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Minimizers of the Landau–de Gennes Energy Around a Spherical Colloid Particle

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Abstract

We consider energy minimizing configurations of a nematic liquid crystal around a spherical colloid particle, in the context of the Landau–de Gennes model. The nematic is assumed to occupy the exterior of a ball B r0, and satisfy homeotropic weak anchoring at the surface of the colloid and approach a uniform uniaxial state as \({|x|\to\infty}\). We study the minimizers in two different limiting regimes: for balls which are small \({r_0\ll L^{\frac12}}\) compared to the characteristic length scale \({L^{\frac 12}}\), and for large balls, \({r_0\gg L^{\frac12}}\). The relationship between the radius and the anchoring strength W is also relevant. For small balls we obtain a limiting quadrupolar configuration, with a “Saturn ring” defect for relatively strong anchoring, corresponding to an exchange of eigenvalues of the Q-tensor. In the limit of very large balls we obtain an axisymmetric minimizer of the Oseen–Frank energy, and a dipole configuration with exactly one point defect is obtained.

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References

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

    Article  MathSciNet  MATH  Google Scholar 

  2. Bethuel F., Brezis H., Hélein F.: Asymptotics for the minimization of a Ginzburg–Landau functional. CalcVar. Partial Differ. Equ. 1(2), 123–148 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bourgain J., Brezis H., Mironescu P.: Lifting in Sobolev spaces. J. Anal. Math. 80, 37–86 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Contreras, A., Lamy, X.: Biaxial escape in nematics at low temperature. arXiv:1405.2055

  6. Cucker F., Gonzalez Corbalan A.: An alternate proof of the continuity of the roots of a polynomial. Am. Math. Mon. 96(4), 342–345 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  7. 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  MATH  Google Scholar 

  8. Hardt R., Kinderlehrer D., Lin F-H.: Existence and partial regularity of static liquid crystal configurations. Commun. Math. Phys. 105(4), 547–570 (1986)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. 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)

    MathSciNet  MATH  Google Scholar 

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

  11. Hardt R., Lin F.H.: A remark on H 1 mappings. Manuscr. Math. 56(1), 1–10 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hardt R., Lin F.H.: Stability of singularities of minimizing harmonic maps. J. Differ. Geom. 29(1), 113–123 (1989)

    MathSciNet  MATH  Google Scholar 

  13. Hardt R., Lin F.H., Poon C.C.: Axially symmetric harmonic maps minimizing a relaxed energy. Commun Pure Appl. Math. 45(4), 417–459 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  14. Helfinstine S.L., Lavrentovich O.D., Woolverton C.J.: Lyotropic liquid crystal as a real-time detector of microbial immune complexes. Lett. Appl. Microbiol. 43(1), 27–32 (2006)

    Article  Google Scholar 

  15. Hussain A., Pina A.S., Roque A.C.A.: Bio-recognition and detection using liquid crystals. Biosens. Bioelectron. 25(1), 1–8 (2009)

    Article  Google Scholar 

  16. 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. (2015). doi:10.1016/j.anihpc.2015.03.007

  17. Ignat, R., Nguyen, L., Slastikov, V., Zarnescu, A.: Stability of point defects of degree \({\pm 1/2}\) in a two-dimensional nematic liquid crystal model. arXiv:1601.02812

  18. 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)

    Article  MathSciNet  MATH  Google Scholar 

  19. 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  MATH  Google Scholar 

  20. Kaiser R., Wiese W., Hess S.: Stability and instability of an uniaxial alignment against biaxial distortions in the isotropic and nematic phases of liquid crystals. J. Non-Equilib. Thermodyn. 17, 153–169 (1992)

    Article  ADS  MATH  Google Scholar 

  21. Kuksenok O.V., Ruhwandl R.W., Shiyanovskii S.V., Terentjev E.M.: Director structure around a colloid particle suspended in a nematic liquid crystal. Phys Rev. E 54(5), 5198–5203 (1996)

    Article  ADS  Google Scholar 

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

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  24. 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  MATH  Google Scholar 

  25. Mkaddem S., Gartland E.C. Jr.: Fine structure of defects in radial nematic droplets. Phys. Rev. E 62(5), 6694 (2000)

    Article  ADS  Google Scholar 

  26. Muševič I., Škarabot M., Tkalec U., Ravnik M., Žumer S.: Two-dimensional nematic colloidal crystals self-assembled by topological defects.. Science 313(5789), 954– (2006)

    Article  ADS  Google Scholar 

  27. 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  MATH  Google Scholar 

  28. Nomizu K.: Characteristic roots and vectors of a differentiable family of symmetric matrices. Linear Multilinear Algebra 1(2), 159–162 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  29. Porenta, T., Čopar, S., Ackerman, P.J., Pandey, M.B., Varney, M.C.M., Smalyukh, I.I., Žumer, S.: Topological switching and orbiting dynamics of colloidal spheres dressed with chiral nematic solitons. Sci. Rep. 4, 7337 (2014). doi:10.1038/srep07337

  30. Poulin P., Stark H., Lubensky T.C., Weitz D.A.: Novel colloidal interactions in anisotropic fluids. Science 275(5307), 1770–1773 (1997)

    Article  Google Scholar 

  31. Ravnik M., Zumer S.: Landau–de Gennes modelling of nematic liquid crystal colloids. Liq. Cryst. 36(10–11), 1201–1214 (2009)

    Article  Google Scholar 

  32. Schoen R., Uhlenbeck K.: Regularity of minimizing harmonic maps into the sphere. Invent. Math. 78(1), 89–100 (1984)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Senyuk B., Liu, Q., He S., Kamien R.D., Kusner R.B., Lubensky T.C., Smalyukh I.I.: Topological colloids. Nature 493(7431), 200–205 (2013)

    Article  ADS  Google Scholar 

  34. Shafrir I.: Remarks on solutions of \({-\Delta u=(1-\vert u\vert ^2)u}\) in R 2. C. R Acad. Sci. Paris Sér. I Math. 318(4), 327–331 (1994)

    MathSciNet  MATH  Google Scholar 

  35. Shiyanovskii, S.V., Schneider, T., Smalyukh, I.I., Ishikawa, T., Niehaus, G.D., Doane, K.J., Woolverton, C.J., Lavrentovich, O.D.: Real-time microbe detection based on director distortions around growing immune complexes in lyotropic chromonic liquid crystals. Phys. Rev. E 71, 020702 (2005)

  36. Stark H.: Director field configurations around a spherical particle in a nematic liquid crystal. Eur. Phys. J. B 10(2), 311–321 (1999)

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  ADS  Google Scholar 

  39. Whitney, H.: Complex Analytic Varieties. Addison-Wesley, Menlo Park, 1972

  40. Woltman S.J., Jay G.D., Crawford G.P.: Liquid-crystal materials find a new order in biomedical applications. Nat. Mater. 6(12), 929–938 (2007)

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. McMaster University, Hamilton, ON, Canada

    Stan Alama & Lia Bronsard

  2. Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22, 04103, Leipzig, Germany

    Xavier Lamy

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

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Communicated by S. Serfaty

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Alama, S., Bronsard, L. & Lamy, X. Minimizers of the Landau–de Gennes Energy Around a Spherical Colloid Particle. Arch Rational Mech Anal 222, 427–450 (2016). https://doi.org/10.1007/s00205-016-1005-z

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  • DOI: https://doi.org/10.1007/s00205-016-1005-z

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