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Landau-de Gennes Corrections to the Oseen-Frank Theory of Nematic Liquid Crystals

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Abstract

We study the asymptotic behavior of the minimisers of the Landau-de Gennes model for nematic liquid crystals in a two-dimensional domain in the regime of small elastic constant. At leading order in the elasticity constant, the minimum-energy configurations can be described by the simpler Oseen-Frank theory. Using a refined notion of \(\Gamma \)-development we recover Landau-de Gennes corrections to the Oseen-Frank energy. We provide an explicit characterisation of minimizing Q-tensors at this order in terms of optimal Oseen-Frank directors and observe the emerging biaxiality. We apply our results to distinguish between optimal configurations in the class of conformal director fields of fixed topological degree saturating the lower bound for the Oseen-Frank energy.

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Notes

  1. More general bulk potentials \(g({\mathrm{tr}}\, Q^2, {\mathrm{tr}}\, Q^3)\) have been studied in the literature; see, for example, [2, 14]. We expect the results presented here to apply more generally to bulk potentials with a unique minimiser (modulo rotations) which is nondegenerate and uniaxial.

References

  1. Anzellotti, G., Baldo, S.: Asymptotic development by \(\Gamma \)-convergence. Appl. Math. Optim. 27(2), 105–123, 1993

    Article  MathSciNet  Google Scholar 

  2. Ball, J.M., Majumdar, A.: Nematic liquid crystals: from Maier–Saupe to a continuum theory. Mol. Cryst. Liq. Cryst. 525(1), 1–11, 2010

    Article  Google Scholar 

  3. Ball, J.M., Zarnescu, A.: Orientability and energy minimization in liquid crystal models. Arch. Ration. Mech. Anal. 202(2), 493–535, 2011

    Article  MathSciNet  Google Scholar 

  4. Bauman, P., Park, J., Phillips, D.: Analysis of nematic liquid crystals with disclination lines. Arch. Ration. Mech. Anal. 1–32 2012

  5. Bethuel, F., Brezis, H., Hélein, F.: Asymptotics for the minimization of a Ginzburg–Landau functional. Calc. Var. Partial Differ. Equ. 1, 123–148, 1993

    Article  MathSciNet  Google Scholar 

  6. Brezis, H.M., Coron, J.-M.: Large solutions for harmonic maps in two dimensions. Commun. Math. Phys. 92(2), 203–215, 1983

    Article  ADS  MathSciNet  Google Scholar 

  7. Canevari, G.: Biaxiality in the asymptotic analysis of a 2d Landau-de Gennes model for liquid crystals. ESAIM: Control Optim. Calc. Var.21(1), 101–137 2015

    MathSciNet  MATH  Google Scholar 

  8. Canevari, G.: Line defects in the small elastic constant limit of a three-dimensional Landau-de Gennes model. Arch. Ration. Mech. Anal. 223(2), 591–676, 2017

    Article  MathSciNet  Google Scholar 

  9. Canevari, G., Ramaswamy, M., Majumdar, A.: Radial symmetry on three-dimensional shells in the landau-de gennes theory. Phys. D: Nonlinear Phenom. 314, 18–34, 2016

    Article  ADS  MathSciNet  Google Scholar 

  10. Canevari, G., Segatti, A.: Defects in nematic shells: a \(\gamma \)-convergence discrete-to-continuum approach. Arch. Ration. Mech. Anal. 1–62 2016

  11. Canevari, G., Segatti, A., Veneroni, M.: Morse’s index formula in vmo for compact manifolds with boundary. J. Funct. Anal. 269(10), 3043–3082, 2015

    Article  MathSciNet  Google Scholar 

  12. de Gennes, P.-G., Prost, J.: The Physics of Liquid Crystals. Oxford University Press, Oxford 1995

    Book  Google Scholar 

  13. Di Fratta, G., Robbins, J.M., Slastikov, V., Zarnescu, A.: Half-integer point defects in the \(Q\)-tensor theory of nematic liquid crystals. J. Nonlinear Sci. 26(1), 121–140, 2016

    Article  ADS  MathSciNet  Google Scholar 

  14. Fatkullin, I., Slastikov, V.: Critical points of the Onsager functional on a sphere. Nonlinearity18(6), 2565, 2005

    Article  ADS  MathSciNet  Google Scholar 

  15. Frank, F.C.: Liquid crystals. on the theory of liquid crystals. Discus. Faraday Soc. 25, 19–28, 1958

    Article  Google Scholar 

  16. Gartland Jr., E.C.: Scalings and limits of the Landau-de Gennes model for liquid crystals: a comment on some recent analytical papers. Math. Model. Anal. 23(3), 414–432, 2018

    Article  MathSciNet  Google Scholar 

  17. Golovaty, D., Montero, J.A.: On minimizers of a Landau-de Gennes energy functional on planar domains. Arch. Ration. Mech. Anal. 213(2), 447–490, 2014

    Article  MathSciNet  Google Scholar 

  18. Hélein, F.: Régularité des applications faiblement harmoniques entre une surface et une variété riemannienne. C. R. Acad. Sci. Paris Sér. I Math. 312(8), 591–596, 1991

    MathSciNet  MATH  Google Scholar 

  19. Henao, D., Majumdar, A., Pisante, A.: Uniaxial versus biaxial character of nematic equilibria in three dimensions. Calc. Var. Partial Differ. Equ.56, 2017

  20. Hopf, E.: On nonlinear partial differential equations. Lecture Series of the Symposium on Partial Differential Equations, Berkeley1957, 1–29, 1955

  21. Ignat, R., Nguyen, L., Slastikov, V., Zarnescu, A.: Symmetry and multiplicity of solutions in a two-dimensional Landau-de Gennes model for liquid crystals (in preparation)

  22. Ignat, R., Nguyen, L., Slastikov, V., Zarnescu, A.: Instability of point defects in a two-dimensional nematic liquid crystal model. Annales de l’Institut Henri Poincaré (C) Non Linear Analysis33(44), 1131–1152, 2015

    ADS  MathSciNet  MATH  Google Scholar 

  23. Ignat, R., Nguyen, L., Slastikov, V., Zarnescu, A.: Stability of the melting hedgehog in the Landau-de Gennes theory of nematic liquid crystals. Arch. Ration. Mech. Anal. 215(2), 633–673, 2015

    Article  MathSciNet  Google Scholar 

  24. Ignat, R., Nguyen, L., Slastikov, V., Zarnescu, A.: Stability of point defects of degree \(\pm \)12 in a two-dimensional nematic liquid crystal model. Calc. Var. Partial Differ. Equ.55(5). cited By 4. 2016

  25. Ignat, R., Nguyen, L., Slastikov, V., Zarnescu, A.: On the uniqueness of minimisers of Ginzburg-Landau functionals 2017. arXiv preprint arXiv:1708.05040

  26. Kitavtsev, G., Robbins, J.M., Slastikov, V., Zarnescu, A.: Liquid crystal defects in the landau-de gennes theory in two dimensions—beyond the one-constant approximation. Math. Mod. Methods Appl. Sci. 26(14), 2769–2808, 2016

    Article  MathSciNet  Google Scholar 

  27. Lamy, X.: Uniaxial symmetry in nematic liquid crystals. In: Annales de l’Institut Henri Poincare (C) Non Linear Analysis vol. 32, Elsevier, pp. 1125–1144 2015

  28. Majumdar, A., Zarnescu, A.: Landau-de Gennes theory of nematic liquid crystals: the Oseen-Frank limit and beyond. Arch. Ration. Mech. Anal. 196(1), 227–280, 2010

    Article  MathSciNet  Google Scholar 

  29. Nguyen, L., Zarnescu, A.: Refined approximation for minimizers of a Landau-de Gennes energy functional. Calc. Var. Partial Differ. Equ. 47(1–2), 383–432, 2013

    Article  MathSciNet  Google Scholar 

  30. Nomizu, K.: Characteristic roots and vectors of a diifferentiable family of symmetric matrices. Linear Multilinear Algebra1(2), 159–162, 1973

    Article  MathSciNet  Google Scholar 

  31. Sandier, E., Shafrir, I.: On the uniqueness of minimizing harmonic maps to a closed hemisphere. Calc. Var. Partial Differ. Equ. 2(1), 113–122, 1994

    Article  MathSciNet  Google Scholar 

  32. Schoen, R., Uhlenbeck, K.: Boundary regularity and the Dirichlet problem for harmonic maps. J. Differ. Geom. 18(2), 253–268, 1983

    Article  MathSciNet  Google Scholar 

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Acknowledgements

Giovanni Di Fratta acknowledges support from the Austrian Science Fund (FWF) through the special research program Taming complexity in partial differential systems (Grant SFB F65) and of the Vienna Science and Technology Fund (WWTF) through the research project Thermally controlled magnetization dynamics (Grant MA14-44). Jonathan M. Robbins and Valeriy Slastikov acknowledge support from EPSRC Grant EP/K02390X/1 and Leverhulme Grant RPG-2014-226, and Jonathan M. Robbins acknowledges support from a Lady Davis Visiting Professorship at the Hebrew University. The work Arghir Zarnescu is supported by the Basque Government through the BERC 2018–2021 program, by Spanish Ministry of Economy and Competitiveness MINECO through BCAM Severo Ochoa excellence accreditation SEV-2017-0718 and through the project MTM2017-82184-R, acronym “DESFLU”, funded by (AEI/FEDER, UE). The authors would like to thank the Isaac Newton Institute for Mathematical Sciences for support and hospitality during the programme “The design of new materials programme” when work on this paper was undertaken. This work was supported by: EPSRC Grant Numbers EP/K032208/1 and EP/R014604/1.

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Authors and Affiliations

  1. Institute for Analysis and Scientific Computing, TU Wien, Wiedner Hauptstraße 8-10, 1040, Wien, Austria

    Giovanni Di Fratta

  2. School of Mathematics, University of Bristol, University Walk, Bristol, BS8 1TW, United Kingdom

    Jonathan M. Robbins & Valeriy Slastikov

  3. IKERBASQUE, Basque Foundation for Science, Maria Diaz de Haro 3, 48013, Bilbao, Bizkaia, Spain

    Arghir Zarnescu

  4. Basque Center for Applied Mathematics (BCAM), Mazarredo 14, 48009, Bilbao, Bizkaia, Spain

    Arghir Zarnescu

  5. “Simion Stoilow” Institute of the Romanian Academy, 21 Calea Griviţei, 010702, Bucharest, Romania

    Arghir Zarnescu

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  1. Giovanni Di Fratta
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  2. Jonathan M. Robbins
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  4. Arghir Zarnescu
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Correspondence to Arghir Zarnescu.

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Communicated by A. Garroni

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Di Fratta, G., Robbins, J.M., Slastikov, V. et al. Landau-de Gennes Corrections to the Oseen-Frank Theory of Nematic Liquid Crystals. Arch Rational Mech Anal 236, 1089–1125 (2020). https://doi.org/10.1007/s00205-019-01488-0

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  • DOI: https://doi.org/10.1007/s00205-019-01488-0

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