Buoyancy and electrically driven convection models in thin-layer electrodeposition

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Recent work demonstrates the inevitable role of convection during the growth of patterns in thin-layer electrochemical deposition. Convection is driven mainly by Coulombic forces due to local charges and by buoyant forces due to concentration gradients that lead to density gradients. Here we study by theoretical and numerical modeling the limiting regimes under which electroconvection or gravitoconvection prevail. The model describes the diffusive, migratory, and convective motion of ions in a fluid subject to an electric field. The equations are written in terms of dimensionless quantities, in particular, the gravity Grashof and the electrical Grashof numbers. The simulations reveal that gravitoconvection becomes increasingly important as the gravity Grashof number increases, while electroconvection becomes increasingly important as the electrical Grashof number increases. For both regimes the model predicts concentration, electric potential, and velocity patterns that are in qualitative agreement with typical electrodeposition experiments. In gravitoconvection, the model predicts the evolution, before collision, of the convection rolls near each electrode growing first as ${t}^{4/5}$ and then slowing down to ${t}^{1/2};$ the same scaling behavior was observed in experiments. After collision, the cathodic and anodic rolls merge into a single roll. In electroconvection, the model predicts the existence of vortex pairs formed by the electrical force on space charge accumulating near the growing filament tip. Such vortex rolls and pairs have been observed in experiments.