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
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
D. Allender, L. Longa, Landau-de Gennes theory of biaxial nematics reexamined. Phys. Rev. E 78(1), 011–704 (2008)
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)
D. Andrienko, Introduction to Liquid Crystals (International Max Planck Research School, Bad Marienberg, 2006)
B. Bahadur, Liquid Crystals: Applications and Uses (World Scientific, 1991)
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)
J.M. Ball, A. Majumdar, Nematic liquid crystals: from Maier-Saupe to a continuum theory. Mol. Cryst. Liq. Cryst. 525(1), 1–11 (2010)
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)
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)
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)
H. Brezis, J.M. Coron, E.H. Lieb, Harmonic maps with defects. Commun. Math. Phy. 107(4), 649–705 (1986)
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)
S. Chandrasekhar, Liquid Crystals (Cambridge University Press, 1992)
J. Chen, C.T. Liu, Technology advances in flexible displays and substrates. Access IEEE 1, 150–158 (2013)
O.J. Dammone, Confinement of colloidal liquid crystals. Ph.D. thesis, University College, University of Oxford, 2013
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)
A.E. Danese, Advanced Calculus, vol. 1 (Allyn and Bacon, 1965)
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)
I. Dozov, M. Nobili, G. Durand, Fast bistable nematic display using monostable surface switching. Appl. Phys. Lett. 70(9), 1179–1181 (1997)
J.L. Ericksen, Liquid crystals with variable degree of orientation. Arch. Ration. Mech. Anal. 113(2), 97–120 (1991)
F.C. Frank, I. liquid crystals. on the theory of liquid crystals. Discuss. Faraday Soc. 25, 19–28 (1958)
P.G. de Gennes, The Physics of Liquid Crystals (Clarendon Press, Oxford, 1974)
E. Grelet, Hexagonal order in crystalline and columnar phases of hard rods. Phys. Rev. Lett. 100, 168–301 (2008)
R. Hardt, D. Kinderlehrer, F.H. Lin, Existence and partial regularity of static liquid crystal configurations. Commun. Math. Phys. 105(4), 547–570 (1986)
A. Jeffrey, D. Zwillinger, Table of Integrals, Series, and Products (Elsevier Science, 2000)
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)
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)
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)
S. Kralj, A. Majumdar, Order reconstruction patterns in nematic liquid crystal wells. Proc. R. Soc. A 470(2169), 20140276 (2014)
S. Kralj, E.G. Virga, Universal fine structure of nematic hedgehogs. J. Phys. A: Math. Gen. 34(4), 829 (2001)
S. Kralj, E.G. Virga, S. Žumer, Biaxial torus around nematic point defects. Phys. Rev. E 60(2), 1858 (1999)
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)
F.M. Leslie, Continuum theory for nematic liquid crystals. Contin. Mech. Thermodyn. 4(3), 167–175 (1992)
A.H. Lewis, Defects in liquid crystals: Mathematical and experimental studies. Ph.D. thesis, University of Oxford, 2016
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)
F. Lin, C. Wang, Recent developments of analysis for hydrodynamic flow of nematic liquid crystals. Philos. Trans. R. Soc. A 372(2029), 20130361 (2014)
F.H. Lin, C. Liu, Static and dynamic theories of liquid crystals. J. Partial Diff. Equ. 14(4), 289–330 (2001)
F.H. Lin, C.C. Poon, On Ericksens model for liquid crystals. J. Geom. Anal. 4(3), 379–392 (1994)
C. Luo, A. Majumdar, R. Erban, Multistability in planar liquid crystal wells. Phys. Rev. E 85, 061–702 (2012)
A. Majumdar, Equilibrium order parameters of nematic liquid crystals in the Landau-de Gennes theory. Eur. J. Appl. Math. 21, 181–203 (2010)
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)
N.J. Mottram, C.J.P. Newton, Introduction to Q-tensor theory. Research report (University of Strathclyde, 2014)
M.J. Stephen, J.P. Straley, Physics of liquid crystals. Rev. Mod. Phys. 46, 617–704 (1974)
I.W. Stewart, The Static and Dynamic Continuum Theory of Liquid Crystals: A Mathematical Introduction (CRC Press, Oxford, 2004)
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)
E.G. Virga, Variational Theories for Liquid Crystals (Chapman and Hall, London, 1994)
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)
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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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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