  We study the linear stability of transient electrodeposition in a charged
random porous medium, whose pore surface charges can be of any sign, flanked by
a pair of planar metal electrodes. Discretization of the linear stability
problem results in a generalized eigenvalue problem for the dispersion relation
that is solved numerically, which agrees well with the analytical approximation
obtained from a boundary layer analysis valid at high wavenumbers. Under
galvanostatic conditions in which an overlimiting current is applied, in the
classical case of zero surface charges, the electric field at the cathode
diverges at Sand's time due to electrolyte depletion. The same phenomenon
happens for positive charges but earlier than Sand's time. However, negative
charges allow the system to sustain an overlimiting current via surface
conduction past Sand's time, keeping the electric field bounded. Therefore, at
Sand's time, negative charges greatly reduce surface instabilities and suppress
dendritic growth, while zero and positive charges magnify them. We compare
theoretical predictions for overall surface stabilization with published
experimental data for copper electrodeposition in cellulose nitrate membranes
and demonstrate good agreement between theory and experiment. We also apply the
stability analysis to how crystal grain size varies with duty cycle during
pulse electroplating.
