The Ericksen Model of Liquid Crystals with Colloidal and Electric Effects

TL;DR

This paper introduces a robust discretization method for the Ericksen model of liquid crystals that incorporates colloidal and electric effects, enabling accurate simulation of defect patterns and proving energy convergence.

Contribution

It presents a novel discretization and minimization algorithm for the Ericksen model with colloidal and electric effects, including a rigorous Gamma-convergence proof.

Findings

Successfully captures complex defect patterns like Saturn rings

Demonstrates energy decreasing property of the quasi-gradient flow algorithm

Validates the method through numerical experiments in 2D and 3D

Abstract

We present a robust discretization of the Ericksen model of liquid crystals with variable degree of orientation coupled with colloidal effects and electric fields. The total energy consists of the Ericksen energy, a weak anchoring (or penalized Dirichlet) energy to model colloids, and an electrical energy for a given electric field. We describe our special discretization of the total energy along with a method to compute minimizers via a discrete quasi-gradient flow algorithm which has a strictly monotone energy decreasing property. Numerical experiments are given in two and three dimensions to illustrate that the method is able to capture non-trivial defect patterns, such as the Saturn ring defect. We conclude with a rigorous proof of the Gamma-convergence of our discrete energy to the continuous energy.