Electrostatic potential distribution boundary conditions

To compute electrostatic potential and field distributions in very complex geometries, this equation, or one of its subsidiaries, can be solved numerically subject to a set of boundary conditions (McAllister et al.,... 

If we neglect the overpotential at the electrodes, then the boundary conditions for solving this problem are the constant electrode potentials. This type of problem has exact analogs in electrostatics, and many generalized solutions for symmetric configurations are available. In this type of problem, the current density is proportional to the potential gradient, and the current distribution can be calculated from Ohm s law ... 

The solution for this equation is the same as that of the constant potential boundary conditions (Dirichlet s problem) and was solved not only for electrostatic fields but also for heat fluxes and concentration gradients (chemical potentials) [10]. The primary potential distribution between two infinitely parallel electrodes is simply obtained by a double integration of the Laplace equation 13.5 with constant potential boundary conditions (see Figure 13.3). The solution gives the potential field in the electrolyte solution and considering that the current and the electric potential are orthogonal, the direct evaluation of one function from the other is obtained from Equation 13.7. 

Large differences between slow coagulation and ion adsorption consist in the chosen geometry and in the boundary conditions of ion (particle) distribution. Slow coagulation is assumed to be an irreversible process and particles which overcame the electrostatic barrier and occupy places in the potential well, are not considered any longer in the particle distribution. The particle distribution only contains mobile particles which can move in any direction, while particles in the potential well are considered as bounded. Thus, Fuchs theory does not describe particles in the coagulated state and the boundary condition chosen sets the particle concentration in the well equal to zero. Consequently, the Eq. (7.4) contains only the bulk concentration c. For the process of ion adsorption a completely different situation exists. The concentration of ion is not assumed to be zero in the subsurface and the ions can move in any direction. The state of ions... 

The exact distribution of charge and potential can be derived from electrostatic theory and the laws of electron distribution statistics, solving the Poisson-equation with the appropriate boundary conditions. 

So by subjecting the soap film to a pressure distribution p/(x, y ) we have a soap film with height z that is related by (6.61) to the electrostatic potential, V, and with coordinates that are simply related to those of the potential problem by (6.63) and (6.64). We must also ensure when setting up the soap film experiment that the boundary conditions satisfied by (6.55) are transferred by (6.61), (6.63) and (6.64) into boundary conditions that are satisfied by the soap film experiment. 

Even when applying the thin boundary layer approximation, the equations required for solving the current and potential distributions in the electrochemical cell yield a nonlinear system requiring iterative solution. The reason is that the boundary conditions incorporate the unknown term (the electrostatic potential or the current density). While this presents no serious hurdle for computer-implemented numerical solutions, analytical solutions of nonlinear systems are difficult and generally require a linearization procedure. To analytically characterize features of the current distribution, some simplifying approximations are frequently applied. These are summarized in Table 1, and are discussed below. 

The surface of the polymer particle is smooth and charges are uniformly distributed over the surface. To satisfy the condition of electroneutrality, the sodium carboxylate moiety resides at the interphase with sodium counterions solubilized in the aqueous phase near the carboxylate coions. The spatial distribution of coions and coimterions form the electronic double layer of 11k thickness. This boundary layer stabilizes the colloid (139). In the 1940s Deijagin, Landau, Verway, and Over-beek suggested that the electrostatic stability (84,85) of latexes could be explained on the basis of three potential energy terms that include repulsive potential... 

---