Pair correlation function integral equations

In any relation given above, the knowledge of the total or direct pair correlation functions yields an equation for the density profile. The domain of integration in Eqs. (14)-(16) must include all the points where pQ,(r) 0. In the case of a completely impermeable surface, pQ,(r) = 0 inside the wall... 

Integral equations can also be used to treat nonuniform fluids, such as fluids at surfaces. One starts with a binary mixture of spheres and polymers and takes the limit as the spheres become infinitely dilute and infinitely large [92-94]. The sphere polymer pair correlation function is then simply related to the density profile of the fluid. 

We could of course write down the explicit form of the general nth. order ring diagrams we prefer however to establish directly an algebraic equation for the whole series and deduce the pair correlation function from its exact solution. Indeed, it is easily verified that the nth order term is derived from the (n — l)th one by adding a loop either on the upper or on the lower line. This leads immediately to the integral equation of Fig. 9b which we now write in an analytic form. 

Eq. (14) for the exchange-correlation energy in terms of the coupling constant integrated pair-correlation function. If we take the functional derivative of this equation we find that we can write as [66]... 

From the many tools provided by statistical mechanics for determining the EOS [36, 173, 186-188] we consider first integral equation theories for the pair correlation function gxp(ra,rp) of spherical ions which relates the density of ion / at location rp to that of a at ra. In most theories gafi(ra,rp) enters in the form of the total correlation function hxp(rx,rp) = gxp(rx,rp) — 1. The Omstein-Zemike (OZ) equation splits up hap(rx,rp) into the direct correlation function cap(ra, rp) for pair interactions plus an indirect term that reflects these interactions mediated by all other particles y ... 

To find the rate of energy loss of the mode, let us insert equation (7) into equation (2). After substitution of the integration variables t = (t, +12)/2, t" = ti — t2, DitJ) = (0 q(t)q(t ) 0) is the displacement pair correlation function. The same time pairings are neglected while they contribute to k — 2, k — 4-phonon transitions and, therefore, result in small change of the anharmonic constants Vk. Note also that the validity of equation (3) with a non-zero value of I ty (f)l means the existence of anomalous correlations (bf(t)) = vitnit, these correlations depend on time. 

Integral Equation Theory for Liquids Pair Correlation Function... 

The integral equation theory consists in obtaining the pair correlation function g(r) by solving the set of equations formed by (1) the Omstein-Zernike equation (OZ) (21) and (2) a closure relation [76, 80] that involves the effective pair potential weff(r). Once the pair correlation function is obtained, some thermodynamic properties then may be calculated. When the three-body forces are explicitly taken into account, the excess internal energy and the virial pressure, previously defined by Eqs. (4) and (5) have to be, extended respectively [112, 119] so that... 

Figure 18. Pair correlation function g(r), at T = 297.6 K, for p = 0.95, 1.37, 1.69, and 1.93 nm 3 (from the bottom to the top), calculated by the ODS integral equation with both the two-body interactions (dotted lines), and the two- plus three-body interactions (solid lines). The curves for p= 1.37, 1.69, and 1.93 nm-3 are shifted upward by 0.5, 1, and 1.5, respectively. The comparison is made with molecular dynamics simulation (open circles). Taken from Ref. [129].

The equations described earlier contain two unknown functions, h(r) and c(r). Therefore, they are not closed without another equation that relates the two functions. Several approximations have been proposed for the closure relations HNC, PY, MSA, etc. [12]. The HNC closure can be obtained from the diagramatic expansion of the pair correlation functions in terms of density by discarding a set of diagrams called bridge diagrams, which have multifold integrals. It should be noted that the terms kept in the HNC closure relation still include those up to the infinite orders of the density. Alternatively, the relation has been derived from the linear response of a free energy functional to the density fluctuation created by a molecule fixed in the space within the Percus trick. The HNC closure relation reads... 

We propose the study of Lennard-Jones (LJ) mixtures that simulate the carbon dioxide-naphthalene system. The LJ fluid is used only as a model, as real CO2 and CioHg are far from LJ particles. The rationale is that supercritical solubility enhancement is common to all fluids exhibiting critical behavior, irrespective of their specific intermolecular forces. Study of simpler models will bring out the salient features without the complications of details. The accurate HMSA integral equation (Ifl) is employed to calculate the pair correlation functions at various conditions characteristic of supercritical solutions. In closely related work reported elsewhere (Pfund, D. M. Lee, L. L. Cochran, H. D. Int. J. Thermophvs. in press and Fluid Phase Equilib. in preparation) we have explored methods of determining chemical potentials in solutions from molecular distribution functions. 

Fluid microstructure may be characterized in terms of molecular distribution functions. The local number of molecules of type a at a distance between r and r-l-dr from a molecule of type P is Pa T 9afi(r)dr where Pa/j(r) is the spatial pair correlation function. In principle, flr (r) may be determined experimentally by scattering experiments however, results to date are limited to either pure fluids of small molecules or binary mixtures of monatomic species, and no mixture studies have been conducted near a CP. The molecular distribution functions may also be obtained, for molecules interacting by idealized potentials, from molecular simulations and from integral equation theories. 

The so-called product reactant Ornstein-Zernike approach (PROZA) for these systems was developed by Kalyuzhnyi, Stell, Blum, and others [46-54], The theory is based on Wertheim s multidensity Ornstein-Zernike (WOZ) integral equation formalism [55] and yields the monomer-monomer pair correlation functions, from which the thermodynamic properties of the model fluid can be obtained. Based on the MSA closure an analytical theory has been developed which yields good agreement with computer simulations for short polyelectrolyte chains [44, 56], The theory has been recently compared with experimental data for the osmotic pressure by Zhang and coworkers [57], In the present paper we also show some preliminary results for an extension of this model in which the solvent is now treated explicitly as a separate species. In this first calculation the solvent molecules are modelled as two fused charged hard spheres of unequal radii as shown in Fig. 1 [45],... 

One still has the problem of relating the potential energy of the system U, which appears in the configurational integral (equation (2.2.32)) and in the pair correlation function (equations (2.4.4) and (2.4.7)) to the intermolecular potential energy between any two molecules, Uy. The potential energy can be expanded in terms of... 

V, N, and T are held constant. In this way, the configurational integral Z is calculated. This immediately allows one to calculate the partition function Q for the ensemble (equation (2.2.31)) and thus the thermodynamic functions for the system. In addition, the pair correlation function for the fiuid is obtained in this calculation. 

In applying the compressibility equation (3.109), care must be exercised to use the pair correlation function g(R) as obtained in the grand canonical ensemble, rather than the corresponding function g(R) obtained in a closed system. Whenever this distinction is important, we use the notation gQ (R) and gc(R) for open and closed systems, respectively. Although the difference between the two is in a term of the order of AT 1 this small difference becomes important when integration over the entire volume is performed as in the definition of the quantity G (equation 3.110). 

Equation (4.12) is a connection between the cross fluctuations in the number of particles of various species, and integrals involving only the spatial pair correlation functions for the corresponding pairs of species a and p. 

As we have expressed the compressibility equation in terms of the integral over the direct correlation function in (C.16), one can write the KB theory in terms of Cy instead of G / the two are equivalent formulations. O Connel (1971) has expressed the view that the formulation in terms of Cy might be more useful for numerical work since the direct correlation function is considered to be shorter range than the pair correlation function. For further applications of this approach, see O Connel (1971), Perry and O Connel (1984), and Hamad et al. (1987, 1989, 1990a, b, 1993, 1997, 1998). 

The history of the search for an integral equation for the pair correlation function is quite long. It probably started with Kirkwood (1935), followed by Yvon (1935, 1958), Born and Green (1946), and many others. For a summary of these efforts, see Hill (1956), Fisher (1964), Rushbrooke (1968), Munster (1969), and Hansen and McDonald (1976). Most of the earlier works used the superposition approximation to obtain an integral equation for the pair correlation function. It was in 1958 that Percus and Yevick developed an integral equation that did not include explicitly the assumption of superposition, i.e., pairwise additivity of the higher order potentials of mean force. The Percus-Yevick (PY) equation was found most useful in the study of both pure liquids as well as mixtures of liquids. 

During the 1950s and the 1960s, two important theories of the liquid state were developed, initially for simple liquids and later applied to mixtures. These are the scaled-particle theory, and integral equation methods for the pair correlation function. These theories were described in many reviews and books. In this book, we shall only briefly discuss these theories in a few appendices. Except for these two theoretical approaches there has been no new molecular theory that was specifically designed and developed for mixtures and solutions. This leads to the natural question why a need for a new book with the same title as Prigogine s ... 

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