Interaction Percus-Yevick

We will describe integral equation approximations for the two-particle correlation fiinctions. There is no single approximation that is equally good for all interatomic potentials in the 3D world, but the solutions for a few important models can be obtained analytically. These include the Percus-Yevick (PY) approximation [27, 28] for hard spheres and the mean spherical (MS) approximation for charged hard spheres, for hard spheres with point dipoles and for atoms interacting with a Yukawa potential. Numerical solutions for other approximations, such as the hypemetted chain (EfNC) approximation for charged systems, are readily obtained by fast Fourier transfonn methods... 

Ben-Naim (1972b, c) has examined hydrophobic association using statistical mechanical theories of the liquid state, e.g. the Percus-Yevick equations. He has also examined quantitative aspects of solvophobic interactions between solutes using solubility data for ethane and methane. The changes in thermodynamic parameters can be calculated when two methane molecules approach to a separation of, 1-533 x 10-8 cm, the C—C distance in ethane, and the solvophobic quantities 8SI/i, s 2 and 8SiS2 can be calculated. In water (solvophobic = hydrophobic) 5si/i is more negative than in other solvents and decreases as the temperature rises both 8s iH%... 

Another possible approach solving the equilibrium distribution for an electric double layer is offered by integral equation theories [22]. They are based on approximate relationships between different distribution functions. The two most common theories are Percus-Yevick [23] and Hypernetted Chain approximation (HNQ [24], where the former is a good method for short range interactions and the latter is best for long-range interactions. They were both developed around 1960, but are still used. The correlation between two particles can be divided into two parts, one is the direct influence of particle j on particle i and the other originates from the fact that all other particles correlate with particle j and then influence particle i in precisely... 

An approximate functional for Ap, is derived in Matubayasi et al. [15,16] and Takahashi et al. [19]. The functional is constructed by adopting the Percus-Yevick-type approximation in the unfavorable region of the solute-solvent interaction and the hypernetted-chain-type approximation in the favorable region. Ap, is then given... 

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). 

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 ). 

Standard approximate methods, e.g., the Percus-Yevick or hyper-chain approximations, are applicable for systems with the Gibbs distribution and are based on the distinctive Boltzmann factor like exp —U r)/ ksT)), where U(r) is the potential energy of interacting particles. The basic kinetic equation (2.3.53) has nothing to do with the Gibbs distribution. The only approximate method neutral with respect to the ensemble averaging is the Kirkwood approximation [76, 77, 87]. 

Using the hard sphere adhesive state equation proposed by Baxter (16), it is possible to calculate the demixing line due to interactions. This state equation corresponds to the exact solution of the Percus-Yevick equation in the case of an hard sphere potential with an infinitively thin attractive square well. In our calculation we assum that the range of the potential is short in comparison to the size of the particles (in fact less than 10 %). 

This result is known as the Percus-Yevick (PY) approximation [17]. It has been applied with considerable success to the evaluation of the properties of a liquid composed of hard spheres. In this case, the interaction energy u(r) is zero so that... 

In order to complete the MSA estimate of Iny,- one must add the hard-sphere contribution, which accounts for the fact that work must be done to introduce the ions as hard spheres into the solution. It is obtained from the Percus-Yevick model for non-interacting hard spheres. For the case that all ions (spheres) have the same radius, the result is (see equation (3.9.22))... 

Both closures have been employed for determining the distribution functions for liquids 182 the Percus-Yevick equation tends to yield better results for nonpolar systems, while the hypemetted-chain equation (with the appropriate renormalization of long-range interactions)113,183 is found to be more appropriate for polar and ionic liquids.184... 

Polydispersity plays an important role for S(q) as well [63]. Here a discussion of the alterations effected to S(q) in the case of hard sphere interaction will suffice. For this purpose S(q) of a system of hard spheres maybe obtained from the Percus-Yevick theory [66] generalized by Vrij and coworkers [67,68] to polydis-perse systems. The solid line in Fig. 6 displays S(q) resulting for a system of hard spheres with a Gaussian size distribution characterized by a standard deviation of 7.5%. The main feature induced by polydispersity is the much weaker side minimum of S(q) as compared to the monodisperse case. Hence, a finite width of the size distribution will tend to smear out the oscillations of S(q) at higher q. 

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]. 

To conclude this section, I would like to add two sets of results on the pair correlation function (PCF) between two simple solutes in an L/ solvent. These results may or may not be relevant to the problem of Hhard-sphere (particles labeled A) solutes in an LJ solvent (particles labeled B) with varying strength of the energy parameter sbb- All the calculations for this and the subsequent demonstration were done by solving the Percus-Yevick integral equations with the following molecular... 

At least in the case of liquid simple metals, a knowledge of the effective pan-potentials describing the interaction between the ions in the liquid metal can also be utilized to calculate g(r) and A K). The most common such method involves the assumption of a hard-sphere potential in the Percus-Yevick (PY) equation its solution provides the hard-sphere structure factor, /4hs( C). (See Ashcroft and Lekner 1966.) The two parameters that must be provided for a calculation of Ahs( ) are the hard-sphere diameter, a, and the packing fraction, x. It is found that j = 0.45 for most liquid metals at temperatures just above their melting points. A hard-sphere solution of the PY equation has also been obtained for binary liquid metal alloys, and provides estimates of the three partial structure factors describing the alloy structure (Ashcroft and Langreth 1967). To the extent that the hard-sphere approximation appears to be valid for the liquid R s, pair potentials should dominate these metals also, at least at short distances. 

The next step in the procedure of evaluation of the mixture properties is the evaluation of the pseudo-radial distribution functions for all i — j interactions in the mixture as well as the mean free-path parameter atj for the unlike interaction. It is consistent with the remainder of the procedure to estimate them from mixing rules based upon a rigid-sphere model. Among the many possible mixing rules for the radial distribution function one that has proved successful is based upon the Percus-Yevick equation for the radial distribution function of hard-sphere mixtures (Kestin Wakeham 1980 Vesovic ... 

FH = Flory-Huggins GF = generalized Flory GFD = generalized Flory dimer HNC = hypemetted chain HTA = high temperature approximation IFJC = ideal freely joined chain ISM = interaction site model LCT = lattice cluster theory MS = Martynov-Sarkisov PMMA = polymethyl methacrylate PRISM = polymer reference interaction site model PVME = polyvinylmethylether PS = polystyrene PY = Percus-Yevick RMMSA = reference molecule mean spherical approximation RMPY = reference molecular Percus-Yevick SANS = small angle neutron scattering SFC = semiflexible chain TPT = thermodynamic perturbation theory. 

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