Percus—Yevick approximation

Baxter R J 1970 Ornstein Zernike relation and Percus-Yevick approximation for fluid mixtures J. Chem. Phys. 52 4559... 

Cummings P T and Stell G 1984 Statistical mechanical models of chemical reactions analytic solution of models of A + S AS in the Percus-Yevick approximation Mol. Phys. 51 253... 

A set of equations (15)-(17) represents the background of the so-called second-order or pair theory. If these equations are supplemented by an approximate relation between direct and pair correlation functions the problem becomes complete. Its numerical solution provides not only the density profile but also the pair correlation functions for a nonuniform fluid [55-58]. In the majority of previous studies of inhomogeneous simple fluids, the inhomogeneous Percus-Yevick approximation (PY2) has been used. It reads... 

However, it is known that the direct correlation functions have an exact long-range asymptotic form, arising due to intramolecular correlations in clusters formed via the association mechanism. This asymptotics is not included in the Percus-Yevick approximation. Other common liquid state approximations also do not provide correct asymptotic behavior of Ca ir). 

The EMSA requires the degree of dimerization A as an input parameter. This is quite disappointing. However, it ehminates the deficiency of the Percus-Yevick approximation, Eq. (38). The EMSA represents a simpHfied version, to obtain an analytic solution, of a more sophisticated site-site extended mean spherical approximation (SSEMSA) [67-69]. The results of the aforementioned closures can be used as an input for subsequent calculations of the structure of nonuniform associating fluids. 

In the limit of zero association, x — 0 the latter equation reduces to the adsorption isotherm of hard spheres, evaluated within the singlet Percus-Yevick approximation, whereas for xx 1 (i-S- the limit of complete association) one obtains the adsorption isotherm of tangent dimers... 

Typical forms of the radial distribution function are shown in Fig. 38 for a liquid of hard core and of Lennard—Jones spheres (using the Percus— Yevick approximation) [447, 449] and Fig. 44 for carbon tetrachloride [452a]. Significant departures from unity are evident over considerable distances. The successive maxima and minima in g(r) correspond to essentially contact packing, but with small-scale orientational variation and to significant voids or large-scale orientational variation in the liquid structure, respectively. Such factors influence the relative location of reactants within a solvent and make the incorporation of the potential of mean force a necessity. 

Tc. The two power-law exponents are not independent but depend on a single parameter, the so-called critical exponent X, which is specific for a given interaction potential (e.g., hard spheres). Actually, the interaction potential enters the MCT equations only indirectly via the structure factor S(q), which fixes the nonlinear coupling in the generalized oscillator equation. It is important to note that the MCT exponents are not universal in contrast to those of second-order phase transitions. In the case of hard spheres, for example, S(q) can be calculated via the Percus-Yevick approximation [26], and the full time and -dependence of < >(q. f) were obtained. As an example, Fig. 10 shows the susceptibility spectra of the hard-sphere system at a particular q. Note that temperature cannot be defined in the hard-sphere system instead, the packing fraction cp is used as a parameter. Above the critical packing fraction 0), which corresponds to T < Tc in systems where T exists, the a-process is absent (frozen) and only the fast dynamics is present. At cp < tpc the a-peak and the concomitant susceptibility minimum shift to lower frequencies with increasing cp, so that the closer cp is to the critical value fast dynamics can be identified (curve c in Fig. 10). 

The relation Eq. (6.56) is key to Percus s derivation of the Percus-Yevick approximation (Percus, 1964). We consider the case where the inhomogeneity is produced by the location of a distinguished particle of type v at the origin, and inquire about the surrounding fluid. The Kirkwood-Salsburg formula Eq. (6.24), p. 130, offers the interpretation of the left side of Eq. (6.56) ... 

To obtain the Percus-Yevick approximation, we neglect second-order and higher contributions, and the right side of Eq. (6.56) is evaluated as... 

Accounting for size differences can also be realized in terms of distribution functions, assuming certain interaction energies. Simply because of size differences between molecules preferential adsorption will take place, i.e. fractionation occurs near a phase boundary. In theories where molecular geometries are not constrained by a lattice, this distribution function is virtually determined by the repulsive part of the interaction. An example of this kind has been provided by Chan et al. who considered binary mixtures of adhesive hard spheres in the Percus-Yevick approximation. The theory incorporates a definition of the Gibbs dividing plane in terms of distribution functions. A more formal thermodynamic description for multicomponent mixtures has been given by Schlby and Ruckenstein ). 

In the physical picture ion-pairs are just consequences of large values of the Mayer /-functions that describe the ion distribution [22], The technical consequence, however, is a major complication of the theory the high-temperature approximations of the /-functions applied, e.g. in the mean spherical approximation (MSA) or the Percus-Yevick approximation (PY) [25], suffice in simple fluids but not in ionic systems. 

Kalyuzhnyi, Yu.V., and Cummings, P.T. Solution of the polymer Percus-Yevick approximation for the multicomponent totally flexible sticky 2-point model of polymerizing fluid. Journal of Chemical Physics, 1995, 103, No. 8, p. 3265-3267. 

Estimate the direct correlation function for liquid argon at 85 K using the hypernetted chain approximation with the data given in problem 3. Compare the result with that found using the Percus-Yevick approximation. 

In Figure 5.27 a curve calculated from Equation 5.216 is compared with the predictions of other studies. The dotted line is calculated by means of the Henderson theory. The theoretical curve calculated by Kjellander and Sarmatf for ( ) = 0.357 and h>2 by using the anisotropic Percus-Yevick approximation is shown by the dashed line the crosses represent grand canonical Monte Carlo simulation results due to Karlstrom. We proceed now with separate descriptions of solvation, depletion, and colloid structural forces. 

Relation (D.18) is often referred to as the Percus-Yevick approximation. If we use (D.18) in the Ornstein-Zernike relation, we get an integral equation for y... 

The most commonly used approach to the problem is to expand the correlation functions and their Fourier transforms in a series of orthogonal functions, usually the spherical harmonics. This approach was pioneered by Chen and Steele in the case of the Percus-Yevick approximation for hard diatomic fluids. More recently, the approach has been generalized to arbitrary... 

Vrij A. Light scattering of a concentrated multicomponent system of hard spheres in the Percus-Yevick approximation. J Chem Phys 1978 69 1742-1747. 

Hard Spheres with Surface Adhesion The Percus-Yevick Approximation and the Energy Equation ... 

THE PERCUS-YEVICK APPROXIMATION AND THE ENERGY EQUATION Table I. Critical Values of the Thermodynamic Properties... 

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