Abstract
The use of the Nernst–Planck and Poisson (NPP) equations allows computation of the space charge density near solution/electrode or solution/ion-exchange membrane interface. This is important in modelling ion transfer, especially when taking into account electroconvective transport. The most solutions in literature use the condition setting a potential difference in the system (potentiostatic or potentiodynamic mode). However, very often in practice and experiment (such as chronopotentiometry and voltammetry), the galvanostatic/galvanodynamic mode is applied. In this study, a depleted stagnant diffusion layer adjacent to an ion-exchange membrane is considered. In this article, a new boundary condition is proposed, which sets a total current density, i, via an equation expressing the potential gradient as an explicit function of i. The numerical solution of the problem is compared with an approximate solution, which is obtained by a combination of numerical solution in one part of the diffusion layer (including the electroneutral region and the extended space charge region, zone (I) with an analytical solution in the other part (the quasi-equilibrium electric double layer (EDL), zone (II). It is shown that this approach (called the “zonal” model) allows reducing the computational complexity of the problem tens of times without significant loss of accuracy. An additional simplification is introduced by neglecting the thickness of the quasi-equilibrium EDL in comparison to the diffusion layer thickness (the “simplified” model). For the first time, the distributions of concentrations, space charge density and current density along the distance to an ion-exchange membrane surface are computed as functions of time in galvanostatic mode. The calculation of the transition time, τ, for an ion-exchange membrane agree with an experiment from literature. It is suggested that rapid changes of space charge density, and current density with time and distance, could lead to lateral electroosmotic flows delaying depletion of near-surface solution and increasing τ.
Highlights
Mathematical modelling based on the Nernst-Planck and Poisson (NPP) equations built in one, two- and three-dimensional geometry is largely used for describing ion transport in electrode and membrane systems
Mathematical modelling of 1D non-stationary ion transfer in ion-exchange membrane systems under an applied direct current (DC) current is carried out using the NPP equations
The problem allows describing the concentrations of cations and anions, as well as the potential, space charge density and current density as functions of time and distance
Summary
Mathematical modelling based on the Nernst-Planck and Poisson (NPP) equations built in one-, two- and three-dimensional geometry is largely used for describing ion transport in electrode and membrane systems. Ion transfer modelling in this mode encounters serious difficulties associated with the method of specifying the current strength, as a parameter determining the electrical regime For this reason, the existing mathematical models for the galvanostatic mode are based either on the inverse problem method or on the simplifying assumption of local electroneutrality (LEN) [21,22,23,24,25,26]. The boundary condition proposed in Reference [27] requires the calculation of the counterion and co-ion concentrations and their gradients, as well as the potential gradient in the membrane Another novelty of our paper is a simplification of the solution of the boundary-value problem, where a numerical solution in one part of the system (comprising the electroneutral zone and the extended SCR) is combined with an approximate analytical solution in the other part (in the equilibrium zone of the EDL). The most computations in literature were carried out for the λ/δ values, which are far from the parameters of real membrane systems (at least in electrodialysis applications)
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