Spheres, Macroscopic Approach

We find that each reflection of the interaction fields at sphere 1 and at sphere 2 gives rise to another factor d(m, 1) (f i/r2i) and d(n, 2)(/ 2/ 2i) respectively. 

The clear derivation of the dispersion energy according to Eqs. (4.32), (4.34) from multipole contributions suggests that it is the macroscopic multipole modes rather than the molecular dipole oscillators of spheres 1 and 2 which undergo fluctuations. This becomes even more obvious if we substitute the macroscopic dielectric permeabilities i(co) and fi2( ) of materials 1 and 2 for the molecular susceptibilities Xi(co) and X2(co) according to the law of Clausius-Mosotti 

j) is the 2 -pole susceptibility of a sphere with dielectric permeability 

Expression (4.34) is built up in a similar manner to expression (3.52), which gives the dispersion energy between particles 1 and 2 in terms of multiplet interactions between their molecules. Multipole nii at sphere 1 interacts with multipole at sphere 2, which in turn interacts with multipole m2 at sphere 1, and so on. The dispersion energy between spheres 1 and 2 can be obtained by drawing all the closed graphs passing forth and back between the multipoles of spheres 1 and 2 as vertices. Multipole m at sphere j yields the factor A m,j)Rj , the line passing from multipole m at sphere 1 to multipole n at sphere 2 yields the factor m + n)l/ m — fi) n +11)1 21 . Finally, we have to sum over all rotational wave numbers. 

Consequently, it is also possible to represent the dispersion energy in a closed form like in expressions (3.48), (3.49). Let us try to find this representation. What have we really done in the preceding sections  

---

In the opposite case, when the separation d = r2i—Pi —P2 of the spheres is small compared with the radii, we find the reduced radii Pi/f2i, R2l 2i to be of the order of one, so that all multipoles make similar contributions to the dispersion energy. We postpone this case to the discussion on the macroscopic approach in the following sections. 

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]). 

In Buckingham s approach [17,18], it is assumed that the solution is composed of small solvent macroscopic spheres (small with respect to the radiation wavelength) comprising a single solute molecule and surrounded by pure solvent each sphere is independent of the others (i.e. the solution is dilute). The ratio between the integrated absorption in solution and in gas phase can be written as ... 

Furthermore, these van der Waals interactions are important only near the interface, where it is unlikely that either Lifshitz or Hamaker approaches are accurate for spheres of molecular sizes. For example, the magnitude of the interaction for Na+ ions at. 5 A from the interface is only approximately 0.02kT (the values of B used in the calculation, Z Na = — 1X10 50 J m3 was obtained from fit by Bostrom et al. [17] and ZJNa= +0.8X10 511 J m3 was calculated by Karraker and Radke [18]). Eq. (8) might provide a convenient way to account for the interfacial interactions, if suitable values for Bt (not related to the macroscopic Hamaker constants) would be selected. 

Koelman and Hoogerbrugge (1993) have developed a particle-based method that combines features from molecular dynamics (MD) and lattice-gas automata (LGA) to simulate the dynamics of hard sphere suspensions. A similar approach has been followed by Ge and Li (1996) who used a pseudo-particle approach to study the hydrodynamics of gas-solid two-phase flow. In both studies, instead of the Navier-Stokes equations, fictitious gas particles were used to represent and model the flow behavior of the interstial fluid while collisional particle-particle interactions were also accounted for. The power of these approaches is given by the fact that both particle-particle interactions (i.e., collisions) and hydrodynamic interactions in the particle assembly are taken into account. Moreover, these modeling approaches do not require the specification of closure laws for the interphase momentum transfer between the particles and the interstitial fluid. Although these types of models cannot yet be applied to macroscopic systems of interest to the chemical engineer they can provide detailed information which can subsequently be used in (continuum) models which are suited for simulation of macroscopic systems. In this context improved rheological models and boundary condition descriptions can be mentioned as examples. 

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. 

---