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This article analyzes modulated phases in liquid crystals, from thelong-established cholesteric and blue phases to the recently discoveredtwist-bend, splay-bend, and splay nematic phases, as well as thetwist-grain-boundary (TGB) and helical nanofilament variations on smecticphases. The analysis uses the concept of four fundamental modes of directordeformation: twist, bend, splay, and a fourth mode related to saddle-splay.Each mode is coupled to a specific type of molecular order: chirality,polarization perpendicular and parallel to the director, and octupolar order.When the liquid crystal develops one type of spontaneous order, the ideal localstructure becomes nonuniform, with the corresponding director deformation. Ingeneral, the ideal local structure is frustrated; it cannot fill space. As aresult, the liquid crystal must form a complex global phase, which may have acombination of deformation modes, and may have a periodic array of defects.Thus, the concept of an ideal local structure under geometric frustrationprovides a unified framework to understand the wide variety of modulatedphases.

Director Deformations, Geometric Frustration, and Modulated Phases in Liquid Crystals