NUI MAYNOOTH  DEPARTMENT OF MATHEMATICAL PHYSICS SEMINAR

Friday 3rd February 2006 at 4pm

Lecture Hall CS1, Callan Building, NUIM North Campus


Dr. Jonathan Robbins
Department of Mathematics, University of Bristol, U.K.

Liquid crystals, polyhedral geometries and harmonic maps

Motivated by prototype designs for bistable displays, we consider nematic liquid crystals, described by a director field, in a polyhedral domain subject to tangent boundary conditions.  Tangent boundary conditions mean that, on the surfaces of the domain, the director must lie flat, but is otherwise unconstrained, and imply that the director field is singular at the vertices.  Otherwise-continuous director fields may be classified by a number of topological invariants.

Equilibrium configurations are minimisers of the Frank-Oseen or elastic energy.  We obtain lower bounds for the equilibrium energy in the one-constant approximation.  These generalise the classical "minimal connection" of Brezis, Coron and Lieb.

For a rectangular prism, we obtain upper bounds for the equilibrium energy by constructing trial configurations from local solutions of the Euler-Lagrange equations.  These scale with the topological parameters in the same way as the lower bounds do, differing from them by factor depending only on the prism geometry.  For certain topologies, this factor is (1 + a^2 + b^2)^{1/2}, where a and b are the aspect ratios of the prism.

We argue that minimisers for the simplest topologies are smooth, while for other topologies we observe a transition between smooth and singular minimisers as the aspect ratios are varied.