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
•   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:
                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
      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
• 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

               ▼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
    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

                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  1    
                       Furthermore, if we consider aspect ratio r>>1, then
                          La  (log( r ) / r 2 )  O (r 2 )
    clay platelets
                       Geometric polarization tensor E(m):
                                                      
                     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:

           1I   1 2 ( 2   1 )(
                                        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.
We show       Ε j  E jiso  E jnema  j

 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 )
       La                             1 2
                1,  e   1I             M  O( 1 2 ),
     1  2                           La
       La                                      2 2
               ~ O(1),    1I 
                                                        M  O( 1 2 ),
     1  2                            1
                                               1  2
                1,  e   1I   2 2M  O( 1 2 ).
     1  2
 where M is the second moment ten of f (m).
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)
                         elongation rate
               Pe 
                    orientation relaxation rate

                       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 

Conductivity                                 N.B. Similar
enhancement                                gains at takeup
                                           location as pure
                 Final fiber radius

  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,
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

          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


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

 Bottom surface
          Nano-Composite Property Fluctuations and
         Heterogeneity During Steady Film Processing
 Max principal
                                                 The degrees of orientation enter
                                                 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.
                                   of level sets The maximum relative
                                   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
       Heterogeneity During Steady Film Processing
Max principal                                            Another spatio-temporal attractor
conductivity                                             arises in steady processing
                                                         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
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 “load sharing”?
•   In failure?
•   In self-healing strategies?
•   Can these features guide new materials
         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,
  resolve them, return to the pipeline of hi-performance materials,
  establish new benchmarks, new open issues, and so it goes.

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