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

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 com-plements 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 -> 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 -> infinity limit, there are at least ((2) (K)) stable states and the 2 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.

Automated summary

We study nematic equilibria in an unbounded domain with a two-dimensional regular polygonal hole of K edges using a reduced Landau-de Gennes framework, which complements our previous work on nematic equilibria confined inside regular polygons. The dimensionless model parameters, lambda and gamma *, represent the ratio of the hole's edge length to the nematic correlation length and the nematic director at infinity, respectively. In the limit of lambda -> 0, the limiting profile exhibits 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 *, while a triangular hole only has a unique interior point defect outside the hole. In the limit of lambda -> infinity, there are at least ((2)(K)) stable states, and the presence of gamma * enhances bistability compared to the interior problem. Our work provides new insights into manipulating the existence, location, and dimensionality of defects.