  Inverse bicontinuous cubic phases are ubiquitous in lipid-water mixtures and
consist of a lipid bilayer forming a cubic minimal surface, thereby dividing
space into two cubic networks of water channels. For small hydrocarbon chain
lengths, the monolayers can be modeled as parallel surfaces to a minimal
midsurface. The bending energy of the cubic phases is determined by the
distribution of Gaussian curvature over the minimal midsurfaces which we
calculate for seven different structures (G, D, P, I-WP, C(P), S and F-RD). We
show that the free-energy densities of the structures G, D and P are
considerably lower than those of the other investigated structures due to their
narrow distribution of Gaussian curvature. The Bonnet transformation between G,
D, and P implies that these phases coexist along a triple line, which also
includes an excess water phase. Our model includes thermal membrane
undulations. Our qualitative predictions remain unchanged when higher order
terms in the curvature energy are included. Calculated phase diagrams agree
well with the experimental results for 2:1 lauric acid/dilauroyl
phosphatidylcholine and water.
