  This article analyzes modulated phases in liquid crystals, from the
long-established cholesteric and blue phases to the recently discovered
twist-bend, splay-bend, and splay nematic phases, as well as the
twist-grain-boundary (TGB) and helical nanofilament variations on smectic
phases. The analysis uses the concept of four fundamental modes of director
deformation: 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 local
structure becomes nonuniform, with the corresponding director deformation. In
general, the ideal local structure is frustrated; it cannot fill space. As a
result, the liquid crystal must form a complex global phase, which may have a
combination of deformation modes, and may have a periodic array of defects.
Thus, the concept of an ideal local structure under geometric frustration
provides a unified framework to understand the wide variety of modulated
phases.
