Yvon-Born-Green equation

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)... 

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... 

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. 

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]. 

Differentiation gives the force between the large spheres. Thus, the force between two colloidal spheres can be calculated from Eq. (71) and some equation like Eq. (70) to yield the pressure [Lozada-Cassou uses what is called the Born-Green-Yvon equation, see Eq (97) below] or from Eqs. (72) and (76). The bridge between the two methods is the Derjaguin approximation, Eq. (64). In principle, either scheme is acceptable. The few calculations made so far suggest that Eqs. (72) and (76) give more accurate results. 

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... 

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. 

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... 

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. 

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. 

Finally, we mention that very recently three other integral equation approaches to treating polymer systems have been proposed. Chiew [104] has used the particle-particle perspective to develop theories of the intermolecular structure and thermodynamics of short chain fluids and mixtures. Lipson [105] has employed the Born-Green-Yvon (BGY) integral equation approach with the Kirkwood superposition approximation to treat compressible fluids and blends. Initial work with the BGY-based theory has considered lattice models and only thermodynamics, but in principle this approach can be applied to compute structural properties and treat continuum fluid models. Most recently, Gan and Eu employed a Kirkwood hierarchy approximation to construct a self-consistent integral equation theory of intramolecular and intermolecular correlations [106]. There are many differences between these integral equation approaches and PRISM theory which will be discussed in a future review [107]. 

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 ... 

The Born-Green-Yvon (BGY) integral equation approach was used [16] for dilute electrolyte solution, with a modification which ensures electroneutrality (BGY-i-EN). Ions were taken as hard charged spheres of the same diameter, corresponding to the closest ionic approach distance and to twice the closest approach distance to the interface (restricted pritive model, see Figure 2.5). 

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. 

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). 

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

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

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

The history of the search for an integral equation for the pair correlation function is quite long. It probably started with Kirkwood (1935), followed by Yvon (1935, 1958), Born and Green (1946), and many others. For a summary of these efforts, see Hill (1956), Fisher (1964), Rushbrooke (1968), Munster (1969), and Hansen and McDonald (1976). Most of the earlier works used the superposition approximation to obtain an integral equation for the pair correlation function. It was in 1958 that Percus and Yevick developed an integral equation that did not include explicitly the assumption of superposition, i.e., pairwise additivity of the higher order potentials of mean force. The Percus-Yevick (PY) equation was found most useful in the study of both pure liquids as well as mixtures of liquids. 

In the following discussions, we use the expressions for gjj and g derived independently by Born and Green, Yvon, Kirkwood, and Bogoliubov in various different forms. These basically equivalent hierarchy of equations (sometimes known as either the BGY or the BBGKY hierarchy) can be expressed as an infinite set of the following integrodifferential equations N- 00) ... 

It will be noted that the procedure used in obtaining Eq. (9) is identical with that followed in obtaining the hierarchy of equations of Bogoliubov, Born and Green, Kirkwood, and Yvon. There, however, the concern is with a distribution function / of low order, referring to one or a small number of molecules, while here the system of interest is of macroscopic size. Actually Eq. (9) could have been written down immediately, since it is known to be the form of the Liouville equation in the presence of non-conservative forces. (Cf. the discussion by Whitaker on integral invariants.)... 

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