Inner potential Poisson-Boltzmann equation

The inner potentials have to be calculated by solving the Poisson-Boltzmann equations for the potentials this is done in Appendix A. 

Modified Gouy-Chapman theory has been applied to soil particles for many years (Sposito, 1984, Chapter 5). It postulates only one adsorption mechanism -the diffuse-ion swarm - and effectively prescribes surface species activity coefficients through the surface charge-inner potential relationship contained implicitly in the Poisson-Boltzmann equation (Carnie and Torrie, 1984). Closed-form... 

For the inner region (avery small as compared with that in the outer region (bspherical Poisson-Boltzmann equation [6.139] with the following planar Poisson-Boltzmann equation ... 

The sign reversal takes place also in the electrophoretic mobility of a non-uniformly charged soft particles, as shown in this section. We treat a large soft particle. The x-axis is taken to be perpendicular to the soft surface with its origin at the front edge of the surface layer (Fig. 21.8). The soft surface consists of the outer layer —d < x < 0) and the inner layer (x < —d), where the inner layer is sufficiently thick so that the inner layer can be considered practically to be infinitely thick. The liquid flow m(x) and equilibrium electric potential i//(x) satisfy the following planar Navier-Stokes equations and the Poisson-Boltzmann equations [39] ... 

Diffuse double layer (DDL) theory as applied to the surface chemistry of soils refers to the description of ion charge and inner potential contained in the Poisson-Boltzmann equation ... 

There are different mathematical models of the EDL which reflect the possible variety of its inner stmcture at different levels of sophistication (Hunter 1993, pp. 379). All of them refer to the approach of Gouy (1909) und Chapman (1913) which neglects the existence of the Stern layer and solely accounts for a diffuse layer in immediate contact with the surface. The potential distribution l/(r) inside the diffuse layer is defined by the Poisson-Boltzmann-equation (PBE) ... 

Close to the zero-charge potential difference the effect of the ion penetration on the interfacial capacitance can be estimated by solving the linearized Poisson-Boltzmann equations in all three regions of the MVN model [41]. For the sake of simplicity, it was assumed [32] that only the ions from the organic solvent phase enter the inner layer, so that their concentrations differ from zero at X2organic solvent phase. In that case, the double-layer capacitance C is [32] ... 

Instead of an exact calculation, Gouy and Chapman have assumed that (4) can be approximated by combining the Poisson equation with a Boltzmann factor which contains the mean electrical potential existing in the interface. (This approximation will be rederived below). From this approach the distribution of the potential across the interface can be calculated as the function of a and from (2) we get a differential capacitance Cqc- It has been shown by Grahame that Cqc fits very well the measurements in the case of low ionic concentrations [11]. For higher concentrations another capacitance in series, Q, had to be introduced. It is called the inner layer capacitance and it was first considered by Stern [1,2]. Then the experimental capacitance Cexp is analyzed according to ... 

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