Topology and ground state degeneracy of tetrahedral smectic vesicles

TL;DR

This paper models tetrahedral smectic vesicles made of block copolymers, revealing their shape, defect structures, degeneracy, and potential for chiral symmetry breaking, with implications for material applications.

Contribution

It introduces a zero-temperature model of tetrahedral smectic vesicles, analyzing their shape, defect arrangements, and degeneracy, highlighting spontaneous chiral symmetry breaking.

Findings

Vesicles assume a faceted tetrahedral shape with topological defects at vertices.

Smectic layer patterns depend on tilt angle, showing degeneracy and spiral defects.

Spontaneous chiral symmetry breaking occurs via spiral layer organization.

Abstract

Chemical design of block copolymers makes it possible to create polymer vesicles with tunable microscopic structure. Here we focus on a model of a vesicle made of smectic liquid-crystalline block copolymers at zero temperature. The vesicle assumes a faceted tetrahedral shape and the smectic layers arrange in a stack of parallel straight lines with topological defects localized at the vertices. We counted the number of allowed states at $T=0$. For any fixed shape, we found a two-dimensional countable degeneracy in the smectic pattern depending on the tilt angle between the smectic layers and the edge of the tetrahedral shell. For most values of the tilt angle, the smectic layers contain spiral topological defects. The system can spontaneously break chiral symmetry when the layers organize into spiral patterns, composed of a bound pair of $+1/2$ disclinations. Finally, we suggest possible applications of tetrahedral smectic vesicles in the context of functionalizing defects and the possible consequences of the spiral structures for the rigidity of the vesicle.