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Chemical design of block copolymers makes it possible to create polymervesicles with tunable microscopic structure. Here we focus on a model of avesicle made of smectic liquid-crystalline block copolymers at zerotemperature. The vesicle assumes a faceted tetrahedral shape and the smecticlayers arrange in a stack of parallel straight lines with topological defectslocalized 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 thesmectic pattern depending on the tilt angle between the smectic layers and theedge of the tetrahedral shell. For most values of the tilt angle, the smecticlayers contain spiral topological defects. The system can spontaneously breakchiral symmetry when the layers organize into spiral patterns, composed of abound pair of $+1/2$ disclinations. Finally, we suggest possible applicationsof tetrahedral smectic vesicles in the context of functionalizing defects andthe possible consequences of the spiral structures for the rigidity of thevesicle.

Topology and ground state degeneracy of tetrahedral smectic vesicles