Skip to main content

Half-Integer Point Defects in the Q-Tensor Theory of Nematic Liquid Crystals

Abstract

We investigate prototypical profiles of point defects in two-dimensional liquid crystals within the framework of Landau–de Gennes theory. Using boundary conditions characteristic of defects of index k/2, we find a critical point of the Landau–de Gennes energy that is characterised by a system of ordinary differential equations. In the deep nematic regime, \(b^2\) small, we prove that this critical point is the unique global minimiser of the Landau–de Gennes energy. For the case \(b^2=0\), we investigate in greater detail the regime of vanishing elastic constant \(L \rightarrow 0\), where we obtain three explicit point defect profiles, including the global minimiser.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Notes

  1. 1.

    The careful reader will note that \(\mathrm{tr}(Q) = 0\) implies that the eigenvalues cannot all be positive. In order to obtain positive lengths for the axes, we add to each eigenvalue a sufficiently large positive constant (we assume the eigenvalues of Q are bounded).

  2. 2.

    The figures represent the numerically computed solutions of (3.7), (3.8) for \(k=\pm 1,\pm 2\).

References

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

    MathSciNet  Article  MATH  Google Scholar 

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

    MathSciNet  Article  MATH  Google Scholar 

  3. Bethuel, F., Brezis, H., Coleman, B.D., Hélein, F.: Bifurcation analysis of minimizing harmonic maps describing the equilibrium of nematic phases between cylinders. Arch. Ration. Mech. Anal. 118(2), 149–168 (1992)

    Article  MATH  Google Scholar 

  4. Brezis, H., Oswald, L.: Remarks on sublinear elliptic equations. Nonlinear Anal. 10, 55–64 (1986)

    MathSciNet  Article  MATH  Google Scholar 

  5. Biscari, P., Virga, E.: Local stability of biaxial nematic phases between two cylinders. Int. J. Nonlinear Mech. 32(2), 337–351 (1997)

    MathSciNet  Article  MATH  Google Scholar 

  6. Canevari, G.: Biaxiality in the asymptotic analysis of a 2-D Landau-de Gennes model for liquid crystals. ESAIM COCV 21, 101–137 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  7. Chandrasekhar, S., Ranganath, G.S.: The structure and energetics of defects in liquid crystals. Adv. Phys. 35, 507–596 (1986)

    Article  Google Scholar 

  8. Cladis, P.E., Kleman, M.: Non-singular disclinations of strength S = + 1 in nematics. J. Phys. 33, 591–598 (1972). (Paris)

    Article  Google Scholar 

  9. Copara, S., Porentab, T., Zumer, S.: Visualisation methods for complex nematic fields. Liq. Cryst. 40, 1759–1768 (2013)

    Article  Google Scholar 

  10. De Gennes, P.G.: The Physics of Liquid Crystals. Clarendon Press, Oxford (1974)

    Google Scholar 

  11. Ericksen, J.L.: Liquid crystals with variable degree of orientation. Arch. Ration. Mech. Anal. 113(2), 97120 (1990)

    MathSciNet  Google Scholar 

  12. Fatkullin, I., Slastikov, V.: Vortices in two-dimensional nematics. Commun. Math. Sci 7, 917–938 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  13. Gartland, E.C., Mkaddem, S.: Instability of radial hedgehog configurations in nematic liquid crystals under Landaude Gennes free-energy models. Phys. Rev. E 59, 563–567 (1999)

    Article  Google Scholar 

  14. Golovaty, D., Montero, A.: On minimizers of the Landau-de Gennes energy functional on planar domains. arXiv:1307.4437 (2013)

  15. Henao, D., Majumdar, A.: Symmetry of uniaxial global Landau-de Gennes minimizers in the theory of nematic liquid crystals. SIAM J. Math. Anal. 44 3217–3241 (2012); 45 3872–3874 (2013) (corrigendum)

  16. Ignat, R., Nguyen, L., Slastikov, V., Zarnescu, A.: Uniqueness result for an ODE related to a generalized Ginzburg-Landau model for liquid crystals. SIAM J. Math. Anal. 46(5), 3390–3425 (2014)

    MathSciNet  Article  MATH  Google 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, 633–673 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  18. Ignat, R., Nguyen, L., Slastikov, V., Zarnescu, A.: Stability of the vortex defect in Landau-de Gennes theory of nematic liquid crystals. C. R. Acad. Sci. Paris Ser. I 351, 533–535 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  19. Ignat, R., Nguyen, L., Slastikov, V., Zarnescu, A.: Instability of point defects in a two-dimensional nematic liquid crystal model, submitted

  20. Kléman, M.: Points, Lines and Walls in Liquid Crystals, Magnetic Systems and Various Ordered Media. Wiley, New York (1983)

    Google Scholar 

  21. Kléman, M., Lavrentovich, O.D.: Topological point defects in nematic liquid crystals. Philos. Mag. 86(25–26), 4117–4137 (2006)

    Article  Google Scholar 

  22. Kralj, S., Virga, E.G., Zumer, S.: Biaxial torus around nematic point defects. Phys. Rev. E 60, 1858 (1999)

    Article  Google Scholar 

  23. 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)

    MathSciNet  Article  MATH  Google Scholar 

  24. Mkaddem, S., Gartland, E.C.: Fine structure of defects in radial nematic droplets. Phys. Rev. E 62, 6694–6705 (2000)

    Article  Google Scholar 

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

    MathSciNet  Article  MATH  Google Scholar 

  26. Shirokoff, D., Choksi, R., Nave, J.-C.: Sufficient conditions for global minimality of metastable states in a class of non-convex functionals: a simple approach via quadratic lower bounds, J. Nonlinear Sci. (2015). doi:10.1007/s00332-015-9234-0

  27. Virga, E.G.: Variational Theories for Liquid Crystals. Chapman and Hall, London (1994)

    Book  MATH  Google Scholar 

Download references

Acknowledgments

GDF, JMR, VS would like to acknowledge support from EPSRC Grant EP/K02390X/1. VS also acknowledges support from EPSRC grant EP/I028714/1. AZ gratefully acknowledges the hospitality of the Mathematics Department at the University of Bristol, through EPSRC grants EP/I028714/1 and EP/K02390X/1.

Author information

Affiliations

Authors

Corresponding author

Correspondence to G. Di Fratta.

Additional information

Communicated by Felix Otto.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Di Fratta, G., Robbins, J.M., Slastikov, V. et al. Half-Integer Point Defects in the Q-Tensor Theory of Nematic Liquid Crystals. J Nonlinear Sci 26, 121–140 (2016). https://doi.org/10.1007/s00332-015-9271-8

Download citation

Keywords

  • Nonlinear elliptic PDE system
  • Singular ODE system
  • Stability
  • Vortex
  • Liquid crystal defects