Journal of Statistical Physics

, Volume 22, Issue 6, pp 709–742 | Cite as

Hard-particle fluids. II. Generaly-expansion-like descriptions

  • Boris Barboy
  • William M. Gelbart
Articles

Abstract

We present a critical discussion of the “y-expansion” approach to the thermodynamics of hard-particle fluids. First we discuss briefly our original formulation for many-component mixtures of anisotropic species, using the usual virial series as a point of departure. Difficulties arising in the case of attractive tails and nonadditive hard-core interactions are exposed. To resolve these problems we suggest a straightforward generalization of the expansion quantityy. Instead of\({{y_\alpha \equiv \rho _\alpha } \mathord{\left/ {\vphantom {{y_\alpha \equiv \rho _\alpha } {\left( {1 - \sum\nolimits_{\gamma = 1}^v {\upsilon _{0\gamma } \rho _\gamma } } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 - \sum\nolimits_{\gamma = 1}^v {\upsilon _{0\gamma } \rho _\gamma } } \right)}}\), whereΝ andργ are the particle volume and number density of theγth species in theΝ-component mixture, we define\({{y_\alpha \equiv \rho _\alpha } \mathord{\left/ {\vphantom {{y_\alpha \equiv \rho _\alpha } {\left( {1 - \sum\nolimits_{\gamma = 1}^v {\psi _\gamma ^\alpha \rho _\gamma } } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 - \sum\nolimits_{\gamma = 1}^v {\psi _\gamma ^\alpha \rho _\gamma } } \right)}}\), where theψγα are determined by optimizing the convergence of the series expressing thermodynamic functions in powers of the yα. This procedure provides in particular a good description of nonadditive binary mixtures of hard spheres withσ22 = 0 andσ12 = (1/2)σ11(1 +δ) (δ ≠ 0, ≥ −1 is the usual nonadditivity parameter.) We present a generalization of the analysis of Widom and Rowlinson whereby such systems are shown to be equivalent topure fluids ofattracting hard spheres. Critical point properties of the pure fluid are determined via this equivalence, using oury-expansion description of the nonadditivemixture. Finally, we present the results ofy-expansion studies of some anisotropic (i.e., orientationally ordered) states of fluids composed of asymmetric hard particles. For the case of rectangular parallelepipeds whose allowed orientations are restricted, we can compare our description of the isotropic-nematic liquid crystal phase transition with those obtained earlier by virial expansions and Padé approximants. Finally, generalization to continuously allowed orientations is discussed.

Key words

y-Expansion hard-core particles nonadditive pair potential Widom-Rowlinson correspondence isotropic-nematic liquid crystal phase transition 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. E. Mayer, Theory of Real Gases, inHandbuch der Physik, S. Flügge, ed. (Springer, Berlin, 1958), Vol. 12.Google Scholar
  2. 2.
    L. Onsager,Ann. N.Y. Acad. Sci. 51:627 (1949).Google Scholar
  3. 3.
    R. W. Zwanzig,J. Chem. Phys. 39:1714 (1963).Google Scholar
  4. 4.
    E. Helfand and F. H. Stillinger, Jr.,J. Chem. Phys. 49:1232 (1968).Google Scholar
  5. 5.
    L. K. Runnels and C. Colvin,J. Chem. Phys. 53:4219 (1970).Google Scholar
  6. 6.
    F. H. Ree and W. G. Hoover,J. Chem. Phys. 46:4181 (1967).Google Scholar
  7. 7.
    J. A. Barker and D. Henderson,Can. J. Phys. 45:3959 (1967).Google Scholar
  8. 8.
    J. A. Barker and D. Henderson,Rev. Mod. Phys. 48:587 (1976), and references therein.Google Scholar
  9. 9.
    B. Barboy and W. M. Gelbart,J. Chem. Phys. 71:3053 (1979).Google Scholar
  10. 10.
    J. K. Percus and G. J. Yevick,Phys. Rev. 110:1 (1958); E. Thiele,J. Chem. Phys. 39:474 (1963); M. S. Wertheim,Phys. Rev. Lett. 10:321 (1963).Google Scholar
  11. 11.
    H. Reiss, H. L. Frisch, and J. L. Lebowitz,J. Chem. Phys. 31:369 (1959); H. Reiss, inStatistical Mechanics and Statistical Methods in Theory and Applications, V. Landman ed. (Plenum, New York, 1977).Google Scholar
  12. 12. (a)
    B. J. Alder and T. E. Wainwright,J. Chem. Phys. 33:1439 (1960);Google Scholar
  13. 12. (b)
    B. J. Alder,J. Chem. Phys. 40:2724 (1964); E. B. Smith and K. R. Lea,Trans. Faraday Soc. 59:1535 (1963).Google Scholar
  14. 13.
    K. W. Kratky,Physica 87A:584 (1977).Google Scholar
  15. 14.
    W. G. Hoover and A. G. DeRocco,J. Chem. Phys. 36:3141 (1962).Google Scholar
  16. 15.
    J. S. Rowlinson,Liquids and Liquid Mixtures (Butterworths, London, 1959).Google Scholar
  17. 16.
    M. Rigby and E. B. Smith,Trans. Faraday Soc. 59:2469 (1963).Google Scholar
  18. 17.
    T. W. Melnyk and B. L. Sawford,Mol. Phys. 29:891 (1975).Google Scholar
  19. 18.
    T. L. Hill,Statistical Mechanics (McGraw-Hill, New York, 1956), p. 210.Google Scholar
  20. 19.
    B. Widom and J. S. Rowlinson,J. Chem. Phys. 52:1670 (1970).Google Scholar
  21. 20.
    T. W. Melnyk, J. S. Rowlinson, and B. L. Sawford,Mol. Phys. 24:809 (1972).Google Scholar
  22. 21.
    J. Vieillard-Baron,Mol. Phys. 28:809 (1974); J. Kushick and B. Berne,J. Chem.Phys. 64:1362 (1976).Google Scholar
  23. 22.
    T. Kihara,Rev. Mod. Phys. 25:831 (1953);Adv. Chem. Phys. 5:147 (1963).Google Scholar
  24. 23.
    A. Isihara,J. Chem. Phys. 19:1142 (1951).Google Scholar
  25. 24.
    J. P. Straley,Mol. Cryst. Liq. Cryst. 24:7 (1973).Google Scholar
  26. 25.
    W.M. Gelbart and B. Barboy,Mol. Cryst. Liq. Cryst. 55:209 (1979); A van der Wools picture of the isotropic-nematic liquid crystal phase transition, preprint.Google Scholar
  27. 26.
    J. P. Straley, M. A. Cotter, Tae-Journ Lie, and B. Widom,J. Chem. Phys. 57:4484 (1972).Google Scholar
  28. 27.
    G. Lasher,J. Chem. Phys. 53:4141 (1970).Google Scholar
  29. 28.
    S. Jen, N. A. Clark, P.S. Pershan, and E. B. Priestley,J. Chem. Phys. 66:4635 (1977).Google Scholar
  30. 29.
    J. P. Straley,J. Chem. Phys. 57:3694 (1972).Google Scholar
  31. 30.
    J. P. Straley,Phys. Rev. A 10:1881 (1974).Google Scholar

Copyright information

© Plenum Publishing Corporation 1980

Authors and Affiliations

  • Boris Barboy
    • 1
  • William M. Gelbart
    • 1
    • 2
  1. 1.Department of ChemistryUniversity of CaliforniaLos Angeles
  2. 2.Camille and Henry Dreyfus Foundation Teacher-ScholarUSA

Personalised recommendations