Pattern formation of three-dimensional electroconvection on a charge selective surface*

Electroconvection (EC) has been in the spotlight for enhancing ion flux in various electrochemical systems, but its dynamics is yet to be probed in three-dimensions. In this paper, we describe the first laboratory observation of 3-D EC on an exchange membrane and its pattern diversification. Combining experiment and scaling analysis, we successfully categorized three distinguished patterns of 3-D EC according to Reynolds number (Re), electric Rayleigh number (\( \mathrm{Ra_E} \)) and Schmidt number (Sc) as i) polygonal, ii) transverse, or iii) longitudinal rolls. If Re increases or \( \mathrm{Ra_E} \) decreases, pure longitudinal vortices are presented. On the other hand, transverse rolls are formed between longitudinal rolls, and two rolls are transformed as polygonal rolls at higher \( \mathrm{Ra_E} \) or lower Re. In this pattern selection scenario, Sc determines the critical electric Rayleigh number (\( \mathrm{Ra^*_E} \)) for the onset of each transverse or polygonal rolls, resulting \( \mathrm{Ra^*_E \sim Re^2 Sc} \). We also verify that convective ion flux by EC (represented by an electric Nusselt number, \( \mathrm{Nu_E} \)) is fitted to a power law, \( \mathrm{Nu_E \sim Ra_E^{0.48} Sc^{0.39}} \).