Monotonic and oscillatory free-convective instability of solution in the space between two plane horizontal electrodes: Solutions containing three types of ions V.M. Volgin a, * , A.D. Davydov b a Chair Physical-Chemical Processes and Technology, Tula State University, Pr. Lenina 92, Tula 300600, Russia b Frumkin Institute of Electrochemistry, Russian Academy of Sciences, Leninskii Pr. 31, Moscow 119071, Russia Received 25 January 2005; received in revised form 12 September 2005; accepted 18 September 2005 Available online 3 November 2005 Abstract The conditions for the onset of natural convection (convective instability) in the electrochemical system consisting of two plane hor- izontal electrodes and solution containing three types of ions between them were analyzed theoretically. The equations of flow of incom- pressible viscous liquid to the Boussinesq approximation and the equations of material balance with the electroneutrality condition were used as the mathematical model. The set of equations for amplitudes of perturbations was solved using the Galerkin method. The depen- dences of critical Rayleigh number on the system parameters are obtained for monotonic and oscillatory instabilities in the case that the electrochemical reaction proceeds under the conditions of limiting current. The examples of instability diagrams are presented. In con- trast to the binary electrolyte, for a solution containing three types of ions, the critical Rayleigh number depends on the charges of ions and the ratio between the diffusion coefficients of ions in the solution. Ó 2005 Elsevier B.V. All rights reserved. Keywords: Natural convection; Rayleigh–Benard convection; Convective instability; Critical Rayleigh number; Electrochemical system; Limiting current 1. Introduction When a current is passed through a stagnant solution between two horizontal electrodes, two states of the system can be observed: (1) Solution remains stagnant in spite of the variation in its density near the electrodes. The buoy- ancy forces are balanced by the viscosity forces. (2) Buoy- ancy force initiates convective instability, initially stagnant solution starts to flow: solution with a higher density, which forms near the upper electrode, flows downward, and solution with a lower density, which forms near the lower electrode, flows upward. Natural convection and convective instability have a pronounced effect on many electrochemical processes, on the processes rates and the distribution of current density over the electrode surface, on the relief of electrodeposit and electropolished surface, on the processes in the batteries, and on the characteristics of electrochemical devices [1–8]. For the limiting-current mode, to the approximation of solution electroneutrality, the problem of Rayleigh– Benard instability for a binary electrolyte is equivalent to the problem of heat convection that have been much studied [9–13]. In this case, only monotonic convective instability can arise [8]. In a solution with a more complex composition, oscillatory instability can arise along with the monotonic one. Several works were devoted to the study of monotonic and oscillatory instabilities in the electrochemical systems with three types of ions [14–16]. In these works, approximate solutions of the problem for the cathodic deposition (anodic dissolution) of metal were obtained. Systems with redox reactions were not 0022-0728/$ - see front matter Ó 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jelechem.2005.09.010 * Corresponding author. Tel.: +7 0872 352452; fax: +7 0872 331305. E-mail address: volgin@uic.tula.ru (V.M. Volgin). www.elsevier.com/locate/jelechem Journal of Electroanalytical Chemistry 586 (2006) 308–315 Journal of Electroanalytical Chemistry