V. M. Volgin

Natural-convective instability of electrochemical systems: A review

Basic theoretical and experimental results of research into natural-convective instability of electrochemical systems are considered.

Simulation of ion transfer under conditions of natural convection by the finite difference method

Abstract The finite difference method of calculation of non-steady-state ion transfer in electrochemical systems under the conditions of natural convection is elaborated. The method is based on the mathematical model involving the continuity equations for electrolyte species, the condition of electroneutrality, and the Navier–Stokes equations for a viscous incompressible liquid with the corresponding initial and boundary conditions. A scheme of decoupling is proposed, which provides successive calculation of the field of hydrodynamic velocities (a stream function), the distribution of electric potential, and the distribution of electrolyte species concentrations subject to the condition of electroneutrality. To enhance the efficiency of the method at large Schmidt numbers, the distribution of electrolyte species concentrations was calculated by the implicit difference scheme. The results of computational experiments are reported.

Finite difference method of simulation of non-steady-state ion transfer in electrochemical systems with allowance for migration

Finite difference methods of the second order of accuracy are elaborated for numerical calculation of non-steady-state ion transfer, which is caused by diffusion, migration, and convection in the unidimensional electrochemical systems. The methods of decoupling a set of coupled continuity equations of the electrolyte species are proposed, which ensures that the discrete equations are consistent with the initial differential equations and the electroneutrality condition is rigorously met. The methods of approximation of the boundary conditions of the second order temporal and spatial accuracy and the method of decoupling the transfer equations in the boundary nodes are elaborated. The explicit, fully implicit, and semi-implicit finite difference schemes are elaborated. For semi-implicit schemes, two versions of difference equation closure are proposed, which assure the unambiguity of determination of the distribution of electrical potential. Comparison analysis of the accuracy of elaborated finite difference methods of calculation of non-steady-state ion transfer is performed.

Numerical Modeling of Non-Steady-State Ion Transfer in Electrochemical Systems with Allowance for Migration

Numerical methods that are used for modeling non-steady-state ion transfer in electrochemical systems and account for the diffusion, migration, convection, and homogeneous chemical reactions are analyzed. It is shown that the violation of the electroneutrality condition (ENC) in the process of numerical solution is due to the difference equations being inconsistent with the initial differential equations. Difference schemes for numerical calculation of transfer processes, which make it possible to split a set of coupled equations, are designed and conditions for their stability are determined. The explicit difference scheme is self-consistent, i.e. it ensures that ENC is rigorously met. In the implicit difference scheme, ENC is probably violated when splitting the set of equations. To restore electroneutrality of the medium, it is proposed to use a physically substantiated analytical relation for the space charge relaxation under the action of a strong electric field.

Template electrodeposition of metals. Review

The works devoted to the electrodeposition of metals, alloys, and semiconductors onto the substrates through the templates of various non-conducting materials with pores of various shapes and sizes are reviewed. The composite materials, nanowires, metal foams, and the parts with nanostructured surface obtained by this method are promising materials for the electrocatalysis, electroanalysis of media, development of various sensors, modern miniature magnetic memory devices, optoelectronics, power sources, etc.

Mass-transfer problems in the electrochemical systems

The problems concerning quantitative analysis of mass-transfer processes in the electrochemical systems (ECS) are briefly reviewed. The interrelation between the mass-transfer problems in the electrochemical and heat systems is considered. Various approaches to the numerical determination of distributions of concentrations of ions of all types and electrochemical potential in the ECS are considered. The methods of allowance for the migration transfer and choosing the boundary conditions for the transfer equations and other problems are considered. Some actual lines of development of the theory of mass-transfer in the ECS are pointed out.

Numerical simulation of natural convection of electrolyte solution with three types of ions in the electrochemical cell with vertical electrodes

The mass transfer in the electrolyte solution with three types of ions in the electrochemical cell of square section with vertical electrodes is studied. The mathematical model of the process involves the Navier-Stokes equations in the Boussinesq approximation, the equations of ionic transfer of electrolyte components, which is caused by diffusion, convection, and migration, and the condition of electroneutrality. It is shown that this problem corresponds to a special case of thermosolutal convection with regard for thermodiffusion (the Soret effect), where the cell boundaries are permeable to an impurity and the flux of impurity through the boundary is proportional to the heat flux. Using the numerical simulation, the distributions of concentration of ions, solution density, local and average mass-transfer rates are studied. The approximate analytical equations for the limiting current are obtained for typical electrochemical systems.

Modeling of metal electrodeposition in the pores of anodic aluminum oxide

The pore filling process in the metal electrodeposition in the anodic aluminum oxide pores under the potentiostatic conditions is studied theoretically. The model takes into consideration the transfer of metal cations inside the pores and in the outer diffusion layer. The thickness of outer diffusion layer is determined by the hydrodynamic conditions. The kinetics of metal electrodeposition is described by the equation of mixed kinetics at large deviations from the equilibrium. In the quasi-steady-state approximation, the problem for the pores of different initial lengths, which are characterized by the normal distribution law, is solved analytically. The results are compared with the data obtained by the numerical method.

Modeling of through-mask electrochemical micromachining

A scheme of numerical modeling of through-mask electrochemical machining, which enables one to predict the shape and sizes of workpiece surface, is developed. The effect of mask parameters (a fraction of unprotected area, mask thickness and unprotected region width) on the distribution of an average current density over the workpiece surface is analyzed. The developed mathematic model and the calculated results can be used to design the operations of electrochemical micromachining with partial insulation of workpiece surface.

Calculation of effective diffusion coefficient in a colloidal crystal by the finite-element method

The work is devoted to the calculation of effective diffusion coefficient of ions from the bulk solution to the electrode through a mask and the calculation of the distribution of the limiting current density over the electrode surface. A colloidal crystal, which is formed by orderly arranged monodispersed spherical particles, serves as a mask. It is shown that the diffusion of electroactive ions in the pores between spherical particles can be simulated by unit cells with rhombic, rectangular, or triangular cross-section. In the latter case, the cell side surface has no periodical boundaries. This simplifies significantly the numerical solution of the Laplace’s equation by the finite-element method. The effective diffusion coefficient in the bulk colloidal crystal is calculated at various values of its porosity. The calculated results agree well with the literature data. It is found that, for close-packed spherical particles, the relative effective diffusion coefficient in the bulk colloidal crystal is 0.16. The thicknesses of transient zones adjacent to the electrode surface and outer boundary of colloidal crystal and the effective diffusion coefficients for these zones are determined. The dependence of effective diffusion coefficient on the number of spherical particle layers in the colloidal crystal is obtained. The distribution of the limiting current density over the electrode surface is analyzed at various numbers of particle layers.