Liquid junction potentials calculated from numerical solutions of the Nernst-Planck and Poisson equations

We present numerical solutions for the one-dimensional Nernst-Planck and Poisson system of equations for steady-state electrodiffusion. Commonly used approximate solutions to these equations invoke assumptions of local electroneutrality (Planck approximation) or constant electric field (Goldman approximation). Calculations were performed to test the ranges over which these approximate theories are valid. For a dilutional junction of a 1:1 electrolyte, separated from adjoining perfectly stirred solutions by sharp boundaries, the Planck approximation is valid for values of kappa dL greater than 10, where 1/kappa d is the Debye length of the more dilute solution. The Goldman approximation is valid for kappa cL less than 0.1 where 1/kappa c is the Debye length of the more concentrated solution. These results suggest that the modeling of electrodiffusive flows in and near membrane ion channels may require numerical solutions of this set of equations rather than the use of either limiting case.