Yvon-Born-Green equation

The Born-Green-Yvon equation can be derived from a simple force-balance argument. First we write 

The Born-Green-Yvon (BGY) integral equation of state [24] was derived and shown to fit PVT experimental data for both small molecules and polymers. The fit parameters were then used to make predictions about thermodynamic properties for the system of interest. In the athermal limit the BGY treatment is equivalent to Guggenheim s approximation of random mixing. The EOS can be written as follows  

After Bogoliubov, Born, Green, Kirkwood, and Yvon, all of whom derived fEquation (172) is called the first hierarchy equation, Eq. (174) the second, etc. tThe virial expansion of the pressure is well-known example of such a series. 

Bardeen potential, 340, 341 Bjerrum parameter, 112 Born-Green-Yvon equation, 22-24, 122 

The Born-Green-Yvon (BGY) hierarchy is truncated by introducinga super position approximation = correlation function 1234. The resulting pair of two simultaneous integral equations including the triplet- and pair-correlation functions 423 S i2> which hereafter 

Under the superposition approximation, the above force-balance expression becomes the Born-Green-Yvon (BGY) equation 

The remaining approximations of liquid phase incompressibility and a discontinuous mass distribution can be removed through the use of the Yvon-Born-Green (YBG) equation (3), which is simply a 

A second, entirely different class of new polymer integral equation theories have been developed by Lipson and co-workers, Eu and Gan, " and Attard based on the site-site version of the Born-Green-Yvon (BGY) equation. The earliest work in this direction was apparently by Whittington and Dunfield, but they addressed only a special aspect of the isolated polymer problem (dilute solution). The central quantity in the BGY approaches is the formally exact expressions that relate two and three (or more) intramolecular and intermolecular distribution functions. The generalized site-site Ornstein-Zernike equations and direct correlation functions do not enter. In the BGY schemes the closure approximation(s) enter as approximate relations between the two- and three-body distribution functions supplemented with exact normalization and asymptotic conditions. In the recent BGY work of Taylor and Lipson a four-point distribution function also enters. 

In this section ve consider the manner in which the radially dependent density distribution, p(r), of the equilibrium droplet can be obtained from the Yvon>Born-Green equation (3) 

One of the important methods of developing a model for the potential energy of the liquid system is that based on integral equations. These include the Kirkwood integral equation, the Born-Green-Yvon equation, and the Omstein-Zernike equation. The last approach leads to the definition of the direct correlation function c(fi2). It is the approach which is most frequently used and is the one which is considered here. 

Sec. 4 is concerned with the development of the theory of inhomogeneous partly quenched systems. The theory involves the inhomogeneous, or second-order, replica OZ equations and the Born-Green-Yvon equation for the density profile of adsorbed fluid in disordered media. Some computer simulation results are also given. 

In contrast to the case of the OZ equation, where only a closure is needed, for the OZ2 equation we need a closure and a relation between the pair functions h( 1, 2) and c( 1, 2) and the singlet function p(l). There are at least two possibilities. One is the Born-Green- Yvon (BGY) equation 

See, e.g., D. C. Grahame, The electrical double layer and the theory of electrocapillarity, Chem. Rev. 41 441 (1947) C. W. Outhwaite, Modihed Poisson-Boltzmann equatioh in electric double layer theory based on the Bogoliubov-Born-Green-Yvon integral equations, J.C.S. Faraday 7/74 1214 (1978) and the references cited therein. 

To the best of our knowledge, there was only one attempt to consider inhomogeneous fluids adsorbed in disordered porous media [31] before our recent studies [32,33]. Inhomogeneous rephca Ornstein-Zernike equations, complemented by either the Born-Green-Yvon (BGY) or the Lovett-Mou-Buff-Wertheim (LMBW) equation for density profiles, have been proposed to study adsorption of a fluid near a plane boundary of a disordered matrix, which has been assumed uniform in a half-space [31]. However, the theory has not been complemented by any numerical solution. Our main goal is to consider a simple model for adsorption of a simple fluid in confined porous media and to solve it. In this section we follow our previously reported work [32,33]. 

To complete our theoretical treatment of liquids, we need a procedure to calculate g(r). It so happens that there is no exact equation for g(r), but there are several accurate approximate equations. Four equations that have had some success for fluids are the Kirkwood equation, the Born-Green-Yvon equation, the Percus-Yevick equation and the Hypernetted chain equation (McQuarrie, 1976). All four of these equations are integral equations for g(r) in terms of u(r). 

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