Weak Anchoring for a Two-Dimensional Liquid Crystal

TL;DR

This paper analyzes the behavior of nematic liquid crystals under weak anchoring conditions in two dimensions, revealing how defect locations depend on the scaling of the anchoring parameter as it tends to infinity.

Contribution

It provides a detailed analysis of defect locations in 2D nematic liquid crystals with weak anchoring, especially for the Landau-De Gennes model reduced to Ginzburg-Landau, including a specific oil droplet example.

Findings

Defects lie on the boundary for certain scaling regimes.

Defects occur inside the domain for other scaling regimes.

Precise defect location descriptions depending on the anchoring parameter scaling.

Abstract

We study the weak anchoring condition for nematic liquid crystals in the context of the Landau-De Gennes model. We restrict our attention to two dimensional samples and to nematic director fields lying in the plane, for which the Landau-De Gennes energy reduces to the Ginzburg--Landau functional, and the weak anchoring condition is realized via a penalized boundary term in the energy. We study the singular limit as the length scale parameter $\varepsilon\to 0$, assuming the weak anchoring parameter $\lambda=\lambda(\varepsilon)\to\infty$ at a prescribed rate. We also consider a specific example of a bulk nematic liquid crystal with an included oil droplet and derive a precise description of the defect locations for this situation, for $\lambda(\varepsilon)=K\varepsilon^{-\alpha}$ with $\alpha\in (0,1]$. We show that defects lie on the weak anchoring boundary for $\alpha\in (0,\frac12)$, or for $\alpha=\frac12$ and $K$ small, but they occur inside the bulk domain $\Omega$ for $\alpha>\frac12$ or $\alpha=\frac12$ with $K$ large.