Regulation of ramified electrochemical growth by a diffusive wave

The role of ion transport during ramified growth in a dilute, binary electrolyte is investigated by obtaining a wave solution to effective equations derived by asymptotic analysis of the Nernst transport equations. The concentration profile exhibits a diffusion layer ahead of the growing tips, in agreement with experiment and theory. The non-Laplacian electric field is stronger in the diffusion layer than in the bulk, and cations are accelerated through the layer toward the tips. The well-known growth speed of the aggregate envelope, roughly equal to the speed of migrating anions in the bulk emerges as the characteristic wave speed of our equations. The ‘‘copper ratio’’ is also predicted and is linked to the regulation of growth by the diffusive wave. Finally, an estimate of the induction time for ramified growth is derived, based on the idea that a critical diffusion layer width must be attained.