Hard-sphere theory application

The smooth hard-sphere theory discussed above has been remarkably successful for monatomic fluids, as exemplified by xenon (see Chapter 10). For application to polyatomic fluids, it is necessary to take into account additional considerations ... 

In view of the success of the methods based on hard-sphere theories for the accurate correlation and prediction of transport properties of single-component dense fluids, it is worthwhile to consider the application of the hard-sphere model to dense fluid mixtures. The methods of Enskog were extended to mixtures by Thome (see Chapman Cowling 1952). The binary diffusion coefficient >12 for a smooth hard-sphere system is given by... 

Erkey, C., Rodden, J.B., Matthews, M.A. Akgerman, A. (1989). Application of rough hard-sphere theory to diffusion in n-alkanes. Int. J. Thermophys., 10,953-962. 

In the application of this rough hard-sphere theory for the interpretation of transport properties of dense pseudo-spherical molecules, it is assumed that equations (10.21) and (10.22) are exact. Reduced quantities for diffusion and viscosity, similar to those defined by equations (10.11) and (10.12), are given by... 

Another important application of perturbation theory is to molecules with anisotropic interactions. Examples are dipolar hard spheres, in which the anisotropy is due to the polarity of tlie molecule, and liquid crystals in which the anisotropy is due also to the shape of the molecules. The use of an anisotropic reference system is more natural in accounting for molecular shape, but presents difficulties. Hence, we will consider only... 

Reiss H 1977 Scaled particle theory of hard sphere fluids Statistical Mechanics and Statistical Methods in Theory and Application ed U Landman (New York Plenum) pp 99-140... 

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. 

Fig. 10(b)). One of the reasons for the differences between both theories is a different form of a hard sphere part of the free energy functional. Segura et al. have used the expression resulting from the Carnahan-Starhng equation of state, whereas the Meister-Kroll-Groot approach requires the application of the PY compressibility equation of state, which produces higher oscillations. 

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. 

Our main focus in this chapter has been on the applications of the replica Ornstein-Zernike equations designed by Given and Stell [17-19] for quenched-annealed systems. This theory has been shown to yield interesting results for adsorption of a hard sphere fluid mimicking colloidal suspension, for a system of multiple permeable membranes and for a hard sphere fluid in a matrix of chain molecules. Much room remains to explore even simple quenched-annealed models either in the framework of theoretical approaches or by computer simulation. 

The extent of the agreement of the theoretical calculations with the experiments is somewhat unexpected since MSA is an approximate theory and the underlying model is rough. In particular, water is not a system of dipolar hard spheres.281 However, the good agreement is an indication of the utility of recent advances in the application of statistical mechanics to the study of the electric dipole layer at metal electrodes. 

Lagues et al. [17] found that the percolation theory for hard spheres could be used to describe dramatic increases in electrical conductivity in reverse microemulsions as the volume fraction of water was increased. They also showed how certain scaling theoretical tools were applicable to the analysis of such percolation phenomena. Cazabat et al. [18] also examined percolation in reverse microemulsions with increasing disperse phase volume fraction. They reasoned the percolation came about as a result of formation of clusters of reverse microemulsion droplets. They envisioned increased transport as arising from a transformation of linear droplet clusters to tubular microstructures, to form wormlike reverse microemulsion tubules. 

Let us consider a hard sphere in a continuous matrix under an electric field. When the relative dielectric constant of particle s2 is larger than that of matrix s1, a point dipole in the particle is formed by application of an electric field. According to the classical theory [51], the point dipole moment is given by... 

The attentive reader will realize that we have strayed rather far from the hard spheres of the Einstein theory to find applications for it. It should also be appreciated, however, that the molecules we are discussing are proteins that-through disulfide bridges and hydrogen bonding —have fairly rigid structures. Therefore the application of the theory —amended to allow for solvation and ellipticity —is justified. This would not be the case for synthetic polymers, which are best described as random coils and for which a different formalism is employed. This is the topic of Section 4.9. 

Lorentz1 advanced a theory of metals that accounts in a qualitative way for some of their characteristic properties and that has been extensively developed in recent years by the application of quantum mechanics. He thought of a metal as a crystalline arrangement of hard spheres (the metal cations), with free electrons moving in the interstices.. This free-electron theory provides a simple explanation of metallic luster and other optical properties, of high thermal and electric conductivity, of high values of heat capacity and entropy, and of certain other properties. 

In recent years, a number of investigators have studied the phase equilibria of simple fluids in pores by the application of density functional theory. Semina] studies were carried out by Evans and his co-workers (1985,1986). Their approach was considered to be the simplest realistic model for an inhomogeneous three-dimensional fluid . The starting point was a model intrinsic Helmholtz free energy functional F(p), with a mean-field approximation for the attractive forces and hard-sphere repulsion. As explained in Section 7.6, the equilibrium density profile of the fluid in a pore was obtained by minimizing the grand potential functional. 

Bjorling, M., Pellicane, G and Caccamo, C On the application of Hory-Huggins and integral equation theories to asymmetric hard sphere mixtures. J. Chem. Phys. Ill, 6884-6889 (1999). 

A conceptually complementary approach to describe hydrophobic effects has been introduced by Pratt and colleagues (78, 96). Their iifformation theory (IT) model is based on an application of Widom s potential distribution theorem (97) combined with the perception that the solvation free energy of a small hard sphere, which is essentially governed by the probability to find an empty sphere, can be expressed as a limit of the distribution of water molecules in a cavity of the size... 

Jog, P.K. and Chapman, W.G., Application of Wertheim s thermodynamic perturbation theory to dipolar hard sphere chains, Mol. Phys., 97(3), 307-319, 1999. 

Agterof, W.G.M., van Zomeren, J.A.J. and Vrij, A. (1976) On the application of hard sphere fluid theory to liquid particle dispersions. Chem. Phys. Lett., 43, 363-367. 

Unfortunately, real molecules differ significantly from hard spheres, so Equation (1.12) to (1.14) are not directly useful for real fluids. Additional correction factors can be added to these equations for fairly realistic spherically symmetric interactions these can represent nonpolar fluids that are roughly spherical, such as the noble gases and CH4. However, most molecules of interest are far from spherical, and kinetic theory is still intractable for molecular interactions that are not spherically symmetric. Therefore, the direct applicability of kinetic theory for calculating transport properties of real fluids is limited. However, kinetic theory plays an important role in guiding the functional form of semiempirical correlations such as those discussed below. 

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