Description of cubic liquid-crystalline structures using simple surface foliations

Abstract

In lyotropic and thermotropic liquid crystals of cubic symmetry the molecular organisation is naturally described in relation to a central partitioning surface, which is frequently assumed to be a minimal surface. Here we suggest a mathematically simpler basis, borrowed from solid-state physics, given by the leading invariants of the three-dimensional Fourier series for the space group. The zero level surface is taken as the central partition, with the complete family of level surfaces then providing the foliation of space either side of it. In particular, we analyse the space group Imm and establish a topological classification of the simplest central partitions and their accompanying families of foliations. Further, we quantitatively compare the particular subclass which shares the topology of the familiar P minimal surface, and find that it yields extremely accurate approximations. Amongst the proposed applications of our analysis, we illustrate the construction of space-filling director fields for ‘blue’ phases.