Comparison and numerical treatment of generalised Nernst-Planck models

In its most widespread, classical formulation, the Nernst–Planck– Poisson system for ion transport in electrolytes fails to take into account finite ion sizes. As a consequence, it predicts unphysically high ion concentrations near electrode surfaces. Historical and recent approaches to an approriate modification of the model are able to fix this problem. Several appropriate formulations are compared in this paper. The resulting equations are reformulated using absolute activities as basic variables describing the species amounts. This reformulation allows to introduce a straightforward generalisation of the Scharfetter-Gummel finite volume discretization scheme for drift-diffusion equations. It is shown that it is thermodynamically consistent in the sense that the solution of the corresponding discretized generalized Poisson–Boltzmann system describing the thermodynamical equilibrium is a stationary state of the discretized time-dependent generalized Nernst–Planck system. Numerical examples demonstrate the improved physical correctness of the generalised models and the feasibility of the numerical approach.