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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 mm1 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(1012 ~ 105 ) 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 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, 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 Heterogeneity During Steady Film Processing 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 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 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 “load sharing”? • 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, resolve them, return to the pipeline of hi-performance materials, establish new benchmarks, new open issues, and so it goes.

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