Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-10T21:54:32.534Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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