Effective surface energies in nematic liquid crystals as homogenised rugosity effects

TL;DR

This paper investigates how microscopic surface roughness influences nematic liquid crystals by using homogenisation techniques to replace complex boundary effects with an effective surface energy, providing quantitative convergence results.

Contribution

It introduces a general framework for analyzing boundary rugosity effects in nematic liquid crystals and derives homogenised surface energies applicable across various liquid crystal theories.

Findings

Homogenised surface energy effectively models boundary rugosity effects.

Quantitative convergence rates are established in simplified settings.

The approach is versatile across different liquid crystal models.

Abstract

We study the effect of boundary rugosity in nematic liquid crystalline systems. We consider a highly general formulation of the problem, able to simultaneously deal with several liquid crystal theories. We use techniques of Gamma convergence and demonstrate that the effect of fine-scale surface oscillations may be replaced by an effective homogenised surface energy on a simpler domain. The homogenisation limit is then quantitatively studied in a simplified setting, obtaining convergence rates.