  We present theoretical models for the time-dependent voltage of an
electrochemical cell in response to a current step, including effects of
diffuse charge (or "space charge") near the electrodes on Faradaic reaction
kinetics. The full model is based on the classical Poisson-Nernst-Planck
equations with generalized Frumkin-Butler-Volmer boundary conditions to
describe electron-transfer reactions across the Stern monolayer at the
electrode surface. In practical situations, diffuse charge is confined to thin
diffuse layers (DLs), which poses numerical difficulties for the full model but
allows simplification by asymptotic analysis. For a thin quasi-equilibrium DL,
we derive effective boundary conditions on the quasi-neutral bulk electrolyte
at the diffusion time-scale, valid up to the transition time, where the bulk
concentration vanishes due to diffusion limitation. We integrate the thin DL
problem analytically to obtain a set of algebraic equations, whose (numerical)
solution compares favorably to the full model. In the Gouy-Chapman and
Helmholtz limits, where the Stern layer is thin or thick compared to the DL,
respectively, we derive simple analytical formulae for the cell voltage versus
time. The full model also describes the fast initial capacitive charging of the
DLs and super-limiting currents beyond the transition time, where the DL
expands to a transient non-equilibrium structure. We extend the well-known Sand
equation for the transition time to include all values of the super-limiting
current beyond the diffusion-limiting current.
