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.
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Alama, S., Bronsard, L., Galvão Sousa, B.: Weak anchoring for a two-dimensional liquid crystal. Nonlinear Anal. 119, 74–97 (2015)
Alama, S., Bronsard, L., Lamy, X.: Analytical description of the saturn-ring defect in nematic colloids. Phys. Rev. E 93, 012705 (2016a)
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)
Alama, S., Bronsard, L., Golovaty, D., Lamy, X. (in preparation)
Ball, J.M., Zarnescu, A.: Orientability and energy minimization in liquid crystal models. Arch. Ration. Mech. Anal. 202(2), 493–535 (2011)
Bauman, P., Park, J., Phillips, D.: Analysis of nematic liquid crystals with disclination lines. Arch. Ration. Mech. Anal. 205(3), 795–826 (2012)
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)
Canevari, G.: Line defects in the small elastic constant limit of a three-dimensional Landau-de Gennes model. arXiv:1501.05236 (2016)
Contreras, A., Lamy, X.: Biaxial escape in nematics at low temperature. J. Funct. Anal. 272(10), 3987–3997 (2017)
Contreras, A., Lamy, X., Rodiac, R.: On the convergence of minimizers of singular perturbation functionals. Indiana Univ. Math. J. (2016)
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)
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)
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)
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)
Gu, Y., Abbott, N.: Observation of saturn-ring defects around solid microspheres in nematic liquid crystals. Phys. Rev. Lett. 85, 4719–4722 (2000)
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)
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)
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)
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)
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)
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)
Stark, H.: Physics of colloidal dispersions in nematic liquid crystals. Phys. Rep. 351(6), 387–474 (2001)
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)
Sternberg, P.: Vector-valued local minimizers of nonconvex variational problems. Rocky Mt. J. Math. 21, 799–807 (1991)
Acknowledgements
We thank E. C. Gartland for useful discussions on nondimensionalization. SA and LB were supported by NSERC (Canada) Discovery Grants.
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Communicated by Robert V. Kohn.
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Alama, S., Bronsard, L. & Lamy, X. Spherical Particle in Nematic Liquid Crystal Under an External Field: The Saturn Ring Regime. J Nonlinear Sci 28, 1443–1465 (2018). https://doi.org/10.1007/s00332-018-9456-z
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DOI: https://doi.org/10.1007/s00332-018-9456-z