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

Abstract

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.