Numerical solution surface complexation calculations

Examples of correlations between stability constants of surface complexes (calculated using different adsorption models, cf. Chapter 5) on the one hand, and the first hydrolysis constant and other constants characterizing the stability of solution complexes on the other are more munerous than the studies of correlations involving directly measured quantities. It should be emphasized that there is no generally accepted model of adsorption of ions from solution, and stability constants of surface complexes are defined differently in particular models, thus, the numerical values of these constants depend on the choice of the model. Moreover, some publications reporting the correlations fail to define precisely the model. 

The analytical calculations involve determining the auxiliary functions of complex variables. Except for the simple shapes of WP surfaces, this presents some difficulties. Therefore, along with the analytical methods of solving the inverse ECM problem, numerical methods have evolved. Normally, in the numerical solution, the variables are changed in order to reduce the initial problem to a problem in the region with the known boundary. 

Several physicochemical models of ion exchange that link diffuse-layer theory and various models of surface adsorption exist (9, 10, 14, 15). The difficulty in calculating the diffuse-layer sorption in the presence of mixed electrolytes by using analytical methods, and the sometimes over simplified representation of surface sorption have hindered the development and application of these models. The advances in numerical solution techniques and representations of surface chemical reactions embodied in modem surface complexation mod-... 

Such a method has seldom been used with systems containing an aqueous fluid, probably because the complexity of the solution s free energy surface and the wide range in aqueous solubilities of the elements complicate the numerics of the calculation (e.g., Harvie el al., 1987). Instead, most models employ a procedure of elimination. If the calculation described fails to predict a system at equilibrium, the mineral assemblage is changed to swap undersaturated minerals out of the basis or supersaturated minerals into it, following the steps in the previous chapter the calculation is then repeated. 

For this more complex cavity, the electrostatic interaction energy between the solute charge distribution and the reaction field is calculated numerically. In the present study, the charge distribution was approximated by a distributed set of 150 fractional point charges on the cavity surface. 

Numerically, it is now a common practice to calculate within the dielectric continuum formulation but employing cavities of realistic molecular shape determined by the van der Waals surface of the solute. The method is based upon finite-difference solution of the Poisson-Boltzmann equation for the electrostatic potential with the appropriate boundary conditions [214, 238, 239]. An important outcome of such studies is that even in complex systems there exists a strong linear correlation between the calculated outer-sphere reorganization energy and the inverse donor-acceptor distance, as anticipated by the Marcus formulation (see Fig. 9.6). More... 

The use of an equation as complex as Eq. (67) requires a lot of numerical calculations so that approximate solutions are very favorable. The first model to describe the adsorption at the surface of a growing drop was derived by Ilkovic in 1938 (107). The boundary conditions were chosen such that the model corresponded to a mercury drop in a polarography experiment. These conditions, however, are not suitable for describing the adsorption of surfactants at a liquid-drop surface. Delahay and coworkers (108, 109) used the theory of Ilkovic and derived an approximation suitable for the description of adsorption kinetics at a growing drop. The relationship was derived only for the initial period of the adsorption process ... 

Depending on the volumetric distribution of the source of radiation, the following typical geometries maybe selected for assessment of external doses point source, immersion in a plume, and radioactivity deposited on surfaces. In addition, more complex arrangements of sources and geometries can often be described. For those cases, the point-kernel numeric integration method is performed based on calculations as a point source solution (Wood 1982). 

Given the complexity of the calculations (particularly the relationship between strand cross-sectional area, and width of the surface of the strand), it was decided to integrate this model numerically rather than to attempt analytical solution. 

The theory and practical aspect of potentio- and galvanostatic measurements of selective dissolution of homogeneous alloys are developed quite good [24-27], but not all the possibilities of chronovoltammetry are utilized. This is primarily determined by the fact that a solution of even a simplest diffusion problem for a perfectly smooth electrode at the linear potential trace is given by an integral equation [28, 29]. Obviously, accounting such complex nonlinear effects of non-equilibrium vacancy subsystem relaxation and alloy / solution interface displacement is hardly possible without the use of labor-intensive numerical calculations. At the same time the surface roughness of an electrode, equilibrium solid-phase adsorption accumulation of alloy components before selective dissolution, and ability of the mixed kinetic control can be taken into account in the analytieal solution of the voltammetric problem. 

---