From faceted vesicles to liquid icoshedra: Where topology and crystallography meet

Many common amphiphiles spontaneously self-assemble in aqueous solutions, forming membranes and unilamellar vesicles. While the vesicular membranes are bilayers, with the hydrophilic moieties exposed to the solution, the structure formed by amphiphiles at the oil-water (i.e., alkane-water) interfaces, such as the surface of an oil droplet in water, is typically a monolayer. It has recently been demonstrated that these monolayers and bilayers may crystallize on cooling, with the thermodynamic conditions for this transition set by the geometry of the constituent molecules. While a planar hexagonal packing motif is particularly abundant in these crystals, a hexagonal lattice is incompatible with a closed-surface topology, such as a closed vesicle or the surface of a droplet. Thus, (at least) 12 five-fold defects form, giving rise to a complex interplay between the stretching and the bending energies of these two-dimensional crystals; in addition, a central role is also played by the interfacial tension. This interplay, part of which has been theoretically studied in the past, gives rise to a range of unexpected and counterintuitive phenomena, such as the recently-observed temperature-tunable formation of stable liquid polyhedra, and a tail growing and droplet-splitting akin to the spontaneous emulsification effect.