Texture and shape of two-dimensional domains of nematic liquid crystal

TL;DR

This paper introduces a generalized conformal mapping approach to analyze the shape and internal director field of two-dimensional nematic liquid crystal domains, revealing new regimes and accurate descriptions of domain configurations.

Contribution

It develops a novel method using conformal mappings to compute nematic director fields for diverse 2D domain shapes, including previously uncharacterized regimes.

Findings

Identified a new regime with nearly circular drops, bipolar director fields, and rounded tips.

Bicircular director fields accurately describe the optimal configuration across various shapes.

Mapped phase transitions between different domain shapes and director field configurations.

Abstract

We present a generalized approach to compute the shape and internal structure of two-dimensional nematic domains. By using conformal mappings, we are able to compute the director field for a given domain shape that we choose from a rich class, which includes drops with large and small aspect ratios, and sharp domain tips as well as smooth ones. Results are assembled in a phase diagram that for given domain size, surface tension, anchoring strength, and elastic constant shows the transitions from a homogeneous to a bipolar director field, from circular to elongated droplets, and from sharp to smooth domain tips. We find a previously unaccounted regime, where the drop is nearly circular, the director field bipolar and the tip rounded. We also find that bicircular director fields, with foci that lie outside the domain, provide a remarkably accurate description of the optimal director field for a large range of values of the various shape parameters.