# Modeling the pipeline of high performance_ nano-composite - IMA

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```					   Modeling the pipeline of high
performance, nano-composite
materials and effective properties, II
IMA Workshop
Composites: Where Mathematics Meets Industry

February 8, 2005

M. Gregory Forest
UNC Chapel Hill
Mathematics & Institute for Advanced Materials,
Nanoscience & Technology
Acknowledgements
Collaborators:
•   Qi Wang, Florida State
•   Ruhai Zhou, Old Dominion
•   Xiaoyu Zheng, Eric Choate, Joo Hee Lee,
UNC, Bill Mullins ARO + UNC
•   Robert Lipton, LSU
•   Hong Zhou, Naval Postgraduate School
Research support from:
•   AFOSR, NSF, NASA, ARO
Model the nano-composite pipeline
Anticipate:   Control wrapper around 4 direct solvers
Parametrize control variables      f(m,x,t)
1. Nano-element + Matrix features                       Phase 2
2. Flow type and rate                          Quench into
Nano pdf
3. Confinement effects                          solid film

f(m,x)       Nano pdf
The holy grail              M(f)        2nd moment
(4th moment)

Performance Properties                           Phase 3
Elliptic   Effective anisotropic
E field, Temperature, Strength
Transport, Healing….. For given               property tensor:
Solvers            σ (x)
boundary value problems and        + time       2nd or 4th order
“shaping” of the material
stepper
The nematic polymer, nano-composite, high
performance materials pipeline.

• Lecture 1 focused on why? & what’s possible?
in flow processing of nematic polymer nano-
composites Anisotropy, dynamics, and
heterogeneity of the molecular ensemble
orientational distributions, associated stored
elastic stresses, and hydrodynamic feedback
phenomena.
• Lecture 2: we now map these questions &
answers onto effective property tensors of nano-
composites, either based either on modeling
results or experimental data.
Nematic Polymer, Nano-Composite
Effective Electric Conductivity

Ј=ΣE
▼▪Ј=0
▼X E=0
J is the current
E = ▼φ is the electric field
φ is the electric field potential
Σ(x) is the composite conductivity
tensor
Forget resolving the actual Nano-Composite Elliptic
Problem….. Σ(x) is simply not computable, so
Homogenize up to Effective Conductivity
• The power of homogenization is that we can average the
above elliptic problem and recover an identical
formulation for volume-averaged current ‹ Ј › and
electric field ‹ E ›
• They are related by a new upscaled constitutive relation
‹ Ј › = Σ^(x) ‹ E ›
through the effective, anisotropic conductivity
tensor Σ^
which we either analytically (in asymptotic or moment-
closure limits) or numerically (in general) generate from
results of the previous lecture!
• With Rob Lipton’s guidance, we either use or extend
results to spheroidal inclusions of high aspect ratio
Homogenization formula in low volume fraction
limit of spheroidal inclusions (analogous formulas
for thermal and dielectric properties)

Effective                                        Probability distribution
Volume fraction
conductivity                                    function of nano-inclusions
of nano-inclusions
tensor

 e   I (  ) E(m) f (m)dm O( 2 )
1    2 2      1      S2

Conductivity                                  Orientation-averaged
of pure matrix           Conductivity          Polarization tensor
of nano-inclusions
Geometry of nano-inclusions
allows explicit evaluation of the integrals
Depolarization factor:

abc                        ds
Spheroidal molecules     La  
2 0
( s  a 2 ) ( s  a 2 )(s b 2 )(s  c 2 )
For axisymmetric (spheroidal) molecules, r=a/b,
b=c, the above Integral can be calculated explicitly:
1  2  1  1   
La  2  ln            1,   1  (1 / r )
2

  2  1    
rods
Furthermore, if we consider aspect ratio r>>1, then
La  (log( r ) / r 2 )  O (r 2 )
clay platelets
Geometric polarization tensor E(m):
-1
                          
E(m)  
1 2 1 L mm1 2 1 L nn 1 2 1 L kk 
a   

    1          1 b 

   1   c
  
Exact scaling laws of effective
electrical conductivity tensor
Since La  Lb , E(m) is explicitly invertible as a linear
function of the tensor product mm
This simple observation leads, because of orthogonality of
spherical harmonics, to a remarkable collapse of       E(m)* f(m)
over the sphere as a linear function of the second moment tensor

M(f), leading to an (Zheng, Forest, Lipton et al. Adv Funct Mat. ’05)
Explicit effective conductivity tensor formula:

2
   1I   1 2 ( 2   1 )(
e
I
 2   1  ( 2   1 ) La
( 2   1 )(1  3La )
                                                             M( f ))  O( 22 )
(( 2   1 )  ( 2   1 ) La )( 1  ( 2   1 ) La )

Remember: M (f) encodes vol %, shear rate, and molecule geometry
This formula does not care where you get M from, theory or experiment
Anisotropic conductivity enhancements
Effective conductivity is a tensor, with 3
        
principal directions (eigenvectors           n1 , n2 , n3 )   Inherited from M(f)

and principal values (eigenvalues           1, 2 , 3 )       Explicit from
Order parameters
We define the relative conductivity enhancements                of M(f)

i   1
Ei           , j  1,2,3.
1
We show       Ε j  E jiso  E jnema  j
Pe

Note: enhancements decomposes into a linear sum of enhancements
of the dilute isotropic phase, of the nematic phase above the critical
volume fraction of order transition, and of flow-induced pdf.
Finer scaling properties require ordering of two
inherent asymptotic parameters of nano-composites:
high aspect ratio vs. high conductivity contrast

9         3      1
2       4
r ~ O(10 ~ 10 ), La ~ O(10 ~ 10 ),                           ~ O(1012 ~ 105 )
2
La                             1 2
               1,  e   1I             M  O( 1 2 ),
1  2                           La
La                                      2 2
              ~ O(1),    1I 
e
M  O( 1 2 ),
1  2                            1
La
1  2
La
               1,  e   1I   2 2M  O( 1 2 ).
1  2
where M is the second moment ten of f (m).
sor
Hierarchy of materials & property predictions
for nematic polymer nano-composites

• Bulkmonodomain steady phases, at rest, in pure
extension, in fiber spinning & after shearing Adv.
Functional Materials, to appear; Macromolecular
Symposia, to appear

•Transient, fluctuating properties due to “rheo-
oscillator” response to steady flow (to be submitted,
Feb. 05)

•Heterogeneous film properties from confined
processing flows (American Chemical Society,
Polymer Nano-Composite Symposium, March, ’05)
Property inheritance of hysteresis & bistability
of monodomains at rest and after shearing
Extension-induced bulk phases: Effective
conductivity vs. vol %, uniaxial & biaxial
extension (Zhou et al. Macro. Symposia ’05)
def
elongation rate
Pe 
orientation relaxation rate

conductivity
enhancement
I-N transition

Vol %
Uniaxial extension        Biaxial extension
N.B. order of magnitude gain over shear
Fiber effective conductivity vs final radius
(equiv., 1/takeup speed in fiber process)

 ma x   1
ma x 
1

Conductivity                                 N.B. Similar
enhancement                                gains at takeup
location as pure
extension

From Forest,Wang,Zhou ’00 spinning simulations
Oscillating conductivity in simple shear vs shear
rate: inheritance of strong pdf shape distortions at
T/W transition
Effective conductivity fluctuations in steady shear

Kayaking bulk attractor. N=5.2, Peclet number=4

Property contrasts: σ1=10^(-4), σ2=10^8,
L_a=10^(-6)
Effective conductivity fluctuations in steady shear

Chaotic attractor: N=5.2, Pe=4.044
Molecular morphology & rheology
in confined film flows: strong elasticity limit

Flow feedback              Shear stress
ouette cell schematic      Relative
Stored                  conductivity
normal                   enhancement
stresses                  ma x 
 ma x   1
1

Snapshot ~ 5 minutes after startup
Resolved kinetic simulations (R. Zhou) in plane Couette cells:
Toward realistic Frank elasticity (Er=500), moderate plate speeds
(De=1), Tilted Molecular Anchoring at plate walls

Temporally
Periodic
Structure
Attractor

Peak axis of
PDF strongly
oscillating at
fixed film height
& across the film
at each time

Principal axes of Conductivity Tensor follow the PDF
Now image the principal value, aka the Flory
order parameter of the PDF, across the film        N=500, De=1, Tilted Anchoring

Order parameter distortions, indicating oscillatory
shape deformations of the pdf with “flashing local defect cores”
Significant flow feedback structure associated with molecular and
nematic structure attractor for same run: Er=500, De=1, Tilted

Primary flow velocity distortions due to
stored elastic stresses
Transfer database for the molecular pdf to effective
conductivity principal value thru the film:
flashing local “conductivity voids” near defect structures
Top surface

Level sets of maximum
principal value of the Effective
Film                Conductivity Tensor
height

Bottom surface
Nano-Composite Property Fluctuations and
Max principal
The degrees of orientation enter
conductivity
strongly into the principal
conductivity values. One finds a
dramatic drops in effective
conductivity localized in space
and time around defect cores
Film                    Process
where the PDF goes through strong
height                  clock time
defocusing to mediate layers with
director tumbling versus finite
oscillation (wagging) of the principal
Color scheme axis.
Film
of level sets The maximum relative
height
of maximum    enhancement reaches ~105 %.
conductivity  Thus, at ~1% vol fraction of the
nano-phase, a gain in 5 orders.
Process clock time
Spatial-temporal structure of maximum scalar effective
conductivity with Er=500 , De=4 , for parallel anchoring.
Nano-Composite Property Fluctuations and
Max principal                                            Another spatio-temporal attractor
conditions. The entire gap
experiences finite amplitude
oscillation of the principal axis of
M(f) at each gap height. The
Film                      Process           interesting feature of the PDF of
height                    clock time        wagging oscillations versus
tumbling is that energy shifts into
focusing and defocusing of the
Color scheme PDF rather than rotation. This
Film                                  of level sets
height                                                   implies enhanced oscillation
of maximum         and variability in the principal
conductivity       values of the effective
conductivity tensor & less
Process clock time                         tortuous paths of principal axes
Spatial-temporal structure of maximum scalar effective          Ref: Forest, Zhou, Zheng,
conductivity with Er=500 , De=6 , for parallel anchoring.       Lipton, Wang; invited paper
Amer. Chem. Soc., March 05
Chaotic structure regime (to be submitted)
full kinetic flow-molecular simulations

Post processing of major director dynamic heterogeneity

Polar angle                       Azimuthal angle
UPSHOT: spatial coherence with temporal chaos
Stored normal stress differences & shear stress in chaotic attractor:
Mechanical and conductivity properties inherit these features
through 2nd and 4th moment tensors!!

N1                                                           Shear stress
N2
Degree of orientation, s, which correlates with principal conductivity values
and flow feedback in primary flow component

S=d1-d2                                        Velocity vx
What roles do these dynamic structures play in
multifunctional material properties?

•   In permeability?
•   In failure?
•   In self-healing strategies?
•   Can these features guide new materials
design?
A plug for the benefit to all of relationships
between pure science, mathematics, engineering,
and industrial technologies
• Underlying these technological applications is the remarkable
behavior of nematic polymers in shear-dominated flow. This is a
perfect storm of nonlinear phenomena, which must be understood
from rigorous multiscale, multiphysics theory, modeling and
simulation in order to approach the holy grail of controlling
performance features on demand.
• Indeed, the direct models and solutions reported in these lectures
give guidance toward qualitative phenomena (what is possible in
such nano-composite materials), followed by scaling laws and
bounds on properties across composition, processing, &
confinement space. We have far to go to remove idealizations.
• Fundamental science driven by applications is one mode to be in;
the pendulum is now poised to swing toward technological priorities.
My lectures are meant to convey the interplay, and shifts in
emphasis, as we identify key rate-limiting steps and open issues,