Passage of Diffusion-Migration Current Across Electrode/Membrane/Solution System. Part 1: Short-Time Evolution. Binary Electrolyte (Equal Mobilities)

Abstract

The express-method proposed recently for experimental determination of diffusion coefficients of electroactive ions inside a membrane and their distribution coefficients at the membrane/solution boundary (Russ. J. Electrochem., 2022, 58, 1103) is based on the comparison of the measured non-stationary current for the electrode/membrane/electrolyte solution system upon the applying of a potential step with the theoretical expressions for the current–time dependence. Application of this method for the study of bromide-anion transport across the membrane was performed in the previous work under the condition of the membrane permselectivity where the amplitude of the electric field inside the membrane was suppressed owing to a high concentration of non-electroactive counterions. Then, the coion (bromide anion) transport occurred by the diffusional mechanism, for which the solution was available in an analytical form. The present study considers for the first time a non-stationary electrodiffusional transmembrane transport of two singly charged ions (e.g., background cation М⁺ as the counterion and electroactive anion X^– as the coion) having identical diffusion coefficients where the current passage induced a transient electric field in this space, resulting in a deviation from predictions for the diffusional mechanism. It is found that within the short time interval after the applying of the potential step from the membrane equilibrium state to the limiting current regime (where the thickness of the non-stationary diffusion layer is significantly smaller than that of the membrane) the non-stationary distributions of the ion concentrations and of the electric field strength as a function of two variables (the spatial and temporal ones, x and t) can be expressed via a function of one variable, Z(z), where z = x/(4Dt)^1/2. The form of the expression, depending on the ratio of the surface concentration of component X to the fixed charge density inside the membrane (X_m/C_f) has been found by numerical integration. The limiting current varies with time according to the Cottrell formula (I ~ t^–1/2); the dependence of the dimensionless current amplitude, i, on the X_m/C_f ratio is found by numerical calculation; an approximate analytical formula has also been proposed. In particular, the passing current is shown to be close to the diffusion-limited one for a low coion concentration at the membrane/electrolyte solution boundary as compared with the concentration of immobile charged groups inside the membrane (X_m/C_f \( \ll \) 1), whereas the migration contribution to the ionic fluxes doubles the limiting current when the opposite condition (X_m/C_f \( \gg \) 1) is fulfilled.