A Reduced Study for Nematic Equilibria on Two-Dimensional Polygons

TL;DR

This paper investigates stable nematic equilibria in two-dimensional polygons, analytically and numerically exploring how geometry influences defect structures and stability in a simplified Landau-de Gennes model.

Contribution

It introduces a novel ring solution for nematic equilibria in polygons and analyzes stability and bifurcations as a function of polygon size and shape.

Findings

Analytical ring solution with a central defect for small edge length

Existence of multiple stable equilibrium classes for large edge length

Bifurcation diagrams illustrating defect behavior in pentagons and hexagons

Abstract

We study reduced nematic equilibria on regular two-dimensional polygons with Dirichlet tangent boundary conditions, in a reduced two-dimensional Landau-de Gennes framework, discussing their relevance in the full three-dimensional framework too. We work at a fixed temperature and study the reduced stable equilibria in terms of the edge length, $\lambda$ of the regular polygon, $E_K$ with $K$ edges. We analytically compute a novel "ring solution" in the $\lambda \to 0$ limit, with a unique point defect at the centre of the polygon for $K \neq 4$. The ring solution is unique. For sufficiently large $\lambda$, we deduce the existence of at least $\left[K/2 \right]$ classes of stable equilibria and numerically compute bifurcation diagrams for reduced equilibria on a pentagon and hexagon, as a function of $\lambda^2$, thus illustrating the effects of geometry on the structure, locations and dimensionality of defects in this framework.