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Variational Methods in Molecular Modeling pp 223–254Cite as

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

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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.

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References

  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. D. Allender, L. Longa, Landau-de Gennes theory of biaxial nematics reexamined. Phys. Rev. E 78(1), 011–704 (2008)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  4. D. Andrienko, Introduction to Liquid Crystals (International Max Planck Research School, Bad Marienberg, 2006)

    Google Scholar 

  5. B. Bahadur, Liquid Crystals: Applications and Uses (World Scientific, 1991)

    Google Scholar 

  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. J.M. Ball, A. Majumdar, Nematic liquid crystals: from Maier-Saupe to a continuum theory. Mol. Cryst. Liq. Cryst. 525(1), 1–11 (2010)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  11. H. Brezis, J.M. Coron, E.H. Lieb, Harmonic maps with defects. Commun. Math. Phy. 107(4), 649–705 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  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. S. Chandrasekhar, Liquid Crystals (Cambridge University Press, 1992)

    Google Scholar 

  14. J. Chen, C.T. Liu, Technology advances in flexible displays and substrates. Access IEEE 1, 150–158 (2013)

    Article  Google Scholar 

  15. O.J. Dammone, Confinement of colloidal liquid crystals. Ph.D. thesis, University College, University of Oxford, 2013

    Google Scholar 

  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. A.E. Danese, Advanced Calculus, vol. 1 (Allyn and Bacon, 1965)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  19. I. Dozov, M. Nobili, G. Durand, Fast bistable nematic display using monostable surface switching. Appl. Phys. Lett. 70(9), 1179–1181 (1997)

    Article  Google Scholar 

  20. J.L. Ericksen, Liquid crystals with variable degree of orientation. Arch. Ration. Mech. Anal. 113(2), 97–120 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  21. F.C. Frank, I. liquid crystals. on the theory of liquid crystals. Discuss. Faraday Soc. 25, 19–28 (1958)

    Article  Google Scholar 

  22. P.G. de Gennes, The Physics of Liquid Crystals (Clarendon Press, Oxford, 1974)

    MATH  Google Scholar 

  23. E. Grelet, Hexagonal order in crystalline and columnar phases of hard rods. Phys. Rev. Lett. 100, 168–301 (2008)

    Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  25. A. Jeffrey, D. Zwillinger, Table of Integrals, Series, and Products (Elsevier Science, 2000)

    Google Scholar 

  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. 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. 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. S. Kralj, A. Majumdar, Order reconstruction patterns in nematic liquid crystal wells. Proc. R. Soc. A 470(2169), 20140276 (2014)

    Google Scholar 

  30. S. Kralj, E.G. Virga, Universal fine structure of nematic hedgehogs. J. Phys. A: Math. Gen. 34(4), 829 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  31. S. Kralj, E.G. Virga, S. Žumer, Biaxial torus around nematic point defects. Phys. Rev. E 60(2), 1858 (1999)

    Article  Google Scholar 

  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. F.M. Leslie, Continuum theory for nematic liquid crystals. Contin. Mech. Thermodyn. 4(3), 167–175 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  34. A.H. Lewis, Defects in liquid crystals: Mathematical and experimental studies. Ph.D. thesis, University of Oxford, 2016

    Google Scholar 

  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)

    Article  Google Scholar 

  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. F.H. Lin, C. Liu, Static and dynamic theories of liquid crystals. J. Partial Diff. Equ. 14(4), 289–330 (2001)

    MathSciNet  MATH  Google Scholar 

  38. F.H. Lin, C.C. Poon, On Ericksens model for liquid crystals. J. Geom. Anal. 4(3), 379–392 (1994)

    Google Scholar 

  39. C. Luo, A. Majumdar, R. Erban, Multistability in planar liquid crystal wells. Phys. Rev. E 85, 061–702 (2012)

    Google Scholar 

  40. A. Majumdar, Equilibrium order parameters of nematic liquid crystals in the Landau-de Gennes theory. Eur. J. Appl. Math. 21, 181–203 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  42. N.J. Mottram, C.J.P. Newton, Introduction to Q-tensor theory. Research report (University of Strathclyde, 2014)

    Google Scholar 

  43. M.J. Stephen, J.P. Straley, Physics of liquid crystals. Rev. Mod. Phys. 46, 617–704 (1974)

    Article  Google Scholar 

  44. I.W. Stewart, The Static and Dynamic Continuum Theory of Liquid Crystals: A Mathematical Introduction (CRC Press, Oxford, 2004)

    Google Scholar 

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

    Book  MATH  Google Scholar 

  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)

    Article  Google Scholar 

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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.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK

    Apala Majumdar

  2. Mathematical Institute, University of Oxford, Andrew Wiles Building Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK

    Alexander H. Lewis

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  1. Apala Majumdar
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Correspondence to Apala Majumdar .

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  1. Department of Chemical and Environmental Engineering and Department of Mathematics, University of California, Riverside, California, USA

    Jianzhong Wu

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Majumdar, A., Lewis, A.H. (2017). A Theoretician’s Approach to Nematic Liquid Crystals and Their Applications. In: Wu, J. (eds) Variational Methods in Molecular Modeling. Molecular Modeling and Simulation. Springer, Singapore. https://doi.org/10.1007/978-981-10-2502-0_8

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  • DOI: https://doi.org/10.1007/978-981-10-2502-0_8

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