Defect densities in directional media, mainly liquid crystals

It is possible to define a local frame of reference at each point of a distorted liquid crystal, transformed from a fixed frame in the perfect crystal by a pure rotation. In the local frame the director n has constant components. A curvature tensor Kij is used to describe rotations between neighbouring frames; it obeys the compatibility condition θ = Rot K − M = 0, where M is a tensor formed from the minors of K. M takes into account the non-commutativity of small rotations. The deformation can therefore be described by three types of defect densities. αij = δij K ii − K ji is the source of translation dislocations, similar to the Nye tensor introduced in multiple glide. In a liquid crystal, in contradistinction to a solid, ’glide’ may involve stretching of the lattice. M ij is related to tilt and twist boundaries terminating in the medium, and is analogous to a disclination density. When disclination lines are present, θ ij = 0. There is, moreover, a vectorial density Λi = K ijnj, related to disclination points. Disclination and dislocation densities cannot be imposed at will in a liquid crystal, which constitutes a striking difference from crystals. A relationship between densities is provided by minimizing the Frank energy. This relationship describes a limitation on the processes of viscous relaxation.