Bicontinuous geometries and molecular self-assembly: Comparison of local curvature and global packing variations in genus-three cubic, tetragonal and rhombohedral surfaces

Balanced infinite periodic minimal surface families that contain the cubic Gyroid (G), Diamond (D) and Primitive (P) surfaces are studied in terms of their global packing and local curvature properties. These properties are central to understanding the formation of mesophases in amphiphile and copolymer molecular systems. The surfaces investigated are the tetragonal, rhombohedral and hexagonal tD, tP, tG, rG, rPD and H surfaces. These non-cubic minimal surfaces furnish topology-preserving transformation pathways between the three cubic surfaces. We introduce `packing (or global) homogeneity', defined as the standard deviation Δd of the distribution of the channel diameter throughout the labyrinth, where the channel diameter d is determined from the medial surface skeleton centered within the labyrinthine domains. Curvature homogeneity is defined similarly as the standard deviation ΔK of the distribution of Gaussian curvature. All data are presented for distinct length normalisations: constant surface-to-volume ratio, constant average Gaussian curvature and constant average channel diameter. We provide first and second moments of the distribution of channel diameter for all members of these surfaces complementing curvature data from [A. Fogden, S. Hyde, Eur. Phys. J. B 7, 91 (1999)]. The cubic G and D surfaces are deep local minima of Δd along the surface families (with G more homogeneous than D), whereas the cubic P surface is an inflection point of Δd with adjacent, more homogeneous surface members. Both curvature and packing homogeneity favour the tetragonal route between G and D (via tG and tD surfaces) in preference to the rhombohedral route (via rG and rPD)