Eur. Phys. J. B 7, 91-104 (1999)
https://doi.org/10.1007/s100510050592

Continuous transformations of cubic minimal surfaces

¹ Physical Chemistry 1, Center for Chemistry and Chemical Engineering, University of Lund, Box 124, 221 00 Lund, Sweden
² Department of Applied Mathematics, Research School of Physical Sciences and Engineering, Canberra ACT 0200, Australia

Received: 5 March 1998
Revised: 29 July 1998
Accepted: 31 July 1998
Published online: 15 January 1999

Abstract

Although the primitive (P), diamond (D) and gyroid (G) minimal surfaces form the structural basis for a multitude of self-assembling phases, such as the bicontinuous cubics, relatively little is known regarding their geometrical transformations, beyond the existence of the Bonnet isometry. Here their highest symmetry deformation modes, the rhombohedral and tetragonal distortions, are fully elucidated to provide a unified description of these simplest minimal surface families, with all quantities expressed in terms of complete elliptic integrals. The rhombohedral distortions of the gyroid are found to merge continuously with those which bridge the P and D surfaces, furnishing direct transformations between all three cubics, preserving both topology and zero mean curvature throughout. The tetragonal distortions behave analogously, offering an alternative route from the gyroid to the D surface. The cell axis ratios, surface areas and Gaussian curvature moments of all families are given, supplying the necessary geometrical input to a curvature energy description of cubic and intermediate phase stability.