Elastic anisotropy in the reduced Landau–de Gennes model

Abstract

We study the effects of elastic anisotropy on Landau–de Gennes critical points, for nematic liquid crystals, on a square domain. The elastic anisotropy is captured by a parameter, L 2 , and the critical points are described by 3 d.f. We analytically construct a symmetric critical point for all admissible values of L 2 , which is necessarily globally stable for small domains, i.e. when the square edge length, λ , is small enough. We perform asymptotic analyses and numerical studies to discover at least five classes of these symmetric critical points—the WORS , Ring ± , C o n s t a n t and p W O R S solutions, of which the WORS , Ring + and C o n s t a n t solutions can be stable. Furthermore, we demonstrate that the novel C o n s t a n t solution is energetically preferable for large λ and large L 2 , and prove associated stability results that corroborate the stabilizing effects of L 2 for reduced Landau–de Gennes critical points. We complement our analysis with numerically computed bifurcation diagrams for different values of L 2 , which illustrate the interplay of elastic anisotropy and geometry for nematic solution landscapes, at low temperatures.