Liquid Crystals

This chapter is devoted to a brief review of liquid crystals, providing both a physical background and a mathematical background. Namely, we discuss the various competing theories for nematic liquid crystals: the Oseen–Frank model, Eriksen model, and Landau–de Gennes theory. We describe some of the mathematics used to understand defects and classical director deformations (bend, splay, and twist). We also derive formulas for the effective refractive indices for a nematic liquid crystal, based on both definitions of light ray discussed in the previous chapter. We also derive a formula for the effective refractive index for a biaxial nematic liquid crystal. This chapter is meant to serve as a brief introduction to liquid crystals to those more familiar with the mathematical optics aspects. In particular, we also provide a short review of some recent mathematical results.