Several ways to simulate time dependent liquid junction potentials by finite differences

Several ways of simulating time-dependent migration effects at an electrolytic liquid junction are explored, for a simple uni-univalent electrolyte, both with the full Nernst-Planck-Poisson (NPP) equation set and the reduced set from the electroneutrality condition (ENC) assumption. Using the NPP approach, the system can be simulated using all three variables (method ABψ), the two concentrations and the potential, or the two concentrations and the potential field (method ABE), or in principle by substituting for the potential field, thereby reducing to the two concentration variables (method AB). The two first methods are about equally efficient, whereas the latter method is seen to be quite inaccurate. Results at long times compare very well with the Henderson equation. Using the full NPP set, junction potentials are time-dependent but not when applying the ENC, where the potential rises to the Henderson value immediately. Results for KCl and HCl are presented, with left/right concentration ratios equal to 0.1 in both cases.