Hard spheres systems mixtures

Mixtures of equisized charged spheres were also treated by the MSA. Such a system is then uniquely characterized by the ratio of the critical temperatures of the pure components. Harvey [235] found that a continuous critical curve from the dipolar solvent to the molten salt is maintained until the critical temperature of the ionic component exceeds that of the dipolar component by a factor of about 3.6. This ratio is much higher than theoretically predicted for nonionic model fluids. We recall that for NaCl the critical line is still continuous at a critical temperature ratio of about 5. Thus, the MSA of the charged-hard-sphere-dipolar-hard-sphere system captures, at least in part, some unusual features of real salt-water systems with regard to their critical curves. 

Many of the papers on DFT have focused primarily on the hard-sphere system, and it is for this system that most success has been achieved. However, DFT has also been applied to the Lennard-Jones 12-6 system, binary mixtures, nonspherical molecules, and coulombic systems. We will discuss some of these applications later in the chapter as we review what is known about the phase diagrams of various models systems. 

The application of this approach to the hard-sphere system was presented by Ree and Hoover in a footnote to their paper on the hard-sphere phase diagram. They made a calculation where they used Eq. (2.27) for the solid phase and an accurate equation of state for the fluid phase to obtain results that are in very close agreement with their results from MC simulations. The LJD theory in combination with perturbation theory for the liquid state free energy has been applied to the calculation of solid-fluid equilibrium for the Lennard-Jones 12-6 potential by Henderson and Barker [138] and by Mansoori and Canfield [139]. Ross has applied a similar approch to the exp-6 potential. A similar approach was used for square well potentials by Young [140]. More recent applications have been made to nonspherical molecules [100,141] and mixtures [101,108,109,142]. 

The most successful corresponding-states theory for mixtures is called the van der Waals one-fluid theory. This theory was developed on a molecular basis by Leland and co-workers and follows from an expansion of the properties of a system about those of a hard sphere system. A hard-sphere system is one whose molecules only have repulsive intermolecular potentials with no attractive contributions. The starting equation for the development of the van der Waals one-fluid (known by the acronym VDW-1) theory is a rigorous statistical-mechanical result for the equation of state of a mixture of pair wise-additive, spherically symmetric molecules ... 

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... 

Once the hard sphere fluid has been appropriately defined, the free energy of the repulsive system may be described using the equation of state of hard sphere binary mixtures proposed by Boublik [302] ... 

One approach of this category is to solve the integral equations using the Percus-Yevick closure for the system of adhesive hard sphere (AHS) mixtures (17-22). An adhesive hard sphere is a hard sphere that has attractive sites at surface. The attractive interaction on these attractive sites is infinitely strong and infinitesimally short ranged. The Percus-Yevick closime yields an analytical solution for such systems. The adhesive attraction, which resembles the chemical bonding, is used to build up chains by employing the proper connectivity constraints. 

With regard to real electrolytes, mixtures of charged hard spheres with dipolar hard spheres may be more appropriate. Again, the MSA provides an established formalism for treating such a system. The MSA has been solved analytically for mixtures of charged and dipolar hard spheres of equal [174, 175] and of different size [233,234]. Analytical means here that the system of integral equations is transformed to a system of nonlinear equations, which makes applications in phase equilibrium calculations fairly complex [235]. 

The more obvious and consistent deviations from the hard sphere theory occur, at the low density values, due to the effects of attractive forces in the real system. We can attempt to correct for these effects using a method described previously (27-30) for the analysis of angular momentum correlation times in supercritical CFjj and CFjj mixtures with argon and neon. We replace the hard sphere radial distribution function at contact hs with a function gp (0) which uses the more realistic... 

Recent experimental studies (1-3), on systems of sterically stabilized colloidal particles that are dispersed in polymer solutions, have highlighted the role played by the free polymer molecules. These experiments are particularly relevant because the systems chosen are model dispersions in which the particles can be well approximated as monodisperse hard spheres. This simplifies the interpretation of the data and leads to a better understanding of the intcrparticle forces. DeHek and Vrij (1, 2) have added polystyrene molecules to sterically stabilized silica particles dispersed in cyclohexane and observed the separation of the mixtures into two phases—a silica-rich phase and a polystyrene-rich phase—when the concentration of the free polymer exceeds a certain limiting value. These experimental results indicate that the limiting polymer concentration decreases with increasing molecular weight of... 

The comparisons presented in Figures 7 and 8, show that the RY closure relation works better than HNC, PYand RMSA, for the two kinds of systems with repulsive interactions here considered. Before the introduction of the RY approximation, the picture was that HNC and PY were better approximations to describe the static structure of systems with repulsive long-range and hard-sphere interactions, respectively. The RMSA approximation, however, has been used extensively in the comparison with experimental data for the static structure of aqueous suspensions of polystyrene spheres, mainly because it has an analytical solution even for mixtures [36]. 

Dimitrelis, D and Prausnitz, J.M. Comparison of two Hard-Sphere Reference Systems for Perturbation Theories for Mixtures, Fluid Phase Equilibria. Vol. 31. 1986, pp. 1-21. 

Anderko and Lencka find. Eng. Chem. Res. 37, 2878 (1998)] These authors present an analysis of self-diffusion in multicomponent aqueous electrolyte systems. Their model includes contributions of long-range (Coulombic) and short-range (hard-sphere) interactions. Their mixing rule was based on equations of nonequilibrium thermodynamics. The model accurately predicts self-diffusivities of ions and gases in aqueous solutions from dilute to about 30 mol/kg water. It makes it possible to take single-solute data and extend them to multicomponent mixtures. 

Figure 3. Dielectric constant of the dipolar fluid embedded in a charged matrix quenched at low temperature 3r,e2/a = 1. ROZ vs GCMC results. HNC results for the corresponding equilibrated mixture and ROZ results for an equivalent system with a neutral (hard sphere) matrix are included for comparison.

The volume restriction effect as discussed in this paper was proposed several years ago by Asakura and Oosawa (12,13). Their theory accounted for the instability observed in mixtures of colloidal particles and free polymer molecules. Such mixed systems have been investigated experimentally for decades (14-16). However, the work of Asakura and Oosawa did not receive much attention until recently (17,18). A few years ago, Vrij (19) treated the volume restriction effect independently, and also observed phase separation in a microemulsion with added polymer. Recently, DeHek and Vrij (20) have reported phase separation in non-aqueous systems containing hydrophilic silica particles and polymer molecules. The results have been treated quite well in terms of a "hard-sphere-cavity" model. Sperry (21) has also used a hard-sphere approximation in a quantitative model for the volume restriction flocculation of latex by water-soluble polymers. 

Fig. 14 Normalized Dcoii/Bo of PDMS coated silica suspension with = 0.3 in a symmetric mixture of toluene and heptane (solid circles) along with hard sphere suspension (open squares) at similar volume fraction. The hydrodynamic interactions expressed in H(q) for the two systems (solid squares for the hard sphere suspension) are shown in the inset [101]. This system is crystallized by sedimentation as seen in the photograph...

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