Numerical method for the equilibrium configurations of a Maier-Saupe bulk potential in a Q-tensor model of an anisotropic nematic liquid crystal

TL;DR

This paper introduces a numerical method for modeling anisotropic elastic effects in nematic liquid crystals using a Q-tensor framework with a Maier-Saupe potential, ensuring physical validity of the order parameter.

Contribution

The method extends the Landau-de Gennes Q-tensor model to include elastic anisotropy with eigenvalue constraints, incorporating a microscopic Maier-Saupe potential.

Findings

Successfully models 2D and 3D nematic configurations with elastic anisotropy.

Handles complex defect structures like hedgehogs and disclinations.

Maintains physical eigenvalue bounds of the order parameter.

Abstract

We present a numerical method, based on a tensor order parameter description of a nematic phase, that allows fully anisotropic elasticity. Our method thus extends the Landau-de Gennes $\mathbf{Q}$-tensor theory to anisotropic phases. A microscopic model of the nematogen is introduced (the Maier-Saupe potential in the case discussed in this paper), combined with a constraint on eigenvalue bounds of $\mathbf{Q}$. This ensures a physically valid order parameter $\mathbf{Q}$ (i.e., the eigenvalue bounds are maintained), while allowing for more general gradient energy densities that can include cubic nonlinearities, and therefore elastic anisotropy. We demonstrate the method in two specific two dimensional examples in which the Landau-de Gennes model including elastic anisotropy is known to fail, as well as in three dimensions for the cases of a hedgehog point defect, a disclination line, and a disclination ring. The details of the numerical implementation are also discussed.