Frank–Kasper, quasicrystalline and related phases in liquid crystals

The review covers 3-dimensional and some 2-dimensional self-assembly patterns of supramolecular liquid crystals possessing either quasi-periodic or closely related truly periodic order. Compounds showing such structures are amphiphilic, and most often wedge-shaped, with dendrons being the most common examples. The topology is described in terms of 3D and 2D tiling of a variety of polyhedra or polygons, respectively. Analogy is made with structures of metallic alloys and soap froth. The recently discovered dodecagonal liquid quasicrystal is compared with the different tetrahedrally close packed Frank–Kasper phases in thermotropic and lyotropic liquid crystals. Parallels are also drawn with honeycomb columnar phases with related plane tilings, including that of distorted pentagons. The potential for the creation of nearly isotropic photonic bandgap materials is mentioned.