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Numerics for Liquid Crystals with Variable Degree of Orientation

Published online by Cambridge University Press:  16 February 2015

Ricardo H. Nochetto ,
Shawn W. Walker  and
Wujun Zhang
Ricardo H. Nochetto
Affiliation:
Department of Mathematics, University of Maryland
Shawn W. Walker
Affiliation:
Department of Mathematics, Louisiana State University
Wujun Zhang
Affiliation:
Department of Mathematics, University of Maryland
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Abstract

We consider the simplest one-constant model, put forward by J. Eriksen, for nematic liquid crystals with variable degree of orientation. The equilibrium state is described by a director field n and its degree of orientation s, where the pair (n, s) minimizes a sum of Frank-like energies and a double well potential. In particular, the Euler-Lagrange equations for the minimizer contain a degenerate elliptic equation for n, which allows for line and plane defects to have finite energy. Using a special discretization of the liquid crystal energy, and a strictly monotone energy decreasing gradient flow scheme, we present a simulation of a plane-defect in three dimensions to illustrate our method.

Keywords

simulationliquid crystalfluid
Type
Articles
Information
MRS Online Proceedings Library (OPL) , Volume 1753: Symposium NN – Mathematical and Computational Aspects of Materials Science , 2015 , mrsf14-1753-nn06-11
Copyright
Copyright © Materials Research Society 2015 

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References

Alouges, F.. A new algorithm for computing liquid crystal stable configurations: The harmonic mapping case. SIAM Journal on Numerical Analysis, 34(5):pp. 1708–1726, 1997.CrossRefGoogle Scholar
Ambrosio, L.. Existence of minimal energy configurations of nematic liquid crystals with variable degree of orientation. Manuscripta Mathematica, 68(1):215–228, 1990.CrossRefGoogle Scholar
Ambrosio, L.. Regularity of solutions of a degenerate elliptic variational problem. Manuscripta Mathematica, 68(1):309–326, 1990.CrossRefGoogle Scholar
Araki, T. and Tanaka, H.. Colloidal aggregation in a nematic liquid crystal: Topological arrest of particles by a single-stroke disclination line. Phys. Rev. Lett., 97:127801, Sep 2006.CrossRefGoogle Scholar
Badia, S., Guill´en-Gonz´alez, F. M., and Guti´errez-Santacreu, J. V.. An overview on numerical analyses of nematic liquid crystal flows. Archives of Computational Methods in Engineering, 18(3):285–313, 2011.CrossRefGoogle Scholar
Ball, J. and Zarnescu, A.. Orientability and energy minimization in liquid crystal models. Archive for Rational Mechanics and Analysis, 202(2):493–535, 2011.CrossRefGoogle Scholar
Barrett, J. W., Feng, X., and Prohl, A.. Convergence of a fully discrete finite element method for a degenerate parabolic system modelling nematic liquid crystals with variable degree of orientation. ESAIM: Mathematical Modelling and Numerical Analysis, 40:175–199, 1 2006.CrossRefGoogle Scholar
Bartels, S., Dolzmann, G., and Nochetto, R.. Finite element methods for director fields on flexible surfaces. Interfaces and Free Boundaries, 14:231 – 272, 2012.CrossRefGoogle Scholar
Bartels, S., Dolzmann, G., and Nochetto, R. H.. A finite element scheme for the evolution of orientational order in fluid membranes. ESAIM: Mathematical Modelling and Numerical Analysis, 44:1–31, 1 2010.CrossRefGoogle Scholar
Bauman, P., Calderer, M. C., Liu, C., and Phillips, D.. The phase transition between chiral nematic and smectic a* liquid crystals. Archive for Rational Mechanics and Analysis, 165(2):161–186, 2002.CrossRefGoogle Scholar
Bird, R. B., Armstrong, R. C., and Hassager, O.. Dynamics of Polymeric Liquids - Volume 1: Fluid Mechanics, volume 1 of Wiley Interscience Publication. John Wiley and Sons, 2nd edition, 1987.Google Scholar
Blinov, L.. Electro-optical and magneto-optical properties of liquid crystals. Wiley, 1983.Google Scholar
Brezis, H., Coron, J.-M., and Lieb, E. H.. Harmonic maps with defects. Communications in Mathematical Physics, 107(4):649–705, 1986.CrossRefGoogle Scholar
Calderer, M., Golovaty, D., Lin, F., and Liu, C.. Time evolution of nematic liquid crystals with variable degree of orientation. SIAM Journal on Mathematical Analysis, 33(5):1033–1047, 2002.CrossRefGoogle Scholar
Cohen, R., Lin, S.-Y., and Luskin, M.. Relaxation and gradient methods for molecular orientation in liquid crystals. Computer Physics Communications, 53(13):455 – 465, 1989.CrossRefGoogle Scholar
de Gennes, P. G. and Prost, J.. The Physics of Liquid Crystals, volume 83 of International Series of Monographs on Physics. Oxford Science Publication, Oxford, UK, 2nd edition, 1995.Google Scholar
Ericksen, J.. Liquid crystals with variable degree of orientation. Archive for Rational Mechanics and Analysis, 113(2):97–120, 1991.CrossRefGoogle Scholar
Gonz´alez, F. M. G. and Guti´errez-Santacreu, J. V.. A linear mixed finite element scheme for a nematic ericksenleslie liquid crystal model. ESAIM: Mathematical Modelling and Numerical Analysis, 47:1433–1464, 9 2013.CrossRefGoogle Scholar
Hardt, R., Kinderlehrer, D., and Luskin, M.. Remarks about the mathematical theory of liquid crystals. In Hildebrandt, S., Kinderlehrer, D., and Miranda, M., editors, Calculus of Variations and Partial Differential Equations, volume 1340 of Lecture Notes in Mathematics , pages 123–138. Springer Berlin Heidelberg, 1988.Google Scholar
Larson, R. G.. The Structure and Rheology of Complex Fluids. Oxford University Press, 1999.Google Scholar
Lin, F.-H.. Nonlinear theory of defects in nematic liquid crystals; phase transition and flow phenomena. Communications on Pure and Applied Mathematics, 42(6):789–814, 1989.CrossRefGoogle Scholar
Lin, F. H.. On nematic liquid crystals with variable degree of orientation. Communications on Pure and Applied Mathematics, 44(4):453–468, 1991.CrossRefGoogle Scholar
Lin, S.-Y. and Luskin, M.. Relaxation methods for liquid crystal problems. SIAM Journal on Numerical Analysis, 26(6):1310–1324, 1989.CrossRefGoogle Scholar
Liu, C. and Walkington, N.. Approximation of liquid crystal flows. SIAM Journal on Numerical Analysis, 37(3):725–741, 2000.CrossRefGoogle Scholar
Schoen, R. and Uhlenbeck, K.. A regularity theory for harmonic maps. Journal of Differential Geometry, 17(2):307–335, 1982.CrossRefGoogle Scholar
Tojo, K., Furukawa, A., Araki, T., and Onuki, A.. Defect structures in nematic liquid crystals around charged particles. The European Physical Journal E, 30(1):55–64, 2009.CrossRefGoogle ScholarPubMed
Virga, E. G.. Variational Theories for Liquid Crystals, volume 8. Chapman and Hall, London, 1st edition, 1994.CrossRefGoogle Scholar
Walkington, N. J.. Numerical approximation of nematic liquid crystal flows governed by the ericksen-leslie equations. ESAIM: Mathematical Modelling and Numerical Analysis, 45:523–540, 5 2011.CrossRefGoogle Scholar
Yang, X., Forest, M. G., Li, H., Liu, C., Shen, J., Wang, Q., and Chen, F.. Modeling and simulations of drop pinch-off from liquid crystal filaments and the leaky liquid crystal faucet immersed in viscous fluids. Journal of Computational Physics, 236(0):1 – 14, 2013.CrossRefGoogle Scholar
Yang, X., Forest, M. G., Liu, C., and Shen, J.. Shear cell rupture of nematic liquid crystal droplets in viscous fluids. Journal of Non-Newtonian Fluid Mechanics, 166(910):487 – 499, 2011.CrossRefGoogle Scholar