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Saturn ring defect around a spherical particle immersed in a nematic liquid crystal

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Abstract

We consider a nematic liquid crystal occupying the three-dimensional domain in the exterior of a spherical colloid particle. The nematic is subject to Dirichlet boundary conditions that enforce orthogonal attachment of nematic molecules to the surface of the particle. Our main interest is to understand the behavior of energy-critical configurations of the Landau–de Gennes Q-tensor model in the limit of vanishing correlation length. We demonstrate existence of configurations with a single Saturn-ring defect approaching the equator of the particle and no other line or point defects. We show this by analyzing asymptotics of energy minimizers under two symmetry constraints: rotational equivariance around the vertical axis and reflection across the horizontal plane. Energy blow-up at the ring defect is a significant obstacle to constructing well-behaved comparison maps needed to eliminate the possibility of point defects. The boundary estimates we develop to address this issue are new and should be applicable to a wider class of problems.

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References

  1. Alama, S., Bronsard, L., Lamy, X.: Minimizers of the Landau-de Gennes energy around a spherical colloid particle. Arch. Ration. Mech. Anal. 222(1), 427–450 (2016)

    Article  MathSciNet  Google Scholar 

  2. Alama, S., Bronsard, L., Lamy, X.: Spherical particle in nematic liquid crystal under an external field: the Saturn ring regime. J. Nonlinear Sci. 28(4), 1443–1465 (2018)

    Article  MathSciNet  Google Scholar 

  3. Alouges, F., Chambolle, A., Stantejsky, D.: The Saturn ring effect in nematic liquid crystals with external field: effective energy and hysteresis. In preparation

  4. André, N., and Shafrir, I.: Asymptotic behavior of minimizers for the Ginzburg-Landau functional with weight. I, II. Arch. Rational Mech. Anal. 142, 1, 45–73, 75–98 (1998)

  5. 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 

  6. Bauman, P., Park, J., Phillips, D.: Analysis of nematic liquid crystals with disclination lines. Arch. Ration. Mech. Anal. 205(3), 795–826 (2012)

    Article  MathSciNet  Google Scholar 

  7. Beaulieu, A., Hadiji, R.: On a class of Ginzburg-Landau equations with weight. PanAmer. Math. J. 5(4), 1–33 (1995)

    MathSciNet  MATH  Google Scholar 

  8. Beaulieu, A., Hadiji, R.: A Ginzburg–Landau problem with weight having minima on the boundary. Proc. R. Soc. Edinb. Sect. A 128(6), 1181–1215 (1998)

    Article  MathSciNet  Google Scholar 

  9. Bethuel, F., Brezis, H., Hélein, F.: Ginzburg–Landau vortices. In: Progress in Nonlinear Differential Equations and their Applications, vol. 13. Birkhäuser Boston Inc, Boston (1994)

  10. Bethuel, F., Chiron, D.: Some questions related to the lifting problem in Sobolev spaces. In: Perspectives in Nonlinear Partial Differential equations, vol. 446 of Contemp. Math. Amer. Math. Soc., Providence, RI, pp. 125–152 (2007)

  11. Brezis, H., Coron, J.-M., Lieb, E.H.: Harmonic maps with defects. Commun. Math. Phys. 107(4), 649–705 (1986)

    Article  MathSciNet  Google Scholar 

  12. 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)

    Article  MathSciNet  Google Scholar 

  13. 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 

  14. COMSOL Multiphysics® v. 5.3. http://www.comsol.com/. COMSOL AB, Stockholm, Sweden

  15. Contreras, A., Lamy, X.: Singular perturbation of manifold-valued maps with anisotropic energy. Preprint (2019)

  16. Contreras, A., Lamy, X., Rodiac, R.: On the convergence of minimizers of singular perturbation functionals. Indiana Univ. Math. J. 67(4), 1665–1682 (2018)

    Article  MathSciNet  Google Scholar 

  17. Dipasquale, F., Millot, V., Pisante, A.: Torus-like solutions for the Landau-de Gennes model. Part II: Topology of \({\mathbb{S}}^1\)-equivariant minimizers. arXiv:2008.13676

  18. Dipasquale, F., Millot, V., Pisante, A.: Torus-like solutions for the Landau-de Gennes Model. Part I: The Lyuksyutov Regime. Arch. Ration. Mech. Anal. 239(2), 599–678 (2021)

    Article  MathSciNet  Google Scholar 

  19. Fukuda, J., Stark, H., Yoneya, M., Yokoyama, H.: Dynamics of a nematic liquid crystal around a spherical particle. J. Phys.: Condens. Matter 16(19), S1957 (2004)

    Google Scholar 

  20. Fukuda, J., Yokoyama, H.: Stability of the director profile of a nematic liquid crystal around a spherical particle under an external field. Eur. Phys. J. E 21(4), 341–347 (2006)

    Article  Google Scholar 

  21. Fukuda, J.-I., Yoneya, M., Yokoyama, H.: Nematic liquid crystal around a spherical particle: investigation of the defect structure and its stability using adaptive mesh refinement. Eur. Phys. J. E 13(1), 87–98 (2004)

    Article  Google Scholar 

  22. 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 

  23. Gu, Y., Abbott, N.: Observation of Saturn-ring defects around solid microspheres in nematic liquid crystals. Phys. Rev. Lett. 85, 4719–4722 (2000)

    Article  Google Scholar 

  24. Hardt, R., Kinderlehrer, D., Lin, F.H.: Stable defects of minimizers of constrained variational principles. Ann. Inst. H. Poincaré Anal. Non Linéaire 5(4), 297–322 (1988)

    Article  MathSciNet  Google Scholar 

  25. Hardt, R., Kinderlehrer, D., Lin, F.H.: The variety of configurations of static liquid crystals. In: Variational methods (Paris, 1988), vol. 4 of Progr. Nonlinear Differential Equations Appl. Birkhäuser Boston, Boston, pp. 115–131 (1990)

  26. Hardt, R., Lin, F.-H.: A remark on \(H^1\) mappings. Manuscripta Math. 56(1), 1–10 (1986)

    Article  MathSciNet  Google Scholar 

  27. Jerrard, R.L.: Lower bounds for generalized Ginzburg–Landau functionals. SIAM J. Math. Anal. 30(4), 721–746 (1999)

    Article  MathSciNet  Google Scholar 

  28. Kralj, S., Virga, E.G.: Universal fine structure of nematic hedgehogs. J. Phys. A 34(4), 829–838 (2001)

    Article  MathSciNet  Google Scholar 

  29. Lamy, X.: Bifurcation analysis in a frustrated nematic cell. J. Nonlinear Sci. 24(6), 1197–1230 (2014)

    Article  MathSciNet  Google Scholar 

  30. Lavrentovich, O.D.: Transport of particles in liquid crystals. Soft Matter 10(9), 1264–1283 (2014)

    Article  Google Scholar 

  31. Loudet, J.C., Poulin, P.: Application of an electric field to colloidal particles suspended in a liquid-crystal solvent. Phys. Rev. Lett. 87, 165503 (2001)

    Article  Google Scholar 

  32. Lubensky, T.C., Pettey, D., Currier, N., Stark, H.: Topological defects and interactions in nematic emulsions. Phys. Rev. E 57, 610–625 (1998)

    Article  Google Scholar 

  33. 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 

  34. Muševič, I.: Liquid Crystal Colloids. Springer, New York (2017)

    Book  Google Scholar 

  35. 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 

  36. Palais, R.S.: The principle of symmetric criticality. Commun. Math. Phys. 69(1), 19–30 (1979)

    Article  MathSciNet  Google Scholar 

  37. Sandier, E.: Lower bounds for the energy of unit vector fields and applications. J. Funct. Anal. 152(2), 379–403 (1998)

    Article  MathSciNet  Google Scholar 

  38. Sandier, E., Shafrir, I.: On the symmetry of minimizing harmonic maps in \(N\) dimensions. Differ. Integral Equ.| 6(6), 1531–1541 (1993)

    MathSciNet  MATH  Google Scholar 

  39. Stark, H.: Physics of colloidal dispersions in nematic liquid crystals. Phys. Rep. 351(6), 387–474 (2001)

    Article  Google Scholar 

  40. Stark, H.: Saturn-ring defects around microspheres suspended in nematic liquid crystals: An analogy between confined geometries and magnetic fields. Phys. Rev. E 66, 032701 (2002)

    Article  Google Scholar 

  41. Struwe, M.: On the asymptotic behavior of minimizers of the Ginzburg–Landau model in \(2\) dimensions. Differ. Integ. Equ. 7(5–6), 1613–1624 (1994)

    MathSciNet  MATH  Google Scholar 

  42. Terentjev, E.M.: Disclination loops, standing alone and around solid particles, in nematic liquid crystals. Phys. Rev. E 51, 1330–1337 (1995)

    Article  Google Scholar 

  43. Yu, Y.: Disclinations in limiting Landau–de Gennes theory. Arch. Ration. Mech. Anal. 237(1), 147–200 (2020)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

S.A. and L.B. are supported by NSERC (Canada) Discovery Grants, D.G. was supported in part by NSF grant DMS-1729538, and X.L. by ANR project ANR-18-CE40-0023. We also thank the Erwin Schrödinger International Institute for Mathematics and Physics (Vienna) for their kind hospitality in December 2019, when this work was being completed. Finally, we thank the referees for their careful reading and helpful suggestions.

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Authors and Affiliations

  1. Department of Mathematics & Statistics, McMaster University, Hamilton, ON, L8S 4K1, Canada

    Stan Alama & Lia Bronsard

  2. Department of Mathematics, The University of Akron, Akron, OH, 44325, USA

    Dmitry Golovaty

  3. Institut de Mathématiques de Toulouse, UMR5219 - Université de Toulouse, CNRS - UPS IMT, 1062, Toulouse Cedex 9, France

    Xavier Lamy

Authors
  1. Stan Alama
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  2. Lia Bronsard
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  4. Xavier Lamy
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Correspondence to Stan Alama.

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Communicated by J. M. Ball.

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Alama, S., Bronsard, L., Golovaty, D. et al. Saturn ring defect around a spherical particle immersed in a nematic liquid crystal. Calc. Var. 60, 225 (2021). https://doi.org/10.1007/s00526-021-02091-6

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