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

Abstract

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 continuum models [P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, Clarendon Press, Oxford, 1993; G. Marrucci, Macromolecules, 24 (1991), pp. 4176--4182; G. Marrucci and F. Greco, Flow behavior of liquid crystalline polymers, in Advances in Chemical Physics, Vol. 86, Wiley, New York, pp. 331--404] and a variety of mesoscopic, second-moment orientation tensor models [T. Tsuji and A. D. Rey, J. Non-Newt. Fluid Mech., 73 (1997), pp. 127--152; R. Kupfermann, M. Kawaguchi, and M. M. Denn, J. Non-Newt. Fluid Mech., 91 (2000), pp. 255--271; G. Sgalari, G. L. Leal, and J. Feng, J. Non-Newt. Fluid Mech., 102 (2002), pp. 361--382; D. Grecov and A. D. Rey, Phys. Rev. E, 68 (2003), 061704; M. G. Forest et al., J. Rheol., 48 (2004), pp. 175--192; Z. Cui, M. G. Forest, and Q. Wang, On weak plane Couette and Poiseuille flows of nematic polymers, SIAM J. Appl. Math., submitted]. The model consists of a Smoluchowski equation for the space-time evolution of the orientational probability distribution function, coupled with a momentum flow balance equation, a constitutive equation for the extra stress, and the continuity equation. The Smoluchowski equation is first reduced to a finite set of partial differential equations in time and space for spherical harmonic amplitudes. Then we discretize the spatial variables (by the method of lines) using high-order finite differences, which reduces the full system to a large set of ordinary differential equations. Adaptive grid generation techniques are implemented. To provide an accurate and stable time integration, we employ the newly developed spectral deferred correction algorithm. We close with an application of the kinetic theory and numerical code to explore steady state flow-molecular structures in slow planar Couette experiments (low Deborah number) and low Ericksen number, where distortional elasticity dominates short-range excluded volume effects.