The S'-S' ' and H'-T Minimal Surfaces and their Application to Structural Modelling of Intermediate Phases

A geometric basis is presented for the analysis of possible structures of anisotropic liquid-crystalline phases of surfactant-water mixtures between the hexagonal and lamellar phase regions. As a starting point the candidates among the triply periodic minimal surfaces partitioning symmetrically distinct labyrinths are considered. The two simplest examples of this type of bicontinuous geometry are the tetragonal S'-S^{' '} and hexagonal H'-T surfaces of genus 4. Using the exact parametrizations, their cell dimensions, vertex positions, areas and volumes are calculated. These details of the minimal surfaces are useful both in assessing the possibility of such bicontinuous intermediate phases and for generating the corresponding constant mean-curvature families which embrace other topologies, including the interesting mesh structures.