Elastic energy of liquid crystals in convex polyhedra

TL;DR

This paper investigates the elastic energy of nematic liquid crystals confined in convex polyhedra, deriving bounds and analyzing how the minimal energy state transitions from smooth to singular configurations as the aspect ratio varies.

Contribution

It provides new bounds for the elastic energy based on topological invariants and introduces test configurations using local conformal solutions for convex polyhedra.

Findings

Derived lower bounds for elastic energy in convex polyhedra.

Constructed upper bounds using local conformal solutions.

Identified a sharp transition in minimal-energy states from smooth to singular as aspect ratio changes.

Abstract

We consider nematic liquid crystals in a bounded, convex polyhedron described by a director field n(r) subject to tangent boundary conditions. We derive lower bounds for the one-constant elastic energy in terms of topological invariants. For a right rectangular prism and a large class of topologies, we derive upper bounds by introducing test configurations constructed from local conformal solutions of the Euler-Lagrange equation. The ratio of the upper and lower bounds depends only on the aspect ratios of the prism. As the aspect ratio is varied, the minimum-energy conformal state undergoes a sharp transition from being smooth to having singularities on the edges.