Symmetry, topology and faceting in bicontinuous lyotropic crystals.

Phase diagrams of phytantriol/ethanol/water and phytantriol/DSPG/ethanol/water systems are explored and experiments on facetings of Pn3m-in-L1 and Im3m-in-L1 crystals are performed. Observed crystal habits do not agree with the Friedel-Donnay-Harker rules. We argue that this paradox can be explained in terms of constraints imposed on Pn3m/L1 and Im3m/L1 interfaces by the bicontinuous topology of the cubic phases. We point out that when free edges of the surfactant bilayer are prohibited at these interfaces, the two labyrinthes separated by the bilayer cannot anymore be equivalent. The corresponding [Formula: see text] and [Formula: see text] symmetry breakings are unveiled by the abnormal facetings.