Shape minimization problems in liquid crystals

TL;DR

This paper introduces a finite element algorithm for solving liquid crystal free-boundary problems, enabling simultaneous determination of equilibrium shapes and internal configurations, validated through applications to tactoids and flexible capacitors.

Contribution

It develops a novel finite element method with dynamic mesh control for liquid crystal free-boundary problems, integrating an auxiliary functional to improve mesh quality.

Findings

Successfully applied to liquid crystal tactoids and flexible capacitors.

Results agree with theoretical predictions and experimental data.

Demonstrates effectiveness of the algorithm in complex free-boundary problems.

Abstract

We consider a class of liquid crystal free-boundary problems for which both the equilibrium shape and internal configuration of a system must simultaneously be determined, for example in films with air- or fluid-liquid crystal interfaces and elastomers. We develop a finite element algorithm to solve such problems with dynamic mesh control, achieved by supplementing the free energy with an auxiliary functional that promotes mesh quality and is minimized in the null space of the energy. We apply this algorithm to a flexible capacitor, as well as to determine the shape of liquid crystal tactoids as a function of the surface tension and elastic constants. These are compared with theoretical predictions and experimental observations of tactoids from the literature.