Numerical Solution of the Coupled Nernst-Planck and Poisson Equations for Ion-Selective Membrane Potentials
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AA8.11.1
Numerical Solution of the Coupled Nernst-Planck and Poisson Equations for Ion-Selective Membrane Potentials Peter Lingenfelter*, Tomasz Sokalski and Andrzej Lewenstam Process Chemistry Group (www.abo.fi/instut/pcg/), c/o Centre for Process Analytical Chemistry and Sensor Technology (ProSens; www.abo.fi/fak/ktf/prosens/), Åbo Akademi University, Biskopsgatan 8, FIN-20500 Åbo/Turku, Finland *
Corresponding author: Peter Lingenfelter, Tel: +358 2 215 3247, Fax: +358 2 215 4479, email: [email protected] ABSTRACT A numerical model is presented for analyzing the propagation of ionic concentrations and electrical potential in space and time in the solutionion-exchanging membrane system. 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 non-linear differential equations that is solved numerically. As a result of the physicochemical properties of the system, both the contact/boundary potential and the diffusion potential contribute to the overall membrane potential. It is shown that interpreting the electrical potential of ion-exchanging membranes exclusively in terms of boundary potential at steady-state is incorrect. The NernstPlanck-Poisson (NPP) model is general and applies to ions of any charge in space and time domains. INTRODUCTION The prevalent approach in ion-selective membrane electrode (ISE) modeling is to relate the processes of ion exchange and membrane transport to the operationally defined electrical potential of the ion-selective electrode. This restricts the theoretical modeling to interpretations based on, and biased towards, empirical postulates in the form of empirical equations.1,2 A general equation (postulate) of this type, accepted widely in the ISE field, is the NikolskiiEisenman (N-E) equation (eq 1).3,4 E = E0 +
zM R ⋅T ln aM + ∑ K Mpot, N aN z N zM ⋅ F
(1)
where E is electrode potential, E0 is a constant, a M + and a N+ are the activities of the main and interfering ions in the solution, z M and z N their respective charges, and K Mpot, N the selectivity coefficient. This practice is in stark contrast to the theoretical routines in membrane science or biophysics, in which membranes and membrane channels are regularly modeled with the Nernst-PlanckPoisson (NPP) system of partial differential equations.5-18 The first publication to herald the
AA8.11.2
application of these methods to the NPP system of equations was the historical work by Cohen and Cooley.19 Hafemann then went on to simulate liquid-junction potentials and ion concentration profiles over the liquid-junction as a function of time.20 These efforts made possible the first numerical solution of the NPP equation system – relevant to then current ISE research – by Brumleve and Buck, who solved the NPP system via a fully implicit iterative Newton-Raphson method coupled to Gaussian elimination
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