European Journal of Applied Mathematics

Table of Contents - February 2012 - Volume 23, Special Issue 01 (Liquid Crystals)  

Papers

The radial-hedgehog solution in Landau–de Gennes' theory for nematic liquid crystals

APALA MAJUMDARa1

a1 Oxford Centre for Collaborative Applied Mathematics, University of Oxford, UK email: majumdar@maths.ox.ac.uk

Abstract

We study the radial-hedgehog solution in a three-dimensional spherical droplet, with homeotropic boundary conditions, within the Landau–de Gennes theory for nematic liquid crystals. The radial-hedgehog solution is a candidate for a global Landau–de Gennes minimiser in this model framework and is also a prototype configuration for studying isolated point defects in condensed matter physics. The static properties of the radial-hedgehog solution are governed by a non-linear singular ordinary differential equation. We study the analogies between Ginzburg–Landau vortices and the radial-hedgehog solution and demonstrate a Ginzburg–Landau limit for the Landau–de Gennes theory. We prove that the radial-hedgehog solution is not the global Landau–de Gennes minimiser for droplets of finite radius and sufficiently low temperatures and prove the stability of the radial-hedgehog solution in other parameter regimes. These results contain quantitative information about the effect of geometry and temperature on the properties of the radial-hedgehog solution and the associated biaxial instabilities.

(Received July 05 2011)

(Revised July 15 2011)

(Accepted July 18 2011)

(Online publication September 06 2011)

Key words:

  • Defects;
  • Landau–de Gennes;
  • Ginzburg–Landau