A Continuum Approach to Liquid Crystals—The Ericksen–Leslie–Parody Formulation

  • Kolumban HutterEmail author
  • Yongqi Wang
Part of the Advances in Geophysical and Environmental Mechanics and Mathematics book series (AGEM)


Liquid crystals (LCs) are likely the most typical example of a polar medium of classical physics, in which the balance of angular momentum is a generic property, not simply expressed as a symmetry requirement of the Cauchy stress tensor. They were discovered in the second half of the nineteenth century. Liquid crystals are materials, which exhibit fluid properties, i.e., they possess high fluidity, but simultaneously exhibit crystalline anisotropy in various structural forms. We present an early phenomenological view of the behavior of these materials, which conquered a tremendous industrial significance in the second half of the twentieth century as liquid crystal devices (LCD) (Sect. 25.1). The theoretical foundation as a continuum of polar structure was laid in the late 1950s to 1990s by Ericksen, Leslie, Frank , and Parodi , primarily for nematic LCs by postulating their general physical conservation laws, hydrostatics, and hydrodynamics, thus, illustrating their connection with nontrivial balance laws of angular momentum (Sect. 25.2). This is all done by treating nematics as material continua equipped with continuous directors (long molecules), which by their orientation induce a natural anisotropy. The thermodynamic embedding (Sect. 25.3) is performed by employing an entropy balance law with nonclassical entropy flux and the requirement of Euclidian invariance of the constitutive quantities, which are assumed to be objective functions of the density, director, its gradient, and velocity, as well as stretching, vorticity, temperature, and temperature gradient. This is specialized for an incompressible LC with directors of constant length (Sect. 25.4). Constitutive parameterizations with an explicit proposal of the free energy as a quadratic polynomial of the director and its gradient (according to Frank ) are reduced to obey objectivity. Based on this, the objective form of the free energy is derived (Appendix 25.A), as are the linear dissipative Cauchy stress, director stress, and heat flux vector for the cases that the Onsager relations are fulfilled. The chapter ends with the presentation of shear flow solutions in a two-dimensional half-space and in a two-dimensional channel.


Phenomenology Balance laws Hydrodynamics Thermodynamics of LCs Directors Nematics Free energy Parodi relation Onsagerism 


  1. 1.
    Beatty, M.F.: On the foundation principles of general classical mechanics. Arch. Rational Mech. Anal. 24, 264–273 (1967)CrossRefGoogle Scholar
  2. 2.
    Beris, A.N., Edwards, B.J.: Thermodynamics of Flowing Systems with Internal Microstructure. Oxford Engineering Science Series, vol. 36. Oxford University Press, Oxford (1994)Google Scholar
  3. 3.
    Busch, K.W., Busch, M.A.: Chiral Analysis. Elsevier, Amsterdam (2006)Google Scholar
  4. 4.
    Capriz, G.: Continua with Microstructure. Springer, New York (1989)CrossRefGoogle Scholar
  5. 5.
    Castellano, J.A.: Liquid Gold: The Story of Liquid Crystal Displays and the Creation of an Industry. World Scientific Publishing (2005). ISBN 978-238-956-8Google Scholar
  6. 6.
    Chandrasekhar, S.: Liquid Crystals. Cambridge University Press, Cambridge (1977)Google Scholar
  7. 7.
    Chistyakov, I.G., Schabischev, L.S., Jarenov, R.I., Gusakova, L.A.: The polymorphism of the smectic liquid crystal. Mol. Cryst. Liq. Cryst. 7, 279 (1969) and Liq. Cryst. 2, Part II, Proceedings of the Second International Conference, G.H. Brown (ed.) Gorden and Breach, New York, p. 813 (1969)Google Scholar
  8. 8.
    Collings, P.J., Hird, M.: Introduction to Liquid Crystals. Taylor and Francis, Bristol (1997). ISBN 0-7484-0643-3Google Scholar
  9. 9.
    Currie, P.K.: Parodi’s relation as a stability condition for nematics. Mol. Cryst. Liq. Cryst. 28, 335–338 (1974)CrossRefGoogle Scholar
  10. 10.
    de Gennes, P.G.: Short range order effects in the isotropic phase of nematics and cholesterics. Mol. Cryst. Liq. Cryst. 12, 193–214 (1971)CrossRefGoogle Scholar
  11. 11.
    de Gennes, P.G.: The Physics of Liquid Crystals. Oxford University Press (Clarendon), London (1974)Google Scholar
  12. 12.
    de Gennes, P.G., Prost, J.: The Physics of Liquid Crystals. Clarendon Press, Oxford (1993). ISNB 0-19-852024-7Google Scholar
  13. 13.
    de Gennes, P.G., Sarma, G.: Tentative model for the smectic B phase. Phys. Lett. A38, 219–220 (1972)CrossRefGoogle Scholar
  14. 14.
    De Groot, S.R., Mazur, P.: Non-equilibrium Thermodynamics. Dover Publications Inc.; Dover Ed, New York (1985). ISBN-10: 0486647412, ISBN-13: 978-0486647418Google Scholar
  15. 15.
    de Vries, A.: Evidence for the existence of more than one type of nematic phase. Mol. Cryst. Liq. Cryst. 10, 31–37 (1970)CrossRefGoogle Scholar
  16. 16.
    Dierking, I.: Textures of Liquid Crystals. Wiley-VCH, Weinheim (1993). ISBN 3-527-30725-7 (1993)Google Scholar
  17. 17.
    Doi, M.: Molecular dynamics and rheological properties of concentrated solutions of rodlike polymers in isotropic and liquid crystalline phases. J. Polym. Sci. Polym. Phys. 19(2), 229–243 (1981)CrossRefGoogle Scholar
  18. 18.
    Duhem, P.M.: Introduction a la Mecanique Chimique. Kessinger Publishing, Whitefish (1893). ISBN-13: 978-1168396204Google Scholar
  19. 19.
    Dunmur, D., Sluckin, T.: Soap, Science, and Flat-Screen TVs: A History of Liquid Crystals. Oxford University Press, Oxford (2011). ISBN 978-0-19-954940-5Google Scholar
  20. 20.
    Durand, G., Leger, L., Rondelez, F., Veyssie, M.: (Orsay Liquid Crystal Group) Quasielastic Rayleigh scattering in nematic liquid crystals. Phys. Rev. Lett. 22, 1361 (1969); Erratum Phys. Rev. Lett. 23, 208 (1969)Google Scholar
  21. 21.
    Ehrentraut, H.: A unified mesoscopic continuum theory of uniaxial and biaxial liquid crystals. Ph.D. thesis, TU Berlin, Department of Physics (1996)Google Scholar
  22. 22.
    Ericksen, J.L.: Anisotropic fluids. Arch. Rational Mech. Anal. 4, 231 (1960)CrossRefGoogle Scholar
  23. 23.
    Ericksen, J.L.: Conservation laws for liquid crystals. Trans. Soc. Rheol. 5, 23–34 (1961)CrossRefGoogle Scholar
  24. 24.
    Ericksen, J.L.: Hydrostatic theory of liquid crystals. Arch. Rational Mech. Anal. 21, 371–378 (1962)CrossRefGoogle Scholar
  25. 25.
    Ericksen, J.L.: Inequalities in liquid crystal theory. Phys. Fluids 9(9), 1205–1207 (1966)CrossRefGoogle Scholar
  26. 26.
    Ericksen, J.L.: On equations of motion for liquid crystals. Q. J. Mech. Appl. Math. 29, 203–208 (1976)CrossRefGoogle Scholar
  27. 27.
    Ericksen, J.L.: Liquid crystals with variable degree of orientation. Arch. Rational Mech. Anal. 113, 97–120 (1991)CrossRefGoogle Scholar
  28. 28.
    Ericksen, J.L.: Introduction to the Thermodynamics of Solids. Applied Mathematical Sciences, vol. 131. Springer, Berlin (1998)Google Scholar
  29. 29.
    Forster, D.: Microscopic theory of flow alignment in nematic liquid crystals. Phys. Rev. Lett. 32, 1161 (1974)CrossRefGoogle Scholar
  30. 30.
    Frank, F.C.: On spontaneous asymmetric synthesis. Biochem. Biophys. Acta 11, 459 (1953)CrossRefGoogle Scholar
  31. 31.
    Frank, F.C.: I. Liquid crystals. On the theory of liquid crystals. Discuss. Faraday Soc. 25, 19–28 (1958)CrossRefGoogle Scholar
  32. 32.
    Friedel, G.: Mesomorphic states of matter. Ann. Phys. (Paris) 19, 273 (1922)Google Scholar
  33. 33.
    Gray, G.W.: Molecular Structure and the Properties of Liquid Crystals. Academic Press, New York (1962)Google Scholar
  34. 34.
    Green, A.E., Rivlin, R.S.: Simple force and stress multipoles. Arch. Rational Mech. Anal. 16, 325–354 (1964)CrossRefGoogle Scholar
  35. 35.
    Green, A.E., Naghdi, P.M., Rivlin, R.S.: Directors and multipolar displacements in continuum mechanics. Int. J. Eng. Sci. 2, 611–620 (1965)CrossRefGoogle Scholar
  36. 36.
    Hess, S.: Irreversible thermodynamics of non-equilibrium alignment phenomena in molecular liquids and in liquid crystals. Z. Naturforsch. 30a, 728 (1975)Google Scholar
  37. 37.
    Hess, S.: Pre- and post-transitional behavior of the flow alignment and flow-induced phase transition in liquid crystals. Z. Naturforsch. 31a, 1507–1513 (1976)Google Scholar
  38. 38.
    Hess, S.: Transport phenomena in anisotropic fluids and liquid crystals. J. Non-Equilib. Thermodyn. 11, 175–193 (1986)CrossRefGoogle Scholar
  39. 39.
    Hess, S., Pardowitz, Z.: On the unified theory for non-equilibrium phenomena in the isotropic and nematic phases of a liquid crystal – spatially homogeneous alignment. Z. Naturforsch. 36a, 554–558 (1981)Google Scholar
  40. 40.
  41. 41.
    Hutter, K., Jöhnk, K.: Continuum Methods of Physical Modeling. Springer, Berlin (2004), 635 ppGoogle Scholar
  42. 42.
    Hutter, K., Wang, Y.: Fluid and Thermodynamics. Advanced Fluid Mechanics and Thermodynamic Fundamentals, vol. 2, p. 633. Springer, Berlin (2016).
  43. 43.
    Kleinert, H., Maki, K.: Lattice textures in cholesteric liquid crystals. Fortschritte der Physik 29(5), 219–259 (1981)CrossRefGoogle Scholar
  44. 44.
    Lehmann, O.: Über fließende Kristalle. Zeitschrift für Physikalische Chemie 4, 462–472 (1889)Google Scholar
  45. 45.
    Lehmann, O.: Flssige Kristalle: sowie Plastizität von Kristallen im allgemeinen, molekulare Umlagerungen und Aggregatzustandsänderungen. Wilhelm Engelmann, Leipzig (1904)Google Scholar
  46. 46.
    Leslie, F.M.: Some constitutive equations for anisotropic fluids. Q. J. Mech. Appl. Math. 19, 357–370 (1966)CrossRefGoogle Scholar
  47. 47.
    Leslie, F.M.: Some constitutive equations for liquid crystals. Arch. Rational Mech. Anal. 28, 265–283 (1968)CrossRefGoogle Scholar
  48. 48.
    Leslie, F.M.: Continuum theory for nematic liquid crystals. Contin. Mech. Thermodyn. 4, 167–175 (1992)CrossRefGoogle Scholar
  49. 49.
    Levelut, A.M., Lambert, M.: Structure des cristaux liquides smectic B. C. R. Acad. Sci. B 272, 1018–1021 (1971)Google Scholar
  50. 50.
    Madsden, L.A., Dingemans, T.J., Nakata, M., Samuelski, E.T.: Thermotropic nematic liquid crystals. Phys. Rev. Lett. 92(14), 145505 (2004).
  51. 51.
    Maffettone, P.L., Sonnet, A.M., Virga, E.G.: Shear-induced biaxiality in nematic polymers. J. Non-Newton. Fluid Mech. 90, 283–297 (2000)CrossRefGoogle Scholar
  52. 52.
    Mainzer, K.: Symmetries of Nature - A Handbook for Philosophy of Nature and Science. Walter de Gruyter, Berlin (1996)Google Scholar
  53. 53.
    Müller, I.: On the entropy inequality. Arch. Rational Mech. Anal. 26, 118–141 (1967)CrossRefGoogle Scholar
  54. 54.
    Müller, M.: Twisted nematic Flüssigkeiten (2006).
  55. 55.
    Olmsted, P.D., Goldbart, P.: Theory of the nonequilibrium phase transition for nematic liquid crystals under shear flow. Phys. Rev. A41, 4578–4581 (1990)CrossRefGoogle Scholar
  56. 56.
    Olmsted, P.D., Goldbart, P.: Isotropic-nematic transition in shear flow: state selection, coexistence, phase transitions, and critical behavior. Phys. Rev. A46, 4966–4993 (1992)CrossRefGoogle Scholar
  57. 57.
    Onsager, L.: Reciprocal relations in irreversible processes I. Phys. Rev. 37, 405 (1931); II, Phys. Rev. 38, 2265 (1931)Google Scholar
  58. 58.
    Oseen, C.W.: The theory of liquid crystals. Trans. Faraday Soc. 29(140), 883–885 (1933)CrossRefGoogle Scholar
  59. 59.
    Parodi, O.: Stress tensor for a nematic liquid crystal. Le journal de Physque 31, 581–584 (1970)CrossRefGoogle Scholar
  60. 60.
    Pereira-Borgmeyer, C., Hess, S.: Unified description of the flow alignment and viscosity in the isotropic and nematic phases of liquid crystals. J. Non-Equilib. Thermodyn. 20, 359–384 (1995)CrossRefGoogle Scholar
  61. 61.
    Qian, T., Sheng, P.: Generalized hydrodynamic equations for nematic liquid crystals. Phys. Rev. E 58, 7475 (1998)CrossRefGoogle Scholar
  62. 62.
    Reinitzer, F.: Beiträge zur Kenntniss des Cholesterins. Monatshefte für Chemie (Wien) 9, 421–441 (1888)CrossRefGoogle Scholar
  63. 63.
    Saupe, A.: Das Protonenresonanzspektrum von orientiertem Benzol in nematisch-kristallinflüssiger Lösung. Z. Naturforschung 20a, 572–580 (1965)Google Scholar
  64. 64.
    Seideman, T.: The liquid-crystalline blue phases. Rep. Prog. Phys. 53(6), 659–705 (1990).
  65. 65.
    Slukin, T.J., Dunmur, D.A., Stegmeyer, H.: Crystals that Flow - Classic Papers from the History of Liquid Crystals. Taylor and Francis, London (2004)Google Scholar
  66. 66.
    Smith, G.E.: On isotropic integrity bases. Arch. Rational Mech. Anal. 18, 282–292 (1965)CrossRefGoogle Scholar
  67. 67.
    Sonnet, A.M., Virga, E.G.: Dynamics of dissipative ordered fluids. Phys. Rev. E 64(3), 031705-1-10 (2001)Google Scholar
  68. 68.
    Sonnet, A.M., Virga, E.G.: Dissipative Ordered Fluid: Theories for Liquid Crystals. Springer Science & Business Media LLC, New York (2012)Google Scholar
  69. 69.
    Sonnet, A.M., Maffettone, P.L., Virga, E.G.: Continuum theory for nematic liquid crystals with tensorial order. J. Non-Newton. Fluid Mech. 119, 51–59 (2004)CrossRefGoogle Scholar
  70. 70.
    Sparavigna, A.C.: James Fergason, a pioneer in advancing of liquid crystal technology (2006).
  71. 71.
    Spencer, A.J.M.: Theory of invariants. In: Eringen, A.C. (ed.) Continuum Physics. Mathematics, vol. 1, pp. 239–353. Academic Press, New York (1971)Google Scholar
  72. 72.
    Stephen, M.J., Straley, J.P.: Physics of liquid crystals. Rev. Mod. Phys. 46(4), 617–703 (1974).
  73. 73.
    Truesdell, C.A.: Rational Thermodynamics, 2nd edn. Springer, New York (1984).
  74. 74.
    Truesdell, C.A., Noll, W.: Non-linear field theories of mechanics, vol. III/3. Handbuch der Physik edited by Siegfried Flügge (1965)Google Scholar
  75. 75.
    Vertogen, G.: The equations of motion for nematics. Z. Naturforsch. 38a, 1273 (1983)Google Scholar
  76. 76.
    Vertogen, G., de Jeu, W.H.: Thermotropic Liquid Crystals, Fundamentals. Springer Series in Chemical Physics, vol. 45. Springer, Berlin (1988)Google Scholar
  77. 77.
    Wang, C.C.: A new representation for isotropic functions. Parts I and II. Arch. Rational Mech. Anal. 36, 166–197, 198–223 (1970)Google Scholar
  78. 78.
    Wang, C.C.: Corrigendum to my recent paper on ‘Representations for isotropic functions’. Arch. Rational Mech. Anal. 43, 392–395 (1971)CrossRefGoogle Scholar
  79. 79.
    Wu, H., Xu, X., Liu, C.: On the general Ericksen-Leslie system: Parodi’s relatation, well posedness and stability (2012). arXiv:1105.2180v6 [math.AP]

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Authors and Affiliations

  1. 1.c/o Laboratory of Hydraulics, Hydrology, GlaciologyETH ZürichZürichSwitzerland
  2. 2.Department of Mechanical EngineeringDarmstadt University of TechnologyDarmstadtGermany

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