Spheres, Microscopic Approach

Let us now assume that particle 2 is a sphere as well, with radius R2 and center r2- Then, for the screened interaction tensor Tj of a dipole j in 2 with an external dipole k, we find analogous to Eq. (4.19) 

The center of the multipole V rj-rj) can be transposed from i-j to 1-2 by means of the addition theorem 

Inserting Eq. (4.24) into Eq. (4.23) we simultaneously change the sign of (pj in order to keep up with the inverted coordinates at position rj. By performing the surface integral (4.20), we obtain 

We obtain Sit in terms of a double multipole expansion with respect to tf — around axis 2 — and with respect to — r2 around the inverted axis — T2- 

The final step for finding the van der Waal energy between spheres 1 and 2 is to integrate the trace of Sjj, ESjjtSui. over sphere 1. We use 

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Macroscopic experiments allow determination of the capacitances, potentials, and binding constants by fitting titration data to a particular model of the surface complexation reaction [105,106,110-121] however, this approach does not allow direct microscopic determination of the inter-layer spacing or the dielectric constant in the inter-layer region. While discrimination between inner-sphere and outer-sphere sorption complexes may be presumed from macroscopic experiments [122,123], direct determination of the structure and nature of surface complexes and the structure of the diffuse layer is not possible by these methods alone [40,124]. Nor is it clear that ideas from the chemistry of isolated species in solution (e.g., outer-vs. inner-sphere complexes) are directly transferable to the surface layer or if additional short- to mid-range structural ordering is important. Instead, in situ (in the presence of bulk water) molecular-scale probes such as X-ray absorption fine structure spectroscopy (XAFS) and X-ray standing wave (XSW) methods are needed to provide this information (see Section 3.4). To date, however, there have been very few molecular-scale experimental studies of the EDL at the metal oxide-aqueous solution interface (see, e.g., [125,126]). 

Beam condensers, by using a pair of ellipsoid mirrors, produce very small images of the Jacquinot stop or the entrance slit at the sample position. The size of these images may be even further reduced by making use of a Weierstrass sphere. Weierstrass (1856) showed that a spherical lens has two aplanatic points . If a sphere of a glass with a refractive index n is introduced into an optical system which has a focus at a distance of r n from its center, then the beam is focused inside the sphere at a distance of r/n from the center (Fig. 3.5-9). In this case the angle O in Eq. 3.4-5 may approach 90°. Thus, a sample with a very small area can fully fit the optical conductance of the spectrometer (Fig. 3.4-2d). Microscopes usually have an optical conductance which is considerably lower than that of spectrometers. In this case, the sample and the objective are the elements limiting the optical conductance (Schrader, 1990 Sec. 3.5.3.3). 

Depending on the geometrical model (Figure 17) used and on the theoretical approach taken, different relationships exist for the approximation of adhesion by van der Waals forces. The best known equations are those developed by Hamaker based on the microscopic theory of London-Heitler. For the model sphere/plane. Figure 17(a), a distance a< 100 nm, and the particle diameter x, the adhesion force A, i is... 

The properties of the ions and the solvent which are ignored are similar to those ignored in the Debye-Hiickel treatment. These are very important properties at the microscopic level, but it would be a thankless task to try to incorporate them into the treatment used in the 1957 equation. Furthermore, Stokes Law is used in the equations describing the movement of the ions. This law applies to the motion of a macroscopic sphere through a structureless continuous medium. But the ions are microscopic species and the solvent is not structureless and use of Stokes Law is approximate in the extreme. Likewise, the equations describing the motion also involve the viscosity which is a macroscopic property of the solvent and does not include any of the important microscopic details of the solvent structure. The macroscopic relative permittivity also appears in the equation. This is certainly not valid in the vicinity of an ion because the intense electrical field due to an ion will cause dielectric saturation of the solvent immediately around the ion. In addition, alteration of the solvent stmcture by the ion is an important feature of electrolyte solutions (see Section 13.16). However, solvation is ignored. As in the Debye-Hiickel treatment the physical meaning of the distance of closest approach, i.e. a is also open to debate. 

The primary reason for the discrepancy is the fact that as two particles approach one another, it is necessary for solvent molecules between the particles to be moved out of the way. This process is accounted for by the viscosity term in Equation (10.20) for large distances of separation, but at smaller distances, of the order of molecular dimensions, the simple viscosity relationships no longer strictly apply, so that the mutual diffusion coefficient D is no longer equal to Di + D2 (see Chapter 4). One could say that the microscopic viscosity of the solvent increases so that diffusion is slowed and the particles approach at a reduced velocity. The exact calculation of this hydrodynamic effect represents a difficult problem in fluid dynamics. However, a relatively simple formula for two spheres of equal diameter is... 

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