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Equilibrium order parameters of nematic liquid crystals in the Landau-de Gennes theory

Published online by Cambridge University Press:  25 February 2010

APALA MAJUMDAR
APALA MAJUMDAR*
Affiliation:
Mathematical Institute, University of Oxford, 24–29 St.Giles, Oxford, OX1 3LB, UK email: [email protected]
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Abstract

We study equilibrium liquid crystal configurations in three-dimensional geometries, within the continuum Landau-de Gennes theory. We obtain explicit bounds for the equilibrium scalar order parameters in terms of the temperature and material-dependent constants. We explicitly quantify the temperature regimes where the Landau-de Gennes predictions match and the temperature regimes where the Landau-de Gennes predictions do not match the probabilistic second-moment definition of the Q-tensor order parameter. The regime of agreement may be interpreted as the regime of validity of the Landau-de Gennes theory since the Landau-de Gennes theory predicts large values of the equilibrium scalar order parameters – larger than unity, in the low-temperature regime. We discuss a modified Landau-de Gennes energy functional which yields physically realistic values of the equilibrium scalar order parameters in all temperature regimes.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 21 , Issue 2 , April 2010 , pp. 181 - 203
Copyright
Copyright © Cambridge University Press 2010

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References

[1]Bahadur, B. (1991) Liquid Crystal: Applications and Uses, World Scientific.CrossRefGoogle Scholar
[2]Ball, J. M. (2007) Mathematical theories of liquid crystals. Graduate Lecture Course. Available on request.Google Scholar
[3]Ball, J. M. & Majumdar, A. (2009) Nematic liquid crystals: from Maier-Saupe to a continuum theory. Accepted for publication in Molecular Crystals and Liquid Crystals, Proceedings of the ECLC 2009.Google Scholar
[4]Bethuel, F., Brezis, H. & Hélein, F. (1993) Asymptotics for the minimization of a Ginzburg–Landau functional. Calc. Var. Partial Differ. Equ. 1 (2), 123–148.CrossRefGoogle Scholar
[5]Chandrasekhar, S. (1992) Liquid Crystals, Cambridge University Press.CrossRefGoogle Scholar
[6]Dacorogna, B. (1989) Direct methods in the calculus of variations. In Applied Mathematical Sciences, ed. Antman, S. S., Marsden, J. E., Sirovich, L. Vol. 78, Springer.Google Scholar
[7]Davis, T. & Gartland, E. (1998) Finite element analysis of the Landau–de Gennes minimization problem for liquid crystals. SIAM J. Numer. Anal. 35, 336–362.CrossRefGoogle Scholar
[8]De Gennes, P. D. & Prost, J. (1974) The Physics of Liquid Crystals, Clarendon Press, Oxford, UK.Google Scholar
[9]Evans, L. (1998) Partial Differential Equations. American Mathematical Society, Providence, RI.Google Scholar
[10]Forest, M. G., Wang, Q. & Zhou, H. (2000a) Homogeneous pattern selection and director instabilities of nematic liquid crystal polymers induced by elongational flows. Phys. Fluids 12 (3), 490–498.CrossRefGoogle Scholar
[11]Forest, M. G., Wang, Q. & Zhou, H. (2000b) Exact banded patterns from a Doi-Marruci-Greco model of nematic liquid crystal polymers. Phys. Rev. E 61 (6), 6655–6662.Google ScholarPubMed
[12]Gilbarg, D. & Trudinger, N. (1977) Elliptic partial differential equations of second order. In Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics, 2nd ed., Vol. 224, Springer.Google Scholar
[13]Katriel, J., Kventsel, G. F., Luckhurst, G. R. & Sluckin, T. J. (1986) Free energies in the Landau and molecular field approaches. Liq. Cryst. 1, 337–55.CrossRefGoogle Scholar
[14]Lin, F. H. & Liu, C. (2001) Static and dynamic theories of liquid crystals. J. Partial Differ. Equ. 14 (4), 289–330.Google Scholar
[15]Mkaddem, S. & Gartland, E. C. (2000) Fine structure of defects in radial nematic droplets. Phys. Rev. E 62 (5), 6694–6705.Google ScholarPubMed
[16]Mottram, N. J. & Newton, C. (2004) Introduction to Q-tensor Theory. Research Report 10. Department of Mathematics, University of Strathclyde.Google Scholar
[17]Priestley, E. B., Wojtowicz, P. J. & Sheng, P. (1975) Introduction to Liquid Crystals, Plenum, New York.CrossRefGoogle Scholar
[18]Stephen, M. J. & Straley, J. P. (1974) Physics of liquid crystals. Rev. Mod. Phys. 46, 617–701.CrossRefGoogle Scholar