Computational Aspects of Energy Minimization of the Landau-de Gennes Model for Liquid Crystals

Abstract

From a mathematically rigorous foundation, energy minimizing configurations are numerically computed using the Landau-de Gennes energy expression for a variety of liquid crystal domains in order to compare them with previous experimental and numerical results as well as present new results. The method of manufactured solutions is used to validate numerical computations in two- and three-dimensional domains before computing the minimizers. Many configurations and defect structures seen in experiments were reproduced numerically and studied carefully, with three notable results: First, in the two-dimensional disc domain with normal Dirichlet boundary conditions, the 'split core' defect is shown to be a product of improper mesh refinement. Second, for all concentric cylinder domains with planar radial Dirichlet boundary conditions, biaxial regions were present, which is a marked difference from previous results using the Oseen-Frank model. Third, for a single colloidal particle with normal Dirichlet boundary conditions suspended in a cylinder with everywhere vertical boundary conditions, the core of the Saturn ring defect is shown to be uniaxial with negative scalar order parameter. These three results more accurately describe experimental and theoretical results than those using the vector-based models and motivate the use of the Landau-de Gennes model for future numerical computations.