Density direct correlation function

Most integral equations are based on the Ornstein-Zernike (OZ) equation [3-5]. The idea behind the OZ equation is to divide the total correlation function h ri2) iiito a direct correlation function (DCF) c r 12) that describes the fact that molecules 1 and 2 can be directly correlated, and an indirect correlation function 7( 12), that describes the correlation of molecule 1 with the other molecules that are also correlated with molecule 2. At low densities, when only direct correlations are possible, 7(r) = 0. At higher densities, where only triplet correlations are possible, we can write... 

Let us begin our discussion from the model of Cummings and Stell for heterogeneous dimerization a + P ap described in some detail above. In the case of singlet-level equations, HNCl or PYl, the direct correlation function of the bulk fluid c (r) represents the only input necessary to obtain the density profiles from the HNCl and PYl equations see Eqs. (6) and (7) in Sec. II A. It is worth noting that the transformation of a square-well, short-range attraction, see Eq. (36), into a 6-type associative interaction, see Eq. (39), is unnecessary unless one seeks an analytic solution. The 6-type term must be treated analytically while solving the HNCl... 

The density functional approach of Refs. 91, 92 introduces a correction to the wall-particle direct correlation function resulting from the HNCl approximation (see Eqs. (32)-(34)). A correction to Eq. (34) reads (we drop the species label because the model is one-component)... 

We apply the singlet theory for the density profile by using Eqs. (101) and (103) to describe the behavior of associating fluids close to a crystalline surface [120-122], First, we solve the multidensity OZ equation with the Percus-Yevick closure for the bulk partial correlation functions, and next calculate the total correlation function via Eq. (68) and the direct correlation function from Eq. (69). The bulk total direct correlation function is used next as an input to the singlet Percus-Yevick or singlet hypernetted chain equation, (6) or (7), to obtain the density profiles. The same approach can be used to study adsorption on crystalline surfaces as well as in pores with walls of crystalline symmetry. 

The equlibrium between the bulk fluid and fluid adsorbed in disordered porous media must be discussed at fixed chemical potential. Evaluation of the chemical potential for adsorbed fluid is a key issue for the adsorption isotherms, in studying the phase diagram of adsorbed fluid, and for performing comparisons of the structure of a fluid in media of different microporosity. At present, one of the popular tools to obtain the chemical potentials is an approach proposed by Ford and Glandt [23]. From the detailed analysis of the cluster expansions, these authors have concluded that the derivative of the excess chemical potential with respect to the fluid density equals the connected part of the fluid-fluid direct correlation function (dcf). Then, it follows that the chemical potential of a fluid adsorbed in a disordered matrix, p ), is... 

The simplest choice for Fex is to approximate the nonuniform fluid direct correlation function in Eq. (47) with the uniform fluid direct correlation at some bulk density, that is, to set... 

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 ... 

One of the few attempts to tackle the problem of ionic criticality more quantitatively was made by Hafskjold and Stell in 1982 [36], and was later taken up by H0ye and Stell [17, 302, 303]. Based on a comparative analysis of the correlation functions for nonionic and ionic fluids, these authors asserted that the critical point of the RPM is Ising-like. To this end, they argued that the density-density correlation function hpp(r) and the associated direct correlation function cpp(r) obey essentially the same OZ equation and closure as that of a single-component, nonionic fluid. It was assumed that this analogy suffices to ensure that the critical exponents are Ising-like. 

Another factor that contributes to the decoupling is the two particle direct correlation function. The product cn q)F q, t) defines the modified structure of the solvent probed by the solute. The value of the direct correlation function is less for smaller solutes at all wavevectors. The smaller the value of the two-particle direct correlation function, the lesser will be the contribution of the density mode to the total friction. 

Deeper insight into the consequences of counterion condensation is gained by an effective monomer-monomer and counterion-counterion potential, respectively. The idea is to reduce the multicomponent system (macromolecules + counterions) to effective one-component systems (macromolecules or counterions, respectively). We define the simplified model in such a way that the effective potential between the counterions or monomers, respectively, of the new system yields exactly the same correlation function (gcc, gmm) as found in the multicomponent case at the same density. Starting from the correlation function gcc -respectively gmm-of the multi-component model we calculate an effective direct correlation function cefy via the one-component Ornstein-Zernike equation. An effective potential is then obtained from the RLWC closures of the one- and multicomponent models [24]. For low and moderate densities the effective potential is well approximated by... 

In this framework, we present the repercussions on the physical properties of a renormalized indirect correlation function y (r) conjugated with an optimized division scheme. All the units are expressed in terms of the LJ parameters, that is, reduced temperature T = kBT/e and reduced density p = pa3. In order to examine the consequences of a renormalization scheme, the direct correlation function c(r) calculated from ZSEP conjugated with DHH splitting is compared in Fig. 7 to those obtained with the WCA separation. For high densities, the differences arise mainly in the core region for y(r) and c(r) [77]. These calculated quantities are in excellent agreement with simulation data. The reader has to note that similar results have been obtained with the ODS scheme (see Ref [80]). Since the acuracy of c(r) can be affected by the choice of a division scheme, the isothermal compressibility is affected too, as can be seen in Table III for the pkBTxT quantity. As compared to the values obtained with... 

Practically, the values of f c(r, p )r2dr are fitted in a wide range of densities by a polynomial of order 3 in density. Then, ppex is obtained by analytically integrating the resulting polynomial as a function of density. It should be stressed that this method involves only the direct correlation function c(r), but neither B (r) nor B r), which are known to be the keys of the IETs. It must be stressed that such a thermodynamic integration process is performed along an isotherm T. This method is only accurate for supercritical temperatures, but is not at all for lower temperatures. Furthermore, it is not adapted to a predictive scheme. 

Since experiments for Kr have been performed at small angle neutron scattering for some low density states, we present the results of the Fourier transform of the direct correlation function, c(q) = (S(q) — 1 )/p.S (7/), rather than those of the structure factor S(q). Figure 20 shows the curves of c q). As it can be seen, the theoretical results, obtained by HMSA+WCA and MD with the AS plus AT potentials, are in excellent agreement with the experimental data [12]. While the AT contribution is included by means of an effective pair potential in the SCIET, it is used under its original form owing to Eqs.(l 19) and... 

If the structure is decided by an effective potential ues (r), it was demonstrated in the mean spherical approximation (MSA) that the direct correlation function c(r) should rapidly approach — pMerr (r) for large r (see Section IE). According to Reatto and Tau [131], this relationship, which is asymptotically exact for large distance and low density, holds quite well when the long-range dispersion term of the AS potential, - Cg / r6, and the AT triple-dipole potential, < m3 (r) > (8n/3)vp/r6, are considered, so that the direct correlation function reads... 

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 theory reflects the solvent properties through the thermody-namic/hydrodynamic input parameters obtained from the accurate equations of state for the two solvents. However, the theory employs a hard sphere solute-solvent direct correlation function (C12), which is a measure of the spatial distribution of the particles. Therefore, the agreement between theory and experiment does not depend on a solute-solvent spatial distribution determined by attractive solute-solvent interactions. In particular, it is not necessary to invoke local density augmentation (solute-solvent clustering) (31,112,113) in the vicinity of the critical point arising from significant attractive solute-solvent interactions to theoretically replicate the data. 

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