Hard sphere solvents, scaled particle theory

The molecular size of a solvent can be characterized in several ways. One of them is to assign the solvent a molecular diameter, as if its molecules were spherical. From a different aspect, this diameter characterizes the cavity occupied by a solvent molecule in the liquid solvent. From a still further aspect, this is the mean distance between the centers of mass of two adjacent molecules in the liquid. The diameter plays a role in many theories pertaining to the liquid state, not least to those treating solvent molecules as hard spheres, such as the scaled particle theory (SPT, see below). Similar quantities are the collision diameters a of gaseous molecules of the solvent, or the distance characterizing the minimum in the potential energy curve for the interaction of two solvent molecules. The latter quantity may be described, e.g., according to the Lennard-Jones potential (Marcus 1977)... 

As discussed above, solvation free energy is t3q)ically divided into two contributions polar and nonpolar components. In one popular description, polar portion refers to electrostatic contributions while the nonpolar component includes all other effects. Scaled particle theory (SPT) is often used to describe the hard-sphere interactions between the solute and the solvent by including the surface free energy and mechanical work of creating a cavity of the solute size in the solvent [148,149]. 

Go is the partial molar Gibbs function associated with cavity formation and Gi is the partial molar Gibbs function for the solute-solvent interaction. Pierotti used an expression for Go derived from scaled-particle theory assuming a hard-sphere potential. Go is thus a function of pi, the solvent number density, and of the diameters of the solvent and of the cavity. Ui and hence Gi were estimated from the expression ... 

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 ... 

In the preceding section we discussd the work required to create a cavity in the liquid. This concept is fundamental in the study of the solvation of solutes in any solvent. The simplest solute is a hard-sphere (HS) particle, and the simple solvent also consists of HS particles. We shall see in section 6.14 that the solvation of any solute in any solvent can always be decomposed into two parts, creating a suitable cavity and then turning on the other parts of the solute-solvent interaction. The scaled-particle theory (SPT) provides an approximate procedure to compute the work required to create a cavity. 

The simplest nontrivial solvation phenomenon is the case of a hard-sphere (HS) particle in a fluid of HS particles. This case was discussed in connection with the scaled-particle theory in section 5.11. Here we note that the solvation Gibbs energy of an HS solvaton in an HS solvent is always positive, and it increases monotonically as a function of the size of the HS solvaton or, equivalently, the radius of the corresponding cavity (see section 5.11). 

The scaled particle model (SPM) was the first essentially molecular theory of hydrophobicities (see Scaled Particle Theory), It derived from an earlier scaled particle theory, a successful theoretical calculation of the thermodynamic properties of the hard sphere liquid. Pierotti then adapted the scaled particle theory to produce a solubility model for realistic liquids by a natural replacement of the hard sphere pressure with the measured pressure of the real solvent of interest. With attractive solute-solvent interactions treated perturba-tively this scaled particle model was remarkably successful. The SPM is the molecular theory of hydrophobicities most widely considered among biomolecular modelers. However, its success is somewhat fortuitous.For example, though the SPM predicts a reasonable value for the surface tension for the water liquid-vapor interface at room temperature, the predicted temperature dependence is wrong. Since entropies and temperature dependencies are special goals of theories of hydrophobic effects, this incorrect temperature dependence is important. 

The scaled particle theory of fluids is a theory that offers an excellent conceptual framework with which to understand solvent effects. The theory was originally designed to give the thermodynamic properties of a fluid of hard spheres, but experimentally determined parameters can be introduced to treat real liquids. The treatment of simple liquids is better than the treatment of structured liquids such as water. The theory has been applied extensively to certain aspects of hydrophobic solvation and interactions. In some studies of hydrophobic solvation based on molecular dynamics or Monte Carlo simulations, scaled particle theory has provided useful guidelines for analysis of the results. ... 

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... 

Scaled Particle Theory for Hard Sphere Solvents... 

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