Weak Anchoring for a Two-Dimensional Liquid Crystal

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 $\epsilon \to 0$, assuming the weak anchoring parameter $\lambda =\lambda (\epsilon )\to \infty$ 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 $\lambda (\epsilon )=K{\epsilon }^{-\alpha }$ with $\alpha \in (0,1]$. We show that defects lie on the weak anchoring boundary for $\alpha \in (0,\frac{1}{2})$, or for $\alpha =\frac{1}{2}$ and $K$ small, but they occur inside the bulk domain $\Omega$ for $\alpha >\frac{1}{2}$ or $\alpha =\frac{1}{2}$ with $K$ large.

appears in Nonlinear Analysis A 2014