A Finite Element Method for a Phase Field Model of Nematic Liquid Crystal Droplets

TL;DR

This paper introduces a new finite element method for simulating nematic liquid crystal droplets using a phase field model, capturing complex interfacial and elastic behaviors with proven energy stability and convergence.

Contribution

It presents a novel finite element discretization for a phase field model of nematic droplets, including a discrete gradient flow with proven energy decreasing and convergence properties.

Findings

The method effectively models droplet coalescence and break-up.

Discrete energy minimizers converge to continuous minimizers.

Numerical experiments demonstrate dynamic behaviors of droplets.

Abstract

We develop a novel finite element method for a phase field model of nematic liquid crystal droplets. The continuous model considers a free energy comprised of three components: the Ericksen's energy for liquid crystals, the Cahn-Hilliard energy representing the interfacial energy of the droplet, and a weak anchoring energy representing the interaction of the liquid crystal molecules with the surface tension on the interface (i.e. anisotropic surface tension). Applications of the model are for finding minimizers of the free energy and exploring gradient flow dynamics. We present a finite element method that utilizes a special discretization of the liquid crystal elastic energy, as well as mass-lumping to discretize the coupling terms for the anisotropic surface tension part. Next, we present a discrete gradient flow method and show that it is monotone energy decreasing. Furthermore, we show that global discrete energy minimizers $\Gamma$-converge to global minimizers of the continuous energy. We conclude with numerical experiments illustrating different gradient flow dynamics, including droplet coalescence and break-up.