How the fine structure of the electric double layer and the flow affect morphological instability in electrodeposition

We analyze the effects of the fine structure of the electric double layer and of the electrode reaction rate upon wavelength selection of the growing perturbation mode in morphological instability in cathodic electrodeposition from a dilute electrolyte solution. The metal cathode is modeled as a cation exchanger with a fixed charge density equal to the average concentration of free electrons that is several orders of magnitude higher than the typical ionic concentration in a solution. This invokes the need to consider the steric modifications in the modeling of the electric double layer at the solution/electrode interface. Accordingly, the results for Bikerman's steric model are compared to those for the classical Nernst-Planck-Poisson-Stokes model valid for pointlike ions. For both models, the diffuse electric double layer splits into a thin inner portion commensurate with the width of the Stern layer and the steric exclusion width, and a much thicker outer portion scaling with the Debye length. For quiescent cathodic electrodeposition, this fine structure combined with a finite electrode reaction rate regularizes the short-wave singularity in morphological instability of the electrodeposition front. This regularization selects a finite range of unstable perturbation modes, with critical wavelength corresponding to the fastest-growing mode. This wavelength scales with the geometric average of the width of the highly charged inner portion of the electric double layer and the reaction-diffusion length, defined as the ratio of cation diffusivity to the electrode reaction rate. The fluid flow induced by the motion of the solid/liquid interface has a negligible effect on morphological instability. At the same time, the emerging electroconvective (electroosmotic) flow in the nonequilibrium regime with its related electroosmotic instability, has a major effect, selecting the width of the cathodic diffusion layer as the dominant length scale for morphological instability and the emerging dendrites. It is observed that the steric correction considerably lowers the voltage threshold for the onset of the nonequilibrium electroosmotic instability.