Differential equations for the ultra-fast transient migration in electrolytic dissolution

Laplace equation has been a reliable tool to describe the static and primary current distribution in electrolytic cells. This brief reveals a set of pertaining equations for transient migration, so far not available in a form of partial differential equation that is applicable to a general 3D electrolytic cell. The new transient model includes the effects of charging up the double layer as well as building up the activation overpotentials on both electrodes. It focuses on the transient migration that is often omitted but becomes significant when extremely short electrical pulses (<50 ns) are applied. An analytic solution for a one-dimensional case is given to exhibit the contributions of the process parameters to the dynamic current distribution. The transient model enhances the understanding of electrolytic dissolution with ultra-short pulses, recently applied in micro and even nano-processing.