Percus-Yevick theory

Henderson D, Sokolowski S, Wasan DT (1997) Second order Percus-Yevick theory for a confined hard sphere fluid. J Stat Phys 89 233-247... 

Lots of ideas, many suboptimal, coexisted prior to the availability of the clear data that simulations provided. It was less that the simulations suggested new ideas than that the new simulation data served to alleviate the confusion of unclear ideas and to focus effort on the fruitful approaches. The theory of simple liquids treated by those simulations promptly made progress that we recognize, from our historical vantage point, as permanent. For example, the Percus-Yevick theory, proposed in 1957, was solved analytically for the hard-sphere case in 1963 (Wertheim, 1963 Thiele, 1963). 

Percus-Yevick theory giving the approximate PY of a hard sphere fluid... 

Packing fractions are conveniently measured in relative separations = (- 0c)/ to the glass transition point, which for this model of hard spheres lies at 0c = 0.516 [38, 72]. Note that this result depends on the static structure factor 5 (g), which is taken from Percus-Yevick theory, and that the experimentally determined value 0expt. somewhat higher [13, 14]. The wavevector integrals were dis-... 

The equation of state and free energy of a system of hard spheres with surface adhesion are calculated from the internal energy of the fluid as given by the Percus-Yevick theory. A first-order phase change occurs. Further, the liquid-vapor coexistence curve, which cannot be found by the more usual routes to the equation of state, is calculated. It is found that the equation of state exhibits van der Waals type sigmoid isotherms in the region in which the Percus-Yevick theory has solutions. 

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. 

As a last physical approach we mention, but do not further consider, the scaled-particle-theory (SPT) which was developed about the same time as the Percus-Yevick theory. It gives good results for the thermodynamic properties of hard molecules (spheres or convex molecules). It is not a complete theory (in contrast to the integral equation and perturbation theories) since it does not yield the molecular distribution functions (although they can be obtained for some finite range of intermolecular separations). 

Lamperski, S. and Outhwaite, C.W., A non primitive model for the electrode/electrolyte interface based on the Percus-Yevick theory, J. Electroanal. Chem., 460, 135-143, 1999. 

The interpolation between the low and high density limits, which is inherent to this variational approach, leads in a very natural way to the scaled particle theory for the structure and thermodynamics of isotropic fluids of hard particles. This unifies, for the first time the Percus Yevick theory, which is based on diagram expansions, and the scaled particle theory of Reiss, Frisch and Lebowitz, and, at the same time yields the analytical expressions of the dcf conformal to those of the hard spheres. It provides an unified derivation of the most comprehensive analytic description available of the hard sphere thermodynamics and pair distribution functions as given by the Percus Yevick and scaled particle theories, and yields simple explicit expressions for the higher direct direct correlation functions of the uniform fluid. 

Faraday Soc. Discussion (1978). Chandler (1978 and 1982) has reviewed some of this work. Specific examples are Lowden and Chandler (197 ), Hsu et al. (1976) and Hsu and Chandler (1978). In its simplest version, that is, when it is applied to single site spherical particles, the RISM theory for hard core molecules reduces to the Percus-Yevick theory for the hard sphere fluid. 

When the smoothed or nonlocal density approximation (or NL-DFT model) is used, the weighting function is chosen so that the hard-sphere direct pair-correlation function is well described for the uniform fluid over a wide range of densities. One example of such a weighting function is the model proposed by Tarazona [69], which uses the Percus-Yevick theory for approximating the correlation function over a wide range of density. In this case, the weighting function is expanded as a power series of the smoothed density. The use of a smoothed density in NL-DFT provides an oscillating density profile expected of a fluid adjacent to a sohd surface, the existence of which is corroborated by molecular simulation results [17,18]. 

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... 

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. 

Scattering and Disorder. For structure close to random disorder the SAXS frequently exhibits a broad shoulder that is alternatively called liquid scattering ([206] [86], p. 50) or long-period peak . Let us consider disordered, concentrated systems. A poor theory like the one of Porod [18] is not consistent with respect to disorder, as it divides the volume into equal lots before starting to model the process. He concludes that statistical population (of the lots) does not lead to correlation. Better is the theory of Hosemann [158,211], His distorted structure does not pre-define any lots, and consequently it is able to describe (discrete) liquid scattering. The problems of liquid scattering have been studied since the early days of statistical physics. To-date several approximations and some analytical solutions are known. Most frequently applied [201,212-216] is the Percus-Yevick [217] approximation of the Ornstein-Zernike integral equation. The approximation offers a simple descrip-... 

Roe and co-workers (Nojima et al. 1990 Rigby and Roe 1984,1986 Roe 1986) have used SAXS to characterize micelles formed by PS-PB diblocks at low concentrations in blends with low-molecular-weight PB. Micellar dimensions and association numbers were determined for symmetric and asymmetric diblocks (Rigby and Roe 1984,1986). The effective hard sphere radius, the core radius and volume fraction of hard spheres were determined using the Percus-Yevick model (Rigby and Roe 1986). These results were compared (Roe 1986) to the predictions of the theory of Leibler et al (1983). The theory qualitatively reproduced the observed trend for the cmc to increase with temperature for blends containing a particular diblock. The cmc was found to decrease at a fixed temperature... 

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... 

Dabei scheint fiir Metalle die Bom-Green-Theorie besser zu stimmen, als die von Percus-Yevick. Der Zusammenhang zwischen der Struktur ge-schmolzener Elemente und der Struktur der festen Phase wird besprochen in (158). 

Figure 7.12 Excess chemical potential of the hard-sphere fluid as a function of density. The open and filled circles correspond to the predictions of the primitive quasi-chemical theory and the self-consistent molecular field theory, respectively. The solid and dashed lines are the scaled-particle (Percus-Yevick compressibility) theory and the Carnahan-Starling equation of state, respectively (Pratt and Ashbaugh, 2003).

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 ). 

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