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A Model for the Calculation of Thermodynamic Properties of a Fluid Using Hard-Sphere Perturbation Theory and the Zero-Kelvin Isotherm of the Solid GERALD I. KERLEY Los Alamos National Laboratory, Applied Theoretical Physics Division, Los Alamos, NM 87545 The CRIS model of fluids is reviewed and calculations using the theory are compared with experimental data. The equation of state is computed from an expansion about a hard-spherereference system, in which the optimum hard-sphere diameter is chosen by a variational principle. All information about the intermolecular forces is obtained from the zero-Kelvin isotherm of the solid. Calculations for the rare gases, for the hydrogen isotopes and other polyatomic molecules, and for liquid iron are shown to agree well with experiment. Liberman's model for the electronic structure of a compressed atom is used to calculate contributions from thermal electronic excitation to the equation of state. These terms are shown to be important in explaining shock-wave data for xenon.
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E V E R A L E X C E L L E N T T H E O R I E S recently have been developed for calculating the thermodynamic properties of fluids from specified pair potentials (1-10). Barker and Henderson (1) showed that hard-sphere perturbation methods are very accurate, even at low temperatures, when the hard-sphere diameter is defined in an optimum fashion. Subsequently, Mansoori and Canfield (2) and Rasaiah and Stell (3) developed the variational principle for choosing the hard-sphere diameter. Anderson et al. (4) showed that perturbation theories succeed because repulsive forces, or effects of excluded volume, play the principal role in determining the equilibrium structure of dense fluids (for spherically symmetric molecules). Approaching the problem from a different point of view, Rosenfeld and Ashcroft (5) developed an accurate integral equation method that relies on the universality of the short-range structure in dense fluids. Other important developments include applications of fluid theory to nonspherical molecules (1, 6, 7) and to liquid metals (8-10).
0065-2393/83/0204-0107$09.00/0 © 1983 American Chemical Society In Molecular-Based Study of Fluids; Haile, J., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1983.
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MOLECULAR-BASED STUDY O F FLUIDS
Unfortunately, applications of these accurate theories to problems of practical interest are often hampered by lack of knowledge about intermolecular forces. For this reason, we have developed the CRIS model (11,12), a perturbation theory offluidsin which explicit knowledge of the interaction potentials is not required. Our model retains the key concepts of fluid structure that are essential to the success of the other perturbation theories. However, the energy of a fluid molecule in the cage formed by its neighbors is estimated from the zero-Kelvin isotherm of the solid. This cold curve is usually easier to compute or measure than is an effective pair potential. In this chapter we discuss the theoretical model and review the results of several calculations. First, the theory for the case of spherical molecules in the ground state is considered. The model is shown to agree with computer simulation studies on systems where the pair potentials are known (13). We then show how other degrees of freedom can be included in calculating equations of state. In particular, an electronic structure model due to Liberman (14) is useful for computing contributions from thermal electronic excitation. Rotational ordering and other perturbations of intramolecular motions are not considered in this paper. Additional theoretical problems, including treatment of vaporization, melting, and shock waves, are then discussed. The rest of the chapter compares calculations using the model with experimental data for rare gases, molecular fluids, and liquid metals. Because of space limitations, only an outline of the main theoretical ideas is presented here. Detailed and rigorous discussions are given in the literature cited. We also note that Rosenfeld (15) has derived the first-order CRIS model by a method different from ours.
Outline of the CRIS Model Consider a system of N spherical molecules, having no internal degrees of freedom, in a volume V at temperature T. The thermodynamic properties of the system are determined by the potential energy function (J). Although 4> is a function of the positions of all N molecules, only the short range structure is important for perturbation theories. To see this fact, define coordinates {q } that specify the positions of all molecules relative to an origin fixed at the center of mass of molecule k. We write k
= £
(i)
where (q ), the potential energy of molecule k in the field of its neighbors, includes all pair, triplet, and higher-order interactions (11). This k
In Molecular-Based Study of Fluids; Haile, J., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1983.
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KERLEY
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Calculation of Thermodynamic Properties
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function depends only on the local structure of the fluid, i.e., the coordinates q of nearby molecules relative to the one under consideration. For spherical molecules, the structure of dense fluids is determined primarily by the effects of excluded volume, and it is useful to express the Helmholtz free energy as a perturbation expansion about a model system, the hard-sphere fluid (11). A^(V,T,N) = A (V,T,N;cr) + ) , the first order correction, is an average of O taken in the hard-sphere system. By definition, AA^ contains all remaining contributions to A^; these corrections are caused by differences between the structure of the real fluid and that of the hard-sphere system. The term AA^ can be made quite small by making an optimum choice for o\ It can be shown that the first-order approximation gives an upper bound to the true free energy of the system (2, 3). 0
0
A * = A + ) = \ (qr) n0(q) dq1 dq2 . . . 0
(4)
where n0(q) is a hard-sphere distribution function. It specifies the probability density that a molecule in the fluid will have neighbors located within differential elements dqx, dq2, . . . , at positions qx, q2, . . . (11). Because (g) depends upon the short range structure of the fluid, the position of the first shell of neighbors is the most important quantity specifying the local configuration. In the CRIS model, the nearest neighbors are assumed to lie on a spherical shell, of radius R, that varies from molecule to molecule (12). We further assume that the coordination number v varies with R so that the volume per molecule is fixed at the macroscopic value V/N. If there are 12 nearest neighbors in a close-
In Molecular-Based Study of Fluids; Haile, J., et al.; Advances in Chemistry; American Chemical Society: Washington, DC, 1983.
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MOLECULAR-BASED STUDY O F FLUIDS
packed configuration, it can be shown that v = 6 V 2 NR /V
(5)
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In this approximation, only one variable, R, is required to specify the local arrangement of neighbors about a particular molecule. Equation 4 becomes
>o =
\ 4> (R, v) n (R) 4TTR dR
(6)
2
0
Furthermore, the distribution function n (R) is given by 0
(v/N)n (R) = (N/V) g (R)
(7)
0
0
where g (R) is the contribution from the nearest neighbor shell to the radial distribution function for the hard-sphere fluid. A satisfactory working definition of this quantity can be obtained from the first peak in the radial distribution function (12). The potential energy function c()(R,v) can be estimated from the zero-Kelvin isotherm of the solid in the following way. Let E (V ) be the electronic contribution to the energy per molecule for the close-packed solid at volume V and zero temperature. (Note that this definition does not include any contribution from the zero-point lattice vibrations, which are not part of the intermolecular forces.) In the solid, there are 12 nearest neighbors on a sphere of radius R, given by 0
C
S
s
V
s
= NR3/V2
(8)
In the fluid, a molecule has the same potential energy as it would have in the solid phase at the same nearest neighbor distance, except that the coordination number is reduced from 12 to v. Hence
