanism is no longer purely one- dimensional involving only vz (fig. 3.12.2). A thermal fluctuation causes a fluctuation vz as before, but because of the incompressibility of the fluid this, in turn, causes a velocity fluctuation vy that contributes the major destabilizing torque Fx~ju2vyz for Ka > 0. Again with a stabilizing magnetic field Hz there is a linear relationship between A Tc and H\. A field Hy favours rolls with axes normal to y, but in the field-free case the rolls degenerate into a square pattern that may be regarded as a linear superposition of crossed convection rolls. When AT is increased well beyond A Tc a complex hexagonal structure is found with a nematic- isotropic interface if the temperature of the upper plate is large enough. The studies outlined here are the most important ones that established the fundamental of principles of thermal instability in nematics. A number of theoretical and experimental investigations on these and other geo- metries have since been reported. (155) A particularly interesting study is that of Lekkerkerker(168) who predicted that a homeotropic nematic heated from below (which, it will be recalled, is stable against stationary convection) should become unstable with respect to oscillatory convection. The phenomenon was demonstrated experimentally by Guyon et al.a69t 170) 3.13 Flexoelectricity 3.13.1 Determination of the flexoelectric coefficients If the molecule possesses shape polarity in addition to a permanent electric dipole moment then the possibility exists that a splay or bend deformation 206 3. Continuum theory of the nematic state (c) id) Fig. 3.13.1. Meyer's model of curvature electricity. The nematic medium composed of polar molecules is non-polar in the undeformed state {{a) and (c)) but polar under splay (b) or bend (d). (After Meyer.(171)) (b) Fig. 3.13.2. Interpretation of the origin of flexoelectricity in an assembly of quadrupoles: (a) in the undeformed state the symmetry is such that there is no bulk polarization, (b) a splay deformation causes the positive charges to approach the upper plane and to be pushed away from the lower one. This dissymmetry gives rise to a dipole moment. will polarize the material, and conversely that an electric field will induce a deformation (fig. 3.13.1). This phenomenon, which is somewhat analogous to the piezoelectric effect in solids and is therefore termed as the 3.13 Flexoelectricity 207 Fig. 3.13.3. A hybrid aligned cell for the determination of the anisotropy of the flexoelectric coefficients. In this geometry, the director has a splay-bend distortion which gives rise to a flexoelectric polarization Px. On applying an electric field Ey, the director is twisted by an angle oc (e1 — e3) which can be measured optically.
(Dozov, Martinot-Lagarde and Durand. (174) )

flexoelectric effect, was first proposed by Meyer.(171) Subsequently Prost
and Marcerou(172) showed that electric quadrupole moments can also
contribute to this effect (fig. 3.13.2). Since, as a rule, all molecules
have non-zero quadrupole moments, it follows that flexoelectricity is a
universal property of nematics. The observation of flexoelectricity in a
nematic composed of symmetric, non-polar molecules(173) has confirmed
the existence of a quadrupolar contribution, which turns out to be
comparable in magnitude to that from molecular dipoles.
In a first order theory, the polarization P should be proportional to the
distortion:
)] + e8[n-Vn],              (3.13.1)
where ex and e3 are the flexoelectric coefficients corresponding to splay and
bend respectively. In an external electric field E, the flexoelectric effect
leads to a free energy density F = — P E .
A simple method of measuring (e1 — e3) is as follows.(174) The nematic is
taken in a ' hybrid' aligned cell with the director oriented homogeneously
on one surface, and homeotropically on the other. The anchoring at the
boundaries is assumed to be firm, and in the absence of an external electric
field the director is confined to the xz plane (fig. 3.13.3). The director field
in such a cell has a splay-bend distortion which gives rise to a flexoelectric
polarization along x. If, now, a DC electric field E is applied along y, the
nematic acquires a twist about z due to the action of E on P. (The influence
of dielectric anisotropy can be neglected as long as the field strength is
208                  3. Continuum theory of the nematic state

<

-30
-50                         0
Electric field (V mm"1)
Fig. 3.13.4. ^(0) versus is for two twin domains in the sample. The linearity of the
slope confirms the validity of (3.13.2). (Dozov, Martinot-Lagarde and Durand. (174) )

small, since the dielectric free energy is proportional to E2.) The twist angle
is maximum close to the bottom plate (fig. 3.13.3) where the curvature is
maximum and is given by
Ed
(3.13.2)

where k is the elastic constant (in the one-constant approximation) and d
the sample thickness. The angle ^(0) can be determined by measuring the
rotation of the plane of polarization of linearly polarized light incident
along z (see §4.1.1). (Strictly speaking, the adiabatic approximation is not
valid close to the bottom plate of fig. 3.13.3, for in this region the director
orientation is homeotropic and the effective birefringence for light
propagation along z is small. However, the authors have shown that for
thick samples the resulting error is negligible.) A plot of ^(0) versus E is a
straight line (fig. 3.13.4), the slope of which yields (e1 — es). The value was
found to be 1 x 10"4 dyn* for MBBA.
The sum of the flexoelectric coefficients (ex + e3) was first measured by
Prost and Pershan.(175) A homeotropically aligned sample is taken between
two glass plates and a periodic electrostatic potential is applied by means
of interdigitated electrodes coated on one of the plates, as shown in fig.
3.13.5. The flexoelectric effect being linearly proportional to the applied
voltage, the resulting distortion has a periodicity 2d, where d is the spacing
between neighbouring electrodes. On the other hand, the distortion due to
3.13 Flexoelectricity                           209

(a)

(b)

Fig. 3.13.5. A periodic electrostatic potential applied by means of interdigitated
electrodes coated on one of the plates gives rise to (a) SLflexoelectricdistortion
having a periodicity Id, where d is the spacing between the electrodes and (b) a
dielectric distortion having a periodicity d. (Prost and Pershan.(175))

dielectric alignment, which is proportional to the square of the applied
voltage, has a periodicity d. By optical diffraction it was possible to
distinguish between the two types of distortion. Further, by using optical
heterodyne detection techniques it was shown that the diffracted intensity
due to flexoelectric distortion increases linearly with voltage, as expected.
By evaluating the distortion of the uniformly aligned nematic under the
action of the field gradient, the magnitude of (e1 + es) was estimated to be
2.5 x 10~4 dyn* for MBBA.
A method of determining (e1 + e2) both in magnitude and sign was
devised by Dozov et al.(176) The sample, homeotropically aligned with firm
anchoring at the boundaries, was placed in a quadrupolar field distribution
obtained with two pairs of electrodes. The electric field gradient causes a
tilt of the director at the centre of the sample and hence a tilt in the
conoscopic pattern seen between crossed polarizers. A reversal of the field
gradient reverses the tilt, and therefore the sign of (e1 + e^) could be
determined unambiguously. For MBBA (e1 + e3) was found to be
— 1 x 10~4 dyn*. Methods requiring weak anchoring boundary conditions
have also been used. (177178) It may be noted, however, that there are some
inherent difficulties, e.g., the large screening effect due to the ions, which
introduce considerable uncertainty in all these determinations.
210                3. Continuum theory of the nematic state

d = 35 /^m
/ c = 120 Hz
4   -

Normal
rolls

50                     90
Frequency (Hz)

Fig. 3.13.6. The formation of oblique convective rolls in electrohydrodynamic
instability. Below the point M, which may be called a 'triple point', the transition
takes place directly to oblique rolls ( / = 10 Hz). Beyond M, normal rolls are
obtained first, followed by undulatory rolls which then change continuously into
oblique rolls ( / = 60 Hz). (Ribotta et al.a81))

3.13.2 Influence offlexoelectricity on electrohydrodynamic instability
In our discussion of electrohydrodynamic instability (§3.10) we have
throughout assumed that the convection rolls (or Williams domains) are
normal to the initial orientation of the director n. However, in many
experiments oblique rolls, whose wavevector makes an acute angle a with
n0, have been observed at low frequencies. (179~81) Detailed studies by
Ribotta, Joets and Lei(181) have shown that these oblique rolls appear at the
threshold up to a critical frequency/ 0 in the conduction regime (fig. 3.13.6).
F o r / > / 0 normal rolls (a = 0) are obtained at the threshold, but on
increasing the field strength these normal rolls first become undulatory and
then oblique. Solutions corresponding to oblique rolls can be obtained by
extending the Orsay model to three dimensions and imposing boundary
conditions at the two surfaces, but the predicted values of a and f0 are
3.13 Flexoelectricity                          211
about an order of magnitude smaller than the experimental values.(182)
Later investigations*183"7* have shown that flexoelectricity plays an im-
portant role in the formation of oblique rolls.
Flexoelectricity enters the problem in two ways. Firstly, the flexoelectric
polarization arising from the periodic distortion at the onset of instability
gives rise to additional space charges, and secondly the applied electric field
theory now involves three linear coupled equations.(184) It turns out that
two regimes of instability - the conduction and dielectric regimes - are
possible as in the Orsay model (§3.10.4). For frequencies greater than a
certain value f0 (which is much lower than the cut-off frequency fc
separating the conduction and dielectric regimes) one regains the normal
rolls at threshold. Calculations*185"7* including boundary conditions have
confirmed that the model is able to explain the occurrence of oblique rolls
and its dependence on frequency. The critical frequency/ 0 is proportional
to the average conductivity of the sample, which probably accounts for the
fact that while some observers have reported oblique rolls in MBBA,(181)
others have not.<154)

3.13.3 Order electricity
Meyer's idea of flexoelectricity has been generalized to include a con-
tribution due to the gradient of the orientational order parameter/ 188 ' 189)
The polarization in this case arises not from the curvature distortion of the
director but from the spatial variation of the degree of orientational order
of the molecules. In a simple first order theory, one may take P oc Vs,
where s is the order parameter as defined in §2.3.1. This effect has been
termed as 'order electricity'.
Order electricity may be expected to manifest itself at the nematic-
isotropic (or air) interface where, as discussed in §2.7, the order
parameter changes rapidly across the transition zone from one phase to the
other. Let us make the simple assumption that at the N - I interface the
gradient of the order parameter ~s/£, where £ is a coherence length. If Pz
is the component of the polarization normal to the interface created by the
order parameter gradient, and the director at the interface is tilted at an
angle 9 with respect to z, the dielectric energy due to order electricity will
be proportional to (189)
212              3. Continuum theory of the nematic state
where e is the dielectric constant (taken to be isotropic). Other factors may
come into play, but neglecting their contributions, it is seen that the energy
is minimum when the tilt assumes the 'magic' angle 0 ~ 53° such that
cos2 0 = \. Interestingly, this is in very good agreement with the observed
value of 52° for some cyanobiphenyl compounds. (190)
4
Cholesteric liquid crystals

4.1 Optical properties
The unique optical properties of the cholesteric phase were recognized by
both Reinitzer and Lehmann at the time of their early investigations which
culminated in the discovery of the liquid crystalline state. When white light
is incident on a 'planar' sample (whose optic axis is perpendicular to the
glass surfaces) selective reflexion takes place, the wavelengths of the
reflected maxima varying with angle of incidence in accordance with
Bragg's law. At normal incidence, the reflected light is strongly circularly
polarized; one circular component is almost totally reflected over a
spectral range of some 100 A, while the other passes through practically
unchanged. Moreover, contrary to usual experience, the reflected wave has
the same sense of circular polarization as that of the incident wave.
Along its optic axis, the medium possesses a very high rotatory power,
usually of the order of several thousands of degrees per millimetre. In the
neighbourhood of the region of reflexion, the rotatory dispersion is
anomalous and the sign of the rotation opposite on opposite sides of the
reflected band. The behaviour is rather similar to that of an optically active
molecule in the vicinity of an absorption. Following the theoretical work
of Mauguin, (1) Oseen(2) and de Vries(3) these remarkable properties can
now be explained quite rigorously in terms of the spiral structure
represented schematically in fig. 1.1.4.

4.1.1 Propagation along the optic axis for wavelengths <^ pitch
Basic theory
We shall first consider the propagation of light along the optic axis for
wavelengths much smaller than the pitch so that reflexion and interference

213
214                    4. Cholesteric liquid crystals

effects may be neglected. The problem was investigated by Mauguin (1) with
a view to explaining the optical rotation produced by twisting a nematic
about an axis perpendicular to the preferred direction of the molecules. He
used the Poincare sphere(4) and ' rolling cone' method, but we shall adopt
an identically equivalent formalism, viz, the Jones calculus/ 5 ' 6)
The basic principle underlying the Jones method is that any elliptic
vibration can be represented by the column vector

where Ax and A2 are the resolved components of the electric displacement
vector D along x and y, and are, in general, complex quantities. The
intensity is |y41|2 + |^42|2, while the complex ratio A1/A2 describes its
polarization state. The azimuth A (i.e., the angle which the major axis of
the ellipse makes with x) and the ellipticity co are related as follows: if
tana = |^41|/|^42| and A is the phase difference between Ax and A2,

tan 2 A = cos A tan 2a,
sin 2co = sin A sin 2a.

The effect of an optical system - comprised, say, of birefringent, absorbing
and dichroic plates - is to change A1 and A2, so that

D = JD,

where J is a 2 x 2 matrix with complex elements. In what follows, we shall
adopt the convention of describing the optical system as viewed by an
observer looking at the source of light.
Our aim is to evaluate the matrix J for the cholesteric liquid crystal when
light is incident along the helical axis.(7) As demonstrated by Jones, (8) a
problem of this type can be solved by regarding the medium as being
composed of a large number of infinitesimally thin sections, each section
representing an optical element, in this case a linearly birefringent or
retardation plate. For the purposes of this calculation, therefore, we treat
the liquid crystal (fig. 1.1.4) as a pile of very thin birefringent (quasi-
nematic) layers with the principal axes of the successive layers turned
through a small angle /?.
Let the principal axes of the first layer be inclined at an angle /? with
respect to x,y. If light is incident normal to the layers, i.e., along z, the
4.1 Optical properties                           215
Jones retardation matrix for the first layer referred to its own principal axes
is given by
G[exp(-iy)              0
[     0        exp(iy)
where y is half the phase difference between the waves linearly polarized
along the principal axes after passing through a single layer of thickness /?,
i.e., y = ndnp/k, Sn = na — nb being the layer birefringence and I the
vacuum wavelength. The retardation matrix with respect to x9y is then
Jx = SGS"1,
where

and S"1 is the inverse of S so that SS"1 = S-1S = E, the unit matrix.
If Do is the complex column vector with respect to x, y representing the
incident light, the emergent light after passing through the first layer is

where, as we are interested at present only in the state of polarization
of the emergent beam, we neglect the phase factor exp( — ir/), where
rj = n(na + nb)p/A. (Throughout, we follow the convention of representing
the phase factor at any point +z by exp( — \2nnz/X).) Let Dx be now
incident on a second birefringent layer whose principal axes are inclined
at 2/? with respect to x,y. The Jones matrix for this layer is

and the emergent vector is
D2 = S2GS 2DX = S2GS 2SGS            ^
= Sa(GS-1)aD0 = J 2 D 0 ,
where J2 = S2(GS~1)2 is the appropriate Jones matrix for this system of two
layers. In general, if we have a pile of m layers, where the principal axis of
the sth layer is inclined at sfi with respect to x, y (s = 1,2,..., m), the Jones
matrix for the pile is evidently

J m = Sm(GS~T = [l *] say.                          (4.1.1)
It can be shown from the theory of matrices<7'9) that

E,          (4.1.2)
where
cos 6 = cos p cosy.
216                       4. Cholesteric liquid crystals

Now, as stated earlier, the layer thickness is assumed to be very small, say
a few A, while the pitch P is taken to be at least a few wavelengths of light,
so that both ft ( = 2np/P) and y are small quantities. Therefore

(4.1.3)
From (4.1.1) and (4.1.2)

_ tan/? .    _.      _ . sinra# .    ,
a = cos rap cos    rat/H -sin rap sinrat/— i ——— sinycos(ra + l)p,
tan 6                 sin 6
(4.1.4)
tan/?      _.  _ .   _    _ .sinra# .  . ,
b=       -cosmpsmmu — smmpcosmu — I—-—— sinysin(ra + l)p,
tan 6                         sin 6
(4.1.5)
c = — b* and                     d=a*,
where a* and &* are respectively the complex conjugates of a and b.
It is a standard result in optics (6) that such a system can, in general, be
replaced by a rotator and a retarder. If p is the rotation produced by the
system, 2(p the phase retardation and y/ the azimuth of the principal axes of
the retarder,

fcos y/   — siny/ifcos/?                — sin/?l
m
[sm^      c o s ^ J Lsin/?              cos/? J

ip)          0 1 _ r               _ r
L       0                 exp (i#>) J [ — sin y/   cos^J

From (4.1.1) and (4.1.6)

c = -6*            and                   rf=fl*.             (4.1.9)
Equating the real and imaginary parts of (4.1.4) and (4.1.7) and of (4.1.5)
and (4.1.8), we obtain after simplification(10)
p = m(J3- 6') radians,                                     (4.1.10)
1         2           2
(p = cos" (sec rafl'/sec m6)\                                  (4.1.11)
y/ = \[{m+ \)p—p],                                     (4.1.12)
where

m
4.1 Optical properties                            217

a

0        1    2       3         4   5      6
I/A2 (Mm-2)
Fig. 4.1.1. Rotatory dispersion of a cholesteric liquid crystal for wavelengths <^
pitch. Solution of poly-y-benzyl-L-glutamate (PBLG) in chloroform (18 g/100 g).
(12)
(After Robinson. ).

In these equations m represents the total number of layers in the system.
Since the layer thickness is taken to be a few A, it turns out in actual
practice that even with the thinnest specimens employed, m is usually a
very large number. We shall therefore assume m to be large throughout this
discussion.
Optical rotatory power
When /? > y, 0' « 6. This condition is satisfied when \PSn <^ A, i.e., when
the pitch is not too large. The optical rotation produced by m layers is then
p = m(p-0) = m[fi
and the phase retardation 2(p « 0. Thus the system behaves in effect as a
pure rotator. If m is the number of layers per turn of the helix, rnfi = 2n and
mp = P, so that the rotatory power in radians per unit length
p = -n(dn)2P/4l\                          (4.1.13)
the negative sign indicating that the sense of the rotation is opposite to that
of the helical twist of the structure. (11) Typically, dn « 0.05 for a cholesteric;
218                       4. Cholesteric liquid crystals

0.1         0.2          0.3        0.4        0.5
Inverse pitch (jim"
Fig. 4.1.2. Optical rotation per unit length in a twisted nematic film versus inverse
pitch for light of wavelength 0.5 jum. The incident linear polarization is parallel to
the director on the entrance side of the film. Layer birefringence Sn = 0.1. Film
thickness (a) 1.0 /urn, (b) 1.25 jum and (c) 1.5 jum.

taking P = 5 jum and X = 0.5 jum, p ~ 2000° mm" 1. This equation has been
verified experimentally in considerable detail by Robinson (12) in lyotropic
systems and by Cano and Chatelain (13) in thermotropic systems.
Robinson discovered that solutions of some polypeptides in organic
solvents, for example, poly-y-benzyl-L-glutamate (PBLG) in dioxan,
methylene chloride, chloroform etc., spontaneously adopted the choles-
teric mesophase above a certain concentration. Under suitable conditions
the solutions exhibited equi- spaced alternate bright and dark lines when
observed through a microscope (see fig. 4.2.7). The lines may be interpreted
as a view of the structure at right angles to the screw axis, so that the
periodicity of the lines is equal to half the pitch. Robinson confirmed this
interpretation by observations between crossed polaroids and also by the
use of a quartz wedge; the retardation plotted against distance in a
direction perpendicular to the lines had an oscillating value, as is indeed to
be expected from the structure. The pitch for any given polypeptide
depended on factors such as concentration, solvent, temperature etc. When
viewed along the screw axis no lines were seen, but a very high optical
rotatory power was present. The rotation in every solution, with a very
wide range of values of P was found to be proportional to \/k2 (fig. 4.1.1).
Robinson substituted the observed values of p and P in (4.1.13), and
4.1 Optical properties                           219
calculated the layer birefringence Sn per volume fraction of the polypeptide
in solution. The birefringence was remarkably constant despite the widely
varying values of p and P. He then prepared a solution with equal
quantities of the D and L forms (PBDG and PBLG) which too under
certain conditions adopted the spontaneously birefringent phase, only, in
this case, it was not the twisted cholesteric, but the untwisted nematic
structure. He was therefore able to measure the birefringence directly and
the value agreed well with that derived from (4.1.13).
Similar studies have been carried out by Cano and Chatelain(13) on
mixtures of nematic and cholesteric liquid crystals. The birefringence of the
nematic being very large, Sn of the mixture could be assumed without
sensible error to be equal to that of the nematic itself. The pitch P of the
mixture was measured directly from the Grandjean-Cano steps formed in
a wedge (fig. 4.2.9). The values of Sn and P when inserted in (4.1.13) gave
a rotatory power in quantitative accord with observations.
When P is comparable to or less than y9 the system is no longer a pure
rotator. For large values of the pitch, (4.1.1) may be expressed as(14)

fcos mp    — sin ra/TI    Texp(-im(9)          0   1
I sin mp         /?
cosra/    J   L    0           exp (im6)\

+ sin m6
\sin 9              [Sl exp( -iy)]           .   (4.1.14)

\sin0      ^

When /? and ft/y are extremely small,         y and (4.1.14) reduces to

Tcosra/?    — sin m/TI fexp (— imy)       0
m                                                             (4.1.15)
~ [sinm/?      cosmfi J[      0         exp (imy)J'
Equation (4.1.15) implies that at any point in the medium there are two
linear vibrations polarized along the local principal axes. The polarization
directions of these two vibrations rotate with the principal axes as they
travel along the axis of twist and the phase difference between them is the
same as that in the untwisted medium. This result was first derived by
Mauguin(1) and is sometimes referred to as the adiabatic approximation. It
is this property that is made use of in the twisted nematic device discussed
in §3.4.2.
Fig. 4.1.2 illustrates the theoretical variation of the optical rotation with
220                     4. Cholesteric liquid crystals

pitch and sample thickness derived from (4.1.14) and (4.1.15) for very thin
cholesteric or twisted nematic films (the sample thickness being smaller
than the pitch). (1415) Here the rotation is defined as the quantity that is
most conveniently measured, viz, the angle between the incident plane of
polarization, assumed to be parallel to the director axis on the entrance
side of the film, and the major axis of the emergent ellipse. Unlike p given
by (4.1.10), this rotation is not independent of the azimuth of the plane of
polarization of the incident light/ 16 ' 17) but the general conclusions that we
may draw from these curves are still valid. We observe firstly that in
contrast to an ordinary optically active substance the optical rotation per
unit length in the twisted nematic (or cholesteric) is a function of the
thickness. A noteworthy feature is the reversal of the sign of rotation as the
pitch is varied.(17) In the neighbourhood of the Mauguin limit, the rotation
has the same sign as that of the helical structure. With decrease of P the
rotation drops to zero and then reverses sign; thus for lower values of P the
sense of the rotation is opposite to that of the helix, in conformity with
(4.1.13). With increase of sample thickness the peaks in the rotation
increase in height and become sharper. The predicted trends have been
confirmed experimentally.(18)

Absorbing systems: circular dichroism
When linearly dichroic dye molecules are dissolved in a cholesteric liquid
crystal the medium exhibits circular dichroism because of the helical
arrangement of the solute molecules in the structure.(19) The theory
developed above can be extended to take into account the effect of
absorption by treating the layers as both linearly birefringent and linearly
dichroic.(19'20) Assuming that the principal axes of linear birefringence and
linear dichroism are the same, the Jones matrix of any layer with reference
to its principal axes is

G     i— *-v WJ       0 l[exp(-A: a /?)           0    1
p(iy)J|_
exp(iy)J   0               exp(-kbp)\

exp(-iy)        0
0

where ka, kb are the principal absorption coefficients of the layer, and

y=     y-lKK-
4.1 Optical properties                        221
Proceeding as before, the Jones matrix for m layers is

j m = Sm(GS"1)ro.                      (4.1.16)

If Xx and X2 are the eigenvalues of (GS)" 1, it can be shown that

r = ^—f-GS-'-AxA/ 1 .                  7 E,      (4.1.17)
where
^ = exp (-<![) exp (i0),
A2 = exp(-£)exp(-i/9),
and
cos^ = cos 7 cos )8.

Using (4.1.16) and (4.1.17)

J^exp(-^S-[^GS-^Sin(m-1)'El.                             411 )
(..8
L sin 6   sin 9 J
Such a system can be resolved uniquely into a rotator, a retarder, a
circularly dichroic plate and a linearly dichroic plate. The unique matrix
resolution is given by

where
_ [cos ^    - sin y/1
[sin y/    cos y/ J '

_ [cos/?    — sin/7
[sin/7    cos/7

[ cosh a/2     i sinh a/2l
[ — i sinh a/2 cosh a/2 J'

[     0       exp (icp) J'

K =   [exp(-/c)        0 1
L     0       exp(fc)J'
Here p is the rotation, a the circular dichroism, 2#? the linear phase
retardation, 2K the linear dichroism, / the attenuation coefficient and y/ the
222                    4. Cholesteric liquid crystals

azimuth of the retardation plate (or, equivalently, of the linearly dichroic
plate). From (4.1.18) and (4.1.19),

-id = m(p — U),

_x/sec2^
-IK =        COS"1      r m0/             (4.1.20)
Vsec

X=
with

ICLL
j
n           \    tan<9

When/ ? is much larger than y
= K
and
f
Hence

and
a = -myju/p.
Therefore, the linear dichroism of the layers not only results in circular
dichroism but also makes a small contribution to the optical rotation
which is opposite in sign to that due to linear birefringence. However, this
contribution is usually negligibly small. Another consequence of the theory
is that o changes sign whenever /? or ju changes sign. Further, the parameter
o exhibits a marked dependence on pitch and sample thickness. These
predictions are in qualitative agreement with observations.(18"20)

4.1.2 Propagation along the optic axis for wavelengths ~ pitch: analogy
with Darwin's dynamical theory of X-ray diffraction
When the wavelength is comparable to the pitch, the optical properties are
modified profoundly. Before discussing the rigorous electromagnetic
treatment of the problem it is instructive to examine it first from the
standpoint of X-ray diffraction theory/ 7 ' 21) Since the dynamical theory of
X-ray diffraction from perfect crystals and its applications are now quite
thoroughly understood, this approach may be useful in elucidating the
optical behaviour of cholesterics and in looking for new optical analogues
of certain well established X-ray effects. An example of a new phenomenon
is the Borrmann effect in cholesterics.(22)
4.1 Optical properties                                    223

Kinematical theory of reflexion
The theory discussed in §4.1.1 shows that as long as the pitch is not too
large compared with the wavelength, i.e., when \Pdn <^ 1, the liquid crystal
can be treated to a good approximation as a pure rotator for light
propagating along the helical axis. In other words, right- and left-circular
waves| travel without change of form but at slightly different velocities.
The refractive indices for the two components are respectively

and the rotatory power
(4.1.21)
where Sn = na — nb and n = \{na + nb).
We shall now give a simple interpretation of how under certain
conditions reflexion of one of the circularly polarized components takes
place. Let right-circular light given by D o = [*] referred to x,y be incident
along z. We shall suppose that the structure is right-handed, i.e., ft is
positive. To calculate the reflexion coefficient at the boundary between the
(s+ l)th and (s + 2)th layers, we resolve the incident light vector along the
principal axes of the (s+ l)th layer which are inclined at angle (s+ 1)/? with
respect to x, y. The resolved components are

i                                                 (4.1.22)

where cps+1 = 2nnn(s+ \)p/X, p being the thickness of each layer. At the
boundary, the £ vibration emerges from a medium of refractive index
na and the rj vibration from a medium of refractive index nh. Qualitatively,
it is obvious that since the principal axes of the (s + 2)th layer are rotated
slightly with respect to those of the (s+ l)th layer, one of the components
of (4.1.22) will on emerging from the (s+ l)th layer meet a 'rarer' medium
while the other will meet a ' denser' medium. One component therefore gets
reflected without any change of phase and the other with a phase change of
n. Thus, in contrast to reflexion from a normal dielectric, the sense of
circular polarization remains the same after reflexion. Applying the
f Right- and left-circular polarizations are denned from the point of view of an observer
looking at the source of light. If the electric vector rotates clockwise with progress of time,
then it is right-circular. Thus, at any instant of time, the tip of the electric vector forms a
right-handed screw in space for right-circular polarization.
224                     4. Cholesteric liquid crystals
standard formulae for reflexion at normal incidence from the surface of a
non-aborbing anisotropic crystal, the reflected components £', rj\ referred
to the principal axes of the (s + 2)th layer, are

rf

where q = fidn/2n. We make the approximation here that sin/? « /?, since ft
is assumed to very small (~ 10" 2 rad). On reflexion a very slight ellipticity
is introduced in the transmitted beam but we shall ignore it in the present
discussion. Transforming back to x,y the reflected wave on reaching the
surface of the liquid crystal will be

GI--UJ
which represents a right-circular vibration travelling in the negative
direction of z. Clearly the phase difference between this wave and that
reflected at the boundary between the first and second layers is 2(sp—(p s).
When h = nnP, we have 2nnnp/A = /? and <ps = s/3 (since mp = P and
mp = 2n, where m is the number of layers per turn of the helix). Hence the
phase factor exp[2i(s/—(p 8)] becomes unity irrespective of the value of s, and there results a strong interference maximum. For a left-handed structure, /? is negative and (s^—(p s) does not vanish; therefore, the waves from the different layers will not be in phase and the vibration will be transmitted practically unchanged. Using the kinematical approximation, i.e., neglecting multiple reflexions within the m layers, the reflexion coefficient per turn of the helix is |g| = m\q\ = ndn/n. (4.1.23) Dynamical theory of reflexion The complete solution of the problem has to take into account the effect of multiple reflexions. This can be done by setting up difference equations closely similar to those formulated by Darwin (23) in his dynamical theory of X-ray diffraction. For the purposes of this theory we shall regard the liquid crystal as consisting of a set of parallel planes spaced P apart. Each plane therefore replaces the m layers per turn of the helix. We ascribe a reflexion coefficient — \Q per plane for right-circular light at normal incidence. Assuming a kinematical approximation for the m layers, Q is given by 4.1 Optical properties 225 ,., | f Sr+1 r+1 Fig. 4.1.3. Notation for the primary (T) and reflected (S) waves in the dynamical theory. (4.1.23). We can then write the difference equations in a simple manner because, as stated earlier, circularly polarized waves travel practically without change of form, so that the interference of the multiply reflected waves with one another and with the primary wave can be evaluated directly. We shall suppose, as before, that the structure is right-handed and that right-circular light is incident normally. Let Tr and Sr be the complex amplitudes of the primary and reflected waves at a point just above the rth plane, the topmost plane being designated by the serial number zero (fig. 4.1.3). Neglecting absorption, the difference equations may be written as Sr = - i f i r r + e x p ( - i p ) S r + 1 , (4.1.24) Tr+1 = exp ( - iq>) Tr - xQ exp ( - 2ip) S r+1 , (4.1.25) where cp = 2nnnP/A. The reflexion coefficient is here taken to be the same on both sides of the plane. Replacing r by r— 1 in (4.1.24) and (4.1.25), substituting and simplifying, we obtain Tr^ + Tr_x=yTr, (4.1.26) Sr+1 + Sr_1=ySr, (4.1.27) where y = exp (icp) + exp ( - i(p) + Q2 exp ( — \<p). (4.1.28) Suppose that the liquid crystal is a film consisting of v planes. Putting Sv = 0, we have from (4.1.27) Sv_2 = ySv_!, and -i v-2.._, ,(v-4)(v-3) _ (4.1.29) 226 4. Cholesteric liquid crystals Similarly from (4.1.25), (4.1.26) and (4.1.28) ^v-3 = [ ( / - 1) exp (icp) -y] Tv, etc. and To = (fly) exp (i(p) -fUy)) Tv. (4.1.30) Since, from (4.1.24) Sv_, = -iQTv_x = - i e e x p ( i ^ ) Tv9 the ratio of the reflected to the incident amplitudes is Let us assume a solution in the form Tr+l = xTr, (4.1.32) where x is independent of r. Hence x must satisfy =y= We have seen that the reflexion condition is nR P = 20 or q>0 = 2n. Accordingly, we may write where which is a small quantity in the neighbourhood of the reflexion. Therefore x + (\/x) = exp(i^) + exp(-i^) + g 2 exp(-ie). (4.1.33) This suggests that in the neighbourhood of the reflexion we may put x = exp(-£)exp(-i<? 0 ) = exp(-£), (4.1.34) where £ is small and may be complex. From (4.1.33) and (4.1.34) When y = exp ( 0 + exp (-£) = 2cosh£, the series in (4.1.29) is given by(24) 4.1 Optical properties 227 1.0 :J 0.8 (a) 0.6 I 9S 0.4 - I 0.2 - 8 o.o o ^ '% 1.0 - <S ib) 0.8 - 0.6 - A _ 1 0.4 - 0.2 - 00 0.45 0.47 0.49 0.51 0.53 0.55 Wavelength (jum) Fig. 4.1.4. Reflexion coefficient 0t at normal incidence versus wavelength for a non- absorbing cholesteric: (a) semi-infinite medium, (b) film of thickness 25P, where P is the pitch. Curves are derived from the dynamical theory; circles represent values computed from the exact theory (§4.1.3) assuming that the medium external to the cholesteric (e.g., glass) has a refractive index 1.5. The parameters used in the calculations are n = 1.5, Sn = 0.07, X0 = nP = 0.5 jum. (After reference 21.) Substituting in (4.1.31) and simplifying So_ -iQexpjie) (4.1.36) or + cf cothV For the semi-infinite medium, v = oo in (4.1.36) and So Q (4.1.37) To e±i(Q2-ey When - Q < e < Q, £ is real and o 2 01= — = 1. 228 4. Cholesteric liquid crystals 0.75 0.50 0.25 0 0.70 0.68 0.66 0.64 0.62 0.60 0.58 0.56 Wavelength (jum) Fig. 4.1.5. Reflexion spectrum from a monodomain cholesteric film at normal incidence. Full curve: experimental spectrum for a mixture of cholesteryl nonanoate, cholesteryl chloride and cholesteryl acetate in weight ratios 21:15:6 at 24 °C (intensity in arbitrary units). Broken curve: spectrum computed from the exact theory for a film thickness of 21.0 //m and pitch 0.4273 jum. (After Dreher et The reflexion is total within this range. The spectral width of total reflexion is therefore A/I = Qkjn. Using (4.1.23), A/I = Pdn, (4.1.38) (3) a result first derived by de Vries. Outside this range, the reflexion decreases rapidly on either side. When X > Ao, e is negative and hence the negative value of the square root in the denominator of (4.1.37) has to be taken because @ can never exceed unity; when X < l0 the positive root has to be taken. Illustrative curves of M as a function of wavelength are shown in fig. 4.1.4. The semi-infinite medium gives the familiar flat topped curve of the dynamical theory, while the thin film gives a principal maximum accompanied by subsidiary fringes. The fringes are somewhat difficult to observe as even slight inhomogeneities in the specimen and small variations in its thickness tend to obliterate them, but careful experiments by Dreher, Meier and Saupe(25) have confirmed that they do occur (fig. 4.1.5). 4.1 Optical properties 229 Primary extinction and anomalous rotatory dispersion If reflexions are neglected, the optical rotation per thickness P of the liquid crystal is \{(pn — (pL) and the rotatory power is given by (4.1.21). Near the region of reflexion, the dynamical theory shows that the right-circular component suffers anomalous phase retardation and, under certain circumstances, attenuation as it travels through the medium. Left-circular light, on the other hand, exhibits normal behaviour throughout, and as a consequence the rotatory dispersion is anomalous around the reflecting region. (i) We shall consider first the semi-infinite case. According to (4.1.32), the amplitude of the right-circular wave as it passes from one plane to the next is given by T — xT where x = exp(-£)exp(-i<p 0 ), Inside the totally reflecting range, £ is real and the wave is strongly attenuated. In X-ray diffraction theory, this phenomenon is referred to as primary extinction. The extinction length, defined as the distance over which the amplitude of the incident wave decreases to l/e of its value, is P/Q at the centre of the reflexion band. Outside the range of total reflexion, £ is imaginary and primary extinction vanishes. Fig. 4.1.6 gives a plot of the wavevectors Kn ( = 2nnn/X) and KL ( = innJX) for right- and left-circular polarizations respectively, as functions of the wavelength. The real part of Kn shows a gap within the reflexion band - analogous to the familiar band gap in solid state physics - while the imaginary part grows rapidly in the same region. When e2 > Q2, ^ — i(e2 — Q2)* and may be positive or negative. The optical rotation per pitch is clearly and hence the rotatory power in radians per unit length n(5n)2P niX — X P= 7TT- + 5- 230 4. Cholesteric liquid crystals 17.0 - 18.0 - i | 19.0 - & 20.0 - 21.0?- 0.45 0.47 0.49 0.51 0.53 0.55 Wavelength (jim) Fig. 4.1.6. The wavevectors Kn and K^ of the normal waves as functions of wavelength in a semi-infinite non-absorbing medium. Curves are derived from the dynamical theory; circles represent values computed from the exact theory, n, Sn and XQ same as in fig. 4.1.4.(21) When Q2 > e2, i.e., within the region of total reflexion, p given by (4.1.39) becomes complex, showing that the medium is now circularly dichroic. The real part which represents the rotatory power is n{Sn)2P F (4.1.40) ~ U2 ' Pk ' (ii) For a thin film, we have from (4.1.30) and (4.1.35) sinh(v *r« £ cosech v^ k 4- £ coth v£ (4.1.41) and - 1. Consequently the oscillations which appear in the reflexion curve should 4.1 Optical properties 231 0.45 0.47 0.49 0.51 0.53 0.55 Wavelength (jim) Fig. 4.1.7. Rotatory power versus wavelength for a non-absorbing cholesteric: (a) semi-infinite medium, (b) film of thickness 25P. Curves are derived from the dynamical theory; circles represent values computed from the exact theory. «, Sn and Xo same as in fig. 4.1.4.(21) also be seen in transmission and in circular dichroism. Equation (4.1.41) may be expressed as where tan vy/ = The optical rotation per thickness P is therefore and n{8nfP yj-e P ~ W~ ~^p~' The theoretical variation of p with X is shown in fig. 4.1.7. As observed 232 4. Cholesteric liquid crystals -30 r -20 0.8 0.6 -10 0.4 0.2 U 0.0 o 10 20 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Wavelength (//m) Fig. 4.1.8. Experimental circular dichroism (open circles) and rotatory dispersion (closed circles) of cholesteric cinnamate at 177 °C. Sample thickness ~ 3 jum. (After reference 26.) already in §4.1.1 the rotatory power of a cholesteric liquid crystal, unlike that of an ordinary optically active substance, is a function of the sample thickness. Some measurements(26) of the optical rotation right through the reflexion band using thin films are represented in fig. 4.1.8. The oscillations in the theoretical curve for p appear to be smeared out, probably owing to slight imperfections in the sample, but the trends are in agreement with theory. There is also some evidence of subsidiary maxima in the circular dichroism which again is to be expected theoretically. Absorbing systems: the Borrmann effect The Borrmann effect is the anomalous increase in the transmitted X-ray intensity when a crystal is set for Bragg reflexion.(27) An analogous optical effect in absorbing cholesteric media in the vicinity of the reflexion band has been predicted and confirmed experimentally/ 22 ' 2829) The origin of the effect can be readily understood by extending the dynamical theory to include absorption. However, in contrast to the X-ray case, the polar- ization of the wave field and the linear dichroism play an essential part. Suppose that the birefringent layers are also linearly dichroic and that the principal axes of linear birefringence and linear dichroism are the same. All the equations obtained for the non-absorbing medium hold good in 4.1 Optical properties 233 -1.0 0.45 0.47 0.49 0.51 0.53 0.55 Wavelength (jim) Fig. 4.1.9. (<z) Reflexion coefficient 0t at normal incidence, (Z?) imaginary parts of Kn and ATL, (c) rotatory power /? (the real part of />), plotted against the wavelength for an absorbing semi-infinite medium calculated from the dynamical theory, K = 0.02, SK = 0.028 and other parameters same as in fig. 4.1.4. (After reference 21.) this case also except that Q, q>n, etc. have to be replaced by the corresponding complex quantities: Q = nSn/n, 2nnnP 2nnP n(Sn)2P2 <Pn = X X 4/1 2 2nnP n(Sn)2P2 4X2 KR = innJX, KL = Innjk, etc., with yi yi \rr yt yi i -is n a — na 1K a> U b ~ n b lK b> where Ka,Kb are the principal absorption coefficients. Fig. 4.1.9 gives the reflexion coefficient 0t and the dependence of the real part of p and the 234 4. Cholesteric liquid crystals 0.00 - 0.45 0.47 0.49 0.51 0.53 0.55 Wavelength (jjm) Fig. 4.1.10. Transmission coefficients, 5"R and ^ L for right- and left-circular waves calculated for a film of thickness 25P; (a) non-absorbing and (b) absorbing. Parameters same as in fig. 4.1.9. The enhanced transmission for the right-circular component in (b) is the analogue of the Borrmann effect. (After reference 21.) imaginary parts of Kn and K^ on wavelength. Here Sn, n and P are taken to be the same as for the non-absorbing case (see fig. 4.1.4) and in addition it is assumed that K = \{Ka + Kb) = 0.02 and SK = Ka — Kb = 0.028. The interesting result is obtained that on the shorter wavelength side Im (K n) is less than Im(^ L ), i.e., the right-circular component is less attenuated than the left component, while on the longer wavelength side the opposite is true. To observe this effect thin films have to be used. For an absorbing film of thickness vP, £ cosech v£ The theoretical dependence of 3Tn and ^ L on X is shown in fig. 4.1.10 for both the non-absorbing and absorbing cases. The structure being right- handed, the right-circular wave is reflected, and hence in a non-absorbing film (K = SK = 0) ^"R is always less than «^L. On the other hand, in the 4.1 Optical properties 235 -0.8 0.31 0.33 0.35 0.37 0.39 0.41 Wavelength (//m) Fig. 4.1.11. Circular dichroism versus wavelength computed for different K and SK. Sample thickness 25/>, n = 1.5, Sn = 0.07 and X0 = 036jiim. The absorption coefficients were assumed to be Gaussian curves having a maximum at 0.36 jum and of halfwidth 0.06 jum. The maximum values of K and SK are, respectively, as follows: (a) 0.0125, 0.0157; (b) 0.0250, 0.0314; (c) 0.0375, 0.0471; (d) 0.0500, 0.0628; (e) 0.0625, 0.0785. (After reference 28.) absorbing case 3T^ shows an enhanced value on the short wavelength side of the reflexion band, which is the analogue of the Borrmann effect. The phenomenon is shown up even more convincingly in the circular dichroism curves (fig. 4.1.11). For a left-handed structure (i.e., negative)?), «yL exhibits an anomalous increase and for SK < 0, the peak transmission occurs on the long wavelength side of the reflexion. 236 4. Cholesteric liquid crystals -0.4 - 0.2 - 0.1 • 0.0 -0.1 - -0.2 - -0.3 -0.4 0.32 0.34 0.36 0.38 Wavelength (//m) Fig. 4.1.12. Experimental circular dichroism curves versus wavelength, (a) Pure cholesteryl nonanoate (CN), (b) CN + 0.98 per cent by weight of PAA. Sample thickness in both cases 6.5 jum. (After reference 28.) The first experiments demonstrating the effect were conducted on cholesteryl nonanoate in which was dissolved small quantities of PAA or n-/7-methoxybenzylidene-/?-phenylazoaniline.(22'28) The temperature of the system was adjusted so that the reflexion band overlapped with the strongly linearly dichroic absorption band of the solute molecule. Under these circumstances, the circular dichroism exhibits the features predicted by theory (fig. 4.1.12). Similar measurements were reported subsequently by Aronishidze et a/.(29) (fig. 4.1.13). 4.1 Optical properties 237 4.1.3 Exact solution of the wave equation for propagation along the optic axis: Mauguin-Oseen-de Vries model We next consider the exact solution of the wave equation for propagation along the optic axis. The complete theory is contained in the papers of Mauguin,(1) Oseen(2) and de Vries,(3) and has since been presented in various forms by other authors. (30~3) We shall discuss an elegant treatment of the theory developed by Kats, (30) and by Nityananda. (31) We represent the dielectric tensor by a 'spiralling ellipsoid' whose principal axis Oc is always parallel to z; the other two principal axes Oa and Ob (with principal values sa and eb) spiral around z with a twist angle q = 2n/P per unit length. If Oa, Ob are taken to be along x, y at the origin, the tensor s at any point z may be expressed with respect to x, y as fcos^z — sin#zir£ a 01 f cos^z singzl [sin#z cos qz J L 0 e j [ —singz cosgzj ocsin2qz , (AAA2) [ asin2#z _ e — acoslqz •]• where sa = n2a, sb = n\, e = %ea + eb), a = ea-eb) = \{na + nb)(na-nb) = nSn. The wave equation for propagation along z is e2E 2=--jeE. (4.1.43) 8? c We introduce the variables E" = 2~\Ex-xEy\ E' is right-circular and E" left-circular for propagation along -fz and vice versa for propagation along — z. Substituting in (4.1.43) we obtain co2[ e aexp(2ig. ,22 l « ^ ^ / -»:^_\ » Ilr7//I' (4.1.44) c [aexp( — 2\qz) e The solution of (4.1.44) is of the form [A'cxp[i(k-q)z]\' which is a superposition of two waves of opposite circular polarizations 238 4. Cholesteric liquid crystals -0.5 \JLJLJLJ I 500 600 700 A (nm) Fig. 4.1.13. Circular dichroism in a mixture of 91.5% nematic ROTN 103 (of Hoffman-La Roche) +7.5 % optically active L-menthol + 1 % dye/?-nitrobenzene- bis(benzalazo)-/?'-dimethylaniline. Sample thickness 11.3/mi. Crosses represent experimental values and the solid curve gives the theoretical variation. (After Aronishidze et al.{29)) with wavevectors differing by 2q. It is readily verified that when substituted into (4.1.44) the (k + q) component of (4.1.45) suffers a wavevector shift of 2q and is converted into a(k — q) wave, and vice versa, so that together the two components form a closed set. Equation (4.1.45) therefore represents a true normal wave which can satisfy (4.1.44) with a proper choice of A' and A". From (4.1.44) and (4.1.45), we have (4.1.46) (k-qf-t where K = 2n/X and Km = InnIX, X being the wavelength in vacuo. The condition for (4.1.46) to yield non-vanishing solutions of A' and A" is [(k + qY- Kl] [(k -qf-Kl\- o?K* = 0, whose roots are kt, k, = (4.1.47) 4.1 Optical properties 239 Corresponding to the kx and k2 solutions we have respectively A' *K2 A" and } (4.1.48) A" A' -qf-Kl') When a is small, (4.1.47) and (4.1.48) give for the k2 solution A' I A" ~ a, and for the k± solution A" I A' ~ a. In other words, each normal wave is made up of two oppositely polarized circular components with one of the components generally dominating. The mixing of these two components with wavevectors differing by 2q is a consequence of the Bragg reflexion. Equation (4.1.45) may conveniently be rewritten as U_ T exp(i^z) U I exp(iK2z) ]' where Kx = k1+q = q + [K2m + q2-(4Klq* + a*K^]K (4.1.51) 2 2 2 K2 = K ~ q = - q + [Kl + q + ( 4 ^ q + a K*)i\\ (4.1.52) The fact that the wavevectors Kx and K2 are different is responsible for the optical activity of the medium, the optical rotation per unit length being p = K ^ — ^g) rad. However, as emphasized in previous sections the phe- nomenon is not identical with natural optical activity because the normal waves are not pure circular waves. The de Vries equation If now we make the approximation that (Kx — K2)/q <^ 1, or that the rotation per pitch is small compared with n, which is certainly valid in most cholesterics, P= where 240 4. Cholesteric liquid crystals When x2 < a2K*, p becomes a complex quantity. The real part gives the rotatory power and the imaginary part the circular dichroism. Since no dissipative mechanism is built into the model it follows that the imaginary part of p is associated with the reflexion of one of the components. The reflexion band is centred at x = 0, i.e., at Km = q or Xo = nP where Xo is the wavelength in vacuo. The range of reflexion extends from x = +aK2 to x = —OLK2, i.e., from + q\dn/ri) to -q\dn/ri). Since 3x = S(KJ2 = 2Km(SKm) ~ 2q(SKJ, the spectral width of total reflexion is given by AX = PSn (4.1.54) as was first shown by de Vries(3) (see (4.1.38)). When x21> a 2 ^ 4 , which is not valid close to or inside the reflexion band, n{dnfP (4.1.55) This is known as the de Vries equation. The sign of the rotatory power reverses on crossing the reflexion band (Ao). When X <^ Ao, (4.1.55) reduces to (4.1.21), and when X > 20, p tends asymptotically to 0. The behaviour on either side of the reflexion band has been confirmed experimentally(26) (fig. 4.1.8). Thin films (16) Nityananda and Kini have applied the theory to obtain exact solutions for reflexion and transmission by a plane parallel film bounded on either side by an isotropic medium. The treatment allows for the contribution due to reflexion at the cholesteric-isotropic interface. In general, for each circular polarization at normal incidence the reflected and transmitted waves consist of both circular polarizations. Four coefficients, two for reflexion and two for transmission, are required to describe the problem fully and the solution consists of matching the incoming and reflected waves on one side of the slab with four waves within the slab (two in the forward direction and two in the backward direction) and the transmitted wave on the other side. An extension of the treatment to absorbing media yields the theory of the Borrmann effect.(22) 4.1 Optical properties 241 Some calculations for the semi-infinite medium and for the thin film are shown in figs. 4.1.4, 4.1.6 and 4.1.7. In these calculations, the isotropic medium external to the liquid crystal is assumed to have a refractive index equal to n = \{na + nb) so that the contribution of the ordinary Fresnel reflexion coefficient at the cholesteric-isotropic interface is eliminated. It is clear from these figures that the results of the exact theory differ only slightly from those of the dynamical model, indicating that the latter is probably adequate for most practical calculations. However, the simple formulation of the dynamical model presented in §4.1.2 does have some inherent limitations: (i) it is valid only for integral values of the pitch, (ii) it is developed for small e (= —2U{X — XQ)/X) and therefore does not give exact values for wavelengths away from the reflexion band, and (iii) it fails when the film thickness is very small, or when the extinction length is of the order of a pitch. These limitations arise primarily because of the kinematical approximation made for the reflexion from the m layers per turn of the helix. We shall now show that when multiple reflexions within the m layers are also included, the simple difference equations become matrix difference equations and the resulting solutions turn out to be fully equivalent to the exact electromagnetic treatment. Proofs of this result have been given by Joly(32) and by Nityananda. (3134) We shall follow the latter's treatment(34) which is simpler. 4.1.4 Equivalence of the continuum and the dynamical theories We go back to the model discussed in §4.1.1, viz, a twisted stack of thin birefringent layers. The principal refractive indices of each are na and nb, and the angle of twist between the successive layers is /?. Let ra, ta, rb and tb be the reflexion and transmission coefficients for a single layer for light linearly polarized along its principal axes at normal incidence. For polarization along the a axis, rn = -- /„ = („„+!)*-(„„-1)* T J' where xa — exp (ina Kp), K is the wavevector in vacuo and p the layer thickness.(35) Exactly similar expressions may be written for the other polarization also. We define Is and Js as the amplitudes incident on the sih layer in + z and 242 4. Cholesteric liquid crystals s s+\ Fig. 4.1.14. Notation for the incident and reflected waves. Is and Js are the amplitudes of the waves incident on the 5th layer in the positive and negative directions respectively, and Es and Ss the amplitudes emerging from the 5th layer in these two directions. — z directions respectively, and similarly Es and Ss as the amplitudes emerging from the sth layer in the + z and — z directions respectively (fig. 4.1.14). Therefore, for the sth layer we have a _ f ja l l s ~ a s (4.1.56) The first two equations of (4.1.56) can be combined and written as The second two lines can also be similarly combined. However, in what follows we shall write them in the form (4.1.57) with the understanding that E, I, S and J* are each column vectors ( 2 x 1 matrices), and that r and t are each 2 x 2 diagonal matrices. Now, the emergent wave E s in the + z direction from the ^sth layer is physically the same as I s+1 , the wave incident on the (s+l)th layer, but (4.1.57) will apply to the (s+ l)th layer only if I s+1 is referred to its own 4.1 Optical properties 243 principal axes which are rotated through /? with respect to those of the sth layer. We therefore write IS+1 = SES (4.1.58) and similarly J^S-1^, (4.1.59) where cosy? sin/?l — sin/? cos/?J Strictly these equations should include a phase factor allowing for the air gaps between the layers, but the gap may be taken to be infinitesimally small compared with the thickness of the layer, which itself tends to zero. Using (4.1.58), (4.1.59) and (4.1.57) Because of the periodicity with respect to s, the difference equations can be solved in the form Es = Eexp(i^). This yields Now the total field in the gap is F = E +«/, so that F - J ^ = tSexp(-i^)(F-j^) + rJ^, (4.1.60) Jf = S" 1 rS(F-./) + S- 1 texp(ip)./. (4.1.61) From (4.1.60) we get Using this to eliminate £ from (4.1.61) {[l+S- 1 rS-S- 1 texp(i(^)][l+r-tSexp(-i^)]- 1 x [1 - tS exp ( - \(p)\ - S'^S} F = 0. (4.1.62) As before we effect a change of variables: F±=2-kFa±\Fb), i.e., M -3EMS-* 244 4. Cholesteric liquid crystals The matrices in the difference equations should also be transformed to the new variables. For example r should be replaced by i = [i(',. + O K ' a - O i r r Sr/2] ~YWa-rb) Wa + rb)\ [Sr/2 f \ where r = \{ra + rb) and Sr = ra — rb9 and S by ASA"1 = i/0 0 " I 0 exp(iy9)J' Since r and t are functions of the thickness p of the layer, we expand them in powers of/?. It is sufficient to retain the first power in each case: F=l(s-l)iKp, dr/2 = \xaKp, dt/2 = (ta - Q/2 = \i<xKp, where as before e = |(£a + eb) and a = |(e o — £„). Writing fi = qp and q> = kp (and remembering that since r is of order p, rS ~ r to this order) (4.1.62) may be expressed as (4.1.63) where R simplifies to (4.1.64) below: -1 K Uk + qf-eK* -*K* 1 2 ( 6 ) 1 -a* (k-q)*-eK'\- Since the first matrix on the right-hand side is non-singular we may premultiply by its inverse to obtain (4.1.63) in the form Uk + qf- [ -<xK2 (k-q)2-sK2\[F_\ ' which is precisely the same as (4.1.46). The dynamical theory applied in this 4.1 Optical properties 245 manner to a twisted pile of birefringent layers is then exactly valid for any arbitrary thickness of the sample and for the entire range of wavelengths. 4.1.5 Oblique incidence The theory of propagation inclined to the optic axis is, of course, very much more complicated, and analytical solutions have not so far been found.<36) The first attempt at solving the problem numerically was by Taupin,(37) but the most complete calculations are those of Berreman and Scheffer<38) who also carried out a precise experimental study of reflexion from monodomain samples at oblique incidence. Fig. 4.1.15 presents their observed reflexion spectra for two polarizations. Berreman used a 4 x 4 matrix multiplication method. Assuming the incident and reflected wavevectors to be in the xz plane, z being along the helical axis of the cholesteric, the dependence on the y coordinate may be ignored altogether. Writing Ex = E°xexp [i(cot — kx)] etc., it is easily verified that Maxwell's equations can be reduced to the matrix form .kceT — 1- 0 Ex CO CO \HV -e 0 iHy c b • x l Ey kcY _ 0 CO CO 1 or QZ where, assuming a spiralling ellipsoid model, the components of the dielectric tensor are given by exx = eyy = e — occos2qz, and all other components are zero (see (4.1.42)). 246 4. Cholesteric liquid crystals ([ r 0.2 - 00 , ji^ Second 0.2 — order I j First order Observed spectra 0.6 _ V i I 1 1.0 1.4 0.6 - Computed spectra Fig. 4.1.15. First and second order reflexion spectra of a cholesteric liquid crystal film (0.45:0.55 mole fraction mixture of 4/-bis(2-methylbutoxy)-azoxybenzene and 4,4'-di-n-hexoxyazoxybenzene) 15 pitch lengths or 11.47 jum thick. Angle of incidence 45°. Polarizer and analyser are parallel to the plane of reflexion for 0tn and normal to it for 0to measurements. The small oscillations are interference fringes from the two cholesteric-glass interfaces. (After Berreman and Scheffer.(38)) 4.1 Optical properties 247 To a first order of approximation where 3P is a 4 x 4 propagation matrix and E is the unit 4 x 4 matrix. Repeated matrix multiplication in very small steps Sz gives the matrix for the total film, and by taking into account the appropriate boundary conditions on either side (glass) one can work out the transmitted and reflected waves. Of course, cyclic and other symmetry properties of 3?(z) reduce the number of matrix multiplications to a reasonable number in practical cases. In the actual calculation, Berreman and Scheffer included a second order term subject to the symmetry property &>(z,Sz) = 0>-1(z,-dz). The results of their computations are shown in fig. 4.1.15. The agreement with the experimental spectra can be seen to be good. There is a difference in the intensities, which may conceivably be due to thin regions near the surface with anomalous dielectric properties or due to the neglect of absorption. An important fact that emerged from this study is that the observed features were best reproduced when the local dielectric ellipsoid was taken to be a prolate spheroid, with the principal axis Oc parallel to z and sc = s — a. Thus the assumption that is generally made that the local dielectric ellipsoid is uniaxial would appear to be valid to a very good approximation as far as optical calculations are concerned (see, however, §4.10). 4.1.6 Propagation normal to the optic axis When light is incident normal to the optic axis polarized diffraction maxima are seen in transmission.(39) For the electric vector polarized parallel to z the refractive index is independent of the z coordinate, whereas for the vector polarized perpendicular to it the refractive index varies periodically from na to nb. The wavefront having the latter polarization therefore suffers changes of phase which vary along z with a periodicity equal to half the pitch. There can also be changes of amplitude as parallel rays tend to acquire a slight curvature when travelling in a medium in which the gradient of refractive index is normal to the direction of propagation. (40) This periodicity in the phase and amplitude gives rise to polarized diffraction effects. Approximate expressions for the intensity of the maxima may be derived by applying the Raman-Nath theory of the diffraction of light by ultrasonic waves.(41) 248 4. Cholesteric liquid crystals 0.3 r 0.2 0.1 £ -0.1 -0.2 -0.3 30 34 38 42 46 50 54 Temperature (°C) Fig. 4.1.16. Variation of inverse pitch with temperature in a 1.75:1 weight mixture of right-handed cholesteryl chloride and left-handed cholesteryl myristate as determined by laser diffraction. The mixture becomes nematic at 42 °C. (Sackmann et Sackmann et al.(39) have investigated the temperature variation of the pitch of a mixture of right-handed cholesteryl chloride and left-handed cholesteryl myristate by this method. At a certain temperature (7^) there is an exact compensation of the two opposite helical structures and the sample becomes nematic. At this temperature only the central spot (zero order) is observed, while at the other temperatures, polarized diffraction maxima of higher order make their appearance. The inverse pitch varies almost linearly with temperature passing through zero at 7^ (fig. 4.1.16). 4.2 Defects 4.2.1 %-disclinations We now consider defect structures in the cholesteric liquid crystal. Treating the cholesteric as a spontaneously twisted nematic, nx = cos 0, ny = sin #, nz = 0, 6 = qz, q = 2n/P, P being the pitch. The free energy density is then expressible as (4.2.1) 4.2 Defects 249 Fig. 4.2.1. Director patterns for .s = £ and 1 /-screw disclinations in a cholesteric. In the one-constant approximation, (4.2.2) and V26> = 0. disclinations Such disclinations are closely analogous to nematic wedge disclinations (§3.5.1). The singular line is along the z axis (parallel to the twist axis) and the director pattern is given by nx = cos 9, ny = sin 6, nz = 0 = s tan" 1 (y/x) + qz (4.2.3) s = ±N/2, where N is an integer. The presence of the disclination does not alter the pitch. Fig. 4.2.1 illustrates the director patterns for disclinations of strength s = 1 and 1. Many of the conclusions arrived at in §3.5.1 are valid in this case as well, and in the one constant approximation the expressions for the energies and interactions are the same as for nematic wedge disclinations. As already indicated briefly in §3.5.8 the effect of elastic anisotropy has some interesting implications for cholesterics, especially for long-pitched structures. We have seen that disclination pairs in nematics have angular forces in the presence of elastic anisotropy. For all practical purposes, the solutions that were obtained for nematics will hold good for each nematic- like cholesteric layer, except that the layers now twist continuously in the 250 4. Cholesteric liquid crystals (a) Fig. 4.2.2. A pair of like /-screw disclinations forming a stable double helix in a cholesteric: (a) a pair of s = \ disclinations (after Cladis, White and Brinkmann(42)), (b) a pair of s = 1 disclinations (after Rault(43)); see fig. 3.5.24. medium. Therefore, for a pair of disclinations, angular forces will lock the line joining the disclinations at the same orientation with respect to the local director n (of the defect-free sample) in every layer. Hence while single disclinations may be straight, pairs of disclinations in a cholesteric may be expected to have a helical configuration (see fig. 3.5.24). For a pair of like disclinations, mutual repulsion increases the separation between them, but in the helical state this will be opposed by the line tension in each disclination. In the end the two opposing processes should balance to result in a stable double helix. This is indeed found to be the case experimentally 4.2 Defects 251 & Fig. 4.2.3. A helical %(s = — 1) disclination wound round a straight %(s = +1) disclination in a cholesteric liquid crystal (Rault(44)); see fig. 3.5.24. (fig. 4.2.2). On the other hand, if the individual disclinations of the pair have different line tensions (as is the case for 1 and — 1, both of which are escaped structures) then the disclination with a lower tension should wind around the one with the higher tension. This again appears to be in conformity with experimental observations (fig. 4.2.3). The escaped configurations for 1 and — 1 in the cholesteric are rather complex,(45) which may perhaps account for the fact that the two do not annihilate one another. 252 4. Cholestenc liquid crystals H H -I h- I- h- I- h- Fig. 4.2.4. The director pattern for s = \ /-edge disclination in a cholesteric. Dots signify that the director is normal to the plane of the diagram, dashes that it is parallel to and nails that it is tilted. X~edge disclinations In this case the singular line is perpendicular to the twist axis. On going round this line, one gains or loses an integral number of half-pitches. The director pattern around the /-edge disclination was first worked out by de Gennes(46) who proposed a nematic twist disclination type of solution: nx = cos 0, ny = sin 0, nz = 0,> 9 = s tan"1 (z/x) + qz, (4.2.4) s = N/2. The cholesteric pitch is altered around the singular line where T is anV integer. The pattern for s = \ is shown in fig. 4.2.4. Again, the energies and interactions in the one-constant approximation are the same as for nematic twist disclinations. A somewhat more elaborate treatment of this model has been presented by Scheffer(47) and the effect of elastic anisotropy has been investigated by Caroli and Dubois-Violette.(48) The Volterra process for creating these disclinations is the same as for nematic disclinations. For the screw disclination the plane of cut is parallel to the cholesteric twist axis while for the edge disclination it is perpendicular to it. 4.2.2 Lattice disclinations The cholesteric may also be regarded as having a layered structure with a periodicity of P/2 along the z axis. This lattice can have disclinations just as in smectic A and the Volterra process for creating them is also essentially the same (see §5.4.4). If the cut is such that the line L is along the local molecular axis, the disclinations so created are designated as X+ and A~, {X standing for ' longitudinal' and the plus (or minus) sign indicating that 4.2 Defects 253 (a) (b) \ •• N id) + Fig. 4.2.5. The configurations for (a) A", (6) A , (c) T" and (d) T+ disclinations in a cholesteric. Dots, dashes and nails have the same significance as in fig. 4.2.4. (a) (d) Fig. 4.2.6. Examples of pairing of X and z disclinations of opposite signs in a cholesteric. Edge disclinations composed of (a) X' and A+, (b) T~ and T+, (C) T~ and A+, (d) X~ and T+ and (e) pincement composed of z+ and T~. material has to be removed (or added) to arrive at the final configuration) but if L is perpendicular to the local molecular axis they are designated as T+ and T~ (fig. 4.2.5) (T standing for 'transverse'). The Is are coreless and therefore have lower energies than the is which do have cores. 254 4. Cholesteric liquid crystals As or rs of opposite signs occur in pairs to form dislocations and pincements (fig. 4.2.6). Such pairing can be observed directly in the fingerprint textures which are exhibited by cholesterics of large pitch when the helical axis is parallel to the plates (figs. 4.2.7 and 4.2.8). 4.2.3 Dislocations Because of the layered structure, defects in the cholesteric can be likened in many respects to those in smectic A. Both of them exhibit focal conic textures(52) and both allow for the existence of screw and edge dislocations. To discuss these similarities we employ a 'coarse-grained' approximation in which the cholesteric distortions are considered to be small and to vary slowly over a pitch. In this approximation the free energy of distortion may be expressed in terms of layer displacement u parallel to the twist axis: where B = k22 q2 and K = |fc33 (see (4.6.22)). This expression first derived by de Gennes is exactly of the same form as (5.3.3) for smectic A. We shall now consider some applications of this model.(53) Screw dislocations The deformation around a screw dislocation may be written as NP u = -^Uur1(y/x)9 (4.2.5) V where PQ is the pitch and T is an integer. The singular line is along the twist axis. A circuit around the singular line results in a displacement of the layer along the helical axis by an integral number of half-pitches. Equation (4.2.5) can be recast to give the director orientation, and it is easily verified that it becomes identical to (4.2.3) for the /-screw disclination and leads to same results for the energies and interactions. Edge dislocations Let the layers be in the xy plane and the line of singularity along y. Following the theory for smectic A (§5.4.2), one may write, p Y If 00 H/r 1 u(x,z) = -£ 1 + - T-exp(-Aic a |z|+iK*) where Po is the pitch of the undistorted cholesteric and X= 4.2 Defects 255 (a) (b) (iv) Fig. 4.2.7. Disclinations and dislocations in (a) fingerprint textures of cholesterics. (b) interpretation of (i), (ii) and (iv) of (a). (After Bouligand and Kleman.(49)) 256 4. Cholesteric liquid crystals Fig. 4.2.8. Fingerprint textures of cholesterics showing pincements, (a) after Robinson, Ward and Beevers,(50) and (b) after Bouligand.(51) 4.2 Defects 257 (P0/4n)(3kS3/2k22)i. From this one gets the layer tilt 6 (with respect to the z axis) and the layer dilatation S = du/dz as <4 2 6) )\ - - The energy of a single dislocation in an infinite medium is then 0.6nkP h where Ec is the energy of the core and C the core radius, which in this model can probably be assumed to be of the order of Po. Hence the energy turns out to be finite and does not diverge logarithmically with sample size as does a nematic-lilce solution of the form (4.2.4). If there are two like dislocations, one at (0,0) and the other at (x0, z0) the forces Fx and Fz (along and perpendicular to the layers) acting between them are „ kPl For a pair of like dislocations Fx is always repulsive, while Fz is repulsive for x\ < 2A0z0 and attractive for x\ > 2Aozo. Thus, as in the case of smectic A (see fig. 5.4.7), there can result a clustering of like edge disclinations to form a 'grain boundary'. Such clustering is often observed in fingerprint textures (fig. 4.2.7). When a cholesteric sample is prepared in the form of a thin wedge between two glass plates inclined at a small angle, and the twist axis is approximately normal to the plates, regular striations are seen running across the film when viewed under a polarizing microscope (fig. 4.2.9(a)). These striations, usually referred to as the Grandjean-Cano pattern, are due to edge dislocations. If 9 is the wedge angle, the number of edge dislocations per unit length that are required to minimize the energy is d'1 = 26/P0 where d is the average distance between dislocations. The darker lines in the photograph are dislocations having double the Burgers vector of the weaker ones. They are composed of /l~A+ pairs, which as we have seen already are devoid of cores.(55) When a magnetic field is applied 258 4. Cholesteric liquid crystals Fig. 4.2.9. Grandjean-Cano pattern of edge dislocations in a cholesteric wedge formed between a cylindrical lens and a flat plate (a) without a magnetic field and (b) with a magnetic field normal to the spiral axis and the dislocation line. The ' double' dislocations - the darker lines - become zig-zag in the presence of the magnetic field while the 'single' dislocations remain stable. (The Orsay Liquid Crystal Group.(54)) normal to the helical axis and to the dislocation line, the 'double' dislocations get distorted (fig. 4.2.9 (b)). This is because a cholesteric composed of molecules of positive diamagnetic anisotropy prefers to have its helical axis normal to the field, but the region between the disclination pair has its axis along the field. Energetically, therefore, it is favourable for the lines to adopt a zig-zag shape. Qualitative arguments suggest that the lowering of the energy is greater the larger the separation between the disclinations. This probably explains why the weaker lines, which are composed of Xx pairs, remain stable. 4.3 Leslie's theory of thermomechanical coupling The fact that the properties of the cholesteric liquid crystal are not invariant with respect to reflexion introduces some additional complexity into the equations of the continuum theory. The possibility now exists of 4.3 Leslie's theory of thermomechanical coupling 259 a coupling between thermal and mechanical effects which symmetry considerations rule out automatically from the theory of the nematic state. Thus translational or rotational motion of the fluid can, in principle, cause heat transfer, and equally a thermal gradient can create motion. (56) Consider an incompressible cholesteric fluid with a non-polar director of constant magnitude. The basic equations developed in §3.1 need to be modified slightly because of the absence of a plane of symmetry. Allowing for heat flux and thermal gradients the conservation laws are /> = 0, (4.3.1) U=Q-qi,i + tjidti + njiNtj-giNi, (4.3.3) where Q is the heat supply per unit volume, qt the heat flux vector per unit area per unit time and the other symbols have the usual meanings (see §3.3). If /?^ is the entropy flux vector per unit area per unit time, it is convenient to introduce a vector cpi such that The entropy inequality may then be written as t^ + nttNv-gtNt-PtTt-F-St-v^ > 0, (4.3.5) where F = U— TS is the free energy function. Making use of the fact that Ti9 dtj and nu can be chosen arbitrarily and independently, we obtain the following equations: with <Pt = ^m^(Nk + dkpnp), where a is a constant. The inequality (4.3.5) then becomes 260 4. Cholesteric liquid crystals Resolving tip gt and pt into their static and hydrodynamic parts, we have ~~^nk,p (4.3.8) P°t = 0 , (4.3.9) and (4.3.6) reduces to t'iidii-g',N<-(p'i+%j!) T{ > 0. (4.3.10) Also fn + gi^ = t'ji + g'jni, (4.3.11) and the entropy generation per unit volume TS = Q-qitt + <KtjJ£n£Nk + dkpnp)li + fJtdv-g'iNt. (4.3.12) The hydrodynamic components of the stress tensor etc., are t'n = Mi nk % dkv nt n, + ju2 N< n, + //3 N, nt + //4 dn + ju5 dik nk n5 + ^edjknkni + M7etpqTQnjnp + /i8eJpqTqntnp9 (4.3.13) g\ = k1Ni + A2 ^ rij + /l3 ^ M /i p Tq9 (4.3.14) ft = ^ r , + ^ 2 /ifc r f c ^ + ^ 3 etpq np NQ + A:4 eipq nv dqr nr, (4.3.15) (4.3.16) where // 15 ...,// 6 are the six viscosity coefficients already defined in the continuum theory of the nematic state; ju7 and ju8 are two additional viscosity coefficients coupling thermal and mechanical effects; A^1?...,AT4 are the coefficients of thermal conductivity. The free energy of elastic deformation per unit volume is given by (3.2.7). Martin, Parodi and Pershan(57) and Lubensky(58) developed a general 4.3 Leslie's theory of thermomechanical coupling 261 Fig. 4.4.1. Lehmann's diagrams depicting the rotation phenomenon in open cholesteric droplets heated from below. (After Lehmann.(59)) hydrodynamic theory of layered systems, and showed that symmetry permits a thermomechanical coupling between the phase of the layers and the temperature gradient. For a cholesteric, the phase is determined by the azimuthal angle of the director, and this again leads to the same conclusions as Leslie's theory. 262 4. Cholesteric liquid crystals 4.4 The Lehmann rotation phenomenon An example of this type of thermomechanical coupling appears to have been observed by Lehmann(59) in cholesteric liquid crystals very soon after their discovery. He found that droplets of the material when heated from below seemed to be rotating violently, but from optical studies he concluded that it was not the drops themselves but the structure that was rotating. Fig. 4.4.1 shows a few of the many sketches that he made depicting his observations. Leslie's equations(56) offer a simple explanation of the phenomenon; because of the absence of mirror symmetry, an applied field, which is a polar vector, can result in a torque, which is an axial vector. Let the cholesteric film be bounded between the planes z = 0 and z = /*, and let there be a temperature gradient along the screw axis z. The components of the director in a right-handed cartesian coordinate system are {cos#(z, t), sin#(z, i), 0}. We assume that there are no heat sources within the liquid crystal, no external body forces and that the velocity vector is zero. Hence T = T{z)Ji = G{ = dtj = wtj = 0. Thus from (4.3.4) A _ 87* +. (4A) ^ M* *) 820 _ 60 8/ 7 80\ and from (4.3.12) ^)-{x^-x^k> (4A2) where and Fo is the free energy in the absence of elastic deformation. In static situations, 820/8f2 = dO/dt = 0, and (4.4.1) and (4.4.2) yield de Af -k k \-)idT-o 2 22 3 dz\ dz) 6z giving CT Jo 4.4 The Lehmann rotation phenomenon 263 and K ^ Jo 22 \ Jo / Jo ^22 where ^4, 5 , C, D are constants which can be determined from the boundary conditions. If T(0) = To, T(h) = T± and the director is assumed to be completely free at the bounding surfaces (i.e., the torques TU = eijkntnlk at the boundaries are zero), ^—k 2 + oc) = [k22-—k 2 + oc) =0. One can then work out a simple solution if the material constants are assumed to be independent of temperature. Let 9 = cot +/(z) and T = g(z); (4.4.1) and (4.4.2) then reduce to ^ 2 2 0 - ^ +^ = 0, (4.4.3) a2? a? oz oz Equation (4.4.4) gives g = (V^s) ^ + ^ exp (A8 G > Z / ^ ) + 5, and (4.4.3) then gives / = (Kx A/cok22) exp (A3 coz/KJ + Cz + D. From the boundary conditions, one obtains finally r=ro+[(r1-ro)z//i], (4.4.5) e = eo + ^ + [(*a-a)z/fc22], (4.4.6) where 60 is the orientation at z = 0, and . (4.4.7) Thus the director rotates about Oz with an angular velocity co, which explains Lehmann's observations. In the absence of a temperature gradient (4.4.6) reduces to 0 = 60 + [(k2-a)z/k22] which describes the normal cholesteric structure with a pitch P = 2nk22/(k2 — a). The coefficient a has not yet been determined experimentally 264 4. Cholesteric liquid crystals Fig. 4.4.2. A spherical drop of a long-pitched cholesteric liquid crystal showing the characteristic/-line (from Robinson(62)). The structure of this defect was explained by Frank and Pryce(61). and it is not known whether its contribution is of practical significance. According to the Oseen-Zocher-Frank elasticity equations a = 0, and in the absence of any evidence to the contrary it is generally neglected in most discussions. The Lehmann rotation phenomenon has never been reproduced since its discovery. However, the experiment has been successfully repeated in the author's laboratory (60) using a DC electric field instead of a thermal gradient. Difficulties arising from anchoring effects at the boundaries were eliminated by forming cholesteric drops suspended in the isotropic phase. This was achieved in the following manner. The material was carefully chosen to avoid other electrical effects on the orientation of the director. A binary mixture of alkoxyphenyl-rra«s-alkylcyclohexyl carboxylates (supplied by Merck) was prepared to give a room temperature nematic with dielectric anisotropy «— 1. The mixture did not exhibit any electrohydrodynamic instabilities up to DC voltages of ~ 8 V. Addition of ~ 5 wt % of cholesteryl chloride resulted in a left-handed cholesteric liquid crystal with pitch P % 5//m, while ~ 10% of methylbutylbenzoyloxy- heptyloxycinnamite yielded a right-handed cholesteric with the same P. 4.4 The Lehmann rotation phenomenon 265 Fig. 4.4.3. Photographs demonstrating the Lehmann rotation effect in cholesteric drops under the action of a DC electric field. (a)-(c) taken about 30 s apart, illustrate a clockwise rotation of the structure, and (d)—(f) an anticlockwise rotation when the voltage is reversed. The material used was a binary nematogenic mixture of alkoxyphenyl-fraws-alkylcyclohexyl carboxylates (supplied by Merck) to which was added ~ 5 wt% of cholesteryl chloride. (After reference 60). The cholesteric material was then doped with a small quantity of a non- mesomorphic epoxy compound, Lixon, which lowered the cholesteric- isotropic transition temperature and gave rise to a broad two-phase region. Because the glass plates have greater affinity for Lixon than for the liquid crystal compound, the cholesteric drops were surrounded on all sides by the isotropic phase. With thick cells, spherical drops with the characteristic /-line of strength 2 were formed (fig. 4.4.2). In thin cells (~ 8 jum thick) flattened drops were obtained in which the central portion 266 4. Cholesteric liquid crystals had an essentially planar structure with the helical axis normal to the flat region and the /-line was confined to the periphery. Since the anchoring energy at the cholesteric-isotropic interface may be expected to be negligible, these flat drops proved to be most suitable for studying the phenomenon. In principle, any transport current can produce a cross- coupling effect and hence a DC electricfieldwas used for convenience. At 2 V DC, the dark brushes emanating from the /-line became curved, and then the whole structure started to rotate, apparently without any further distortion. Fig. 4.4.3 shows the photographs of the rotating drops, which can be seen to be closely similar to Lehmann's diagrams reproduced in fig. 4.4.1. Systematic observations on a number of drops established the following results: (a) all the drops rotate in the same direction for a given sense of the field: the right-handed helix has an anticlockwise rotation when viewed along the field direction. When the voltage is reversed, the curvature of the dark brushes and the sense of rotation of the structure are reversed; (b) the angular velocity increases linearly with applied voltage up to ~ 3.5 V, beyond which the structure of the drop changes and the rotational velocity becomes a non-linear function of the applied voltage (fig. 4.4.4); (c) nematic drops do not rotate under the action of E; (d) when the handedness of the helix is reversed, the angular velocity also reverses sign for any given sense of the field E; (e) the angular velocity does not depend on the radius of the drop, showing that it is a rotation of the structure rather than a rigid body rotation of the drop as a whole. All these observations are in conformity with the theory. Though the angular velocity was approximately the same for all drops, some drops which had dust particles attached to them rotated with a lower velocity. For the sake of completeness, even these values have been plotted in fig. 4.4.4. The extrapolated angular velocity becomes zero for V « 1.9 V (fig. 4.4.4). The last point indicates that the DCfieldis totally screened up to « 1.9 V and that the redox potential of at least one of the components in the mixture is about 1.9 V. For a defect-free planar structure, the angular velocity in the presence of an electric field E is in analogy with (4.4.7), where vE is the electromechanical coupling coefficient. However, in the actual experiment a line defect at the periphery of the drop also rotates with the structure. The effective friction coefficient for the slow motion of a nematic line singularity has been estimated by Imura and Okano(63) and 4.5 Flow properties 267 0.6 D OX 0.5 - 0.4 - A 0.3 - 0.2 - 0.1 J^ ? T 1 . 1 1 2 3 4 5 Voltage (V) Fig. 4.4.4. The rotational velocity against applied voltage. The different symbols denote measurements on different drops. The angular velocity was noticeably less for drops which accidentally had dust particles attached to them (see the circles in thefigure),but only the data for the fastest rotating drops were considered in the calculations. Between 3.5 and 5 V there was visible disturbance within the drop and measurements were not possible. At 5 V and above, the drop regained a uniform texture. (After reference 60.) by de Gennes.(64) Extending this theory to the case of the /-line of strength 2, it turns out that in the presence of the line defect rotating with the structure (60) Using the slope of the linear part of the |o>| against |E| curve (fig. 4.4.4), and putting Xx = 0.7 P, it was found that |vE| = 0.28 cgs, and vE/q = — 2 x 10~5 cgs for the material used in the experiment. 4.5 Flow properties The flow properties of a cholesteric liquid crystal are surprisingly different from those of a nematic. Its viscosity increases by about a million times as the shear rate drops to a very low value(65) (fig. 4.5.1). One of the difficulties in interpreting this highly non-Newtonian behaviour is the uncertainty in the wall orientation which cannot be controlled as easily as in the nematic case. Some careful measurements of the apparent viscosity ^ app in Poiseuille flow have been made by Candau, Martinoty and Debeauvais (66) of a 268 4. Cholesteric liquid crystals 105 Is 1O4 103 102 1 Cholesteric r Isotropic lO"1 I io- 2 ^ - — - ^ 1 I 1 I i l l 1 100 110 120 130 140 Temperature (°C) Fig. 4.5.1. Apparent viscosity of cholesteryl acetate versus temperature. Capillary shear rate (s"1): open inverted triangle, 10; cross, 50; open square, 100; filled inverted triangle, 1000; filled square, 5000. Rotational shear rate (s"1) open triangle, 104; open circle, high shear rate and normal liquid behaviour. (After Porter, Barrall and Johnson.(65)) nematic-cholesteric mixture whose pitch could be varied by changing the composition. Microscopic examination revealed that the helical axes were oriented radially in the capillary so that the cholesteric layers were rolled up in the form of coaxial cylinders. The flow direction was therefore normal to the helical axis at every point. In this geometry there is a slight dependence of the viscosity on the shear rate and on the pitch, but the significant fact emerges that //app is approximately of the same order of magnitude as that for a nematic even at low shear rates (figs. 4.5.2 and 4.5.3). The application of the Ericksen-Leslie equations to cholesteric flow is less straightforward than in the case of nematics and no detailed solutions have so far been possible even for simple geometries. However, the behaviour in certain limiting situations can be explained qualitatively. 4.5 Flow properties 269 0.30 - 0.28 0.26 0.24 fe) 0.22 0.25 0.50 0.75 1.00 1.25 Shear rate (103 s"1) Fig. 4.5.2. Apparent viscosity in Poiseuille flow as a function of shear rate. Flow normal to the helical axis (see text). Pitch (//m) = (a) 1.9, (b) 2.6, (c) 3, (d) 3.9, (e) 6, (/) 9.1 and (g) oo. (After Candau et a/.<66 1.5 V- 1.0 0.5 10 15 20 lAP 2 (10 6 cm- 2 ) Fig. 4.5.3. Threshold of shear rate above which the fluid becomes non-Newtonian plotted against 1/P2, where P is the undistorted value of the pitch. The arrows represent the upper and lower limits of the shear rate (see fig. 4.5.2). (After Candau 66 et 270 4. Cholesteric liquid crystals Fig. 4.5.4. Helfrich's model of'permeation' in a cholesteric liquid crystal. At low shear ratesflowtakes place along the helical axis without the helical structure itself moving. 4.5.1 Flow along the helical axis Helfrich(67) proposed a simple physical mechanism, which he called permeation, to account for the very high apparent viscosity at low shear rates. He suggested that flow takes place along the helical axis without the helical structure itself moving owing to the anchoring effects at the walls (fig. 4.5.4) and that the velocity profile is flat rather than parabolic. Under these circumstances, the translational motion of the fluid along the capillary can be directly related to the rotational motion of the director. The energy gained by the motion in the pressure gradient should be equal to that dissipated by the rotational motion. Now the viscous torque exerted by the director T = n x g', where g' is given by (3.3.13). In the absence of velocity gradients this torque is evidently — Xxqv, where v is the linear velocity and q = 2n/P is the cholesteric twist per unit length. In nematics, Xx < 0; we shall assume this to be true in the present case also. Thus where dp/dz is the pressure gradient. The quantity of fluid flowing per second is Q = _nR\dp/dz) where R is the radius of the capillary. Applying Poiseuille's law, / X /app = \ • (4.5.1) Typically, R ~ 500 jum and P = 2n/q - l / / m s o that ?/app - - 10% which explains the very high viscosity at low pressure gradients. 4.5 Flow properties 271 We shall now show that the essential features of Helfrich's model can be derived on the basis of the Ericksen-Leslie theory. (68) Flow between parallel plates We shall consider flow between two parallel plates, caused by a pressure gradient. Choosing a right-handed cartesian system such that the plates occupy x = ±h, we seek solutions of the form nx = cos (qz + (p) cos 0, ny = sin (qz + q>) cos #, nz = sin 0, vx = 0, vy = 0, vz = w, with 0 = 6(x,z), cp = (p(x,z) and w = w(x). This gives a cholesteric of pitch P = 2n/q with the helical axis along z for 0 = cp = 0. Strictly speaking, a general theory should allow for non-zero values of vx and vy but as we shall see shortly the present approximation is valid except in a negligibly thin layer very near the boundary. We consider very low pressure gradients and retain only the first powers of 0, cp and w. Then nx = C—cpS, ny — S+cpC, nz = 6, where C = cos qz and S = sin qz. Neglecting director inertia and product terms involving w6, w X6 i9 6 XX9, w X6 x etc., (4.3.4) reduces to ezx(k11-k22s2)-(pxx(k11s+k33sc2)-lp^k22s-d^(k!>3+^2)qsc -2<p ^k^qC-^wqS+yiC-Sg?) = 0, (4.5.2) 2S2)q] = 0, (4.5.3) 0. (4.5.4) 2 In the above equations 9 x = d9/dx, 9 xt = 9 ^/8x9z etc. Similarly, under the same approximations, (4.3.2) becomes P,x = -%(»*+Hz)+ vAqSCwtX, (4.5.5) py = [M.C'-^ + lfi.iC'-S^qw^ (4.5.6) P,z = S^ik^-k^qS-cp^ik^S^k^C^q-y^k^q -^, I (A; 22 + fc33)^C + > ^ 4 + ^6-/u2)C2]. (4.5.7) From (4.5.2) and (4.5.3) we get and from (4.5.7) and (4.5.8) ^ 2 2 + k33)qC-A, Wq = 0, (4.5.8) ^ - p . , = 0. (4.5.9) 272 4. Cholesteric liquid crystals We make a 'coarse-grained' approximation and replace |[// 4 + (JU5—JU2) C2] by an average value if and rewrite (4.5.9) as nw.xx + n*i42-P.z = 0. (4.5.10) A solution of (4.5.10) with the boundary conditions w(±h) = 0 is(68) w(x) = T-i£2 l rrr ' (4.5.11) ^ \ cosh Kh) where K2 = — Xx q2/ff. The velocity is symmetric about x = 0. The amount of fluid flowing per second in the z direction is Q= w(x)dx = 2 w(x)dx Hence the apparent viscosity /app 3Q 3[1 Taking 2h = 100 jum and P= 1 jum, the velocity attains 0.99 of the maximum value within a thickness of 0.5 //m of the boundary. Thus in all practical situations the velocity profile is flat over most of the region between the plates and * a p p « - ^ (4.5.14) which is the analogue of (4.5.1). Poiseuille flow In cylindrical polar coordinates, we seek solutions of the form nr = cos(qz—y/ + <p)cos8, n¥ = sin(qz—y/ + (p), nz = sin 0, where 9 and (p are functions of r, y/ and z. Considering very small pressure gradients we obtain to a first order in 6 and (p, where C = cos(qz—y/) and S = sin(qz—y/). For the velocity field we assume vr = 0, v¥ = 0, vz = w(r). 4.5 Flow properties 273 Proceeding as before we obtain p z = \w rr [// 4 + (ju5 - /z2) C2] ^ ^ w q \ (4.5.15) Again replacing the coefficients of wrr and w r by an average value Wrr + IWr + ^ _ ^ = 0. (4.5.16) A well behaved solution of (4.5.16) is(60) where A is a constant, / 0 is the modified zero order Bessel function of the first kind and K2 = — Xx q2/fj. Using the boundary condition w(R) = 0, A = ~ffIQ(KRY and where R is the radius of the capillary. The quantity of liquid crystal flowing per second KI0{KR)\ where Ir is the modified first order Bessel function of the first kind. Hence /aPP 8{ x _ [2ii(KR)/KRI0(KR)]}' Again, in practical situations the velocity profile is flat except very near the boundary and which is identical with (4.5.1). Thus Helfrich's idea of permeation along the helical axis of a cholesteric can be justified in terms of the Ericksen-Leslie equations. 274 4. Cholesteric liquid crystals 4.5.2 Flow normal to the helical axis The general theory of shear flow normal to the helical axis has been discussed by Leslie.(69) An interesting feature that comes out of this analysis is that a shear in the xz plane can give rise to secondary flow along y. (See §3.6.5; in principle secondary flow should occur in the Helfrich configuration also, but only in a negligibly thin layer very near the boundary where the velocity profile is not flat.) We shall now present a simplified version of Leslie's theory ignoring thermomechanical coupling. Consider a cholesteric film between two plane parallel plates, one of which is moving with constant velocity V in its own plane. The plates occupy the planes z = ±h. We examine solutions of the form nz = cosOcosp, ny = cos 6 sin <p, nz = sin0, (4.5.17) vx = u(z), vy = v(z), vz = 0, (4.5.18) where 6 = 6{z) and (p = cp{z). Then, tzx = a (constant shear), tzy = 0 and tzz = —p (an arbitrary constant). Using (4.3.7) and (4.3.13), we get (H1 + H2 cos 2 cp) £ + H2 rj sin cp cos cp = a, (4.5.19) H2 £ sin cp cos cp + (H1 + H2 sin 2 (p) rj = 0 , (4.5.20) where 2£ = dw/dz, 2r] = dv/dz, and H2 = (2//x sin2 6 + jus + ju6) cos2 6. From (4.3.4) we obtain d20 XdFJddY \dF2(d(p\2 „ . n Ay dz2 2 dO \dzj 2 d6 \dz) dz h A2 cos 20) (<!; cos p + iy sin p) = 0, (4.5.21) and i ^ ^ 2 uifl 1/COS (/~— sin (p-rj cos (p) = 0, (4.5.22) where F± = kxl cos2 6 +fc33sin2 6, F2 = (k22 cos2 9 + &33 sin2 0) cos2 0. From the symmetry of the problem it is clear that 6 and v should be even 4.5 Flow properties 275 functions of z, while u — f Kand cp are odd functions of z. Equations (4.5.19) and (4.5.20) yield f = 41 +(// 2 /// 1 )sin>]/(i/ 1 + // 2 ), (4.5.23) rj = -aKHJHJsinvcosvy^ + HJ, (4.5.24) which immediately give the velocity profiles II = 2 P <Jdz, r = - 2 7 7 dz. (4.5.25) It is seen that v 4= 0 even though the shear is confined to the zx plane; in other words, secondary flow occurs. Using (4.5.23) and (4.5.24), (4.5.21) and (4.5.22) can be simplified to U f 1 Uij I KXIS 1 1 VJ.J. 2 I \J. \ \dz/ d^? 2 = 0 (4.5.26) dz and 2 2 sin#cos#-;—al = 0, (4.5.27) dz2 d# dz dz dz where Q = ( ^ _|_ ^ 2 cos20)/(/f 1 + /f2) and P = (A 2 -.^ s i n ^ c o s ^ / ^ . Leslie assumed the following boundary conditions: fd<p\ / d ^ fc2 } (4.5.28) dz)h \dz)_h k22 Now V=4\ Jo and the apparent viscosity _ a 17app ~ K/2A 2 (4.5.29) 276 4. Cholesteric liquid crystals 1.2 l.l 100 1000 Fig. 4.5.5. Theoretical variation of the apparent viscosity //app with pitch P = 2n/q for flow normal to the helical axis of a cholesteric (or twisted nematic) at low shear rates. Plot ofrjapp(q)/rjapp(0) versus P for twisted PAA. The separation between the plates = 100 jum. The horizontal dashed line corresponds to f] (co)/rfapp(0). (After ref. 70.) 3.7 3.3 2.9 T 2.5 I '50 51 52 53 54 T(°C) Fig. 4.5.6. Evidence of oscillatory behaviour in the variation of the apparent viscosity with temperature in a cholesteryl chloride-cholesteryl myristate mixture in the neighbourhood of the nematic point 7^ (see fig. 4.1.16). (After Bhattacharya, Hong and Letcher.<72)) 4.6 Distortions of the structure by external fields 277 Detailed analytical and numerical calculations have been presented by K i n i (7o,7i) w h e n a i s It is seen that for pitch values of the order of the sample thickness, ?/app should exhibit oscillatory behaviour with varying pitch because of the term sin qh/qh in (4.5.30). A representative theoretical curve is presented in Fig. 4.5.5. This prediction has been verified qualitatively. (72) Measurements of the apparent viscosity on the cholesteryl chloride-cholesteryl myristate mixture, whose pitch, as we have seen earlier, is sensitive to temperature (fig. 4.1.16), showed evidence of oscillations as the temperature was varied (fig. 4.5.6). Often, in practical cases, qh is so large that sin qh/qh is negligibly small and ?/app approaches a maximum limiting value which is independent of the pitch or gap width. This value is several orders of magnitude less than that for flow along the helical axis and is comparable to that for a nematic. When a is sufficiently large q> = 0 and 0 = 00 = i c o s - 1 ^ / ^ ) , i.e., the helix is unwound completely except in a layer of thickness of the order of the q~x at the boundaries, and which is again independent of the pitch or gap width. This lower limit is reached when a & k22 q2 or more. All these predictions are in qualitative agreement with the observations of Candau et #/.(66) (see fig. 4.5.2). 4.6 Distortions of the structure by external fields 4.6.1 Magnetic field normal to the helical axis: the cholesteric-nematic transition When a magnetic field is applied at right angles to the helical axis of an unbounded cholesteric liquid crystal composed of molecules of positive diamagnetic anisotropy (xa = X\\~X± > 0) the structure gets distorted as illustrated schematically in fig. 4.6.1. As the field strength approaches a certain critical value Hc the pitch increases logarithmically; for H > Hc the helix is destroyed completely and the structure becomes nematic. (73) The dependence of the pitch on the field strength was calculated by de Gennes(74) and by Meyer.(75) 278 4. Cholesteric liquid crystals >H >H H=0 0<H<Hc H>HC Fig. 4.6.1. Schematic representation of the effect of a magnetic fieldapplied normal to the helical axis of a cholesteric liquid crystal composed of molecules of positive diamagnetic anisotropy. For H > Hc the cholesteric is transformed into a nematic. Taking the helical axis to be along z, H = (H, 0,0) and n = (cos cp, sin q>, 0), the free energy of the system is = ((Fdz = 1 J[(^f- -/aH2 sin2 J d p + constant, where q0 = 2n/P0, Po being the pitch of the undistorted structure in the absence of a magnetic field. The equation of equilibrium is therefore X H sin COS 9 = * * f which yields z = 2AZK(A), where 22/ and 72 K(A) = is the complete elliptic integral of the first kind; A is a constant which can 4.6 Distortions of the structure by external fields 279 1.5 1.0 H/Hc Fig. 4.6.2. Dependence of the pitch P on the magnetic field strength H in PAA mixed with a small quantity of cholesteryl acetate. Curve represents the theoretical variation predicted by de Gennes's equation (4.6.1). (After Meyer.(76)) be determined from the condition that J^ should be a minimum; z = P/2 the half-pitch of the distorted structure. It is assumed that the sample is sufficiently thick for boundary effects to be neglected. Therefore where rnr. 4 =2 ?)2 dcp = — Jo A and E(A) = is the elliptic integral of the second kind. The condition = 0 leads to the relations 280 4. Cholesteric liquid crystals H t V///////////////////////A W7/7///////////////////. 2* Fig. 4.6.3. Deformation of a planar structure due to a magnetic field acting along the helical axis of cholesteric liquid crystal composed of molecules of positive diamagnetic anisotropy. A similar deformation superposed in an orthogonal direction results in the square-grid pattern (see fig. 4.6.4). (Helfrich. (79)) Putting z0 = n/q0 = |P 0 , we have (4.6.1) and (4.6.2) 2E(A)' When l, E(A) -> 1, K(A) -> oo and H-> Hc, so that h = n/2 or Hc = Inqo(k22/XJK (4.6.3) which is the critical field at which the structure becomes nematic. The variation of pitch with magnetic field strength predicted by (4.6.1) has been verified experimentally (76'77) (fig. 4.6.2). It has also been confirmed that Hc is inversely proportional to Po the pitch of the undistorted structure.(77) It turns out that with the usually available magnetic field strengths, the experiment is conveniently performed only with cholesterics of relatively large pitch. For example, in a typical measurement using nematic PAA doped with a small amount of cholesteryl acetate Hc was 8.3 kG for PQ = 26 jum. 4.6 Distortions of the structure by external fields 281 4.6.2 Magnetic field along the helical axis: the square grid pattern We next examine the effect of a magnetic field acting along the helical axis of a cholesteric film having a planar texture. If / a > 0 and boundary constraints are absent, there is a possibility of a 90° rotation of the helical axis because \{X\\+X±) > X±- I£ o n t n e other hand, boundary effects are such as to maintain the orientation of the helix, an expected type of deformation is for the director at every point to be tilted towards the field, i.e., a conical distortion/ 75 ' 78) However, it was pointed out by Helfrich(79) that yet another type of deformation can set in, viz, a corrugation of the layers (fig. 4.6.3). This has since been confirmed experimentally using both magnetic(80) and electric fields/81'82) and takes place at a much lower threshold. It results in the so-called square grid pattern (fig. 4.6.4), the theory of which was first proposed by Helfrich(79) and subsequently elaborated by Hurault. (84) We shall discuss first the magnetic field case. For the unperturbed cholesteric n = (cos# 0 z,sin# 0 z,0), where we take z normal to the film. For small perturbations we may put nx = cos(q0z + (p) « cosq 0 z — cpsinq0z, ny = sin (q0 z + (p) « sin q0 z + q> cos q0 z, (4.6.4) nz = 9 cos qoz. Substituting in (4.2.1) the local energy density M\ , • * ?>d( n d<p\ ^ + smqozcosqoz—J +A: 33 cos 4 ?0 z^—J j , (4.6.5) and the total energy #- = \FdV. Now -^-= \AdV and ^—= \BdV, (4.6.6) 282 4. Cholesteric liquid crystals Fig. 4.6.4. The square grid pattern in a cholesteric liquid crystal induced by (a) a magnetic field and (b) an electric field. (Rondelez.(83)) 4.6 Distortions of the structure by external fields 283 where (ffm dQ A = - (fc u sin 2 q0 z + fc33 cos 2 q0 z) ( ^ + q0 — 0 dx 47 (A) and H 8> c)2Q O A- rvn wS 2 ^ 0 Z—r. (4.6.8) oz oz For a given #, ^ is a minimum when cU^/8^ = Oov A = 0. We consider the perturbations 6 and ^ to be dependent on z and x (where x is any arbitrary direction in the plane of the layers) and write them in the form 6 = 00 sin kx x cos kz z, (4.6.9) <p = (<p0 + cpx c o s 2q0 z) c o s kx x sin kz z, where kz = mn/d, d being the film thickness and m an integer. We shall confine our discussion to m = 1. In practice, K<kx<<lo (4.6.10) and we shall assume this to be true in what follows. The condition • = 0 yields - ^ i i ) ^ > i = 0, (4.6.11) Kl^ql <P, - q0 K 0O)+K*S3 - *n) ( ^ ^o - q» K e0)+ife+*n) /c2 ^ = o, (4.6.12) where we make a 'coarse-grained' approximation, i.e., take into con- sideration only the slowly varying parts of A. The minimum energy density O 00 ~ K <Po) (*11 + ^ -ko K <Pi(2k22+^33 - ^11)] cos K z s i n K x- ( 4 - 6 - 13 ) 284 4. Cholesteric liquid crystals From (4.6.11) and (4.6.12) we get ^i^Z^o (4.6.14) and 1 ffo ^o ~ \i(ic +L. \ + k ^±\±(k +k } - (k k ^*1* ~ 1 because of (4.6.10). Therefore Thus, in the coarse-grained approximation the energy density becomes \ (4.6.16) An alternative derivation of (4.6.16) has been given by de Gennes. We consider the cholesteric to be a quasi-layered structure and write the energy density in terms of the displacement u(r) of each plane in the following form: where B is an elastic coefficient associated with the compression of the layers. Terms involving (du/dx)2 and (du/dy)2 are not included as they correspond to a uniform rotation of the layers and do not contribute to the free energy. Introducing a unit vector h along the helical axis, (4.6.17) may be expressed as ( P where P is the local value of the pitch. We know that the twist energy of deformation of the cholesteric structure can be written in terms of the pitch as \k22(q — q0)2 where q = 2n/P and q0 = 2n/P0. Comparing this with the first term on the right-hand side of (4.6.18), it is clear that B*k22q*. (4.6.19) In order to evaluate K, one may use the following argument. Suppose that 4.6 Distortions of the structure by external fields 285 the film is rolled up into a cylinder, i.e., the screw axes are oriented radially about the cylinder axis and the layers form coaxial cylinders. In cylindrical coordinates, the components of the director are now nr = 0, riy = cos 0(r), and nz = sin 6{r) and the local free energy is given by sin0cos0\ 2 1f cos 4 0 , „X + j ^-^^- (4-6.20) The optimum value of 6{r) compatible with the periodicity 6{r) = 6{r + Po) corresponds to d0 1 . . . —- = # n + - s m 0 c o s 0 . dr r Averaging over cos4 0, K = \k33, (4-6.21) so that We may take u as our variable and write U — UQ COS kx x cos kz z, (4.6.23) where kz = n/d. It is seen at once that (4.6.22) becomes equivalent to (4.6.16) if we replace 0 by kx u. In the presence of a magnetic field applied normal to layers the total free energy where (£)V (4-6.24) ^ a = ^ ~x± being the anisotropy of diamagnetic susceptibility. Applying the condition dF/du = 0 and using (4.6.22), (4.6.23) and (4.6.24) = 0. (4.6.25) Thus H^oo when kx^co as also when kx->0. This is because the perturbation u is made up of two components, bend and twist: kx -> oo 286 4. Cholesteric liquid crystals excites the bend mode while kx -> 0 excites the twist mode whose amplitude diverges in the limit. The optimum wavevector corresponds to an admixture of both modes and is given by 8 7 4 ^ 2 2 2 / 2 tA<LK\ kx = — ^ q\ k\, (4.6.26) or kxK(Pod)-K (4.6.27) and the threshold field 1 i H^ = —(6k 22 k 33 ) 2 q 0 k z , (4.6.28) /a or 77HocOP0 </)-". (4.6.29) It is interesting to note that this threshold field is lower than that for a conical distortion or that for cholesteric-nematic (unwinding) transition. For a conical distortion, the theory is closely similar to that discussed in §3.4.2 and has been treated by Leslie (78); the critical field is given by HI = —(k, J /a For the cholesteric-nematic transition, we have from (4.6.3) G ~ A ^ 2 2 V/0> Aa and in view of (4.6.10), 77H is much less than HF or HG. The experimentally observed square grid pattern corresponds to two such distortions, which are orthogonal. 4.6.3 Electric field along the helical axis Electric field effects are more complicated because of conduction. (84) The Carr-Helfrich instability, which occurs in nematics of negative dielectric anisotropy (see §3.10.2), may be expected to take place in this case too, only the bend and twist distortions are now coupled. Moreover, the fluid motion along z can occur only by the process of permeation (§4.5.1). We shall consider the DC case first (neglecting, of course, any charge injection at the boundaries). If al]h and alh are the conductivities along and 4.6 Distortions of the structure by external fields 287 perpendicular to the helical axis (which, as before, is taken to be parallel to z), the electric current Jx = °±*Ex-(°\\*-Ort)E^ (4.6.30) which should be zero (V-J = 0). Here Eo is the applied field and Ex that caused by the Carr-Helfrich mechanism. Therefore OX (Tl The charge density is given by , c 4v ( £ E) = An An e. ^ (4.6.32) and the electric force fc=PeE0. (4.6.33) The dielectric torque An -°\E0 U^ from which it follows that the contribution to the vertical force is ox <+n alh ox The net electric force = /elec /c ' /diel- From (4.6.22) the elastic restoring force The threshold field is obtained by setting / e l e c +/ e l a s = 0. The optimum wave vector kx is given by (4.6.26) and the thresholdfieldby /72 ^- *h - - r3/r ^ "jirp //V 1 ^th ~~ ^2^22 ^ 3 3 / V^0u/ _87T 3 <7 | | +a 1 )-\ (4.6.34) where crn and aL are the conductivities parallel and perpendicular to the 288 4. Cholesteric liquid crystals preferred molecular direction and el{ and e± are similarly defined. It may be noted here that we have neglected the contribution of the viscous torque altogether. This is because, as remarked earlier, fluid motion takes place only by permeation, and, moreover, the distortions are infinitesimal and of such long wavelength {kx <4 q0) that the effect of shear flow will be very small indeed. The dependence of the spatial periodicity of the pattern and the threshold field on d and Po is similar to the magnetic field case, and is borne out by experiments/ 81 ' 8285) Hurault(84) has extended the treatment to AC fields. The method is somewhat analogous to the one discussed in §3.10 for nematic instabilities and leads, in the conduction regime, to the following threshold for distortion: _8n3e]]+s± 1+Q) 2 T 2 ^(PdY1 (4 6 35) where and T is the dielectric relaxation time given by For negative dielectric anisotropy (£j— £± < 0) the conduction regime occurs when co < coc, where These results are also in quantitative agreement with observations.(85) However, the behaviour at higher frequencies does not appear to be fully understood. In the case of dielectrically negative molecules, the square grid pattern changes as the voltage is raised above the threshold and the helical axis rotates by 90°. (The tilted structure is metastable and relaxes to the planar texture if the field is switched off but only after a long time.) At even higher voltages turbulence sets in and the system goes over to the dynamic scattering mode. On switching off, the liquid crystal relaxes to the focal conic texture and the scattering persists. This has been described as the storage mode or the memory effect in cholesteric liquid crystals.(86) An 4.7 Anomalous optical rotation 289 audio frequency (~ 10 kHz) pulse then restores it to the planar texture. In combination with a photoconductor, this effect has been made use of to construct image storage panels. (87) 4.7 Anomalous optical rotation in the isotropic phase We have seen in §2.5 that the orientational correlations between the molecules give rise to certain remarkable pretransitional effects in the isotropic phase of nematic liquid crystals. Similar correlations exist in the cholesteric phase as well, except that by virtue of the chiral nature of the interactions the local order lacks a centre of inversion. Cheng and Meyer (88) discovered that these correlations give rise to an enhancement of the optical activity in the isotropic phase. The correlation length increases as the temperature approaches the isotropic-cholesteric transition point, the local helical ordering builds up and the optical rotation increases accordingly. The magnitude of this effect is just barely observable in most cholesteric materials as the anisotropy of molecular polarizability of these compounds is usually rather small. For this reason Cheng and Meyer used a specially synthesized nematogen with an optically active end group: p- ethoxybenzal-/?'-(/?-inethylbutyl)aniline. This molecule forms a cholesteric phase and has the advantage of having a high anisotropy. Cheng and Meyer's experimental results are shown in fig. 4.7.1. The natural optical activity of the molecule (in the absence of correlations), determined by measuring the rotatory power of dilute solutions of varying concentration and extrapolating to 100 per cent concentration, was just about 1° cm" 1 as compared with a total rotation of nearly 40° cm" 1 close to the transition, proving that correlations play the predominant role. The theory of Cheng and Meyer is rather elaborate and will not be discussed in detail here. We shall merely indicate the major steps in the calculations. We consider a system of identical molecules and for simplicity neglect the contribution of the natural optical activity to the total rotation. If Eo is the externally applied field, the net field F, acting on a molecule at x, is F(x) = E o + T P ( x ) •G(x - x', k0) d V , J V where P(x) = 7Va(x) • F(x) is the polarization (or the dipole moment per unit volume) at x, a is the polarizability of the molecule at x, G(x — x', k0) a tensor representing the field at x due to a dipole at x', k0 the wavevector of the incident radiation, v the volume of the Lorentz cavity which is not supposed to contribute to the effective field and V the total volume. 290 4. Cholesteric liquid crystals 35 40 35 30 25 I 20 o 60.5 61.0 61.5 62.0 62.5 63.0 63.5 "S3 15 Temperature (°C) 60 65 70 75 80 85 90 95 100 105 110 115 Temperature (°C) Fig. 4.7.1. Anomalous optical rotation in the isotropic phase of a cholesteric liquid crystal. Open and closed circles are measurements on two different samples appropriately normalized. Cholesteric-isotropic transition temperature 60.57 °C. X = 0.6328 /an. (After Cheng and Meyer.(88)) Writing ) + <SP(x). where Po is the polarization in the absence of correlations and SP a small correction term, 2 f dV<<5a(x)-G-<Ja(x')>-F(x'), J V neglecting higher powers of SOL. Assuming a Lorentz-Lorenz type of relationship for the polarization field in the medium, expressing Sen in terms of the tensor order parameters s (see, for example, §2.3.1) the susceptibility d3/?G(R, k0) • (mk0 •R)<s(0) • exp s(R)> , where n is the refractive index of the isotropic phase, a the mean molecular polarizability and aa = a,, — a ± the polarizability anisotropy. Expressing the integral in terms of the Fourier components, 0 + q) •<s*(q) •s(q)> J. 4.7 Anomalous optical rotation 291 Hence the dielectric tensor may be expressed as where A is a tensor of the form In order that the medium be non-absorbing we must have so that for an isotropic medium 8 = -iA;; S+A;, iA;; . L-iAL -iA;' 2 e + A^ Such a system will exhibit circular birefringence, or an optical rotation The rotation exists because of the correlations (s%asyfiy which are non- vanishing in the case of a cholesteric. The averages may be evaluated on the basis of de Gennes's model(89) (see §2.5). To allow for the non- centrosymmetric ordering in the cholesteric, we include an additional term of the form s • V x s in the free energy of the isotropic phase /? c = /5 N + 2 t f 0 L ' ^ ^ - i f e , (4.7.1) oxy where FIN is the free energy per unit volume in the isotropic phase of a nematic given by (2.5.15) and (2.5.22), q0 is a pseudo-scalar and L is a constant. The average can then be worked out. For example 292 4. Cholesteric liquid crystals BPIII y BPII Disordered - * = = -- BPI Helicoidal Inverse pitch Fig. 4.8.1. Schematic phase diagram, showing the three experimentally observed blue phases (BP). where A = a(T-T*), l\ = LJA, and L l 5 L 2 are the constants occurring in (2.5.22). Therefore the optical rotation increases rapidly as the temperature approaches T*. 4.8 The blue phases The blue phases occur in cholesteric systems of sufficiently low pitch, less than about 5000 A. They exist over a narrow temperature range, usually ~ 1 °C, between the cholesteric liquid crystal phase and the isotropic liquid phase (see (1.3.5)). The first observation of a blue phase was described by Reinitzer(90) himself in his historic letter to Lehmann as follows: 'On cooling (the liquid phase of cholesteryl benzoate) a violet and blue phenomenon appears, which then quickly disappears leaving the substance cloudy but still liquid.' Although Lehmann (91) recognized it as a stable phase, not until the 1970s was it generally accepted that the blue phases are thermodynamically distinct phases. The nature of these phases has now become a subject of considerable interest to condensed matter physicists. Fig. 4.8.2. (a) Faceted single crystals of BP I in equilibrium with the isotropic phase (58 wt% mixture of cyano-4-methylbutylbiphenyl (CB15) in nematic ZLI 1840 of Merck). (From Cladis, Pieranskio and Joanicot(95).) (b) Optical Kossel diagram of BP I, {110} direction, X = 5290 A (42.5 wt% mixture of CB15 in nematic E9 of BDH). (From Cladis, Garel and Pieranski.(96)) 293 (a) Fig. 4.8.2. For legend see facing page. 294 4. Cholesteric liquid crystals Fig. 4.8.3. Unit cells of BP disclination lattices. O2 is simple cubic, O5, O 8 + and O8 — are body-centred cubic. The tubes represent disclination lines whose cores are supposed to be isotropic (liquid) material. (From Berreman.(98)) Three distinct blue phases have been identified: BP I, BP II and BP III, occurring in that order with increasing temperature (fig. 4.8.1). All of them are optically active but isotropic. (They may have colours other than blue, but are still referred to as blue phases). From observations of optical Bragg reflexions(92) and other studies, it is found that BP I is a body-centred cubic lattice (crystallographic space group I4X32 or O8), BP II a simple cubic lattice (P4232 or O2) and BP III probably amorphous. The suggestion has been made that BP III, called the blue fog, may be quasi-crystalline.(93) 4.8 The blue phases 295 Fig. 4.8.4. A representative phase diagram calculated from the Landau theory.(110) The normalized ordinate t is proportional to T— T* and the normalized abscissa K is proportional to 1/P, where P is the cholesteric pitch. The diagram illustrates how different BPs can occur by changing the chirality. (From Crooker.(104)) Striking confirmation of the cubic structures of BP I and BP II was obtained by Onusseit and Stegemeyer(94) and others, who succeeded in growing beautiful single crystals of up to a few hundred microns in size(95) (fig. 4.8.2(#)). Optical Kossel diagrams, analogous to the Kossel lines observed in X-ray diffraction from crystals, have confirmed their sym- metry (96) ( f i g 4.8<2(6)). Topologically, it turns out that the helical structure of the cholesteric cannot be deformed continuously to produce a cubic lattice without creating defects. Thus BP I and BP II are unique examples in nature of a regular three-dimensional lattice composed of disclination lines. (97) Poss- ible unit cells of such a disclination network, arrived at by minimizing the Oseen-Frank free energy, are shown in fig. 4.8.3.(98) The tubes in the diagram represent disclination lines, whose cores are supposed to consist of isotropic (liquid) material. Precisely which of these configurations represents the true situation is a matter for further study. The occurrence of the BPs can also be described in terms of the Landau theory. (99103) The free energy expansion contains the usual nematic terms 296 4. Cholesteric liquid crystals and an additional term of the form s - V x s to allow for the non- centrosymmetric ordering (see (4.7.1)). The order parameter is expressed as a Fourier series, and by choosing the appropriate (hkl) reciprocal lattice vectors for the Fourier components and minimizing the free energy, one can generate phase diagrams. A representative phase diagram illustrating how the different phases can occur by changing the chirality is given in fig. 4.8.4. Calculations show that there can be stable body-centred cubic and simple cubic structures, and in the presence of an electric field an hexagonal phase as well. All these predictions are in broad agreement with the experimental facts. There has been a surge of activity on various aspects of the BP problem - phase stability and its dependence on pitch, phase diagrams, electric field effects, single crystal morphology, growth rate, mechanical properties, etc.(104) 4.9 Some factors influencing the pitch In this section, we present a brief survey of experimental studies on the dependence of the pitch on temperature, composition, etc. Dependence of pitch on temperature: applications to thermography In most pure cholesteric materials, the pitch is a decreasing function of the temperature. An elementary picture of the temperature dependence of the pitch can be given in analogy with the theory of thermal expansion in crystals.(105) Assuming anharmonic angular oscillations of the molecules about the helical axis, the mean angle between successive layers where A is the coefficient of the cubic anharmonicity term, (106) a>0 the angular frequency and / the moment of inertia of the molecule. Thus the pitch P (oc 1/(6}) may be expected to decrease slightly with temperature. However, in many substances the rate of variation is extremely high. It is now established that if the cholesteric phase is preceded by a smectic phase at a lower temperature, the pitch increases very rapidly as the sample is cooled to the smectic-cholesteric transition point (see fig. 5.5.3). The strong temperature dependence of the pitch has practical appli- cations in thermography, as was first demonstrated by Fergason. (107108) The material has to be so chosen that the pitch is of the order of the wavelength of visible light in the temperature range of interest. This is achieved by preparing suitable mixtures. Small variations of temperature 4.9 Some factors influencing the pitch 297 are shown up as changes in the colour of the scattered light and can be used for visual display of surface temperatures, (109) imaging of infrared(110) and microwave(111) patterns, etc. Dependence of pitch on pressure Pollmann and Stegemeyer(112) investigated the effect of pressure on the pitch of cholesteryl oleyl carbonate (COC) mixed with cholesteryl chloride and found that the pitch increases very rapidly with pressure, the effect being more pronounced the greater the concentration of COC. This appears at first quite surprising, but, in fact, the explanation is straight- forward.(113) We have emphasized that the pitch diverges as the temperature approaches the cholesteric-smectic transition point. Now pure COC exhibits a smectic A phase below 14 °C at atmospheric pressure. This temperature may, of course, be somewhat lower in the case of the mixture. However, as the pressure is raised the temperature of transition goes Updi4,ii5) s o ^ a ^- t k e p^ch at room temperature may be expected to rise accordingly. Mixtures: dependence of pitch on composition We have seen in §4.1.6 that a mixture of right- and left-handed cholesterics adopts a helical structure whose pitch is sensitive to temperature and composition. This result was first described by Friedel. (116) For a given composition, there is an inversion of the rotatory power as the temperature is varied, indicating a change of handedness of the helix. The inverse pitch exhibits a linear dependence on temperature, passing through zero at the nematic point where there is an exact compensation of the right- and left- handed forms (fig. 4.1.16). A similar reversal of handedness takes place as the composition is varied.(117) The inverse pitch shows a nearly linear relationship with composition around the nematic point, but there are significant departures when one of the components has a smectic phase at a lower tem- perature. (118) The anomaly may again be attributed to smectic-like short- range order. It is well known that a nematic liquid crystal readily adopts a helical configuration if a small amount of a cholesteric is added to it. For low concentrations of the cholesteric, the inverse pitch is a linear function of the concentration, but at higher concentrations the linear law is not obeyed.(119) The curve attains a maximum at a certain concentration, beyond which it decreases. It turns out, for example, that the twist per unit length of pure cholesteryl propionate is actually less than that of its 298 4. Cholesteric liquid crystals mixture with a small amount of MBBA. Saeva and Wysocki (120) have observed that a compensation occurs even in an MBBA + cholesteryl chloride mixture when the MBBA concentration is about 30 per cent by weight. They suggest that certain optically inactive molecules may become chiral in the helical environment of a cholesteric liquid crystal. In this case, the MBBA in the mixture would appear to have a chirality opposite to that of the cholesteryl chloride. A small quantity of a non-mesomorphic optically active compound may also transform a nematic into a cholesteric. (121) However, the handedness of the helix does not seem to be directly related to the absolute configuration of the solute molecule, as has been shown by Saeva. (122) For example, (S)-s-amyl-/?-aminocinnamate and (S)-2-(octyl)-/?-amino- cinnamate, both of which have the same absolute configuration, result in helices of opposite senses when dissolved in MBBA. Impurities, indeed even the vapour of an organic liquid coming into contact with a cholesteric,(108) have a profound influence on the pitch. 4.10 Molecular models The first attempt to develop a statistical model of the cholesteric phase was by Goossens(123) who extended the Maier-Saupe theory to take into account the chiral nature of the intermolecular coupling and showed that the second order perturbation energy due to the dipole-quadrupolar interaction must be included to explain the helicity. However, a difficulty with this and some of the other models that have since been proposed <124) is that in their present form they do not give a satisfactory explanation of the fact that in most cholesterics the pitch decreases with rise of temperature. The local order in a cholesteric may be expected to be very weakly biaxial.(125) The director fluctuations in a plane containing the helical axis are necessarily different from those in an orthogonal plane and result in a 'phase biaxiality'. (126) Further, there will be a contribution due to the molecular biaxiality as well. It turns out that the phase biaxiality plays a significant role in determining the temperature dependence of the pitch. Goossens(127) has developed a general model taking this into account. The theory now involves four order parameters; the pitch depends on all four of them and is temperature dependent. However, a comparison of the theory with experiment is possible only if the order parameters can be measured. Interesting evidence in regard to the factors responsible for the helical 4.10 Molecular models 299 molecular arrangement has been reported by Coates and Gray.(128) They have demonstrated that a hydrogen-nleuterium asymmetry in a molecule is sufficient to produce a cholesterogen. An example is given below: O H II I N = C — Q — C H = N — ® — C H = C H — C —O— C —CH 2 CH 2 CH 3 I X X = H, nematic; X = D, cholesteric. The C—H and C—D bond lengths are the same, 1.085 A,(129) when one takes into account the anharmonicity of the vibrations. Thus it would seem that steric effects are not essential for the helical arrangement. 5 Smectic liquid crystals 5.1 Classification of the smectic phases Present classification of smectic liquid crystals is based largely on the optical and miscibility studies of Sackmann and Demus.(1) The miscibility criterion relies on the postulate that two liquid crystalline modifications which are continuously miscible (without crossing a transition line) in the isobaric temperature-concentration diagram have the same symmetry and therefore can be designated by the same symbol. It is not clear whether this criterion is valid regardless of the differences in the molecular shapes and dimensions of the two components, but empirically Sackmann and Demus have found that in no case does a phase of a given symbol mix continuously with a phase of another symbol. The method is simple and has been used for the identification of a number of new phases, but, of course, it does not throw light on the precise nature of the molecular order in these phases. Systematic X-ray investigations have been carried out during the last decade at several laboratories, and particularly with the availability of synchrotron X-ray sources, considerable progress has been made in elucidating the structures/ 2 ' 3) The notation of Sackmann and Demus is according to the order of the discovery of the different phases and bears no relation to the molecular packing. The broad structural features of these phases are summarized in table 5.1.1. A more detailed description of these structures may be found in the excellent reviews by Pershan(2) and by Leadbetter.(3) 300 5.7 Classification of the smectic phases 301 Table 5.1.1. Structural classification of smectic liquid crystals Smectic A(SA) Liquid-like layers with the molecules upright on the average (fig. 1.1.5(a)); negligible in-plane and interlayer positional correlations. Thus the structure may be described as an orientationally ordered fluid on which is superimposed a one- dimensional density wave. A number of polymorphic types of smectic A have been discovered (see §5.6). Smectic B(SB) Two distinct types of smectic B have been identified: {a) Crystal B - three-dimensional crystal, hexagonal lattice with upright molecules. Though the structure has three- dimensional long-range positional order, the interlayer ordering is extremely weak energetically because of the weak interlayer forces, (b) Hexatic B - stack of interacting 'hexatic' layers with in-plane short-range positional correlation, negligible interlayer positional correlation and long-range three-dimensional six-fold bond-orientational order (see fig. 5.7.1). Here, the term 'bond' signifies the line joining the centres of mass of the nearest neighbours. Smectic C(SC) Liquid-like layers, as in SA, but with the molecules inclined with respect to the layer normal (fig. 1.1.5 (/?)). Smectic C*(SC*) Chiral S c with twist axis normal to the layers. Smectic D(D) Cubic lattice with about 103 molecules per unit cell. (46) The detailed molecular arrangement is not known, but is generally assumed to be of the micelle type. Thus this phase should probably be labelled as ' D ' rather than 'Smectic D \ At present only four compounds are known to exhibit this phase, 4'-n-hexadecyloxy- and 4 /-n-octadecyloxy-3 /-nitrobiphenyl-4- carboxylic acid and two similar acids with CN replacing NO 2 . Interestingly, D occurs between S c and SA or between S c and the isotropic phase. The manner in which such a structural rearrangement takes place is yet to be resolved. Smectic E(S E) Three-dimensional crystal, orthorhombic with interlayer herringbone arrangement of the molecules. (2) Smectic F(SF) C-centred monoclinic (a > b) with in-plane short-range positional correlation and weak or no interlayer positional correlation: tilted hexatic. Smectic F*(SF.) Chiral SF with twist axis normal to the layers. Smectic G(SG) Three-dimensional crystal, C-centred monoclinic (a > b). Smectic H(S H) Three-dimensional crystal, monoclinic (a > b), herringbone structure.(2) Smectic H*(SH*) Chiral SH with twist axis normal to the layers. Smectic I(ST) C-centred monoclinic (b > a), tilted hexatic with slightly greater in-plane correlation than SF. Smectic I*(SIA) Chiral Sz with twist axis normal to the layers. Smectic J(ST) Three-dimensional crystal, C-centred monoclinic (b > a). Smectic J*(SJ#) Chiral Sj with twist axis normal to the layers. Smectic K(SK) Three-dimensional crystal, monoclinic (b > a), herringbone structure.(2) Smectic K*(SK*) Chiral SK with twist axis normal to the layers. 302 5. Smectic liquid crystals 5.2 Extension of the Maier-Saupe theory to smectic A: McMillan's model McMillan(7) proposed a simple and elegant description of smectic A by extending the Maier-Saupe theory to include an additional order par- ameter for characterizing the one-dimensional translational periodicity of a layered structure. A similar but somewhat more general treatment, based on the Kirkwood-Monroe theory of melting,(8) was developed inde- pendently by Kobayashi(9), but McMillan's approach lends itself more easily to numerical calculations and comparison with experiment. The anisotropic part of the pair potential is conveniently taken in the form 3cos 2 0 1 2 -l __,. ^ — , (5.2.1) where the exponential term reflects the short-range character of the interaction, r12 is the distance between the molecular centres and r0 of the order of the length of the rigid part of the molecule. If the layer thickness is d, we may write the self-consistent single particle potential, retaining only the leading term in the Fourier expansion, as follows: ^(z,cos/9) = - F0[^ + (7acos(27rz/^)]|(3cos26>-l), (5.2.2) where (7rr0A02]. (5.2.3) As will be seen later (§5.3.1), experiments have confirmed that the density wave in smectic A is, in fact, very well represented by a sinusoidal function, indicating that higher terms in the Fourier expansion can be neglected. The form of the potential (5.2.2) ensures that the energy is a minimum when the molecule is in the smectic layer with its axis along z; s and o are order parameters which we shall define presently. The single particle distribution function is then /i(z, cos 0) = exp [ - Fx(z, cos 0)/kB T] (5.2.4) and self consistency demands that _ /3cos2fl-l\ 5.2 Extension of the Maier-Saupe theory to smectic A 303 1.0 s • — • — - — — - . _ ^ ^ 0.5 - I 1 4 - - - '20 < o opi tic is - 15 3 - o B n § 2 - 10 > 1 - - 5 0.8 0.9 1.0 Reduced temperature Fig. 5.2.1. Order parameters s and a, entropy S and specific heat cv versus reduced temperature kB T/0.2202 Vo predicted by the model for a = 1.1 showing afirstorder smectic A-isotropic transition. S and cv are expressed in terms of Ro, the gas constant. (After McMillan.(7)) 3 cos2 0 - 1 o = /cos(27iz/d)f : (5.2.6) where the angular brackets denote statistical averages over the distribution fv The parameter s defines the orientational order, exactly as in the Maier-Saupe theory, while a is a new order parameter which is a measure of the amplitude of the density wave describing the layered structure. The last two equations can be solved numerically to obtain the following types of solutions: (i) o = s = 0 (isotropic phase) 1 (ii) a = 0, s == 0 (nematic phase) = (iii) a H 0, s ^F 0 (smectic phase). 304 5. Smectic liquid crystals 0.8 0.9 1.0 Reduced temperature Fig. 5.2.2. Order parameters s and a, entropy S and specific heat cv versus reduced temperature for a = 0.85 showing first order smectic A-nematic and nematic- isotropic transitions. (After McMillan.(7)) The free energy of the system can be calculated in the usual manner: F=U- TS, where (5.2.7) and 1 -TS = NV0(s2 T dz T d(cos 0)/^, cos 0)1. Jo Jo J (5.2.8) The two parameters characterizing the material are Vo, which determines 5.2 Extension of the Maier-Saupe theory to smectic A 305 1.0 0.5 )I 1 < 4 ~ - o / 1 .a a, o - 20 •s o? 3 1 / - 1 / I - 15 - 10 1 - - 5 0 0.8 0.9 1.0 Reduced temperature Fig. 5.2.3. Order parameters s and cr, entropy S and specific heat cv versus reduced temperature for a = 0.6 showing a second order smectic A-nematic transition and a first order nematic-isotropic transition. (After McMillan. (7)) the nematic-isotropic transition temperature, and a, a dimensionless interaction strength, which can vary between 0 and 2. Experimentally the layer thickness d is of the order of the molecular length. Neglecting the odd-even effect (see §2.3.3) the energy associated with smectic ordering tends to increase if a (and hence d) is larger. Thus a increases with increasing chain length of the alkyl tails. Curves of the order parameters, entropy and specific heat for three representative values of a are presented in figs. 5.2.1, 5.2.2 and 5.2.3. For a > 0.98, the smectic A transforms directly into the isotropic phase, while for a < 0.98 there is a smectic A-nematic (A-N) transition followed by a nematic-isotropic transition at higher temperature. For a < 0.70 and 87 t n e m ^AN/^NI < 0- °d e l predicts a second order A-N transition. Hence 306 5. Smectic liquid crystals l.l Isotropic / 1.0 Nematic / Smectic A 0.9 - Transition temperature \. Isotropic Nematic\. 0.8 - or \ ^ ^ cholesteric^^"^^^^-^^ / Smectic A I i 1 Alkyl chain length 0.7 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 Model parameter a Fig. 5.2.4. Phase diagram for theoretical model parameter a. Inset: typical phase diagram for homologous series of compounds showing transition temperatures versus length of the alkyl end-chains. (After McMillan.(7)) a = 0.7 (and TA^/TNI = 0.87) corresponds to a tricritical point (10) at which the line of first order transition goes over to a line of second order transition. The phase diagram of transition temperature versus a or alkyl chain length is shown in fig. 5.2.4. There is broad agreement with the trends in thermodynamic data, though the theoretical A - N transition entropy versus TAN/Tm is somewhat higher than the observed values (fig. 5.2.5). To improve the agreement, McMillan used, in a later paper (11), the modified pair potential K12(r12,cos012) = _ (5.2.9) There are now three model-potential parameters which are fixed by requiring the theory to fit 7^N, Tm and 5 AN . The results are essentially the 5.2 Extension of the Maier-Saupe theory to smectic A 307 1.5 r 1.0 0.5 0.85 0.90 0.95 1.00 Fig. 5.2.5. Smectic A-nematic (or cholesteric) transition entropy versus ratio of transition temperatures TAN/Tm. Solid line is theoretical curve taken from fig. 5.2.4; open circles are experimental values of Davis and Porter (MoL Cryst. Liquid Cryst., 10, 1 (1970)); open triangles are data of Arnold (Z. Physik. Chem. (Leipzig), 239, 283 (1968); ibid., 240, 185 (1969)). (After McMillan.(7)) ~ 200 (a) •S 100 Smectic A Isotropic I 60 70 80 90 Temperature (°C) Fig. 5.2.6. Measured intensity of X-ray scattering at the Bragg angle versus temperature for cholesteryl myristate. The dashed line is the calculated diffuse scattering and fluctuation scattering contribution. The full lines represent the theoretical curves for the total intensity due to Bragg, diffuse and fluctuation scattering derived from (a) the simpler model potential and (b) the refined one. The theoretical intensity has been adjusted to be equal to the experimental value at the lowest temperature. (After McMillan. (11)) 308 5. Smectic liquid crystals 0.8 0.6 (a) 0.4 0.8 Z. 0.6 0.4 0.8 0.6 (c) 0.4 0.8 0.9 1.0 Reduced temperature Fig. 5.2.7. Experimental values of the orientational order parameter s obtained from NMR measurements for the 4-n-alkoxy-benzylidene-4/-phenylazoaniline series, (a) C14; (b) C10 (open circles), C7 (X), C3 (filled circles); (c) C2. The solid curves give the values predicted by McMillan's model. (After Doane et al.(22)) same as those obtained with the simpler model but there are some quantitative improvements. A number of other refinements and extensions have been proposed, (1221) but McMillan's model remains the simplest which brings out all the qualitative features of the A-N and A-I transitions. A direct method of studying the translational order (or the amplitude of the density wave) is by measuring the intensity of the Bragg scattering from the smectic planes. McMillan's experimental results on cholesteryl myr- istate(11) are shown in fig. 5.2.6 and as can be seen there is excellent agreement with the refined model. The X-ray intensities reveal an appreciable pretransitional smectic-like behaviour in the cholesteric (nematic) phase. This aspect of the problem will be dealt with in a later section. The orientational order parameters in the smectic and nematic phases, studied by magnetic resonance and other techniques also follow the 5.2 Extension of the Maier-Saupe theory to smectic A 309 1.5 - N==C O O-C 8 H 17 1.0 0.5 1.2 - 90 °C 0 J 50 60 70 80 90 100 110 Temperature T (°C) Fig. 5.2.8. Temperature dependence of the orientational order parameter s determined by 14 N quadrupolar splitting measurements for CBOOA. The inset shows the discontinuity in slope at the smectic A-nematic transition. (After Cabane and Clark. (23) ) predicted type of behaviour as the length of the alkyl end-chain is increased. In particular, a continuous change of s at 7^N, as expected of a second order transition, has been found (within experimental limits) for a number of cases. Figs 5.2.7 and 5.2.8 present the data for n-/?-ethoxy- benzylidene-/7-phenyl-azoaniline(22) and 4-cyano-benzylidene-4 /-octyloxy- aniline (CBOOA) (23). Thus this simple molecular model correctly leads to the existence of a tricritical point, but it makes no predictions regarding the critical exponents. Being a mean field theory, it may be expected to yield y = 1, V,| = v± = 0.5, which, as we shall see later, is not in accord with the experimental results (§5.5). 310 5. Smectic liquid crystals 5.3 Continuum theory of smectic A 5.5.7 The basic equations The stratified structure of a smectic liquid crystal imposes certain restrictions on the types of deformation that can take place in it. A compression of the layers requires considerable energy - very much more than for a curvature elastic distortion in a nematic - and therefore only those deformations are easily possible that tend to preserve the interlayer spacing. Consider the smectic A structure in which each layer is, in effect, a two-dimensional fluid with the director n normal to its surface. Assuming the layers to be incompressible, the integral i-dr (5.3.1) represents the number of layers crossed on going from A to B, where d is the layer thickness.(24) In a dislocation-free sample, this number should be independent of the path chosen so that Vxn = 0 and hence - n-Vxn = 0 I and \ (5.3.2) n x V x n = 0.J In other words, both twist and bend distortions are absent, leaving only the splay term in the Oseen-Frank free energy expression (3.3.7). It is seen from fig. 5.3.1, that by merely bending or corrugating the layers a splay deformation can be readily achieved without affecting the layer thickness. A more complete description of smectic A needs to take into account the compressibility of the layers, though, of course, the elastic constant for compression may be expected to be quite large. The basic ideas of this model were put forward by de Gennes.(24) We consider an idealized structure which has negligible positional correlation within each smectic layer and which is optically uniaxial and non-ferroelectric. For small displacements u of the layers normal to their planes, the free energy density in the presence of a magnetic field along z, the layer normal, takes the form /6M 8w\ ..... (5 3 3) (a?+# '- where the first term is the elastic energy for the compression of the layers 5.3 Continuum theory of smectic A 311 nTTTTTTT Fig. 5.3.1. Flexibility of smectic A layers: only such deformations as preserve the interlayer spacing take place readily. and/ a the anisotropy of diamagnetic susceptibility. When H = 0, there will be no terms in (du/dx)2 or (du/dy)2 as a uniform rotation about y or x does not affect the free energy. The last two terms are usually negligible and may be omitted. Also, the physically reasonable assumption is made that twist and bend distortions given by (5.3.2) are not allowed despite the fact that V x n does not strictly vanish when the layers are compressible. 5.3.2 The Peierls-Landau instability Equation (5.3.3) is analogous to the Peierls-Landau (25) free energy expression for a two-dimensional crystal, and leads to a logarithmic divergence of the mean square fluctuation <w2> as H->0. Writing the free energy in terms of the Fourier components of u uq = u(r) exp (iq • r) dr, (5.3.4) and substituting in (5.3.3) we get in the harmonic approximation, F = \ Yu \uq\2 [Bg2 + £n(#i + £~2) #j.L (5.3.5) 2 1 /xJiH' where q\ = q x + ql and £ = {klx/ is the magnetic coherence length. From the equipartition theorem kBT (5.3.6) from which the mean square fluctuation h T <u2(r)} = (5.3.7) where d is the layer spacing, assuming the sample to be infinitely large. As 312 5. Smectic liquid crystals I n , l III 1 , 1 , 1 , 1 , Fig. 5.3.2. A diagram depicting the molecular arrangement in (a) the nematic and (b) the smectic A phases. In the nematic phase the molecules are randomly distributed so that any horizontal line intersects the same number of molecules. In the smectic A phase the number of molecules intersected by a line varies sinusoidally with the position of the line, being 50 per cent more than the average at the positions indicated by the arrows and 50 per cent less than the average half way between. This represents a density wave whose amplitude is 50 per cent of the mean density, which is far greater than what actually occurs near the A-N transition in a real system. (After Schaetzing and Litster.(26)) i/-*0, <w2>->oo showing that such a structure cannot be stable. For a sample of dimension L kBT L (5.3.8) in the absence of a magnetic field. The layer fluctuations therefore diverge logarithmically with sample size. These results imply that smectic A does not possess true long-range translational order. Hence the conventional picture of the smectic A structure with the molecules forming well defined layers (fig. 1.1.5 (a)), though useful conceptually, is far from accurate. Fig. 5.3.2 gives a more realistic representation of the actual situation. X-ray experiments confirm 5.3 Continuum theory of smectic A 313 that the density wave is, in fact, very well described by a sinusoidal function; the higher order diffraction maxima are either absent or, when they do occur, extremely weak, about 10~3 or 10~4 times weaker than the intensity of the principal peak. (27) In the harmonic approximation, the displacement-displacement cor- relation function may be defined as G(r) = exp{-[^ 2 <l«W-t/(0)| 2 >]}. (5.3.9) For the crystalline lattice, this is the familiar Debye-Waller factor. In the case of smectic A, one gets from (5.3.4) (5.3.10) This was evaluated by Caille (28): for d <^ \r\ < L, the correlation functions for the two principal directions vary as and G(r)ocz- r± = ( * • + , • ) * - 0,1 ( 5 3 U ) 2 G(r) oc rL \ z~0, J where n = ~~zi, (5-3.12) which is usually a small number ( ~ 0.1-0.4). The Fourier transform of G(r) yields the intensity of scattering: lK\qz-q0\-*+i, q± = 0, (5.3.13) Ioc\q±\-*^, qz = 0, (5.3.14) where q0 = In Id. The sharp Bragg peak characteristic of the long-range ordered crystal lattice is now replaced by strong thermal diffuse scattering with a ' power- law' singularity.(29) The excellent high resolution X-ray measurements of Als-Neilsen et al.m) have confirmed this prediction (fig. 5.3.3). 533 The Helfrich deformation If a magnetic field is applied parallel to the smectic planes in a homeotropically aligned sample and / a > 0, one may expect a Helfrich 314 5. Smectic liquid crystals 10° : / \ 6 10"1 \\ v M \ \ \ \ ry = 0.38 A \\ V O ^ = 0.17 \ 10"2 - \ \ 1 1 \ 1 1 I -1 Fig. 5.3.3. X-ray scattering intensity profile from the smectic A phase of 80CB at two reduced temperatures, ( r A N - T)/TAN = 9 x 10"4 (filled circles) and 10"6 (open circles). The dashed line is the experimental resolution function as would be seen if smectic A had true long-range order. The full lines are the best fits of the theoretical line shape \qz — qo\~2+n folded with the resolution function. The values of rj so obtained agree with those calculated from (5.3.12) using experimentally determined values of q0, B and k1±. (Als-Nielson et al.(m) type of deformation to set in above a critical field, as in the case of cholesterics (fig. 4.6.3). Assuming a distortion of the form u(x, z) = u0 sin kz z cos kx x, where kz = n/L and L is the sample thickness, and evaluating the average free energy, one obtains F =- (5.3.15) where X= (5.3.16) is a characteristic length of the material of the order of the layer thickness. Proceeding as in §4.6.2, the optimum value of the distortion wavevector is k\ = kjk = n/kL. (5.3.17) 5.3 Continuum theory of smectic A 315 In other words, the spatial periodicity of the deformation is proportional to the geometric mean of the layer thickness and the sample thickness. The critical field given by is, however, rather large compared to that for cholesterics. For example, taking k = 20 A and L = 1 mm, Hc ~ 50 kG, and thus far no experimental studies of this effect appear to have been carried out. The same type of distortion can be achieved more easily by mechanical means, i.e., by increasing the separation between the glass plates (3132) (fig. 5.3.4). In the absence of a magnetic field, and taking u to be independent of y, (5.3.3) reduces to (5.3.18) However, in the present case we note that a bending of the layers alters the effective layer spacing along z and therefore makes a second order contribution to the layer dilatation in that direction. Hence (5.3.18) needs a correction term: ir = i^r^_I/ r ^yi 2 _ h fc n ^Y (5.3.19) The displacement u is now given by u = sz + u0 sin kz z cos kx x, (5.3.20) where s = A/L, A being the plate displacement. The problem is analogous to the magnetic case except that Bs replaces z a / / 2 . The threshold value of the strain sc = 2nk/L (5.3.21) and the plate displacement A = Ink ~ 150 A, which is easily realized in practice. Experiments have been done on CBOOA to verify these conclusions/ 31 ' 33) The plate separation was increased in a controlled manner by piezoelectric ceramics. When the dilatation reached a certain value, there appeared two transient bright spots in the laser diffracted beam confirming the onset of a spatially periodic distortion above a threshold strain. A transient periodic pattern was also visible under a microscope. It was verified that kx cc L'1 in accordance with (5.3.17) (fig. 5.3.5). The measurements yielded k = 22 ± 3 A for CBOOA at 78 °C. Also, assuming kxx ~ 10~6 dyn, B was estimated to be 2 x 107 cgs. 316 5. Smectic liquid crystals AA V/////////////////7/A Y/////. 777/77//A Y////X 277/*, . Fig. 5.3.4. The Helfrich deformation in a smectic A film subjected to a mechanical dilatation A. 50 100 150 Fig. 5.3.5. Dependence of the wavevector of the Helfrich deformation in the smectic A phase of CBOOA on the sample thickness d. The slope yields a value of X = 2 2 ± 3 A. (After Durand. (33) ) 5.3 Continuum theory of smectic A 317 5.3.4 Fluctuations and Rayleigh scattering If a smectic A sample is homeotropically aligned between two glass plates having a separation L, the boundary conditions require that qz = mn/L, where m is an integer. We shall confine the discussion to m = 1. When H = 0 we have from (5.3.6) (\uQ\2y = ^B T/iBql + k^q^). (5.3.22) We know that the intensity of light scattering is proportional to the mean square fluctuation of the director (see §3.9): k T » — (5.3.23) The elastic energy is minimized when qL = qc, which represents the optimum wavevector. When qz = 0, <|<N2> = KT/kuq\ (5.3.24) which is a large quantity as in a nematic liquid crystal, since it involves only the splay coefficient. On the other hand, when qz and q± are comparable, <53 25) - is quite small; consequently, in certain geometries, e.g., when a homeo- tropically aligned sample is held against an extended source of light and viewed normally, the medium will not appear very turbid. Strictly speaking, one should take into account the contributions of k 22 and k33, since the layers are assumed to be compressible. Following the procedure outlined in §3.9, the fluctuations may be decomposed into two modes, and choosing the wavevector q in the xz plane, one gets the general expressions(34) ,,*„ , , s = kBT[l+(B/D)(qJqLn xl? l j ^ k q i + k^ + iB/DHD +k q l +k ^ D i q J q r where D is the elastic constant associated with the fluctuations of the director away from the layer normal. 318 5. Smectic liquid crystals Fig. 5.3.6. Distortion caused by an irregularity on the glass surface. The bending of the layers can be relaxed only by compression, which requires considerable energy, so that the surface distortion extends to appreciable depths inside the specimen. In a perfect sample of smectic A, the most important contribution to light scattering comes from the undulation mode with the wavevector parallel to the layers (qz = 0). However, unless very special precautions are taken, it is difficult to observe this scattering, for even small irregularities on the surfaces of the glass plates cause static distortions of the layers which extend to appreciable depths inside the specimen (3532) (fig. 5.3.6). The reason for the high penetration length of a surface distortion is easily understood. A corrugation of the layer which involves a nematic-like splay elastic constant and requires very little energy can be relaxed only by compression, which needs considerable energy and therefore occurs over a large distance. If a static undulation of this type is written as u = u0 exp ( - z / 0 cos qx, where C is the 'attenuation' length, substitution in the free energy expression (5.3.18) gives or C = (V) (whereas in nematics, £ ~ q x ). These long-range static undulations give rise to an intense scattering of light which completely swamps that due to the thermal fluctuations. Indeed in the first experiments*36'37) it was the static effects that were observed as was confirmed by a temporal analysis of 5.3 Continuum theory of smectic A 319 105 r- 104 103 77 °C 102 § S 10 79 °C 74 C 0.1 I 10" io 101 102 1 Shear rate (s" ) Fig. 5.3.7. Apparent viscosity versus shear rate for cholesteryl myristate at different temperatures: full line, smectic A; dashed line, cholesteric; chain line, isotropic. (After Sakamoto, Porter and Johnson.(41)) the scattered light, using a laser beat spectrometer and correlator. However, in subsequent studies(38) the scattering due to thermal fluctu- ations were detected by choosing optical glass plates of very high quality. The temporal analysis showed an exponentially decaying correlation function characteristic of a dynamic undulation. These modes are highly damped, the relaxation time T being of the order of IO"3 s. To discuss the damping quantitatively we have to consider the hydrodynamics of smectic A. The most general formulation of the theory is due to Martin, Parodi and Pershan(39) but we shall present the relevant equations in a simplified form using de Gennes's notation. 320 5. Smectic liquid crystals 5.3.5 Damping rate of the undulation mode We write the energy density in a more general form(24) to include volume dilatation 6: For static isothermal deformations, F may be minimized with respect to 6 to give Aodz so that B = B0-(2C20/A0) (5.3.29) and (5.3.28) reduces to the simpler form (5.3.3) assumed previously. If vt is the velocity of the particle, the equation of motion is />*i = -P.i + gi + t'Jij> (5.3.30) where gt is the force on the layers and t'n the viscous stress tensor which, in contrast to the nematic case, is assumed to be symmetrical and dependent on the velocity gradients only. In the problem under consideration we have (5.3.31) and g = -SF/Su. (5.3.32) The force g normal to the layers will be associated with permeation effects. The idea of permeation was put forward originally by Helfrich(40) to explain the very high viscosity coefficients of cholesteric and smectic liquid crystals at low shear rates (see figs. 4.5.1 and 5.3.7). In cholesterics, permeation falls conceptually within the framework of the Ericksen-Leslie theory(42) (see §4.5.1), but in the case of smectics, it invokes an entirely new mechanism reminiscent of the drift of charge carriers in the hopping model for electrical conduction (fig. 5.3.8). The rate of entropy production may be written as (see §3.1.4) TS=t'ndij+g(u-vz), (5.3.33) where dtj = \{v{ j + Vj t), and (u — vz) describes the permeation. Treating t'n 5.5 Continuum theory of smectic A 321 Fig. 5.3.8. Helfrich's model of permeation in smectic liquid crystals. The flow takes place normal to the layers in a manner similar to the drift of charge carriers in the hopping model for electrical conduction. 10 -- A \ <»\ L \ \ | \ \ J \ 5 T T ^ ^ f I 1 1 10 15 20 25 q2 (108 cm- 2 ) Fig. 5.3.9. Damping rate of the undulation mode in the smectic A phase of CBOOA determined by laser beat spectroscopy for two different sample thicknesses, 200 and 800 /zm. Solid lines represent the theoretical curves calculated from (5.3.40) and (5.3.41). (After Ribotta, Salin and Durand.(38)) and g as fluxes and dtj and (u — vz) as forces, and expanding t'n one obtains the following relations: ^ = Mi sn + M2 Sjz dzi + Siz dzj) + // 5 Sjz Si (5.3.34) u-vz = v p g, (5.3.35) where //15 ...,// 5 are five viscosity coefficients and vp the permeation coefficient. 322 5. Smectic liquid crystals As far as the highly damped undulation modes are concerned, the volume dilatation can justifiably be neglected and the isothermal approxi- mation is probably satisfactory. The equation of motion then reduces to pvz = -rjq2vz + g, (5.3.36) where g = -kliq*u = (u-vz)/vv (5.3.37) and * = SG<s + / O . (5.3.38) Neglecting the inertial term in (5.3.36) and eliminating vz, For small q, the last term may be neglected (except very near the boundary, but we shall ignore the boundary layer). This results in a purely damped mode whose relaxation rate is 1 k n2 l L= h±3_. (5.3.39) T rj More generally, taking into account the thickness L of the sample one may write where qc is defined in (5.3.23). The relaxation rate should have a minimum value ^ = 2 ^ TC t]lL rj where 1± = qc = (n/lL)i (5.3.42) Experiments with specimens of different thicknesses have confirmed this prediction(38) (fig. 5.3.9). At very low q±, the boundary effects quench the fluctuations and T"1 increases sharply. At high q±9 T"1 becomes a linear function of q2± from the slope of which klx/rj may be determined. This value is comparable to that for a nematic ( ~ 2 x 10~6 cgs) as is to be expected. The relaxation time TC decreases linearly with decreasing sample thickness as predicted by (5.3.41), and the wavevector qc shows the expected dependence on L. The measurements yield a value of I which is in reasonable agreement with that obtained from (5.3.21). 53 Continuum theory of smectic A 323 5.3.6 Ultrasonic propagation and Brillouin scattering For an arbitrary direction of the wavevector there are two acoustic modes.(24) Neglecting viscous effects and permeation, putting y = du/dz, and using the conservation law divv + 0 = 0, (5.3.43) we obtain the following equations: y and 6 being the independent variables. This leads to the secular equation co{\pco2 - (Bo + Co) q% (po? - Ao q -(Au + Co) q\{paf + Co q\)} = 0. (5.3.45) We are interested here in propagating modes and ignore the case co = 0. Solving for the velocities c = co/q, we get c\ + c\ = p-\AQ cos2 <p + (Ao + Bo + 2C0) sin2 p], (5.3.46) 2 2 2 2 2 2 c c =/7- (^0^0-C )sin cos ^, (5.3.47) where cp is the angle between q and its projection on the layer. Ultrasonic velocity measurements have been reported on oriented smectic A samples of 4,4/-azoxydibenzoate. The first studies by Lord(43) for two values of (0° and 90°) established the anisotropy of the velocity of propagation. Subsequently, Miyano and Ketterson(44) investigated the angular dependence of the sound velocity and from a least squares fit with (5.3.45) were able to determine the elastic coefficients Ao, Bo and Co. By analogy with the elasticity theory of solids(45), we may write C —C — —C ^33 Their estimates show that Ao is much larger than Bo or Co (fig. 5.3.10). This enables us to give a simple physical interpretation of the two branches 324 5. Smectic liquid crystals 1.6 r 1.5 1.4 110 115 120 Temperature (°C) 0.3 r 0.2 X 0.1 i A 8- -8- 0 I I no 115 120 Temperature (°C) Fig. 5.3.10. Temperature dependence of the elastic constants of the smectic A phase of diethyl 4,4/-azoxydibenzoate determined by ultrasonic velocity measurements: open circles, 2 MHz; filled circles, 5 MHz; triangles, 12 MHz; crosses, 20 MHz. (After Miyano and Ketterson.(44)) described by (5.3.46) and (5.3.47): the density and layer oscillations are, in effect, uncoupled. Hence one of the branches corresponds to the normal longitudinal wave whose velocity can be seen from (5.3.46) to be Cl « (AJp)\ (5.3.48) which is practically independent of the direction of propagation. The other branch corresponds to changes in the layer spacing, without appreciable density changes and may be compared with the phonon branch in superfluids known as second sound.m) The velocity of this mode c 2 « (B0/p)* sin q> cos <p (5.3.49) is strongly orientation dependent. It becomes zero for propagation along 5.3 Continuum theory of smectic A 325 the layers as well as perpendicular to them. When the wavevector is parallel to the layers (qz = 0), it becomes the highly damped undulation mode which we have already examined in detail. When q is normal to the layers permeation effects set in and the wave is again strongly damped.(39) Neglecting density changes for this mode, it is clear from (5.3.43) that for q along z,vz = 0. Thus from (5.3.35) u = vpg (5.3.50) or using (5.3.24) u = vpBd2u/dz2. (5.3.51) Therefore, when the wavevector of second sound is normal to the layers, it becomes a purely dissipative mode with a relaxation rate \/T = vvBq\ (5.3.52) Furthermore, second sound is a critical mode, i.e., its velocity goes to zero as r - TAN. A direct confirmation of the existence of these two branches has been found by Liao, Clark and Pershan(47) from their Brillouin scattering experiments on a monodomain sample of ^-methyl butyl /?((/?-methoxy- benzylidene)amino) cinnamate. This compound shows the nematic, smectic A and smectic B phases. Choosing both the incident and the scattered light to be polarized either as ordinary or extraordinary waves, they observed two peaks corresponding to the two modes, the angular dependence of which is in excellent agreement with the theory (fig. 5.3.11). 5.3.7 Breakdown of conventional hydrodynamics Theoretical developments*48'49) in the early 1980s showed that the nonlinear interaction of thermally excited layer undulations, which as we have seen have large amplitudes because of the Peierls-Landau instability, leads to interesting new effects in the hydrodynamics of smectic A at small wavevector and frequency. We present below a very brief outline of the physical arguments involved.(50) In writing the expression for the free energy density (5.3.18) we neglected terms involving (du/dx)2 and (du/dy)2 since a uniform rotation of the layers does not cost energy (assuming, of course, that there is no externally applied magnetic field). Consequently in the harmonic approximation, <|wJ2> takes the form (5.3.22). However, while discussing the undulation instability in smectic A (§5.3.3), we observed that a bending of the layers alters the effective layer spacing along z and therefore makes a second 326 5. Smectic liquid crystals 2.0 H 1.5 1.0 0.5 (a) 0 1.0 2.0 2.0 1.5 1.0 (b) 0.5 0 1.0 2.0 Sound velocity (105 cm s"1) Fig. 5.3.11. Dependence of the sound velocities on the polar angle q> from Brillouin scattering experiments on ^-methyl butyl /?((/>-methoxy-benzylidene)amino) cin- namate. (a) Smectic A (T = 60.7 °C), (b) smectic B (T = 48.1 °C). The dashed lines are calculated from theory. The presence of a third component in (b) indicates that the shear modulus does not vanish in smectic B at these very high frequencies. Circles, triangles and squares represent measurements at different scattering angles. (After Liao, Clark and Pershan.(47)) order contribution to the layer dilatation in that direction. The corrected form of the free energy density is given by (5.3.19), which we write here as follows: a (say), (5.3.53) where the subscripts h and a stand for the harmonic and anharmonic parts 5.4 Defects in smectic A 327 of F, and V± and A± are the gradient operator and Laplacian respectively in the xy plane. Using a renormalization procedure, Grinstein and Pelcovits(48) showed that the effect of Fa at non-zero temperature is to change (5.3.22) into a form where the elastic constants are replaced by q dependent functions with for asymptotically small q. This surprising result prompted Mazenko, Ramaswamy and Toner(49) to examine the anharmonic fluctuation effects in the hydrodynamics of smectics. We have already shown that the undulation modes are purely dissipative with a relaxation rate given by (5.3.39). To calculate the effect of these slow, thermally excited modes on the viscosities, we recall that a distortion u results in a force normal to the layers given by (5.3.32). This is the divergence of a stress, which, from (5.3.53), contains the non-linear term 82(V± u)2. Thus, there is a non-linear contribution (V± uf to the stress. Now the viscosity at frequency co is the Fourier transform of a stress autocorrelation function,(51) so that Ar/(co), the contribution of the undulations to the viscosity, can be evaluated. It was shown by Mazenko et al.m) that Arj(co) ~ \/co. In other words, the damping of first and second sounds in smectics, which should go as rj{co) co2, will now vary linearly as co at low frequencies. A similar calculation for discotics(52) yields An(co) ~ co~*. The original work of Mazenko et al.m) argued that one of the shear viscosities should also diverge, but Milner and Martin(53) showed that this was not the case. This remarkable l/co divergence of the viscosities of a smectic at low frequencies is now confirmed by several independent experiments(54) using ultrasonic attenuation and second sound resonance. 5.4 Defects in smectic A 5.4.1 Focal conic textures We have noted that the smectic A layers are flexible and easily distorted, but tend to preserve the interlayer spacing (fig. 5.3.1). Moreover, as the layers can slide over one another, the structure adjusts itself readily to surface conditions. For example, when there is a centre of attachment at the glass surface the molecules adopt a radiating or a fan-like arrangement and the layers form a family of equi-spaced surfaces normal to the molecular directions. Under a polarizing microscope such distortions give 328 5. Smectic liquid crystals (a) (b) Circle L(. Hyperbola Lh to Ellipse Le Fig. 5.4.1. (a) Smectic layers in concentric cylinders to form a myelin sheath with a singular line L along the axis; (b) the cylinders are closed to form tori: there are two singular lines, a circle Lc and a straight line Ls; (c) the general case when the smectic layers form Dupin cyclides: the circle becomes an ellipse L e and the straight line a hyperbola Lh. rise to beautiful optical patterns known as focal conic textures. These were studied in considerable detail by Friedel, (55) to whom we owe the explanation of their origin. We shall now examine the structures of these focal conic domains. (56) (a) The simplest case is shown in fig. 5.4.1 (a). The layers are wrapped round in concentric cylinders to form a myelin sheath and there is a singular line L along the axis, (b) When the myelin sheath is bent and closed to form tori, the singular line L becomes a circle Lc (fig. 5.4.1 (b)). In addition, the structure now has another singular line Ls - a straight line through the centre of the circle and perpendicular to it. Such a structure is 5.4 Defects in smectic A 329 {a) (b) Fig. 5.4.2. A rare example of a pair of singular lines, one almost straight and the other almost circular in a toric domain in the (a) smectic A and (b) smectic C phases. Additional disclination lines develop near the centre in the smectic C phase for reasons discussed in §5.8.3. (From A. Perez, M. Brunet and O. Parodi, J. de Physique Lettres, 39, 353 (1978)). observed only very rarely (fig. 5.4.2). (c) In the most general case, the smectic layers lie on Dupin cyclides and not on tori. The circle Lc is then transformed to an ellipse Le and the straight line into a hyperbola Lh (fig. 5.4.1 (c)). Le and Lh are conic sections in perpendicular planes, one going through the focus of the other (figs. 5.4.3 and 5.4.4). It is these structures that are most commonly seen under the polarizing microscope (fig. 1.1.7 (a)). Based on the fact that the layers are bent into Dupin cyclides, one can derive expressions for the principal curvatures ox and cr2, and thus work out the energy, f 330 5. Smectic liquid crystals (a) Fig. 5.4.3. (a) Geometry of a pair of focal conies. (After Friedel.(55)). (b) A section of (a) in the plane of the hyperbola, showing parts of Dupin cyclides. (After Bragg.(56)) For the general focal conic structure, Kleman (57) has shown that where e is the eccentricity of the ellipse, p its perimeter, a its semi-major axis, and rc the core radius. This leads to the interesting result that E decreases with increasing e. In other words, the circle (e = 0) and straight- line combination has higher energy than the ellipse and hyperbola combination. Hence the possibility of Dupin cyclides degenerating into tori is not energetically feasible. This is in accord with observations, for the most commonly observed texture in smectic A is the one in which the focal lines are ellipses and hyperbolas. 5.4 Defects in smectic A 331 (a) Fig. 5.4.4. (a) Arrangement of a set of cones within a pyramid. The hyperbolas belonging to the ellipses meet at the vertex of the pyramid, (b) The base of the pyramid and the axial directions radiating from the foci at the base of the hyperbolas. (After Bragg.(56)) In principle, the focal lines can both be parabolas. Such parabolic focal conies have been observed by Rosenblatt et al.{58) but they are not common. Kleman(57) has shown that in this case 4/rJ' where/is the focal length of the parabola and R the sample radius. The energy increases with R and diverges as f-+ 0. Thus, in general, such structures are not energetically favourable. However, it turns out that when R < 3/, they can have smaller energies and Kleman argues that this may explain the observations of Rosenblatt et al A remarkable property of the smectic liquid crystal is that an open 332 5. Smectic liquid crystals Fig. 5.4.5. Optical interferometric patterns showing terraces on the surface of a homeotropically aligned smectic liquid crystal formed by a solution of potassium oleate in aqueous methyl alcohol, k = 0.5893 jum. (Reference 59.) 5.4 Defects in smectic A 333 droplet, homeotropically aligned on a perfectly clean glass plate or on a fresh cleaved mica sheet, forms terraces (fig. 5.4.5). This is known as the stepped drop, goutte a gradins, and was discovered by Grandjean.(60) From surface energy considerations it can be argued that the occurrence of steps is a consequence of the layered structure.(61) Under sufficient magnification the terraces are seen to be fringed by a chain of focal conic patterns indicating that the layers are crumpled at the edges. When the isotropic melt is cooled, the smectic phase often makes its appearance in the form of needle-shaped particles showing evidence of focal conic structure. Examples of such needles, termed bdtonnets, can be seen in fig. 2.1.15 (b). 5.4.2 Edge dislocations Fig. 5.4.6 shows a simple edge dislocation in smectic A with the dislocation line parallel to the smectic planes. In fig. 5.4.6(a) the extra half-plane is to the right of the observer looking down the dislocation line L, and in fig. 5.4.6(b) it is to the left. Consider now the contour integral -cpn-dr = TV dj along a closed path about L in the anticlockwise direction. Clearly N = + \ in fig. 5.4.6 (a) and — 1 in fig. 5.4.6 (b). Accordingly, we may refer to the two cases as positive and negative edge dislocations of unit strength re- spectively. Topologically these defects are very similar to their counterparts in crystals, but they differ significantly from the point of view of energetics, as we shall see presently. From the free energy expression (5.3.18), we obtain the equation of equilibrium where A. = (k11/B)* and u = u(z, x). We now proceed to work out the singular solutions of (5.4.1). For an edge dislocation with the extra half-plane on the side x > 0, the boundary conditions are m \0 for i < 0 "(0, x) = i [b/2 for x > 0 . 334 5. Smectic liquid crystals n n (a) (b) Fig. 5.4.6. (a) Positive and (b) negative edge dislocations of unit strength. Also, u(z,x) = u( — z,x). We look for a solution of the form<62) b b f , x fexp (i<7.x)l , ( 5 A 2 ) We then have (5.4.3) The tilt in the layer normal 6 = du/dx, and the dilatation in the layer spacing S = Ad/d = —Xffl/dx; 6 and 5 can be derived explicitly from the form of u: |= J b__ex i__x' (5.4.4) x Xb (5.4.5) From (5.4.5) it is seen that there is a compression of the layers for x > 0 and an extension for x < 0. The stresses associated with the dislocation cannot be derived uniquely. However, one acceptable set of values can be obtained from the equation of motion. (63) They are Bb -x 17-3 e x P -: (5.4.6) 2 72 XBb -x (5.4.7) 2 AX\z exp - AX\z\ /15Z? z I x (5.4.8) 2X\z exp - 4A|zj Apart from the exponentially decaying term, a33 varies as l/n, GX1 and a13 as \/n. In contrast, all these stress components vary as 1 /r in the case of crystal 5.4 Defects in smectic A 335 edge dislocations. The displacement u around the crystal dislocation is given by where v is Poisson's ratio. Again the form of this expression can be seen to be quite different from (5.4.2) for u in smectic A. The energy of a single edge dislocation The total energy is given by E= I \Fdxdz. (5.4.10) Using (5.3.18) and (5.4.2) and integrating, where rc is the core radius, which is expected to be of the order of the interlayer spacing d, and Ec the core energy. The total energy of an isolated edge dislocation is therefore independent of the sample size for smectic A, whereas it increases logarithmically with sample size for a crystal edge dislocation. Interaction between parallel edge dislocations Since (5.4.1) is linear in w, we may invoke the superposition principle to evaluate the interaction between dislocations.* 63'64) We superpose the displacements due to one dislocation line at (0,0) and another at (x o ,z o ). The total displacement is for z < 0, u = - 2 ^ l £ + £ 4^ \{exp[-(iqxo + Aq\)]-e}[exp(iqx + kq2z)]j?-, (5.4.12) for 0 < z < z0, u = 2 <*i.-«+ 4^ K e x P ( ~ V ^ ) + £exp[-i#x o + Aq2(z0-z)]}[exp(iqx)]jt, (5.4.13) for z > z0, u = - 3± >£ + — I {1 - s exp ( - iqx0 + A#2z0)}[exp (iqx - lq*zj\ -^-. (5.4.14) 336 5. Smectic liquid crystals Here e is 1 for like dislocations and — 1 for unlike ones. The interaction energy Ex= J \Fdxdz-E0, (5.4.15) where Eo is the self energy of two isolated dislocations. Therefore |[exp(-Vkol)]cos(^ 0 )d^. (5.4.16) This enables us to calculate dislocation interactions but it must be borne in mind that the theory is not strictly valid for distances of the order of the core radius. When the two edge dislocations are in the same horizontal plane, i.e., z0 = 0, the interaction is negligible. When z0 > rl/4n2l, i.e., the vertical separation between the dislocations is much greater than the core size, Ex can be simplified to ^ ( ^ f f ^ ) (5.4.17) From this we get two forces Fx and Fz along and normal to the smectic planes: Since k.JX2 = b, Fx = -ebaS3 (5.4.20) and Fz = ebals. (5.4.21) Here cr33 and <713 are the stresses at the origin, where the first dislocation is situated, due to the presence of the other dislocation at (x o,zo). This is known as the Peach-Koehler force(65) on a dislocation arising from the stress field of the other. This can be generalized to mean that under application of a stress oip a dislocation experiences a force Fi whose exact relationship is given by the above expressions. It is seen from (5.4.18) that Fx is always repulsive for a pair of like dislocations and always attractive for an unlike pair. On the other hand, Fz can be attractive or repulsive depending on the relative positions of the two dislocations. For example, Fz is repulsive for like dislocations when xl < 2Xz and attractive when x\ > 2Az0. This provides a mechanism for the 5.4 Defects in smectic A 337 Fig. 5.4.7. Clustering of like edge dislocations in smectic A to form a domain wall or 'grain boundary'. (After Pershan.(63)) clustering of dislocations to form a domain wall or a grain boundary (63) (fig. 5.4.7). Another interesting implication of the theory is that under an applied stress <T33 = Bdu/dz, which can be generated by stretching or compressing the smectic lattice along the layer normal, the dislocation experiences a force Fx parallel to the layers. The force changes sign with the sign of the edge dislocation. This effect is similar to the magnus force acting on vortices in superfluids. In the case of smectic A, a constant stress induces a constant flow of dislocations from the boundaries of the container. The dislocations move to release the stress. As a result, if the dislocations are not pinned the applied stress will always be expelled by the movement of dislocations. This has been termed as ' dielasticity' by analogy to diamagnetism. Any measurement of the intrinsic elastic constants of smectic A should therefore be carried out at frequencies (or other experimental conditions) such that the dislocations are immobile. The above phenomenon has a bearing on the behaviour of a dislocation loop under an applied stress. For a given dislocation loop, the extra half- plane may be either (i) inside the loop or (ii) outside it. Consequently, subjecting the system to a tensile stress (normal to the smectic layers) results in an expansion of the loop in situation (i) and a shrinking of it in situation (ii). Loops can also occur spontaneously to relax an applied stress. Pershan(63) has investigated the interaction of edge dislocations with different kinds of boundaries in various geometries. Williams and Kleman(66) have observed, by phase contrast microscopy, walls of dislocations (probably of large Burgers vector) which are strongly attracted to the boundaries (the glass substrate or the smectic-solid interface). 338 5. Smectic liquid crystals (a) vllllIMIIIII ^11111111111 1 III 1 r 111111 1 1111 , 1111 1 1,.,,,.. dmmmii II HIM (c) (d) Fig. 5.4.8. Disclinations in smectic A: (a) £la(n); (b) Qb(n); (c) Qo( - n); (</) Qb( - n). 5.4.3 Screw dislocations These are exactly analogous to screw dislocations in crystals. There is a spiral arrangement of the smectic layers around the dislocation line L which is normal to the layers. The associated deformation is given by u = —tan"1 r- 2n \x where u is the layer displacement along z, and b the Burgers vector is equal to an integral multiple of the layer spacing d. For a positive screw dislocation one gains a step of height b on going once round the line L in the anticlockwise direction, while for a negative dislocation one loses a step. A noteworthy aspect of the above solution is that it does not involve either lattice dilatation du/dz or layer undulation V-n. Therefore, within the approximations of the linear theory considered here, screw dislocations in smectic A have no self energy (apart from the core), nor do they interact amongst themselves. In this respect they are entirely different from screw dislocations in crystals. 5.4.4 Disclinations The geometrical process for creating a disclination in smectic A is as follows.(67) Cut the material by a semi-infinite plane that runs parallel to the layers, the limit of the cut being the disclination line L. Rotate the two faces 5.4 Defects in smectic A 339 ^\\\\ II II 1111 1111 1 II MI 1 1 111 i i n i rr IT 1, ^X//lll 11II i i II I I I II 1 IN sttt Y l III I I I I III -UrrU II i 41 El III! I I I II («) (b) -rrfiSl) 111 ' i IIII III II ,1,1,1 111 II n1 i 1111 HIM II {d) (c) Fig. 5.4.9. Edge dislocations in smectic A composed of disclination pairs: (a) a(-n); (b) na(n) + nb(-n); (c) Qb(n) + « 4 (-n); (d) Q.b(n)+ ila(-n). (a) (b) Fig. 5.4.10. 'Pincements' in smectic A composed of disclination pairs: (a) n a (-7r) + Q » ; (b) Qb(-7r) of the cut about L through a relative angle + Nn, N being an integer. Fill in the voids (or remove overlapping material) to get positive (or negative) disclinations, and allow the system to relax. The structures so obtained depend on whether the cut is made at the extreme end of the molecule or through its middle. The resulting disclinations, designated as Q a and Q&, are illustrated in fig. 5.4.8. It is obvious that Q a and Qb are not the same - their core energies are different, and they also have different con- figurations at large distances. The creation of Q disclinations also gives rise to disclinations of strength s = ± | in the smectic director n. The energy of an isolated disclination being large, disclinations of opposite signs may be expected to occur in pairs to form edge dislocations and what Bouligand has called 'pincements' (see figs. 4.2.6 and 4.2.8). Some possibilities are shown in figs. 5.4.9 and 5.4.10. 340 5. Smectic liquid crystals 5.5 The smectic A-nematic transition 5.5.7 Phenomenological theory of the smectic A-nematic transition We now proceed to consider a Landau type of phenomenological description of the A-N transition. This approach to the problem was due to de Gennes (6869) and to McMillan,(11) both of whom recognized the analogy with phase transitions in superfluids. We shall follow de Gennes's treatment. We start with the density wave in the smectic phase p(z) = ^ [ 1 + 2 % ! cos (? s z-?>)], (5.5.1) where p0 is the mean density, \y/\ the amplitude and qs = 2n/d the wavevector of the density wave, d the interlayer spacing and (p a phase factor which gives the position of the layers. Thus the smectic order can be fully specified by the complex parameter y/ = Hexp(i<p). (5.5.2) Near the transition, the free energy density may be expanded in powers of y/ as follows .... (5.5.3) From symmetry considerations it is clear that only even powers of y/ may be included. The coefficient /? is always positive. At a certain temperature 7"*, which is the second order transition point, a = 0. Accordingly, as explained in §2.5.1, we set in the mean field approximation, a = a(T-T*). To allow for the coupling between \y/\ and the orientational order parameter s, the free energy is expressed as(69) F= «\¥\*+P\y\* + [(dsy/2x\-C\w\>ds, (5.5.4) where Ss is the change in the nematic order brought about by layering of amplitude y/, x a response function which depends on the degree of saturation of nematic order, and C a positive constant. Minimization with respect to Ss leads to 2 F \ where For a wide nematic range x(TAN) is small, /?' > 0, and the transition is of second order. For a small nematic range x(TAN) is large, /?' < 0, and the transition is of first order; in this case one must add a positive sixth order 5.5 The smectic A-nematic transition 341 term in the free energy to ensure stability. The tricritical point occurs when /?' = 0, i.e., x(TAN) = 2/3/C2. This is in agreement with the prediction of the microscopic theory, according to which the tricritical point occurs at 7; N /r NI = 0.87 (see §5.2). /(7^ N ) may be altered by varying the length of the end-chain, or by preparing mixtures, or by the application of pressure.(70) 5.5.2 Pretransition effects in the nematic phase In §5.2, we referred briefly to McMillan's X-ray evidence(11) for the growth of smectic-like short-range order in the nematic (or cholesteric) phase near the A-N transition (see fig. 5.2.6). This pretransition effect manifests itself more strikingly in the temperature dependence of the elastic properties of the nematic phase, as was first demonstrated by Gruler.(71) The twist and bend distortions, which are normally disallowed in the smectic phase, become increasingly difficult as the smectic clusters build up and these two elastic constants rise much more rapidly than expected from the simple s2 law discussed in §2.3.5. To investigate these properties, we expand the free energy in powers of y/ and its gradients. We shall suppose for the present that the A-N transition is of second order. For a fixed orientation of the director, In the smectic phase (T < T*), the amplitude of the density wave may be taken to be constant, so that only q> varies. The gradient terms of F therefore become Comparing this with (5.3.3) it is at once clear that cp is related to the layer displacement u:\q>\2 = q2\u\2 and B =\y/\2 ql/Mv. The terms d<p/dx and dcp/dy represent the tilt of the layers with respect to the director. If the director orientation is not fixed, it is the relative tilt between the layers and the director that should be considered and therefore (5.5.6) takes the generalized form ^(^)\± (5.5.7) 2Mv\dz) 2M T ' where VT is the gradient operator in the plane of the layers. This equation 342 5. Smectic liquid crystals is reminiscent of the Landau-Ginsburg expression(72) for the free energy of superconductors; n corresponds to the vector potential A, V x A being the local magnetic field. The analogy may be extended further. By including the Frank elastic free energy terms in (5.5.6), we may define (as already shown in §5.3.3) a characteristic length X = (k/B)*. Making use of the condition dF/dy/0 = 0 and ignoring the difference between MT and Mv, where k is an appropriate elastic constant. For a twist or bend, both of which involve V x n, X may be interpreted as the depth to which the distortion penetrates into the smectic material. Thus X is equivalent to the penetration depth of the magnetic field in superconductors. Above 7"*, we may ignore the term involving the fourth power in y/ and from the equipartition theorem obtain in the usual manner —, , (5.5.9) where the half-widths or the associated coherence lengths may be related to Mv, MT and a as follows : £2=l/2MFa, (5.5.10)1 = l/2MT(X.                           (5.5.11)
Since a ^ 0 as T^ T*, <|^(#)|2>, ^y and £L diverge near the transition. The
variation of {\y/(q)\2} reveals itself in the intensity of the Bragg scattering
(see fig. 5.2.6).
Critical divergence of the elastic constants
To discuss the critical behaviour of the twist and bend elastic constants in
the nematic phase, we observe that the Frank free energy expression should
include the contribution due to smectic short-range order:
F=\k\Vnf      + Fs{yy\                     (5.5.12)
where k° is the usual nematic elastic constant in the absence of smectic-like
order, and Fs(y/) when averaged over all y/ is of the form

<f(Vn)2,                    (5.5.13)
1V1

where we have replaced Sn by £Vn and ignored the difference between Mv
5.5 The smectic A-nematic transition                 343
and MT. For a correlated region of volume £3, it can be shown, using
(5.5.9), that in the mean field approximation
<M 2 >oc/; B 7yacf.                     (5.5.14)
Thus from (5.5.12), (5.5.13) and (5.5.14), the effective elastic constant for
twist or bend will be k° + Sk, where

Skoc^t                               (5.5.15)

Since the coherence lengths diverge rapidly near the A-N transition, the
elastic constants for twist and bend should also show critical behaviour. In
the mean field approximation
£ oc ( r - r * ) ~ 1 / 2 .              (5.5.16)
However, invoking the analogy with superfluids, de Gennes predicted
£ oc (T- T*)~2/\                        (5.5.17)
Using scaling arguments, Jahnig and Brochard(73) have shown that in
the anisotropic case
<J* M ocft/£,,                   (5.5.18)
<**„«£„,                            (5.5.19)
and that, below T* (i.e., in the smectic phase),
Bazit/Zl,                           (5.5.20)
Dccl/£r                            (5.5.21)
The theory continues to be valid even if the A-N transition is weakly first
order, except that T* represents a hypothetical second order transition
temperature slightly below 7^N.
The first X-ray scattering experiments of McMillan(74) on /?-n-octyl-
oxybenzylidene-/?-toluidine, which exhibits a first order A-N transition,
agreed with the mean field theory. On the other hand, CBOOA showed
appreciable anisotropy in the temperature variation of the longitudinal
and transverse coherence lengths. Later studies on a number of materials
have confirmed that anisotropy is a general feature exhibited by all of
them.
The increases in k22 and fc33 in the nematic phase near the A-N transition
over and above that given by the usual s2 law are now well established. Fig.
5.5.1 presents the data of Cheung, Meyer and Gruler(75) for CBOOA; &33
344                           5. Smectic liquid crystals
70 r-

60

50

£« 40
Ax
30

20

10

,, 1       I   I
0.1                       1.0                     10           30
r-r A
Fig. 5.5.1. The temperature dependence of the splay and bend elastic constants, kxl
(crosses) and k33 (circles) respectively, in the nematic phase of CBOOA prior to the
smectic A-nematic transition. The values are plotted as &../A/ where A/ is the
anisotropy of the diamagnetic susceptibility of the nematic. The full line shows the
order parameter s, normalized to fit fcn/A/ at higher temperatures. The splay
constant klx deviates only very slightly from ordinary nematic behaviour while the
bend constant kZ2> exhibitsa critical increase near 7^N due to pretransition
fluctuations. (After Cheung, Meyer and Gruler. (75))

diverges rapidly while fcn exhibits normal behaviour. Similarly, it has been
verified that the layer compressibility constant B shows a critical variation
in the vicinity of the transition.

Critical divergence of the viscosity coefficients
Certain nematic viscosity coefficients also exhibit critical behaviour. The
origin of this effect may be explained physically as follows. Take, for
example, the frictional torque associated with the twist viscosity coefficient
Xx defined by (3.3.14). The formation of smectic clusters results in an extra
torque due to the flow of the liquid normal to the smectic planes. This extra
torque increases as the clusters become larger and longer lived, and in
consequence there is a net enhancement of the effective Xx. To estimate this
enhancement/ 73 ' 76) consider a slowly rotating magnetic field having an
angular velocity Q. We have seen in § 3.6.2 that the director follows the field
at a constant inclination, the torque being given by — 2XQ. Now, the
layered structure of the smectic-like regions makes an additional con-
5.5 The smectic A-nematic transition                   345
tribution to the torque, say F s . If the angle between the layer normal and
the director is 0, then
6= QV                             (5.5.22)
where TV is the relaxation time of \y/\. The torque Ts may be derived from
(5.5.13) with 0 = <jVn:

< w 2 >       a             (5 5 23)
M             ^                   - -

Using the value of \i//\2 from (5.5.14), the excess viscosity
8X1 oc rji.                            (5.5.24)

The temperature dependence of SX1 depends on both z¥ and £. The
theory has been worked out in detail by McMillan(76) using the mean field
approximation and by Brochard(73) assuming dynamical scaling laws. The
critical exponents for the divergence of the visosity as predicted by the two
theories are different:

S^ ~ (T- r*)-° 5 0     (mean          field)    (5.5.25)
33
SXX ~ (T- r*)-°-        (dynamical scaling).          (5.5.26)

Another difference between these two approaches lies in the behaviour of
the director relaxation time r. For example, for a twist deformation we
know that (§3.8.1)
T~1 =      -k22q2/Al.

It is seen that in the mean field approach T is independent of temperature
since near T* both k22 and lx diverge similarly. On the other hand, with
helium-like exponents
T-1 oc (T- r*)-° 3 3 -
An anomalous increase of Xx has been observed but the principal
experimental difficulty in determining the critical exponent accurately is
that the normal nematic viscosity (in the absence of correlations) is itself
strongly temperature dependent and the anomalous part forms only a
relatively small contribution. However, careful measurements(77) have
indicated a mean field behaviour.

Comparison between theory and experiment
Halperin, Lubensky and Ma(78) have argued that when director fluctu-
ations are taken into account, the A-N transition should be at least
346                        5. Smectic liquid crystals

Reduced temperature [t = (T/Tc) -1]
Fig. 5.5.2. Temperature dependence of the susceptibility (x) and the correlation
lengths ^ and £± for smectic A short-range order in the nematic phase of
butoxybenzylidene-octylaniline. Open circles were deduced from X-ray scattering,
and filled circles from light scattering. The smectic densityowavevector q0 remains
essentially constant, but f(| increases from about 100 A to 1.5 jum over the
temperature range shown in the diagram. Here Tc represents the A-N transition
point. (After Litster and Birgeneau.(88))

weakly first order. The shift in the transition point due to this effect,
relative to the second order transition point T*, was estimated to be of
the order of 10 mK. Whether or not the A-N transition can, in principle,
be of second order has been the subject of some discussion. (7981) From
high resolution experimental studies it appears that the second order
nature of the transition has been more or less established in at least a
few cases(82) (though Anisimov et al.m) have claimed to have observed
the 'Halperin-Lubensky-Ma effect' in three different systems). A
number of theories have been proposed, which have been reviewed in
authoritative articles,(84~6) but a precise description of the A-N transition
still remains elusive. We shall give a very brief summary of the current
situation.
5.5 The smectic A-nematic transition                           347

Table 5.5.1. The A-N transition: summary of experimental data(S9           9

Compound                    a        v
n       v±       y     a + 2v± + v|| Reference
T8              0.66       0.07    0.70    0.65     1.22    1.93 + 0.15   92
T7              0.71       0.05    0.69    0.61     1.22    2.06 + 0.15   92
40.7            0.926    -0.007    0.78    0.65     1.46    2.07 + 0.1    89,93
8S5             0.936      0       0.83    0.68     1.53    2.19 + 0.16   94,95
CBOOA           0.94       0.15    0.70    0.62    ]1.30    2.09 + 0.14   96,97
40.8            0.958      0.15    0.70    0.57     1.31    1.99 + 0.18   98
80CB            0.963      0.2     0.71    0.58     1.32    2.07 + 0.17   96,99
9S5             0.967      0.22    0.71    0.57     1.31    2.01 ±0.18    95, 100
8CB             0.977      0.31    0.67    0.51    11.26    2.00 + 0.13   101,82
10S5            0.983      0.45    0.61    0.51    ]1.10    2.08 + 0.18   95, 100
9CB             0.994      0.50    0.57    0.39    11.10    1.85 + 0.18   100, 82
XY model                 -0.007    0.669   0.669   ] .316   2.0           102, 103
Tricritical                0.5     0.5     0.5     11.0     2.0

Tn = 4-alkoxybenzoyloxy-4/-cyanostilbene n = 7,8
40.m = butyloxybenzylidene alkylaniline m = 7,8
nS5 = 4-n-pentylphenylthiol-4/-alkyloxybenzoate n = 8,9,10

Reduced temperature: t =        (T-TAN)/TAN.
Specific heat at constant pressure: Cv = A\t\~
Correlation length,
(5.5.27)
longitudinal (parallel to n): ^ = ^\t\'vw
transverse (perpendicular to n):        = £°±\
7
Susceptibility: x = Xo\*\~
Anisotropic hyperscaling relation,
vy+2v_L + a = 2.                            (5.5.28)
Measurements of the intensity and the width of the X-ray scattering
yield the susceptibility (x) and the correlation lengths (£) respectively.
Light scattering, which gives the divergence of the elastic constants, can
also be used to measure £. The values of £ determined by the two methods
are in excellent agreement (fig. 5.5.2). Table 5.5.1 presents the relevant
exponents derived from the best available measurements for nine com-
pounds. It is seen that the data for the different materials are by no means
'universal'.
Present theories of the A-N transition indicate two possibilities. Firstly,
that it belongs to the 'inverted' XY class(103) which leads to the critical
exponents(80)
a = -0.007,                = 0.67    and      y =1.32.
348                       5. Smectic liquid crystals
Secondly, that it belongs to an anisotropic class with(104)
v(| = 2v±,
(though some doubts have been raised as to the validity of this
relation(105)). Another proposal for the anisotropic class is that (106)
v,, = ivxy = 0.8,   v± = lvxy = 0.53.
We note from Table 5.5.1 that v is anisotropic for all the materials, v|( — v±
being about 0.13. In no case is v(| = 2v±. Also, all the materials satisfy the
anisotropic hyperscaling relation (5.5.28) to within experimental limits.
For two compounds, 407 and 8S5, a agrees with the predicted value of
-0.007, but the vs do not. Brisbin et al.{95) have suggested that the A-N
transition for some of the compounds, e.g., 10S5 and 9CB, is near a
tricritical point as the values of TAN/TNl are high for these compounds. This
may well be true, for y, a and (v)( + 2v ± )/3 for both 10S5 and 9CB are close
to the tricritical values of 1.0, 0.5 and 0.5 respectively. But this still leaves
the question of anisotropy unexplained.
From (5.5.20) and (5.5.21) it is seen that B oc t*9 where ^ = 2 v ± - v,, and
D oc fw. If V|| = 2v1? B should be finite at the transition temperature.
However, experimentally/ 107'108) it appears that B at the transition is
almost vanishingly small within experimental limits. Few measurements
are available on D to draw any definite conclusions. In any case, as pointed
out earlier, the exponents are neither universal nor do they agree with the
predictions of any of the theoretical models. Vithana et al.ao9) have
suggested that the widely differing values of the exponents for the different
compounds may be a consequence of the fact that one is measuring
effective values associated with crossover effects between the XY class and
a tricritical point. A further complication is that the experiments of
Evans-Lutterodt et al.(92) appear to indicate that the occurrence of different
forms of smectic A (see §5.6) may have a bearing on the nature of the A-N
transition.
In summary, therefore, it would be fair to conclude that no theory
predicts all the exponents correctly for any of the systems. Thus the A-N
transition, which is probably the most extensively studied transition in
liquid crystals, still remains a major unsolved problem in condensed matter
physics.
The smectic A-cholesteric transition
Lubensky(110) has shown that, in principle, the smectic A-cholesteric
transition should always be of first order, in analogy with the behaviour of
a superconductor in an external magnetic field. The relative shift of the
5.5 The smectic A-nematic transition                 349

0.40

0.35

0.30

0.25

72          76        80          84
Temperature (°C)
Fig. 5.5.3. Pitch versus temperature in cholesteryl nonanoate prior to the smectic
A-cholesteric transition (74 °C); the crosses are values obtained from observations
of the Grandjean-Cano walls and the circles from the wavelengths of maximum
reflexion. (After Kassubek and Meier.(111))

transition temperature AT/T* due to this effect is estimated to be ~ 10~3,
but no experimental study of the effect appears to have been reported.
Near the transition the free energy of the cholesteric may be written as
F =\k\2q2 — k\2qQq + FJ<y/)                  (5.5.29)
omitting the ' background' term, where q0 is the equilibrium value of the
twist per unit length in the absence of smectic-like short range order and q
the actual value. Therefore
(5.5.30)
Minimizing with respect to q, we get

(5.5.31)
/Coo I* Ofcn

which decreases rapidly as the temperature drops to the smectic A-chol-
esteric transition point, or, in other words, the pitch P = 2n/q increases.
Fig. 5.5.3 presents the temperature dependence of the pitch of cholesteryl
nonanoate, (111) a compound which shows the smectic A phase below 74 °C.
350                      5. Smectic liquid crystals
Pindak, Huang and Ho (112) have reported that dk22 oc r 0 6 7 for this
compound. The high temperature sensitivity of the pitch of such materials
has applications in thermography, as we have already seen in §4.9.

5.6 Smectic A polymorphism
5.6.1 Smectic A phases of strongly polar molecules
We have so far regarded the molecules to be symmetric and non-polar, and
therefore assumed the smectic A layer spacing (d) to be approximately
equal to the molecular length (/). However, if the molecules have a strong
longitudinal dipole moment there will be near-neighbour antiparallel
correlations, as discussed in §2.6.2, and this can result in subtle changes in
the structure.
The first evidence that more than one form of smectic A exists came from
an observation of Sigaud et a/.(113) of a phase transition that was detected
by calorimetry, but could not be observed optically. X-ray studies revealed
that this was a transition between two forms of the A phase: the higher
temperature phase was characterized by a pair of reflexions corresponding
to a layer spacing d ~ /, and the lower temperature one by two pairs of
reflexions corresponding to d ~ I and 2/ respectively. The former type of
smectic A is called the monolayer (Ax) phase and the latter the bilayer (A2)
phase. A third type which has a layer spacing intermediate between / and
2/, has been identified and is called the partially bilayer (Ad) phase. The
structures of these phases are represented schematically in fig. 5.6.1.
Prost,(114) and later Prost and Barois,(115) developed a phenomenological
theory of the A phases in which the free energy density is expressed in terms
of two coupled order parameters, p(r) describing the mass density wave,
and ^(r), the dipolar density wave arising from the antiparallel associations
of neighbouring molecules. The total free energy is written as

(5.6.1)
where

F2 = \{A2p2 + \B2p* + C2[(V2 + k22)p}\               (5.6.3)
F12 = A12pj> + D12<l>*p + D21p^ + \B12p^\               (5.6.4)
Ax = a-^T—Tj), A2 = a2(T—T2), and T19 T2 are the mean field transition
temperatures for the two types of ordering.
The elastic terms C1[(V2 + ^ ) ^ ] 2 and C2[(V2 + k22)p]2 describe spatial
modulations of <f> and p with wavevectors k1(= 2n/l') and k2(= 2n/l)
5.6 Smectic A polymorphism                                 351
A^

i   t       i

A

in,               !!
1
!                   n
Ad                        1                             I      — I I P II
1                                            II
•i
ii
1
I
1           1    i
II

II
ji
i   i       I
• K • vi
r                 i         if    " "i
h

\i
i
}        \   \           \
'!    . •            n
i, - .                                   •   ,        i
i, , , ,          i        • >;      \       11
ii                I           l> '                i
II                i           i;
i                ii          ii                  i

Fig. 5.6.1. The different forms of smectic A composed of polar molecules:
monolayer A1? bilayer A2, partially bilayer A d and A phases, the last being similar
to A2 but with a transverse modulation of the structure.

respectively. The importance of the different coupling terms in (5.6.4)
depends on the relative values of/' and /. When /' « 2/, it is readily shown
that A12p(r)(/>(r) and D21p*(r)(p(r) can be neglected. It then follows that
=
|
p = 0, <f> = 0 corresponds to the monolayer Ax phase in which the dipoles
can point up or down with equal probability within each layer (fig. 5.6.1);
t
=         N
p = 0, </> = 0 with k2 = 2k1 represents A2 which has an antiferroelectric type
of ordering; and <j> ^> p yields A d . The elastic terms favour modulations
of (f> and p with wavevectors kx and k2 which are incommensurate, while
the coupling terms favour a lock-in of kx and k2 such that 2kx = k2, which
as we have already seen leads to the A 2 phase. The system gains energy
through the coupling term at the expense of the elastic energy, and vice
versa, and the competition between these two causes ' frustration' in the
system. Different ways of satisfying these opposing tendencies result in the
A phase which has a transverse modulation of the structure (fig. 5.6.1) or
in a smectic A phase with two collinear incommensurate periodicities. Two
types of incommensurate phases are envisaged, the weakly coupled one in
which the two interpenetrating density waves are almost independent of
each other (fig. 5.6.2(a)) and the strongly coupled one in which the phases
352                         5. Smectic liquid crystals

Soliton region

D3I     II—'7T-1I

I
Locked phase
region

JIT"
Wi Wn'fi                                                     nx.
Soliton region -
(a)

Fig. 5.6.2. Schematic representation of the molecular arrangement in (a) the weakly
coupled incommensurate smectic A, and (b) the strongly coupled incommensurate
smectic A. (After Prost and Barois. (115))

of the two waves are modulated (fig. 5.6.2(b)) exactly in accordance with
the theory proposed originally by Frank and Van der Merwe. (116) In the
latter case, the structure consists of large regions of A 2 separated
periodically by defect walls or phase solitons. Experimentally, however,
the occurrence of incommensurate smectic A phases has yet to be
established conclusively. (117~119)
Barois, Prost and Lubensky(120) have used the phenomenological model,
within the framework of the mean field theory, to construct phase diagrams
involving the polymorphic forms of the A phase. They have predicted three
kinds of critical points, the A d -A 2 critical point, the A x -A 2 tricritical point
and the A 1 -N-A d bicritical point, the salient features of which are
summarized below.
Since Ad and A2 have the same symmetry, it is to be expected that a first
order A d -A 2 phase boundary should terminate at a critical point. This has
been confirmed(121) in a binary liquid crystal system. Plots of wavevectors
against temperature for different concentrations (X) in the vicinity of the
critical point are shown in fig. 5.6.3. It is seen that the first order A d -A 2
transition manifests itself as a jump in the wavevector, accompanied by a
two-phase coexistence region. On increasing X the magnitude of the jump
as well as the width of the two-phase region decrease, until finally only a
5.6 Smectic A polymorphism                                 353
124      125     126      127

119      120      121      122     123     124      125
Temperature (°C)
Fig. 5.6.3. Plots of the magnitudes of wavevectors q0 and q'Q (corresponding to A2
and Ad periodicities respectively) against temperature (7") for mixtures of 4-n-
undecyloxyphenyl-4/-(4//-cyanobenzyloxy)benzoate (11OPCBOB) and 4-n-nonyl-
oxybiphenyl-4'-cyanobenzoate near the A d -A 2 critical point (CP). The mole
fractions of 11OPCBOB in the mixtures are (a) 0.550, (b) 0.571, (c) 0.597, (d) 0.619,
(e) 0.642, (/) 0.715, (g) 0.80 and (h) 1.0. The first order A d -A 2 transition is identified
for plots (a)-(d) by a two-phase region consisting of both Ad and A2 periodicities
(shown as closed circles) and also by a jump in the wavevector. As CP is
approached the two-phase region shrinks with an accompanying decrease in the
difference between q0 and q'o. At CP (plot (e)) only a vertical inflexion is seen.
Beyond CP, Ad evolves continuously into the A2 phase (plots (/)-(//)). The
behaviour is similar to that seen in the vicinity of the gas-liquid critical point.
(After reference 121.)

vertical inflexion point is seen. Beyond this critical point, Ad evolves
continuously into the A2 phase. The analogy with the density curves near
a gas-liquid critical point is obvious. However, Park et al.(122) have pointed
out that when the coupling of the order parameter with the elastic degrees
of freedom and with the fluctuations in the layer spacing is taken into
account, the A d -A 2 critical point may belong to a new universality class
with the upper marginal dimensionality dc = 6.
The transition between the Ax and A2 phases can be of first or second
order because of the exact doubling of the layer periodicity (see Landau
and Lifshitz,(25) p. 468). Thus the A x -A 2 phase boundary - say in a binary
354                       5. Smectic liquid crystals

120 —
I
5              10              15
Concentration (molar % TBBA)
Fig. 5.6.4. Binary phase diagram for mixtures of 4-n-hexylphenyl-4/-cyano-
benzoyloxybenzoate and terephthal-to-butylaniline (TBBA) showing the A^-Ag
tricritical point (TCP). The solid line represents the first order phase boundary
and the dashed lines represent the second order phase boundaries. (After Chan
et   tf/.(123))

system in the temperature (^-concentration (X) plane-can exhibit a
tricritical point. This has, in fact, been observed by the high resolution
X-ray experiments of Chan et al.(123) The phase diagram is shown in fig.
5.6.4. It was found that the correlation length exponents (associated with
the divergence of the A2-like fluctuations in the Ax phase) are isotropic, i.e.,
v|t = v± = 0.74. This value is reasonably consistent with the Fisher-
renormalized Ising exponent, indicating that the A x -A 2 transition probably
belongs to the Ising universality class.
The mean field theory predicts a bicritical point when the second order
N-A 1 and N-A d phase boundaries meet a first order A d -A 1 boundary.
However, when the effect of fluctuations is taken into account, the
existence of such a point becomes doubtful. (124) On the experimental side,
phase diagrams with an A 1 -N R -A d point have been reported, (125) (where
N R is the re-entrant nematic, see §5.6.2) the topology of these diagrams
resembling that of the magnetic bicritical point. (126) But high-resolution
experiments(127) carried out subsequently in the immediate vicinity of an
5.6 Smectic A polymorphism                            355

52            53               54                        56
Concentration, X (mol %)
Fig. 5.6.5. High resolution temperature-concentration (T-X) diagram for binary
mixtures of 4-n-octyloxy- and 4-n-decyloxy-phenyl-4 / -nitrobenzoyloxybenzoate in
the vicinity of the A 1 - N R - A d point. The solid lines denote first order phase
boundaries. The critical end point (CEP) for the A d - N R b o u n d a r y and the
approximate location of the tricritical point (TCP) for the A X - N R boundary are
indicated in the diagram. (After reference 127.)

A 1 -N R -A d point show that the bicritical point has, in fact, split into a
tricritical point (for the A X -N R boundary) and a critical end point (for
the A d -N R boundary, fig. 5.6.5). This new result, though predicted
theoretically for magnetic systems,(128) has not been envisaged in any of
the theories of frustrated smectics.

5,6.2 The phenomenon of re-entrance
When a compound exhibits both nematic and smectic phases, then, as a
rule, the nematic occurs at a higher temperature. Exceptions to the rule
were discovered by Cladis(129) in certain strongly polar materials. The first
observations were on binary mixtures of two cyano compounds: over a
range of composition the sequence of transitions on cooling was as follows :
iso -> N -> SA -> N R -> crystal,
where N stands for the usual nematic and N R for a second nematic, called
356                         5. Smectic liquid crystals

1.4

1.0

0.6

Smectic A                       Isotropic

0.2

50               70          90            110
Temperature (°C)
Fig. 5.6.6. Experimental pressure-temperature diagram for 80CB showing the re-
entrant nematic phase. (After Cladis et al.a30))

C 9 H 19 O                 OOC                   OOC

Fig. 5.6.7. The structural formula of 4-nonyloxyphenyl-4 /-nitrobenzoyloxy-
benzoate, a compound that shows multiple re-entrant behaviour.

the re-entrant nematic, which appears at a lower temperature. The SA phase
occurring between two nematics was identified to be the partially bilayer
A d . Later, a similar effect was reported by Cladis et al.am in a pure
compound at elevated pressures (fig. 5.6.6).
These observations stimulated a great deal of interest, and much more
5.6 Smectic A polymorphism                            357

(a)                                          (b)
Fig. 5.6.8. Schematic representation of (a) a dimer unit consisting of two
antiparallel molecules, (b) the mechanism of destabilization of the smectic A phase.

complex examples of re-entrance have since been found. (125131) Probably
the most spectacular case is the compound (132) whose molecular structure
is shown in fig. 5.6.7, which exhibits three nematic, four smectic A and two
smectic C phases in the following sequence :<133)
iso -> N -> Ad -> N R -> Ad -> N R -> Ax -> C -> A2 -> C2 -> crystal,
where C and C2 are two different forms of S c (fig. 5.8.3). It turns out that
re-entrant behaviour and the closely related phenomenon of smectic A
polymorphism are both extremely sensitive to the molecular structure. For
example, the reversal of the sense of the longitudinal component of the
dipole moment of just one of the bridging groups of the molecule, relative
to that of the end group, may suppress completely the occurrence of the
A-A transition or re-entrance or both. (134)
From the molecular point of view, only an approximate qualitative
explanation of the phenomenon has been possible. Fig. 5.6.8 illustrates the
elements of a simple model proposed originally by Cladis. (135) The basic
idea underlying this model is that because of antiparallel correlations, the
molecules form dimers, which are assumed to be somewhat bulgy in the
middle (fig. 5.6.8 (a)). Once the smectic phase is formed, the bulgy parts are
lined up in a plane, but the alkyl chains cannot fill the rest of the space.
With increasing dimer formation (i.e., with decreasing temperature) and
also possibly with the stiffening of the end-chains, the packing becomes so
unfavourable that the A d phase is destabilized and the nematic re-enters
(fig. 5.6.8 (b)).
A number of other more elaborate models,(136~9) which take into account
attractive forces and hard core repulsions, (136) dipole-induced dipole
interactions (which, interestingly, can favour a parallel configuration at
small intermolecular separations (139)), etc., have been proposed sub-
358                       5. Smectic liquid crystals
sequently, but as remarked earlier, only a qualitative description of the
effect has emerged until now.(140) The frustrated spin-gas model of Indekeu
and Berker(137) is noteworthy in that it is able to account for multiple re-
entrance of the kind shown by the compound of fig. 5.6.7 and the sensitive
dependence of re-entrant polymorphism on molecular chain length and
pressure.(133)
In principle re-entrant phases can occur in non-polar systems as
we U U36,i4i) Dowdi^D has discussed this possibility by treating the
molecules as hard rigid cores with semiflexible tails and interacting via
segmental hard repulsions. In the SA phase, the molecules are segregated
into layers. Calculations show that as the tails become more rigid at lower
temperatures the lamellar packing may become disadvantageous and the
SA phase can be destabilized. Experimental evidence of re-entrant phases in
non-polar systems has been presented by the Halle group.(142)
Another example of a special type of molecular interaction concerns the
'induced' smectic phase, i.e., the appearance of a smectic phase in a binary
mixture even though neither component shows this phase in the pure state.
This effect occurs most commonly in mixtures with one component having
a strongly polar terminal group and the other a non-polar terminal
group. (14356) Obviously, dipole-induced dipole interaction plays a part in
phase induction. There is also evidence of charge transfer complex
formation, the polar molecule acting as the acceptor and the other as the
donor. (143151157) However, phase induction has been observed in other
types of mixtures as well. (15860) For example, mixtures of two cyano
compounds have been found to give rise to an induced S c phase. Thus no
generalizations are possible as yet and the precise nature of the molecular
interactions and correlations responsible for promoting phase induction is
not clear.

5.6.3 Smectic A* or the twist grain boundary phase
A significant result that emerged from the phenomenological theory of the
A-N transition discussed in §5.5.2 was that the director n in smectic A
plays the same role as the magnetic vector potential in a superconductor.
Recognizing this analogy, de Gennes(68) predicted that a twist or bend
distortion should depress the A-N transition point, 7^N, relative to that of
the curvature-free sample. The nature of the phase diagram depends on
whether the material is of type I or type II, i.e., on whether the
Landau-Ginsburg parameter, i / £ , is less or greater than I / A / 2 , where X is
the penetration depth defined by (5.5.8) and £ the coherence length. If the
5.6 Smectic A polymorphism                            359

C 13 H 27 O                                            -COOC*H(CH 3 )C 6 H 13

81.6 °C    88.3 °C    94.1 °C
K           *>C*       * A*

33.7 °C

s4 <     s,
43.7 °C
Fig. 5.6.9. The structural formula of (R)- and (S)-l-methylheptyl 4'-(((4"-n-
alkoxyphenyl)propioloyl)oxy)biphenyl-4-carboxylates, which show the smectic A*
phase.

Fig. 5.6.10. The structure of smectic A* or the twist grain boundary phase.

A-N transition is of second order, or nearly so, the material may be
expected to be of type I.(69) Measurements(161) on CBOOA, which has a
nearly second order A-N transition, have confirmed that the variation of
7^N with the twist per unit length does indeed reproduce the theoretically
predicted trend for A/f < \/y/2.
For a type II material, de Gennes envisaged the possibility of a regular
network of dislocations being formed in smectic A under an imposed twist
or bend distortion. A model for such a network, similar to the Abrikosov
flux lattice phase(162) in a type II superconductor subjected to an external
magnetic field, was discussed by Renn and Lubensky.(163) Soon afterwards,
a phase consisting of a regular array of dislocations was discovered by
Goodby et al.am who designated it as smectic A*.
Smectic A* is composed of optically active molecules. The molecular
structure of one of the compounds studied by Goodby et #/.(164) and the
phase transitions occurring in it are given in fig. 5.6.9. The structure of
smectic A* is depicted schematically in fig. 5.6.10. Unlike smectic C* whose
helical axis is normal to the layers (see fig. 5.10.1) smectic A* has its helical
axis parallel to them. Hence the structure may be looked upon as a series
of smectic A blocks or grains separated by twist grain boundaries. The
director (which in this case is the layer normal) is rotated by a constant
angle on going from one grain to the next. From observations of the
360                       5. Smectic liquid crystals
reflexion of circularly polarized light incident along the helical axis,(165) it
is found that the pitch is of the order of 1 jum and from X-ray
measurements(165) that the grain size is about 180 A. Renn and Lubensky
have called this phase the 'twist grain boundary' (TGB) phase. The
discovery of this phase has raised a number of fundamental questions of
considerable interest to condensed matter physicists.

5.7 The hexatic phase
(166)
Kosterlitz and Thouless         proposed that the unbinding of dislocation
pairs can lead to a continuous melting transition from a two-dimensional
solid to a two-dimensional liquid. Halperin and Nelson(167) then worked
out the details of a defect-mediated melting theory in two dimensions and
predicted the existence of an intermediate phase between the two-
dimensional solid and liquid. This phase, referred to as the hexatic phase,
has short-range positional order, but long-range 'bond-orientational'
order (bond, in this case, implying the line joining the molecular centres of
nearest neighbours). Its structure is illustrated in fig. 5.7.1. Based on this
idea, Birgeneau and Litster<168) suggested the possibility of a three-
dimensional liquid crystal phase consisting of a stack of two-dimensional
hexatic layers which interact to produce three-dimensional bond-orienta-
tional order. Experimental evidence of this type of phase was presented
by Leadbetter, Frost and Mazid,(169) but its existence was established
conclusively only later by the high resolution X-ray studies of Pindak
et al.(170) on free standing films. This phase, called the hexatic B phase,
occurs between SA (which is a fluid phase with negligible in-plane and
interlayer positional correlations) and a three-dimensional long-range
translationally ordered 'liquid crystal' (e.g., crystal B or E). It has three-
dimensional sixfold bond-orientational order, short-range in-plane po-
sitional correlation and no interlayer positional correlation.
Fig. 5.7.2 presents the experimental results of Pindak et al.(170) on free
standing films. The /-scan profile in the SA phase is a diffuse ring
independent of the angle /- As hexatic ordering develops, the ring exhibits
a sixfold modulation, which eventually breaks up into six diffuse spots.
The degree of hexatic ordering can be defined quantitatively(171) by
fitting the /-scans to a Fourier series of the form

c
Six) = I0\c0+ t      *ncos6,2(90-/)l + /bg,             (5.7.1)

where Co is a constant and Ihg the background intensity. The coefficients
5.7 The hexatic phase                              361

(a)

Orientational long-range order, positional short-range order

0'

(b)          .

Orientational short-range order, positional short-range order
Fig. 5.7.1. (a) The structure of the hexatic phase of a two-dimensional lattice. The
orientation of the lattice vectors a and b is preserved over a long range, but the
molecular positions are correlated only over a short distance fp. (b) The
orientational order as well as the positional order are short range. (After Litster
and Birgeneau.(88))

C6w measure the degree of 6«-fold ordering. An application of (5.7.1) to
study the bond-orientational order across the Sj-S c transition(172) will be
described in §5.8.1.
The hexatic B-S A transition is expected to belong to the XY universality
class,(171) but high resolution calorimetric investigations(173) appear to
indicate that this may not be the case. Tilted analogues of hexatic B,
namely SF and SI5 have also been identified.(174)
362                        5. Smectic liquid crystals

4 -                                     (a)

7=68.1 °C

I o

15
(c)

10

r=63.9°C
ir/6
Rotation angle, x
Fig. 5.7.2. A 60° segment of a/-scan about an axis normal to the smectic layers of
the intensity of X-ray scattering from n-hexyl-4'-n-pentyloxybiphenyl-4-carb-
oxylate exhibiting the smectic A-hexatic B phase transition, (a) The scattering in
the A phase is a diffuse ring independent of/. For temperatures below the smectic
A-hexatic B transition the ring develops a sixfold modulation (Z>), which
eventually breaks up into six diffuse spots (c). (From Pindak et al.ai0))

5.8 Smectic C
5.8.1 Description of the structure
In S c the molecules are disordered within the layers, as in SA, but inclined
with respect to the layer normal. The structure has biaxial symmetry of the
monoclinic class, with a plane of symmetry as indicated in fig. 5.8.1. The
origin of the biaxiality may be explained as follows. The tilt angle 6 is
directly coupled with the layer thickness whereas the azimuthal angle <p is
not. Therefore, at any given temperature, the amplitudes of the 6
oscillations of the director are small compared with those of the (/>
5.8 Smectic C                             363

^ Symmetry plane
' Molecular axis
'///////////////////A
V///////////////////,
*        V////////////////A

Fig. 5.8.1. The structure of smectic C.

oscillations, with the result that the uniaxial symmetry about the mean
molecular direction disappears, and there exists only a plane of symmetry.
Because of this, and also because of possible anisotropic polarization field
effects, S c is optically biaxial. The optic axial angle is generally quite small
(-10°).
S c is often followed by SA at a higher temperature, and in such a case the
tilt angle decreases to zero gradually with rise of temperature. If the C-A
transition is of second order, the tilt 6 decreases smoothly to zero, whereas
if it is of first order 6 drops abruptly from a finite value to zero at the
transition point.
If S c undergoes a transition directly to the nematic phase, 6 is generally
found to be temperature independent and usually about 45°. According to
the Landau rules, the C-N transition can be continuous, but when
fluctuations(175) are taken into account it is predicted to be of first order.(176)
Experimentally, only first order C-N transitions have been observed.
Some compounds exhibit transitions from S c to the isotropic phase.
Interestingly, a slight increase of 6 with increasing temperature has been
reported for two such compounds.(177)
As in the SA phase, Sc lacks true long-range translational order because
of the Peierls-Landau instability. Theoretically, the existence of molecular
tilt implies that there must be a certain degree of bond-orientational order
in the S c phase. (171178) This has been verified experimentally by a high
resolution synchrotron X-ray study of the transition from S c to Sx in a
monodomain freely suspended film.(172) It will be recalled that Sx is a tilted
hexatic phase. The bond-orientational order parameter C6 (as defined in
(5.7.1), but appropriately modified to allow for the fact that the molecules
are tilted) is plotted as a function of temperature infig.5.8.2. The weak
bond-orientational order in S c evolves continuously into ST showing
364                        5. Smectic liquid crystals

"2
o

75           77             79            81
Temperature (°C)
Fig. 5.8.2. The hexatic order parameter C6 as a function of temperature in the S c
and Sj phases of racemic 4-(2-methylbutyl) phenyl 4 / -(octyloxy)-(l,r)-biphenyl-4-
carboxylate. The weak bond-orientational order in S c evolves continuously into Sx
showing that thermodynamically the two phases are not distinct in this compound.
(After Brock et a/.(172))

that thermodynamically the two phases are not distinct. Tilt angle
measurements (179) (using X-ray methods) have confirmed that there is
indeed a continuous evolution of S c into SI5 as the temperature is lowered.
Generally speaking, compounds exhibiting the S c phase have transverse
components of permanent electric dipole moments. A number of molecular
statistical models (including hard rod theories for systems composed of
oblique cylinders) have been developed. (1808) Goossens(189) has proposed a
model composed of ellipsoidal molecules with attractive interactions
arising from anisotropic dispersion forces as well as permanent quadrupole
moments. His calculations show that the interaction between the per-
manent quadrupole moments can produce a tilting of the molecules, but a
detailed comparison of the predictions with experimental data has yet to be
When the molecule has a strong longitudinal dipole moment, several
modifications of the S c structure have been identified. They are shown in
fig. 5.8.3, and as can be seen they are closely analogous to the SA phases of
polar molecules (fig. 5.6.1). However, the theoretical situation for S c is not
as clear as for SA. Another modification of S c is the chiral S c or the Sc*
phase which will be discussed in §5.10.
5.8 Smectic C                                 365

Fig. 5.8.3. The different forms of smectic C composed of polar molecules:
monolayer C1? bilayer C2, partially bilayer Cd and C phases, the last being similar
to C2 but with a transverse modulation of the structure. These are analogous to A1?
A2, Ad and A phases depicted in fig. 5.6.1.

5.8.2 Continuum theory of smectic C
We define a unit vector c to represent the preferred orientation of the
projection of the molecules on the basal (xy) plane. This vector is referred
to as the c-director. It is clear that c is somewhat analogous to the nematic
director n in a homogeneously aligned sample. For example, as already
pointed out, the orientational fluctuations of c (in the xy plane) can be
large, and the S c phase appears quite turbid in certain geometries. Also S c
exhibits schlieren textures similar to, though not exactly like, those seen in
a nematic. Again, in principle, a Freedericksz transition should be
observable in S c .
To construct a continuum theory of S c , we have to take into account
firstly, the orientational fluctuations of the director about the layer normal
(z axis), and secondly, as in smectic A, the distortions of the layers
themselves. Expressions for the former were given by Saupe, (190191) but the
complete theory including the latter contributions and the coupling
between the two was derived by the Orsay group.(192) We chose a cartesian
366                         5. Smectic liquid crystals

(arbitrary units)

\

Fig. 5.8.4. Plot of qxl* versus qzl^ from light scattering measurements in the
smectic C phase of di(4-n-decyloxybenzal)-2-chloro-l-4-phenylene diamine. The
wavevector q relative to the layers is indicated in the inset at the top left-hand
corner. The experimental points lie on an ellipse as expected theoretically. However,
the minor axis of the ellipse does not coincide exactly with the molecular axis,
which is assumed to be inclined at 45° to the layer normal. (After Galerne et a/.(193))

coordinate system such that the projection of the mean molecular direction
on the basal (xy) plane is along x. If the layer displacement along z is
represented by w, we observe that

du/dy = £lx,

where Q x and Qy represent rotations about the x and y axes respectively.
Therefore

(5.8.1)
5.8 Smectic C                               367
Making use of (5.8.1), and assuming the layers to be incompressible, the
free energy of elastic distortion may be written in the form

Here the A terms describe curvature distortions of the smectic planes, the
B terms the distortions of the director when the smectic planes are
unperturbed, and the C terms the coupling between these two types of
distortions. All the coefficients are approximately of the same order of
magnitude as the nematic elastic constants. A term of the type \B(<du/dz)2
may also be included to allow for the compression of the layers, but we
shall neglect it in the present discussion.
The fluctuations of the director are evidently related to fluctuations in Q.
From (5.8.2) and the equipartition theorem, we obtain for a general
wavevector q

Light scattering studies using monodomain samples of smectic C have
been reported by Galerne et al.(193) They have confirmed that the intensity
of scattering arising from the director fluctuations in the vertical
(scattering) plane (the ke, k'e configuration, e standing for extraordinary) is
extremely weak, whereas it is quite large due to fluctuations normal to the
scattering plane (the ke,k'o or ko,k'e configuration). Now, for the ke,k'o
configuration, we may write

(BlCos2 6 + B2sm20 + 2Blssin0 cos 6) q2I=B(6)q2I=         const,    (5.8.4)

where 6 = tan" 1 (qz/qx) and / the intensity. Thus a plot of qx I* versus qz I*
should give an ellipse. This has been verified to be true (fig. 5.8.4). The
damping time of these modes also shows a similar angular dependence.
From an analysis of the data Galerne et al have estimated that the
viscosity coefficients are an order of magnitude larger than for a typical
nematic.
Again, as in SA, the undulation mode (qz = 0) may be expected to make
an important contribution to the scattering.
The hydrodynamical properties of S c have been discussed by Martin,
5. Smectic liquid crystals

0)                                        (ii)
Fig. 5.8.5. Disclinations in the odirector field of smectic C. (a) 5 = 1 wedge
disclination with a radial configuration: (i) sink, (ii) source and (iii) meridian
section of (i). (b) s = 1 wedge disclination with a circular configuration: (i) vortex,
(ii) antivortex and (iii) meridian section of (i); the nails signifying that the director
is tilted with respect to the plane of the paper, (c) (i) s = — 1 wedge disclination and
(ii) s = 1 twist disclination.

Parodi and Pershan (MPP), (39) and by others.(194) Leslie, Stewart and
Nakagawa (195) have formulated a general non-linear theory, which in the
static case becomes identical with that proposed by the Orsay group. The
full implications of the dynamical aspects of this theory and its comparison
with the MPP theory have yet to be worked out.
5.8 Smectic C                                369

5.8.3 Defects in smectic C
Disclinations in the c-director field
As mentioned earlier, schlieren textures, very similar to those of a nematic,
are seen in planar samples of S c . However, since c and — c correspond to
opposite tilts, they are not equivalent; consequently, as explained in § 3.5.1,
only defects of integral strength (i.e., four brush disclinations) can occur in
S c . Moreover, as the c-director is confined to the xy plane, the usual escape
mechanism for nematic disclinations of unit strength is prohibited in this
case (see, however, §5.8.4).
The polarity of the c-director introduces some additional complications.
For example, the s = 1 disclination with a radial configuration can now
have two independent solutions, the source and the sink. Also, s = — 1 no
longer has four-fold symmetry (fig. 5.8.5). The structures are closely
similar to defects in ferromagnetic planar spin systems.
The c-director field can also have twist disclinations, the structures of
which are like those shown in fig. 3.5.8, except that the director is polar. (196)
The director pattern for s = 1, c = 0 is shown in fig. 5.8.5.
S c can have lattice disclinations as well, but they are perfect and
energetically favoured only along the twofold axis (which is normal to the
plane containing the layer normal and the c-director). (197) Focal conic
textures are also seen, though they are always accompanied by additional
singular lines of disclination, which arise because of the molecular tilt (fig.
5.8.6).
Single edge dislocations have been observed by simple optical micro-
scopic techniques(199) at temperatures near the C-A transition. Fig. 5.8.7
shows a photograph of an array of parallel dislocations in a sample
prepared in the form of a wedge. The reason for the appearance of these
lines is the following. We know that around an edge dislocation in SA there
is a stress field, the components of which are given by (5.4.6)-(5.4.8). On
one side of the dislocation line (the side having an extra half-plane) there
is compression, while on the other side there is dilatation. This can give rise
to a phase transition, as discussed in §5.8.4. The sample being at a
temperature close to the C-A transition point, one may expect that in the
region of compression the material has already undergone transition to S c ,
while in the region of dilatation it still remains S A. As the phase change is
accompanied by a molecular tilt it immediately reveals itself optically.
370                         5. Smectic liquid crystals

(a)                                         (b)
Fig. 5.8.6. Disposition of the molecules and layers in elliptical domains in (a)
smectic A and {b) smectic C. The mismatch of the molecular orientations in smectic
C gives rise to additional singular lines. (Bourdon, Sommeria and Kleman. (198))

Fig. 5.8.7. Edge dislocations in smectic C at a temperature very close to the smectic
C-smectic A transition. The sample, prepared in the form of a wedge, consists of
two domains of opposite molecular tilts. (Lagerwall and Stebler. (199))

5.8.4 The smectic C-smectic A transition
The order parameter for the C-A transition has two components and may
be written as(69)
).                    (5.8.5)
This at once brings out the analogy with the superfluid transition in
helium (the XY model).(103) Therefore, for T < TCA one expects the order
parameter
0=\¥\~\t\'9      /f = 0.35,
where t = (T- TCA)/TCA, and a = -0.007, y = 1.32 and v = 0.67, a, y and
v having the same meanings as in (5.5.27).
Though evidence of helium-like critical behaviour has been reported, (200)
high resolution experiments indicate that the transition is of the mean field
type. Safinya et al.(201) have argued that the bare correlation length of the
5.8 Smectic C                          371
tilt fluctuations is so large that the true critical region, as defined by the
Ginsburg criterion, should be very small and only a mean field behaviour
should be observable. Huang and Viner(202) and Birgeneau et al.(20S) have
demonstrated that this is, in fact, the case, and that the transition is well
described by a mean field expression but with an unusually large sixth
order term.
We write the free energy as
F= Fo + atO2 + b6* + c6% + . . . .                (5.8.6)
Minimizing with respect to 6
0 = 0,         T>TCA

K
3A1/2     -11/2
1--J         - l j , T<TCA,              (5.8.7)
where
R = (b/3cr\
t0 = b*/ac.
Substituting (5.8.7) in (5.8.6) and making use of the standard relation

we get
fC                              T>T
CA
C,=         "                                        (5.8.8)
l.C 0 H-^ii(i m —7 J        ,   i < i CA
where
A = ^3/2 /[2(3c) 1/2 r3/A2],

x = |^,       r>rCA.                           (5.8.9)
The parameter /0 represents the full width at half maximum of the excess
specific heat curve. It also represents the cross-over temperature from
mean field to tricritical behaviour, for
9 - t1/2, t < t0 (mean field)
6 ~ J1/4,     t > t0 (tricritical).
Fig. 5.8.8 presents the experimental data for the tilt angle, specific heat,
and the susceptibility (the last being measured from the magnetoclinic
effect, the magnetic analogue of the electroclinic effect, see §5.10.1) along
with the curves fitted to (5.8.7), (5.8.8) and (5.8.9). It is seen that the
agreement is very satisfactory.
372                           5. Smectic liquid crystals

1.0          0.6         -0.2     0       0.2                 1.0
T-TC{K)
Fig. 5.8.8. (a) Tilt angle 6 versus temperature (open circles and filled circles
representing two different determinations) in butyloxybenzylidene heptylaniline
(40.7). The solid line is a fit to (5.8.7) with t0 = 1.3 x 10"3. The triangles are the
reciprocal of the susceptibility / in arbitrary units measured for two different
samples, (b) Heat capacity near the C-A transition in 40.7. The dashed curve is the
background scaled from 40.8 and the solid line is the fit to (5.8.8). (After Birgeneau
et a/.(203))

C-A transition induced by layer compression
Ribotta, Meyer and Durand (204) observed that a compression applied
normal to the layers of SA induces a transition to S c when the stress exceeds
a threshold value. The effect is particularly easy to observe very near T CA.
If s is the strain normal to the layers, the energy density may be written as
F = \B(s2 + 2ocs92) + a62 + b0\                       (5.8.10)
where B is the elastic constant for compression. It is assumed that the
reduced layer thickness due to the tilting of the molecules is dil—aO 2),
where d is the normal layer thickness and a a constant which depends on
5.8 Smectic C                              373
the molecular axial ratio. For a second order C-A transition b > 0, and in
the high temperature phase a > 0, and 6 = 0. Clearly for a large enough
compression (i.e., large negative s), there is a finite tilt angle
0 = [-(a + asB)/2b]*                       (5.8.11)
when s is greater than a threshold value
sth = -a/a/l.                        (5.8.12)

C-A transition near the core of a disclination
Let us suppose that the C-A transition is continuous. Near TCA, i.e., for
small 0, one may write the free energy density as(69)

(5.8.13)
where a = oc0(T— TCA), k119 k22 and A:33 are the elastic constants for in-plane
distortions, nx = 0cos(f>9 ny = 0sin(/> and nz&\. Assuming the one-
constant approximation and that the distortions are such that there is no
z dependence of 0 and <fi, the equations of equilibrium become
\k[V20-0(V(f>)2]-(x0-2p03        =0            (5.8.14)
and
= 0.                  (5.8.15)
Equation (5.8.15) allows solutions of the type
cj) = ± Nco,

where co = tan -1 (y/x) and N is an integer. These describe the disclinations
of integral strength in the odirector (nx,ny) field. However, it is seen from
(5.8.14) that 0 depends on r, the distance from the centre of the singularity.
The variation of 0 with r (for \N\ = 1) is given by(205)

/ 2 ) = o            (5 8 i6)
'          - -
2   2
where / = 0/0o, £ = r/£0, r = (x +y )*, 0O is the tilt angle at r-> oo, and
£o = { — k/2(xf is the coherence length. Equation (5.8.16) is of the same
form as the Ginsburg-Pitaevskii equation for a quantized vortex filament
in a superfluid or a magnetic fluxoid in a type II superconductor. The tilt
angle drops to zero at the centre of the singularity, most of the variation
taking place over a distance of the order of <J0 from the centre (fig. 5.8.9).
374                       5. Smectic liquid crystals
This result appears to account for the experimental observation (206) that
there is an appreciable region near the centre of the + 1 singularity over
which 6 is very nearly zero, i.e., the material is smectic A-like.

5.9 The nematic-smectic A-smectic C multicritical point
The nematic-smectic A-smectic C (NAC) point is the point of intersection
of the N-A, A-C and N - C phase boundaries in a thermodynamic plane -
say, e.g. in the temperature (T) against pressure (P) or concentration (X)
diagram. It is termed a multicritical point because close to it all three phase
transitions are continuous, and at the point itself the three phases are
indistinguishable. (207) It has been observed in the T-X diagram of binary
liquid crystal mixtures(208) as well as in the P-T diagram of a single-
component liquid crystal system.(209) High resolution studies have been
made in both T-X(210) and P-T(209) planes, and it is now well established
that the topology of the phase diagram in the vicinity of this point exhibits
universal behaviour (fig. 5.9.1).
Another type of multicritical point, namely the re-entrant nematic-
smectic A-smectic C (RNCA) point has also been observed. In the first
studies,(211~13) it was noted that the singularities in the phase boundaries of
the kind seen near the NAC phase were conspicuously absent near the
point. However, it was demonstrated subsequently (214) that this was
because the RNAC point was far away from the N-I transition, so much
so that the Brazovskii fluctuations had a negligible influence near the
multicritical point and hence the critical region was unobservably small. By
investigating different binary systems and 'tuning' the experimentally
observable critical region it was shown that singularities do appear in the
immediate vicinity of the RNCA point as well. Analysis of the topology of
these phase diagrams confirmed that the RNCA point also exhibits the
same universal behaviour.
Earlier, Hornreich, Luban and Shtrikman (215) had proposed a new type
of higher order critical point, which they called the Lifshitz point, by
considering a Landau-Ginsburg free energy expansion in terms of a scalar
order parameter M for a magnetic system
F = ^ 2 M 2 + « 4 M 4 + « 6 M 6 + ... + c1(VM)2 + c2(V2M)2.   (5.9.1)
The phase diagram involves the ferromagnetic, paramagnetic and heli-
coidal phases, the last being the modulated form of the ordered phase
described by the gradient terms. The Lifshitz point occurs when the
coefficients a2 and cx vanish, and is characterized by the fact that the
5.9 The nematic-smectic A-smectic C multicritical point                375

Fig. 5.8.9. Variation of the superfluid order parameter with distance from the
centre of a quantized vortex filament as given by the Ginsburg-Pitaevskii equation
(5.8.16). The same curve describes the variation of the tilt angle in the core region
of the smectic C disclination.(206)

wavevector of the modulated phase increases from zero as the system
moves away from this point. Soon afterwards Chen and Lubensky(216)
predicted that the point at which the nematic, smectic A and smectic C
phases meet and become identical is a possible realization of a Lifshitz
point in a liquid crystal system. A noteworthy feature of their model is that
there is a single order parameter defined as

.8exp(ikT)/>(k)d8fc,                (5.9.2)
-/.    87T

where /?(k) is the Fourier transform of the centre of mass density p(r). The
domain of integration D takes into account the fact that the X-ray
scattering in the S c phase will be a ring defined by (qp q±cos <fi, q±sin (/>). The
general form of the free energy is written as

C2

+ D±[V2y/-n•        (n                       3
+ Fel d r,        (5.9.3)

where Fel is the Frank elastic energy for distortions of the nematic director.
The coefficients C , D and D± are positive and a = ao(T— 7^ A )/7^ A . The
type of fluctuations in the nematic phase is thus determined by the
coefficient C ± , i.e.,
if C ± > 0, SA-like fluctuations
if CL < 0, Sc-like fluctuations.
376                                 5. Smectic liquid crystals

D

4 -

3 -

<•>
N

U
1-                                                     A
/

X-
0 -
/DvM   MX   X*A*

-1

(a)
-2

_2
(4 data sets)

-1
l               1           1
*\  1         x\

151.4

150.6     -

149.8     -

(b)
275                       295       305          315           325

Pressure (bar)
Fig. 5.9.1. For legend see facing page.
5.9 The nematic-smectic A-smectic C multicritical point              311
Hence the CL = 0 point is a fluctuation cross-over point. The locus of such
points (a > 0, C± = 0) in a phase diagram, say the T-X diagram, gives the
Lifshitz line, and the point where both a and CL are zero is the Lifshitz
point.
The X-ray scattering intensity in the nematic phase can be obtained by
applying the equipartition theorem to (5.9.3):

k T

For C± > 0 this gives a peak at k = ±q]{ with a Lorentzian profile and
corresponds to SA-like fluctuations, and
q, = (q/2/),,)*.
For C± < 0, (5.9.4) can be rewritten as

k T

where the transverse wavevector
q\ = \C±\/2D±9
a = a'0(T-T
and

Hence, when C± is negative, I(k) has a maximum on the two rings
k = (±qpq±cos(/>,q±sm(/>) as expected for Sc-like fluctuations in the
nematic phase.
At the Lifshitz point 7^N = 7^N and C ± = 0, and (5.9.4) reduces to

Fig. 5.9.1. Topology of phase diagrams in the vicinity of the nematic-smectic
A-smectic C (NAC) multicritical point, (a) The temperature-concentration (T-X)
data for four binary liquid crystal systems: (1) 4-n-pentylphenylthiol-4 /-nonyl-
benzoate and 7S5, (2) 4-propionylphenyl-trans-(4-n-pentyl)cyclohexane carb-
oxylate (XC) and 4-n-octyloxyphenyl-4 /-n-octyloxybenzoate, (3) XC and 7S5 and
(4) 80CB and 2/?-n-heptyloxybenzylidene aminofluorenone (Brisbin et al.(210)); (b)
the pressure-temperature (P—T) data for a single component system: 4-n-
heptacylphenyl-4/-(4//-cyanobenzoyloxybenzoate). Analysis of the phase bound-
aries yields identical exponents for both the T-X and P—T diagrams showing that
the NAC point exhibits universal behaviour. (After reference 209.)
378                       5. Smectic liquid crystals
The transverse mass density fluctuations then become purely quartic. Thus
the model makes the important prediction that there will be a change from
Lorentzian to quartic in the scattering profile near the Lifshitz point.
All these predictions are generally borne out by high resolution X-
(208 22 2)
ray( 2i7-i9) calorimetric     ' °- and light scattering studies. (223"5)
Near the A-N transition, the present theory leads to the same
conclusions as de Gennes's model discussed in §5.5.2; the divergent
contributions to k22 and k33 in the nematic phase are given by (5.5.18) and
(5.5.19). Near the C-N transition, all three Frank constants diverge:

T

(5.9.7)

k T

At the Lifshitz point itself, there is a drastic change and Sk33 oc £, while dklx
and Sk22 oc In £.
Grinstein and Toner(226) applied the renormalization group technique
and presented a dislocation loop model for the NAC point. The most
striking conclusion of this theory is that a biaxial nematic phase N b should
intervene between the N and S c phases. Thus four phases, N, N b , SA and
S c , meet at a point giving rise to a tetracritical point topology. A
fluctuation corrected mean field theory developed by Lubensky(227)
supports this prediction, but points out that it may be difficult to detect the
presence of the N b phase through X-ray and calorimetric measurements.
All experimental studies but one have so far failed to detect the N b phase.
The one study is by Wen, Garland and Wand(228) whose high resolution
specific heat measurements have revealed anomalous variations near the
C-N transition very close to the NAC point, which they suggest may be
related to biaxial fluctuations. An attempt has been made to explain the
universal topology of the phase diagram in the vicinity of this multicritical
point.(258)
5.10 Ferroelectric liquid crystals
5.70.7 The properties of smectic C*
An interesting modification of smectic C is the Sc* phase, which has a twist
axis normal to the layers (fig. 5.10.1 (a)). The possibility of such a structure
being formed by the addition of optically active molecules to the ordinary
5.10 Ferroelectric liquid crystals                    379

\\L
11/111/111]                                      urn ii II
1111111111
miiiiui
Helical                  inn IIui
WWWY
pitch, p
mmmi
\
IIIIIIIIII
iiii/wmr
im/yini                                         mil! mi'
{a)                                            (b)
Fig. 5.10.1. (a) Helicoidal structure of the ferroelectric smectic C* phase, (b) a
' poled' sample with the helix unwound by an electric field applied normal to the
helical axis.

smectic C was envisaged by Saupe,(190) but the first identification of this
chiral phase in pure optically active compounds was by Helfrich and
Ok (229) J^Q optical properties of this structure are similar to those of a
cholesteric, though there are some obvious differences. In the cholesteric,
the local dielectric properties can be represented by an ellipsoid of
=
revolution with ea 4 sb = sc, the principal axis Oc being parallel to the
t
=    =
helical axis Oz. In Sc* the local dielectric ellipsoid is triaxial, ea = eb 4 sc,
with Oa making an angle 6 with Oz. As far as light propagation along the
twist axis is concerned, Sc* is identical to the cholesteric, but at oblique
incidence some additional features may be expected/ 230 ' 231)
Meyer et al.(232) demonstrated that the Sc* phase has ferroelectric
properties. The first studies were carried out on DOBAMBC (p-
decyloxybenzylidene-/7/-amino-2-methylbutylcinnamate) which shows the
following transitions

crystal -                         isotropic

Both Sc* and Sj* (the chiral forms of S c and Sx) were shown to be
ferroelectric.
380                        5. Smectic liquid crystals
In SA the molecules are upright, and since there is no head-to-tail
ordering (the director being apolar) there is no polarization normal to the
layers. Moreover, even if the molecules themselves are chiral, there is equal
probability of their assuming any orientation about their long axes. Hence
the transverse component of the dipole moment is averaged out and there
is no net polarization parallel to the layers.
In the Sc* phase, on the other hand, the molecules are tilted, and their
rotation about their long axes is biased. The symmetry plane of the
ordinary S c structure (fig. 5.8.1) is now absent because the molecules are
chiral. The only symmetry element that remains is a twofold axis parallel
to the layers and normal to the long molecular direction. This allows the
existence of a permanent dipole moment parallel to this axis. (Of course,
these arguments apply to the Sx* phase as well.)
Thus in Sc* each layer is spontaneously polarized. Since the structure has
a twist about the layer normal, the tilt and the polarization direction rotate
from one layer to the next (fig. 5.10.1 (a)). This implies that there is a
constant bend around the helical axis, which gives rise to a flexoelectric
contribution to the polarization.
When an electric field E is applied normal to the helical axis, the helix
gets distorted in a manner somewhat analogous to that depicted in fig.
4.6.1 for the cholesteric case. Above a critical field given by

where p is the pitch of the undistorted helix and k the 'twist' elastic
constant (which may be expected to vary rapidly with the tilt angle 6), the
helix is completely unwound and the sample is poled (fig. 5.10.1 (£)). The
molecules are then aligned in a plane perpendicular to E with a tilt 6 with
respect to the layer normal. When the field is reversed, the polarization P
reverses direction, and because of the coupling between polarization and
tilt, the molecular orientation switches from 6 to —6. As we shall see later,
this effect finds important applications in electro-optic display devices.
The value of the spontaneous polarization in these materials is quite
small, usually between 10 and 1000 nC cm" 2, i.e, about one or two orders
of magnitude less than that for a solid ferroelectric like KH 2 PO 4 .
The coupling between P and 6 manifests itself even above the C*-A
transition: an electric field induces a tilt in the A phase as well. This is called
the electro clinic effect, and was first demonstrated by Garoff and Meyer.(233)
Induced tilt angles as high as 10° have been observed in high polarization
materials. Due to its submicrosecond response and its linear dependence
5.10 Ferroelectric liquid crystals                      381
60   r

15
-r(K)
Fig. 5.10.2. Variation of the polarization P with relative temperature in the smectic
C* phase of DOBAMBC. The curve represents thefitobtained with the generalized
Landau theory. (After Dumrongrattana and Huang.(236))

15

Fig. 5.10.3. Variation of the tilt angle 6 and the ratio P/0 with relative temperature
in the smectic C* phase of DOBAMBC. The curves represent thefitsobtained with
the generalized Landau theory. (After Dumrongrattana and Huang.(236))

on the applied field, the electroclinic effect has proved to be useful in
making optical modulator devices.
The C*-A transition may be of first or second order/ 234 ' 235) The
polarization, tilt and pitch are, of course, temperature dependent and drop
to zero at the transition. The experimental curves for these three parameters
for DOBAMBC. which exhibits a second order C*-A transition, are
382                         5. Smectic liquid crystals
10

Is

f 5

5                10              15
TC.A-T(K)
Fig. 5.10.4. Variation of the pitch with relative temperature in the smectic C* phase
of DOBAMBC. (Ostrovskii et al.(237)). The curve represents the fit obtained with
the generalized Landau theory. (After Dumrongrattana and Huang. (236))

20   -

T(°C)
Fig. 5.10.5. The temperature variation of the field-induced tilt (or the electroclinic
effect) in the smectic A phase of 4-(3-methyl-2-chlorobutanoyloxy)-4 /-heptyloxy
biphenyl. (After Bahr and Heppke. (238))

shown in figs. 5.10.2, 5.10.3 and 5.10.4 and the variation of the electroclinic
effect with temperature in a high polarization material is shown in fig.
5.10.5. The static dielectric constant (for measuring field parallel to the
layers) increases slightly with temperature initially, but drops rapidly close
to 7^*A (fig. 5.10.6). From the dielectric relaxation one can study the two
important director modes, viz, the symmetry-recovering Goldstone mode
and the soft mode, in the vicinity of the C*-A transition. The Goldstone
mode is associated with the fluctuations of the azimuthal angle (<f>) of the
director, and the soft mode with the tilt (9) fluctuations. The former owes
its origin to the helical structure, and hence its frequency and strength
5.10 Ferroelectric liquid crystals                    3
300
0.3 kHz

240 -

180
~"           0.4 kHz

120
0.5 kHz

60 -

10 kHz
0
1                 i           i           i    i           i ,
-        3        -       2   -       1   0              1     2
r-r c . A (K)
Fig. 5.10.6. The dielectric constant for measuring field parallel to the layers as a
function of temperature in the smectic C* phase of [S]-4'-(2-chloro-4-methyl-
pentanoyloxy)phenylftYtfw-4//-n-decyloxycinnamate.The variation of the di-
electric constant with frequency is a consequence of the Goldstone mode relaxation.
(After reference 239.)

vanish in the SA phase (fig. 5.10.7(<z)). As for the soft mode, its frequency
decreases and strength increases on approaching the transition point from
either side (fig. 5.10.7 (b)). The coefficients of viscosity, y^ and y0, associated
with these modes are also strongly temperature dependent (fig. 5.10.8).
By virtue of their symmetry, ferroelectric smectics are piezoelectric.
Polarization can be induced by mechanical shear (2413) and, conversely, an
electric field can produce shear flow. They also possess pyroelectric
properties/ 244 ' 245)
An antiferroelectric smectic phase has been identified recently. (246) The
structure is expected to have an interlayer herringbone arrangement.

5.10,2 The smectic C*-smectic A transition
As the difference in the transition temperatures between the chiral and
racemic forms of the same compound is found to be only of the order of
1 °C, it is clear that the transition is driven by intermolecular forces
responsible for the tilted smectic phase and not by the dipole-dipole
384                                    5. Smectic liquid crystals
640
o       1.0
o o o 0 O Oo                                                 iI»                o
480                     oo                             OOQQ
DC
#                #
•             #
• •# •                                                    0.5
•
|320                      «.—

0
160 -                                                                                  1
%
(a) 0
-6
1                         1
-4         -2
1
1
0
T- TC*A (K)
0.16                                                                                        • 200
•        •

0.12                                                                                        o—      150
w                              o
•                                                     o

kHz)
•                   /                           o
< 0.08 -                      <                                         o                              100
o                     1

0.04                             V                      o°
o
-   50
o ooc, • 6>°
-       0
b) °      ~            1                i             1             1
-0.5                      0            0.5        1.0                   15
T--T         A(K)
Fig. 5.10.7. The variation of the frequency (open circles) and dielectric strength
(filled circles) of (a) the Goldstone mode and (b) the soft mode in the vicinity of the
C*A transition. Material same as in Fig. 5.10.6. (After reference 239.)

10

fe, 1.0

0.1

90                                           80                                            70
T(°C)
Fig. 5.10.8. The temperature variation of the Goldstone mode and soft mode
viscosity coefficients, y^ and y0, in the smectic C* phase of DOBAMBC. (After
Pozhidayev et a/.(240))
5.10 Ferroelectric liquid crystals                  385
(ferroelectric) coupling. Liquid crystalline ferroelectrics may therefore be
classified as 'improper' ferroelectrics: the primary order parameter is the
tilt (0) and not the polarization (P). Meyer (247) proposed a simple
phenomenological model that accounts for some of the observed phenom-
ena.
We shall suppose that the transition is of second order, and consider
first the spatially homogeneous case (i.e., the helix unwound completely,
q = 27r/pitch = 0), ignoring the effect of the spontaneous helical torsion
that appears below the transition. The free energy may then be written
as (247)

^eErj0P,               (5.10.2)
oil

where A = a(T— TCA). The first two terms are the leading terms of the usual
Landau expansion, the next three terms describe the electrostatic free
energy and the last term expresses the coupling between P and 6.
Minimizing with respect to P and 0,
P = X(E+r,6),                          (5.10.3)
e = t,XE/A,                           (5.10.4)
and hence,

^                ±^!iy                 (5.10.5)
We may draw the following conclusions: (a) from (5.10.5), we see that
the P—B coupling produces a divergent component in the dielectric
constant, (b) from (5.10.3), that polarization can be induced by shear stress
in the absence of an electric field; this is the piezoelectric effect discussed
earlier, and (c) from (5.10.4), that the field-induced tilt, or the electroclinic
effect in the SA phase should diverge as the temperature approaches 7^*A.
However, this treatment is inadequate in many respects. For example,
fig. 5.10.5 shows that the dielectric constant does not exhibit a divergence
at all on approaching TC*A. In order to give a better description of the
properties of the Sc* phase, Pikin and Indenbom (248) proposed the following
free energy expression:

F = \a92 + \b6* + \kq292 - A02q + ^- P2-juP9q - CP6.       (5.10.6)

Here a = ao(T— To), To being the transition temperature for the cor-
responding racemate, b > 0 and is temperature independent, k the elastic
constant, s the high temperature dielectric constant (i.e., in the absence of
ferroelectricity), ju and C are the constants describing respectively the
386                       5. Smectic liquid crystals
flexoelectric and piezoelectric types of coupling between P and 0; A is
called the Lifshitz invariant parameter characteristic of chiral molecules
and produces the helicoidal structure. Minimizing with respect to q, P
andfl,
A + suC
q = — -^r- = constant,
2
k-sju

(5.10.7)

The main predictions of the model are: (i) the transition temperature of the
chiral compound is higher than that of the racemate, (ii) the tilt angle 0 and
hence P (which is proportional to 6) exhibit a power law variation with
temperature with the mean field exponent of 0.5, and (iii) the pitch and the
ratio P/0 are independent of temperature.
While this theory is an improvement over the previous one, its
predictions are still not quite in agreement with observations. The exponent
governing the 6 variation is generally found to be smaller than the mean
field value. Further, the theory is unable to account for the non-monotonic
temperature variation of the pitch and the anomalous behaviour of P/0.
More elaborate models have been proposed(249236) involving additional
phenomenological coefficients, including a sixth order term in 0 and a term
involving P292. A 'generalized' expansion of the form
F = \aO2 + \b6* + \c6« - AqO2 + \Kzq202

+ ^P2-HqP6-CP6-\ClP262            + \riP*-dq0*     (5.10.8)

has been shown to be necessary to give a satisfactory explanation of the
various properties of the ferroelectric phase(236'250) (see figs. 5.10.2. 5.10.3
and 5.10.4).

5.10.3 Applications of ferroelectric liquid crystals
Smectic C* is proving to be of great practical importance as a material for
fast electrooptical switching.(251) The principle of operation of this switch
can be readily understood on the basis of the following simplified model.
The ferroelectric liquid crystal is taken in a thin cell, usually about 2-3 jum
thick. As long as the cell thickness is less than the pitch of the helix, the
strong surface forces align the molecules parallel to the substrate and the
5.10 Ferroelectric liquid crystals                    387

\\\\\\\\\\\\\\\\\\\\\\\                  11 ii mi mini mi
wwwwwwwwwww
\\\\\\\\\\\\\\\\\\\\\
wwwwwwwwwwww
Fig. 5.10.9. The SSFLC cell: The bookshelf geometry of a thin film of smectic C*
sandwiched between two glass plates (a) field ' up' (normal to the plane of the
diagram) and (b) field 'down'.

helix is unwound. The molecules are then arranged in the so-called
'bookshelf geometry, all inclined one way at an angle of about 20° with
respect to the layer normal as illustrated in fig. 5.10.9 (a). The spontaneous
polarization of every layer will therefore point in the same direction (say
upwards). If now a reverse electrical pulse is applied across the film, the
polarization will switch over to the opposite direction (downwards) and
the molecular tilt will change by 26 ~ 40° (fig. 5.10.9 (b)). By using a pair
of polarizers and adjusting the cell thickness so that, in effect, the liquid
crystal film becomes a half-wave plate, high optical contrast can be
achieved. The remarkable feature of the device - which is known as the
SSFLC (surface stabilized ferroelectric liquid crystal) device - is that the
switch on and switch off times are just a few tens of microseconds, i.e.,
about 1000 times faster than the twisted nematic device. For a given cell
thickness, the material parameters that determine the switching speed are
the polarization and viscosity. The system is bistable and has memory as
well.
However, in practice, it is found that the molecules adopt a chevron type
of arrangement(252) rather than the bookshelf geometry and this, of course,
affects the contrast ratio slightly. The mechanism of reorientation from
one stable state to the other on reversing the voltage is a complex one
involving the formation of defects.(252) Though the SSFLC cell is more
difficult to fabricate than the twisted nematic or supertwisted nematic cell,
its very fast switching time makes it extremely useful for large area, high
information content displays.(253~6)
Another phenomenon that has potential applications (257) is the field-
induced tilt or the electroclinic effect. Unlike the SSFLC device, this effect
does not possess bistability but it has a faster (submicrosecond) response.
By using the same bookshelf geometry and a suitable polarizer and
retarder arrangement, the electroclinic effect can be used for modulating a
light signal with a transmitted intensity linearly proportional to the applied
voltage or as a tunable colour filter.
6
Discotic liquid crystals

6.1 Description of the liquid crystalline structures
Since the early investigations of Lehmann(1) and others(2) elucidating the
relationship between liquid crystalline behaviour and chemical consti-
tution, the accepted principle was that the molecule has to be rod-like in
shape for thermotropic mesomorphism to occur, but it has emerged during
the last decade that compounds composed of disc-like molecules may also
exhibit stable mesophases. (310) Some typical discotic molecules are shown
in fig. 6.1.1. Generally speaking, they have flat (or nearly flat) cores with six
or eight (or sometimes four) long chain substituents, commonly with ester
or ether (but rarely more complex) linkage groups. Available experimental
evidence indicates that the presence of these side chains is crucial to the
formation of discotic liquid crystals. The mesophases whose structures are
clearly identified fall into two distinct categories, the columnar and the
nematic. A smectic-like (lamellar) phase has also been reported but the
precise arrangement of the molecules in each layer is not yet fully
understood.
The basic columnar structure is as illustrated in fig. 1.1.8(a); it is
somewhat similar to the hexagonal phase of soap-water and other
lyotropic systems (fig. 1.2.2). However, a number of variants of this
structure have been found. Fig. 6.1.2 presents the different two-di-
mensional lattices of columns that have been identified; here the ellipses
denote discs or, more precisely, cores (216) that are tilted with respect to the
column axis. Table 6.1.1 gives the space groups of the columnar structures
formed by some derivatives of triphenylene. (These are planar space groups
that constitute the subset of the 230 space groups when symmetry elements
relating to translations along one of the axes, in this case the column axis,
are absent.)

388
6.1 Liquid crystalline structures                      389

RO

RO         I = Cu                                          M = Cu

(g)                                             (h)
Fig. 6.1.1. Examples of disc-shaped mesogens: (a) hexa-n-alkanoates of benzene,(3)
(b) hexakis ((4-octylphenyl)ethynyl)benzene,(11) (c) hexa-n-alkanoates of scyllo-
inositol,(12) (d) hexa-n-alkanoates of triphenylene and hexa-n-alkoxytripheny-
lene, (1314) (e) hexa-n-alkyl and alkoxybenzoates of triphenylene, (1516) (/) hexa-n-
alkanoates of truxene, (1718) (g) bis(3,4-nonyloxybenzoyl)methanato copper(II),(19)
(h) octasubstituted metallophthalocyanine.(20)

High resolution X-ray studies have been reported on the columnar
phases of a few compounds. The measurements were made on very well
oriented monodomain samples obtained by preparing freely suspended
liquid crystal strands, typically about 200 jum in diameter and 1.5-2 mm
390                        6. Discotic liquid crystals

Table 6.1.1. Columnar structures formed by some derivatives of
triphenylene^

Lattice           Temperature
R                             Space group      parameters (A)           (°C)
C5HUO                         P6 2/m 2/m     a=   18.95                 ^80
C 7 H 15 O                    P6 2/m 2/m     a=   22.2                  ^80
C 8 H 17 O                    P6 2/m 2/m     a=   23.3                  ^80
C U H 23 COO                  PVa            a=   44.9, b   = 26.4       117
P6 2/m 2/m     a=   26.3                   105
C 7 H 15 COO                  P^/a           a=   37.8,/?   = 22.2       100
P 2x/a         a=   51.8, 6   = 32.6       165
QH^-O-C.H.-COOf               C2/m           a=   30.7, 6   = 28.4       185

f The C 6 H 4 groups are para substituted.

(b)

id)                      (e)
Fig. 6.1.2. (a)-(e) Columnar phases of disc-shaped molecules.(6) Plan views of the
two-dimensional lattice; ellipses denote discs that are tilted with respect to the
column axis: (a) hexagonal (P6 2/m 2/m); (b) rectangular (P2 1 /a); (c) oblique
(P^; (d) rectangular (P 2 /a); (e) rectangular face-centred, tilted columns (C2/m).
(/) The nematic phase.

long, with the column axis parallel to the axis of the strand/ 2 2 6) The results
of these studies are summarized below.
(a) The correlation length of the two-dimensional lattice of the
hexagonal columnar phase of (C 1 3 H 2 7 COO) 6 -truxene (fig. 6.1.1 ( / ) ) is
greater than 4000 A (or approximately 200 columns), the lower limit being
6.1 Liquid crystalline structures                     391

set by the instrumental resolution. (25) Long-range order of the two-
dimensional lattice is, in fact, to be expected theoretically (see §6.3.1).
Within each column, on the other hand, the flat molecular cores form an
oriented, one-dimensional liquid, i.e., they are orientationally ordered but
translationally disordered. In contrast the flexible hydrocarbon chains are
highly disordered and produce a nearly isotropic scattering pattern. This
striking difference between the ordering of the cores and the tails is also
seen in the X-ray scattering from the columnar mesophases of triphenylene
compounds (fig. 6.1.1 (d)), but is not so apparent in hexa-n-octanoate of
benzene (fig. 6.1.1 (a)), which has a much smaller core. (2324)
Deuterium NMR spectroscopy of the disco tic phase of hexa-n-hexyloxy
triphenylene has led to similar conclusions. (27) Spectra of two selectively
deuterated isotopic species, one in which all aromatic positions are
substituted and the other in which only the a-carbon side chains are
substituted, bring out the difference between the order parameters of the
cores and the tails. Fig. 6.1.3 gives the quadrupole splittings of the
aromatic and the a-aliphatic deuterons versus temperature in the meso-
phase region. It is seen that the rigid core is highly ordered, the
orientational order parameter s ranging from 0.95 to 0.90, whereas the
a-aliphatic chains are in a disordered state.
These studies emphasize the fact that any realistic theory of the statistical
mechanics of discotic phases cannot treat the molecules as rigid discs, but
has to take into account the conformational degrees of freedom of the
hydrocarbon chains.
(b) Triphenylene hexa-n-dodecanoate exhibits two columnar phases,
designated as D h and D r , the subscripts h and r standing for hexagonal and
rectangular respectively. The D h -D r transition, which is weakly first order,
is associated with a small distortion of the lattice, consistent with a
herringbone arrangement in the rectangular structure with only the core of
the molecule tilted with respect to the column axis. However, high resol-
ution synchrotron X-ray studies on monodomain discotic strands (2324)
have established that the tilt of the molecular core persists in the D h
phase as well, except that the tilts in neighbouring columns are now
rotationally uncorrelated, i.e., they are free to assume different azimuthal
angles with reference to the column axis (neglecting possible orientational
short-range order). Thus the D h -D r transition may be looked upon as an
orientational order-disorder transition.
(c) Hexa-hexyl thiotriphenylene (fig. 6.1.1 (d) with R = SC 6H13) shows
a transition from an hexagonal 'ordered' phase to an hexagonal 'dis-
ordered' one. The former is a phase in which there is regularity in the
392                        6. Discotic liquid crystals

0.40
10         0
rc-r(°c)
Fig. 6.1.3. Quadrupole splittings for the aromatic v™ and a-aliphatic VQ deuterons
of deuterated hexa-n-hexyloxytriphenylene (THE6) as functions of temperature
(T— Tc) in the mesophase region, where Tc is the mesophase-isotropic transition
point. The open circles correspond to measurements on neat THE6-ard 6 and
THE6-ad 12 separately, while the filled circles correspond to a 2:1 mixture of the
two isotopic species. The scale on the upper right-hand side gives the orientational
order parameter of the aromatic part. The curve at the bottom gives the ratio of the
quadrupole splittings for the a-aliphatic and aromatic deuterons (Goldfarb, Luz
and Zimmermann(27)).

stacking of the triphenylene cores in each column, and the latter one in
which the column is liquid-like. X-ray studies using freely suspended
strands,(26) have shown that in the ordered phase there is a helicoidal
stacking of the triphenylene cores within each column, the helical period
being incommensurate with the intermolecular spacing.(28) In addition, a
three-column superlattice develops as a result of the frustration caused by
molecular interdigitation in triangular symmetry. It may be argued that if
there is no intercolumn interaction true long-range translational order
6.1 Liquid crystalline structures                     393
RO                  .OR

OC

R              R0-    -C-N                     -OR
0                 0
.CO
R =C9H19COO-
R                                        RO                  OR

R
=C12H25°-          -coo-
(a)                                         (b)
Fig. 6.1.4. (a) Cyclotricatechylene hexaesters, the cores of which are cone-shaped
(Malthete and Collet(29)), (b) hexa-(/?-n-dodecyloxybenzoyl) derivative of macro-
cyclic polyamines which is hollow at the centre (Lehn, Malthete and Levelut(33)).
Both (a) and (b) show columnar mesophases; the latter mesophase has been
described as 'tubular'.

cannot exist within a column because of the Peierls-Landau instability.
The existence of a regular periodicity in the stacking in each column
therefore implies that neighbouring columns must be in register. Thus,
ordered columnar phases, of which several have been reported, can
probably be compared with the highly ordered smectic phases of rod-like
molecules, e.g., smectics B, E, G, H, etc., which possess three-dimensional
positional order. However, more high resolution studies are necessary
before general conclusions can be drawn about the true nature of these
phases.
Columnar mesophases are also formed when the flat core of the
molecule is replaced by a conical one (2932) as in the cyclotricatechylene
hexaesters (fig. 6.1.4(a)). With macrocyclic molecules, which are hollow at
the centre (fig. 6.1.4(6)), the columns are in the form of tubes; these
mesophases have been described as ' tubular \ (33)
The nematic phase (N D) is exhibited by relatively few compounds;
examples are hexakis((4-octylphenyl)ethynyl)benzene (fig. 6.1.1(6)) and
the hexa-n-alkyl and alkoxybenzoates of triphenylene (fig. 6.1.1 (e)). The
N D phase has an orientationally ordered arrangement of the discs with no
long-range translational order (fig. 6.1.2(/)). Unlike the usual nematic of
rod-like molecules, N D is optically negative, the director n now representing
the preferred axis of orientation of the disc normal. The properties of this
phase will be discussed in greater detail in §6.5. A twisted nematic (or
cholesteric) phase, with the helical axis normal to the director, has also
been identified.(34)
394                      6. Discotic liquid crystals
The hexa-n-alkanoates of truxene (fig. 6.1.1 (/)) show an unusual
sequence of transitions. For the higher homologues the phase sequence on
cooling is: isotropic->hexagonal columnar->rectangular columnar^
discotic nematic-> reentrant hexagonal columnar^crystal/ 1 7 ' 1 8 ) It has
been suggested that the truxene molecules are probably associated in pairs,
and that these pairs break up at higher temperatures, which might explain
this extraordinary behaviour. However, this conjecture still remains to be
proved.
i?w(/?-n-decylbenzoyl)methanato-copper(II), which is similar to the
molecule depicted in fig. 6.1.1 (g), but with four chains instead of eight, has
been reported to exhibit a smectic-like lamellar mesophase. (357) A tilted
smectic C type of structure has been proposed, (36) but the disposition of the
molecules in each layer does not appear to have been resolved. It is worth
noting that these copper complexes were the first paramagnetic mesogens
to be synthesized.
Molecules which combine the features of the rod and the disc may be
expected to form new types of mesophases. (38) An example is the biaxial
nematic phase reported in thermotropic systems (see §6.6). Malthete
ettf/.(39>40)have prepared an interesting series of mesogens shaped like
stick insects called ' phasmids' (fig. 6.1.5 (a)). Some of them form columnar
mesophases; the structure proposed for the hexagonal phase is shown
schematically in fig. 6.1.5(6).

6.2 Extension of McMillan's model of smectic A to discotic liquid
crystals
Transitions between the columnar (D) and the nematic (N D) phases occur
in some compounds. Available data for a few homologous series of
compounds indicate that the lower members of the series show only the N D
phase, the next few members show both D and N D phases, the temperature
range of the latter phase diminishing with increasing chain length, while for
the higher members D transforms directly into the isotropic (I) phase.
Broadly the trend is reminiscent of the behaviour of the smectic
A-nematic-isotropic sequence of transitions in systems of rod-like
molecules. This suggests that one should be able to give a qualitative
description of the D-ND—I transitions by extending McMillan's mean field
model of smectic A (see §5.2) so that the density wave is now periodic in
two dimensions. (413)
The hexagonal phase can be described by a superposition of three
density waves with wavevectors
6.2 Extension of McMillan's model                      395

H25C,2C>      O                                    O

12H25

Fig. 6.1.5. (a) Hexa-n-alkoxy terepthal-to-(4-benzoyloxyaniline) and (b) the
structure of the hexagonal columnar mesophase formed by this compound. The
mesophase has been described as 'phasmidic'. (Malthete, Levelut and Tinh.(39))

4TT   .
V 3d
V3
R             \

C = A + B,
where d is the lattice constant.
The appropriate single particle potential, which depends on the
orientation of the short axis of the molecule as well as the position r of its
centre of mass, may be written in the mean field approximation as
r)         r)
V^x, y9 cos 0) = -Vo P2(cos 8) {s + a<r[cos (A • + cos (B • + cos (C •r)]},
(6.2.1)
retaining only the leading terms in the Fourier expansion. Here Vo is the
396                      6. Disco tic liquid crystals
interaction energy which determines the nematic-isotropic transition, a is
the McMillan parameter given by
2exp[-(27rro/V3</)2],                       (6.2.2)
r0 being the range of interaction which is of the order of the size of the
aromatic core,                s   = </»(cos0)>                         (6.2.3)
is the usual orientational order parameter, and
+          +               2(cos
a = |<[cos (A • r) cos (B • r) cos (C • r)] i> 9)}            (6.2 A)
is an order parameter coupling the orientational and the translational
ordering. The angular brackets represent the statistical average over the
distribution function derived from the potential (6.2.1), the spatial
integrations being carried out over a primitive cell of the hexagonal lattice.
This form of the potential ensures that the energy of the molecule is
minimum when the disc is centred in the column with its plane normal to
the z axis.
The free energy can then be calculated using standard arguments:

(6.2.5)

Numerical calculations yield three possible solutions to the equations:
(1) s = a = 0       (isotropic phase),
=
(2) s 4 0, a = 0 (nematic phase),
(3) s 4 0,(7 41 0 (hexagonal columnar phase).
=
For any given values of a and T, the stable phase is the one which
minimizes the free energy (6.2.5). Interpreting a as a measure of the chain
length as in McMillan's model, the phase diagram as a function of a (fig.
6.2.1) reflects the observed trends for a homologous series of compounds.
The temperature range of the nematic phase decreases with increasing a,
and for a > 0.64 the D phase transforms directly to the isotropic phase. It
should be noted that since the wavevectors of the hexagonal lattice satisfy
the relation A + B — C = 0, the columnar-nematic transition is always of
first order: a Landau expansion of the free energy contains a non-
vanishing cubic term in the order parameter o.
The theory has been extended using a variational principle to solve the
problem with the full potential rather than the one truncated to the first
Fourier component.(44) This method, which is closely analogous to the
extension of McMillan's model by Lee ettf/.,(45)leads to some qualitative
6.2 Extension of McMillan's model                      397

Isotropic
1.0

I0-8               Nematic

I
c

Columnar
I
0.4

I                    J
0        0.2         0.4           0.6      0.8
Model parameter (a)
Fig. 6.2.1. Theoretical plot of the reduced transition temperature against the model
parameter a showing the hexagonal, nematic and isotropic phase boundaries. All
the transitions are of first order.

improvements in the phase diagram for a homologous series. A simpler
mean field version of this theory has been presented which yields essentially
similar results.(46)
For the rectangular lattice, the nature of the phase diagram depends on
the axial ratio b/a.m) When b/a is only slightly different from y/3, the
phase diagram is similar to that for the hexagonal structure. As the
asymmetry of the lattice is increased, one of the density waves disappears
before the other to give rise to a smectic (or lamellar) phase, which, in turn,
transforms to the N D phase at a higher temperature. Evidence of a smectic-
like disco tic phase has been reported, (357) but whether this corresponds to
the lamellar phase predicted by theory is a matter for further study.
As mentioned in §6.1, transitions are observed between different types of
columnar phases. In particular, the transition from the rectangular (D r ) to
the hexagonal (Dh) structures has been established experimentally to be an
orientational order-disorder transition. The tilts of the molecular cores
with respect to the column axis in neighbouring columns are ordered or
correlated to give rise to a rectangular packing with a herringbone
structure in the D r phase, whereas they are free to assume different
398                         6. Discotic liquid crystals
azimuthal angles with reference to the column axis in the D h phase.
Theoretical models have been developed to describe this transition. (479)

6.3 The columnar liquid crystal: applications of the continuum theory
6.3.1 Fluctuations
We now discuss the fundamental question of fluctuations in the columnar
phase. Let us suppose that the liquid-like columns are along the z axis and
that the two-dimensional lattice (assumed to be hexagonal) is parallel to
the xy plane. The two basic deformations in such a structure are (i) the
curvature deformation (or bending) of the columns without distortion of
the lattice and (ii) lattice dilatation (or compression) without columnar
curvature. There can also be coupling between the two types of distortion
but, as shown by Kleman and Oswald,(50) the coupling term merely rescales
the bend elastic constant of the columns. We shall consider only the
vibrations of the lattice in its own plane. The free energy may be written

2 \ 9x   dy )       2 \_\ dx   dy

where B and D are the elastic constants for the deformation of the two-
dimensional lattice in its own plane; ux and uy are the displacements along
x and y at any lattice point, and /r33 is the Frank constant for the curvature
deformation (bending) of the columns. (In the notation of the standard
crystal elasticity theory, B = | ( c n + c12) and D = \{clx — c12).) We neglect
here the splay and twist deformations because they give rise to a distortion
of the lattice and involve considerable energy. (51) We also neglect any
contributions from the surface of the sample. Writing the displacement u
in terms of its Fourier components,
H(r) = X>(q)exp(iq-r);                     (6.3.2)
Q

substituting in (6.3.1), we get, in the harmonic approximation

and from the equipartition theorem
<ul)=kBT/(

where Bo = B + 2D,k9 = 2k3S and qL =
6.3 The columnar liquid crystal                      399

The mean square displacement at any lattice point is given by

where L is the length of the columns, L the linear dimension of the lattice
in the xy plane and d its periodicity. Assuming that U > L
<W2> = [kB T/4B£Xd\$\ [l-(d/L%                       (6.3.3)
(5>52>53)
where X = (ko/Bo)* is a characteristic length.               The structure is
therefore stable as L -> oo. As is well known from the classical work of
Peierls(54) and Landau (55) , the two-dimensional lattice itself is an unstable
system with <w2> diverging as lnL. ( 5 6 5 7 ) Hence the curvature elasticity of
the liquid-like columns stabilizes the two-dimensional order in the
columnar liquid crystal. This result was first proved by Landau, who
observed: ' Thus bodies having such a structure could in theory exist, but
it is not known whether they do in fact exist in Nature.' (58)

6.3.2 X-ray scattering
The ' structure factor' for the intensity of X-ray scattering may be written
as
5(K)=

The second exponential term on the right-hand side is the familiar
Debye-Waller factor exp(— W). Now

where p = (x2+y2)* and U is the confluent hypergeometric Kummer
function.(59) For z > (Ap)*9 the second term of (6.3.4) varies as

(6.3.5)
400                         6. Discotic liquid crystals
Thus, in contrast to smectic A and the two-dimensional lattice, for which
the displacement-displacement correlation is of logarithmic form, the
columnar liquid crystal gives the usual Bragg reflexions. As noted in §6.1,
this is borne out by high resolution X-ray experiments on well oriented
monodomain discotic strands.(25)
So far it has been assumed that the length (Z/) of the liquid columns
is much larger than L. One may similarly consider the opposite situation
Z/ <^ L. In this case, it turns out that for the bounded sample

const.    for very small L

— In Z/   for larger L.

If the surfaces are free <V> will have additional terms, which may be
analogous to those for a two-dimensional lattice. In any case, it is clear that
the mean square fluctuations of the lattice as well as the Debye-Waller
factor may be expected to show a certain dependence on the linear
dimensions of the sample.

6.3.3 Light scattering
The mean square amplitude of the orientational fluctuations of the director
and the corresponding intensity of light scattering may be worked out in a
similar fashion. For example, when the incident and scattered beams are
both polarized in the plane perpendicular to the optic axis (or column axis)
the scattered intensity is given by

Tco*

where q± = (ql + qffi and co is the angular frequency of the light.(41) The
scattering is similar to that from smectic A in certain geometries (see
§5.3.4), but is, of course, small compared to that from a nematic.

6.3.4 Mechanical instabilities
If the elasticity theory of the columnar phase is developed to the next
higher order, one finds that ux x, uxy and uz z of the linear theory have to
be replaced as follows:
U
x,x^Ux,x     +        \Ul,x~\Ul,z~Ux,zUz,x
Ux, y ~> I K , y + Uy x) - \{UX xUyx + Ux yUyy    + Ux z + Uy z)
6.3 The columnar liquid crystal                       401

where ux x = dux/dx, ux y = dux/dy, etc. As a result, a lattice strain like ux x
gets coupled to ux z which represents the tilt of the columns. This type of
coupling is present in smectic A as well and gives rise to an undulation
instability which has been studied in detail (see §5.3.3). An equivalent type
of instability may be expected to occur in the columnar liquid crystal as
well and can be analysed exactly as in the case of smectic A. (50) The
columns are parallel to the glass plates, and the separation (d) between the
plates is increased (see fig. 5.3.4). If ux = ax, (a being positive) then at a
critical value a = ac given by

kS3   i

a spatially periodic distortion should set in with a wavevector qz given by

Another type of instability arises from the coupling between uz z and ux z
or uy z. This may be described as a buckling instability. The analysis is very
similar except that a compressive stress is applied parallel to the columns.
At a critical value of the stress given by

the columns buckle. This has actually been observed, (60) but there is a
significant discrepancy between theory and experiment which has not been
explained satisfactorily.

6.3.5 Acoustic wave propagation
The basic hydrodynamic equations for the hexagonal columnar phase

w            (
l T+ «8^
402                         6. Discotic liquid crystals

dt
dK
At
dK

6e

where
8

^2             ^2       ^2         ^2

A              +        8

2 = heat density; 9 = volume strain; p = density; Vi = components of the
velocity; KpK± = thermal conductivity parallel and perpendicular to the
column axis; ut = components of the column displacement; P = pressure;
e = energy density expressible as F-\~\A62 + C6(ux x-\-uy y), where F is
given by (6.3.1), A is the bulk modulus and C a coupling constant; C =
permeation coefficient; £ = thermomechanical coefficient; rj ijkl = viscosity
coefficients.
In the absence of a temperature gradient, and assuming that permeation
is negligible, i.e., u = Vi9 and that damping is weak, it turns out that under
adiabatic conditions there are three propagating acoustic modes for any
arbitrary direction not coinciding with an axis of symmetry. Of these, one
is the familiar longitudinal wave arising from density fluctuations, the
velocity (Fx) of which is practically independent of the direction of
propagation. Another is a transverse wave which is exactly analogous to
the second sound of smectic A (see §5.3.6). The velocities Vl and V2 of these
two waves are given by the roots of the equation

where y/ is the angle between the direction of propagation and the column
axis. We have ignored the coupling between the lattice and the curvature
deformation of the columns. The third mode is a transverse wave whose
polarization is orthogonal to that of second sound. This wave propagates
6.4 Defects in the columnar liquid crystal                    403

Fig. 6.3.1. Dependence of the sound velocities on the polar angle y/ in the columnar
liquid crystal.

because the two-dimensional lattice can sustain a shear. Its velocity for any
arbitrary direction is given by

K3 = (/)//>)* sin <p.

Fig. 6.3.1 depicts the variation of the velocities of the three waves with
polar angle. This angular dependence can be studied by Brillouin
scattering, as has been demonstrated for smectic A (see fig. 5.3.11). In the
present case, there should be one, two or three pairs of Brillouin
components depending on the orientation, but no experiments have yet
been reported.

6.4 Defects in the columnar liquid crystal
The symmetry elements in the hexagonal structure composed of liquid-like
columns are shown in fig. 6.4.1. The director n is parallel to the column
axis; a (or equivalently a' or a") is the lattice vector of the two-dimensional
hexagonal lattice; L2, T2 and 92 are two-fold axes of symmetry; L 3 is a
three-fold axis and L6 a six-fold axis. Any point on L2 or L6 is a centre of
symmetry. Any planes normal to n and the planes (L6, T2) and (L6,62) are
planes of symmetry. Defects in such a structure have been investigated in
detail by Bouligand(61) and by Kleman and Oswald.(50)
404                         6. Discotic liquid crystals

Fig. 6.4.1. The hexagonal columnar phase with the symmetry elements in the
structure.

Fig. 6.4.2 gives photographs of the columnar liquid crystal appearing in
the isotropic phase as the sample is cooled slowly. The growth pattern
reflects the hexagonal symmetry of the columnar mesophase.

6.4.1 Dislocations
The Burgers vector b of a dislocation in the columnar structure is normal
to the columnar axis n. In general

b = /a + ma',

where / and m are positive or negative integers. For edge dislocations, b is
perpendicular to the dislocation line L. Two types of edge dislocations are
possible: longitudinal edge dislocations (L parallel to n, figs. 6.4.3(a) and
(b)) and transverse edge dislocations (L perpendicular to n, figs. 6.4.3 (c)
and (d)). For screw dislocations b is parallel to L (figs. 6.4.3 (e) and (/)). A
hybrid composed of screw and edge dislocations is shown in fig. 6.4.3 (g).

Longitudinal edge dislocations
In this case, the dislocation line L is along n, the z axis. Let the Burgers
vector be along y. Minimization of the free energy (6.3.1) gives

{B + D)   9 (%£          -DA(%*-%*) J = 0
ox\ax     oy      oy\ox oy
6.4 Defects in the columnar liquid crystal                405

Fig. 6.4.2. The discotic liquid crystal appearing in the isotropic phase when a
sample is cooled very slowly. The growth pattern is diagnostic of the hexagonal
symmetry of the columnar structure: (a) Queguiner, Zann and Dubois (62), (b)
Bouligand(61).
406                         6. Discotic liquid crystals
b= a                           ,M          b

m

Fig. 6.4.3. Dislocations in the columnar phase: (a) and (Z?) longitudinal edge
dislocations; (c) and (d) transverse edge dislocations; (e) and (/) screw dis-
locations ; (g) a hybrid of screw and edge dislocations. It should be noted that the
Burgers vector b = a for (a), (c), (e) and (g), and b = a —a' for (b), (d) and (/).
(Bouligand.(61))

and

These are exactly like the equations for edge dislocations in crystals.(63) The
solutions are
B y2\
6.4 Defects in the columnar liquid crystal             407

where 0 = tan" 1 (y/x). The energy of the dislocation is

where Ec and rc are the energy and radius of the core. The radial and
angular components of the force of interaction between two edge
dislocations are
Db1b2
fr = 2n(\-v)r'

/« TTrV s i n 2 a '
2n(\— v)r
where v is Poisson's ratio.
Transverse edge dislocations
The dislocation line is now normal to n. Let it be along the x axis, so that
the only component of the displacement that needs to be considered is u y.
Minimization of the free energy yields

dy2         33
dz* '

which is exactly of the same form as that for edge dislocations in smectic A
(see §5.4.2). The solution is

b   b

and the energy

where X = [k33/(B + D)]*. The properties of these dislocations are essen-
tially the same as those of smectic A edge dislocations.

Screw dislocations
Let the dislocation line be along x. The relevant component of the
displacement is therefore ux, and the equation of equilibrium becomes

X     1^         X
33
8/ ~           8z4 '
408                          6. Disco tic liquid crystals

(a)                                         (b)

o° o S o °o
Oo>o O nO°
o
o                                       o    oo
° o o oo oo
0    o o oo                                  o oo ooo
o ooo                                     o o o oo
o o oo
(c)

Fig. 6.4.4. Disclinations in the columnar phase: (a) and (b) —n/3 and n/3
longitudinal wedge disclinations about the sixfold axis L6; (c) and (d) n transverse
wedge disclinations about the binary axes T2 and 62 respectively; (e) —n transverse
wedge disclination leading to the formation of walls; (/) two n disclinations at right
angles to each other, one about T2 and the other about 92. (Bouligand.(61))

which is again of the same form as the equation for an edge dislocation in
smectic A. The solution is

b b
ux = 7 + —
x
4     An

and the energy
6.4 Defects in the columnar liquid crystal                    409

Fig. 6.4.5. Photograph indicating the presence of two n disclinations at right angles
to each other as depicted in fig. 6.4.4 (/). (Oswald.(64))

where in this case X = (k33/D)k In contrast, the screw dislocation in smectic
A has no self energy (apart from the core) under the same approximation.
Thus the energies and interactions of screw dislocations in the columnar
phase are entirely different from those of their counterparts in smectic A or
the crystal.

6.4.2 Disclinations
Longitudinal wedge disclinations are the standard crystal disclinations in
a hexagonal lattice. The rotation vector is L 6 or L3 or L2, parallel to the
columns. Two examples of such defects are shown in figs. 6.4.4 (a) and (b).
The lattice gets compressed near a positive disclination and stretched near
a negative one. Generally, most of these disclinations have prohibitively
large energies, and hence they occur as unlike pairs at the core of a
dislocation.
In transverse wedge disclinations the rotation vector is 62 or T2, normal
to n. Examples of ± n disclinations are shown in figs. 6.4.4(c), (d) and (e).
Two transverse wedge disclinations may occur in association as shown
in fig. 6.4.4 (/). The angle between the rotation axis may be 90°, 60° or 30°.
Such defects have been observed experimentally. (64) Fig. 6.4.5 presents a
photograph which may be interpreted as arising from two n disclinations
410                           6. Discotic liquid crystals

(a)

Fig. 6.4.6. Developable domains in the columnar phase, (a) The developable
surface is degenerated into a straight line S common to the planes P. The columnar
axes C form coaxial circles about S. (b) The developable surface D is a cylinder. The
columns (or the layers in the case of smectic A) are a set of parallel and equispaced
involutes of a circle, (c) A Reimann surface generated by half-tangents to a helix.
The columns are normal to the half-tangents. (Bouligand.(61))

parallel to the glass slides and at right angles to each other as depicted in
fig. 6.4.4 (/).
The symmetry of the columnar phase also permits the occurrence of
twist disclinations in the hexagonal lattice and of hybrids consisting of a
twist disclination in the hexagonal lattice and a wedge disclination in the
director field. According to Bouligand these defects are not likely to exist.

6.4.3 Developable domains
We shall now consider the most general deformations that can occur
without distorting the hexagonal lattice and involving only the bending of
the columns. The problem reduces to one of finding physical planes P
6.5 The discotic nematic phase                       411
perpendicular to the columns, the envelope of these planes defining the
'developable' surface D. We give below some specific examples/ 61 ' 65)

(i)   If the developable surface degenerates into a straight line, then the
columns form a set of coaxial circles around this line (fig. 6.4.6(a)).
The line need not necessarily be a twofold axis, 02 or 7L The structure
of this defect is similar to the 2n disclinations of smectics.
(ii) If the developable surface is a cylinder, then the columns are along the
involutes of a circle in any plane normal to the axis of the cylinder (fig.
6.4.6 (b)). This defect is similar to a disclination of unit strength. It is
seen that this is an obvious generalization of the solutions for smectic
A. Structures similar to this have been observed by Oswald.
(iii) If the developable surface is a Reimann surface generated by half-
tangents to a helix, then the columns are normal to the half-tangents
and are involutes of the helix (fig. 6.4.6 (c)). This defect has the
features of a dislocation and a disclination, and can therefore be
described as a dispiration.

6.5 The discotic nematic phase
Theoretical calculations had shown, even before the discovery of disco tics,
that a transition from the isotropic to the nematic phase is possible, in
principle, in an assembly of plate-like particles.(66"9) On the experimental
side, Brooks and Taylor(70) observed that there is a mesophase trans-
formation at high temperatures during the carbonization of certain
graphitizable organic materials such as petroleum and coal tar pitches. The
transformation is an irreversible reaction which proceeds from the
isotropic melt to a nematic type of mesophase with increasing temperature.
The mesophase, referred to as the 'carbonaceous' phase, is a complex
multicomponent system composed of large plate-like polynuclear aromatic
molecules having a wide range of molecular weights (with a mean
molecular weight around 2000) and has a transient existence. We shall not,
however, discuss the carbonaceous phase any further, but confine our
attention to stable nematic liquid crystals formed by relatively simple
compounds.
Fig. 6.1.2(/) gives a schematic illustration of the structure of the discotic
nematic (N D ): in contrast to the classical nematic of rod-shaped molecules,
the director n now represents the preferred orientation of the short
molecular axis (or the disc norm~ ,. The symmetry of the two kinds of
nematics is the same, and identical types of defects - the schlieren texture,
412                        6. Disco tic liquid crystals

Fig. 6.5.1. Schlieren texture in a discotic nematic film. (Destrade et al.a5))

umbilics, e t c . - a r e seen in both cases. Fig. 6.5.1 is a photograph of a
typical schlieren pattern exhibited by a thin N D film when viewed under a
polarizing microscope.
The N D phase is optically and diamagnetically negative. However, its
dielectric anisotropy Ae may be positive or negative depending on the de-
tailed molecular structure. For example, hexa-n-heptyloxybenzoate of tri-
phenylene (fig. 6.1.1 (V)) and hexa-n-dodecanoyloxytruxene (fig. 6.1.1 (/))
are both dielectrically positive in the nematic phase/ 71 ' 72) This is because
while the contribution of the electronic polarizability of these disc-shaped
molecules to Ae is negative, the permanent dipoles associated with the six
ester linkage groups make a stronger positive contribution, so that Ae > 0
for the two compounds. On the other hand, hexakis ((4-octylphenyl)-
ethynyl)benzene (fig. 6.1.1 (&)), which does not have any permanent dipoles,
does, in fact, show negative Ae, as expected/ 73 ' 74)
Very few quantitative measurements of the physical properties have
been reported. The Frank constants for splay and bend have been
determined using the Freedericksz method.(71"5) Interestingly the values are
of the same order as for nematics of rod-like molecules. The twist constant
k22 has not yet been measured. The fact that the diamagnetic anisotropy is
negative makes it somewhat more difficult to measure these constants by
6.5 The discotic nematic phase                         413

(a)                                (b)

Fig. 6.5.2. Flow alignment of the director in nematic liquid crystals, (a) For rod-
shaped molecules the alignment angle 9 with respect to the flow direction lies
between 0° and 45°, while (b) for disc-shaped molecules it lies between —90° and
-45° (or equivalent^ between 90° and 135°).

the Freedericksz technique. However, by a suitable combination of electric
and magnetic fields it is possible, in principle, to determine all three
constants.
The hydrodynamic equations of the classical nematic (§3.1) are
applicable to the N D phase as well. There are six viscosity coefficients (or
Leslie coefficients) which reduce to five if one assumes Onsager's reciprocal
relations. A direct estimate of an effective value of the viscosity of N D from
a director relaxation measurement(71'73) indicates that its magnitude is
much higher than the corresponding value for the usual nematic.
We have already seen in §3.6.4 that in ordinary nematics of rod-like
molecules, the Leslie coefficients ju2 and ju3 are both negative and \ju2\ > |//3|.
Under planar shear flow, the director assumes an equilibrium orientation
60 given by
tan 2 0o =

where 60 lies between 0° and 45° with respect to the flow direction (fig.
6.5.2(a)). In practice 00 is usually a small angle. In certain nematics, it is
found that //3 > 0 at temperatures close to the nematic-smectic A transition
point. Under these circumstances there is no equilibrium value of #0, and
in the absence of an orienting effect due to the walls or a strong external
field, the flow becomes unstable.(76)
It has been suggested that in the N n phase, the disc-like shape of the
molecule may have a significant effect on ju2 and //3.(77'78) The stable
orientation of the director under planar shear will now be as shown in fig.
6.5.2(b). Thus it can be argued that both ju2 and jus should be positive, and
the flow alignment angle 60 should lie between —45° and —90°. It then
follows that when ju2 < 0, the director tumbles and the flow becomes
414                       6. Discotic liquid crystals

Fig. 6.6.1. Two examples of molecules which exhibit the biaxial nematic phase:
(a) (bis-1 -(/?-n-decylbiphenyl)-3-(/?-ethoxyphenyl) propane-1,3-dionatocopper
(H))<8i,82) (hereafter referred to as complex A); (b) l,12-te(pentakis((4-pentyl-
phenyl)ethynyl)pheny loxy)dodecane.(83)

unstable. As yet, however, no experimental studies have been carried out
to verify any of these ideas.

6.6 The biaxial nematic phase
The biaxial nematic (Nb) phase was first identified by Yu and Saupe(79) in
a ternary amphiphilic system composed of potassium laurate, 1-decanol
and D 2 O. In such systems the constituent units are molecular aggregates,
called micelles, whose size and shape are sensitive to the temperature and
concentration; the N b phase was found to occur over a range of
temperature/concentration. There are obvious advantages in having a
single-component, low-molar-mass thermotropic N b phase. The sugges-
6.6 The biaxial nematic phase                         415

Fig. 6.6.2. Conoscopic figure demonstrating the biaxiality of the nematic phase of
complex A. (8284)

tion was made(38) that a convenient way of achieving this would be by
'bridging the gap between rod-like and disc-like mesogens', i.e., by
preparing a mesogen that combines the features of the rod and the disc.
This has proved to be useful and the N b phase has been observed in some
relatively simple compounds.(80) Two examples are given in fig. 6.6.1. The
biaxiality of the nematic phase was established by optical observations,
supported by X-ray evidence (826) (figs. 6.6.2 and 6.6.3).
The difference between the structures of the uniaxial and biaxial nematics
is illustrated schematically in fig. 6.6.4. The N b phase is depicted here as an
orthorhombic fluid whose preferred molecular orientation is described by
an orthonormal triad of director fields. (In principle, nematics of lower
symmetry are possible, but none of the N b phases identified to date have
been reported to be other than orthorhombic.) The structure, therefore,
gives rise to an additional pair of diffuse (liquid-like) X-ray diffraction
peaks(87-9) (fig. 6.6.3(6)).
A number of important ideas concerning the N b phase have been
discussed theoretically - molecular statistical and phenomenological
theories/ 69 ' 90103) continuum theories, (10417) topological theories of de-
fects/118-25* etc. For example, Saupe(104) and Kini(109) who used different
theoretical approaches, have both concluded that the incompressible
orthorhombic nematic has 12 curvature elastic constants (excluding three
which contribute only to the surface torque) and 12 viscosity coefficients.
416                                                6. Discotic liquid crystals

(i)                       H                                         (ii)            H

\         T   I   .   I   .   I       .   I     .   i   ,   i   .   i   .      i    ,   i   .   i   .   i    .    i    i    i
-12 - 6                 0           6           12 - 2 4 - 1 8 - 1 2 - 6                  0       6       12        18        24
(a)                                                         26 (deg)

(i)                                                                 (ii)
H

-12          -6                               12       -24 -18 -12                    -6                        12        18        24
20 (deg)

Fig. 6.6.3. Raw microdensitometer scans of the X-ray intensity plotted against
diffracting angle (26) for magnetically aligned nematic samples: (a) the uniaxial
nematic phase of 80CB at 77 °C; (b) the biaxial nematic phase of complex A at
168.5 °C, (i) meridional scan (parallel to / / ) , (ii) equatorial scan (perpendicular to
H). M represents the diffraction peaks from the mylar film which covered the
windows of the sample holder and heater assembly. (8586)

Again, remarkable predictions have been made on the basis of the
homotopy theory regarding the properties of defects in the N b phase. The
usual law of coalescence of defects (see §3.5.1) breaks down; the
combination rule is now non-Abelian. Moreover, there can be an
entanglement of disclination lines, which, in turn, can give rise to what
6.6 The biaxial nematic phase                     417

(a)                              (b)
Fig. 6.6.4. Schematic diagram of the molecular order in (a) the uniaxial nematic
phase and (b) the biaxial nematic phase.

Toulouse(118) describes a s ' topological rigidity'. These and other ideas have
yet to be investigated experimentally. The availability of the N b phase in
simple thermotropic systems is likely to make it conveniently possible to
test some of these predictions.
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130 B. R. Ratna, M. S. Vijaya, R. Shashidhar and B. K. Sadashiva in
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135 N. V. Madhusudana and S. Chandrasekhar, Sol. State Commun., 13, 377
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153 A. J. Leadbetter, R. M. Richardson and C. N. Colling, J. de Physique, 36,
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155 A. Ferguson and S. J. Kennedy, Phil. Mag., 26, 41 (1938).
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36    An analytical solution in the two-wave approximation has been given by
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85 H. Arnould-Netillard and F. Rondelez, Mol. Cryst. Liquid Cryst., 26, 11
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97 S. Meiboom, J. P. Sethna, P. W. Anderson and W. F. Brinkman, Phys. Rev.
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98 D. W. Berreman, in Liquid Crystals and Ordered Fluids, Vol. 4 (eds. A. C.
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99 S. A. Brazovskii and S. G. Dmitriev, Sov. Phys.-JETP, 42, 497 (1976).
100 S. A. Brazovskii and V. M. Filev, Sov. Phys.-JETP, 48, 573 (1978).
101 R. M. Hornreich, M. Luban and S. Shtrikman, Phys. Rev. Lett., 35, 1678
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102 H. Grebel, R. M. Hornreich and S. Shtrikman, Phys. Rev., A 30, 3264
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103 For a comprehensive review of the Landau theory of the blue phases, see,
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104 P. P. Crooker, Mol. Cryst. Liquid Cryst., 98, 31 (1983).
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105 P. N. Keating, Mol. Cryst. Liquid Cryst., 8, 315 (1969).
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107 J. L. Fergason, N. N. Goldberg and R. J. Nadalin, Mol. Cryst., 1, 309
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108 J. L. Fergason, Scientific American, 211, 77 (1964).
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110 J. Hansen, J. L. Fergason and A. Okaya, Appl. Opt., 3, 987 (1964).
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112 P. Pollmann and H. Stegemeyer, Chem. Phys. Lett., 20, 87 (1973).
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128 D. Coates and G. W. Gray, Mol. Cryst. Liquid CrySt., 24, 163 (1973).
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Chapter 5
1 H. Sackmann and D. Demus, Mol. Cryst. Liquid Cryst., 21, 239 (1973).
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2 P. S. Pershan, Structure of Liquid Crystal Phases, World Scientific Lecture
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3 A. J. Leadbetter, in Thermotropic Liquid Crystals (ed. G. W. Gray), p. 1,
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4 S. Diele, P. Brand and H. Sackmann, Mol. Cryst. Liquid Cryst., 17, 163
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5 A. Tardieu and J. Billard, / . de Physique Colloq., 37, C3-79 (1976).
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7 W. L. McMillan, Phys. Rev., A 4, 1238 (1971).
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26 R. Schaetzing and J. D. Litster, Advances in Liquid Crystals, Vol. 4 (ed.
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438                         References:   Chapter 5

27 J. Als-Nielsen, R. J. Birgeneau, M. Kaplan, J. D. Litster and C. R. Safinya,
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28 A. Caille, C. R. Acad. ScL, Paris, 274B, 891 (1972).
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56 W. H. Bragg, Nature, 133, 445 (1934).
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71 H. Gruler, Z. Naturforsch., 28a, 474 (1973).
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88 J. D. Litster and R. J. Birgeneau, Physics Today, 35, 26 (1982).
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Chapter 6
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38 S. Chandrasekhar, Plenary Lecture, X International Liquid Crystals
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63 See for example, J. Friedel, Dislocations, Pergamon, Oxford (1967).
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75 T. Warmerdam, D. Frenkel and J. J. R. Zijlstra, / . de Physique, 48, 319
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76 See reveiw by S. Chandrasekhar and U. D. Kini, 'Instabilities in low
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78 G. E. Volovik, JETP Lett., 31, 273 (1980).
79 L. J. Yu and A. Saupe, Phys. Rev. Lett., 45, 1000 (1980).
80 See review by K. Praefcke, B. Kohne, B. Gundogan, D. Singer, D. Demus,
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81 S. Chandrasekhar, B. K. Sadashiva, S. Ramesha and B. S. Srikanta,
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82 S. Chandrasekhar, B. K. Sadashiva, B. R. Ratna and V. N. Raja, Pramana,
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their X-ray study of the N b phase of a sanidic discotic polymer. Evidence of
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Finkelmann(88) and by Windle et a/.<89>
88 F. Hessel and H. Finkelmann, Polymer Bull., 15, 349 (1986).
89 A. H. Windle, C. Viney, R. Golombok, A. M. Donald and G. R. Mitchell,
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97 P. H. Keyes, Proceedings of the Eighth Symposium on Thermophysical
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450                        References: Chapter 6
100 J. W. Doane, NMR of Liquid Crystals (ed. J. W. Emsley), p. 441, Riedel,
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101 S. R. Sharma, P. Palffy-Muhoray, B. Bergersen and D. A. Dunmur, Phys.
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102 D. W. Allender and M. A. Lee, Mol. Cryst. Liquid Cryst., 110, 331 (1984).
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103 T. C. Lubensky, Mol. Cryst. Liquid Cryst., 146, 55 (1987).
104 A. Saupe, J. Chem. Phys., 75, 5118 (1981).
105 M. Liu, Phys. Rev., A 24, 2720 (1981).
106 H. Brand and H. Pleiner, Phys. Rev., A 24, 2777 (1981).
107 E. A. Jacobsen and J. Swift, Mol. Cryst. Liquid Cryst., 78, 311 (1981).
108 G. E. Volovik and E. I. Kats, Zh. Eksp. Teor. Fiz., 81, 240 (1981).
109 U. D. Kini, Mol. Cryst. Liquid Cryst., 108, 71 (1984).
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110 E. Govers and G. Vertogen, Phys. Rev., A 30, 1998 (1984).
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114 D. Baalss, Z. Naturforsch., 45a, 7 (1990).
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117 D. Monselesan and H. R. Trebin, Phys. Stat. Sol. (b), 155, 349 (1989).
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119 V. Poenaru and G. Toulouse, / . de Physique, 8, 887 (1977).
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124 H. R. Trebin, Adv. Phys., 31, 195 (1982).
125 M. Kleman, Points, Lines and Walls, Wiley, New York, (1983).
Index

anisaldazine                                      nematic-nematic coexistence, 51
crystal structure, 18                           order parameter of individual
orientational order parameter, 27                  components, 51
surface tension, 81                           blue phases, 5-6, 16
antiferroelectric short range order in            Landau theory of, 295-6
nematics, 75-80                              optical Kossel diagrams, 292
antiparallel correlations in smectic A,           single crystals, 292
350-8                                        unit cells of blue phases I and II as cubic
smectic A polymorphism                     bond orientational order, 6, 360-2, 3 6 3 ^ ,
liquid crystal display devices (LCD):         Borrmann effect in cholesterics, 232-8
active matrices, 111; colour displays,     Bragg-Williams approximation, 20
111; dye displays, 51, 112; dynamic        Brillouin scattering, 325-6, 402, 404
scattering mode, 178, 288; multiplexed
displays, 111; storage mode, 288;          Carr-Helfrich instability, 181-2, 286-7
supertwisted nematic (STN), 111-12;        CBOOA
surface-stabilized ferroelectric LCD,        correlation lengths, 343, 347
380, 386-7; twisted nematic (TN),            critical exponents associated with the
106-11; twisted nematic, dynamical              smectic A-nematic transition, 347
characteristics of, 161-2; twisted           damping rate of undulation mode, 321-2
nematic, optical bounce and reverse          Helfrich deformation, 316
twist effects in, 112-13, 167                influence of twist distortion on the
optical modulator, 381, 387                        smectic A-nematic transition
thermography, 296-7                                temperature, 305
tunable colour filter, 387                      layer compressibility, 315
p-azoxyanisole, see PAA                           orientational order parameter, 309
p-azoxyphenetole, see PAP                         penetration depth, 315-16, 322
splay and bend constants, 344
backflow, 162-7                                 chevron patterns, 178, 180, 387
Bethe approximation, 71                         cholesteric pitch
biaxial nematics, 3, 378, 414-17, 449             compensated mixtures of right- and left-
(reference 87)                                  handed cholesterics, 218, 248, 276-7
biaxiality:                                       critical divergence of, 349
in nematics, see biaxial nematics               dependence on
in smectic C, 362-3                                composition (in mixtures) 297-8;
in smectic C*, 379                                 magnetic field strength normal to the
molecular, 37, 47-8, 298                           helical axis, 277-80; pressure, 297;
bicritical point, 354                                temperature, 296-7
binary nematic mixtures, 51                       twist per unit length, 97

451
452                                          Index
cholesteryl nonanoate:                                 equations, 97-8; thermal instability
blue phases, 16                                      201-5; twisted nematic and
Borrmann effect, 236                                 supertwisted nematic devices 106-15;
critical divergence of pitch, 349                    viscosity coefficients, 92-3
molecular structure, 16                           of smectic A:
transitions, 16                                      breakdown of conventional
Clausius-Clapeyron equation, 27                        hydrodynamics, 325; damping rate of
conductivity:                                          undulation mode, 322; elastic free
electrical, 177, 181                                 energy density, 310; energetics of
thermal, 202-5                                       defects, 327-39; fluctuations, 317-19;
consistency relation                                   light scattering, 317-19; mechanical
Chang, 72                                            instability, 314—16; Peierls-Landau
Krieger-James, 72                                    instability 312-14; permeation, 321;
Maier-Saupe, 44                                      ultrasonic propagation, 323-5;
continuum theory                                       viscosity coefficients, 321
of cholesterics:                                  of smectic C:
coefficients of thermal conductivity,             elastic free energy density, 367;
260; distortion by external fields,               fluctuations, 367; light scattering, 367
277-88; elastic free energy density, 97,    convection
284-5; energetics of defects, 248-58;          oscillatory, 205
flow along the helical axis                    stationary, 202-5
(permeation), 270-3; flow normal to         critical end point, 355
the helical axis, 274-6;                    critical point, 353
thermomechanical coupling, 258-67;          curvature elasticity, see elastic constants
torque induced by electric field            4-cyanobenzylidene-4/-octyloxyaniline, see
(electromechanical coupling), 264-7;              CBOOA
torque induced by heat flux (the            cyanobiphenyls (4/-n-alkyl-4-
Lehmann rotation phenomenon),                     cyanobiphenyl) nCB
262-3; viscosity coefficients, 260             5CB (pentyl compound):
of columnar liquid crystals:                         antiparallel local order, 80;
acoustic wave propagation, 401-2;                 applications, 110; dielectric anisotropy
elastic free energy density, 397;                 79; molecular structure, 15; surface
energetics of defects, 402-8;                     tension, 81; transitions, 15; X-ray
fluctuations, 398; light scattering, 400;         studies, 79
mechanical instabilities, 400-1;               6CB:
Peierls-Landau instability, 398; X-ray            electric and magnetic birefringence, 66
scattering, 399                                7CB:
of nematics (including discotic                      applications, 110; X-ray studies, 79
nematics):                                     8CB:
backflow and kickback effects, 162-7;             critical exponents associated with the
distortions due to external fields,               smectic A-nematic transition, 347;
98-117; elastic free energy density, 57,          molecular structure, 15; transitions, 15
97, 116; electrohydrodynamic                   9CB:
instabilities, 177-95; energetics of              critical exponents associated with the
defects, 117-44; Ericksen-Leslie                  smectic A-nematic transition, 347
equations, 85-94; flexoelectricity,            12CB:
205-12; flow properties, 144-59,                  surface-induced smectic order in the
413-14; Freedericksz effect, static               isotropic phase, 84; X-ray specular
theory, 98-106; Freedericksz effect,              reflectivity from free surface, 84
dynamics of, 161-7; hydrodynamic               80CB (octyloxycompound):
instabilities, 155-7, 195-201,413-14;             critical exponents associated with the
light scattering, 162-77;                         smectic A-nematic transition, 347;
Oseen-Zocher-Frank elasticity                     P—T diagram, 356; reentrant
equations, 94-7; Parodi's relation, 94;           behaviour at elevated pressures, 356;
periodic distortions in highly                    surface-induced smectic order in
anisotropic media, 113-15; reflexion              nematic phase, 83; X-ray specular
of shear waves, 159-61; summary of                reflectivity from free surface, 83
Index                                       453
cybotactic groups, 2, 4, 83                       discotic liquid crystals
columnar structures, 10, 390
Darwin's theory of X-ray diffraction, see           columnar phases of conical, macrocyclic,
optical properties of cholesterics               phasmidic molecules, 393—4
defects                                             discotic lamellar phase, 9, 394, 397
in cholesterics:                                 discotic nematic, 10, 393, 411-14
X (chi)-edge disclinations, 252; / (chi)-      disc-shaped molecules, 389
screw disclinations, 249-51;                   extension of McMillan's model to
disclination pairs, 250-1; dislocations,          discotics, 394-7
254-8; edge dislocations, 254-7;               high resolution X-ray studies on freely
fingerprint textures, 254-6;                      suspended strands, 389-93
Grandjean-Cano pattern, 257-8;                 ordering of the cores and chains, 390-2
lattice disclinations, 252^4;                  reentrant behaviour, 394
pincements, 254-6; screw dislocations,         structure and classification of
254                                               mesophases, 8-10, 388-93
in columnar liquid crystals, 402-11               twisted nematic (or cholesteric), 393
disclinations, 408-9; dislocations,               discotic nematics, discotic polymers
403-8; longitudinal dislocations,           discotic, nematics, 10, 390, 393, 411-14
406-7; longitudinal wedge                      diamagnetic anisotropy, 412
disclinations, 408; screw dislocations,        dielectric anisotropy, 412-13
408; transverse dislocations, 407;             director, 390, 412
transverse wedge disclinations, 408            Frank (elastic) constants, 413
in nematics:                                      hydrodynamical properties, 413-14
angular forces between disclinations,          optical anisotropy, 412
140-3; Brochard-Leger walls, 136;          discotic polymers, 10-12
consequences of elastic anisotropy,            columnar nematic, 11-12
139—43; core structure, 143—4; director       columnar phase, 11-12
field in the neighbourhood of, 119-20;         sanidic nematic, 11-12
disclination loops, 127-8; energies and     dispersion forces, 41-2
interactions, 120-2; Helfrich walls,        donors and acceptors, 12, 358
135; non-singular structures, 123-6;       Dupin cyclides, 329, see focal conic
schlieren textures in discotic nematics,          textures
137-9; threads and schlieren textures,           applications
7, 117; twist disclinations, 126-8;
umbilics, 136-7; wedge disclinations,       elastic constants
117-28                                         nematics:
in smectic A:                                        critical divergence of, 3 4 2 ^ ;
disclinations, 328-9; edge dislocations,          dependence on order parameter,
333-7; energies and interactions,                 59-60; experimental determination of
335-7; focal conic textures, 327-33;              splay, twist and bend (or Frank)
screw dislocations, 338                           constants, 98-106, 172; permanent
in smectic C:                                        splay and twist, 96, 117
core structure 3 7 3 ^ ; disclinations in      smectic A:
the odirector field, 368-9; edge                  critical divergence of twist, bend and
dislocations, 369-70; toric domains,              layer compressibility constants, 343,
329                                               348; experimental determination of
dielectric anisotropy, theory of, 51-7, 76-8            layer compressibility, 315, 322
dielectric dispersion, 56                         electric birefringence (Kerr effect), 55,
4,4'-di-n-alkoxyazoxybenzenes                           63-6
dielectric dispersion, 56                      electric field induced distortions
odd-even effect, 48-51                            cholesterics:
structure, 50                                        Carr-Helfrich instability, 286-7;
director, definition of, 85-6                           288; square grid pattern, 282, 286-8;
disclinations, see defects                              storage mode, 288
454                                           Index

nematics:                                         soft mode and Goldstone mode, 382-3
Freedericksz transition, 106; principle        structure of smectic C* and the origin of
of the twisted and supertwisted                   ferroelectricity, 379-80
electrohydrodynamic instabilities,             contribution of flexoelectricity to
flexoelectricity                                  electrohydrodynamic instability,
smectic C*:                                          210-11
electroclinic effect, 380, 387; the            determination of the flexoelectric
surface-stabilized ferroelectric display          coefficients, 205-9
device, 386-7; unwinding of helix,          flow birefringence, 69-71
380                                         flow properties
electroclinic effect, 380, 382                       of cholesterics:
application as optical modulators and                flow along helical axis, 271-3; flow
tunable colour filters, 387                       normal to the helical axis, 274-7;
electrohydrodynamic instabilities in                    Helfrich's model of permeation,
nematics                                          270—1; non-Newtonian behaviour,
Carr-Helfrich mechanism, 181^1                       268; oscillatory variation of apparent
chevron pattern, 178, 180                            viscosity with pitch, 276-7; secondary
conduction and dielectric regimes, 178,              flow, 274
183                                           of nematics:
dynamic scattering, 178                              backflow and kickback effects, 162-7;
flow and orientation patterns, 182                   flow alignment in discotic nematics,
Helfrich's theory, 181-6                             413-14; flow alignment in ordinary
influence of flexoelectricity, 210-11                nematics, 150, 157; instability in
threshold for DC and AC excitation,                  cybotactic nematics near the smectic A
186-7, 187-94                                    transition point, 157; Miesowicz's
Williams domains, 178-82                             experiment, 144-5; Poiseuille flow,
electromechanical coupling coefficient in               148-52; shear flow, 152-7; transverse
cholesterics, 266-7                               pressure and secondary flow, 157-9;
electron spin resonance, 39, 146                        Tsvetkov's experiment, 145-8; velocity
enantiotropic transition, definition of, 14             and orientation profiles, 152-3, 164,
end chains, their contribution to                       166; viscosity (or Leslie) coefficients,
hard rods with semiflexible tails, 37                instabilities
influence of chain length on                      of smectic A:
the columnar-nematic-isotropic phase              non-Newtonian behaviour, 319;
sequence, 394-7; the smectic                      Helfrich's model of permeation, 321
A-nematic-isotropic phase sequence,         fluctuation-dissipation theorem, 174
306                                         fluctuations
odd-even effect in homologous series,             director fluctuations
48-51                                             in columnar liquid crystals, 400; in
ordering of the cores and chains in the              nematics, 167-70; in smectic A,
columnar mesophase, 391-2                         317-19; in smectic C, 367
stiffening of the chains as a mechanism           lattice fluctuations in the columnar
for reentrance, 357-8                             phase, 397-8
Ericksen-Leslie theory, 85-94                        layer fluctuations in smectic A, 312-14
equal areas principle, 22-4, 37, 73                  Peierls-Landau instability, 312-14, 398
pretransitional fluctuations
ferroelectric liquid crystals, 378-87                   near the cholesteric-isotropic
applications, 386-7                                  transition, 289-93; near the nematic-
experimental studies on the polarization,            isotropic transition, 66-8; near the
tilt angle, pitch, dielectric properties,         smectic A-cholesteric transition, 349;
381^                                              near the smectic A-nematic transition,
generalized Landau theory, 385-6                     342; near the smectic C-nematic
liquid crystalline ferroelectrics as                 transition, 378; near the smectic C -
improper ferroelectrics, 385                      smectic A transition, 371
rotational viscosities, 384                    Frank constants, see under elastic constants
Index                                       455
Freedericksz effect                            Landau-de Gennes model, 61-3
in nematics                                 Landau-Ginsburg free energy, 342
dynamical theory, 161-7; static          Landau-Ginsburg parameter, 358
theory, 98-101                           laser light beating spectroscopy, 177, 319,
in smectic C, 365                                321,367
free energy of elastic deformation, see        Lennard-Jones potential, 21, 22
continuum theory                         Leslie coefficients, 97-8, 260
free standing monodomain smectic films,        Lifshitz point, 374—5
studies on, 360-4                        light scattering
freely suspended monodomain discotic              columnar liquid crystals, 400
strands, studies on, 389-93                 nematics
dependence on temperature, 59-60;
Ginsburg criterion, 371                             depolarization, 171; determination of
Ginsburg-Pitaevskii equation, 373                   elastic constants, 172, 347;
Goldstone mode in ferroelectric (smectic            determination of viscosity coefficients,
C*) liquid crystals, 382-4                       176; eigenmodes and power spectrum
Grandjean-Cano pattern, 219, 257-8, 349              172-7; intensity and angular
dependence, 170-2; orientational vs
hard particle theories                              density fluctuations, 171; scattering
Andrew's method, 37                               from the free surface, 82-3; scattering
Flory's lattice model, 36                         from the isotropic phase, 66-8
hard rods with semiflexible tails, 37           smectic A
Onsager's theory, 31                              contribution of the undulation modes,
scaled particle theory, 36-7                      317-22; determination of the damping
stiffening of the tails as a mechanism for        rate, 320-2
reentrance, 357-8                            smectic C
summary of theories, 37, 411                      angular dependence of the intensity
Zwanzig's model, 31-6                             and damping rate, 366-7
Helfrich deformation in                        liquid crystal display devices, see
cholesterics (square grid pattern), 281-9         applications
columnar liquid crystals, 400                living systems, 14
smectic A, 314-15                            lyotropic systems:
Helfrich's model of permeation, 270-1,            aerosol-OT-water system, 12
321                                             plant virus preparations, 8
Helfrich's theory of electrohydrodynamic          poly-y-benzyl-glutamate in organic
instabilities in nematics, 181-7               solvents, 13, 113-14,217-18
Helfrich walls, 135                               sufractant-water compositions, soaps, 8,
hexatic phases, 6                                    12
normal and tilted types, 301                    types of molecular packing, 12-14
X-ray studies of the degree of bond-
orientational ordering, 360^4             magnetic birefringence (Cotton-Mouton
homeotropic orientation, 6, 106                    constant), 62-7
homogeneous (or planar) orientation, 6,        magnetic coherence length, 101, 135-7,
104-5                                        194, 278, 312
orienting effect of grooves, 104-5           magnetic field effects in
hybrid models: hard rods with a                  cholesterics
superposed attractive potential, 60-1        cholesteric-nematic (unwinding)
hydrodynamics of liquid crystals                   transition, 277-80; square grid
andPershan, 319,367-8                        induced distortions
hydrodynamics of nematics, cholesterics,     nematics:
smectics and discotics, see continuum        dynamic effects, 161-7; effect on
theory, flow properties                      electrohydrodynamic, hydrodynamic
and thermal instabilities, 179, 194,
induced discotic phases, 12                        196, 201, 205; rotating magnetic field,
induced smectic phases, 358                        144-8; static effects, 98-104, 106-9,
isotherms. 2 2 ^ . 37. 73                          152-5
456                                      Index

Maier-Saupe theory of nematics, 38-51,            end chains
70-1                                     molecular statistical theories:
applications                                 cholesterics, 298
binary mixtures, 51; dependence of        discotics, 394-7
elastic constants on the order            nematics, 17-61, 71-80
parameter, 57-60; dielectric              smectic A, 302-9
anisotropy, 51-6; odd-even effect,        smectic C, 364
47-51                                    monotropic transition, 14, 16, 379
extension of the theory                     multicritical point, nematic-smectic
to cholesterics, 298; to discotics,          A-smectic C (NAC), 374-8
394-7; to smectic A (McMillan's           Chen-Lubensky model, 375-8
model), 302-9                             high resolution X-ray calorimetric and
limitations of the theory, 47-9, 70-71          light scattering studies, 378
Mauguin-Oseen-de Vries model, 237^41           topology of phase diagram in the
MBBA                                              temperature-concentration and
backflow effects, 165                           pressure-temperature planes, 374, 376
disclination patterns, 122
electrohydrodynamic distortions, 180,       near neighbour correlations
183^                                      antiparallel (or antiferroelectric)
flexoelectric coefficients, 207-9                correlation, 75-8
flow birefringence, 71                        Bethe approximation, 71
heats of transition, 26                       correlation lacking a centre of inversion,
light scattering studies                         289
in nematic phase, 60; in the isotropic     Krieger-James approximation, 72-3
phase, 67-8                                short range order parameter, 75
magnetic birefringence, 64                  nematic-isotropic transition
mixtures with cholesterics, dependence of     Bethe approximation, 71-80
pitch on composition, 298                  effect of pressure, 27-30, 46-7
molecular orientation at free surface,        Landau-de Gennes model, 61-3
83                                         latent heat, 15-16,26,47
oblique rolls in electrohydrodynamic          Maier-Saupe theory, 38-48
patterns, 211                              molecular statistical theories, 17^8
roll instability threshold, 198               pressure-induced mesomorphism, 28-30
thermal instability, 202, 205                 short range order effects, 61-70
torque in a rotating magnetic field, 148      volume change, 26, 45
transverse pressure and secondary flow,     nematic liquid crystals, 1-3, 8-10
158                                      nematics as solvents, 39, 146
viscosity (or Leslie) coefficients, 155     nuclear magnetic resonance (NMR)
volume change at transition, 26                  studies, 39, 61, 308-9, 392
McMillan's model of smectic A, 302-9
melting of                                    odd-even effect, 48-51
inert gas crystals, Lennard-Jones and       Onsager's reciprocal relations, 69, 94
Devonshire model, 17-19                  Onsager's theory of dielectric polarization,
molecular crystals, Pople-Karasz model,          52
19-27                                    Onsager's theory of the hard rod fluid, 31
mesomorphic groups, Hermann's                 optical properties of cholesterics
classification, 8                          absorbing systems, 220-2, 232-6
mesomorphism, thermotropic and                  analogy with Darwin's dynamical theory
lyotropic, 1                                  of X-ray diffraction, 222-232
metallomesogens, 389, 414                       Borrmann effect, 232-6
4-methoxybenzylidene-4/-butylaniline, see       equivalence of the continuum and
MBBA                                          dynamical theories, 241-5
Miesowicz's viscosity coefficients, 144-5,      exact solution of the wave equation: the
155                                           Mauguin-Oseen-de Vries model,
miscibility criterion, 300                         237^1
molecular field, 38                             oblique incidence, 245-7
Index                                        457
propagation along the helical axis,         PAA
213^5                                        comparison of properties with hard rod
propagation normal to the helical axis,           and hybrid model calculations, 37,
247-8                                          60
rotatory power, reflectivity and circular     dielectric constants, 53
dichroism as functions of wavelength,       dielectric dispersion, 56
pitch and sample thickness, 217-20,         dipole moment, 55
227-32, 239^1                               electrohydrodynamic distortions, 178
optical textures                                 flow birefringence, 69-71
cholesterics:                                  heat of transition and volume change,
droplets with a /-line of strength 2,          26
442; fingerprint texture, 255-6;            Leslie coefficients, 150
Grandjean-Cano pattern, 258; helical        light scattering studies, 59, 66-8, 83, 172,
configurations of disclination pairs,          176
143, 249-51                                lines of constant order parameter vs. In V
columnar liquid crystals:                         and In T, 46
texture showing two orthogonal n            magnetic and electric birefringence, 62-5
disclinations, 410                          Miesowicz coefficients, 146
nematics:                                      molar volume at the transition, 44
disclination loops, 127-8; schlieren        molecular orientation at the free surface,
textures {structures a noyaux), 6, 30,         83
117-20, 365, 412; singular points,         molecular structure, 15
129-32; threads, 7-8; umbilics,            orientational order parameter, 27, 40
136-8                                      PAA -I- cholesteryl nonanoate, Borrmann
smectic A:                                          effect in, 236;
batonnets, 30, 333; fan-shaped and          Poiseuille flow, 150-3
polygonal textures, 9, 331; focal conic     P-T diagram and other pressure studies,
textures, 9, 30, 327-33; stepped drop          28,46
(goutte a gradins), 332, toric domains,     surface tension, 81
329                                         torque in a rotating magnetic field
smectic C:                                        (Tsvetkov's experiment), 148
edge dislocations, 370; toric domains,      transitions, 15
329                                         twist elastic constant, 104
complex order parameter                      PAP
smectic A, 340; smectic C, 370              heat of transition and volume change,
orientational order parameter                     26
curves, 40, 45, 75, 77, 303-5, 308-9;       light scattering studies, 59
definition, 38—41; experimental             Miesowicz coefficients, 146,
determination of <P2>, 39-^0, 308-9;        orientational order parameter, 27
<.P4> from polarized Raman studies,         twist elastic constant, 104
74—5, 77; variation near an interface,    Parodi's relation, 93, 98
82,211                                    penetration depth, 314, 322, 342, 398
positional order parameter in melting        phase diagrams, 28-9, 291, 295, 306, 353-6,
theory, 23-4                                   376, 397
translational order parameter                planar structure, see homogeneous
amplitude of one-dimensional density           orientation
wave (smectic A), 303-5, 307, 340;        plastic crystals, 22
density wave in two dimensions            Poiseuille flow, 148-52, 157-9, 268-9,
(columnar phase), 395                          270-3
orientation distribution function in MBBA      polymorphism in
derived from Raman studies, 49              smectic A, 350-2
Oseen's equation of motion for the                smectic C, 364-5
director, 87                                thermotropics, 14-16
Oseen-Zocker-Frank elasticity equations,       pressure studies, 27-9, 46-7, 297, 356, 372,
94-7                                           374-6
458                                        Index
pretransition effects in the vicinity of          molecular models, 357
cholesteric-isotropic transition:               multiple reentrance, 357
anomalous optical rotation in the          rotational transitions in crystals, 19
isotropic phase, 289-93;
coherence lengths, 293                     second sound, 324-6, 402, 404
nematic-isotropic transition:                 shear flow, 69, 152-7, 195-201, 274-7, 383
Bethe approximation, 71-80;                shear waves, reflexion of, 159-61
coherence length, 68; flow                 short-range order, see near neighbour
birefringence, 69-71; Landau-de                  correlations, pretransition effects
Gennes model, 61-3; light scattering       smectic A, 6, 301, 341
studies, 66-8; magnetic and electric         absence of true long range translational
neighbour correlations                       high resolution X-ray scattering studies,
smectic A-cholesteric transition:                   313-14
analogy with the behaviour of a              polymorphic forms of smectic A and
superconductor in a magnetic field,              transitions between them, 351-5,
348-9; critical divergence of the pitch,     see also under continuum theory, defects,
349-50                                           optical textures
smectic A-nematic transition:                 smectic A* or the twist grain boundary
critical divergence of twist and bend            phase:
elastic constants, 343; critical             example of a compound which shows
divergence of viscosity coefficients,            this phase, 359
345; critical exponents, 347; high           structure of smectic A*, 359
resolution X-ray, light scattering,          X-ray and optical studies, 359-60
specific heat and other studies, 347;      smectic A-nematic transition:
Landau-Ginsburg description, 341-2;          complex order parameter, 340
longitudinal and transverse coherence        influence of twist and bend distortions
lengths, 343                                     on the transition, 358-9
smectic C-nematic transition:                   McMillan's molecular model, 302-9
Chen-Lubensky model, 374—8; critical         phenomenological theory, 340-1
divergence of the splay, twist and bend      review of theoretical situation and its
constants, 378                                   comparison with experiment, 347
smectic C-smectic A transition:                 tricritical point, 309, 341
transition induced by layer                smectic C:
compression, 372; variation of tilt          molecular models, 364
angle, specific heat, magnetoclinic          optical biaxiality, 363
effect near the transition, 370-2            polymorphic forms in polar systems, 365
smectic C*-smectic A transition:                structure and symmetry, 362
critical variation of the frequencies,       temperature dependence of tilt angle 363
dielectric strengths and viscosities         tilt and bond orientational order, 3 6 3 ^
associated with the soft mode and            see also under continuum theory, defects,
Goldstone mode, 384; generalized                 textures
Landau theory, 386; temperature            smectic C-smectic A transition:
dependence of tilt angle, polarization,      complex order parameter and the helium
pitch, electroclinic effect, dielectric          analogy, 370
constant, 381-3                              Landau mean field theory, 370-1
Raman scattering studies, determination of      smectic C*:
the orientational order parameter            applications
<^4>, 48-9                                       optical modulators, 386-7; surface
reentrant phenomenon:                                 stabilized ferroelectric LCDs, 386-7;
in binary mixtures of polar compounds,              tunable colour filters, 386-7
355                                          description of the structure and the
in discotics, 394                                   origin of ferroelectricity, 378-80
in non-polar materials, 358                     director modes, the soft mode and the
in pure polar compounds at elevated                 Goldston mode, 382
pressures, 356                               electric field effects, 380
Index                                         459

electroclinic effect, 380                      twist grain boundary (TGB) phase, see
measurements of the tilt angle,                     smectic A*
spontaneous polarization, pitch,
dielectric constant and dispersion,         Ultrasonic propagation:
381-3                                         angular variation of Brillouin scattering,
optical properties, 379                             326, 404
rotational viscosity measurements,               in columnar phase, 401-2
382-3                                         in smectic A, 323-5
smectic C*-smectic A transition,                    second sound, 324, 402
generalized Landau theory, 386; see         undulation mode in
also under pretransition effects              smectic A, 317, 322
smectic liquid crystals: structural                 smectic C, 367
classification of smectics A-K, and the
chiral modifications of some of them,       virial coefficients, 34
stepped drop (goutte a gradins), 332                    properties)
supertwisted nematic device, 111-12, see             cholesterics
also under applications, continuum                Leslie coefficients, 260; viscosity
theory                                            coefficients coupling thermal and
surface-stabilized ferroelectric liquid crystal         mechanical effects, 260, 262-3
applications                                   nematics:
surface studies:                                        experimental determination of, 144-8,
interferometric study of the free surface            155-7, 161-2, 177; Leslie coefficients,
of a smectic, the stepped drop, 332-3             97-8; Miesowicz coefficients, 145;
light scattering and optical reflectivity            twist viscosity and its critical
from free surface 82-3                           divergence, 147, 161, 177, 345
surface-induced orientational order, 82           smectic A
surface-induced smectic order 83^4                   permeation, 321; viscosity coefficients,
surface tension measurements, 81-2                   321
variation of the order parameter in the        viscous stress tensor, 92, 97, 260, 321
liquid-vapour transition zone, 82, 211      viscous torque, 93, 147, 1 6 3 ^ , 183, 270,
X-ray reflectivity from free surface, 83^4           345
volume change at transition, 26, 45
tetracritical point, 378
threshold                                              electrohydrodynamic instabilities
cholesteric-nematic transition, 280
electrohydrodynamic instabilities, 187,        X-ray studies
194,211                                       absence of true long range translational
Freedericksz transition, 99-101                     order in smectic A, 313
Helfrich deformation in smectics, 315            amplitude of density wave in smectic A
hydrodynamic instabilities, 196, 201                and pretransitional smectic-like order,
mechanical instabilities in columnar                307-8
liquid crystals, 400-1                        correlation length of two-dimensional
smectic C-smectic A transition through              lattice in the columnar phase, 390
layer compression, 373                        diffraction from
square grid pattern in cholesterics, 281            biaxial nematic, 416; cybotactic
thermal instability, 204                            nematic, 4; ordinary nematic, 3
twisted nematic cell, 109-10                     evidence of interdigitated antiparallel
translational order parameter, see under               ordering in 5CB and 7CB, 79-80
order parameter                               grain size in TGB phase, 360
tricritical point, 309, 341, 347-8, 371             hexagonal-rectangular transition in
triple point, 28-9                                     columnar phase, 391
Tsevetkov's experiment, 145-8                       hexatic ordering, 360-2
applications, and under continuum                polymorphic forms of smectic A, 350;
theory                                           reentrant phases, 355-7
460                                            Index
location of                                      structure factor and Debye-Waller
critical end point, 355; critical point           factor for the columnar structure, 399
353; tricritical point, 354                   structures of columnar phases, 390-3
longitudinal and transverse correlation          studies in the vicinity of the NAC
lengths and susceptibility near the               multicritical point, 377-8
smectic A-nematic transition, 343,            tilt angle measurements in smectics C
346-7                                             and C*, 372, 381
ordering of cores and tails in columnar         XY model, 347, 361, 370
phase, 391
reflectivity from free surface showing          Zwanzig's model, 31-6
surface induced smectic order.


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