Far-Field Expansions for Harmonic Maps and the Electrostatics Analogy in Nematic Suspensions

Abstract

For a smooth bounded domain $G\subset {\mathbb{R}}^3$ we consider maps $n:{\mathbb{R}}^3\setminus G\to {\mathbb{S}}^2$ minimizing the energy $E(n)={\int }_{{\mathbb{R}}^3\setminus G}|\nabla n{|}^2+F_s(n_{\lfloor \partial G})$ among ${\mathbb{S}}^2$-valued map such that $n(x)\approx n_0$ as $|x|\to \infty$. This is a model for a particle $G$ immersed in nematic liquid crystal. The surface energy $F_s$ describes the anchoring properties of the particle, and can be quite general. We prove that such minimizing map $n$ has an asymptotic expansion in powers of $1/r$. Further, we show that the leading order $1/r$ term is uniquely determined by the far-field condition $n_0$ for almost all $n_0\in {\mathbb{S}}^2$, by relating it to the gradient of the minimal energy with respect to $n_0$. We derive various consequences of this relation in physically motivated situations: when the orientation of the particle $G$ is stable relative to a prescribed far-field alignment $n_0$; and when the particle $G$ has some rotational symmetries. In particular, these corollaries justify some approximations that can be found in the physics literature to describe nematic suspensions via a so-called electrostatics analogy.

18 pages, no figures. Final version to appear in J. Nonlin. Sc

References in corpus (2)

• Design of nematic liquid crystals to control microscale dynamics
• The saturn ring effect in nematic liquid crystals with external field: effective energy and hysteresis

Cited by in corpus (1)

• Exact and approximate solutions for elastic interactions in a nematic liquid crystal