Numerical Minimization of the Landau-de Gennes Free Energy: Defects in Cylindrical Capillaries

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Abstract
In order to study the structure of defects in nematic liquid crystals, we have constructed a numerical procedure that minimizes the Landau-de Gennes free energy model. Using a new representation, a finite-element discretization, and a direct minimization scheme based on Newton's method and successive overrelaxation, this procedure determines the order parameter tensor field in three dimensions for a general physical problem with Dirichlet boundary conditions. As a sample problem, we have considered a nematic liquid crystal in a cylindrical capillary with boundary conditions that necessarily give rise to at least one defect. We find that for our parameters, two biaxial defects appear near the ends of a capillary with homeotropic alignment, and that near the center, the director is perpendicular to the axis of the cylinder.