The micellar cubic phases of lipid-containing systems: Analogies with foams, relations with the infinite periodic minimal surfaces, sharpness of the polar apolar partition

Of the 7 cubic phases clearly identified in lipid-containing systems, 2 are bicontinuous; 4 micellar. 3 of these are of type I: one (Q(223)) consists Of two types of micelles, the two others of identical quasi-spherical micelles close-packed in the face-centred (Q(225)) or the body-centred mode (Q(229)). These structures, much like foams, can be described as systems of space-filling polyhedra: distorted 12- and 14-hedra in Q(223), rhombic dodecahedra. in Q(225), truncated octahedra in Q(229). In foams the geometry of the septa and of their junctions are generally assumed to obey Plateau's conditions, at least at vanishing water content: these conditions are satisfied in Q(223), be satisfied in Q(229) by introducing subtle distortions in the hexagonal faces, but cannot be satisfied in Q(225). Alternatively, these structures can be represented in terms of infinite periodic minimal surfaces (IPMS) since it is found that two types of IPMS, F-RD in Q(225) and I-WP in Q(229), almost coincide with one particular equi-electron-density surface of the 3D electron density maps. These IPMS partition 3D space into two non-congruent labyrinths: in the case of the lipid phases one of the labyrinths contains the hydrated micelles, the other is filled by water. If interfacial interactions are associated with these surfaces, then the surfaces being minimal, the interactions may also be expected to be minimal. Another characteristic of the micellar phases is that the dimensions of their hydrophobic core, computed assuming that headgroups and water are totally immiscible with the chains, often are incompatible with the fully extended length of the chains. This paradox is evaded if headgroups and chains are allowed to be partially miscible with each other.