  The kinetic behavior of a phase field model of electrochemistry is explored
for advancing (electrodeposition) and receding (electrodissolution) conditions
in one dimension. We described the equilibrium behavior of this model in [J. E.
Guyer, W. J. Boettinger, J.A. Warren, and G. B. McFadden, ``Phase field
modeling of electrochemistry I: Equilibrium'', cond-mat/0308173]. We examine
the relationship between the parameters of the phase field method and the more
typical parameters of electrochemistry. We demonstrate ohmic conduction in the
electrode and ionic conduction in the electrolyte. We find that, despite making
simple, linear dynamic postulates, we obtain the nonlinear relationship between
current and overpotential predicted by the classical ``Butler-Volmer'' equation
and observed in electrochemical experiments. The charge distribution in the
interfacial double layer changes with the passage of current and, at
sufficiently high currents, we find that the diffusion limited deposition of a
more noble cation leads to alloy deposition with less noble species.
