The weirdest martensite: Smectic liquid crystal microstructure and Weyl-Poincar\'e invariance

TL;DR

This paper explores the microstructure of smectic liquid crystals, revealing compatibility rules for domain formation and connecting these to Weyl-Poincaré invariance, with implications for understanding complex material patterns.

Contribution

It introduces compatibility conditions for smectic domains and links their symmetry to the Weyl-Poincaré group, advancing the understanding of liquid crystal microstructures.

Findings

Compatibility conditions match numerical simulations of smectic interfaces.

Domains are well-described by clustering algorithms based on these conditions.

The approach generalizes to three-dimensional smectic structures with Weyl-Poincaré symmetry.

Abstract

Smectic liquid crystals are remarkable, beautiful examples of materials microstructure, with ordered patterns of geometrically perfect ellipses and hyperbolas. The solution of the complex problem of filling three-dimensional space with domains of focal conics under constraining boundary conditions yields a set of strict rules, which are similar to the compatibility conditions in a martensitic crystal. Here we present the rules giving compatible conditions for the concentric circle domains found at two-dimensional smectic interfaces with planar boundary conditions. Using configurations generated by numerical simulations, we develop a clustering algorithm to decompose the planar boundaries into domains. The interfaces between different domains agree well with the smectic compatibility conditions. We also discuss generalizations of our approach to describe the full three-dimensional smectic domains, where the variant symmetry group is the Weyl-Poincar\'e group of Lorentz boosts, translations, rotations, and dilatations.