International Journal of Thermophysics

, Volume 11, Issue 1, pp 97–107 | Cite as

A new method for the numerical solution of integral equation approximations

  • P. T. Cummings
  • P. A. Monson


A new numerical technique for solving the Ornstein-Zernike equation is described. It is particularly useful in solving the Ornstein-Zernike equation for approximations and pair potentials (such as the Percus-Yevick and mean spherical approximations for finite ranged potentials) which imply a finiteranged direct correlation function since for such approximations the numerical technique is essentially exact. The only approximation involved in such cases is the discretization of direct and total correlation functions over the finite range on which the direct correlation function is nonzero. Thus, the new method avoids truncation of the total correlation function and should permit the critical point and spinodal curve to be mapped out with greater accuracy than is permitted by existing methods. Preliminary explorations on the stability and accuracy of the method are described.

Key words

critical phenomena integral equation approximations numerical methods Ornstein-Zernike equation 


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  1. 1.
    R. C. Reid, J. M. Prausnitz, and T. K. Sherwood, The Properties of Gases and Liquids (McGraw-Hill, New York, 1977).Google Scholar
  2. 2.
    J. S. Rowlinson and F. L. Swinton, Liquids and Liquid Mixtures, 3rd ed. (Butterworths, London, 1982).Google Scholar
  3. 3.
    M. E. Paulaitis, J. M. L. Penninger, R. D. Gray, Jr., and P. Davidson, eds., Chemical Engineering at Supercritical Fluid Conditions (Ann Arbor Science, Ann Arbor, Mich., 1983).Google Scholar
  4. 4.
    J. L. Shelton and L. Yarborough, J. Petrol. Tech. Sept.: 1171 (1977); J. W. Gardner, F. M. Orr, and P. D. Patel, J. Petrol. Tech. Nov.: 2067 (1981); R. L. Henry and R. S. Metcalfe, Soc. Petrol. Eng. J. Aug.: 595 (1983).Google Scholar
  5. 5.
    J. R. Fox, Fluid Phase Equil. 14:45 (1983).Google Scholar
  6. 6.
    S. Torquato and G. Stell, Ind. Eng. Chem. Fund. 21:202 (1982).Google Scholar
  7. 7.
    A. Sivaraman, J. Magee, and R. Kobayashi, Fluid Phase Equil. 16:1 (1984).Google Scholar
  8. 8.
    W. W. Lincoln, J. J. Kozak, and K. D. Luks, J. Chem. Phys. 62:2171 (1975).Google Scholar
  9. 9.
    K. U. Co, J. J. Kozak, and K. D. Luks, J. Chem. Phys. 64:2197 (1976); K. A. Green, K. D. Luks, and J. J. Kozak, Phys. Rev. Lett. 42:985 (1979); K. A. Green, K. D. Luks, E. Lee, and J. J. Kozak, Phys. Rev. A 21:356 (1980).Google Scholar
  10. 10.
    G. L. Jones, J. J. Kozak, E. Lee, S. Fishman, and M. E. Fisher, Phys. Rev. Lett. 46:795 (1981).Google Scholar
  11. 11.
    S. Fishman, Physica (Utrecht) A 109:382 (1981); M. E. Fisher and S. Fishman, J. Chem. Phys. 78:4227 (1983).Google Scholar
  12. 12.
    K. A. Green, K. D. Luks, G. L. Jones, E. Lee, and J. J. Kozak, Phys. Rev. A 25:1060 (1982); G. L. Jones, E. E. Lee, and J. J. Kozak, Phys. Rev. Lett. 48:447 (1982); G. L. Jones, E. K. Lee, and J. J. Kozak, J. Chem. Phys. 79:459 (1983).Google Scholar
  13. 13.
    S. Fishman and M. E. Fisher, Physica (Utrecht) A 108:1 (1981).Google Scholar
  14. 14.
    J. J. Brey, A. Santos, and L. F. Rull, Phys. Rev. A 26:2993 (1982); J. J. Brey and A. Santos, J. Chem. Phys. 79:4652 (1983).Google Scholar
  15. 15.
    F. Gallerani, G. Lo Vecchio, and L. Reatto, Unpublished results (1985); F. Gallerani, G. Lo Vecchio, and L. Reatto, Phys. Rev. A 31:511 (1985).Google Scholar
  16. 16.
    A. Parola and L. Reatto, Physica (Utrecht) 125A:255 (1984); A. Parola and L. Reatto, Phys. Rev. Lett. 53:2417 (1984).Google Scholar
  17. 17.
    G. L. Jones, J. J. Kozak, and E. K. Lee, J. Chem. Phys. 80:2092 (1984).Google Scholar
  18. 18.
    A. Weyland, Phys. Lett. 98A:113 (1983).Google Scholar
  19. 19.
    P. T. Cummings and P. A. Monson, J. Chem. Phys. 82:4303 (1985).Google Scholar
  20. 20.
    P. A. Monson and P. T. Cummings, Int. J. Thermophys. 6:573 (1985).Google Scholar
  21. 21.
    J. Kerins, H. T. Davis, and L. E. Scriven, Adv. Chem. Phys. 65:215 (1986).Google Scholar
  22. 22.
    L. Mier y Terán and E. Fernández-Fassnacht, Phys. Lett. A 117:43 (1986).Google Scholar
  23. 23.
    J. J. Brey and A. Santos, Mol. Phys. 57:149(1986).Google Scholar
  24. 24.
    M. I. Guerrero, G. Saville, and J. S. Rowlinson, Mol. Phys. 29:1941 (1975).Google Scholar
  25. 25.
    S. M. Foiles and N. Ashcroft, Phys. Rev. A 24:424 (1981).Google Scholar
  26. 26.
    J.-P. Hansen and I. R. McDonald, Theory of Simple Fluids (Academic Press, London, 1976).Google Scholar
  27. 27.
    J. K. Percus and G. J. Yevick, Phys. Rev. 110:1 (1958).Google Scholar
  28. 28.
    R. J. Baxter, J. Chem. Phys. 49:2770 (1968).Google Scholar
  29. 29.
    P. T. Cummings and G. Stell, J. Chem. Phys. 78:1917 (1983).Google Scholar
  30. 30.
    R. O. Watts, J. Chem. Phys. 48:50 (1968); 50:984 (1969); 50:1358 (1969).Google Scholar
  31. 31.
    D. L. Jolly, B. C. Freasier, and R. J. Bearman, Chem. Phys. 15:237 (1976); M. Dixon and P. Hutchinson, Mol. Phys. 33:1663 (1977).Google Scholar
  32. 32.
    M. J. Gillan, Mol. Phys. 38:1781 (1979).Google Scholar
  33. 33.
    J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic Press, New York, 1970).Google Scholar
  34. 34.
    E. Waisman, Mol. Phys. 25:45 (1973); P. T. Cummings and E. R. Smith, Mol. Phys. 38:997 (1979); P. T. Cummings and E. R. Smith, Chem. Phys. 42:241 (1979).Google Scholar
  35. 35.
    L. Mier y Terán, A. H. Falls, L. E. Scriven, and H. T. Davis, Proceedings of the 8th Symposium on Thermophysical Properties, Vol. I (American Society of Mechanical Engineers, 1982), p. 45.Google Scholar
  36. 36.
    R. J. Baxter, Phys. Rev. 154:170 (1967).Google Scholar
  37. 37.
    R. O. Watts, in Statistical Mechanics, Vol. 1, K. Singer, ed. (Chemical Society Specialist Periodical Reports, London, 1973).Google Scholar
  38. 38.
    R. J. Baxter, Aust. J. Phys. 21:563 (1968).Google Scholar

Copyright information

© Plenum Publishing Corporation 1990

Authors and Affiliations

  • P. T. Cummings
    • 1
  • P. A. Monson
    • 2
  1. 1.Department of Chemical EngineeringUniversity of VirginiaCharlottesvilleUSA
  2. 2.Department of Chemical EngineeringUniversity of MassachusettsAmherstUSA

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