Numerical Solution of the Coupled Nernst−Planck and Poisson Equations for Liquid Junction and Ion Selective Membrane Potentials

A new numerical model is presented for analyzing the propagation of ionic concentrations and electrical potential in space and time in the liquid junction and in the solution/ion-exchanging membrane system. In this model, diffusion and migration according to the Nernst−Planck (NP) flux equation govern the transport of ions, and the electrical interaction of the species is described by the Poisson (P) equation. These two equations and the continuity equation form a system of partial nonlinear differential equations that is solved numerically. This yields the Nernst−Planck−Poisson (NPP) model that we exploit in this paper. Notably, as a result of the physicochemical properties of the system, which are clearly defined in this paper, both the contact/boundary potential and the diffusion potential contribute to the overall membrane potential. Previously, only the boundary potential at steady state was considered due to some arbitrary and clearly untested assumptions. By accessing space and time domains, it is shown that interpreting the electrical potential of ion-exchanging membranes exclusively in terms of boundary potential at steady state is incorrect. The NPP model is general and applies to ions of any charge in space and time domains. It is shown for the first time that the paradigmatic equations for every open circuit measurement, such as the Henderson liquid junction equation or the Nikolskii−Eisenman equation, are special cases of the NPP model. The NPP model is not only more rigorous but also more complete than previous models, and it is proposed to be a more appropriate and updated platform for dealing with the theory of ion selective membrane electrodes for analytical applications.