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

Published online by Cambridge University Press:  06 September 2011

APALA MAJUMDAR
APALA MAJUMDAR*
Affiliation:
Oxford Centre for Collaborative Applied Mathematics, University of Oxford, UK email: [email protected]
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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.

Keywords

DefectsLandau–de GennesGinzburg–Landau
Type
Papers
Information
European Journal of Applied Mathematics , Volume 23 , Issue 1: Liquid Crystals , February 2012 , pp. 61 - 97
Copyright
Copyright © Cambridge University Press 2011

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