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

TL;DR

This paper develops a mathematical framework for understanding the far-field behavior of harmonic maps modeling particles in nematic liquid crystals, establishing asymptotic expansions and relating them to physical properties and approximations.

Contribution

It proves that minimizing maps have a 1/r asymptotic expansion and links the leading term to the far-field condition, justifying physics-based electrostatics analogies.

Findings

Asymptotic expansion in powers of 1/r for minimizing maps.

Leading order term uniquely determined by far-field condition.

Justification of electrostatics analogy in nematic suspensions.

Abstract

For a smooth bounded domain $G\subset\mathbb{R}^3$ we consider maps $n\colon\mathbb R^3\setminus G\to\mathbb S^2$ 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.