A Simple Model for Non-Topological Defects in Sheared Nematic Polymer Monodomains

Abstract

Nematic fluids have a propensity for orientational defects. In Leslie-Ericksen theory for small molecule liquid crystals, defects are topological in nature and easily visible in textural snapshots. However, kinetic and mesoscopic theories of nematic polymers possess a richer class of defects, including non-topological defects in which the second moment tensor of the orientational probability distribution fails to uniquely define a major director. These defects include the isotropic phase, where the major director lies anywhere on the unit sphere, and so-called oblate defect phases where the major director lies on a circle in the plane normal to a well-defined minor axis. These isotropic and oblate defect phases are independent of the physical space dimension, and while they have appeared in dynamics of heterogeneous attractors [5, 12, 13], a simple monodomain model of defect phases has not previously been developed. Here we present such a model by restricting to the in-plane monodomain response to an imposed shear flow. The orientational dynamics are described by a three-dimensional dynamical system on a compact domain. We show that the set of isotropic and oblate defect phases separates this compact orientational space into two sets of well-defined orientational distributions, one parallel and another orthogonal to the vorticity axis. We identify orbits that pass through the oblate defect set, and how often they do so. Finally, we observe intriguing behavior at the tumbling-wagging bifurcation where limit cycles pass infinitely often through the oblate defect set. 2000 Mathematics Subject Classification. 76A15, 82D60.