Orientability and energy minimization in liquid crystal models

TL;DR

This paper compares the Oseen-Frank and Landau-de Gennes theories for nematic liquid crystals, analyzing how domain topology and regularity influence their agreement and differences in modeling predictions.

Contribution

It provides a detailed analysis of the conditions under which the two theories coincide or differ, especially focusing on domain topology and regularity classes.

Findings

The two theories coincide on simply-connected domains in the natural energy class.

Differences between theories depend on domain topology and regularity.

Complete characterization of discrepancies in planar domains.

Abstract

Uniaxial nematic liquid crystals are modelled in the Oseen-Frank theory through a unit vector field $n$. This theory has the apparent drawback that it does not respect the head-to-tail symmetry in which $n$ should be equivalent to -$n$. This symmetry is preserved in the constrained Landau-de Gennes theory that works with the tensor $Q=s\big(n\otimes n- \frac{1}{3} Id\big)$.We study the differences and the overlaps between the two theories. These depend on the regularity class used as well as on the topology of the underlying domain. We show that for simply-connected domains and in the natural energy class $W^{1,2}$ the two theories coincide, but otherwise there can be differences between the two theories, which we identify. In the case of planar domains we completely characterise the instances in which the predictions of the constrained Landau-de Gennes theory differ from those of the Oseen-Frank theory.