Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-27T05:58:48.108Z Has data issue: false hasContentIssue false
  • English
  • Français

A novel Landau-de Gennes model with quartic elastic terms

Part of: Partial differential equations Optimality conditions Foundations, constitutive equations, rheology Elliptic equations and systems

Published online by Cambridge University Press:  24 March 2020

DMITRY GOLOVATY Open the ORCID record for DMITRY GOLOVATY [Opens in a new window] ,
MICHAEL NOVACK  and
PETER STERNBERG
DMITRY GOLOVATY
Affiliation:
Department of Mathematics, University of Akron, Akron, OH44325, USA email: dmitry@uakron.edu
MICHAEL NOVACK
Affiliation:
Department of Mathematics, Indiana University, Bloomington, IN 47405, USA emails: mrnovack@indiana.edu; sternber@indiana.edu
PETER STERNBERG
Affiliation:
Department of Mathematics, Indiana University, Bloomington, IN 47405, USA emails: mrnovack@indiana.edu; sternber@indiana.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Within the framework of the generalised Landau-de Gennes theory, we identify a Q-tensor-based energy that reduces to the four-constant Oseen–Frank energy when it is considered over orientable uniaxial nematic states. Although the commonly considered version of the Landau-de Gennes theory has an elastic contribution that is at most cubic in components of the Q-tensor and their derivatives, the alternative offered here is quartic in these variables. One clear advantage of our approach over the cubic theory is that the associated minimisation problem is well-posed for a significantly wider choice of elastic constants. In particular, this quartic energy can be used to model nematic-to-isotropic phase transitions for highly disparate elastic constants. In addition to proving well-posedness of the proposed version of the Landau-de Gennes theory, we establish a rigorous connection between this theory and its Oseen–Frank counterpart via a Г-convergence argument in the limit of vanishing nematic correlation length. We also prove strong convergence of the associated minimisers.

Keywords

Nematic liquid crystalGamma-convergenceLandau-de Gennes theoryQ-tensordisparate elastic constantsnematic-to-isotropic phase transition

MSC classification

Primary: 35J50: Variational methods for elliptic systems
Secondary: 35B36: Pattern formation 49K20: Problems involving partial differential equations 76A15: Liquid crystals
Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Issue 1 , February 2021 , pp. 177 - 198
Check for updates
Copyright
© The Author(s) 2020. Published by Cambridge University Press

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.)

Footnotes

†

D. G. acknowledges the support from NSF DMS-1729538. M. N. and P. S. acknowledge the support from a Simons Collaboration grant 585520.

References

Ball, J. M. (2010) The Q-tensor theory of liquid crystals. https://people.maths.ox.ac.uk/ball/teaching/benin%20summer%20school-%202010.pdf.Google Scholar
Ball, J. M. (2017) Liquid crystals and their defects. Mathematical Thermodynamics of Complex Fluids, Lecture Notes in Math., 2200, Fond. CIME/CIME Found. Subser., Springer, Cham, pp. 1–46.CrossRefGoogle Scholar
Ball, J. M. (2017) Mathematics and liquid crystals. Mol. Cryst. Liq. Cryst. 647(1), 1–27.CrossRefGoogle Scholar
Ball, J. M. & Majumdar, A. (2010) Nematic liquid crystals: from Maier-Saupe to a continuum theory. Mol. Cryst. Liq. Cryst. 525(1), 1–11.CrossRefGoogle Scholar
Ball, J. M. & Zarnescu, A. (2011) Orientability and energy minimization in liquid crystal models. Arch. Ration. Mech. Anal. 202(2), 493–535.CrossRefGoogle Scholar
Bauman, P. & Phillips, D. (2016) Regularity and the behavior of eigenvalues for minimizers of a constrained Q-tensor energy for liquid crystals. Calc. Var. Partial Differ. Equ. 55(4), Article No. 81, 22.Google Scholar
Berreman, D. W. & Meiboom, S. (1984) Tensor representation of Oseen-Frank strain energy in uniaxial cholesterics. Phys. Rev. A 30, 1955–1959.CrossRefGoogle Scholar
Davis, T. A. & Gartland, Jr ., E. C. (1998) Finite element analysis of the Landau-de Gennes minimization problem for liquid crystals. SIAM J. Numer. Anal. 35(1), 336–362.CrossRefGoogle Scholar
de Gennes, P. G. (1971) Short range order effects in the isotropic phase of nematics and cholesterics. Mol. Cryst. Liq. Cryst. 12(3), 193–214.CrossRefGoogle Scholar
de Gennes, P. G. & Prost, J. (1995) The Physics of Liquid Crystals. International Series of Monogr. Clarendon Press, Oxford.CrossRefGoogle Scholar
Dickmann, S. (1995) Numerische Berechnung von Feld und Molekülausrichtung in Flüssigkristallanzeigen. PhD thesis, University of Karlsruhe.Google Scholar
Ericksen, J. L. (1966) Inequalities in liquid crystal theory. Phys. Fluids 9(6), 1205–1207.CrossRefGoogle Scholar
Fatkullin, I. & Slastikov, V. (2008) On spatial variations of nematic ordering. Physica D 237(20), 2577–2586.CrossRefGoogle Scholar
Frank, F. C. (1958) I. Liquid crystals. On the theory of liquid crystals. Discuss. Faraday Soc. 25, 19–28.CrossRefGoogle Scholar
Gartland, Jr ., E. C. (2013) An overview of the Oseen-Frank elastic model. https://www.newton.ac.uk/files/seminar/20130108090009401-153462.pdf.Google Scholar
Gartland, Jr ., E. C. (2018) Scalings and limits of Landau–de Gennes models for liquid crystals: a comment on some recent analytical papers. Math. Model. Anal. 23(3), 414–432.CrossRefGoogle Scholar
Golovaty, D., Kim, Y.-K., Lavrentovich, O. D., Novack, M. & Sternberg, P. (2019) Phase transitions in nematics: textures with tactoids and disclinations. arXiv e-prints arXiv:1902.06342. Google Scholar
Golovaty, D., Novack, M., Sternberg, P. & Venkatraman, R. (2018) A model problem for nematic-isotropic transitions with highly disparate elastic constants. arXiv e-prints arXiv:1811.12586. Google Scholar
Golovaty, D., Sternberg, P. & Venkatraman, R. (2019) A Ginzburg-Landau type problem for highly anisotropic nematic liquid crystals. SIAM J. Math. Anal. 51(1), 276–320.CrossRefGoogle Scholar
Hardt, R., Kinderlehrer, D. & Lin, F.-H. (1986) Existence and partial regularity of static liquid crystal configurations. Comm. Math. Phys. 105(4), 547–570.CrossRefGoogle Scholar
Iyer, G., Xu, X. & Zarnescu, A. D. (2015) Dynamic cubic instability in a 2D Q-tensor model for liquid crystals. Math. Models Methods Appl. Sci. 25(8), 1477–1517.CrossRefGoogle Scholar
Katriel, J., Kventsel, G. F., Luckhurst, G. R. & Sluckin, T. J. (1986) Free energies in the landau and molecular field approaches. Liq. Cryst. 1(4), 337–355.CrossRefGoogle Scholar
Kim, Y.-K., Shiyanovskii, S. V. & Lavrentovich, O. D. (2013) Morphogenesis of defects and tactoids during isotropic–nematic phase transition in self-assembled lyotropic chromonic liquid crystals. J. Phys.: Condens. Matter 25(40), 404202.Google ScholarPubMed
Longa, L., Monselesan, D. & Trebin, H.-R. (1987) An extension of the Landau-Ginzburg-de Gennes theory for liquid crystals. Liq. Cryst. 2(6), 769–796.CrossRefGoogle Scholar
Longa, L. & Trebin, H. (1989) Structure of the elastic free energy for chiral nematic liquid crystals. Phys. Rev. A 39(4), 2160–2168.CrossRefGoogle ScholarPubMed
Luckhurst, G. R., Naemura, S., Sluckin, T. J., Thomas, K. S. & Turzi, S. S. Molecular-field-theory approach to the landau theory of liquid crystals: uniaxial and biaxial nematics. Phys. Rev. E 85, 031705.CrossRefGoogle Scholar
Majumdar, A. (2010) Equilibrium order parameters of nematic liquid crystals in the landau-de gennes theory. Eur. J. Appl. Math. 21(2), 181–203.CrossRefGoogle Scholar
Majumdar, A. & Zarnescu, A. (2010) Landau-De Gennes theory of nematic liquid crystals: the Oseen-Frank limit and beyond. Arch. Ration. Mech. Anal. 196(1), 227–280.CrossRefGoogle Scholar
Mottram, N. J. & Newton, C. J. P. (2014) Introduction to Q-tensor theory. ArXiv e-prints.Google Scholar
Nakagawa, M. (1997) On the relation between tensor and vector approaches of nematodynamics. Liq. Cryst. 23(4), 561–567.CrossRefGoogle Scholar
Oseen, C. W. (1933) The theory of liquid crystals. Trans. Faraday Soc. 29, 883–899.CrossRefGoogle Scholar
Virga, E. G. (1994) Variational Theories for Liquid Crystals, Applied Mathematics and Mathematical Computation, Vol. 8. Chapman & Hall, London.Google Scholar
Zhou, S., Neupane, K., Nastishin, Y. A., Baldwin, A. R., Shiyanovskii, S. V., Lavrentovich, O. D. & Sprunt, S. (2014) Elasticity, viscosity, and orientational fluctuations of a lyotropic chromonic nematic liquid crystal disodium cromoglycate. Soft Matter 10, 6571–6581.CrossRefGoogle ScholarPubMed
Zocher, H. (1933) The effect of a magnetic field on the nematic state. Trans. Faraday Soc. 29, 945–957.CrossRefGoogle Scholar