Eur. Phys. J. E (2018) 41: 143
https://doi.org/10.1140/epje/i2018-11755-y

Regular Article

Topology and ground-state degeneracy of tetrahedral smectic vesicles

¹ Physics Department and Syracuse Soft and Living Matter Program, Syracuse University, 13244, Syracuse, NY, USA
² Kavli Institute for Theoretical Physics, University of California, 93106, Santa Barbara, CA, USA
³ The James Franck and Enrico Fermi Institutes and The Department of Physics, The University of Chicago, 60637, Chicago, IL, USA

^* e-mail: fserafin@syr.edu
^** e-mail: bowick@kitp.ucsb.edu
^*** e-mail: srnagel@uchicago.edu

Received: 15 July 2018
Accepted: 16 November 2018
Published online: 11 December 2018

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 . 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.