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

TL;DR

This paper develops exact and approximate methods to analyze elastic interactions among multiple bodies in a two-dimensional nematic liquid crystal, accounting for shape, position, and anchoring effects, with numerical and analytical results.

Contribution

It introduces a complex variables approach for modeling multi-body elastic interactions in nematic liquid crystals, including defect positioning and surface anchoring effects.

Findings

Numerical determination of defect positions minimizes elastic energy.

Analytical expressions for two-body far-field interactions.

Application to various shapes and configurations of inclusions.

Abstract

Anisotropic fluids appear in a diverse array of systems, from liquid-crystal displays to bacterial swarms, and are characterized by orientational order. Large colloidal particles immersed in such environments disturb the medium's orientational order, however, resulting in a stored elastic energy within the bulk. As a consequence, multiple immersed bodies interact at equilibrium through fluid-mediated forces and torques, which depend on the bodies' positions, orientations, and shapes. We provide the equilibrium configuration of a model nematic liquid crystal with multiple immersed bodies or inclusions in two-dimensions, as well as the associated body forces, torques, and surface tractions. A complex variables approach is taken which leans on previous work by Crowdy (2020) for describing solutions with multiply-connected domains. Free periods of a complex director field, which correspond to topological defect positioning and net topological charge, are determined numerically to minimize a global stored elastic energy, including a contribution of a weak (finite) anchoring strength on the body surfaces. Finally, a general, analytical description of two-body far-field interactions is provided, along with examples using two cylindrical inclusions of arbitrary position and size, and two triangles of arbitrary position and orientation.