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Many solid materials and liquid crystals exhibit geometric frustration,meaning that they have an ideal local structure that cannot fill up space. Forthat reason, the global phase must be a compromise between the ideal localstructure and geometric constraints. As an explicit example of geometricfrustration, we consider a chiral liquid crystal confined in a long cylinderwith free boundaries. When the radius of the tube is sufficiently small, thedirector field forms a double-twist configuration, which is the ideal localstructure. However, when the radius becomes larger (compared with the naturaltwist of the liquid crystal), the double-twist structure cannot fill space, andhence the director field must transform into some other chiral structure thatcan fill space. This space-filling structure may be either (1) a cholestericphase with single twist, or (2) a set of double-twist regions separated by adisclination, which can be regarded as the beginning of a blue phase. Weinvestigate these structures using theory and simulations, and show how therelative free energies depend on the system size, the natural twist, and thedisclination energy. As another example, we also study a cholesteric liquidcrystal confined between two infinite parallel plates with free boundaries.

Explicit Demonstration of Geometric Frustration in Chiral Liquid Crystals