Chuck Gartland
Department of Mathematics and Computer Science
Kent State University
Abstract: We investigate the structure of defects in nematic liquid crystals confined in spherical droplets and subject radial boundary conditions. Equilibrium configurations of the order-parameter tensor field in a Landau-de Gennes free energy are numerically modeled using a finite-element package. Within the class of axially symmetric fields, we find three distinct solutions: the familiar radial hedgehog and the small ring/loop disclination, plus a new "split core" solution, which consists of a short disclination line segment along the rotational symmetry axis terminating in isotropic end points. Phase a bifurcation diagrams are constructed to illustrate how the three competing configurations are related. They confirm that the transition from hedgehog to the ring structure is first order. The split-core configuration is metastable (in our symmetry class) and forms an alternate solution branch bifurcating off the radial hedgehog branch at the temperature below which the hedgehog ceases to be metastable. Motivated by this numerical investigation, we have constructed an analytical argument to show that the radial hedgehog configuration must loose its metastability at sufficiently low temperatures in droplets of sufficiently large radii for all but a very limited range of elastic-constant ratios. The analysis is complicated by the fact that no analytical solution is available for the hedgehog configuration (under Landau theory).