Models for biaxial nematic liquid crystals

In the Landau-de Gennes theory, the order parameter describing a biaxial nematic liquid crystal is, at each point of the region occupied by the system, a symmetric, traceless 3 × 3 matrix with three distinct eigenvalues. In the constrained case of matrices with constant eigenvalues, the order parameter space identifies with an eightfold quotient of the 3-sphere, and a configuration of a biaxial nematic liquid crystal is described by a map into such a quotient. We express the (simplest form of the) Landau-de Gennes elastic free-energy density of biaxial nematics as a density on maps into the 3-sphere, whose functional dependence is restricted by the requirements that it is well-defined on the class of configuration maps (residual symmetry) and is independent of arbitrary superposed rigid rotations (frame indifference). This is a report on joint work with D. Mucci [2012 Arch. Ration. Mech. Anal. 206 853–884], to which we refer for more details and applications