Abstract: We study the equilibrium properties of liquid crystal configurations in confined geometries, within the continuum Landau-De Gennes theory. Liquid crystals are intermediate phases of matter that exhibit partial ordering in the orientation and/or positions of their constituent molecules. In the simplest liquid crystal phase, the nematic phase, the constituent rod-like molecules translate freely as in a conventional liquid but whilst flowing, their molecular axes tend to align along certain locally preferred directions. A liquid crystal configuration is mathematically described by a symmetric, traceless 2-tensor, the Q-tensor order parameter whose eigenvectors and eigenvalues contain information about the preferred directions and degree of alignment respectively. From a statistical mechanics point of view, the state of alignment of a liquid crystal is described by a probability distribution function; this description imposes lower and upper bounds on the eigenvalues of the Q-tensor order parameter.
We work within the continuum Landau-De Gennes theory whereby there are no a priori bounds on the eigenvalues. The liquid crystal energy is then given by the Landau-De Gennes energy functional, which typically consists of a non-convex bulk potential with multiple minima and an elastic energy, which penalizes spatial inhomogeneities. The equilibrium configurations are then given by (global and local) minimizers of the Landau-De Gennes energy subject to the imposed boundary conditions. We derive explicit bounds for the eigenvalues of equilibrium Q-tensor configurations, using the energy-minimizing property. These bounds are a function of the temperature, material constants, the confining geometry and the boundary conditions. We compare these bounds with the statistical mechanics predictions. These bounds illustrate the analogies between the statistical mechanics approach and the continuum Landau-De Gennes theory. In particular, it indicates the temperature regimes over which the two theories are in qualitative agreement, and also the regimes over which the two theories exhibit discrepancies. We also briefly comment on some modified Landau-De Gennes type of energy functionals, which account for the statistical mechanics constraints.