Cavities nonspherical

In any liquid or mixture of liquids we can introduce a hard particle of any form at some fixed position and orientation. Associated with this process there is a well-defined pseudochemical potential. This hard particle, by its definition, excludes the centers of all other particles from some region which we denote by Clearly, if the solvent particles are not spherical, or if they have different sizes, then the excluded volume of the hard particle will be different for different orientations of the solvent particles or for particles having different sizes. If we are interested in the work required to create a cavity equivalent to the hard particle, we must take the intersection of all possible with respect to all orientations and all solvent species. 

As an example, consider a hard sphere of diameter hs placed in a mixture of hard spheres of diameters cr and cjh. The excluded volumes with respect to the two particles are 

The cavity equivalent to placing the hard sphere at some fixed position is the intersection of and i.e., n In the case of spherical particles this intersection is simply the largest of the two excluded volumes. Similar considerations apply for particles of different shapes and sizes. 

In general, the cavity produced by a hard particle a in any solvent is the intersection of all the excluded volumes of a with respect to all solvent particles approaching from all possible orientations. This excluded volume is denoted by Va. The probability of finding such an excluded volume empty is simply obtained by generalization of (5.10.5), namely 

In general, a larger volume of the cavity does not necessarily lead to lower probability. For two excluded volumes for which 

---

The most popular of the SCRF methods is the polarized continuum method (PCM) developed by Tomasi and coworkers. This technique uses a numerical integration over the solute charge density. There are several variations, each of which uses a nonspherical cavity. The generally good results and ability to describe the arbitrary solute make this a widely used method. Flowever, it is sensitive to the choice of a basis set. Some software implementations of this method may fail for more complex molecules. 

There are three types of preexisting nuclei we need to consider (1) free approximately spherical bubbles, which generally will be coated to a greater or lesser extent with surface active molecules, (2) similar bubbles, attached to a surface, and (3) portions of gas or vapor trapped in crevices, cracks, pits, and other geometrically nonspherical cavities. Such bubbles may be filled with air... 

Extension of SPT to nonspherical cavities met with difficulties from the very beginning. Thus, many authors looked... 

The empirical parameters are obtained by linear regression analysis, fitting equations (36) and (37) to a set of TCFs for cavities with different shapes (i.e., cube, sheet, chain). The reference values are calculated with a numerical procedure for nonspherical cavities derived from SPT (see below), and the empirical coefficients equations (36) and (37) are taken as linear functions of temperature ... 

Claverie proposes a procedure (III) to calculate Gc for cavities made up of fused hard spheres, by summing up work contributions from individual atomic spheres which compose the nonspherical cavity. Gq required to build that part of the cavity which hosts atom i is obtained by multiplying... 

SPT extension to nonspherical cavities can be done by starting with the expression of solvent pressure, p, on the wall of a spherical cavity ... 

Solvent pressure is not uniform on a nonspherical cavity made of fused hard spheres, and depends on the local curvature of the cavity wall. Hence, the total force exerted by the solvent can be decomposed into components, Fjp, acting on parts of cavity surface A,- with different curvatures (Figure 4b) ... 

Figure 5 Growing a nonspherical cavity by budding. Work is spent to extend the cavity after step i...

A second comparison is given in Figure 7 between I, the method for fused hard sphere cavities based on SPT (III) and the numerical and analytical (IV and V) methods for nonspherical cavities. Figure 7 shows the influence of cavity shape (Figure 8) on TCF (chain 7a sheet 7b cube 7c). For chain- and sheet-like cavities, IV and V give similar results and the difference between their values and those obtained with I and III increases with the increase in cavity volume. I gives quite close results to IV and V for chain-like cavities made up of less then twenty spheres, which explains SPT success for small nonspherical solutes. Overall, III gives the least accurate results. 

Table 4 shows a comparison between experimental and calculated solvation thermodynamics for -alkanes in water obtained by SPT, the method based on SPT for fused hard sphere cavities and the one for nonspherical cavities. Cvdw was calculated with Pierotti s method for all TCF variants. 

The calculation of TCF in multicomponent systems has been done only for spherical cavities with the formalism developed by Lebowitz et al. Methods IV and V can in principle be extended for TCF calculation for nonspherical cavities in multicomponent systems. An artificial binary system (benzene-water) was selected here to illustrate the computational methodology. In practice, these two solvents mix very little, and their mixture can be of little interest, but they are quite different in their chemical nature and this makes such a system interesting. The method is by no means limited to certain mixtures and is universally applicable to any mixture if the molecular and physical parameters of the pure components are known (hard sphere diameter, number density, thermal expansion coefficient, dielectric constant). Figure 9 displays TCF calculated as a function of solvent mole fractions for a spherical cavity of cyclohexane size created in a hypothetical water-benzene mixture. Gc (Figure 9) increases with the increase of water mole fraction, but there is little difference between pure benzene and a mixture containing around 50% water as far as solvation of cyclohexane is concerned. 

By the last two assumptions the theory, strictly speaking, is only applicable to the monatomic gases A, Kr, Xe, to a somewhat lesser extent to the almost spherical molecules CH4, CF4, SFe, and perhaps to nonpolar diatomic molecules. The rotation of even slightly nonspherical molecules like Q2 and N2 will not be free in the entire cavity when such a molecule comes close to the wall of its cage it will have to orient itself parallel to this wall. Furthermore, some of the cavities are somewhat oblate (cf. Section I.B), and thus the rotation of relatively large, oblong molecules may be seriously... 

A number of theoretical models for solvation dynamics that go beyond the simple Debye Onsager model have recently been developed. The simplest is an extension of Onsager model to include solvents with a non-Debye like (dielectric continuum and the probe can be represented by a spherical cavity. Newer theories allow for nonspherical probes [46], a nonuniform dielectric medium [45], a structured solvent represented by the mean spherical approximation [38-43], and other approaches (see below). Some of these are discussed in this section. Attempts are made where possible to emphasize the comparison between theory and experiment. 

Figure 2.9b is an end-on, side view of the cell rhombus. In the figure, are outlines of three 51268 cavities (labeled A, B, and C) are shown with the vertical borders of the rhombus at centroids of each 51268. The fourth 51268 of Figure 2.9a is aligned behind the middle 51268 cavity in Figure 2.9b. This view shows both the nonspherical nature of the 51268 cavities and their nonplanar, strained hexagonal faces in contrast to the almost planar hexagonal faces in si and sll.

While the quantum mechanical calculation of the AG i, is reasonably well established, the term with the greatest uncertainty, which therefore limits the accuracy of the method, is the cavity formation term AG, . Various approximations have been made for this term, including the use of spheres and nonspherical shapes to represent the solute with atomic radii empirically adjusted or based on crystal structure. However, as this term is less important for mixture properties (as for a solute in its pure liquid and in the solvent to some extent cancel), PCM has been more useful for the calculation of activity coefficients, octanol-water partition coefficients... 

In the following we discuss various experimental data, with the aim of showing that use of nonspherical geometries for holes can in some cases supply better agreement between free-volume fractions /and h than can the spherical model. Furthermore, results in some polymers suggest a nonisotropic growth of the cavities with temperature. 

Several methods have been proposed to calculate AGsoiv,cav Tlte scaled-particle-theory formula (derived by Reiss, Frisch, Helfand, and Lebowitz and first applied to a solute in solution by Rerotti) calculates AG°(, y om the radii of the solute and solvent molecules (assumed spherical), the number of solvent molecules per unit volume, and the temperature and pressure [H. Reiss et al.,7. Chem. Phys.,32,119 (I960)].For solvation methods that use a nonspherical molecular cavity, the scaled-particle theory spherical-cavity formula for AGsoiv,cav was modified by Claverie to give what is often called the Rerotti-Claverie formula. The available methods give quite different results for AG°oiv,cavt and which method is best is unclear, since AG°oiv,cav is not a measurable quantity. Monte Carlo simulations of liquid water indicate that the Rerotti-Claverie formula may be significantly in error [F. M. Floris et al., /. Chem. Phys., 107,6353 (1997)]. For details, see J. Tomasi and M. Persico, Chem. Rev., 94,2027 (1994), Section V.B. 

Nonspherical, surface-imprinted magnetic PMMA (see Fig. 28) nanoparticles could be prepared by Tan et al. [177, 178]. A miniemulsion process was used to prepare magnetite/PMMA nanoparticles on which proteins were either immobilized by adsorption (RNAse A) [178] or covalently (bovine serum albumin, BSA) [177]. After creating a shell of PMMA, the proteins were removed, leaving cavities on the particles surface. The BSA-imprinted nanoparticles showed superparamagnetic properties and exhibited a high rebinding capacity for BSA. 

FIGURE 5.20. A cavity formed by a hard nonspherical particle. 

Scaled particle theory has also been used to understand and analyze surface areas, surface tension, and curvature as they relate to solubility and hydrophobic interactions. The accessible surface of an arbitrary hard sphere solute is the same as the surface of its cavity of radius r. Cavity potentials of nonspherical molecules have been related to those of spherical cavities of the same area. The relationship between accessible surface area, molecular surface area, and curvature has been examined. 

---