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A Theoretician’s Approach to Nematic Liquid Crystals and Their Applications

  • Apala MajumdarEmail author
  • Alexander H. Lewis
Chapter
Part of the Molecular Modeling and Simulation book series (MMAS)

Abstract

This chapter is a self-contained but not exhaustive account of continuum modelling approaches for nematic liquid crystals. Nematic liquid crystals are partially ordered liquids or anisotropic liquids with long-range orientational order. This chapter contains an overview of the celebrated Landau-de Gennes, Ericksen and Oseen-Frank theories for nematic liquid crystals, with emphasis on the mathematical modelling of static equilibrium phenomena. The chapter has a section devoted to a case study of a multistable nematic device. The case study focuses on two different modelling approaches to this device—the simple Oseen-Frank framework and the more elaborate Landau-de Gennes approach. The case study describes the corresponding mathematical frameworks, the methodology and the model predictions with comparisons to numerical simulations and experimental results. In particular, the case study elucidates how multistability in nematic devices can be controlled and manipulated by temperature, material properties and boundary effects. The Conclusions section summarizes the chapter content with future perspectives.

Keywords

Liquid Crystal Nematic Liquid Crystal Cholesteric Liquid Crystal Molecular Alignment Elastic Energy Density 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

A.M. is supported by an EPSRC Career Acceleration Fellowship EP/J001686/1 and EP/J001686/2, an OCIAM Visiting Fellowship, support from the Bath Internationalization Grant schemes and the Bath Institute for Mathematical Innovation. A. L. is supported by an Engineering Physical Sciences Research Council studentship. The authors are grateful to Peter Howell, Dirk Aarts and Samo Kralj for fruitful discussions and suggestions.

References

  1. 1.
    Timeline: The Early History of the Liquid Crystal Display. Available via Spectrum. http://spectrum.ieee.org/static/timeline-the-early-history-of-the-liquid-crystal-display. Cited 29 Apr 2016
  2. 2.
    D. Allender, L. Longa, Landau-de Gennes theory of biaxial nematics reexamined. Phys. Rev. E 78(1), 011–704 (2008)Google Scholar
  3. 3.
    M. Ambrožič, F. Bisi, E.G. Virga, Director reorientation and order reconstruction: competing mechanisms in a nematic cell. Contin. Mech. Thermodyn. 20(4), 193–218 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    D. Andrienko, Introduction to Liquid Crystals (International Max Planck Research School, Bad Marienberg, 2006)Google Scholar
  5. 5.
    B. Bahadur, Liquid Crystals: Applications and Uses (World Scientific, 1991)Google Scholar
  6. 6.
    J.M. Ball, Function spaces for liquid crystals (2015). https://people.maths.ox.ac.uk/ball/Teaching/lyon2015.pdf. (Winter school, Nonlinear Function Spaces in Mathematics and Physical Sciences, Lyon)
  7. 7.
    J.M. Ball, A. Majumdar, Nematic liquid crystals: from Maier-Saupe to a continuum theory. Mol. Cryst. Liq. Cryst. 525(1), 1–11 (2010)CrossRefGoogle Scholar
  8. 8.
    G. Barbero, G. Durand, On the validity of the Rapini-Papoular surface anchoring energy form in nematic liquid crystals. J. de Phys. 47(12), 2129–2134 (1986)CrossRefGoogle Scholar
  9. 9.
    E. Barry, D. Beller, Z. Dogic, A model liquid crystalline system based on rodlike viruses with variable chirality and persistence length. Soft Matter 5, 2563–2570 (2009)Google Scholar
  10. 10.
    F. Bethuel, H. Brezis, F. Hélein, Asymptotics for the minimization of a Ginzburg-Landau functional. Calc. Var. Partial Differ. Equ. 1(2), 123–148 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    H. Brezis, J.M. Coron, E.H. Lieb, Harmonic maps with defects. Commun. Math. Phy. 107(4), 649–705 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    G. Carbone, G. Lombardo, R. Barberi, I. Muševič, U. Tkalec, Mechanically induced biaxial transition in a nanoconfined nematic liquid crystal with a topological defect. Phys. Rev. Lett. 103(16), 167–801 (2009)Google Scholar
  13. 13.
    S. Chandrasekhar, Liquid Crystals (Cambridge University Press, 1992)Google Scholar
  14. 14.
    J. Chen, C.T. Liu, Technology advances in flexible displays and substrates. Access IEEE 1, 150–158 (2013)CrossRefGoogle Scholar
  15. 15.
    O.J. Dammone, Confinement of colloidal liquid crystals. Ph.D. thesis, University College, University of Oxford, 2013Google Scholar
  16. 16.
    O.J. Dammone, I. Zacharoudiou, R.P.A. Dullens, J.M. Yeomans, M.P. Lettinga, D.G.A.L. Aarts, Confinement induced splay-to-bend transition of colloidal rods. Phys. Rev. Lett. 109(10), 108–303 (2012)Google Scholar
  17. 17.
    A.E. Danese, Advanced Calculus, vol. 1 (Allyn and Bacon, 1965)Google Scholar
  18. 18.
    T.A. Davis, E.C. Gartland Jr., Finite element analysis of the Landau-de Gennes minimization problem for liquid crystals. SIAM J. Numer. Anal. 35(1), 336–362 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    I. Dozov, M. Nobili, G. Durand, Fast bistable nematic display using monostable surface switching. Appl. Phys. Lett. 70(9), 1179–1181 (1997)CrossRefGoogle Scholar
  20. 20.
    J.L. Ericksen, Liquid crystals with variable degree of orientation. Arch. Ration. Mech. Anal. 113(2), 97–120 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    F.C. Frank, I. liquid crystals. on the theory of liquid crystals. Discuss. Faraday Soc. 25, 19–28 (1958)CrossRefGoogle Scholar
  22. 22.
    P.G. de Gennes, The Physics of Liquid Crystals (Clarendon Press, Oxford, 1974)zbMATHGoogle Scholar
  23. 23.
    E. Grelet, Hexagonal order in crystalline and columnar phases of hard rods. Phys. Rev. Lett. 100, 168–301 (2008)Google Scholar
  24. 24.
    R. Hardt, D. Kinderlehrer, F.H. Lin, Existence and partial regularity of static liquid crystal configurations. Commun. Math. Phys. 105(4), 547–570 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    A. Jeffrey, D. Zwillinger, Table of Integrals, Series, and Products (Elsevier Science, 2000)Google Scholar
  26. 26.
    J. Katriel, G.F. Kventsel, G.R. Luckhurst, T.J. Sluckin, Free energies in the Landau and molecular field approaches. Liq. Cryst. 1(4), 337–355 (1986)Google Scholar
  27. 27.
    A.V. Kityk, M. Wolff, K. Knorr, D. Morineau, R. Lefort, P. Huber, Continuous paranematic-to-nematic ordering transitions of liquid crystals in tubular silica nanochannels. Phys. Rev. Lett. 101(18), 187–801 (2008)Google Scholar
  28. 28.
    S. Kralj, G. Cordoyiannis, A. Zidanšek, G. Lahajnar, H. Amenitsch, S. Žumer, Z. Kutnjak, Presmectic wetting and supercritical-like phase behavior of octylcyanobiphenyl liquid crystal confined to controlled-pore glass matrices. J. Chem. Phys. 127(15), 154–905 (2007)Google Scholar
  29. 29.
    S. Kralj, A. Majumdar, Order reconstruction patterns in nematic liquid crystal wells. Proc. R. Soc. A 470(2169), 20140276 (2014)Google Scholar
  30. 30.
    S. Kralj, E.G. Virga, Universal fine structure of nematic hedgehogs. J. Phys. A: Math. Gen. 34(4), 829 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    S. Kralj, E.G. Virga, S. Žumer, Biaxial torus around nematic point defects. Phys. Rev. E 60(2), 1858 (1999)CrossRefGoogle Scholar
  32. 32.
    J.P.F. Lagerwall, An Introduction to the Physics of Liquid Crystals, ed. by A. Fernandez-Nieves. Soft Materials—generation, physical properties and fundamental applications (John Wiley & Sons, 2014)Google Scholar
  33. 33.
    F.M. Leslie, Continuum theory for nematic liquid crystals. Contin. Mech. Thermodyn. 4(3), 167–175 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    A.H. Lewis, Defects in liquid crystals: Mathematical and experimental studies. Ph.D. thesis, University of Oxford, 2016Google Scholar
  35. 35.
    A.H. Lewis, I. Garlea, J. Alvarado, O.J. Dammone, P.D. Howell, A. Majumdar, B.M. Mulder, M.P. Lettinga, G.H. Koenderink, D.G.A.L. Aarts, Colloidal liquid crystals in rectangular confinement: theory and experiment. Soft Matter 10, 7865–7873 (2014)CrossRefGoogle Scholar
  36. 36.
    F. Lin, C. Wang, Recent developments of analysis for hydrodynamic flow of nematic liquid crystals. Philos. Trans. R. Soc. A 372(2029), 20130361 (2014)Google Scholar
  37. 37.
    F.H. Lin, C. Liu, Static and dynamic theories of liquid crystals. J. Partial Diff. Equ. 14(4), 289–330 (2001)MathSciNetzbMATHGoogle Scholar
  38. 38.
    F.H. Lin, C.C. Poon, On Ericksens model for liquid crystals. J. Geom. Anal. 4(3), 379–392 (1994)Google Scholar
  39. 39.
    C. Luo, A. Majumdar, R. Erban, Multistability in planar liquid crystal wells. Phys. Rev. E 85, 061–702 (2012)Google Scholar
  40. 40.
    A. Majumdar, Equilibrium order parameters of nematic liquid crystals in the Landau-de Gennes theory. Eur. J. Appl. Math. 21, 181–203 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    A. Majumdar, A. Zarnescu, Landau-de Gennes theory of nematic liquid crystals: the Oseen-Frank limit and beyond. Arch. Ration. Mech. Anal. 196(1), 227–280 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    N.J. Mottram, C.J.P. Newton, Introduction to Q-tensor theory. Research report (University of Strathclyde, 2014)Google Scholar
  43. 43.
    M.J. Stephen, J.P. Straley, Physics of liquid crystals. Rev. Mod. Phys. 46, 617–704 (1974)CrossRefGoogle Scholar
  44. 44.
    I.W. Stewart, The Static and Dynamic Continuum Theory of Liquid Crystals: A Mathematical Introduction (CRC Press, Oxford, 2004)Google Scholar
  45. 45.
    C. Tsakonas, A.J. Davidson, C.V. Brown, N.J. Mottram, Multistable alignment states in nematic liquid crystal filled wells. Appl. Phys. Lett. 90(11), 111–913 (2007)Google Scholar
  46. 46.
    E.G. Virga, Variational Theories for Liquid Crystals (Chapman and Hall, London, 1994)CrossRefzbMATHGoogle Scholar
  47. 47.
    M.R. Wilson, Molecular simulation of liquid crystals: progress towards a better understanding of bulk structure and the prediction of material properties. Chem. Soc. Rev. 36, 1881–1888 (2007)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Singapore 2017

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of BathClaverton Down, BathUK
  2. 2.Mathematical Institute, University of OxfordOxfordUK

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