Ruhai Zhou

Kinetic Structure Simulations of Nematic Polymers in Plane Couette Cells. I: The Algorithm and Benchmarks

The Doi--Hess theory coupled with an anisotropic Marrucci--Greco distortional elasticity potential provides a kinetic, mean field description of the coupling between hydrodynamics, molecular orientation by excluded volume, and elastic distortions of flowing nematic liquid crystalline polymers (LCPs). In this paper we provide the first numerical algorithm and implementation of kinetic-scale models for structure formation in confined, planar Couette cells. The model and algorithm extend kinetic simulations of Larson and Ottinger [Macromolecules, 24 (1991), pp. 6270--6282], Faraoni et al. [J. Rheol., 43 (1999), pp. 829--843], Grosso et al. [Phys. Rev. Lett., 86 (2001), pp. 3184--3187], and Forest et al. [Rheol. Acta, 43 (2004), pp. 17--37; Rheol. Acta, 44 (2004), pp. 80--93] for homogeneous nematic polymers in imposed shear flows. The focus here is one-dimensional flow and LCP structures that form between oppositely moving, parallel plates, a classical model problem that has been studied in detail with conti...

Full-tensor alignment criteria for sheared nematic polymers

The shear problem for nematic polymers consists in characterizing all stable stationary orientational distributions, steady and unsteady, versus shear rate and material parameters. Continuum theory [cf. Leslie (1968), Ericksen (1960)] provides formulas for the shear response of liquid crystals in terms of a single viscosity ratio, the Leslie tumbling parameter λL. Kuzuu and Doi (1983, 1984) developed a weak-flow asymptotic analysis of kinetic theory, which gives a molecular basis for all continuum theory parameters. In this paper, we develop a mesoscopic extension of the Kuzuu–Doi method, applicable to any tensor model. Our method yields orientational and rheological features of nematic polymers in weak shear with explicit formulas, parametrized by the parameters of the second-moment tensor model. This provides an explicit mesoscopic theory solution to the problem posed by Marrucci and Greco (1993) of how orientational degeneracy of quiescent nematic equilibria breaks in weak shear, leaving a finite set o...

Structure scaling properties of confined nematic polymers in plane Couette cells: The weak flow limit

One of the confounding issues in laminar flow processing of nematic polymers is the generation of molecular orientational structures on length scales that remain poorly characterized with respect to molecular and processing control parameters. For plane Couette flow within the Leslie–Ericksen continuum model, theoretical results since the 1970s yield two fundamental predictions about the length scales of nematic distortion: a power law scaling behavior, Er−p, 14⩽p⩽1, where Er is the Ericksen number (ratio of viscous to elastic stresses); the exponent p varies according to whether the structure is a localized boundary layer or an extended structure. Until now, comparable results which incorporate molecular elasticity (i.e., distortions in the shape of the orientational distribution) have not been derived from mesoscopic Doi–Marrucci–Greco (DMG) tensor models. In this paper, we derive asymptotic, one-dimensional gap structures, along the flow-gradient direction, in “slow” Couette cells, which reflect self-consistent coupling between the primary flow, in-plane director (nematic) and order parameter (molecular) elasticity, and confinement conditions (plate speeds, gap height, and director anchoring angle). We then read off the small Deborah number, viscoelastic structure predictions: The flow is simple shear. The orientation structures consist of: two molecular-elasticity boundary layers with the Marrucci scaling Er−1/2, which are amplified by tilted plate anchoring; and a nonuniform, director-dominated structure that spans the entire gap, with Er−1 average length scale, present for any anchoring angle. We close with direct numerical simulations of the DMG steady, flow-nematic boundary-value problem, first to benchmark the small Deborah number structure formulas, and then to document onset of new flow-orientation structures as the asymptotic expansions become disordered.One of the confounding issues in laminar flow processing of nematic polymers is the generation of molecular orientational structures on length scales that remain poorly characterized with respect to molecular and processing control parameters. For plane Couette flow within the Leslie–Ericksen continuum model, theoretical results since the 1970s yield two fundamental predictions about the length scales of nematic distortion: a power law scaling behavior, Er−p, 14⩽p⩽1, where Er is the Ericksen number (ratio of viscous to elastic stresses); the exponent p varies according to whether the structure is a localized boundary layer or an extended structure. Until now, comparable results which incorporate molecular elasticity (i.e., distortions in the shape of the orientational distribution) have not been derived from mesoscopic Doi–Marrucci–Greco (DMG) tensor models. In this paper, we derive asymptotic, one-dimensional gap structures, along the flow-gradient direction, in “slow” Couette cells, which reflect self-c...

A Kinetic Theory for Solutions of Nonhomogeneous Nematic Liquid Crystalline Polymers With Density Variations

The kinetic theory developed in [1] for solutions of nonhomogeneous nematic liquid crystalline polymers (LCPs) of spheroidal molecular configurations is extended to account for the translational diffusion and the related spatial density variation. The new theory augments the effect of the density variation to the intermolecular potential, Smoluchowski equation and the elastic stress. It accounts for the molecular aspect ratio as well as the finite range molecular interaction so that it is applicable to liquid crystals ranging from rodlike liquid crystals at large aspect ratios to discotic ones at small aspect ratios. It also exhibits enhanced shape effects in the viscous stress and warrants a positive entropy production, thereby, the second law of thermodynamics. Moment averaged, approximate, mesoscopic theories for complex flow simulations are obtained via closure approximations. In the limit of weak distortional elasticity, weak translational diffusion, and weak flows, the theory yields the torque balance equation of the well-known Ericksen-Leslie theory.

Explicit Flow-Aligned Orientational Distribution Functions for Dilute Nematic Polymers in Weak Shear

Flow-alignment of sheared nematic polymers occurs in various flow-concentration regimes. Analytical descriptions of shear-aligned nematic monodomains have a long history across continuum, mesoscopic and mean-field kinetic models, sacrificing precision at each finer scale. Continuum Leslie-Ericksen theory applies to highly concentrated, weak flows of small molecular weight polymers, giving an explicit macroscopic alignment angle formula dependent only on Miesowicz viscosities. Mesoscopic tensor models apply at all concentrations and shear rates, but explicit “Leslie angle” formulas exist only in the weak shear limit (Cocchini et. al, 90; Bhave et. al, 93; Wang, 97; Rienacker and Hess, 99; Maffettone et. al, 00; Forest and Wang, 02; Forest et. al, 02c; Grecov and Rey, 02), with distinct behavior in dilute versus concentrated regimes. Exact probability distribution functions (pdf’s) of kinetic theory do not exist for highly concentrated nematic states, even without flow, although appealing flow-aligned approximations have been derived (Kuzuu and Doi, 83; Kuzuu and Doi, 84; Semenov, 83; Semenov, 86; Archer and Larson, 95; Kroger and Seller, 95), which offer a molecular theory basis for the Leslie alignment angle. A simpler problem concerns the dilute concentration regime where the unique quiescent equilibrium is isotropic, corresponding to a constant pdf, and whose weak shear deformation is robust to mesoscopic closure approximation (Forest and Wang, 02; Forest et. al, 02c): steady, flow-aligning, weakly anisotropic, and biaxial. The purpose of this paper is to explicitly construct the weakly anisotropic branch of stationary pdf’s by a weak-shear asymptotic expansion of kinetic theory. A second-moment pdf projection confirms mesoscopic model predictions, and further yields explicit Leslie angle and degree of alignment formulas in terms of molecular parameters and normalized shear rate.Copyright