Search Results (48)

Modeling ion transport through membrane channels is crucial for understanding cellular processes, and Poisson–Nernst–Planck (PNP) equations provide a fundamental continuum framework for such ionic fluxes. We investigate a quasi-one-dimensional steady-state PNP system for two oppositely charged ion species, focusing on how large permanent charges within the channel and realistic boundary conditions impact ion transport. In contrast to classical models that impose ideal electroneutrality at the channel ends (a simplification that eliminates boundary layers near the membrane interfaces), we adopt relaxed neutral boundary conditions that allow small charge imbalances at the boundaries. Using asymptotic analysis treating the large permanent charge as a singular perturbation, we derive explicit first-order expansions for each ionic flux, incorporating boundary layer parameters $(\sigma ,\rho )$ to quantify slight deviations from electroneutrality. This analysis enables a qualitative characterization of individual cation and anion flux behaviors. Notably, we identify two critical transmembrane potentials, $V_{1c}$ and $V_{2c}$, at which the cation and anion fluxes, respectively, vanish, signifying flux-reversal thresholds that delineate distinct monotonic regimes in the flux-voltage response; these critical values depend on the permanent charge magnitude and the boundary layer parameters. We further show that both ionic fluxes exhibit saturation: as the applied voltage becomes extreme, each flux approaches a finite limiting value, with the saturation level modulated by the degree of boundary charge imbalance. Moreover, allowing even small boundary charge deviations reveals non-intuitive discrepancies in flux behavior relative to the ideal electroneutral case. For example, in certain parameter regimes, a large permanent charge that enhances an ionic current under strict electroneutral conditions will instead suppress that current under relaxed-neutral conditions (and vice versa). This new analytical framework exposes subtle yet essential nonlinear dynamics that classical electroneutral assumptions would otherwise obscure. It provides deeper insight into the interplay between large fixed charges and boundary-layer effects, emphasizing the importance of incorporating such realistic boundary conditions to ensure accurate modeling of ion transport through membrane channels. Numerical simulations are performed to provide more intuitive illustrations of our analytical results. Full article

This article proposes a 3D mathematical model of the influence of electrical heterogeneity of the ion exchange membrane surface on the processes of salt ion transfer in membrane systems with axial symmetry; in particular, we investigate an annular membrane disk in the form of a coupled system of Nernst–Planck–Poisson and Navier–Stokes equations in a cylindrical coordinate system. A hybrid numerical–analytical method for solving the boundary value problem is proposed, and a comparison of the results for the annular disk model obtained by the hybrid method and the independent finite element method is carried out. The areas of applicability of each of these methods are determined. The proposed model of an annular disk takes into account electroconvection, which is understood as the movement of an electrolyte solution under the action of an external electric field on an extended region of space charge formed at the solution–membrane boundary under the action of the same electric field. The main regularities and features of the occurrence and development of electroconvection associated with the electrical heterogeneity of the surface of the membrane disk of the annular membrane disk are determined; namely, it is shown that electroconvective vortices arise at the junction of the conductivity and non-conductivity regions at a certain ratio of the potential jump and angular velocity and flow down in the radial direction to the edge of the annular membrane. At a fixed potential jump greater than the limiting one, the formed electroconvective vortices gradually decrease with an increase in the angular velocity of rotation until they disappear. Conversely, at a fixed value of the angular velocity of rotation, electroconvective vortices arise at a certain potential jump, and with its subsequent increase gradually increase in size. Full article

This article is devoted to numerically modeling the effect of the geometric modification of the surfaces of ion-exchange membranes in electromembrane systems (EMSs) on the salt ion transport using a 2D mathematical model of the transport process in the desalination channel based on boundary value problems for the coupled system of Nernst–Planck–Poisson and Navier–Stokes equations. The main patterns of salt ion transport are established taking into account diffusion, electromigration, forced convection, electroconvection, and the geometric modification of the surface of ion-exchange membranes. It is shown that the geometric modification of the surface of ion-exchange membranes significantly changes both the formation and development of electroconvection. A significant combined effect of electroconvection and geometric modification of the surface of ion-exchange membranes in the desalination channel on the salt ion transport is shown, as well as a complex, nonlinear, and non-stationary interaction of all the main effects of concentration polarization in the desalination channel. Full article

This article investigates the boundary value problem for an extended stationary system of Nernst–Planck–Poisson equations, corresponding to a mathematical model of the influence of changes in the equilibrium coefficient on the transport of ions of a binary salt in the diffusion layer. Dimensionless variables were introduced using characteristic parameter values. As a result, a dimensionless boundary value problem was obtained, which is singularly perturbed, containing a small parameter in the derivative of the Poisson equation and, additionally, another regular small parameter. A similarity theory was developed: trivial and non-trivial similarity criteria and their physical meaning were determined, which allowed for the identification of general properties of the solutions. A numerical investigation of the boundary value problem was conducted using the finite element method. With an increase in the initial solution concentration, the value of the small parameter entering singularly decreases, reaching values on the order of 10⁻¹² and below, leading to computational difficulties that prevent a comprehensive analysis of the influence of changes in the equilibrium coefficient on salt ion transport. In this regard, an analytical solution to the problem was constructed, based on dividing the solution domain into several subdomains (regions of electroneutrality, extended space charge region, quasi-equilibrium region, recombination region, intermediate layer), in each of which the problem is solved differently, followed by matching these solutions. Verification of the analytical solution was carried out by comparing it with the numerical solution. The advantage of the obtained analytical solution is the possibility of a comprehensive analysis of the influence of the dissociation/recombination reaction of water molecules on salt ion transport over a wide range of real changes in the concentration and composition of the electrolyte solution and other input parameters. This boundary value problem serves as a benchmark for constructing asymptotic solutions for other singularly perturbed boundary value problems in membrane electrochemistry. Full article

This article proposes various new approximate analytical solutions of the boundary value problem for the non-stationary system of Nernst–Planck–Poisson (NPP) equations in the diffusion layer of an ideally selective ion-exchange membrane at overlimiting current densities. As is known, the diffusion layer in the general case consists of a space charge region and a region of local electroneutrality. The proposed analytical solutions of the boundary value problems for the non-stationary system of Nernst–Planck–Poisson equations are based on the derivation of a new singularly perturbed nonlinear partial differential equation for the potential in the space charge region (SCR). This equation can be reduced to a singularly perturbed inhomogeneous Burgers equation, which, by the Hopf–Cole transformation, is reduced to an inhomogeneous singularly perturbed linear equation of parabolic type. Inside the extended SCR, there is a sufficiently accurate analytical approximation to the solution of the original boundary value problem. The electroneutrality region has a curvilinear boundary with the SCR, and with an unknown boundary condition on it. The article proposes a solution to this problem. The new analytical solution methods developed in the article can be used to study non-stationary boundary value problems of salt ion transfer in membrane systems. The new analytical solution methods developed in the article can be used to study non-stationary boundary value problems of salt ion transport in membrane systems. Full article

A time-dependent electrochemical impedance spectroscopy (EIS) model is presented using the finite element method (FEM) to simulate a 2D interdigitated electrode in an aqueous NaCl electrolyte. Developed in COMSOL Multiphysics, the model incorporates ion transport, electric field distribution, Stern layer effects, and electrode sheet resistance, governed by the Poisson and Nernst–Planck equations. This model can predict the transient current response to an applied excitation voltage, which gives information about the dynamics of the electrochemical system. The simulation results are compared with the experimental data, reproducing key features of the measurements. The transient current response indicates the need for multiple excitation cycles to stabilize the impedance measurement. At low frequencies (<1 kHz), the voltage drop at the Stern layer is significant, while at higher frequencies (>100 kHz), the voltage drop due to sheet resistance dominates. Moreover, the amplitude of the excitation voltage influences the EIS measurement, higher amplitudes (above 0.1 V) lead to non-linear impedance behavior, particularly at low ion concentrations. Discrepancies at low frequencies suggest that Faradaic processes may need to be incorporated for improved accuracy. Overall, this model provides quantitative insights for optimizing EIS sensor design and highlights critical factors for high-frequency and low-concentration conditions, laying the foundation for future biosensing applications with functionalized electrodes. Full article

Pulsed electric field (PEF) modes of electrodialysis (ED) are known for their efficiency in mitigating the fouling of ion-exchange membranes. Many authors have also reported the possibility of increasing the mass transfer/desalination rate and reducing energy costs. In the literature, such possibilities were theoretically studied using 1D modeling, which, however, did not consider the effect of electroconvection. In this paper, the analysis of the ED desalination characteristics of PEF modes is carried out based on a 2D mathematical model including the Nernst–Planck–Poisson and Navier–Stokes equations. Three PEF modes are considered: galvanodynamic (pulses of constant electric current alternate with zero current pauses), potentiodynamic (pulses of constant voltage alternate with zero voltage pauses), and mixed galvanopotentiodynamic (pulses of constant voltage alternate with zero current pauses) modes. It is found that at overlimiting currents, in accordance with previous papers, in the range of relatively low frequencies, the mass transfer rate increases and the energy consumption decreases with increasing frequency. However, in the range of high frequencies, the tendency changes to the opposite. Thus, the best characteristics are obtained at a frequency close to 1 Hz. At higher frequencies, the pulse duration is too short, and electroconvective vortices, enhancing mass transfer, do not have time to develop. Full article

The paper considers the a posteriori error estimates for fully discrete approximations of time-dependent Poisson–Nernst–Planck (PNP) equations, which provide tools that allow for optimizing the choice of each time step when working with adaptive meshes. The equations are discretized by the Backward Euler scheme in time and conforming finite elements in space. Overcoming the coupling of time and the space with a full discrete solution and dealing with nonlinearity by taking G-derivatives of the nonlinear system, the computable, robust, effective, and reliable space–time a posteriori error estimation is obtained. The adaptive algorithm constructed based on the estimates realizes the parallel adaptations of time steps and mesh refinements, which are verified by numerical experiments with the time singular point and adaptive mesh refinement with boundary layer effects. Full article

This article considers a theoretical analysis of the influence of the main coupled effects and spacers on the transfer of salt ions in electromembrane systems (EMS) using a 2D mathematical model of the transfer process in a desalting channel with spacers based on boundary value problems for the coupled system of Nernst–Planck–Poisson and Navier–Stokes equations. The basic patterns of salt ion transport have been established, taking into account diffusion, electromigration, forced convection, electroconvection, dissociation/recombination reactions of water molecules, as well as spacers located inside the desalting channel. It has been shown that spacers and taking into account the dissociation/recombination reaction of water molecules significantly change both the formation and development of electroconvection. This article confirms the fact of the exaltation of the limiting current studied by Harkatz, where it is shown that the current (flux) of salt ions increases when the dissociation reaction begins by a certain value called the exaltation current, which is proportional to the flow of water dissociation products. A significant combined effect of electroconvection and dissociation/recombination reactions as well as the spacer system in the desalting channel on the transport of salt ions are shown. The complex, nonlinear, and non-stationary interaction of all the main effects of concentration polarization and spacers in the desalting channel are also considered in the work. Full article

Electromembrane processes underlie the functioning of electrodialysis devices and nano- and microfluidic devices, the scope of which is steadily expanding. One of the main aspects that determine the effectiveness of membrane systems is the choice of the optimal electrical mode. The solution of this problem, along with experimental studies, requires tools for the theoretical analysis of ion-transport processes in various electrical modes. The system of Nernst–Planck–Poisson and Navier–Stokes (NPP–NS) equations is widely used to describe the overlimiting mass transfer associated with the development of electroconvection. This paper proposes a new approach to describe the electrical mode in a membrane system using the displacement current equation. The equation for the displacement current makes it possible to simulate the galvanodynamic mode, in which the electric field is determined by the given current density. On the basis of the system of Nernst–Planck, displacement current and Navier–Stokes (NPD–NS) equations, a model of the electroconvective overlimiting mass transfer in the diffusion layer at the surface of the ion-exchange membrane in the DC current mode was constructed. Mathematical models based on the NPP–NS and NPD–NS equations, formulated to describe the same physical situation of mass transfer in the membrane system, differ in the peculiarities of numerical solution. At overlimiting currents, the required accuracy of the numerical solution is achieved in the approach based on the NPP–NS equations with a smaller time step than the NPD–NS equation approach. The accuracy of calculating the current density at the boundaries parallel to the membrane surface is higher for the model based on the NPD–NS equations compared to the model based on the NPP–NS equations. Full article

Microfluidic devices are increasingly utilized in numerous industries, including that of medicine, for their abilities to pump and mix fluid at a microscale. Within these devices, microchannels paired with microelectrodes enable the mixing and transportation of ionized fluid. The ionization process charges the microchannel and manipulates the fluid with an electric field. Although complex in operation at the microscale, microchannels within microfluidic devices are easy to produce and economical. This paper uses simulations to convey helpful insights into the analysis of electrokinetic microfluidic device phenomena. The simulations in this paper use the Navier–Stokes and Poisson Nernst–Planck equations solved using COMSOL to determine the maximum attainable fluid velocity with an electric potential applied to the microchannel and the most suitable frequency or voltage to use for transporting the fluid. Alternating current electroosmosis (ACEO) directs and provides velocity to the ionized fluid. ACEO can also mix the fluid at low frequencies for the purpose of dispersing particles. DC electroosmosis (DCEO) applies voltage along the microchannel to create an electric field that ionizes fluid within the microchannel, making it a cost-effective method for transporting fluid. This paper explores a method for an alternate efficient utilization of microfluidic devices for efficient mixing and transportation of ionized fluid and analyzes the electrokinetic phenomena through simulations using the Navier–Stokes and Poisson Nernst–Planck equations. The results provide insights into the parameters at play for transporting the fluid using alternating current electroosmosis (ACEO) and DC electroosmosis (DCEO). Full article

For a theoretical analysis of mass transfer processes in electromembrane systems, the Nernst–Planck and Poisson equations (NPP) are generally used. In the case of 1D direct-current-mode modelling, a fixed potential (for example, zero) is set on one of the boundaries of the considered region, and on the other—a condition connecting the spatial derivative of the potential and the given current density. Therefore, in the approach based on the system of NPP equations, the accuracy of the solution is significantly affected by the accuracy of calculating the concentration and potential fields at this boundary. This article proposes a new approach to the description of the direct current mode in electromembrane systems, which does not require boundary conditions on the derivative of the potential. The essence of the approach is to replace the Poisson equation in the NPP system with the equation for the displacement current (NPD). Based on the system of NPD equations, the concentration profiles and the electric field were calculated in the depleted diffusion layer near the ion-exchange membrane, as well as in the cross section of the desalination channel under the direct current passage. The NPD system, as well as NPP, allows one to describe the formation of an extended space charge region near the surface of the ion-exchange membrane, which is important for describing overlimiting current modes. Comparison of the direct-current-mode modelling approaches based on NPP and NPD showed that the calculation time is less for the NPP approach, but the calculation accuracy is higher for the NPD approach. Full article

In electromembrane systems, the transfer of ions near ion-exchange membranes causes concentration polarization, which significantly complicates mass transfer. Spacers are used to reduce the effect of concentration polarization and increase mass transfer. In this article, for the first time, a theoretical study is carried out, using a two-dimensional mathematical model, of the effect of spacers on the mass transfer process in the desalination channel formed by anion-exchange and cation-exchange membranes under conditions when they cause a developed Karman vortex street. The main idea is that, when the separation of vortices occurs on both sides in turn from the spacer located in the core of the flow where the concentration is maximum, the developed non-stationary Karman vortex street ensures the flow of the solution from the core of the flow alternately into the depleted diffusion layers near the ion-exchange membranes. This reduces the concentration polarization and, accordingly, increases the transport of salt ions. The mathematical model is a boundary value problem for the coupled system of Nernst–Planck–Poisson and Navier–Stokes equations for the potentiodynamic regime. The comparison of the current–voltage characteristics calculated for the desalination channel with and without a spacer showed a significant increase in the intensity of mass transfer due to the development of the Karman vortex street behind the spacer. Full article

The electric AC response of electrolytic cells with DC bias is analyzed solving numerically the Poisson–Nernst–Planck equations and avoiding the commonly used infinite solution approximation. The results show the presence of an additional low-frequency dispersion process associated with the finite spacing of the electrodes. Moreover, we find that the condition of fixed ionic content inside the electrolytic cell has a strong bearing on both the steady-state and the frequency response. For example: the characteristic frequency of the high-frequency dispersion decreases when the DC potential increases and/or the electrode spacing decreases in the closed cell case, while it remains essentially insensitive on these changes for open cells. Finally, approximate analytic expressions for the dependences of the main parameters of both dispersion processes are also presented. Full article

The significance of ion activity in transport through a porous concrete material sample with steel rebar in its center and bathing solution is presented. For the first time, different conventions and models of ion activity are compared in their significance and influence on the ion fluxes. The study closes an interpretational gap between ion activity in a stand-alone (stagnant) electrolyte solution and ion transport (dynamic) through concrete pores. Ionic activity models developed in stationary systems, namely, the Debye–Hückel (DH), extended DH, Davies, Truesdell–Jones, and Pitzer models, were used for modeling the transport of ions driven through the activity gradient. The activities of ions are incorporated into a frame of the Nernst–Planck–Poisson (NPP) equations. Calculations were done with COMSOL software for a real concrete microstructure determined by X-ray computed tomography. The concentration profiles of four ions (Na⁺, Cl⁻, K⁺, OH⁻), the ionic strength, and the electric potential in mortar (with pores) and concrete samples (with aggregates and pores) are presented and compared. The Pitzer equation gave the most reliable results for all systems studied. The difference between the concentration profiles calculated with this equation and with the assumption of the ideality of the solution is negligible while the potential profiles are clearly distinguishable. Full article