Perturbation theory of fluids

The term perturbation theory is very general, but when applied to the theory of fluids it usually means a theory that separates the intermolecular potential into two parts. 

In perturbation theory calculations, the properties of the reference system are usually known in advance. In particular, sdo (value of sd for the reference system) and go(xi, X2) (the function g for the reference system) are assumed to be known. 

In Eqs. (23)-(25) we see how the separation of the potential leads naturally to the definition of fo and rp bonds. Equations (26) and (27) then express si and g in terms of these new bonds. Our object is to collect together all terms in these series that are of zeroth order in all of first order, etc. We also wish to transform them so that they no longer look like expansions in powers of the density. One step that helps to accomplish these goals is to perform a topological reduction using an ho bond to eliminate the fo bond, as we did in Section 5.3. 

Vsi = Ks/o+sum of all topologically different irreducible graphs that have no root points, two or more field points, and at most one ho and any number of q bonds between any pair of points at least one q bond and no pair of reducible points with a residual that, when regarded as a graph with two roots, has only ho bonds and one or more field points (68) 

In the series for si there are only two diagrams with one p bond. They each have two field points. One diagram has an ho bond and the other does not. If we neglect diagrams with two or more p bonds, we obtain the following approximation  

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In perturbation theories of fluids, the pair total potential is divided into a reference part and a perturbation... 

The properties of a fluid are determined largely by short-range repulsive forces. The long-range attractive forces can be considered to be perturbations. Using these concepts, a perturbation theory of fluids is developed. In addition, the relationship of empirical equations of state to the perturbation theory is examined. The major weakness of most empirical equations is the use of the van der Waals free-volume term, (V-Nb) 1, to represent the contributions of the repulsive forces. Replacement of this term by more satisfactory expressions results in better agreement with experiment. 

F. Hirata and K. Arakawa, The computation of the thermodynamic properties aqueous electrolyte solutions by means of the perturbation theory of fluids, BuU. Chem. Soc. Jpn. 48, 2139 (1975). 

This volume begins with a chapter on modem cluster methods in equilibrium statistical mechanics and shows how topological reduction can be used to renormalize bonds. A general discussion of renormalization methods is given and the formalism is applied to the study of polar gases, ionic solutions, perturbation theory of fluids, hydrogen-bonded fluids, and integral equations. 

Perhaps the simplest application of the statistical mechanical perturbation theory of fluids is a derivation of the van der Waals equation. To derive the van der Waals equation, we first write the two-body intermolecular potential as the summation of a hard sphere part Ujjg(r) and an attractive part u r),... 

These equations provide a convenient and accurate representation of the themrodynamic properties of hard spheres, especially as a reference system in perturbation theories for fluids. 

Kirkwood derived an analogous equation that also relates two- and tlnee-particle correlation fiinctions but an approximation is necessary to uncouple them. The superposition approximation mentioned earlier is one such approximation, but unfortunately it is not very accurate. It is equivalent to the assumption that the potential of average force of tlnee or more particles is pairwise additive, which is not the case even if the total potential is pair decomposable. The YBG equation for n = 1, however, is a convenient starting point for perturbation theories of inliomogeneous fluids in an external field. 

As we have already pointed out, the theoretical basis of free energy calculations were laid a long time ago [1,4,5], but, quite understandably, had to wait for sufficient computational capabilities to be applied to molecular systems of interest to the chemist, the physicist, and the biologist. In the meantime, these calculations were the domain of analytical theories. The most useful in practice were perturbation theories of dense liquids. In the Barker-Henderson theory [13], the reference state was chosen to be a hard-sphere fluid. The subsequent Weeks-Chandler-Andersen theory [14] differed from the Barker-Henderson approach by dividing the intermolecular potential such that its unperturbed and perturbed parts were associated with repulsive and attractive forces, respectively. This division yields slower variation of the perturbation term with intermolecular separation and, consequently, faster convergence of the perturbation series than the division employed by Barker and Henderson. 

The theoretical techniques used most frequently in this context are perturbation theories of the type discussed in Section III.C based upon spherical reference systems. Thompson et al. focus upon perturbation expansions of the molecular density, p(r, pair potential or the Mayer function to parametrize the expansion. If the pair potential is used, then to first order they obtain... 

The statistical-associated fluid theory (SAFT) of Chapman et al. [25, 26] is based on the perturbation theory of Wertheim [27]. The model molecule is a chain of hard spheres that is perturbed with a dispersion attractive potential and association potential. The residual Helmholtz energy of the fluid is given by the sum of the Helmholtz energies of the initially free hard spheres bonding the hard spheres to form a chain the dispersion attractive potential and the association potential,... 

The perturbed-hard-ehain (PHC) theory developed by Prausnitz and coworkers in the late 1970s was the first successful application of thermodynamic perturbation theory to polymer systems. Sinee Wertheim s perturbation theory of polymerization was formulated about 10 years later, PHC theory combines results fi om hard-sphere equations of simple liquids with the eoneept of density-dependent external degrees of fi eedom in the Prigogine-Flory-Patterson model for taking into account the chain character of real polymeric fluids. For the hard-sphere reference equation the result derived by Carnahan and Starling was applied, as this expression is a good approximation for low-molecular hard-sphere fluids. For the attractive perturbation term, a modified Alder s fourth-order perturbation result for square-well fluids was chosen. Its constants were refitted to the thermodynamic equilibrium data of pure methane. The final equation of state reads ... 

A recent study examined the effect on molecular dynamics of neglecting the long-range attractive part of the Lennard-Jones potential (spherical and ellipsoidal). This study was motivated by the successful static perturbation theory of simple Lennard-Jones fluids developed by Weeks et a/. The latter theory decomposed the spherical Lennard-Jones potential... 

Kleiner and Gross subsequently considered the effect of molecular polarizability and induced dipole interactions through the eombination of PCP-SAFT and the renormalized perturbation theory of Wertheim however, for the 36 polar fluids and their mixtures with hydroearbons studied only slight improvements in the agreement with experimental data was observed compared to the results obtained from the PCP-SAFT equation alone. ... 

Experimental data including the acidic species in the vapor phase within the above concentration range are scarce. Only very few publications of VLE data in that range are available [168, 173]. In contrast, numerous vapor pressure curves are accessible in literature. Chemical equilibrium data for the polycondensation and dissociation reaction in that range (>100 wt%) are so far not published [148]. However, a starting point to describe the vapor-Uquid equilibrium at those high concentratirMis is given by an EOS which is based on the fundamentals of the perturbation theory of Barker [212, 213]. Built on this theory, Sadowski et al. [214] have developed the PC-SAFT (Perturbed Chain Statistical Associated Fluid Theory) equation of state. The PC-SAFT EOS and its derivatives offer the ability to be fuUy predictive in combination with quantum mechanically based estimated parameters [215] and can therefore be used for systems without or with very little experimental data. Nevertheless, a model validation should be undertaken. Cameretti et al. [216] adopted the PC-SAFT EOS for electrolyte systems (ePC-SAFT), but the quality for weak electrolytes as phosphoric... 

The statistical associating-fluid theory (SAET) developed by Chapman et al. [63, 64] is based on the thermodynamic perturbation theory of Wertheim [69]. Since it first appeared, many different versions of SAET have been published. The different SAFT versions and their application have been reviewed by Muller and Gubbins [70]. 

More modem approaches borrow ideas from the liquid state theory of small molecule fluids to develop a theory for polymers. The most popular of these is the polymer reference interaction site model (PRISM) theory " which is based on the RISM theory of Chandler and Andersen. More recent studies include the Kirkwood hierarchy, the Bom-Green-Yvon hierarchy, and the perturbation density functional theory of Kierlik and Rosinbeig. The latter is based on the thermodynamic perturbation theory of Wertheim " where the polymeric system is composed of very sticky spheres that assemble to form chains. For polymer melts all these liquid state approaches are in quantitative agreement with simulations for the pair correlation functions in short chain fluids. With the exception of the PRISM theory, these liquid state theories are in their infancy, and have not been applied to realistic models of polymers. 

Barker J and Henderson D 1967 Perturbation theory and equation of state for a fluids II. A successful theory of liquids J. Chem. Phys. 47 4714... 

Smith W R 1972 Perturbation theory in the classical statistical mechanics of fluids Specialist Periodical Report vol 1 (London Chemical Society)... 

The fugacity coefficient of thesolid solute dissolved in the fluid phase (0 ) has been obtained using cubic equations of state (52) and statistical mechanical perturbation theory (53). The enhancement factor, E, shown as the quantity ia brackets ia equation 2, is defined as the real solubiUty divided by the solubihty ia an ideal gas. The solubiUty ia an ideal gas is simply the vapor pressure of the sohd over the pressure. Enhancement factors of 10 are common for supercritical systems. Notable exceptions such as the squalane—carbon dioxide system may have enhancement factors greater than 10. Solubihty data can be reduced to a simple form by plotting the logarithm of the enhancement factor vs density, resulting ia a fairly linear relationship (52). 

We would like to recall that Xa p) is the fraction of molecules not bonded at an associative site now it is a function of the averaged density p(r). A generalization of the perturbational theory allows us to define Xa p) similar to the case of bulk associating fluids. Namely... 

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