Abstract
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.
Similar content being viewed by others
References
R. C. Reid, J. M. Prausnitz, and T. K. Sherwood, The Properties of Gases and Liquids (McGraw-Hill, New York, 1977).
J. S. Rowlinson and F. L. Swinton, Liquids and Liquid Mixtures, 3rd ed. (Butterworths, London, 1982).
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).
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).
J. R. Fox, Fluid Phase Equil. 14:45 (1983).
S. Torquato and G. Stell, Ind. Eng. Chem. Fund. 21:202 (1982).
A. Sivaraman, J. Magee, and R. Kobayashi, Fluid Phase Equil. 16:1 (1984).
W. W. Lincoln, J. J. Kozak, and K. D. Luks, J. Chem. Phys. 62:2171 (1975).
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).
G. L. Jones, J. J. Kozak, E. Lee, S. Fishman, and M. E. Fisher, Phys. Rev. Lett. 46:795 (1981).
S. Fishman, Physica (Utrecht) A 109:382 (1981); M. E. Fisher and S. Fishman, J. Chem. Phys. 78:4227 (1983).
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).
S. Fishman and M. E. Fisher, Physica (Utrecht) A 108:1 (1981).
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).
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).
A. Parola and L. Reatto, Physica (Utrecht) 125A:255 (1984); A. Parola and L. Reatto, Phys. Rev. Lett. 53:2417 (1984).
G. L. Jones, J. J. Kozak, and E. K. Lee, J. Chem. Phys. 80:2092 (1984).
A. Weyland, Phys. Lett. 98A:113 (1983).
P. T. Cummings and P. A. Monson, J. Chem. Phys. 82:4303 (1985).
P. A. Monson and P. T. Cummings, Int. J. Thermophys. 6:573 (1985).
J. Kerins, H. T. Davis, and L. E. Scriven, Adv. Chem. Phys. 65:215 (1986).
L. Mier y Terán and E. Fernández-Fassnacht, Phys. Lett. A 117:43 (1986).
J. J. Brey and A. Santos, Mol. Phys. 57:149(1986).
M. I. Guerrero, G. Saville, and J. S. Rowlinson, Mol. Phys. 29:1941 (1975).
S. M. Foiles and N. Ashcroft, Phys. Rev. A 24:424 (1981).
J.-P. Hansen and I. R. McDonald, Theory of Simple Fluids (Academic Press, London, 1976).
J. K. Percus and G. J. Yevick, Phys. Rev. 110:1 (1958).
R. J. Baxter, J. Chem. Phys. 49:2770 (1968).
P. T. Cummings and G. Stell, J. Chem. Phys. 78:1917 (1983).
R. O. Watts, J. Chem. Phys. 48:50 (1968); 50:984 (1969); 50:1358 (1969).
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).
M. J. Gillan, Mol. Phys. 38:1781 (1979).
J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic Press, New York, 1970).
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).
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.
R. J. Baxter, Phys. Rev. 154:170 (1967).
R. O. Watts, in Statistical Mechanics, Vol. 1, K. Singer, ed. (Chemical Society Specialist Periodical Reports, London, 1973).
R. J. Baxter, Aust. J. Phys. 21:563 (1968).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Cummings, P.T., Monson, P.A. A new method for the numerical solution of integral equation approximations. Int J Thermophys 11, 97–107 (1990). https://doi.org/10.1007/BF00503862
Issue Date:
DOI: https://doi.org/10.1007/BF00503862