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.
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Notes
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).
References
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)
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)
Brezis, H., Oswald, L.: Remarks on sublinear elliptic equations. Nonlinear Anal. 10, 55–64 (1986)
Biscari, P., Virga, E.: Local stability of biaxial nematic phases between two cylinders. Int. J. Nonlinear Mech. 32(2), 337–351 (1997)
Canevari, G.: Biaxiality in the asymptotic analysis of a 2-D Landau-de Gennes model for liquid crystals. ESAIM COCV 21, 101–137 (2015)
Chandrasekhar, S., Ranganath, G.S.: The structure and energetics of defects in liquid crystals. Adv. Phys. 35, 507–596 (1986)
Cladis, P.E., Kleman, M.: Non-singular disclinations of strength S = + 1 in nematics. J. Phys. 33, 591–598 (1972). (Paris)
Copara, S., Porentab, T., Zumer, S.: Visualisation methods for complex nematic fields. Liq. Cryst. 40, 1759–1768 (2013)
De Gennes, P.G.: The Physics of Liquid Crystals. Clarendon Press, Oxford (1974)
Ericksen, J.L.: Liquid crystals with variable degree of orientation. Arch. Ration. Mech. Anal. 113(2), 97120 (1990)
Fatkullin, I., Slastikov, V.: Vortices in two-dimensional nematics. Commun. Math. Sci 7, 917–938 (2009)
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)
Golovaty, D., Montero, A.: On minimizers of the Landau-de Gennes energy functional on planar domains. arXiv:1307.4437 (2013)
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)
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)
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)
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)
Ignat, R., Nguyen, L., Slastikov, V., Zarnescu, A.: Instability of point defects in a two-dimensional nematic liquid crystal model, submitted
Kléman, M.: Points, Lines and Walls in Liquid Crystals, Magnetic Systems and Various Ordered Media. Wiley, New York (1983)
Kléman, M., Lavrentovich, O.D.: Topological point defects in nematic liquid crystals. Philos. Mag. 86(25–26), 4117–4137 (2006)
Kralj, S., Virga, E.G., Zumer, S.: Biaxial torus around nematic point defects. Phys. Rev. E 60, 1858 (1999)
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)
Mkaddem, S., Gartland, E.C.: Fine structure of defects in radial nematic droplets. Phys. Rev. E 62, 6694–6705 (2000)
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)
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
Virga, E.G.: Variational Theories for Liquid Crystals. Chapman and Hall, London (1994)
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.
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Communicated by Felix Otto.
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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
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DOI: https://doi.org/10.1007/s00332-015-9271-8