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On the Disclination Lines of Nematic Liquid Crystals

Published online by Cambridge University Press:  01 February 2016

Yucheng Hu ,
Yang Qu  and
Pingwen Zhang
Yucheng Hu
Affiliation:
Zhou Pei-yuan Center for Applied Mathematics, Tsinghua University, Beijing, China
Yang Qu
Affiliation:
Laboratory of Mathematics and Applied Mathematics, School of Mathematical Sciences, Peking University, Beijing, China
Pingwen Zhang*
Affiliation:
Laboratory of Mathematics and Applied Mathematics, School of Mathematical Sciences, Peking University, Beijing, China
*
*Corresponding author. Email addresses:[email protected](Y. Hu), [email protected] (Y. Qu), [email protected] (P. Zhang)
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Abstract

Defects in liquid crystals are of great practical importance and theoretical interest. Despite tremendous efforts, predicting the location and transition of defects under various topological constraint and external field remains to be a challenge. We investigate defect patterns of nematic liquid crystals confined in three-dimensional spherical droplet and two-dimensional disk under different boundary conditions, within the Landau-de Gennes model. We implement a spectral method that numerically solves the Landau-de Gennes model with high accuracy, which allows us to study the detailed static structure of defects. We observe five types of defect structures. Among them the 1/2-disclination lines are the most stable structure at low temperature. Inspired by numerical results, we obtain the profile of disclination lines analytically. Moreover, the connection and difference between defect patterns under the Landau-de Gennes model and the Oseen-Frank model are discussed. Finally, three conjectures are made to summarize some important characteristics of defects in the Landau-de Gennes theory. This work is a continuing effort to deepen our understanding on defect patterns in nematic liquid crystals.

Keywords

Liquid crystalsdefectsdisclination linesLandau-de Gennes model
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 2 , February 2016 , pp. 354 - 379
Copyright
Copyright © Global-Science Press 2016 

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References

[1]Allender, D. W., Crawford, G. P., and J. W. DOANE, Determination of the liquid-crystal surface elastic constant k24, Phys. Rev. Lett., 67 (1991), pp. 1442–1445.Google Scholar
[2]Avriel, M., Nonlinear programming: analysis and methods, Courier Dover Publications, 2003.Google Scholar
[3]Ball, J. M. and Zarnescu, A., Orientability and energy minimization in liquid crystal models, Arch. Ration. Mech. Anal., 202 (2011), pp. 493–535.Google Scholar
[4]Bauman, P., Park, J., and Phillips, D., Analysis of nematic liquid crystals with disclination lines, Arch. Ration. Mech. Anal., 205 (2012), pp. 795–826.CrossRefGoogle Scholar
[5]Biscari, P. and Peroli, G. G., A hierarchy of defects in biaxial nematics, Comm. Math. Phys., 186 (1997), pp. 381–392.CrossRefGoogle Scholar
[6]Callan-Jones, A. C., Pelcovits, R. A., Slavin, V. A., Zhang, S., Laidlaw, D. H., and Loriot, G. B., Simulation and visualization of topological defects in nematic liquid crystals, Phys. Rev. E, 74 (2006), p. 061701.Google Scholar
[7]Cheng, H. and Zhang, P., A tensor model for liquid crystals on a spherical surface, Sci. China Math., 56 (2013), pp. 2549–2559.Google Scholar
[8]De Gennes, P. G. and Prost, J., The physics of liquid crystals, Oxford University Press, Oxford, second ed., 1995.Google Scholar
[9]Di Fratta, G., Robbins, J., Slastikov, V., and Zarnescu, A., Profiles of point defects in two dimensions in Landau-de Gennes theory, arXiv:1403.2566, (2014).Google Scholar
[10]Do Carmo, M., Riemannian geometry, Springer, Boston, 1992.Google Scholar
[11]Eisenberg, M. and Guy, R., A proof of the hairy ball theorem, Am. Math. Mon., 86 (1979), pp.571–574.Google Scholar
[12]Erdmann, J. H., Žumer, S., and Doane, J. W., Configuration transition in a nematic liquid crystal confined to a small spherical cavity, Phys. Rev. Lett., 64 (1990), pp. 1907–1910.Google Scholar
[13]Ericksen, J. L., Liquid crystals with variable degree of orientation, Arch. Ration. Mech. Anal., 113 (1991), pp. 97–120.Google Scholar
[14]Gartland, E. C. and Mkaddem, S., Instability of radial hedgehog configurations in nematic liquid crystals under Landau-de Gennes free-energy models, Phys. Rev. E, 59 (1999), pp. 563–567.Google Scholar
[15]Gupta, J. K., Sivakumar, S., Caruso, F., and Abbott, N. L., Size-dependent ordering of liquid crystals observed in polymeric capsules with micrometer and smaller diameters, Angew. Chem. Int. Ed. Engl., 48 (2009), pp. 1652–1655.Google Scholar
[16]Ignat, R., Nguyen, L., Slastikov, V., and Zarnescu, A., Stability of the melting hedgehog in the Landau-de Gennes theory ofnematic liquid crystals, arXiv: 1404.1729, (2014).CrossRefGoogle Scholar
[17]Kralj, S., Virga, E. G., and Žumer, S., Biaxial torus around nematic point defects, Phys. Rev. E, 60 (1999), pp. 1858–1866.Google Scholar
[18]Lavrentovich, O., Defects in liquid crystals: surface and interfacial anchoring effects, in Patterns of Symmetry Breaking, Springer, 2003, pp. 161–195.Google Scholar
[19]Lin, F. H. and Liu, C., Static and dynamic theories of liquid crystals, J. Partial Differ. Equ., 14 (2001), pp. 289–330.Google Scholar
[20]Lopez-Leon, T. and Fernandez-Nieves, A., Drops and shells of liquid crystal, Colloid. Polym. Sci., 289 (2011), pp. 345–359.Google Scholar
[21]Majumdar, A., Equilibrium order parameters ofnematic liquid crystals in the Landau-de Gennes theory, Eur. J. Appl. Math, 21 (2010), pp. 181–203.Google Scholar
[22]Majumdar, A., The radial-hedgehog solution in Landau-de Gennes’ theory for nematic liquid crystals, Eur. J. Appl. Math., 23 (2012), pp. 61–97.Google Scholar
[23]Majumdar, A. and Zarnescu, A., Landau-de Gennes theory of nematic liquid crystals: the Oseen-Frank limit and beyond, Arch. Ration. Mech. Anal., 196 (2010), pp. 227–280.CrossRefGoogle Scholar
[24]Miller, D. S., Wang, X., and Abbott, N. L., Design of functional materials based on liquid crystalline droplets, Chem. Mater., 26 (2013), pp. 496–506.Google Scholar
[25]Mkaddem, S. and Gartland, E. Jr, Fine structure of defects in radial nematic droplets, Phys. Rev. E, 62 (2000), p. 6694.CrossRefGoogle ScholarPubMed
[26]Nelson, D. R., Toward a tetravalent chemistry of colloids, Nano Lett., 2 (2002), pp. 1125–1129.Google Scholar
[27]Nguyen, L. and Zarnescu, A., Refined approximation for a class of landau-de gennes energy minimizers, arXiv:1006.5689, (2010).Google Scholar
[28]Porenta, T., Ravnik, M., and Žumer, S., Effect offlexoelectricity and order electricity on defect cores in nematic droplets, Soft Matter, 7 (2011), pp. 132–136.Google Scholar
[29]Schopohl, N. and Sluckin, T., Hedgehog structure in nematic and magnetic systems, J. Phys., 49 (1988), pp. 1097–1101.Google Scholar
[30]Sonnet, A., Kilian, A., and Hess, S., Alignment tensor versus director: Description of defects in nematic liquid crystals, Phys. Rev. E, 52 (1995), pp. 718–722.Google Scholar
[31]Tasinkevych, M., Silvestre, N., and Telo Da Gama, M. M., Liquid crystal boojum-colloids, New J. Phys., 14 (2012), p. 073030.CrossRefGoogle Scholar
[32]Virga, E. G., Variational theories for liquid crystals, vol. 8, CRC Press, 1995.Google Scholar
[33]Zernike, F., Diffraction theory of the cut procedure and its improved form, the phase contrast method, Physica, 1 (1934), pp. 689–704.Google Scholar
[34]Zhang, W.-Y., Jiang, Y., and Chen, J. Z. Y., Onsager model for the structure of rigid rods confined on a spherical surface, Phys. Rev. Lett., 108 (2012), p. 057801.Google Scholar