Wulff Construction for Deformable Media

TL;DR

This paper extends the Wulff construction to deformable media with XY-like order, deriving exact expressions for domain shapes and showing that such domains develop cusps related to phase transitions.

Contribution

It introduces a generalized Wulff construction for calculating shapes of domains with orientation-dependent energies in deformable media.

Findings

Domains develop cusps under general conditions.

Cusp formation is mathematically linked to phase transitions.

Exact expressions for order parameter textures in circular domains.

Abstract

A domain in a Langmuir monolayer can be expected to have a shape that reflects the textural anisotropy of the material it contains. This paper explores the consequences of XY-like ordering. It is found that an extension of the Wulff construction allows for the calculation of two-dimensional domain shapes when each segment of the perimeter has an energy that depends both on its orientation and its location. This generalized Wulff construction, and newly-derived exact expressions for the order parameter texture in a circular domain, lead to results for the shape of a large domain. The most striking result is that under general conditions such domains will inevitably develop cusps. We show that the onset of a cusp is mathematically related to a phase transition. The present approach is equivalent to a Landau mean-field version of the theory.