A rhombohedral family of minimal surfaces as a pathway between the P and D cubic mesophases

A medial surface (MS) analysis of the rhombohedral infinite periodic minimal surface family rPD is presented. The rPD family of bicontinuous surfaces has been suggested as a pathway for transitions between its two cubic members, the P and the D surface, in mesophases in liquid-crystalline self-assembly. The MS is a representation of a labyrinth as a centered 2D skeleton. By providing a definition of a pointwise channel diameter, the MS allows for an analysis of homogeneity of such surfaces including chain stretching frustration. For the rPD surface, variations of this channel diameter are locally minimal for the D surface, and a horizontal inflection point for the P surface. This may have implications for the phase stability of the corresponding liquid-crystalline mesophases. The MS can be further reduced to a 1D line graph. For the rPD surface, this graph contains curved edges and cannot be deduced from symmetry considerations alone.