Advertisement

Archive for Rational Mechanics and Analysis

, Volume 206, Issue 3, pp 1039–1072 | Cite as

Statistical Foundations of Liquid-Crystal Theory. I: Discrete Systems of Rod-Like Molecules

  • Brian Seguin
  • Eliot FriedEmail author
Article

Abstract

We develop a mechanical theory for systems of rod-like particles. Central to our approach is the assumption that the external power expenditure for any subsystem of rods is independent of the underlying frame of reference. This assumption is used to derive the basic balance laws for forces and torques. By considering inertial forces on par with other forces, these laws hold relative to any frame of reference, inertial or noninertial. Finally, we introduce a simple set of constitutive relations to govern the interactions between rods and find restrictions necessary and sufficient for these laws to be consistent with thermodynamics. Our framework provides a foundation for a statistical mechanical derivation of the macroscopic balance laws governing liquid crystals.

Keywords

Torque Liquid Crystal Constitutive Relation Nematic Liquid Crystal Momentum Balance 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Oseen C.W.: The theory of liquid crystals. Trans. Faraday Soc. 29, 883–899 (1933)CrossRefGoogle Scholar
  2. 2.
    Ericksen J.L.: Conservation laws for liquid crystals. Trans. Soc. Rheol. 5, 23–34 (1961)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Leslie F.: Some constitutive equations for anisotropic fluids. Q. J. Mech. Appl. Math. 19, 357–370 (1966)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    de Gennes P., Prost J.: The Physics of Liquid Crystals. Oxford University Press, New York (1995)Google Scholar
  5. 5.
    Chandrasekhar S.: Liquid Crystals. Cambridge University Press, New York (1992)CrossRefGoogle Scholar
  6. 6.
    Virga E.G.: Variational Theories of Liquid Crystals. Chapman & Hall, London (1994)Google Scholar
  7. 7.
    Stephen M.J., Straley J.P.: Physics of liquid crystals. Rev. Mod. Phys. 46, 617–704 (1974)ADSCrossRefGoogle Scholar
  8. 8.
    Ericksen J.L.: Equilibrium theory of liquid crystals. Adv. Liq. Cryst. 2, 233–298 (1976)Google Scholar
  9. 9.
    Jenkins J.T.: Flows of nematic liquid crystals. Annu. Rev. Fluid Mech. 10, 197–219 (1978)ADSCrossRefGoogle Scholar
  10. 10.
    Leslie E.M.: Theory of flow phenomena in liquid crystals. Adv. Liq. Cryst. 4, 1–81 (1979)Google Scholar
  11. 11.
    de Gennes P.G.: Phenomenology of short-range-order effects in the isotropic phase of nematic materials. Phys. Lett. A 30, 454–455 (1969)ADSCrossRefGoogle Scholar
  12. 12.
    Landau, L.D. Theory of phase transformations. I. Physikalische Zeitschrift der Sowjetunion 11, 26–47 (1937)Google Scholar
  13. 13.
    Landau, L.D.: Theory of phase transformations. II. Physikalische Zeitschrift der Sowjetunion 11, 545–555 (1937)Google Scholar
  14. 14.
    Ericksen J.L.: Anisotropic fluids. Arch. Rational Mech. Anal. 4, 231–237 (1959)MathSciNetADSCrossRefGoogle Scholar
  15. 15.
    MacMillan E. A Theory of Anisotropic Fluids. Dissertation, The University of Minnesota, Minneapolis, (1987)Google Scholar
  16. 16.
    Ericksen J.L.: Liquid crystals with variable degree of orientation. Arch. Rational Mech. Anal. 113, 97–120 (1991)MathSciNetADSzbMATHCrossRefGoogle Scholar
  17. 17.
    Sonnet A.M., Maffettone P.L., Virga E.G.: Continuum theory for nematic liquid crystals with tensorial order. J. Non-Newtonian Fluid Mech. 119, 51–59 (2004)zbMATHCrossRefGoogle Scholar
  18. 18.
    Irving J.H., Kirkwood J.G.: The statistical mechanical theory of transport processes. IV. The equations of hydrodynamics. J. Chem. Phys. 18, 817–829 (1950)MathSciNetADSCrossRefGoogle Scholar
  19. 19.
    Noll, W.: Die Herleitung der Grundgleichungen der Thermomechanik der Kontinua aus der statistischen Mechanik. J. Rational Mech. Anal. 4, 627–646 (1955). Translated into English by Lehoucq, R.B., Lilienfeld-Toal, A.: J. Elast. 100, 5–24 (2010)Google Scholar
  20. 20.
    Lubensky T.C.: Molecular description of nematic liquid crystals. Phys. Rev. A 6, 2497–2514 (1970)ADSCrossRefGoogle Scholar
  21. 21.
    Müller I.: Thermodynamics. Pitman, London (1985)zbMATHGoogle Scholar
  22. 22.
    Capriz C., Podio Guidugli P.: Discrete and continuous bodies with affine structure. Ann. Mat. Pur. Appl. 4, 195–211 (1976)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Fried E.: New insights into the classical mechanics of particle systems. Discrete Contin. Dyn. Syst. 28, 1469–1504 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Noll, W.: La Mécanique Classique, Basée sur un Axiome d’Objectivité. “La Méthode Axiomatique dans les Mécaniques Classiques et Nouvelles” (Colloque International, Paris, 1959). Gauthier-Vallars, 47–56, 1963Google Scholar
  25. 25.
    Beatty M.F.: On the foundation principles of general classical mechanics. Arch. Rational Mech. Anal. 24, 246–273 (1967)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Green A.E., Rivlin R.S.: On Cauchy’s equations of motion. Z. Angew. Math. Phys. 15, 290–292 (1964)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics I: Functional Analysis. Academic Press, San Diago, 1980Google Scholar
  28. 28.
    Gurtin M.E., Williams W.O.: On the first law of thermodynamics. Arch. Rational Mech. Anal. 42, 77–92 (1971)MathSciNetADSzbMATHCrossRefGoogle Scholar
  29. 29.
    Rao A.V.: Dynamics of Particles and Rigid Bodies. Cambridge University Press, Cambridge (2006)Google Scholar
  30. 30.
    Landau, L., Lifchitz, E.: Mecanique. Editions Mir, Moscow, 1969Google Scholar
  31. 31.
    Noll, W.: Five contributions to natural philosophy. Published as B1 on the website www.math.cmu.edu/~wn0g/noll (2004)
  32. 32.
    Maier W., Saupe A.: Eine einfache molekulare Theorie des nematischen Kristallinflüssigen Zustands. Z. Naturforsch. 13a, 564–566 (1958)ADSGoogle Scholar
  33. 33.
    Coleman B.D., Noll W.: The thermodynamics of elastic materials with heat conduction and viscosity. Arch. Rational Mech. Anal. 13, 167–178 (1963)MathSciNetADSzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMontrealCanada
  2. 2.Department of Mechanical EngineeringMontrealCanada

Personalised recommendations