We study the gradient flow model for the Landau--de Gennes energy functional for nematic liquid crystals at the nematic-isotropic transition temperature on prototype geometries. We study the dynamic model on a three-dimensional droplet and on a disc with Dirichlet boundary conditions and different types of initial conditions. In the case of a droplet with radial boundary conditions, a large class of physically relevant initial conditions generate dynamic solutions with a well-defined nematic-isotropic interface which propagates according to mean curvature for small times. On a disc, we make a distinction between “planar” and “nonplanar” initial conditions, and “minimal” and “nonminimal” Dirichlet boundary conditions. Planar initial conditions generate solutions with an isotropic core for all times, whereas nonplanar initial conditions generate solutions which escape into the third dimension. Nonminimal boundary conditions generate solutions with boundary layers, and these solutions can either have a largely ordered interior profile or an almost entirely disordered isotropic interior profile. Our examples suggest that while critical points of the Landau--de Gennes energy typically have highly localized disordered-ordered interfaces, the transient dynamics exhibits observable interfaces of potential experimental relevance.
Majumdar, A., Milewski, P. A., & Spicer, A. (2016). Front Propagation at the Nematic-Isotropic Transition Temperature. SIAM Journal on Applied Mathematics, 76(4), 1296-1320. https://doi.org/10.1137/15M1039250