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

Abstract

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.

Computer simulation of ion-selective membrane electrodes and related systems by finite-difference procedures

“…A series of numerical simulations have been reported in the past for membranes of ion-selective electrodes [18][19][20][21][22][23][24][25], for liquid junctions of reference electrodes [22,24] and for diffusion layers of other electrode systems [26][27][28][29]. In an early contribution, Buck's group introduced digital simulations for investigating the electrical behavior of membranes based on liquid ionexchangers [18,19].…”

“…Buck also pioneered a more general treatment that accounts for space-charge influences according to the Poisson equation [21]. Recently, Lewenstam's group presented a new version of this space-charge approach and reported different applications to membranes and liquid junctions [22,24,25].…”

“…The resulting net charge concentrations are usually well below the range of the total ion concentrations, except for extremely thin membranes [6,31]. A further limitation of space-charge approaches is their need for highly involved, implicit mathematical procedures, which may become a critical factor [22,24,25]. Even more seriously, none of the earlier attempts of simulating ISEs accounted for the presence of a stagnant solution film adhering to the electrode.…”

Numerical simulations of the full set of the time-dependent Nernst-Planck and Poisson equations modeling electrodiffusion on a simple ion channel.

“…Rudolph [16] simulated electrical migration and diffuse double-layer effects using a fast implicit finite difference algorithm, which was developed as a part of a general tool for modeling electrochemical processes related to electroanalytical chemistry. Liquid junction and ion selective membrane potentials were studied using a numerical model based on the time-dependent NPP equations by Sokalski et al [17]. Bieniasz [18] extended his numerical technique based on the finite-difference patch-adaptive strategy to time-dependent models involving electrodiffusion transport described by NPP equation systems in one-dimensional space geometry.…”

A simple currentless method of determination of ion fluxes to and within electroactive ion-exchange membranes

“…A significant progress in the improvement of analytical parameters of these sensors, concerning lower detection limit, higher sensitivity and selectivity, is owing to new constructions and membrane compositions as well as precise tailoring of ion fluxes in the membrane and at the membrane-sample solution interface [1][2][3]. Theoretical studies, based initially on simplified models taking into account a steady state diffusion [4][5][6] and then on more advanced models [7][8][9][10] as well as those based on Nernst-Planck-Poisson equations [11,12], are related to this issue.…”