We study the gradient flow model of the Landau-de Gennes energy functional for nematic liquidcrystals at the isotropic-nematic transition temperature on prototype geometries. We focus onthe three-dimensional droplet, the disc and the square with Dirichlet boundary conditions anddifferent types of initial conditions, with the aim of observing interesting transient dynamics which may be of practical relevance. We use a fourth-order Landau-de Gennes bulk potential which admits isotropic and uniaxial minima at the transition temperature. For a droplet with radial boundary conditions, a large class of physically relevant initial conditions generate dynamic solutions with a well-defined isotropic-nematic front which propagates according to mean curvature for significant times. We introduce radially symmetric obstacles into the droplet and prove the existence of pulsating wave solutions of the gradient flow model in certain parameter regimes. The average velocity of the pulsating wave is determined by some critical forcing which can be verified numerically. On the unit disc, we make a distinction between planar and non-planar initial conditions and minimal and non-minimal Dirichlet boundary conditions. Planar initial conditions generate solutions with an isotropic core for all times whereas non-planar initial conditions generate solutions that escape into the third dimension. Non-minimal boundary conditions result in solutions with boundary layers. These solutions can have either a largely nematic interior profile or a largely isotropic interior profile, depending on the initial conditions. On the square, we provide an analytic description of the Well Order Reconstruction solution first reported numerically by Kralj and Majumdar in 2014. We interpret the Well Order Reconstruction solution as a critical point of a related scalar variational problem and prove that the solution is globally stable on small domains. We use the gradient flow model of the Landau-de Gennes energy to numerically study the emergence of new solution branches from the Well Order Reconstruction solution. We conclude this thesis by studying a triple phase Landau-de Gennes model with a sixth-order bulk potential which admits isotropic, uniaxial and biaxial minima at a special temperature known as the triple point temperature. For some model problems, we can use asymptotic methods to prove that isotropic-uniaxial, uniaxial-biaxial and isotropic-biaxial fronts propagate according to mean curvature and to prove an angle condition that holds when the fronts intersect at a triple junction. We corroborate our formal calculations with a numerical investigation of the full Landau-de Gennes gradient flow system.
|Date of Award||25 Apr 2017|
|Supervisor||Apala Majumdar (Supervisor)|