K. A. Lebedev

Modelling the transport of carbonic acid anions through anion-exchange membranes

Electrodiffusion of carbonate and bicarbonate anions through anion-exchange membranes (AEM) is described on the basis of the Nernst � /Planck equations taking into account coupled hydrolysis reactions in the external diffusion boundary layers (DBLs) and internal pore solution. The model supposes local electroneutrality as well as chemical and thermodynamic equilibrium. The transport is considered in three layers being an anion exchange membrane and two adjoining diffusion layers. A mechanism of

Space charge effect on competitive ion transport through ion-exchange membranes

A mathematical model of the competitive electro-transport of two counter-ions through an ion exchange membrane based on the Nernst-Planck and Poisson equations is developed. A three-layer system is considered: the membrane and two adjacent diffusion layers. Concentration profiles in the three layers, effective transport numbers as functions of the current and current-voltage characteristics are calculated. Deviation from the local electroneutrality in space charge region near the depleted solution/membrane interface is taken into account. It is shown that the space charge region grows with the voltage applied. However the fluxes of the competitive counter-ions at over-limiting currents are determined by their transfer through the electroneutral part of the depleted diffusion layer.

Electroconvection in systems with heterogeneous ion-exchange membranes

The results of studying the surface morphology of heterogeneous cation-(MK-40) and anion-exchange (MA-40) membranes and calculating the structure of electroconvective vortices generated by the electric body force are shown. The body force and its distribution are estimated by taking into account real parameters of the membrane surface morphology. The calculations of vortices were carried out by solving the Navier-Stokes equation with the no-slip boundary condition and the preset body force distribution. It is shown that the body force induced by the flowing current can generate pairs of electroconvective vortices (electroosmosis of the second kind), where the size of induced vortices is comparable with the intermembrane gap in electrodialysis cells.

Steady-state Ion Transport through a Three-Layered Membrane System: A Mathematical Model Allowing for Violation of the Electroneutrality Condition

A relatively simple mathematical model based on the Poisson equation is considered. The model is intended for modeling transport through multilayered ion-exchange membranes operating at overlimiting currents. The boundary-value problem is solved by a numerical method of parallel shooting and by an approximate method based on the assumption that the charge density is distributed quasi-uniformly. Concentration profiles in diffusion layers and membranes, current–voltage curves, and dependences of effective transport numbers on the current density are examined.

Effect of the chemical nature of the ionogenic groups of ion-exchange membranes on the size of the electroconvective instability region in high-current modes

The size of the electroconvective instability region on the membrane-solution boundary at currents exceeding the limiting diffusion current was measured by laser interferometry. The influence of the chemical nature of the ionogenic groups of ion-exchange membranes on the development of electroconvective instability was studied. The thickness of the electroconvection region decreased as the catalytic activity of the ionogenic groups of commercial and pilot membrane samples with respect to the heterolytic water dissociation increased. The maximum size of the electroconvective instability region and the minimum currents at which it was recorded for the anion-exchange membranes under study were determined for the highly basic modified anion-exchange membrane MA-41M with an almost completely suppressed water dissociation function. A correlation was found between the size of the convective instability region and the characteristic points on the current-voltage curves.

Mathematical Model for the Overlimiting State of an Ion-Exchange Membrane System

A three-layered mathematical model is proposed for describing the overlimiting state in an ion-exchange membrane system. The model’s prominent feature is the allowance for the space-charge region; the water dissociation reaction, which is catalyzed by active ionogenic groups; and the coupled gravitational and electroosmotic convection, which leads to the emergence of dependence of the effective diffusion layer thickness on the electric current density. The model is used for calculating, on the basis of known initial current-voltage curves and dependences of effective transport numbers on the current density, such internal characteristics of the system as the diffusion layer thickness, distribution of concentration of ions, space charge, and electric-field strength at various current densities.

A mathematical model for the bi-ionic potential

The influence of the salt concentration (c0), the diffusion boundary layers, and the ratio of the counter-ions diffusion coefficients on the bi-ionic potential (BIP) values have been analysed theoretically on the basis of a simple mathematical model based on the Nernst-Planck equations and on the TMS model. For c0 < 0.1 M, the numerical solution of the boundary-value problem for an ideal permselective membrane gives a theoretical BIP vs c0 curves similar to those obtained experimentally by Dammak et al. The diffusion coefficient ratio in the membrane plays an important role in determining the BIP values.

Correction of pH of diluted solutions of electrolytes by electrodialysis with bipolar membranes

The current efficiencies of the water dissociation water and the voltage-current characteristics of the bipolar (asymmetric bipolar) membranes were measured in a two-chamber electrochemical cell. The cell was formed of an MB-3 bipolar membrane or an asymmetric bipolar membrane, which is an MA-40 heterogeneous membrane with a thin surface layer in the form of a cation-selective homogeneous film and MA-40 and MA-41 heterogeneous monopolar membranes. The dissociation of water on MA-40 in 0.01 M sodium chloride decreased the current efficiency of the acid and alkali both in the channel with a bipolar membrane and in the channel with an asymmetric bipolar membrane. The effective ion transport numbers across MA-40 and MA-41 at different pH values were determined. The water dissociation rate on MA-40 decreased at pH > 9.5. A kinetic model of the electrodialysis of a dilute solution of sodium chloride in a two-chamber unit cell with a bipolar and anionite membranes was suggested.

A mathematical model describing voltammograms and transport numbers under intensive electrodialysis modes

A three-layer mathematical model of overlimiting state is developed. A reactive layer with a thickness depending on the current density is introduced into the model. A decrease in the thickness of diffusion layer, which donates the counterions, with increasing current density as a result of electroconvection is also taken into account. A boundary-value problem is formulated within the Nernst-Planck and Poisson’s model in the three-layer region with the boundary conditions of constant concentrations in the bulk solution. It is shown that an increase in the reactive layer thickness with increasing current density determines the behavior of effective transport numbers in the overlimiting state of ion-exchange electromembrane system. In the current range under consideration (from 1 to 20 limiting currents), the reactive layer thickness is several tens nanometers and reaches 70 nm at a 100-fold excess over the limiting current. To calculate the voltammograms, the dependence of effective thickness δN of diffusion layer on the current density is required. This dependence can be obtained by solving an inverse problem, from the laser interferometry experiments, or calculated by the Navier-Stokes hydrodynamic model. The model enables one to calculate the distribution of electric field strength, potential, concentrations in the diffusion layers and membrane.

Electric Double Layer at the Membrane/Solution Interface in a Three-Layered Membrane System

Transport of ions through a three-layered membrane system at overlimiting currents is simulated mathematically. To allow for the space charge, the Poisson equation is used in both the first diffusion layer and the membrane. Two modes are shown to exist at overlimiting currents: a quasi-equilibrium state of the interface and a Schottky mode.