Percolation-induced exponential scaling in the large current tails of random resistor networks

There is a renewed surge in percolation-induced transport properties of diverse nano-particle composites (cf. RSC Nanoscience & Nanotechnology Series, Paul O'Brien Editor-in-Chief). We note in particular a broad interest in nano-composites exhibiting sharp electrical property gains at and above percolation threshold, which motivated us to revisit the classical setting of percolation in random resistor networks but from a multiscale perspective. For each realization of random resistor networks above threshold, we use network graph representations and associated algorithms to identify and restrict to the percolating component, thereby preconditioning the network both in size and accuracy by filtering {\it a priori} zero current-carrying bonds. We then simulate many realizations per bond density and analyze scaling behavior of the complete current distribution supported on the percolating component. We first confirm the celebrated power-law distribution of small currents at the percolation threshold, and second we confirm results on scaling of the maximum current in the network that is associated with the backbone of the percolating cluster. These properties are then placed in context with global features of the current distribution, and in particular the dominant role of the large current tail that is most relevant for material science applications. We identify a robust, exponential large current tail that: 1. persists above threshold; 2. expands broadly over and dominates the current distribution at the expense of the vanishing power law scaling in the small current tail; and 3. by taking second moments, reproduces the experimentally observed power law scaling of bulk conductivity above threshold

Kinetic Structure Simulations of Nematic Polymers in Plane Couette Cells. II: In-plane structure transitions

Nematic, or liquid crystalline, polymer (LCP) composites are composed of large aspect ratio rod-like or platelet macromolecules. This class of nanocomposites exhibits tremendous potential for high performance material applications, ranging across mechanical, electrical, piezoelectric, thermal, and barrier properties. Fibers made from nematic polymers have set synthetic materials performance standards for decades. The current target is to engineer multifunctional films and molded parts, for which processing flows are shear-dominated. Nematic polymer films inherit anisotropy from collective orientational distributions of the molecular constituents and develop heterogeneity on length scales that are, as yet, not well understood and thereby uncontrollable. Rigid LCPs in viscous solvents have a theoretical and computational foundation from which one can model parallel plate Couette cell experiments and explore anisotropic structure generation arising from nonequilibrium interactions between hydrodynamics, molecular short- and long-range elasticity, and confinement effects. Recent progress on the longwave limit of homogeneous nematic polymers in imposed simple shear and linear planar flows [R. G. Larson and H. Ottinger, Macromolecules, 24 (1991), pp. 6270-6282], [V. Faraoni, M. Grosso, S. Crescitelli, and P. L. Maffettone, J. Rheol., 43 (1999), pp. 829-843], [M. Grosso, R. Keunings, S. Crescitelli, and P. L. Maffettone, Phys. Rev. Lett. 86 (2001), pp. 3184-3187], [M. G. Forest, Q. Wang, and R. Zhou, Rheol. Acta, 43 (2004), pp. 17-37], [M. G. Forest, Q. Wang, and R. Zhou, Rheol. Acta, 44 (2004), pp. 80-93], [M. G. Forest, Q. Wang, R. Zhou, and E. Choate, J. Non-Newtonian Fluid Mech., 118 (2004), pp. 17-31], [M. G. Forest, R. Zhou, and Q. Wang, Phys. Rev. Lett., 93 (2004), 088301] provides resolved kinetic simulations of the molecular orientational distribution. These results characterize anisotropy and dynamic attractors of sheared bulk domains and underscore limitations of mesoscopic models for orientation of the rigid rod or platelet ensembles. In this paper, we apply our resolved kinetic structure code [R. Zhou, M. G. Forest, and Q. Wang, Multiscale Model. Simul., 3 (2005), pp. 853-870] to model onset and saturation of heterogeneity in the orientational distribution by coupling a distortional elasticity potential (with distinct elasticity constants) and anchoring conditions in a plane Couette cell. For this initial study, the flow field is imposed and the orientational distribution is confined to the shear deformation plane, which affords comparison with seminal [T. Tsuji and A. D. Rey, Phys. Rev. E (3), 62 (2000), pp. 8141-8151] as well as our own mesoscopic model simulations [H. Zhou, M. G. Forest, and Q. Wang, J. Non-Newtonian Fluid Mech., submitted], [H. Zhou and M. G. Forest, Discrete Contin. Dyn. Syst. Ser. B, to appear]. Under these controlled conditions, we map out the kinetic phase diagram of spatiotemporal attractors of a Couette cell film in the two-parameter space of Deborah number (normalized shear rate) and Ericksen number (relative strength of elasticity potentials). © 2005 Society for Industrial and Applied Mathematics

Microscopic-Macroscopic Simulations of Rigid-Rod Polymer Hydrodynamics: Heterogeneity and Rheochaos

Rheochaos is a remarkable phenomenon of nematic (rigid-rod) polymers in steady shear, with sustained chaotic fluctuations of the orientational distribution of the rod ensemble. For monodomain dynamics, imposing spatial homogeneity and linear shear, rheochaos is a hallmark prediction of the Doi-Hess theory [M. Doi, J. Polym. Sci. Polym. Phys. Ed., 19 (1981), pp. 229-243; M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, Oxford University Press, London, New York, 1986; S. Hess, Z. Naturforsch., 31 (1976), pp. 1034-1037. The model behavior is robust, captured by second-moment tensor approximations G. Rienäcker, M. Kröger, and S. Hess, Phys. Rev. E (3), 66 (2002), 040702; G. Rienäcker, M. Kröger, and S. Hess, Phys. A, 315 (2002), pp. 537-568; M. G. Forest and Q. Wang, Rheol. Acta, 42 (2003), pp. 20-46 and high-order Galerkin simulations of the Smoluchowski equation for the orientational probability distribution function (PDF) [M. Grosso, R. Keunings, S. Crescitelli, and P. L. Maffettone, Phys. Rev. Lett., 86 (2001), pp. 3184-3187; M. G. Forest, Q. Wang, and R. Zhou, Rheol. Acta, 43 (2004), pp. 17-37; M. G. Forest, Q. Wang, and R. Zhou, Rheol. Acta, 44 (2004), pp. 80-93, and persistent up to critical thresholds of coplanar extensional flow M. G. Forest, R. Zhou, and Q. Wang, Phys. Rev. Lett., 93 (2004), 088301; M. G. Forest, Q. Wang, R. Zhou, and E. Choate, J. Non-Newt. Fluid Mech., 118 (2004), pp. 17-31; S. Heidenreich, P. Ilg, and S. Hess, Phys. Rev. E (3), 73 (2006), 061710] and magnetic fields [M. G. Forest, Q. Wang, H. Zhou, and R. Zhou, J. Rheol., 48 (2004), pp. 175-1921, as well as fluctuating shear rates [S. Heidenreich, P. Ilg, and S. Hess, Phys. Rev. E (3), 73 (2006), 061710]. To be experimentally relevant, rheochaos of the Doi-Hess theory must persist amid heterogeneity observed in birefringence patterns [Z. Tan and G. C. Berry, J. Rheol., 47 (2003), pp. 73-104]. Modeling can further shed light on shear bands produced by hydrodynamic feedback which have thus fax eluded measurement. Some numerical evidence supports persistence: a one-dimensional (1D) study [B. Chakrabarti, M. Das, C. Dasgupta, S. Ramaswamy, and A. K. Sood, Phys. Rev. Lett., 92 (2004), 188301] with a second-moment tensor model and imposed simple shear; and a two-dimensional (2D) study [A. Furukawa and A. Onuki, Phys. D, 205 (2005), pp. 195-206] with a second-moment tensor model and flow feedback. Here we stage the micro-macro (Smoluchowski and Navier-Stokes) system so that monodomain rheochaos is embedded in a 1D simulation [R. Zhou, M. G. Forest, and Q. Wang, Multiscale Model. Simul., 3 (2005), pp. 853-870] of a planar shear cell experiment with distortional elasticity. Longtime simulations reveal (i) heterogeneous rheochaos marked by chaotic time series in the PDF, normal and shear stresses, and velocity field at each interior gap height; (ii) coherent spatial morphology in the PDF and stress profiles across the shear gap and weakly nonlinear shear bands in each snapshot; and (iii) consistency between heterogeneous and monodomain rheochaos as measured by Lyapunov exponents and pointwise orbits of the peak orientation of the PDF but with enhancement rather than reduction in Lyapunov exponent values in the flow coupled, heterogeneous system. © 2007 Society for Industrial and Applied Mathematics

Time-Domain Methods for Diffusive Transport in Soft Matter

Passive microrheology [12] utilizes measurements of noisy, entropic fluctuations (i.e., diffusive properties) of micron-scale spheres in soft matter to infer bulk frequency-dependent loss and storage moduli. Here, we are concerned exclusively with diffusion of Brownian particles in viscoelastic media, for which the Mason-Weitz theoretical-experimental protocol is ideal, and the more challenging inference of bulk viscoelastic moduli is decoupled. The diffusive theory begins with a generalized Langevin equation (GLE) with a memory drag law specified by a kernel [7, 16, 22, 23]. We start with a discrete formulation of the GLE as an autoregressive stochastic process governing microbead paths measured by particle tracking. For the inverse problem (recovery of the memory kernel from experimental data) we apply time series analysis (maximum likelihood estimators via the Kalman filter) directly to bead position data, an alternative to formulas based on mean-squared displacement statistics in frequency space. For direct modeling, we present statistically exact GLE algorithms for individual particle paths as well as statistical correlations for displacement and velocity. Our time-domain methods rest upon a generalization of well-known results for a single-mode exponential kernel [1, 7, 22, 23] to an arbitrary M-mode exponential series, for which the GLE is transformed to a vector Ornstein-Uhlenbeck process

Rheological Signatures in Limit Cycle Behaviour of Dilute, Active, Polar Liquid Crystalline Polymers in Steady Shear

We consider the dilute regime of active suspensions of liquid crystalline polymers (LCPs), addressing issues motivated by our kinetic model and simulations in Forest et al. (Forest et al. 2013 Soft Matter 9, 5207-5222 (doi:10.1039/c3sm27736d)). In particular, we report unsteady two-dimensional heterogeneous flow-orientation attractors for pusher nanorod swimmers at dilute concentrations where passive LCP equilibria are isotropic. These numerical limit cycles are analogous to longwave (homogeneous) tumbling and kayaking limit cycles and two-dimensional heterogeneous unsteady attractors of passive LCPs in weak imposed shear, yet these states arise exclusively at semi-dilute concentrations where stable equilibria are nematic. The results in Forest et al. mentioned above compel two studies in the dilute regime that complement recent work of Saintillan & Shelley (Saintillan & Shelley 2013 C. R. Physique 14, 497-517 (doi: 10.1016/j.crhy.2013.04.001)): linearized stability analysis of the isotropic state for nanorod pushers and pullers; and an analytical-numerical study of weakly and strongly sheared active polar nanorod suspensions to capture how particle-scale activation affects shear rheology. We find that weakly sheared dilute puller versus pusher suspensions exhibit steady versus unsteady responses, shear thickening versus thinning and positive versus negative first normal stress differences. These results further establish how sheared dilute nanorod pusher suspensions exhibit many of the characteristic features of sheared semi-dilute passive nanorod suspensions