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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: majumdar@maths.ox.ac.uk
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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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References

[1]Bethuel, F., Brezis, H. & Helein, F. (1994) Ginzburg–Landau Vortices, Progress in Nonlinear Differential Equations and their Applications, Vol. 13, Birkhauser, Boston.Google Scholar
[2]Bethuel, F., Brezis, H. & Orlandi, G. (2001) Asymptotics for the Ginzburg–Landau Equation in Arbitrary Dimensions. J. Funct. Anal. 186, 432–520.CrossRefGoogle Scholar
[3]Brezis, H. (1999) Symmetry in nonlinear PDEs, Differential equations: La Pietra 1996 (Florence). In: Proceedings of Symposia in Pure Mathematics, Florence, vol. 65, pp. 1–12.Google Scholar
[4]Chen, X., Elliott, C. & Tang, Q. (1994) Shooting method for vortex solutions of a complex valued Ginzburg–Landau equation. Proc. Roy. Soc. Edinburgh, Sec. A 124 (6), 1075–1088.CrossRefGoogle Scholar
[5]Chi, D. P. & Park, G. H. (1992) Weak-stability of x/|x| and symmetries of liquid crystals. J. Korean Math. Soc. 29 (2), 251–260.Google Scholar
[6]Davis, T. & Gartland, E. C. Jr, (1998) Finite element analysis of the Landau–de Gennes minimization problem for liquid crystals. SIAM J. Numer. Anal. 35, 336–362.CrossRefGoogle Scholar
[7]DeGennes, P. G. Gennes, P. G. (1974) The Physics of Liquid Crystals, Clarendon Press, Oxford.Google Scholar
[8]Evans, L. (1998) Partial Differential Equations, American Mathematical Society, Providence, RI.Google Scholar
[9]Farina, A. & Guedda, M. (2000) Qualitative study of radial solutions of the Ginzburg–Landau system in (N ≥ 3). Appl. Math. Lett. 13, 59–64.CrossRefGoogle Scholar
[10]Gartland, E. C. Jr & Mkaddem, S. (1999) Instability of radial hedgehog configurations in nematic liquid crystals under Landaude–Gennes free-energy models. Phys. Rev. E 59, 563–567.CrossRefGoogle Scholar
[11]Mkaddem, S. & Gartland, E. C. Jr, (2000) Fine structure of defects in radial nematic droplets. Phys. Rev. E 62, 6694–6705.CrossRefGoogle ScholarPubMed
[12]Gustafson, S. (1997) Symmetric solutions of the Ginzburg–Landau equation in all dimensions. Int. Math. Res. Not. 16, 807–816.CrossRefGoogle Scholar
[13]Henao, D. & Majumdar, A. Radial symmetry of uniaxial minimizers in Landau–de Gennes theory. In preparation.Google Scholar
[14]Herve, R-M. & Herve, M. (1994) Etude qualitative des solutions reelles d'une equation differentielle liee e lequation de Ginzburg–Landau. Ann. Inst. H. Poincar Anal. Non Lineaire 11, 427–440.CrossRefGoogle Scholar
[15]Kinderlehrer, D. & Ou, B. (1992) Second variation of liquid crystal energy at x/|x|. Proc. R. Soc. A: Math., Phys. Eng. Sci. 437, 475–487.Google Scholar
[16]Kralj, S. & Virga, E. (2001) Universal fine structure of nematic hedgehogs. J. Phys. A: Math. Gen. 24, 829–838.CrossRefGoogle Scholar
[17]Kralj, S., Rosso, R. & Virga, E. G. (2010) Finite-size effects on order reconstruction around nematic defects. Phys. Rev. E 81, 021702.CrossRefGoogle ScholarPubMed
[18]Majumdar, A. & Zarnescu, A. (2010) The Landau–de Gennes theory of nematic liquid crystals: The Oseen–Frank limit and beyond. Arch. Ration. Mech. Anal. 196 (1), 227–280.CrossRefGoogle Scholar
[19]Majumdar, A. (2010) Equilibrium order parameters of liquid crystals in the Landau–de Gennes theory. Eur. J. Appl. Math. 21, 181–203.CrossRefGoogle Scholar
[20]Majumdar, A. The Landau–de Gennes theory for nematic liquid crystals: Uniaxiality versus Biaxiality [online]. Under review in Communications in Pure and Applied Analysis.Google Scholar
[21]Millot, V. & Pisante, A. (2010) Symmetry of local minimizers for the three-dimensional Ginzburg–Landau functional. J. Eur. Math. Soc. 12, 1069–1096.CrossRefGoogle Scholar
[22]Mottram, N. J. & Newton, C. (2004) Introduction to Q-tensor Theory, University of Strathclyde Mathematics, Research Report no. 10.Google Scholar
[23]Penzenstadler, E. & Trebin, H.-R. (1989) Fine structure of point defects and soliton decay in nematic liquid crystals. J. Phys. (France) 50, 1027–1040.CrossRefGoogle Scholar
[24]Priestley, E. B., Wojtowicz, P. J. & Sheng, P. (1975) Introduction to Liquid Crystals, Plenum, New York.CrossRefGoogle Scholar
[25]Rosso, R. & Virga, E. (1996) Metastable nematic hedgehogs. J. Phys. A: Math. Gen. 29, 4247–4264.CrossRefGoogle Scholar
[26]Schopohl, N. & Sluckin, T. J. (1988) Hedgehog structures in nematic and magnetic systems. J. Phys. (France) 49, 1097.CrossRefGoogle Scholar
[27]Sonnet, A., Kilian, A. & Hess, S. (1995) Alignment tensor versus director: Description of defects in nematic liquid crystals. Phys. Rev. E 52, 718–722.CrossRefGoogle ScholarPubMed
[28]Sun, D. & Sun, J. (2002) Strong semismoothness of eigenvalues of symmetric matrices and its application to inverse eigenvalue problems. SIAM J. Numer. Anal. 40, 2352–2367.CrossRefGoogle Scholar