  Understanding over-limiting current (faster than diffusion) is a
long-standing challenge in electrochemistry with applications in desalination
and energy storage. Known mechanisms involve either chemical or hydrodynamic
instabilities in unconfined electrolytes. Here, it is shown that over-limiting
current can be sustained by surface conduction in nano pores, without any such
instabilities, and used to control dendritic growth during electrodeposition.
Copper electrode posits are grown in anodized aluminum oxide membranes with
polyelectrolyte coatings to modify the surface charge. At low currents, uniform
electroplating occurs, unaffected by surface modification due to thin electric
double layers, but the morphology changes dramatically above the limiting
current. With negative surface charge, growth is enhanced along the nanopore
surfaces, forming surface dendrites and nanotubes behind a deionization shock.
With positive surface charge, dendrites avoid the surfaces and are either
guided along the nanopore centers or blocked from penetrating the membrane.
