Nematic Liquid Crystals with Tensorial Order Parameters

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


This chapter goes beyond the ELP theory of LCs by modeling the microstructure of the liquid by a number of rank-i tensors \((i=1, \ldots ,n)\) (generally just one) with vanishing trace. These tensors are called alignment tensors or order parameters. When formed as exterior products of the director vector and weighted with a scalar and restricted to just one rank-2 tensor, the resulting mathematical model describes uniaxial LCs. The simplest extensions of the ELP model are theories, for which the number of independent constitutive variables are complemented by a constant or variable order parameter S and its gradient \(\mathrm {grad}\,S\) paired with an evolution equation for it. We provide a review of the recent literature. Two different approaches to deduce LC models exist; they may be coined the balance equations models, outlined already in Chap.  25 for the ELP model, and the variational Lagrange potential models, which, following an idea by Lord Rayleigh (Strutt, Proc Lond Math Soc 4:357–368, 1873, [50]), are extended by a dissipation potential. The two different approaches may lead to distinct anisotropic fluid descriptions. Moreover, it is not automatically guaranteed in either description that the balance law of angular momentum is identically satisfied. The answers to these questions cover an important part of the mathematical efforts in both model classes. Significant conceptual difficulties in the two distinct theoretical concepts are the postulations of explicit forms of the elastic energy W and dissipation function R. Depending upon, how W and R are parametrized, different particular models emerge. Conditions are formulated especially for uniaxial models, which guarantee that the two model classes reduce to exactly corresponding mathematical models.


Liquid crystals of tensorial microstructure Balance law approach Variational formulation Alignment tensor model Uniaxial LC theory 


  1. 1.
    Ajdari, A.: Pierre-Gilles de Gennes (1932–2007). Science 317(5837), 466 (2007). 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, New York (1994)Google Scholar
  3. 3.
    Carlsson, T., Leslie, F.M., Laverty, J.S.: Flow properties of biaxial nematioc liquid crystals. Mol. Cryst. Liq. Cryst. 210, 95–127 (1992)CrossRefGoogle Scholar
  4. 4.
    Capriz, G.: Continua with Microstructure. Springer, New York (1989)CrossRefGoogle Scholar
  5. 5.
    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
  6. 6.
    de Gennes, P.G.: Scaling Concepts in Polymer Physics. Cornell University Press, Ithaca (1979). ISBN 0-8014-1203-XGoogle Scholar
  7. 7.
    de Gennes, P.G., Prost, J.: The Physics of Liquid Crystals. Clarendon Press, Oxford (1993). ISBN 0-19-852024-7Google Scholar
  8. 8.
    de Gennes, P.G., Brochard-Wyart, F., Quéré, D.: Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves. Springer, Berlin (2003). ISBN 0-387-00592-7Google Scholar
  9. 9.
    Diogo, A.C., Martins, A.F.: Order parameter and temperature dependence of the hydrodynamic viscosities of nematics liquid crystals. Journal de Physique 43, 779–782 (1972)CrossRefGoogle Scholar
  10. 10.
    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
  11. 11.
    Ehrentraut, H.: A unified mesoscopic continuum theory of uniaxial and biaxial liquid crystals. Ph.D. thesis, TU Berlin, Department of Physics (1996)Google Scholar
  12. 12.
    Ericksen, J.L.: Conservation laws for liquid crystals. Trans. Soc. Rheol. 5, 23–34 (1961)CrossRefGoogle Scholar
  13. 13.
    Ericksen, J.L.: Hydrostatic theory of liquid crystals. Arch. Ration. Mech. Anal. 21, 371–378 (1962)CrossRefGoogle Scholar
  14. 14.
    Ericksen, J.L.: On equations of motion for liquid crystals. Quart. J. Mech. Appl. Math. 29, 203–208 (1976)CrossRefGoogle Scholar
  15. 15.
    Ericksen, J.L.: Liquid crystals with variable degree of orientation. Arch. Ration. Mech. Anal. 113, 97–120 (1991)CrossRefGoogle Scholar
  16. 16.
    Faraoni, V., Grosso, M., Crescitelli, S., Maffettone, P.L.: The rigid rod model for neematic polymers: an analysis of the shear problem. J. Rheol. 43, 829–843 (1999)CrossRefGoogle Scholar
  17. 17.
    Forster, D.: Microscopic theory of flow alignment in nematic liquid crystals. Phys. Rev. Lett. 32, 1161 (1974)CrossRefGoogle Scholar
  18. 18.
    Green, A.E., Rivlin, R.S.: Simple force and stress multipoles. Arch. Ration. Mech. Anal. 16, 325–354 (1964)CrossRefGoogle Scholar
  19. 19.
    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
  20. 20.
    Grosso, M., Maffettone, P.L., Dupret, F.: A closure approximation for nematic liquid crystals based on the canonical distribution subspace theory. Rheol. Acta 39, 301–310 (2000)CrossRefGoogle Scholar
  21. 21.
    Hess, S.: Irreversible thermodynamics of non-equilibrium alignment phenomena in molecular liquids and in liquid crystals. Z. Naturforsch. 30a, 728 (1975)Google Scholar
  22. 22.
    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
  23. 23.
    Hess, S.: Transport phenomena in anisotropic fluids and liquid crystals. J. Non-Equilib. Thermodyn. 11, 175–193 (1986)CrossRefGoogle Scholar
  24. 24.
    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
  25. 25.
    Hutter, K., Jöhnk, K.: Continuum Methods of Physical Modeling, 635 pp. Springer, Berlin (2004)Google Scholar
  26. 26.
    Leslie, F.M.: Some constitutive equations for anisotropic fluids. Quart. J. Mech. Appl. Math. 19, 357–370 (1966)CrossRefGoogle Scholar
  27. 27.
    Leslie, F.M.: Some constitutive equations for liquid crystals. Arch. Ration. Mech. Anal. 28, 265–283 (1968)CrossRefGoogle Scholar
  28. 28.
    Leslie, F.M.: Continuum theory for nematic liquid crystals. Contin. Mech. Thermodyn. 4, 167–175 (1992)CrossRefGoogle Scholar
  29. 29.
    Leslie, F.M., Laverty, J.S., Carlsson, T.: Continuum theory of biaxial nematic liquid crystals. Quart. J. Mech. Appl. Math. 45, 595–606 (1992)CrossRefGoogle Scholar
  30. 30.
    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
  31. 31.
    Maier, W., Saupe, A.: Eine einfache molecular-statistische Theorie der nematischen kristallflüssigen Phase. Zeitschrift für Naturforschung A 14(10), 882–889 (1959). Scholar
  32. 32.
    Marrucci, G.: Prediction of Leslie coefficients fior rodlike polymer nematics. Mol. Cryst. Liq. Cryst. (Lett.) 72, 153–161 (1982)CrossRefGoogle Scholar
  33. 33.
    Marrucci, G.: The Doi-Edwards model in slow flow. Predictions on the Weissenberg effect. J. Non-Newton. Fluid Mech. 21(3), 319–328 (1986)CrossRefGoogle Scholar
  34. 34.
    Moffettone, P.L., Sonnet, A.M., Virga, E.G.: Shear-induced bi-axiality in nematics polymers. J. Non-Newton. Fluid Mech. 90, 283–297 (2000)CrossRefGoogle Scholar
  35. 35.
    Olmsted, P.D., Goldbart, P.: Theory of the nonequilibrium phase transition for nematic liquid crystals under shear flow. Phys. Rev. A 41, 4578–4581 (1990)CrossRefGoogle Scholar
  36. 36.
    Olmsted, P.D., Goldbart, P.: Isotropic-nematic transition in shear flow: state selection, coexistence, phase transitions, and critical behavior. Phys. Rev. A 46, 4966–4993 (1992)CrossRefGoogle Scholar
  37. 37.
    Parodi, O.: Stress tensor for a nematic liquid crystal. Le journal de Physque 31, 581–584 (1970)CrossRefGoogle Scholar
  38. 38.
    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
  39. 39.
    Qian, T., Sheng, P.: Generalized hydrodynamic equations for nematic liquid crystals. Phys. Rev. E 58, 7475 (1998)CrossRefGoogle Scholar
  40. 40.
    Rienaecker, G., Hess, S.: Oriental dynamics of nematics liquid crystals under a shear flow. Phys. A 267, 294–321 (1999)CrossRefGoogle Scholar
  41. 41.
    Saupe, A.: Das Protonenresonanzspektrum von orientiertem Benzol in nematisch-kristallinflüssiger Lösung. Z. Naturforsch. 20a, 572–580 (1965)Google Scholar
  42. 42.
    Singh, A.P., Rey, A.D.: Theory and simuation of extensional flow-induced biaxiality in duiscotic mesophases. Journal de Physique II(5), 1321–1348 (1995)CrossRefGoogle Scholar
  43. 43.
    Smith, G.E.: On isotropic integrity bases. Arch. Ration. Mech. Anal. 18, 282–292 (1965)CrossRefGoogle Scholar
  44. 44.
    Sonnet, A.M., Virga, E.G.: Dynamics of dissipative ordered fluids. Phys. Rev. E 64(3), 031705-1-10 (2001)CrossRefGoogle Scholar
  45. 45.
    Sonnet, A.M., Maffettone, P.L., Virga, E.G.: Dissipative Ordered Fluid: Theories for Liquid Crystals. Springer Science & Business Media LLC, New York (2012)CrossRefGoogle Scholar
  46. 46.
    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
  47. 47.
    Spencer, A.J.M.: Theory of invariants. In: Eringen, A.C. (ed.) Continuum Physics: Volume 1–Mathematics, pp. 239–353. Academic Press, New York (1971)Google Scholar
  48. 48.
    Stephen, M.J., Straley, J.P.: Physics of liquid crystals. Rev. Mod. Phys. 46(4), 617–703 (1974). Scholar
  49. 49.
    Stark, H., Lubensky, T.C.: Poisson-bracket approach to the dynamics of nematic liquid crystals - The role of spin angular momentum. Phys. Rev. E 72, 051714 (2005)CrossRefGoogle Scholar
  50. 50.
    Strutt, J.W.: (Lord Rayleigh): some general theorems relating to vibrations. Proc. Lond. Math. Soc. 4, 357–368 (1873)Google Scholar
  51. 51.
    Vertogen, G.: The equations of motion for nematics. Z. Naturforsch. 38a, 1273 (1983)Google Scholar
  52. 52.
    Wang, C.C.: A new representation for isotropic functions. Parts I and II. Arch. Ration. Mech. Anal. 36(166–197), 198–223 (1970)CrossRefGoogle Scholar
  53. 53.
    Wang, C.C.: Corrigendum to my recent paper on ‘Representations for isotropic functions’. Arch Ration. Mech. Anal. 43, 392–395 (1971)CrossRefGoogle Scholar
  54. 54.
    Whittaker, E.T.: A Treatize on the Analytical Dynamics of Particles and Rigid Bodies, 4th edn. Cambridge University Press, Cambridge (1937)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

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

Personalised recommendations