Pair correlation function numerical solutions

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... 

In the next two figures we discuss the pair-correlation functions as obtained from the two-density theory and computer simulations. First, in Fig. 3 we compare the counterion-counterion pair-distribution function as obtained theoretically (lines) and from simulations (symbols). The numerical calculations apply to cp = 0.0001 and ce = 0.005 mol dm-3 the results show that the theory underestimates the counterion-counterion correlation. Next, in Fig. 4 the macroion-counterion pair-distribution is shown for the same set of parameters. Finally, in Fig. 5 the macroion-macroion pair-distribution functions are calculated by both theoretical approaches at cp = 0.0001 mol dm-3 solution and for zp = —10 and —30. 

Figure 2.4. Dependence of the pair correlation function g[R) for the Li particles on the number density. The density p is indicated next to each curve in the dimensionless quantity pa3. We choose a — 1 and el kT—0.5 in the Li potential. All the illustrations of g(R) for this book were obtained by numerical solution of the Percus-Yevick equation. See Appendix E for more details.

One of the most important applications of the theory of PS is to biomolecules. There have been numerous studies on the effect of various solutes (which may be viewed as constituting a part of a solvent mixture) on the stability of proteins, conformational changes, aggregation processes, etc., (Arakawa and Timasheff 1985 Timasheff 1998 Shulgin and Ruckenstein 2005 Shimizu 2004). In all of these, the central quantity that is affected is the Gibbs energy of solvation of the biomolecule s. Formally, equation (8.26) or equivalently (8.28), applies to a biomolecule s in dilute solution in the solvent mixture A and B. However, in contrast to the case of simple, spherical solutes, the pair correlation functions gAS and gBS depend in this case on both the location and the relative orientation of the two species involved (figure 8.5). Therefore, we write equation (8.26) in an equivalent form as ... 

Theories based on the solution to integral equations for the pair correlation functions are now well developed and widely employed in numerical and analytic studies of simple fluids [6]. Further improvements for simple fluids would require better approximations for the bridge functions B r). It has been suggested that these functions can be scaled to the same functional form for different potentials. The extension of integral equation theories to molecular fluids was first accomplished by Chandler and Andersen [30] through the introduction of the site-site direct correlation function c r) between atoms in each molecule and a site-site Omstein-Zemike relation called the reference interaction site... 

Substitution of either the exact pair correlation functions or the solutions of the HNC or MSA equations causes all of the A to vanish. The same is true for the Aq, but only for a primitive model symmetrical electrolyte having equal ion diameters. For refined models, the quantity (A > [Eq. (175)] must vanish. Small values of A and (A ) therefore indicate good accuracy in the numerical procedures, whether or not the computed correlation functions accurately represent the assumed Hamiltonian model. 

Numerous methods for characterising the earliest stages of phase separation have been devised. All rely on sampling the data to make estimates of the local composition. The aim is to be able to identify statistically significant fluctuations in solute concentration which indicate the onset of phase separation. A comprehensive review of the key methods, composition frequency distribution, contingency table analyses, pair correlation functions,spatial distribution mapping and the local chemistry approach ° is presented in Marquis and Hyde. Pair correlations. 

The details of the derivation of the Percus-Yevick equation and the numerical procedure for its solution are highly technical and will not be presented here. [Details may be found in Appendices 9-D and 9-E and in Ben-Naim (1971c, 1972c,d).] We note, however, that each pairwise function in (6.122) depends only on three coordinates, which we can choose as follows R is the distance between the centers of the two particles, = I — RJ, and is the angle between the vector ii and the direction of Ry = Rj — Rj, measured counterclockwise. The full pair correlation function is thus a function of three variables, g(R, a2) Because of the special symmetry of the pair potential, it is clear that all of the pairwise functions, such as U, y, or g, will be invariant to a rotation of the particle... 

Baxter (1968b) showed that the Ornstein-Zernike equation could, for some simple potentials, be written as two one-dimensional integral equations coupled by a function q(r). In the PY approximation for hard spheres, for instance, the q(r) functions are easily solved, and the direct-correlation function c(r) and the other thermodynamic properties can be obtained analytically. The pair-correlation function g(r) is derived from q(r) through numerical solution of the integral equation which governs g(r) for which a method proposed by Perram (1975) is especially useful. Baxter s method can also be used in the numerical solution of more complicated integral equations such as the hypernetted-chain (HNC) approximation in real space, avoiding the need to take Fourier transforms. An equivalent set of relations to Baxter s equations was derived earlier by Wertheim (1964). 

Solutions in hand for the reference pairs, it is useful to write out empirical smoothing expressions for the rectilinear densities, reduced density differences, and reduced vapor pressures as functions of Tr and a, following which prediction of reduced liquid densities and vapor pressures is straightforward for systems where Tex and a (equivalently co) are known. If, in addition, the critical property IE s, ln(Tc /Tc), ln(PcVPc), and ln(pcVPc), are available from experiment, theory, or empirical correlation, one can calculate the molar density and vapor pressure IE s for 0.5 < Tr < 1, provided, for VPIE, that Aa/a is known or can be estimated. Thus to calculate liquid density IE s one uses the observed IE on Tc, ln(Tc /Tc), to find (Tr /Tr) at any temperature of interest, and employs the smoothing relations (or numerically solves Equation 13.1) to obtain (pR /pR). Since (MpIE)R = ln(pR /pR) = ln[(p /pc )/(p/pc)] it follows that ln(p7p)(MpIE)R- -ln(pcVpc). For VPIE s one proceeds similarly, substituting reduced temperatures, critical pressures and Aa/a into the smoothing equations to find ln(P /P)RED and thence ln(P /P), since ln(P /P) = I n( Pr /Pr) + In (Pc /Pc)- The approach outlined for molar density IE cannot be used to rationalize the vapor pressure IE without the introduction of isotope dependent system parameters Aa/a. 

In Section 9.1.1 we have introduced a stochastic model for the description of surface reaction systems which takes correlations explicitly into account but neglects the energetic interactions between the adsorbed particles as well as between a particle and a metal surface. We have formulated this by master equations upon the assumption that the systems are of the Markovian type. In the model an infinite set of master equations for the distribution functions of the state of the surface and of pairs of surface sites (and so on) arise. This chain of equations cannot be solved analytically. To handle this problem practically this hierarchy was truncated at a certain level. The resulting equations can be solved numerically exactly in a small region and can be connected to a mean-field solution for large distances from a reference point. 

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