J. Phys. Chem. 1988,92, 7 177-7 185 intermediates in a nonpolar solvent is appealing in terms of the analysis of ion pair dynamics under conditions of minimal influence by neighboring solvent molecules. Acknowledgment. The support of the Department of Energy,
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Office of Basic Energy Sciences, is gratefully acknowledged. Funds for laser facilities were also provided by the Department of Defense through its University Research Instrumentation Program. The authors are also grateful to William A. Haney for technical assistance.
FEATURE ARTICLE Pair and Singlet Correlation Functions of Inhomogeneous Fluids Calculated Using the Ornstein-Zernike Equation Douglas Henderson* IBM Research Division, Almaden Research Center, 650 Harry Road, San Jose, California 951 20-6099
and Michael Plischke* Department of Physics, Simon Fraser University, Burnaby, British Columbia V5A l S 6 , Canada (Received: May 23, 1988)
Following brief review of applications of the Ornstein-Zernike equation for bulk fluids, singlet and pair Ornstein-Zernike equations are formulated for inhomogeneous fluids in which the density is no longer uniform. The singlet theory can be accurate and is quite easy to use. However, its deficiencies appear in wetting problems and in coulomb fluids near highly charged electrodes. The pair theory has been successful in all its applications to date and, in addition, yields information about the pair correlation functions of inhomogeneous fluids that is lacking in most other approaches.
Introduction There has been great progress in the theory of liquids and fluids during the past 2 decades.1%2 Broadly speaking, there are two approaches that have been useful for bulk liquids. The simplest is perturbation theory, and the second is the integral equation approach, usually based on the Ornstein-Zernike equation. Perturbation theory is quite successful for simple fluids composed of spherical molecules interacting with van der Waals forces but has not been notably successful for more complex systems involving orientationally dependent forces or for the surface region of a fluid where there are density inhomogeneities because of the apparent absence of a good reference fluid. On the other hand, the integral equation approach, although often numerically cumbersome, has the virtue of being applicable to a wide range of systems including molecular fluids with orientationally dependent forces and for inhomogeneous fluids. The latter application will be considered here.
p = l / k T (k is Boltzmann’s constant and T is the temperature) and @(rl...rN) is the potential energy of the fluid molecules. In writing eq 1, it has been assumed that the kinetic energy variables (velocity or momenta) are decoupled from the potential energy variables (position and orientation) as is the case for most systems of theoretical interest (liquid hydrogen is a counterexample). For simplicity, we have assumed that the correlation functions are functions of position. For molecular fluids, the correlation functions are functions of position and orientation. Equation 1 would still apply. However, we would have to interpret ri as a generalized position of the ith molecule including spatial and orientation variables. In actual fact, there have been few calculations for inhomogeneous molecular fluids, so generally we shall interpet ri as the spatial position of molecule i. The generalization of eq 1 to mixtures is quite straightforward but is not given so as to avoid notational complexity. The potential energy can be written in the form N
Theory for Bulk Fluids All the integral equations for fluids are equations for the pair correlation function g(rl,r2), which is a particular case (h = 2) of the h-particle correlation function, defined by
where p = N / V , N and V being, respectively, the number of molecules in the fluid and the volume of the fluid. The parameter (1) Barker, J. A.; Henderson, D. Rev. Mod. Phys. 1976, 48, 587. (2) Hansen, J. P.; McDonald, I. R. Theory of Simple Liquids, 2nd ed.; Academic: London, 1986.
0022-3654/88/2092-7177$01.50/0 , I
,
@(rl...rN) =
N
C u(ri,rj) + C w(ri,rj,rk) + ...
i
