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					 From water-fat emulsions to radar cross
section control: A historical perspective on
     the engineered artificial material

                 Rodolfo E. Díaz
    Mechanical and Aerospace Engineering Dept.
          Laser Diagnostics Laboratory/
             Nanomechanics Program
             Arizona State University
                   rudydiaz@asu.edu



                 September 1999
The engineered artificial material, or
 composite, has wide applicability
•Mechanical Engineering (concrete), Thermal management
(pitch fiber), Geophysical modeling (soil modeling for
geosensing), Biomedical applications…
•In particular, the electromagnetic composite has been of
interest in optics and electrical engineering for over a century.
•Over this time, the two engineering tasks of Analysis and
Synthesis have been developing hand in hand.
•The ultimate question: How do we design and manufacture an
artificial material with prescribed advantageous properties?
 Over this time, we see recurring
 motifs in the problems addressed
•Analysis                                •Synthesis
•Explanation of the optical properties   •Artificial dielectric lenses
of emulsions and suspensions             •Hi-K low loss artificial substrates
•Measurement of the butter-fat           for printed antennas.
content of milk or the red blood cell    •Artificial structures as “matching”
count of mammalian blood.                devices (w/ or w/o dispersion).
•EM properties of metal insulator        •Lossy materials for antenna
composites.                              backlobe suppression, anechoic
•Microwave properties of composites      chambers, radar cross section
and exotic materials, and their          control.
measurement.                             •Design of exotic materials (chiral,
•Reverse engineering of the same.        non-reciprocal, non-linear, optically
                                         active salts.)
   The motifs are question that we
      keep asking ourselves...
•How can a measurement of the composite A+B yield the
individual fractions of A and B?
•Can a measurement of a composite A+B of known
fractions of A and B yield the (unknown) properties of A?

•How can a material with an extreme property (high
dielectric constant, high loss, high conductivity, etc.) be
made, reliably, cheaply?
•What are the physically realizable limits of such materials?
The answer might be found in a truly
 general theory of effective media.
•Focusing on the binary material, a practical theory must:
•Include the shape of the components.
•Contain fitting parameters.
•Possess a well defined region of applicability.
•Let’s follow a Historical Perspective to see how close we
have come to this goal:
•The role of particle shape and the effective field.
•The issue of the internal morphology.
•The encompassing nature of the morphology function.
  The history can be divided along
  two principal paths of research.
•The random mixture                •Structured dielectrics
•Spheres in a host                 •Arrays of spheres
•Spheres and spheres               •Arrays of ellipsoids
•Coated spheres                    •Arrays of slabs
•Coated ellipsoids                 •Arays of rods, disks, spirals, etc.
•Randomly oriented ellipsoids      •Arrays of panels of the same…FSS
•Stochastic models of aggregates   •2D Arrays of planar structures
•Fractal clusters                                     (Honeycomb)
•Percolation theory                •2D Arrays of volumetric structures
•Cluster theory                                 (Absorber pyramids)
                                   •3D arrays = PBG
       We will start along the random
        mixture path and end with
           structured dielectrics
•It all started with an attempt at understanding natural
dielectrics, by seeking to relate the macroscopic permittivity to
microscopic dipole moments.
•In a binary medium with polarizable inclusions,
Maxwell(1873), Rayleigh, Clausius-Mossotti(1879), Lorentz-
Lorenz(1880), Maxwell-Garnett(1904), Bottcher(1946) agree
that the simplest expression is:
                       eff          inc
                            1            1
                      host         host
                                f
                      eff          inc
                            2            2
                      host         host
       This CM formula spans the range
          from host to inc uniformly
                                 •The problem is that for natural materials,
           3
       1 10




           100
                                 in which the molecules possess intrinsic
  CM
       i
                                 polarization, it blows up.
            10
                                 •Onsager fixed that in 1936, highlighting
                                 as crucial the issue of the mean field.
               1
                   0   0.5   1
                       p
                        i

•This opened up the way for much of the early work in the
development of a rigorous theory of mixtures.
•CM, LL, Rayleigh addressed the question for “spheres”.
•Then Polder-Van Santen(1946), and Taylor (1965) squarely
addressed it for ellipsoidal shapes.
    But before either of them, there
        was Hugo Fricke (1924)
•He looked at the same kind of problem (a heterogeneous
medium of filler particles in a surrounding host) in a different
way.
•Instead of asking, “what is the mean field seen by an arbitrarily
shaped particle?”, he assumed a typical unit cell of the medium
could be locally represented by a microscopic capacitor inside
which the particle had been placed.
                                                 -------------
                                           ---                   ---
                    ~                  +   +++       +++
                                             +++++++++
By estimating the charges induced
 on the non-spherical particle he
 obtained a physical model that:
•Sidestepped the issue of the mean field.
•And yielded a formula very much like Wiener’s (1912), self-
consistently including the ellipsoidal shape of the particle.
                   eff          inc
                        1            1
                  host         host
                            f
                  eff          inc
                        x            x
                  host         host

•Where x is the shape factor.
   This HF model is as uniform as
   CM but includes the collective
           shape effect.
•The behavior of the suspension              1 10
                                                 3

depends on the shape and the
volume fraction:                      RM
                                        i

•There is enhancement at low
                                                  100
                                      P rolate
                                              i
volume fractions directly             Oblate
                                            i
proportional to the polarizability                10
                                      CM
of the individual particles.            i


•But as the volume fraction
                                                     1
increases, particles start stealing                      0   0.5   1

flux from each other.                                        f
                                                              i
                           Assuming a reasonable model for red corpuscles as oblate
                          spheroids of aspect ratio 1/4.25, he obtained agreement with
                          experimental data of the order of 1% over a range of volume
                                          fractions from 11% to 91%.

                          25




                          20                                   HF                                           0
                                                                                                                     HF




                                                                         Error in predicted Concentration
Cond. Serum/Cond. Blood




                          15                                        CM                                               CM

                                                                                                            10
                          10




                          5

                                                                                                            20

                          0
                               0   0.2   0.4           0.6   0.8     1                                           1                     10                100
                                          Volume Fraction                                                                 Meassured Conductivity Ratio
Fricke’s model illustrates the three
 major requirements to be met by
   an effective medium theory.
(1) It includes the shape of the particle.
(2) This shape is not taken literally, but rather it is allowed to
be a fitting parameter in the model.
(3) The resulting model is valid (plausible) over a useful range
of volume fractions.
•The objection can be raised that, just as CM, it implies 100%
filling is possible. But this is not a rigorous objection since
materials exist (such as sintered ceramics) for which 100%
filling is indeed possible.
•It is, admittedly, an approximation.
 So, how much better can rigorous
          models fare?
•Before Polder and Van Santen’s rigorous mean field theory,
the general approach to rigor was to change the question from,
“what is the mean field?” to “what is the mean environment?”
•Bruggeman(1935), Maxwell-Garnett(1904), Bottcher(1946),
Looyenga(1965), Ping-Sheng(1980) all obtain simulations of
the average medium by considering the particles as coated, to
some extent, by the host, or the average medium, or each
other.
•The question is, what extent is appropriate?
Niklasson and Granqvist illustrate
     these alternatives for us
         Maxwell-Garnett                     Bruggeman Symmetric
         MG                                      Bs
                 h i                                   i,h

         f determines the ratio       f = Prob of being i, 1-f = Prob of being h
         of volumes                   f is built up iteratively from mixture of ih
          Ping Sheng                         Bruggeman Asymmetric
         PS                                                     Ba
                    i,h                          Ba -D
                 h,i                                       i

         1/3 3
  (1-f     ) =Pih= Prob of i w/ shell of h
                    3
  (1-(1-f ) 1/3 ) =Phi= Prob of h w/ shell of i
      To every set of assumptions
       corresponds a new EMT
•The result is that we can have symmetric or asymmetric
theories.
•The validity can depend on the contrast between the particle
and the host properties or it can be independent of it.
•There is no connection between the assumption used and a
physical process that justifies it above any of the others.
•And there may be no accounting for the shape of the particles.
•The variety of assumptions certainly are equivalent to a
variety of fitting parameters, but the range of validity thus
obtained is unknown.
 How much better can we do with
  rigorous mean field theories?
•Sillars (1937) developed a theory for dielectrics containing
semi-conducting prolate spheroid particles following
Wagner’s 1914 work on spheres.
•Polder and Van Santen (1946) obtained a formal solution to
Laplace’s equation on the ellipsoidal boundary, including
randomness, while ignoring correlation effects.
•Taylor(1965) put Polder and Van Santen’s work on the
sound basis of generalized boundary conditions obtaining a
theory valid for complex .
        So, what’s the problem?

  They prove that ellipsoidal particles of high aspect
  ratio have a much stronger effect on the material’s
  properties than an equivalent volume of spheres.
• For this very reason they have a limited range of
  validity: f 2<<1.
• They can contribute to misleading conclusions:
 “There is no doubt that, given a choice of conductivity
  and shape, we can produce a loss of any magnitude at
  any frequency with as small a quantity of conducting
  medium as we please...”                   Sillars
And, for all the labor involved in these
 formulations, none of them meet all
   three of Fricke’s requirements.
 • The effective coating models meet requirement number 2,
   giving us fitting parameters, but they have no guidelines
   on the range of validity for shape and volume fraction.
 • The mean field models meet requirement number 1,
   correctly accounting for shape, but that leaves no fitting
   parameters. Plus, their limited range of validity is a
   drawback.
 • The classic dilemma between rigor and practicality…
 • If this is so, why are these theories so prevalent today?
    The promise of something for
          (almost) nothing
•Sillars quote… A small amount of slender particles (slivers)
can have a large effect on the material properties.


                   i               i
                        h                h

          For f small            For f small
          and i>>h(=1)         and i>>h(=1)
          eff ~ 1+3f            eff ~ 1+fi/3
     The ideal material: a useful
  susceptibility from a vanishingly
      small addition of slivers
•This is misleading because in the case of conductors i can be
virtually infinite, s/w0>>1, so that f can be arbitrarily small.
•But if we try to apply the same reasoning to obtaining
magnetic susceptibility from non-infinite mi we find the
limitation to f 2<<1 extremely restrictive.
•Take Nickel, with mi at microwave frequencies of the order of
30. At f =2%, the maximum effective permeability that high
aspect ratio slivers can give is meff =1.2.
•And even f =2% is pushing the limit of validity...
 A random mixture of any kind of particle
has a finite probability of generating chains
                 and clusters




  •Chains of spheres can behave like high aspect ratio objects
  (Percolation theory - Lagarkov et al 1992)
  •Under dynamic mixing, the lowest energy configuration of
  slivers may be spherical clusters (Doyle and Jacobs 1992 )
And so the discussion of the most useful
 artificial material reaches a surprising
                conclusion
•Even though the morphology of the individual constituent is a
crucial factor, it is the internal morphology of the mixture that
controls the material’s final property.
•And this transition from individual to collective must be a
function of the particle shape and f.                1 10
                                                         3



•Fricke’s model recognized this: At low f,    RM
                                                i
                                                          100
the particles contribute their full           P rolate
                                                      i

                                              Oblate
enhancement, as in the RM limit. But at       CM
                                                    i
                                                          10
                                                i
high f, they switch over to the CM limit.
                                                             1
                                                                 0   0.5   1
                                                                     f
                                                                      i
The collective behavior has its own
   characterizable morphology
                                RM
           •Isolated slivers
           resemble an early
           percolating system


                                 CM
           •Clustered slivers
           resemble a late
           percolating system
   So we must change our focus from the
     morphology of the particle to the
        morphology of the mixture
•Which is unknown… The first approach to guess is Percolation
Theory, regularized to dielectric particles (Sihvola et al (1994),
McLachlan et al (1992))
                          eff   h                             i   h
                                                  f
                 eff  2 h   eff   h            i  2 h  eff   h 

•Merrill(1996) following Lagarkov, where pc is the percolation
threshold.                1 h
                                           3
                                        1  pc  hpc
                                           2

          h   h / i , a                         1   ,b                      1
                               1  (1  p / pc )h   2
                                                               pc  (1  pc )h   2



               p  pc
                                b 2 x 2  (h  ax) 2                eff  x i
                 pc
  The second alternative is the spherical
cluster theory of Doyle and Jacobs (1990)
generalized to dielectrics (Diaz et al 1998)
  •Let p be the volume fraction of particles and pc be the
  percolation limit. Assume that of the p particles, f are clustered
  into percolated spheres and (1-f ) are isolated.
  •Then the average polarizability is
              b  p (1  f )
                                i   h
                                           pf
                                                i   h  h  2i 
                                i  2 h      9 i  h  2 pc  i   h 
                                                                           2



  •And the effective permittivity is  eff  (2b  1) / 1  b
  •Similarly, for nonspherical particles, we start with the b for
  metal particles derived in DJ, JAP 71(8),3926, 1992, and
  proceed in the same way.                    p 1           
                                                             
                                             b  p u0    u0 
                                                       pc  pc 
 The result is that we have two theories
     that could meet Fricke’s three
               requirements
•In DJ, shape is accounted for by the shape factor u0.
•In ML,(McLachlan’s GEM version of Percolation Theory) the
shape is accounted for implicitly since the depolarization factor
is related to the percolation threshold fc.
•In DJ, there are two adjustable (or fitting) parameters: the
shape factor u0 , and the percolation limit pc .
•In ML,the two adjustable factors are the critical exponent t
and the percolation threshold fc .
•In DJ the formulation is valid up to pc .
•In ML the formulation is valid to either side of fc .
Both theories are modeling the internal
    morphology and, of course,...
•They disagree.                                  1 10
                                                     3

•Consider a mixture of                                                              0.63
i=1000 in h=1.



                               Effective Permittivity
•Let fc= pc =0.63,u0=10, and                            100

t=1.8.
•The two theories model two
                                                        10
completely different
behaviors.
•Note the trend from RM to                               1
                                                              0   0.2   0.4        0.6     0.8   1
CM or viceversa.                                                        Volume Fraction
   It is time to draw some practical
               conclusions
•Rigorous Mean Field or Average Environment theories can be
extended into multi-parameter EMTs that are useful (meet all
three of Fricke’s requirements).
•There are at least two (and perhaps more) families of these
multi-parameter theories (exemplified by the DJ and ML
formulations) with different asymptotic limits.
•But, given a material, before knowing which family to apply
to it, and how to select its parameters, we must know
something about its internal morphology.
•Then, why not start with a theory of morphology?
     An analytic framework for the
      modeling of effective media
•Diaz, Merrill, Alexopoulos, JAP 84 (12) 6815, 1998.
•It is shown that binary mixtures have analytic properties that
describe the internal geometry of the mixture and are
independent of the components.
•All EMTs of such materials can be described in terms of
analytic functions in the u=i/(i-1) plane.
•These functions take the form of Debye poles in which the
strength of the pole and its relaxation frequency are equivalent
to the geometric parameters describing a partially filled
capacitor.
 An EMT merely describes the way
    this ideal capacitor is filled.
•The most general partially filled
capacitor contains three types of         1
regions: (a) empty, (b) fully
filled, and (c) partially filled.    h
•These correspond to portions of
the medium where: (a) the host                     1
is fully connected, (b) the
inclusions are percolated, (c) the            i
inclusions and the host are both
disconnected and in series with
each other.
             Why does this work?

•Because the solution to
Laplace’s equation in a
heterogeneous medium is merely
a description of how the Electric
Flux is conducted from one
capacitor plate to the other.
•And when all multipoles are
taken into account the result is a
parallel sum of series capacitors.
             What is it good for?
           A workable framework...
•When all possible factors are taken into account…
 •Inclusion and host (frequency             •Statistical distribution of the same
 dependent) permittivity                    •Including distribution of sizes,
 •Particles’ mean morphology                number of particles included,
 •Statistical dstribution about this mean   packing density.
 •Statistical distribution of the spatial   •Spatial distribution of clusters and
 orientation of the inclusions              individual particles
 •Mean morphology of the clusters           •The effect of volume fraction on all
 and/or chains formed by the inclusions     these parameters...
•“Rigorous” deterministic theories become unworkable.
•The morphology framework is a thermodynamic simplification.
Because it is a consequence of
           causality
•Whatever the process was that was used to produce
the material,
•whatever the inclusion particles and the surrounding
host were,
•regardless of the microscopic complexity of the
finished product,
•The internal morphology of the mixture must be
expressible as a sum of these poles, or partially filled
capacitors.
                    If this is true...
We have three powerful tools for the
 analysis and design of artificial
            materials
•There is a one-to-one correspondence between the frequency
dependence of a material filled with resistive particles and the
internal morphology. Measurement of the first will reveal the
second.
•Once the morphology function is found (the sum of capacitors),
the properties of the medium can be determined trivially for
arbitrary properties of host and inclusion (electric or magnetic).
•An engineering estimate of the flux paths in a structure or
mixture suffice to obtain a practical effective medium
description of the material.
 The morphology function is
ideal for structured dielectrics.
• Many useful materials can be described by periodic arrays
  of unit cells containing dielectric segments or structures.

                                                   Pyramidal
                                                   Absorbers


                                                   Proposed
                                                   Hollow
      Honeycomb                                    Pyramidal
                                                   Absorbers
      A unit cell of the hollow
      pyramid cross section...
• Is a first order model for the effect of the structure on low
  frequency (TEM like) waves impinging normally on an
  anechoic chamber wall.

                                      Applying a Voltage between
          h
                                      these two planes simulates the
                                      low frequency propagation of
PEC            h                     an electromagnetic wave along
                                      the axis.
                    i




               PMC
  The limiting morphologies of
this structure are easy to deduce
• Use the electric displacement flux paths in the two limits
  of  very high and  very close to 1.

                                         Low i
                                         limit
           h

                h
                     i
                                         High i
                                         limit
  For low i (=high values in the
 u-plane) three structures add in
              parallel
        1
                                         h (1  s)
        s                          C1 
                                             1
        x    t                                        2
                 1                  C2 
                                               1         1
                                                     
                                              i t    ht 
                                                          
                                             s  1 s 
                                                      1
p=s 2                                C3 
    2                                          1        1
f= x                                                
t=0.5(s-x)                                   i x    h x 
                                                          
                                             2t   1  2t 
                 Clow eps  C1  C2  C3
For high i (=low values in the
   u-plane) there are four...
  1
  sw
                            h ( sw)
  s                   C2 
                            1 s
   x     t    1
                           i 2( s / 4)
               s/4    C3 
                               2t
   0.6
sw=s
                            i 2t
                      C4 
                              x
        h (1  sw)
  C1 
             1                                1
                      Chi   C1    1
                                      C2    C13  C14
   When these are plotted versus
           ui=1/(i-1)
•The high  limit can be fitted to a Debye     ui 
                                           e1 1  
function that matches its low u behavior:      u1 
•The low  limit can be fitted to a Debye e2 1  ui 
                                                    
function that matches its high u behavior:      u2 
•And these can be analytically continued
into each other by requiring that:
      et 1u1  et 2u 2  e2u 2 _ and _ et 1  et 2  e1
•So that the morphology function is:
                                        u             u 
                      e(u )  et 1 1  j   et 2 1  j 
                                        u1            u2 
                       valid for all values of i
                                                              • For instance let
             1.5
                                                                p=35%, f=0.8p, and
              1
                                                                i =5 - j2/w0
Low _U
      j
             0.5                                                       3
Hi gh _U
        j                                                          
              0
Ud eb_Tot                                                     Re
         j                                                             2

             0.5                                              Im


               1                                                       1
                 4     3                                  3
             1 10 1 10   0.0 1   0.1    1   10   10 0 1 10
                                    iu
                                      j

                                                                        0
                                                                           8    9    10   11   12   13
                                                                       1 10 1 10 1 10 1 10 1 10 1 10
                                                                                       w
                                                                                        k
  The result can be verified because the
structure can also be modeled using two
                  EMTs

• It is easy to show that the polarizability of a thin shelled
  hollow dielectric is given by the Reverse Mossotti limit.
  Then the Clausius-Mossotti eqn. can be used to put this
  polarizability in the unit cell.    3


                                      eh epsk
                    1 1. ( 1 0.2) .
                                      eh 1. epsk                  2
  Hollowk
            epsk.                                    Re To tal
                                      eh epsk                 k
                     1 ( 1 0.2) .
                                    eh 1. epsk       Im To tal
                                                              k
                                                                  1
                               Hollow eh
                                     k
                    1 1. pe.
                               Hollow 1. eh
                                     k
       T otal                                 . eh
            k
                               Hollow eh
                                     k
                    1 pe.                                          0
                                                                      8    9    10   11   12   13
                            Hollow 1. eh
                                  k                               1 10 1 10 1 10 1 10 1 10 1 10
                                                                                 w
                                                                                   k
Similarly, an effective medium model of
 any structured artificial material (such
    as honeycomb) can be derived.

• Present day Radar Cross Section Control materials fall in
  one of three categories:
       Structured dielectric loss materials
       Random mixtures of dielectric loss materials
       Random mixtures of magnetic loss materials
• The key to understanding, modeling, and predicting their
  behavior lies in a correct model of the internal
  morphology produced by the manufacturing process.
• Once that is obtained all “what if” scenarios can be played
  to design an optimized system.
                CONCLUSION:
A review of the history of the artificial
        material has shown:
•That the Rigorous Mean Field and Average Environment
theories have evolved into sophisticated two-parameter EMTs
that can be useful, provided we know a priori the internal
morphology of the material.
•However, to take into acount all reasonable process factors
with such theories would require at least 10 adjustable
parameters… making that approach unworkable.
•A practical alternative is to use the analytic morphology
function: a compact representation of all possible
morphologies.
•Its use on structured artificial dielectrics was demonstrated.

				
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