On a Divergence Penalized Landau-de Gennes Model

TL;DR

This paper introduces a divergence penalized Landau-de Gennes model for nematic liquid crystals with colloids, analyzing the effects of divergence penalization on defect structures and providing numerical solutions.

Contribution

It presents a novel divergence penalized functional for nematic liquid crystals with colloids, including analytical and numerical analysis of defect behaviors.

Findings

Divergence penalization influences Saturn ring defect formation.

Representation formula for solutions in the small particle limit.

Numerical finite element analysis of the model.

Abstract

We give a brief introduction to a divergence penalized Landau-de Gennes functional as a toy model for the study of nematic liquid crystal with colloid inclusion, in the case of unequal elastic constants. We assume that the nematic occupies the exterior of the unit ball, satisfies homeotropic anchoring at the surface of the colloid and approaches a uniform uniaxial state as $|x|\to\infty$. We study the "small particle" limit and obtain a representation formula for solutions to the associated Euler-Lagrange equations. We also present a numerical analysis of these equations based on a finite element approach and discuss the effect of the divergence penalization on the "Saturn ring" defects and on the properties of the $Q$-tensor.