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

Abstract

SynopsisOne 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 Ϫp , 1 4 р p р 1, where Er is the Ericksen number ͑ratio of viscous to elastic stresses͒; 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 distribution͒ have not been derived from mesoscopic Doi-Marrucci-Greco ͑DMG͒ 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 ͑nematic͒ and order parameter ͑molecular͒ elasticity, and confinement conditions ͑plate speeds, gap height, and director anchoring angle͒. 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 Ϫ1/2 , which are amplified by tilted plate anchoring; and a nonuniform, director-dominated structure that spans the entire gap, with Er Ϫ1 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.

On Weak Plane Couette and Poiseuille Flows of Rigid Rod and Platelet Ensembles

“…Note that, as in the isotropic elasticity limit [14], these solvability conditions imply simple shear flow at leading order in De, and yield that the orientational distribution is dominated by nematic (director) distortions. The prefactor (22) yields that the winding number of the major director between the plates is proportional to the Ericksen number, as with the isotropic elasticity limit [14].…”

“…This also applies to the case of normal anchoring, but not to tilted anchoring (ψ 0 = 0, π 2 ) [14]. The governing equations at order O (1) give the equilibrium solution of Q consistent with the boundary anchoring condition; the equations at order O(De) are obtained by solving the following equations for ψ (1) and v (1) x :…”

“…The prefactor (22) yields that the winding number of the major director between the plates is proportional to the Ericksen number, as with the isotropic elasticity limit [14]. The formula (22) yields the scaling law for elastic distortions which are nonuniform across the gap with lengthscale proportional to M −1 , which in turn is proscribed by three material parameters: (a, N, θ).…”

“…Another issue typical of non-Newtonian fluids is flow feedback, where elastic stresses alter apparent viscosity. This paper is a continuation of our systematic studies of mesostructures, both from free space elasticity patterns absent of external fields and boundary anchoring conditions [11,12], and from planar Couette cells moving at prescribed slow speeds [13,14]. We defer to these articles, where a detailed account of analytical results on continuum Leslie-Ericksen-Frank (LEF) models is given, notably by [26,4,5,7,27,23,28].…”

“…In [14], the authors considered a Doi-Marrucci-Greco (DMG) mesoscopic orientation tensor model, allowing a full coupling between flow structure, director (nematic) and order parameter (molecular) distortions, and with imposed plate motion and molecular anchoring conditions. The model has been benchmarked in the longwave, monodomain regime with resolved simulations of the Doi kinetic theory [15,16].…”