Scalings and limits of Landau-de Gennes models for liquid crystals: a comment on some recent analytical papers
Abstract
Some recent analytical papers have explored limiting behaviors of Landaude Gennes models for liquid crystals in certain extreme ranges of the model parameters: limits of “vanishing elasticity” (in the language of some of these papers) and “low-temperature limits.” We use simple scaling analysis to show that these limits are properly interpreted as limits in which geometric length scales (such as the size of the domain containing the liquid crystal material) become large compared to intrinsic length scales (such as correlation lengths or coherence lengths, which determine defect core sizes). This represents the natural passage from a mesoscopic model to a macroscopic model and is analogous to a “London limit” in the Ginzburg-Landau theory of superconductivity or a “large-body limit” in the Landau-Lifshitz theory of ferromagnetism. Known relevant length scales in these parameter regimes (nematic correlation length, biaxial coherence length) can be seen to emerge via balances in equilibrium Euler-Lagrange equations associated with well-scaled Landau-de Gennes free-energy functionals.
Keyword : liquid crystals, Landau-de Gennes model, Oseen-Frank model, large-body limit, low-temperature limit
How to Cite
Gartland, Jr., E. C. (2018). Scalings and limits of Landau-de Gennes models for liquid crystals: a comment on some recent analytical papers. Mathematical Modelling and Analysis, 23(3), 414-432. https://doi.org/10.3846/mma.2018.025
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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[24] R. Rosso and E.G. Virga. Metastable nematic hedgehogs. Journal of Physics A: Mathematical and General, 29(14):4247–4264, 1996.
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[26] A. Sonnet, A. Kilian and S. Hess. Alignment tensor versus director: Description of defects in nematic liquid crystals. Physical Review E, 52(1):718–722, 1995. https://doi.org/10.1103/PhysRevE.52.718
[27] I.W. Stewart. The Static and Dynamic Continuum Theory of Liquid Crystals: A Mathematical Introduction. Taylor & Francis, London, 2004.
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[29] N.J. Walkington. Numerical approximation of nematic liquid crystal flows governed by the Ericksen-Leslie equations. ESAIM: Mathematical Modelling and Numerical Analysis, 45(3):523–540, 2011. https://doi.org/10.1051/m2an/2010065
[30] P. Ziherl and S. Zumer. Fluctuations in confined liquid crystals above nematic-isotropic phase transition temperature. Physical Review Letters, 78(4):682–685, 1997. https://doi.org/10.1103/PhysRevLett.78.682
[2] P. Bauman, J. Park and D. Phillips. Analysis of nematic liquid crystals with disclination lines. Archive for Rational Mechanics and Analysis, 205(3):795–826, 2012. https://doi.org/10.1007/s00205-012-0530-7
[3] F. Bethuel, H. Brezis and F. H´elein. Asymptotics for the minimization of a Ginzburg-Landau functional. Calculus of Variations and Partial Differential Equations, 1(2):123–148, 1993. https://doi.org/10.1007/BF01191614
[4] A. Contreras and X. Lamy. Biaxial escape in nematics at low temperature. Journal of Functional Analysis, 272(10):3987–3997, 2017. https://doi.org/10.1016/j.jfa.2017.01.012
[5] T.A. Davis and E.C. Gartland, Jr. Finite element analysis of the Landaude Gennes minimization problem for liquid crystals. SIAM Journal on Numerical Analysis, 35(1):336–362, 1998. https://doi.org/10.1137/S0036142996297448
[6] G. De Luca and A.D. Rey. Ringlike cores of cylindrically confined nematic point defects. The Journal of Chemical Physics, 126(9):904907, 2007. https://doi.org/10.1063/1.2711436
[7] A. De Simone. Energy minimizers for large ferromagnetic bodies. Archive for Rational Mechanics and Analysis, 125(2):99–143, 1993. https://doi.org/10.1007/BF00376811
[8] J.-I. Fukuda, H. Stark, M. Yoneyama and H. Yokoyama. Interaction between two spherical particles in a nematic liquid crystal. Physical Review E, 69(4):041706, 2004. https://doi.org/10.1103/PhysRevE.69.041706
[9] E.C. Gartland, Jr. Scalings and limits of the Landau-de Gennes model for liquid crystals: A comment on some recent analytical papers. arXiv.org e-Print archive, 2015. Available from Internet: https://arxiv.org/abs/1512.08164v1
[10] E.C. Gartland, Jr. and S. Mkaddem. Instability of radial hedgehog configurations in nematic liquid crystals under Landau-de Gennes free-energy models. Physical Review E, 59(1):563–567, 1999. https://doi.org/10.1103/PhysRevE.59.563
[11] E.C. Gartland, Jr., P. Palffy-Muhoray and R.S. Varga. Numerical minimization of the Landau-de Gennes free energy: Defects in cylindrical capillaries. Molecular Crystals and Liquid Crystals, 199(1):429–452, 1991. https://doi.org/10.1080/00268949108030952
[12] D. Golovaty and J.A. Montero. On minimizers of a Landau-de Gennes energy functional on planar domains. Archive for Rational Mechanics and Analysis, 213(2):447–490, 2014. https://doi.org/10.1007/s00205-014-0731-3
[13] D. Henao, A. Majumdar and A. Pisante. Uniaxial versus biaxial character of nematic equilibria in three dimensions. Calculus of Variations and Partial Differential Equations, 56(2):55, 2017. https://doi.org/10.1007/s00526-017-1142-8
[14] S. Kralj, E.G. Virga and S. Zumer. Biaxial torus around nematic point defects. Physical Review E, 60(2):1858–1866, 1999. https://doi.org/10.1103/PhysRevE.60.1858
[15] I.F. Lyuksyutov. Topological instability of singularities at small distances in nematics. Sov. Phys. JETP, 48(1):178–9, 1987. Translation of Zh. Eksp. Teor. ´ Fiz. 75(1):358–360, 1978.
[16] A. Majumdar and A. Zarnescu. Landau-de Gennes theory of nematic liquid crystals: The Oseen-Frank limit and beyond. Archive for Rational Mechanics and Analysis, 196(1):227–280, 2010. https://doi.org/10.1007/s00205-009-0249-2
[17] S. Mkaddem and E.C. Gartland, Jr. Fine structure of defects in radial nematic droplets. Physical Review E, 62(5):6694–6705, 2000. https://doi.org/10.1103/PhysRevE.62.6694
[18] N.J. Mottram and C.J.P. Newton. Introduction to Q-tensor theory. arXiv.org e-Print archive, 2014. Available from Internet: https://arxiv.org/abs/1409. 3542
[19] L. Nguyen and A. Zarnescu. Refined approximation for minimizers of a Landaude Gennes energy functional. Calculus of Variations and Partial Differential Equations, 47(1-2):383–432, 2013. https://doi.org/10.1007/s00526-012-0522-3
[20] P. Palffy-Muhoray, E.C. Gartland and J.R. Kelly. A new configurational transition in inhomogeneous nematics. Liquid Crystals, 16(4):713–718, 1994. https://doi.org/10.1080/02678299408036543
[21] E. Pensenstadler and H.-R. Trebin. Fine structure of point defects and soliton decay in nematic liquid crystals. J. Phys. France, 50(9):1027–1040, 1989. https://doi.org/10.1051/jphys:019890050090102700
[22] E.B. Priestley, P.J. Wojtowicz and P. Sheng. Introduction to Liquid Crystals. Plenum Press, New York, 1975.
[23] M. Ravnik and S. Zumer. Landau-de Gennes modelling of nematic ˇ liquid crystal colloids. Liquid Crystals, 36(10–11):1201–1214, 2009. https://doi.org/10.1080/02678290903056095
[24] R. Rosso and E.G. Virga. Metastable nematic hedgehogs. Journal of Physics A: Mathematical and General, 29(14):4247–4264, 1996.
[25] N. Schophol and T.J. Sluckin. Defect core structure in nematic liquid crystals. Physical Review Letters, 59(22):2582–2584, 1987. https://doi.org/10.1103/PhysRevLett.59.2582
[26] A. Sonnet, A. Kilian and S. Hess. Alignment tensor versus director: Description of defects in nematic liquid crystals. Physical Review E, 52(1):718–722, 1995. https://doi.org/10.1103/PhysRevE.52.718
[27] I.W. Stewart. The Static and Dynamic Continuum Theory of Liquid Crystals: A Mathematical Introduction. Taylor & Francis, London, 2004.
[28] E.G. Virga. Variational Theories for Liquid Crystals. Chapman & Hall, London, 1994.
[29] N.J. Walkington. Numerical approximation of nematic liquid crystal flows governed by the Ericksen-Leslie equations. ESAIM: Mathematical Modelling and Numerical Analysis, 45(3):523–540, 2011. https://doi.org/10.1051/m2an/2010065
[30] P. Ziherl and S. Zumer. Fluctuations in confined liquid crystals above nematic-isotropic phase transition temperature. Physical Review Letters, 78(4):682–685, 1997. https://doi.org/10.1103/PhysRevLett.78.682