Bicontinuous structures in lyotropic liquid crystals and crystalline hyperbolic surfaces

The idealized structures of cubic bicontinuous mesophases in amphiphile—water systems are well described using minimal surfaces exhibiting cubic translational symmetry. The question of bicontinuity of noncubic mesophases, including some hexagonal and intermediate phases, remains open. Recent work has focused on the nature of phase transformations involving bicontinuous mesophases, leading to some understanding of the energetics that govern mesophase stability, and locations (if any) of neutral surfaces in swelling bilayers. Further advances await, regarding detailed probes of the topological ‘defect’ density in these systems, including channels and pores within the bilayer membrane. In addition to basic physical understanding of lyotropic mesomorphism, probes of topology are required to sort out differences between novel mesoporous solids, often templated using lyotropic liquid crystals. There is evidence for a bicontinuous morphology in ‘folded sheet materials’ derived from kanemite.