Spherical Particle in Nematic Liquid Crystal Under an External Field: The Saturn Ring Regime

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Abstract

We consider a nematic liquid crystal occupying the exterior region in \({\mathbb {R}}^3\) outside of a spherical particle, with radial strong anchoring. Within the context of the Landau-de Gennes theory, we study minimizers subject to an external field, modeled by an additional term which favors nematic alignment parallel to the field. When the external field is high enough, we obtain a scaling law for the energy. The energy scale corresponds to minimizers concentrating their energy in a boundary layer around the particle, with quadrupolar symmetry. This suggests the presence of a Saturn ring defect around the particle, rather than a dipolar director field typical of a point defect.

Keywords

Partial differential equations Calculus of variations Liquid crystals Line defects 

Mathematics Subject Classification

35J50 35Q56 

Notes

Acknowledgements

We thank E. C. Gartland for useful discussions on nondimensionalization. SA and LB were supported by NSERC (Canada) Discovery Grants.

References

  1. Alama, S., Bronsard, L., Galvão Sousa, B.: Weak anchoring for a two-dimensional liquid crystal. Nonlinear Anal. 119, 74–97 (2015)MathSciNetCrossRefMATHGoogle Scholar
  2. Alama, S., Bronsard, L., Lamy, X.: Analytical description of the saturn-ring defect in nematic colloids. Phys. Rev. E 93, 012705 (2016a)CrossRefGoogle Scholar
  3. 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 (2016b)MathSciNetCrossRefMATHGoogle Scholar
  4. Alama, S., Bronsard, L., Golovaty, D., Lamy, X. (in preparation)Google Scholar
  5. Ball, J.M., Zarnescu, A.: Orientability and energy minimization in liquid crystal models. Arch. Ration. Mech. Anal. 202(2), 493–535 (2011)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle 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)MathSciNetCrossRefMATHGoogle Scholar
  8. Canevari, G.: Line defects in the small elastic constant limit of a three-dimensional Landau-de Gennes model. arXiv:1501.05236 (2016)
  9. Contreras, A., Lamy, X.: Biaxial escape in nematics at low temperature. J. Funct. Anal. 272(10), 3987–3997 (2017)MathSciNetCrossRefMATHGoogle Scholar
  10. Contreras, A., Lamy, X., Rodiac, R.: On the convergence of minimizers of singular perturbation functionals. Indiana Univ. Math. J. (2016)Google Scholar
  11. 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)MathSciNetCrossRefMATHGoogle Scholar
  12. 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)CrossRefGoogle Scholar
  13. 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)CrossRefGoogle Scholar
  14. 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)MathSciNetCrossRefMATHGoogle Scholar
  15. Gu, Y., Abbott, N.: Observation of saturn-ring defects around solid microspheres in nematic liquid crystals. Phys. Rev. Lett. 85, 4719–4722 (2000)CrossRefGoogle Scholar
  16. Ignat, R., Nguyen, L., Slastikov, V., Zarnescu, A.: Uniqueness results for an ODE related to a generalized Ginzburg-Landau model for liquid crystals. SIAM J. Math. Anal. 46(5), 3390–3425 (2014)MathSciNetCrossRefMATHGoogle Scholar
  17. 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)MathSciNetCrossRefMATHGoogle Scholar
  18. Ignat, R., Nguyen, L., Slastikov, V., Zarnescu, A.: Instability of point defects in a two-dimensional nematic liquid crystal model. Ann. Inst. H. Poincaré Anal. Non Linéaire 33(4), 1131–1152 (2016a)MathSciNetCrossRefMATHGoogle Scholar
  19. Ignat, R., Nguyen, L., Slastikov, V., Zarnescu, A.: Stability of point defects of degree \(\pm \frac{1}{2}\) in a two-dimensional nematic liquid crystal model. Calc. Var. Partial Differ. Equ. 55(5), 119 (2016b)CrossRefMATHGoogle Scholar
  20. 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)CrossRefGoogle Scholar
  21. 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)MathSciNetCrossRefMATHGoogle Scholar
  22. Stark, H.: Physics of colloidal dispersions in nematic liquid crystals. Phys. Rep. 351(6), 387–474 (2001)CrossRefGoogle Scholar
  23. 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)CrossRefGoogle Scholar
  24. Sternberg, P.: Vector-valued local minimizers of nonconvex variational problems. Rocky Mt. J. Math. 21, 799–807 (1991)MathSciNetCrossRefMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada
  2. 2.Institut de Mathématiques de Toulouse, UMR5219, CNRS, UPS IMTUniversité de ToulouseToulouse Cedex 9France

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