Multistability for a Reduced Nematic Liquid Crystal Model in the Exterior of 2D Polygons

Abstract

We study nematic equilibria in an unbounded domain, with a two-dimensional regular polygonal hole with K edges, in a reduced Landau–de Gennes framework. This complements our previous work on the “interior problem” for nematic equilibria confined inside regular polygons (SIAM Journal on Applied Mathematics, 80(4):1678-1703, 2020). The two essential dimensionless model parameters are lambda -the ratio of the edge length of polygon hole to the nematic correlation length, and an additional degree of freedom, gamma ^*-the nematic director at infinity. In the lambda rightarrow 0 limit, the limiting profile has two interior point defects outside a generic polygon hole, except for a triangle and a square. For a square hole, the limiting profile has either no interior defects or two line defects depending on gamma ^*, and for a triangular hole, there is a unique interior point defect outside the hole. In the lambda rightarrow infty limit, there are at least left( {begin{array}{c}K 2end{array}}right) stable states and the multistability is enhanced by gamma ^*, compared to the interior problem. Our work offers new insights into how to tune the existence, location, and dimensionality of defects.