Triply-Periodic Smectics

TL;DR

This paper analytically constructs and analyzes the energetics of twist-grain-boundary phases in smectic-A liquid crystals, revealing special angles with simplified structures and identifying new defects using elliptic functions.

Contribution

It introduces an analytical method to construct the height function of a specific twist-grain-boundary phase using elliptic functions, advancing understanding of defect structures in smectics.

Findings

Analytical construction of Schnerk's first surface using elliptic functions.

Identification of unknown defects along the pitch axis.

Existence of privileged angles with simpler grain boundary structures.

Abstract

Twist-grain-boundary phases in smectics are the geometrical analogs of the Abrikosov flux lattice in superconductors. At large twist angles, the nonlinear elasticity is important in evaluating their energetics. We analytically construct the height function of a pi/2 twist-grain-boundary phase in smectic-A liquid crystals, known as Schnerk's first surface. This construction, utilizing elliptic functions, allows us to compute the energy of the structure analytically. By identifying a set of heretofore unknown defects along the pitch axis of the structure, we study the necessary topological structure of grain boundaries at other angles, concluding that there exist a set of privileged angles and that the \pi/2 and \pi/3 grain boundary structures are particularly simple.