Poisson-Nernst-Planck method

This section describes the numerical techniques used for solving the set of differential equations that model the electrodiffusion of ions in solution. The method has historically been called the Poisson-Nernst-Planck (PNP) method because it is based on the coupHng of the Poisson equation with the Nernst-Planck equation. The basic equations used in the PNP method include the Poisson equation (Eq. [18]), the charge continuity equation (Eq. [55]), and the current density of the Nemst-Planck equation (Eq. [54]). 

The main difficulty when working with thin conducting polymer membranes is the lack of quantitative theory of ion diffusion within the membrane. Various theoretical schemes and approximations have been suggested, but the most difficult problem seems to be in the analytical solution or even approximation for the boundary problem of the combined Nernst-Planck and Poisson equations. The latter equation comes from the fact that electroneutrality cannot be assumed to prevail inside the thin membrane. Doblhofer et al." have made an attempt to solve the problem numerically, but even then certain initial approximations were made. Also the brute force method of finite differences does not allow to see clearly the influence of different parameters. 

As can be concluded from Eqs. (7.14) and (7.21), the diffusion-migration problem is non-linear. The Newton-Raphson method has been applied successfully to the resolution of the Nernst-Planck-Poisson equation system although the convergence is slower than for the kinetic-diffusion problems studied in Chapter 6. Thus, the unknown vector x corresponds to... 

M. Kh. Urtenov, Methods of Solution of the Nernst-Planck-Poisson Equation System [in russian], Kuban State University, Krasnodar, Russia, 1998. 

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