Advertisement

Inhomogeneous Fluids and the Freezing Transition

  • N. W. Ashcroft
Part of the NATO ASI Series book series (NSSB, volume 337)

Abstract

We are concerned with the theory of non-uniform systems, mainly classical and mainly with short-ranged interactions. Perhaps the simplest realizable case in nature is a classical gas of N identical non-interacting atoms (coordinates \( {\overrightarrow r _i} \), momenta \( \overrightarrow {{p_i}} \), and each of mass m) subjected to a static spatially varying external potential φ(1), but otherwise confined to a volume V.

Keywords

Density Functional Theory Thermodynamic Function External Potential Physical Density Homogeneous Fluid 
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]
    R. Evans, Microscopic theories of simple fluids and their interfaces, in: “Liquids at Interfaces,” J. Charvolin, J.F. Joanny, and J. Zinn-Justin, eds., Elsevier, Amsterdam (1989).Google Scholar
  2. R. Evans, Density functional in the theory of nonuniform fluids, in: “Fundamentals of Inhomogeneous Liquids,” D. Henderson, ed., Marcel Dekker, New York (1992).Google Scholar
  3. [2]
    A.R. Ubbelohde, “Melting and Crystal Structure,” Clarendon Press, Oxford (1965).Google Scholar
  4. [3]
    M. Baus, J. Phys.: Condens. Matter 2:2111 (1990).ADSCrossRefGoogle Scholar
  5. [4]
    J.-L. Barrat and J.-P. Hansen, Theory of inhomogeneous fluids and freezing, in: “Strongly Coupled Plasma Physics,” S. Ichimaru, ed., Elsevier, Amsterdam (1990).Google Scholar
  6. [5]
    A.D.J. Haymet, Freezing, in: “Fundamentals of Inhomogeneous Liquids,” D. Henderson, ed., Marcel Dekker, New York (1992).Google Scholar
  7. [6]
    H. Löwen, Physics Reports, to be published (1994).Google Scholar
  8. [7]
    J.G. Kirkwood and E. Monroe, J. Chem. Phys. 8:845 (1939).ADSCrossRefGoogle Scholar
  9. J.G. Kirkwood and E. Monroe, J. Chem. Phys. 9:514 (1940).ADSCrossRefGoogle Scholar
  10. [8]
    P.C. Hohenberg and W. Kohn, Phys. Rev. B 136:864 (1964).MathSciNetADSCrossRefGoogle Scholar
  11. [9]
    N.D. Mermin, Phys. Rev. A 137:1441 (1965).MathSciNetADSGoogle Scholar
  12. [10]
    R. Lovett, J. Chem. Phys. 88:7739 (1988).ADSCrossRefGoogle Scholar
  13. [11]
    W.F. Saam and C. Ebner, Phys. Rev. A 137:1113 (1976).Google Scholar
  14. [12]
    See, for example, J.-P. Hansen and I.R. McDonald, “Theory of Simple Liquids,” Academic Press, London (1986).Google Scholar
  15. N.W. Ashcroft, Structure and properties, in: “Condensed Matter Physics,” J. Mahanty and M.P. Das, eds., World Scientific, Singapore (1989).Google Scholar
  16. [13]
    The implicit connection between the theories of inhomogeneous and homogeneous liquids has been noted by S.-C. Kim and G.L. Jones, Phys. Rev. A 41:2222 (1990).ADSCrossRefGoogle Scholar
  17. The notion that the theory of inhomogeneous liquids can be advanced by excursions into the study of inhomogeneous liquids has already arisen earlier; see for example, R.L. Henderson and N.W. Ashcroft, Phys. Rev. A 13:859 (1976), where the mean-density approximation is introduced.ADSCrossRefGoogle Scholar
  18. [14]
    J.K. Percus, Phys. Rev. Lett. 8:462 (1962).ADSCrossRefGoogle Scholar
  19. J.K. Percus, The pair distribution function in classical statistical mechanics, in: “The Equilibrium Theory of Classical Fluids,” H.L. Frisch and J.L. Lebowitz, eds., Benjamin, New York (1964).Google Scholar
  20. [15]
    J.M.J. Van Leeuven, J. Groeneveld, and J. de Boer, Physica 25:792 (1959).ADSCrossRefGoogle Scholar
  21. [16]
    Y. Rosenfeld and N.W. Ashcroft, Phys. Rev. A 20:1208 (1979).ADSCrossRefGoogle Scholar
  22. [17]
    A.R. Demon and N.W. Ashcroft, Phys. Rev. A 41:2222 (1990).ADSCrossRefGoogle Scholar
  23. [18]
    D.M. Kroll and B.B. Laird, Phys. Rev. A 42:4806 (1990).ADSCrossRefGoogle Scholar
  24. [19]
    T.F. Meister and D.M. Kroll, Phys. Rev. A 31:4055 (1985).ADSCrossRefGoogle Scholar
  25. [20]
    S. Nordholm, M. Jonson, and B.C. Freasier, Aust. J. Chem. 33:2139 (1980).CrossRefGoogle Scholar
  26. see also M. Jonson and S. Nordholm, J. Chem. Phys. 79:4431 (1983).ADSCrossRefGoogle Scholar
  27. [21]
    P. Tarazona, Mol. Phys. 52:847 (1984).ADSCrossRefGoogle Scholar
  28. [22]
    W. Curtin and N.W. Ashcroft, Phys. Rev. A 32:2909 (1985).ADSCrossRefGoogle Scholar
  29. [23]
    This form was already anticipated by Kirkwood and Monroe, J. Chem. Phys. 8:845 (1940).ADSCrossRefGoogle Scholar
  30. [24]
    O. Gunnarsson, M. Jonson, and B.I. Lundqvist, Phys. Rev. B 20:3136 (1979).ADSCrossRefGoogle Scholar
  31. [25]
    A.R. Denton and N.W. Ashcroft, Phys. Rev. A 39:4701 (1989).ADSCrossRefGoogle Scholar
  32. [26]
    C. Likos and N.W. Ashcroft, Phys. Rev. Lett. 69:316 (1992).ADSCrossRefGoogle Scholar
  33. [27]
    B. Bildstein and G. Kahl, Phys. Rev. E 47:1712 (1993).ADSCrossRefGoogle Scholar
  34. [28]
    Y. Rosenfeld, J. Chem. Phys. 89:4272 (1988).ADSCrossRefGoogle Scholar
  35. Y. Rosenfeld, Phys. Rev. Lett. 63:980 (1989); see also. E. Kierlik and M.C. Rosinberg for a considerable elucidation of the method.ADSCrossRefGoogle Scholar
  36. [29]
    N.W. Ashcroft and N.D. Mermin, “Solid State Physics,” Holt and Saunders, Philadelphia (1976).Google Scholar
  37. [30]
    See reference [29], Appendix N.Google Scholar
  38. [31]
    BJ. Alder, D.A. Young, and M.A. Mark, J. Chem. Phys. 56:3013 (1972), and previous papers of this group referenced there.ADSCrossRefGoogle Scholar
  39. For a cluster-variational theory addressing this issue, see B. Firey and N.W. Ashcroft, J. Chem. Phys. 82:2723 (1985).ADSCrossRefGoogle Scholar
  40. [32]
    G. Baym, H.A. Bethe, C.J. Pethick, Nuclear Physics A 175:225 (1971).ADSCrossRefGoogle Scholar
  41. [33]
    S. Alexander and J. McTague, Phys. Rev. Lett. 41:702 (1978).ADSCrossRefGoogle Scholar
  42. [34]
    C.N. Likos and N.W. Ashcroft, J. Chem. Phys. 99:9090 (1993). C.N. Likos, Thesis, Cornell University (1993).ADSCrossRefGoogle Scholar
  43. [35]
    L. Verlet and J.J. Weis, Phys. Rev. A 45:939 (1972).ADSCrossRefGoogle Scholar
  44. [36]
    B.B. Laird, J. Chem. Phys. 97:2699 (1992).ADSCrossRefGoogle Scholar
  45. [37]
    D. Frenkel and A.J.C. Ladd, Phys. Rev. Lett. 59:1169 (1987).ADSCrossRefGoogle Scholar
  46. [38]
    See, for example, W.G.T. Kranendonk and D. Frenkel, J. Phys.: Condens. Matter 1:7735 (1989).ADSCrossRefGoogle Scholar
  47. [39]
    J.-L. Barrat, M. Baus, and J.-P. Hansen, J. Phys. C 20:1413 (1987).ADSCrossRefGoogle Scholar
  48. see also M. Baus and J.-L. Colot, Mol. Phys. 55:653 (1985).ADSCrossRefGoogle Scholar
  49. [40]
    S.J. Smithline and A.D.J. Haymet, J. Chem. Phys. 86:6486 (1987).ADSCrossRefGoogle Scholar
  50. S.J. Smithline and A.D.J. Haymet, J. Chem. Phys. 88:4104 (1988).ADSCrossRefGoogle Scholar
  51. [41]
    S.W. Rick and A.D.J. Haymet, J. Chem. Phys. 90:1188 (1989).ADSCrossRefGoogle Scholar
  52. [42]
    A.R. Denton and N.W. Ashcroft, Phys. Rev. A 42:7312 (1990).ADSCrossRefGoogle Scholar
  53. See also X.C. Zheng and D.W. Oxtoby, J. Chem. Phys. 93:4357 (1990).ADSCrossRefGoogle Scholar
  54. [43]
    P. Bartlett, R.H. Ottewill, and PN. Pusey, J. Chem. Phys. 93:1299 (1992).ADSCrossRefGoogle Scholar
  55. P. Bartlett and R.H. Ottewill, J. Chem. Phys. 96:3306 (1992).ADSCrossRefGoogle Scholar
  56. [44]
    R. Leidel and H. Wagner, J. Chem. Phys. 98:4142 (1993).ADSCrossRefGoogle Scholar
  57. [45]
    J.T. Chayes, L. Chayes, and E. Lieb, Commun. Math. Phys. 93:57 (1984).MathSciNetADSMATHCrossRefGoogle Scholar
  58. [46]
    TV. Ramakrishnan and M. Youssouff, Phys. Rev. B 19:2775 (1979).ADSCrossRefGoogle Scholar
  59. see also TV. Ramakrishnan, Pramana 22:365 (1984).ADSCrossRefGoogle Scholar
  60. [47]
    A.D.J. Haymet and D. Oxtoby, J. Chem. Phys. 74:2559 (1981).ADSCrossRefGoogle Scholar
  61. [48]
    J.D. McCoy, S.W. Rick, and A.D.J. Haymet, J. Chem. Phys. 90:4622 (1989).ADSCrossRefGoogle Scholar
  62. [49]
    A.R. Demon, P. Nielaba, K.J. Runge, and N.W. Ashcroft, J. Phys.: Condens. Matter 3:593 (1991).ADSCrossRefGoogle Scholar
  63. [50]
    G. Pastore and G. Senatore, Density functional theory of quantum Wigner crystallization, in: “ Strongly Coupled Plasma Physics,” S. Ichimaru, ed., Elsevier, Amsterdam (1990).Google Scholar
  64. S. Moroni and G. Senatore, Phys. Rev. B 44:9864 (1991).ADSCrossRefGoogle Scholar
  65. S. Moroni and G. Senatore, Europhys. Lett. 16:373 (1991).ADSCrossRefGoogle Scholar
  66. [51]
    M. Levy, these proceedings.Google Scholar
  67. [52]
    K. Moulopoulos and N.W. Ashcroft, Phys. Rev. B 41:6500 (1990).ADSCrossRefGoogle Scholar
  68. [53]
    K. Moulopoulos and N.W. Ashcroft, Phys. Rev. B 45:11518 (1992).ADSCrossRefGoogle Scholar
  69. [54]
    E. Lieb and J. Lebowitz, Adv. Math 9:316 (1972).MathSciNetCrossRefGoogle Scholar
  70. [55]
    K. Moulopoulos and N.W. Ashcroft, to be published (1994).Google Scholar
  71. [56]
    E. Lieb and E. Narnhofer, J. Stat. Phys. 14:465 (1976).Google Scholar
  72. [57]
    W. Kohn and L.J. Sham, Phys. Rev. A 140:1133 (1965).MathSciNetADSGoogle Scholar
  73. [58]
    N.W. Ashcroft (to appear).Google Scholar
  74. [59]
    Translations of Clausius’ papers (for example, Annalen der Physik: (Serial 2) 100:353 (1857), are given by S.G. Brush, “Kinetic Theory,” Pergamon, New York (1965).Google Scholar
  75. The quotations in Appendix C are taken from S.G. Brush, “Statistical Physics and the Atomic Theory of Matter from Boyle and Newton to Landau and Onsager,” Princeton University Press, Princeton (1983).Google Scholar

Copyright information

© Springer Science+Business Media New York 1995

Authors and Affiliations

  • N. W. Ashcroft
    • 1
  1. 1.Laboratoire d’Etudes des Proprietes Electroniques des SolidesCentre National de la Recherche ScientifiqueGrenobleFrance

Personalised recommendations