Application of the Scaled Particle Theory

The scaled particle theory SPT) was developed mainly for the study of hard-sphere liquids. It is not an adequate theory for the study of aqueous solutions. Nevertheless, it has been extensively applied for aqueous solutions of simple solutes. The scaled particle theory (SPT) provides a prescription for calculating the work of creating a cavity in liquids. We will not describe the SPT in detail only the essential result relevant to our problem will be quoted. Let aw and as be the effective diameters of the solvent and the solute molecules, respectively. A suitable cavity for accommodating such a solute must have a radius of c ws = ((Tw + cTs) (Fig. 3.20b). The work required to create a cavity of radius a s at a fixed position in the liquid is the same as the pseudo-chemical potential of a hard sphere of radius as. The SPT provides the following approximation for the pseudochemical potential  

Reiss etal. (1959,1960), Helfand etal. (1960), and Reiss (1966), and for more recent developments, Tully-Smith and Reiss (1970), Reiss and Tully-Smith (1971), and Stillinger (1973). 

It should be noted that the SPT is not a pure molecular theory in the following sense. A molecular theory is supposed to provide, say, the Gibbs free energy as a function of T, P, N as well as of the molecular parameters of the system. Once this function is available, the density of the system can be computed from the relation p = (9/x/9 )t (with pi = G/N). The SPT utilizes the effective diameter of the solvent molecules as the only molecular parameter (which is the case for a hard-sphere fluid) and, in addition to the specification of T and P, the solvent density Pw is also used as input in the theory. The latter being a measurable quantity carries with it implicitly any other molecular properties of the system. The first application of the SPT to calculate the thermodynamics of solvation in liquids was carried out by Pierotti (1963, 1965). 

From these results, Pierotti concluded that It is the entropy of cavity formation that gives rise to the large negative entropies associated with aqueous solutions.  

Some general comments regarding the application of the SPT to water are now in order. First, the SPT was originally devised for treating a fluid of hard spheres or simple non-polar fluids. The extension of the theory to complex fluids such as water is questionable. Second, the SPT employs an effective diameter of the solvent as the only molecular parameter. It is very likely that this diameter is temperature dependent. It is not clear, however, which kind of temperature dependence should be assumed for aitj. [This topic was discussed by Ben-Naim and Friedman (1967) and Pierotti (1967).] Finally, we stress again that the SPT is not a pure molecular theory. Even if it can predict the properties of aqueous solutions of gases, it is still incapable of providing an explanation of these properties. This is an inherent drawback of the theory since it cannot tell us why aqueous solutions behave in such a peculiar way as compared with other fluids. 

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The usual explanation given for the high surface tension of water is that at the surface a water molecule cannot form four tetrahedraUy directed hydrogen bonds with other water molecules but only three, hence water tends to minimize the surface area in order to minimize the energetic expense of the loss of hydrogen bonds. Therefore, a value of 5 of Eq. (4.7) of the order of two molecular diameters, 0.5 nm, should have been expected. Anyway, only very tiny droplets would have a surface tension smaller than that of bulk water with a flat surface. However, with regard to the application of the scaled particle theory, Eqs. (4.3) to (4.6), the question of the finite curvature of the cavity remains. 

Relation (4.4.40) suggests a novel application of the scaled particle theory to the problem of H0O interactions. In spite of some serious reservations that one may have regarding the application of the scaled particle theory to fluids such as water, the results computed by the SPT show the same trends as those for H(pO interactions. [A more detailed examination of the application of the scaled particle theory for this problem was... 

Relation (8.83) permits a novel application of the scaled particle theory to the problem of Table 8.8 gives some computed values of for water and various nonaqueous solvents. [See Ben-Naim (1971a).] It is quite clear that the values of zlyUns largest in water, which, by virtue of (8.83), means that in water, the HI is the strongest. In spite of some serious reservations that one may have regarding the application of the scaled particle theory to fluids such as water (see Section 7.3), the results of Table 8.8 show the same trend we witnessed in Section 8.6. [Recently, a more detailed examination of the application of the scaled particle theory for this problem has been reported by Wilhelm and Battino (1972). We have discussed here only spherical solutes, for which one needs a spherical cavity. An extension of the scaled particle theory to particles of arbitrary shape has been reported by Gibbons (1969).]... 

The diameters of solvents play a role in theoretical considerations, such as the application of the scaled particle theory. For gaseous solvent molecules, the collision diameter, o, is related to the Lennard-Jones pair potential energy,... 

In this article, we have surveyed typical properties of isotropic and liquid crystal solutions of liquid-crystalline stiff-chain polymers. It had already been shown that dilute solution properties of these polymers can be successfully described by the wormlike chain (or wormlike cylinder) model. We have here concerned ourselves with the properties of their concentrated solutions, with the main interest in the applicability of two molecular theories to them. They are the scaled particle theory for static properties and the fuzzy cylinder model theory for dynamical properties, both formulated on the wormlike cylinder model. In most cases, the calculated results were shown to describe representative experimental data successfully in terms of the parameters equal or close to those derived from dilute solution data. 

In effect, such a multi-scale analysis resolves a macro-scale heterogeneous system into three meso- to micro-scale subsystems—dense-phase, dilute-phase and inter-phase. Thus, modeling a heterogeneous particle-fluid two-phase system is reduced to calculations for the three lower-scale subsystems, making possible the application of the much simpler theory of particulate fluidization to aggregative fluidization and the formulation of energy consumptions with respect to phases (dense, dilute and inter) and processes (transport, suspension and dissipation). 

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... 

Studies on the application of the theory of statistical moments in the description of grinding in ball mills have been carried out in the Department of Process Equipment, Lodz Technical University [1-3]. The research was carried out in a laboratory scale for selected mineral materials. Results obtained confirmed applicability of the theory of statistical moments in the description of particle size distribution during grinding. 

The contribution "Application of Meso-Scale Field-based Models to Predict Stability of Particle Dispersions in Polymer Melts" by Prasanna Jog, Valeriy Ginzburg, Rakesh Srivastava, Jeffrey Weinhold, Shekhar Jain, and Walter Chapman examines and compares Self Consistent Field Theory and interfacial Statistical Associating Fluid Theory for use in predicting the thermodynamic phase behavior of dispersions in polymer melts. Such dispersions are of quite some technological importance in the... 

The PDT that is a central feature of this book dates from this period (Widom, 1963 Jackson and Klein, 1964), as does the related but separately developed scaled-particle theory (Reiss et al, 1959). Both the PDT and scaled-particle approaches have been somewhat bypassed as features of molecular theory, in contrast to their evident utility in simulation and engineering applications. Scaled-particle theories have been helpful in the development of sophisticated solution models (Ashbaugh and Pratt, 2004). Yet the scaled-particle results have been almost orthogonal to pedagogical presentations of the theory of liquids. This may be due to the specialization of the presentations of scaled-particle theory (Barrat and Hansen, 2003). 

A hybrid approach of the extended scaled particle theory (SPT) and the Poisson-Boltzmann (PB) equation for the solvation free energy of non-polar and polar solutes has been proposed by us. This new method is applied for the hydration free energy of the protein, avian pancreatic polypeptide (36 residues). The contributions form the cavity formation and the attractive interaction between the solute and the solvent to the solvation free energy compensate each other. The electrostatic conffibution is much larger than other terms in this hyelration free energy, because hydrophilic residues are ionized in water. This work is the first step toward further applications of our new method to free energy difference calculation appeared in the stability analysis of protein. 

We have presented the first application of the newly developed method for calculating the solvation free energy to protein, which is based on the extended scaled particle theory and the Poisson-Boltzmann equation. Although the results are still preliminary, it demonstrates a possibility of obtaining the quantity theoretically, which is difficult even for the modem... 

Thus the energy of attraction becomes infinite as the particle approaches a flat surface. For this reason, it is usually assumed that a surface acts as a p>erfect sink in the theory of aerosol diffusion that is, when a particle penetrates to a distance one radiu.s from the surface, the particles adhere. This holds best for submicron particles moving at thermal velocities. Rebound occurs for larger particles moving at high velocities (Chapter 4). This analysis does not lake into account the effects of surface roughness of the scale of the particle size or of layers or patches of adsorbed gases or liquids. Such factors may be important in practical applications. 

Jackson, R. M. Sternberg, M. J. E. (1994). Application of scaled particle theory to model the hydrophobic effect implications for molecular association and protein stability. Prot. Eng. 7, 371-383. 

Wilhelm and Battino have extended their measurements of binary gas diffusion coefficients using the Stefan technique to mixtures of SF with four aliphatic hydrocarbons. Jaster and Kosky have completed a series of measurements of the solubility of SF, in four commercially-available fluorocarbon mixtures and Pierotti has published a recent review on the application of scaled-particle theory to the estimation of gas solubilities. 

The Navier-Stokes (NS) equations can be used to describe problems of fluid flow. Since these equations are scale-independent, flow in the microscale structure of a porous medium can also be described by a NS field. If the velocity on a solid surface is assumed to be null, the velocity field of a porous medium problem with a small pore size rapidly decreases (see Sect. 5.3.2). We describe this flow field by omitting the convective term v Vv, which gives rise to the classical Stokes equation We recall that Darcy s theory is usually applied to describe seepage in a porous medium, where the scale of the solid skeleton does not enter the formulation as an explicit parameter. The scale effect of a solid phase is implicitly included in the permeability coefficient, which is specified through experiments. It should be noted that Kozeny-Carman s formula (5.88) involves a parameter of the solid particle however, it is not applicable to a geometrical structure at the local pore scale. 

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