Function spaces for liquid crystals

TL;DR

This paper explores the mathematical relationships between three liquid crystal theories by analyzing their function spaces, leading to a new model capable of describing defects and discontinuities in nematic and cholesteric systems.

Contribution

It establishes connections between the variational formulations of Oseen-Frank, Ericksen, and Landau-de Gennes theories through function space analysis, and introduces a new model using functions of bounded variation.

Findings

Model describes both orientable and non-orientable defects

Predicts point and surface discontinuities in director fields

Links different liquid crystal theories via function space choices

Abstract

We consider the relationship between three continuum liquid crystal theories: Oseen-Frank, Ericksen and Landau-de Gennes. It is known that the function space is an important part of the mathematical model and by considering various function space choices for the order parameters $s$, ${\bf n}$, and ${\bf Q}$, we establish connections between the variational formulations of these theories. We use these results to derive a version of the Oseen-Frank theory using special functions of bounded variation. This proposed model can describe both orientable and non-orientable defects. Finally we study a number of frustrated nematic and cholesteric liquid crystal systems and show that the model predicts the existence of point and surface discontinuities in the director.