Competition between diffusion and electroconvection at an ion-selective surface in intensive current regimes

Influence of Concentration Gradients on Electroconvection at a Cation-Exchange Membrane Surface

Membrane-based systems, such as electrodialysis, play an important role in desalination and industrial separation processes. Electrodialysis uses alternating anion- and cation-exchange membranes with a perpendicular electric field to generate concentrated and diluate streams from a feed solution. It is known that under overlimiting current conditions, reduced charge and mass transfer at the membrane interface leads to regions of high ion depletion generating instability and vortices termed electroconvection. While electroconvective mixing is known to directly impact the separation efficiency of electrodialysis, the influence of ion concentration gradients across the membrane experienced in a functional electrodialysis system is not known. Here, we report the influence of ion concentration gradients across a cation exchange membrane (Nafion) that is both aligned with and opposed to the applied electric field. Experiments were conducted by coflowing NaCl solutions of different concentrations (0.1-100 mM) on each side of the membrane, and electroconvection was visualized with a fluorescence dye (Rhodamine 6G). We obtained concentration profiles from fluorescence image data and systematically measured the thickness of the depletion boundary layer dBL under different conditions. We found smaller dBL values at a higher flow rate both with and without concentration gradients. Our results show that electroconvection is enhanced when the electric field is opposite to the direction of the concentration gradient.

Ion Transport in Electromembrane Systems under the Passage of Direct Current: 1D Modelling Approaches

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.

Dynamically Interfacial pH-Buffering Effect Enabled by N-Methylimidazole Molecules as Spontaneous Proton Pumps toward Highly Reversible Zinc-Metal Anodes

Aqueous zinc-metal batteries have attracted extensive attention due to their outstanding merits of high safety and low cost. However, the intrinsic thermodynamic instability of zinc in aqueous electrolyte inevitably results in hydrogen evolution, and the consequent generation of OH^- at the interface will dramatically exacerbate the formation of dead zinc and dendrites. Herein, a dynamically interfacial pH-buffering strategy implemented by N-methylimidazole (NMI) additive is proposed to remove the detrimental OH^- at zinc/electrolyte interface in real-time, thus eliminating the accumulation of by-products fundamentally. Electrochemical quartz crystal microbalance and molecular dynamics simulation results reveal the existence of an interfacial absorption layer assembled by NMI and protonated NMI (NMIH⁺ ), which acts as an ion pump for replenishing the interface with protons constantly. Moreover, an in situ interfacial pH detection method with micro-sized spatial resolution based on the ultra-microelectrode technology is developed to probe the pH evolution in diffusion layer, confirming the stabilized interfacial chemical environment in NMI-containing electrolyte. Accordingly, with the existence of NMI, an excellent cumulative plating capacity of 4.2 Ah cm^-2 and ultrahigh Coulombic efficiency of 99.74% are realized for zinc electrodes. Meanwhile, the NMI/NMIH⁺ buffer additive can accelerate the dissolution/deposition process of MnO₂ /Mn²⁺ on the cathode, leading to enhanced cycling capacity.

Theoretical Analysis of Electroconvection in the Electrodialysis Desalination Channel under the Action of Direct Current

The development of electroconvection in electromembrane systems is a factor that increases the efficiency of the electrolyte solution desalination process. The desalination of the solution, manifested by a change in the distribution of the ion concentration, can affect the mechanisms of development of electroconvection. The purpose of this work is to study the electroconvective flow developing in the desalination channel under various desalination scenarios. The study was carried out on the basis of a mathematical model of the transfer of binary electrolyte ions in the desalination channel formed between the anion and cation exchange membranes under the action of DC current. An analytical estimation of the threshold current density reflecting the conditions of the system transition into a quasi-stationary state has been obtained. The chronopotentiograms of the desalination channel and the thickness of the electroconvective mixing layer are calculated for both pre-threshold and supra-threshold values of the current density.

Estimation of activity coefficients for aqueous organic redox-flow batteries: Theoretical basis and equations

The field of aqueous organic redox flow batteries (AORFBs) has been developing fast in recent years, and many chemistries are starting to emerge as serious contenders for grid-scale storage. The industrial development of these systems would greatly benefit from accurate physics-based models, allowing to optimize battery operation and design. Many authors in the field of flow battery modeling have brought evidence that the dilute solution hypothesis (the assumption that aqueous electrolytes behave ideally) does not hold for these systems and that calculating cell voltage or chemical potentials through concentrations rather than activities, while serviceable, may become insufficient when greater accuracy is required. This article aims to provide the theoretical basis for calculating activity coefficients of aqueous organic electrolytes used in AORFBs to provide tools to predict the concentrated behavior of aqueous electrolytes, thereby improving the accuracy of physics-based models for flow batteries.

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This article reviews the use of concentrated solution theory in flow batteries

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The virial matrix method for estimating activity coefficients is proposed

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Its high accuracy and experimental costs are underlined

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A novel method, the reduced virial matrix, is proposed

Experimental and Theoretical Study of an Autowave Process in a Magnetic Fluid

Magnetic fluid (MF) is a colloidal system consisting of ferromagnetic particles (magnetite) with a diameter of ~10 nm suspended in a dispersion medium of a carrier fluid (for example, kerosene). A distinctive feature of magnetic fluid is the fact that when an electric field is applied to it using two electrodes, thin layers consisting of close-packed particles of the dispersed phase are formed in the regions near the surface of both electrodes. These layers significantly affect the macroscopic properties of the colloidal system. In this work, the interpretation of the near-electrode layer is for the first time given as a new type of liquid membrane, in which the particles of the dispersed phase become charged with the opposite sign. On the basis of experimental studies, we propose a physicochemical mechanism of the autowave process in a cell with a magnetic fluid. It is based on the idea of oppositely recharging colloidal particles of magnetite in a liquid membrane. A mathematical model of an autowave process, which is described by a system of coupled partial differential equations of Nernst–Planck–Poisson and Navier–Stokes with appropriate boundary conditions, is proposed for the first time. One-dimensional, two-dimensional, and three-dimensional versions of the model are considered. The dependence of the frequency of concentration fluctuations on the stationary voltage between the electrodes was obtained, and the time of formation of a liquid membrane was estimated. Qualitative agreement between theoretical and experimental results has been established.

Recent Advancements of UF-Based Separation for Selective Enrichment of Proteins and Bioactive Peptides—A Review

Proteins are one of the primary building blocks that have significant functional properties to be applied in food and pharmaceutical industries. Proteins could be beneficial in their concentrated products or isolates, of which membrane-based filtration methods such as ultrafiltration (UF) encompass application in broad spectra of protein sources. More importantly, selective enrichment by UF is of immense interest due to the presence of antinutrients that may dominate their perspicuous bioactivities. UF process is primarily obstructed by concentration polarization and fouling; in turn, a trade-off between productivity and selectivity emerges, especially when pure isolates are an ultimate goal. Several factors such as operating conditions and membrane equipment could leverage those pervasive contributions; therefore, UF protocols should be optimized for each unique protein mixture and mode of configuration. For instance, employing charged UF membranes or combining UF membranes with electrodialysis enables efficient separation of proteins with a similar molecular weight, which is hard to achieve by the conventional UF membrane. Meanwhile, some proposed strategies, such as utilizing ultrasonic waves, tuning operating conditions, and modifying membrane surfaces, can effectively mitigate fouling issues. A plethora of advancements in UF, from their membrane material modification to the arrangement of new configurations, contribute to the quest to actualize promising potentials of protein separation by UF, and they are reviewed in this paper.

Controlling electrochemical growth of metallic zinc electrodes: Toward affordable rechargeable energy storage systems

Scalable approaches for precisely manipulating the growth of crystals are of broad-based science and technological interest. New research interests have reemerged in a subgroup of these phenomena-electrochemical growth of metals in battery anodes. In this Review, the geometry of the building blocks and their mode of assembly are defined as key descriptors to categorize deposition morphologies. To control Zn electrodeposit morphology, we consider fundamental electrokinetic principles and the associated critical issues. It is found that the solid-electrolyte interphase (SEI) formed on Zn has a similarly strong influence as for alkali metals at low current regimes, characterized by a moss-like morphology. Another key conclusion is that the unique crystal structure of Zn, featuring high anisotropy facets resulting from the hexagonal close-packed lattice with a c/a ratio of 1.85, imposes predominant influences on its growth. In our view, precisely regulating the SEI and the crystallographic features of the Zn offers exciting opportunities that will drive transformative progress.

Scaling laws of electroconvective flow with finite vortex height near permselective membranes

In a steady state, the linear scaling laws are confirmed between the intensity characteristics of electroconvective (EC) vortex (including the vortex height and electroosmotic slip velocity) and the applied voltage for the nonshear EC flow with finite vortex height near permselective membranes. This finding in the nonshear EC flow is different from the shear EC flow [Kwak et al., Phys. Rev. Lett. 110, 114501 (2013)10.1103/PhysRevLett.110.114501] and indicates that the local concentration gradient has a significant improvement in the analysis of slip velocity. Further, our study reveals that the EC vortex is mainly driven by the second peak effect of the Coulomb thrust in the extended space-charge layer, and the linear scaling law exhibited by the Coulomb thrust is an essential reason for the linear scaling laws of vortex intensity. The scaling laws proposed in this paper are supported by our direct numerical simulation data and previous experimental observations [Rubinstein et al., Phys. Rev. Lett. 101, 236101 (2008)10.1103/PhysRevLett.101.236101].

Pattern Formation of Three-Dimensional Electroconvection on a Charge Selective Surface

When a charge selective surface consumes or transports only cations or anions in the electrolyte, biased ion rejection initiates hydrodynamic instability, resulting in vortical fluid motions called electroconvection. In this Letter, we describe the first laboratory observation of three-dimensional electroconvection on a charge selective surface. Combining experiment and scaling analysis, we successfully categorized three distinct patterns of 3D electroconvection according to [(Ra_{E})/(Re^{2}Sc)] [electric Rayleigh number (Ra_{E}), Reynolds number (Re), Schmidt number (Sc)] as (i) polygonal, (ii) transverse, or (iii) longitudinal rolls. If Re increases or Ra_{E} decreases, pure longitudinal rolls are presented. On the other hand, transverse rolls are formed between longitudinal rolls, and two rolls are transformed as polygonal one at higher Ra_{E} or lower Re. In this pattern selection scenario, Sc determines the critical electric Rayleigh number (Ra_{E}^{*}) for the onset of each roll, resulting in Ra_{E}^{*}∼Re^{2}Sc. We also verify that convective ion flux by electroconvection (represented by an electric Nusselt number Nu_{E}) is fitted to a power law, Nu_{E}∼[(Ra_{E}-Ra_{E}^{*})/(Re^{2}Sc)]^{α_{1}}Re^{α_{2}}Pe^{α_{3}} [Péclet number (Pe)], where each term represents the characteristics of electroconvection, shear flow, and ion transport.

Potentiodynamic and Galvanodynamic Regimes of Mass Transfer in Flow-Through Electrodialysis Membrane Systems: Numerical Simulation of Electroconvection and Current-Voltage Curve

Electromembrane devices are usually operated in two electrical regimes: potentiodynamic (PD), when a potential drop in the system is set, and galvanodynamic (GD), when the current density is set. This article theoretically investigates the current-voltage curves (CVCs) of flow-through electrodialysis membrane systems calculated in the PD and GD regimes and compares the parameters of the electroconvective vortex layer for these regimes. The study is based on numerical modelling using a basic model of overlimiting transfer enhanced by electroconvection with a modification of the boundary conditions. The Dankwerts’ boundary condition is used for the ion concentration at the inlet boundary of the membrane channel. The Dankwerts’ condition allows one to increase the accuracy of the numerical implementation of the boundary condition at the channel inlet. On the CVCs calculated for PD and DG regimes, four main current modes can be distinguished: underlimiting, limiting, overlimiting, and chaotic overlimiting. The effect of the electric field regime is manifested in overlimiting current modes, when a significant electroconvection vortex layer develops in the channel.

Mathematical Modeling of the Effect of Water Splitting on Ion Transfer in the Depleted Diffusion Layer Near an Ion-Exchange Membrane

Water splitting (WS) and electroconvection (EC) are the main phenomena affecting ion transfer through ion-exchange membranes in intensive current regimes of electrodialysis. While EC enhances ion transport, WS, in most cases, is an undesirable effect reducing current efficiency and causing precipitation of sparingly soluble compounds. A mathematical description of the transfer of salt ions and H⁺ (OH⁻) ions generated in WS is presented. The model is based on the Nernst–Planck and Poisson equations; it takes into account deviation from local electroneutrality in the depleted diffusion boundary layer (DBL). The current transported by water ions is given as a parameter. Numerical and semi-analytical solutions are developed. The analytical solution is found by dividing the depleted DBL into three zones: the electroneutral region, the extended space charge region (SCR), and the quasi-equilibrium zone near the membrane surface. There is an excellent agreement between two solutions when calculating the concentration of all four ions, electric field, and potential drop across the depleted DBL. The treatment of experimental partial current–voltage curves shows that under the same current density, the surface space charge density at the anion-exchange membrane is lower than that at the cation-exchange membrane. This explains the negative effect of WS, which partially suppresses EC and reduces salt ion transfer. The restrictions of the analytical solution, namely, the local chemical equilibrium assumption, are discussed.

Characterization of MK-40 Membrane Modified by Layers of Cation Exchange and Anion Exchange Polyelectrolytes

Coating of ion exchange membranes used in electrodialysis with layers of polyelectrolytes is a proven approach that allows for the increasing of the limiting current, the suppressing of sedimentation, the controlling of the intensity of generation of H⁺ and OH⁻ ions, and also the improving of monovalent selectivity. However, in the case when two materials with the opposite sign of the charge of fixed groups come in contact, a bipolar boundary is created that can cause undesirable changes in the membrane properties. In this work, we used a MK-40 heterogeneous membrane on the surface of which a layer of polyethyleneimine was applied by adsorption from a solution as a model of heterogeneous membranes modified with oppositely charged polyelectrolyte. It was found that, on one hand, the properties of modified membrane were beneficial for electrodialysis, its limiting current did not decrease and the membrane even acquired a barrier to non-selective electrolyte transport. At the same time, the generation of H⁺ and OH⁻ ions of low intensity arose, even in underlimiting current modes. It was also shown that despite the presence of a layer of polyethyleneimine, the surface charge of the modified membrane remained negative, which we associate with low protonation of polyethyleneimine at neutral pH.

Electro-Kinetic Instability in a Laminar Boundary Layer Next to an Ion Exchange Membrane

The electro-kinetic instability in a pressure driven shear flow near an ion exchange membrane is considered. The electrochemical system, through which an electrical potential drop is applied, consists in a polarization layer in contact with the membrane and a bulk. The numerical investigation contained two aspects: analysis of the instability modes and description of the Lagrangian transport of fluid and ions. Regarding the first aspect, the modes were analyzed as a function of the potential drop. The analysis revealed how the spatial distribution of forces controls the dynamics of vortex association and dissociation. In particular, the birth of a counter-clockwise vortex between two clockwise vortices, and the initiation of clusters constituting one or two envelopes wrapping a vortex group, were examined. In regards to the second aspect, the trajectories were computed with the fourth order Runge Kutta scheme for the time integration and with the biquadratric upstream scheme for the spatial and time interpolation of the fluid velocity and the ion flux. The results for the periodic mode showed two kinds of trajectories: the trochoidal motion and the longitudinal one coupled with a periodic transverse motion. For the aperiodic modes, other mechanisms appeared, such as ejection from the mixing layer, trapping by a growing vortex or merging vortices. The analysis of the local velocity field, the vortices’ shape, the spatial distribution of the forces and the ion flux components explained these trajectories.

2D Mathematical Modelling of Overlimiting Transfer Enhanced by Electroconvection in Flow-Through Electrodialysis Membrane Cells in Galvanodynamic Mode

Flow-through electrodialysis membrane cells are widely used in water purification and the processing of agricultural products (milk, wine, etc.). In the research and operating practice of such systems, a significant place is occupied by a galvanodynamic (or galvanostatic) mode. 2D mathematical modelling of ion transfer in the galvanodynamic mode requires solving the problem of setting the average current density equal to a certain value, while the current density distribution in the system is uneven. This article develops a 2D mathematical model of the overlimiting transfer enhanced by electroconvection in a flow-through electrodialysis cell in the galvanodynamic mode. The model is based on the system of Navier–Stokes, Nernst–Planck, Poisson equations and equations for the electric current stream function. To set the electric mode we use a boundary condition, relating the electric field strength and current density. This approach allows us to describe the formation of the extended space charge region and development of electroconvection at overlimiting currents. For the first time, chronopotentiograms and current–voltage characteristics of the membrane systems are calculated for the galvanodynamic mode taking into account the forced flow and development of electroconvection. The behaviors of the calculated chronopotentiograms and current–voltage characteristic coincide qualitatively with experimental data. The effects of the electrolyte concentration, forced flow velocity and channel size on the mass transfer at overlimiting currents are estimated.

Characterization of Chaotic Electroconvection near Flat Inert Electrodes under Oscillatory Voltages

The onset of electroconvective instability in an aqueous binary electrolyte under external oscillatory electric fields at a single constant frequency is investigated in a 2D parallel flat electrode setup. Direct numerical simulations (DNS) of the Poisson–Nernst–Planck equations coupled with the Navier–Stokes equations at a low Reynolds number are carried out. Previous studies show that direct current (DC) electric field can create electroconvection near ion-selecting membranes in microfluidic devices. In this study, we show that electroconvection can be generated near flat inert electrodes when the applied electric field is oscillatory in time. A range of applied voltage, the oscillation frequency and the ratio of ionic diffusivities is examined to characterize the regime in which electroconvection takes place. Similar to electroconvection under DC voltages, AC electroconvection occurs at sufficiently high applied voltages in units of thermal volts and is characterized by transverse instabilities, physically manifested by an array of counter-rotating vortices near the electrode surfaces. The oscillating external electric field periodically generate and destroy such unsteady vortical structures. As the oscillation frequency is reduced to O(10−1) of the intrinsic resistor–capacitor (RC) frequency of electrolyte, electroconvective instability is considerably amplified. This is accompanied by severe depletion of ionic species outside the thin electric double layer and by vigorous convective transport involving a wide range of scales including those comparable to the distance L between the parallel electrodes. The underlying mechanisms are distinctly nonlinear and multi-dimensional. However, at higher frequencies of order of the RC frequency, the electrolyte response becomes linear, and the present DNS prediction closely resembles those explained by 1D asymptotic studies. Electroconvective instability supports increased electric current across the system. Increasing anion diffusivity results in stronger amplification of electroconvection over all oscillation frequencies examined in this study. Such asymmetry in ionic diffusivity, however, does not yield consistent changes in statistics and energy spectrum at all wall-normal locations and frequencies, implying more complex dynamics and different scaling for electrolytes with unequal diffusivities. Electric current is substantially amplified beyond the ohmic current at high oscillation frequencies. Also, it is found that anion diffusivity higher than cation has stronger impact on smaller-scale motions (≲0.1L).

Modelling of Ion Transport in Electromembrane Systems: Impacts of Membrane Bulk and Surface Heterogeneity

Artificial charged membranes, similar to the biological membranes, are self-assembled nanostructured materials constructed from macromolecules. The mutual interactions of parts of macromolecules leads to phase separation and appearance of microheterogeneities within the membrane bulk. On the other hand, these interactions also cause spontaneous microheterogeneity on the membrane surface, to which macroheterogeneous structures can be added at the stage of membrane fabrication. Membrane bulk and surface heterogeneity affect essentially the properties and membrane performance in the applications in the field of separation (water desalination, salt concentration, food processing and other), energy production (fuel cells, reverse electrodialysis), chlorine-alkaline electrolysis, medicine and other. We review the models describing ion transport in ion-exchange membranes and electromembrane systems with an emphasis on the role of micro- and macroheterogeneities in and on the membranes. Irreversible thermodynamics approach, “solution-diffusion” and “pore-flow” models, the multiphase models built within the effective-medium approach are examined as the tools for describing ion transport in the membranes. 2D and 3D models involving or not convective transport in electrodialysis cells are presented and analysed. Some examples are given when specially designed surface heterogeneity on the membrane surface results in enhancement of ion transport in intensive current electrodialysis.

1D Mathematical Modelling of Non-Stationary Ion Transfer in the Diffusion Layer Adjacent to an Ion-Exchange Membrane in Galvanostatic Mode

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.

Modeling electrokinetics in ionic liquids

Using direct numerical simulations, we provide a thorough study regarding the electrokinetics of ionic liquids. In particular, modified Poisson-Nernst-Planck equations are solved to capture the crowding and overscreening effects characteristic of an ionic liquid. For modeling electrokinetic flows in an ionic liquid, the modified Poisson-Nernst-Planck equations are coupled with Navier-Stokes equations to study the coupling of ion transport, hydrodynamics, and electrostatic forces. Specifically, we consider the ion transport between two parallel charged surfaces, charging dynamics in a nanopore, capacitance of electric double-layer capacitors, electroosmotic flow in a nanochannel, electroconvective instability on a plane ion-selective surface, and electroconvective flow on a curved ion-selective surface. We also discuss how crowding and overscreening and their interplay affect the electrokinetic behaviors of ionic liquids in these application problems.