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

TL;DR

This paper investigates nematic liquid crystal equilibria around polygonal holes in an unbounded domain, revealing how defect configurations depend on geometric and boundary parameters, and demonstrating multistability phenomena.

Contribution

It extends previous interior polygon studies to exterior domains, analyzing defect patterns and multistability in a reduced Landau-de Gennes model with new parameter insights.

Findings

In the small bb limit, two interior point defects form outside most polygons.

For square holes, defect presence depends on b3^*, with no or two line defects.

In the large bb limit, multiple stable states emerge, increasing with polygon edges.

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 $\lambda\to 0$ 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 $\lambda\to\infty$ limit, there are at least $\binom{K}{2}$ stable states %differentiated by the location of two bend vertices 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.