Michael Ding with Timothy Atherton

Liquid Crystal Configurations from Conformal Mapping

The method of conformal mapping can be applied to solve Laplace’s equation in complicated 2-D geometries. First approximations for nematic liquid crystals obey Laplace’s equation and so we may apply a conformal map to determine the configuration of liquid crystal, and hence obtain the energy and consequently the phase diagram. Conformal mapping allows for the preservation of angles in the complex plane. Conformal maps for convex polygons have been explored. However, non-convex geometries are more problematic. Our research will be focused on investigating energy distributions of nematic liquid crystals in non-convex n-sided polygons. The Schwarz-Christoffel mapping, specific conformal map, will be applied to these non-convex polygons. Furthermore, we intend to apply a recently discovered formula due to Crowdy to consider nematic liquid crystals confined to multiply connected domains.