Percus-Yevick relation

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

The general equation can be further reduced to the case of infinite dilution limit, a binary mixmre, ionic solutions, and so on. These equations are supplemented by closure relations such as the Percus-Yevick (PY) and hypernetted chain (HNC) approximations. 

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... 

To solve the replica OZ equations, they must be completed by closure relations. Several closures have been tested against computer simulations for various models of fluids adsorbed in disordered porous media. In particular, common Percus-Yevick (PY) and hypernetted chain approximations have been applied [20]. Eq. (21) for the matrix correlations can be solved using any approximation. However, it has been shown by Given and Stell [17-19] that the PY closure for the fluid-fluid correlations simplifies the ROZ equation, the blocking effects of the matrix structure are neglected in this... 

When supplemented with a closure relation, Eq. (7) can be solved for h r) and c r). For example, the Percus-Yevick (PY) closure is given by [89]... 

An efficient method of solving the Percus-Yevick and related equations is described. The method is applied to a Lennard-Jones fluid, and the solutions obtained are discussed. It is shown that the Percus-Yevick equation predicts a phase change with critical density close to 0.27 and with a critical temperature which is dependent upon the range at which the Lennard-Jones potential is truncated. At the phase change the compressibility becomes infinite although the virial equation of state (foes not show this behavior. Outside the critical region the PY equation is at least two-valued for all densities in the range (0, 0.6). 

From the various possible closures, the mean spherical approximation (MSA) [189] has found particularly wide attention in phase equilibrium calculations of ionic fluids. The Percus-Yevick (PY) closure is unsatisfactory for long-range potentials [173, 187, 190]. The hypemetted chain approximation (HNC), widely used in electrolyte thermodynamics [168, 173], leads to an increasing instability of the numerical algorithm as the phase boundary is approached [191]. There seems to be no decisive relation between the location of this numerical instability and phase transition lines [192-194]. Attempts were made to extrapolate phase transition lines from results far away, where the HNC is soluble [81, 194]. 

A very popular closure relation is the Percus-Yevick (PY) approximation [39]. For a generic potential u r), this approximation assumes that... 

One more relation is required to achieve closure, i.e., to determine the two types of correlation functions. The most commonly used relations are the Percus-Yevick (PY) and the hypernetted chain (HNC) approximations [47-49]. From graph or diagram expansion of the total correlation function in powers of the density n(r) and resummation, an exact relation between the total and direct correlation functions is obtained, namely... 

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) ... 

In Eq. (9), the correlation functions for macroparticles do not appear on the RHS of this equation. Thus, applying a particular closure for the macroparticle correlations need not relate to the closures used for the correlation functions of the suspending fluid on the RHS of Eq. (9). This means, that using the Percus-Yevick (PY) approximation to describe the correlations between macrospheres... 

For separations outside the hard core, the direct correlation functions have to be approximated. Classic closure approximations recently applied to QA models axe the Percus-Yevick (PY) closure [301], the mean spherical approximation (MSA) [302], and the hypernetted chain (HNC) closure [30]. None of these relations, when formulated for the replicated system, contains any coupling between different species, and wc can directly proceed to the limit n — 0. The PY closure then implies... 

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... 

To generate the Percus-Yevick (PY) closure relation, we consider the graphical expansion for 02(1 >2) obtained from Eq. (2.1.26). Each graph in this expansion occurs both with and without a/2(l, 2) bond, so we can factor out (1 -l-/2(l,2)) = e2(l,2). This gives... 

Here we have included a density factor of in the Fourier transforms of the site-site correlation functions h (r) and which is more convenient for mixture calculations. is the interamolecular distance between sites a and rj. For mixtures, = 0 when sites a and t] are in molecules of different species. The SSOZ equation simply relates the total correlation functions h (k) to the direct correlation functions c, (/c). A second relation is required to obtain a closed system of equations. Two of the commonly used closure relations are the Percus-Yevick (PY) and hypernetted chain (HNC) approximations. The PY closure is ... 

The relation is referred to as the hypernetted-chain (HNC) approximation. Further linearizing expt(r, r ) in 1.23, one has the Percus-Yevick (PY) approximation,... 

In the calculation, a model of the averaged structure factor for a hard-sphere (HS) interaction potential, S(g) is used [47, 48], which considers the Gaussian distribution of the interaction radius cr for individual monodisperse systems for polydispersity m, and a Percus-Yevick (PY) closure relation to solve Omstein-Zernike (OZ) equation. The detailed theoretical description on the method has been reported elsewhere [49-51]. 

Once potential parameters have been determined, we can start calculation downward following arrow in the figure. The first key quantity is radial distribution function g(r) which can be calculated by the use of theoretical relation such as Percus-Yevick (PY) or Hypemetted chain (HNC) integral equation. However, these equations are an approximations. Exact values can be obtained by molecular simulation. Ifg(r) is obtained accurately as functions of temperature and pressure, then all the equilibrium properties of fluids and fluid mixtures can be calculated. Moreover, information on fluid structure is contained in g(r) itself. 

The Percus-Yevick (PY) equation for spherical particles in three dimensions was found to be very useful for the study of the pair correlation function. For the purposes of the present 2-D system, the corresponding PY equation may be obtained most directly from the Ornstein-Zernike relation ... 

There are three approximate theories of the liquid state in frequent use (see, for example, Enderby and March (1965)). Their common feature is that they attempt to relate the radial distribution function g(r) to the interatomic pair potential 0(r). For convenience we list the theories and the relevant equations below as applied to a pure liquid. The generalisation to include multi-component liquids is straightforward Percus-Yevick (PY) ... 

The expectation that C(r) is a short range, relatively simple function can be exploited to develop a second approximate relationship, or dosure relation, between h(r) and C(r). Based on graph theoretical techniques [5] or functional expansions [5,7], one can deduce the Percus-Yevick approximation ... 

The adaptation of the Percus-Yevick approximation starts with the three Omstein-Zemike equations which relate h , h, and hi, to the set of direct functions Cu, c.h> and Cm, in a homogeneous binary mixture of molecules a and b, which have hard cores but otherwise unspecified pair potentials. The limit is now taken in which the radius of the hard core of b becomes infinite and its concentration goes almost to zero, so that the system comprises a fluid of a molecules in contact with the flat wall of the one remaining b molecule. Only two Omstein-Zemike equations remain, one for h and one for the molecule-wall correlation, These are solved by using the Percus-Yevick approximation,... 

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