Conservation Laws for Crystal of Topological Defects

TL;DR

This paper investigates the conservation laws governing topological defect crystals in nematic liquid crystals, revealing how flux and charge conservation underpin the stability and structure of defect arrays, with implications for self-assembly and biological growth.

Contribution

It introduces the discovery of hidden conservation laws of flux and topological charges in defect crystals, explaining their stability and structure.

Findings

Periodic and stable defect crystal structures depend on zero total flux in a unit cell.

Topological charge summation follows rules similar to homeotropic droplets.

Conservation laws may explain natural tissue and structure growth patterns.

Abstract

Stable array of point defects was generated in nematic liquid crystal. Point defects with topological charge of +1 and -1 (hedgehogs) were generated in vertically aligned liquid crystal cell. The hedgehogs were arranged in square or hexagonal arrays, and the shape of defect, such as radial or circular hedgehogs, were under the control of delicately patterned electrodes. Base on the two-dimensional defect array discovered in experiments, three-dimensional crystals of defects were postulated. The flux of the liquid crystal director field was summarized for each unit cell. The analysis showed that the crystal structure is periodic and stable if the total flux in a unit cell is zero. The surface and bulk topological charges of each unit cells were analyzed. The summation of topological charges of a unit cell obeys the topological rules of a homeotropic droplet. The hidden conservation laws of flux and topological charges in a soft crystal of defects were discovered, which can be implemented in templates for self-assembled micro-structure and tissue design. Most important of all, the conservation laws may answer why natural structures and tissues grow in to a particular shape and form.