The $\mathbf{Q}$-tensor Model with Uniaxial Constraint

TL;DR

This paper reviews continuum models of nematic liquid crystals, focusing on the $ extbf{Q}$-tensor model with uniaxial constraint, and introduces a numerical method justified by $ ext{Gamma}$-convergence for simulating these systems.

Contribution

It develops a $ extbf{Q}$-tensor model with an exact uniaxial constraint by combining Ericksen and Landau-deGennes models, with a rigorous numerical analysis.

Findings

The uniaxial $ extbf{Q}$-tensor model accurately captures nematic LC behavior.

Numerical experiments demonstrate the model's effectiveness and consistency.

The method is justified through $ ext{Gamma}$-convergence analysis.

Abstract

This chapter is about the modeling of nematic liquid crystals (LCs) and their numerical simulation. We begin with an overview of the basic physics of LCs and discuss some of their many applications. Next, we delve into the modeling arguments needed to obtain macroscopic order parameters which can be used to formulate a continuum model. We then survey different continuum descriptions, namely the Oseen-Frank, Ericksen, and Landau-deGennes ($\mathbf{Q}$-tensor) models, which essentially model the LC material like an anisotropic elastic material. In particular, we review the mathematical theory underlying the three different continuum models and highlight the different trade-offs of using these models. Next, we consider the numerical simulation of these models with a survey of various methods, with a focus on the Ericksen model. We then show how techniques from the Ericksen model can be combined with the Landau-deGennes model to yield a $\mathbf{Q}$-tensor model that exactly enforces uniaxiality, which is relevant for modeling many nematic LC systems. This is followed by an in-depth numerical analysis, using tools from $\Gamma$-convergence, to justify our discrete method. We also show several numerical experiments and comparisons with the standard Landau-deGennes model.