Hard-sphere fluids associating

Now, let us consider a model in which the association site is located at a distance slightly larger than the hard-core diameter a. The excess free energy for a hard sphere fluid is given by the Carnahan-Starling equation [113]... 

In Sec. 3 our presentation is focused on the most important results obtained by different authors in the framework of the rephca Ornstein-Zernike (ROZ) integral equations and by simulations of simple fluids in microporous matrices. For illustrative purposes, we discuss some original results obtained recently in our laboratory. Those allow us to show the application of the ROZ equations to the structure and thermodynamics of fluids adsorbed in disordered porous media. In particular, we present a solution of the ROZ equations for a hard sphere mixture that is highly asymmetric by size, adsorbed in a matrix of hard spheres. This example is relevant in describing the structure of colloidal dispersions in a disordered microporous medium. On the other hand, we present some of the results for the adsorption of a hard sphere fluid in a disordered medium of spherical permeable membranes. The theory developed for the description of this model agrees well with computer simulation data. Finally, in this section we demonstrate the applications of the ROZ theory and present simulation data for adsorption of a hard sphere fluid in a matrix of short chain molecules. This example serves to show the relevance of the theory of Wertheim to chemical association for a set of problems focused on adsorption of fluids and mixtures in disordered microporous matrices prepared by polymerization of species. 

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 reference fluid which consists by TPT of non-bonded nitrogen atoms represents the so-called non-associated limit (NAL) of the hard molecular fluid. The nitrogen atoms interact as hard spheres with the diameter ra via the hard sphere pair potential. The density functional theoretical description of the NAL falls back on that which are used by the spherical DFT approach. The latter provides beside other a suitable description for the inhomogeneous hard sphere fluid. 

FIGURE 3.5 Procedure to form a molecule in the SAFT model, (a) The proposed molecule, (b) Initially the fluid is a hard-sphere fluid, (c) Attractive forces are added, (d) Chain sites are added and chain molecules appear, (e) Association sites are added and molecules form association complexes through association sites. (From Fu, Y.-H. and Sandler, S.I., Ind. Eng. Chem. Res., 34, 1897, 1995. With permission.)... 

There is naturally a wealth of publications on aspects of solvation and a comprehensive review would need a whole book. Hence, it is not practical to wade through all the developments in solvent effect theory, especially as other articles in this encyclopedia also deal with some aspects of solvation (see Related Articles at the end of this article). Instead, the focus will be on the methods used for the evaluation of the thermodynamics of cavity formation (TCF), which is a large part of solvation thermodynamics, and in particular on the application of the most successful statistical mechanical theory for this purpose, namely, the scaled particle theory (SPT) for hard sphere fluids (see Scaled Particle Theory). This article gives a brief introduction to the thermodynamic aspects of the solvation process, defines energy terms associated with solvation steps and presents a short review of statistical mechanical and empirical... 

The present chapter is organized as follows. We focus first on a simple model of a nonuniform associating fluid with spherically symmetric associative forces between species. This model serves us to demonstrate the application of so-called first-order (singlet) and second-order (pair) integral equations for the density profile. Some examples of the solution of these equations for associating fluids in contact with structureless and crystalline solid surfaces are presented. Then we discuss one version of the density functional theory for a model of associating hard spheres. All aforementioned issues are discussed in Sec. II. 

Following Ref. 122, we consider the adsorption of associating hard spheres, Eq. (60), with d = 0.45, a = 0.09 in a slit-like pore with crystalline walls. A weakly associated fluid and a highly associated fluid have been studied. The weakly associated fluid was characterized by the coefficient A (cf. Eq. (79)) equal to 1, whereas in the case of the highly associated fluid A = 100. 

In order to derive a practical approximation for the repulsive contribution to vibrational frequency shifts the excess chemical potential, A ig, associated with the formation of a hard diatomic of bond length r from two hard spheres at infinite separation in a hard sphere reference fluid is assumed to have the following form. 

Weingartner, H., Weiss, V.C., and Schroer, W. Ion association and electrical conductance minimum in Debye-Htickel-based theories of the hard sphere ionic fluid. J. Chem. Phys., 2000, 113, p. 762-70. 

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,... 

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