Scaled-particle theories

An expression for the work of insertion W can be obtained from scaled particle theory (SPT) [31]. SPT was developed to derive expressions for the chemical potential and pressure of hard sphere fluids by relating them to the reversible work needed to insert an additional particle in the system. This work W is calculated is by expanding (scaling) the size of the sphere to be inserted from zero to its final size the size of the scaled particle is Act, with X running from 0 to 1. In the limit 2 0, the inserted sphere approaches a point particle. In this limiting case it is very unlikely that the depletion layers overlap. The free volume fraction in this limit can therefore be written as 

In the opposite limit A 1, when the size of the inserted scaled particle is very large, IV to a good approximation is equal to the volume work needed to create a cavity f(A r) and is given by 

As was the original objective of SPT [31], the pressure P of the hard sphere system can be obtained from the reversible work of inserting an identical sphere (g = 1) 

3 Phase Transitions of Hard Spheres Plus Depletants Basics 

In Fig. 3.9 we present a comparison of the free volume fraction predicted by SPT (3.36) and computer simulations [32] on hard spheres plus penetrable hard spheres for q = 0.5 as a function of / . As can be seen the agreement is very good. In the limit of small depletants the 2 and ) terms of (3.30) can be omitted giving  

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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... 

Gibbons R M 1969 Scaled particle theory for particles of arbitrary shape Mol. Phys. 17 81... 

Reiss H and Hammerich ADS 1986 Hard spheres scaled particle theory and exact relations on the existence and structure of the fluid/solid phase transition J. Phys. Chem. 90 6252... 

Stillinger F 1973 Structure in aqueous solutions from the standpoint of scaled particle theory J. Solution Chem. 2 141 Widom B 1967 Intermolecular forces and the nature of the liquid state Sc/e/ ce 375 157 Longuet-Higgins H C and Widom B 1964 A rigid sphere model for the melting of argon Mol. Phys. 8 549... 

Alternative integral equations for the cavity functions of hard spheres can be derived [61,62] using geometrical and physical arguments. Theories and results for hard sphere systems based on geometric approaches include the scaled particle theory [63,64], and related theories [65,66], and approaches based on zero-separation theorems [67,68]. These geometric theories have been reviewed by Stell [69]. 

To exploit the concept of PMF to represent solvent in free energy calculations, practical approximations must be constructed. A common approach is to treat the two components Z H/"P(X) and Z lYelec(X) separately. Approximations for the nonpolar term are usually derived from geometric considerations, as in scaled particle theory, for example [62], The electrostatic contribution is usually derived from continuum electrostatics. We consider these two contributions in turn. 

Stillinger, F., Structure in aqueous solutions of nonpolar solutes from the standpoint of scaled-particle theory, J. Sol. Chem. 1973, 2, 141-158... 

Pierotti, R.A., A scaled particle theory of aqueous and nonaqueous solutions, Chem. Rev. 1976, 76, 717-726... 

Some workers have attempted to treat particular effects more rigorously, e g., by scaled-particle theory [142] or by extending [95, 103] Linder s theory [143] of dispersion interactions to the case of an SCRF treatment of solute-solvent interactions. We will not review these approaches here. 

Finally we address the issue of contributions. In our view it is unbalanced to concentrate on a converged treatment of electrostatics but to ignore other effects. As discussed in section 2.2, first-solvation-shell effects may be included in continuum models in terms of surface tensions. An alternative way to try to include some of them is by scaled particle theory and/or by some ab initio theory... 

They cannot be directly measured because of the chemical reactions of the dissolved molecular components, but must be calculated theoretically or estimated by correlation. Electrostatic theory does not predict negative coefficients, which are characteristic of ammonia with some salts. To us, it appears that scaled particle theory(22) is probably the best method of calculation, but the required parameters (polarizability and ion size) are not available for the salts of interest. 

Masterton, W.L. Lee, T.P. "Salting Coefficients from Scaled Particle Theory," J. Phys. Chem., 1970, 74, 1776-80. 

Prediction of salting out effect based on scaled particle theory... 

Mitragotri S (2002) A theoretical analysis of permeation of small hydrophobic solutes across the stratum corneum based on scaled particle theory. J Pharm Sci 91 744-752. 

This article reviews the following solution properties of liquid-crystalline stiff-chain polymers (1) osmotic pressure and osmotic compressibility, (2) phase behavior involving liquid crystal phasefs), (3) orientational order parameter, (4) translational and rotational diffusion coefficients, (5) zero-shear viscosity, and (6) rheological behavior in the liquid crystal state. Among the related theories, the scaled particle theory is chosen to compare with experimental results for properties (1H3), the fuzzy cylinder model theory for properties (4) and (5), and Doi s theory for property (6). In most cases the agreement between experiment and theory is satisfactory, enabling one to predict solution properties from basic molecular parameters. Procedures for data analysis are described in detail. 

Scaled Particle Theory for Wormlike Hard Spherocylinders.93... 

In the present article, we focus on the scaled particle theory as the theoretical basis for interpreting the static solution properties of liquid-crystalline polymers. It is a statistical mechanical theory originally proposed to formulate the equation of state of hard sphere fluids [11], and has been applied to obtain approximate analytical expressions for the thermodynamic quantities of solutions of hard (sphero)cylinders [12-16] or wormlike hard spherocylinders [17, 18]. Its superiority to the Onsager theory lies in that it takes higher virial terms into account, and it is distinctive from the Flory theory in that it uses no artificial lattice model. We survey this theory for wormlike hard spherocylinders in Sect. 2, and compare its predictions with typical data of various static solution properties of liquid-crystalline polymers in Sects. 3-5. As is well known, the wormlike chain (or wormlike cylinder) is a simple yet adequate model for describing dilute solution properties of stiff or semiflexible polymers. 

Before proceeding to a review of both scaled particle theory and fuzzy cylinder model theory, it would be useful to mention briefly the unperturbed wormlike (sphero)cylinder model which is the basis of these theories. Usually the intramolecular excluded volume effect can be ignored in stiff-chain polymers even in good solvents, because the distant segments of such polymers have little chance of collision. Therefore, in the subsequent reference to wormlike chains, we always mean that they are unperturbed . 

The above formulation for a monodisperse polymer system by the scaled particle theory can be readily extended to a polydisperse polymer system, as described in [17]. The result is... 

In Table 4, terms of the order of b 2(d/L)2 in B3 are neglected.) The scaled particle theory (SPT), Lee s theory, and Khokhlov-Semenov s theory give the same B2 as Onsager s, so that they become identical at infinite dilution. Flory s... 

Now we compare the above osmotic pressure data with the scaled particle theory. The relevant equation is Eq. (27) for polydisperse polymers. In the isotropic state, it can be shown that Eq. (27) takes the same form as Eq. (20) for the monodisperse system though the parameters (B, C, v, and c ) have to be calculated from the number-average molecular weight M and the total polymer mass concentration c of a polydisperse system pSI in the parameters B and C is unity in the isotropic state. No information is needed for the molecular weight distribution of the sample. On the other hand, in the liquid crystal state2, Eq. (27) does not necessarily take the same form as Eq. (20), because p5I depends on the molecular weight distribution. 

Fig. 2. Comparison between the scaled particle theory (solid curves) and experiment (circles and triangles) for osmotic pressure II of PBLG-DMF [56,57], For the samples with M = 6.6 x 104 and 15.5 x 104, the data at T = 15, 30, and 45 °C are plotted with the same symbols...

Fig. 3. Comparison between the scaled particle theory and experiment for reciprocal osmotic compressibility of PHIC-DCM [60]. Vertical segments indicate C. ...

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