Application of Nernst¯Planck and Poisson equations for interpretation of liquidjunction and membrane potentials in real-time and space domains

In this paper, we show a numerical model designed for analysing the propagation of ionic concentrations and electrical potential in space and time in the liquid-junction and in the solutionmid ion-exchangingmembrane system. In this model, diffusion and migration according to the Nernst¯Planck flux equation govern the transport of ions. The electrical interaction of the species is described by the Poisson equation. These two equations and the continuity equation make a system of partial differential equations that is numerically resolved by the finite difference method. Consequently, the contact and/or boundary potential and diffusion potential are presented as a result of the physicochemical properties of the system rather than assumed a priori in order to find an analytical solution in the form of an equation. We show that the paradigmatic equations in potentiometry, such as Henderson and Nikolskii¯Eisenman (N¯E), are special cases in our model. Although we discuss the examples relevant to electroanalytical potentiometry and, in particular, to the field of ion-selective membrane electrodes (ISE), it is evident that the method presented here is a good tool for solving broader problems in membrane biology and electrochemistry with membranes.