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

Abstract

AbstractWe 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 $$\lambda $$ λ -the ratio of the edge length of polygon hole to the nematic correlation length, and an additional degree of freedom, $$\gamma ^*$$ γ ∗ -the nematic director at infinity. In the $$\lambda \rightarrow 0$$ λ → 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 $$\gamma ^*$$ γ ∗ , and for a triangular hole, there is a unique interior point defect outside the hole. In the $$\lambda \rightarrow \infty $$ λ → ∞ limit, there are at least $$\left( {\begin{array}{c}K\\ 2\end{array}}\right) $$ K 2 stable states and the multistability is enhanced by $$\gamma ^*$$ γ ∗ , compared to the interior problem. Our work offers new insights into how to tune the existence, location, and dimensionality of defects.