Origin of Limiting and Overlimiting Currents in Bipolar Membranes

Bipolar membranes (BPMs), a special class of ion exchange membranes with the unique ability to electrochemically induce either water dissociation or recombination, are of growing interest for environmental applications including eliminating chemical dosage for pH adjustment, resource recovery, valorization of brines, and carbon capture. However, ion transport within BPMs, and particularly at its junction, has remained poorly understood. This work aims to theoretically and experimentally investigate ion transport in BPMs under both reverse and forward bias operation modes, taking into account the production or recombination of H⁺ and OH^-, as well as the transport of salt ions (e.g., Na⁺, Cl^-) inside the membrane. We adopt a model based on the Nernst-Planck theory, that requires only three input parameters─membrane thickness, its charge density, and pK of proton adsorption─to predict the concentration profiles of four ions (H⁺, OH^-, Na⁺, and Cl^-) inside the membrane and the resulting current-voltage curve. The model can predict most of the experimental results measured with a commercial BPM, including the observation of limiting and overlimiting currents, which emerge due to particular concentration profiles that develop inside the BPM. This work provides new insights into the physical phenomena in BPMs and helps identify optimal operating conditions for future environmental applications.

Recent advances and prospects in electrochemical coupling technologies for metal recovery from water

With the development of our society, the desire to recover valuable metal resources from metal-containing wastewaters or natural water bodies is becoming increasingly stronger nowadays. To overcome the limitations of single techniques, coupling technologies with synergistic effects are attracting increasing attention regarding metal resource recovery from water with particular interest in electrochemical coupling technologies in view of the advantages of electrochemical methods. This state-of-the-art review comprehensively presented the mechanisms and performance of electrochemical coupling systems for metal recovery from water. To give a clear overview of current research trends, technologies coupled with electrochemical processes can be categorized into six main types: electrochemical techniques, membrane modules, adsorption/extraction techniques, sonication technologies, energy supply techniques and others. The electrochemical coupling system has shown synergistic advantages (e.g., improving metal recovery efficiency, reducing energy consumption) over single technologies. We then discuss the remaining challenges, present corresponding solutions, and put forward future directions for current electrochemical coupled systems towards metal recovery. This review is conducive to broadening the potential applications of electrochemical coupling processes for metal recovery and sustainable water treatment.

Modeling micropollutant removal by nanofiltration and reverse osmosis membranes: considerations and challenges

Organic micropollutants (OMPs) in drinking water constitute a potential risk to human health; therefore, effective removal of these pollutants is required. Nanofiltration (NF) and reverse osmosis (RO) are promising membrane-based technologies to remove OMPs. In NF and RO, the rejection of OMPs depends on the properties and characteristics of the membrane, the solute, and the solution. In this review, we discuss how these properties can be included in models to study and predict the rejection of OMPs. Initially, an OMP classification is proposed to capture the relevant properties of 58 OMPs. Following the methodology described in this study, more and new OMPs can be easily included in this classification. The classification aims to increase the comprehension and mechanistic understanding of OMP removal. Based on the physicochemical principles used to classify the 58 OMPs, it is expected that other OMPs in the same groups will be similarly rejected. From this classification, we present an overview of the rejection mechanisms involved in the removal of specific OMP groups. For instance, we discuss the removal of OMPs classified as perfluoroalkyl substances (e.g., perfluorooctanoic acid, PFOA). These substances are highly relevant due to their human toxicity at extremely low concentration as well as their persistence and omnipresence in the environment. Finally, we discuss how the rejection of OMPs can be predicted by describing both the membrane-solution interface and calculating the transport of solutes inside the membrane. We illustrate the importance and impact of different rejection mechanisms and interfacial phenomena on OMP removal and propose an extended Nernst-Plank equation to calculate the transport of solutes across the membrane due to convection, diffusion, and electromigration. Finally, we show how the theory discussed in this review leads to improved predictions of OMP rejection by the membranes.

Theory of oil fouling for microfiltration and ultrafiltration membranes in produced water treatment

Due to the complexity of oil-in-water emulsions, the existing literature is still missing a mathematical tool that can describe membrane fouling in a fully quantitative manner on the basis of relevant fouling mechanisms.

HYPOTHESIS

In this work, a quantitative model that successfully describes cake layer formation and pore blocking is presented. We propose that the degree of pore blocking is determined by the membrane contact angle and the resulting surface coverage, while the cake layer is described by a mass balance and a cake erosion flux.

VALIDATION

The model is validated by comparison to experimental data from previous works (Dickhout et al. 2019; Virga et al., 2020) where membrane type, surfactant type and salinity were varied. Most input parameters could be directly taken from the experimental conditions, while four fitting parameters were required.

FINDINGS

The experimental data can be well described by the model which was developed to provide insight into the dominant fouling mechanisms. Moreover, where existing models usually assume that pore blocking precedes cake layer formation, here we find that cake layer formation can start and occur while the degree of pore blocking is still increasing, in line with the more dynamic nature of oil droplets filtration. These new conceptual advances in the field of colloid and interface science open up new pathways for membrane fouling understanding, prevention and control.

Salt and Water Transport in Reverse Osmosis Membranes: Beyond the Solution-Diffusion Model

Understanding the salt-water separation mechanisms of reverse osmosis (RO) membranes is critical for the further development and optimization of RO technology. The solution-diffusion (SD) model is widely used to describe water and salt transport in RO, but it does not describe the intricate transport mechanisms of water molecules and ions through the membrane. In this study, we develop an ion transport model for RO, referred to as the solution-friction model, by rigorously considering the mechanisms of partitioning and the interactions among water, salt ions, and the membrane. Ion transport through the membrane is described by the extended Nernst-Planck equation, with the consideration of frictions between the species (i.e., ion, water, and membrane matrix). Water flow through the membrane is governed by the hydraulic pressure gradient and the friction between the water and membrane matrix as well as the friction between water and ions. The model is validated using experimental measurements of salt rejection and permeate water flux in a lab-scale, cross-flow RO setup. We then investigate the effects of feed salt concentration and hydraulic pressure on salt permeability, demonstrating strong dependence of salt permeability on feed salt concentration and applied pressure, starkly disparate from the SD model. Lastly, we develop a framework to analyze the pressure drop distribution across the membrane, demonstrating that cross-membrane transport dominates the overall pressure drop in RO, in marked contrast to the SD model that assumes no pressure drop across the membrane.

Desalination of Complex Multi-Ionic Solutions by Reverse Osmosis at Different pH Values, Temperatures, and Compositions

For a thorough mechanistic understanding of reverse osmosis (RO), data on ion retention obtained by desalination of multi-ionic solutions are needed. In this paper, we show how to obtain such data under controlled laboratory conditions at any nonextreme pH. For that, we propose a simple method where we use N₂ and CO₂ gas control to set the composition of a gas phase in equilibrium with the feedwater solution. By increasing the CO₂ partial pressure, the pH of the solution will decrease and vice versa. We applied this method of CO₂ gas control to extend and validate an existing data set on ion retention of multi-ionic brackish water with 10 different ionic species, whereas conditions in the prior data set were slightly uncontrolled; in our new analysis, we performed experiments at precisely controlled pH and temperature. We run experiments at pH 6.73 and pH 7.11 and in a temperature range of T = 15-31 °C. Our results show that when pH is decreased, or temperature increased, the ion retention of most ions decreases. We also tested the influence of the Na⁺ to Ca²⁺ concentration ratio in this multi-ionic solution on ion retention at pH 6.73 and T ∼ 31 °C. We noticed that this ratio has a larger effect on ion retention for cations than for anions. We compare our data with the earlier reported data and describe similarities and differences. The improved data set will be an important tool for future development of accurate and validated RO ion transport models. Such RO models that describe desalination performance in detail are important for successful commercial application of the RO technology. We also discuss a relevant preparation method for water slightly oversaturated with barely soluble CaCO₃ by solution preparation at high CO₂ pressure, after which the solution is brought to the required pH by the N₂ and CO₂ gas control method.

Multicomponent mass transport modeling of water desalination by reverse osmosis including ion pair formation

Reverse Osmosis (RO) is one of the main membrane technologies currently used for the desalination of seawater and brackish water to produce freshwater. However, the mechanism of transport and separation of ions in RO membranes is not yet fully understood. Besides acid-base reactions (i.e., including the H⁺-ion), at high concentrations, the salt ions can associate and form ion pairs. In this study, we investigate how to include the formation of these ion pairs in the extended Donnan steric partitioning pore model. We study the desalination of a water source where three ion pairs can be formed (NaCl, MgCl⁺, and MgCl₂) and also include water self-dissociation and the carbonate system. The model assumes infinitely fast reactions, which means that the participating ions are locally at chemical equilibrium with one another. A square stoichiometric reaction matrix composed of active species, moieties, and reactions is formulated. As the final constraint equation, we use the charge balance. The model predicts profiles in concentration, flux, and reaction rates across the membrane for all species and calculates the retention per group of ions. Ion pair formation has an influence on the fluxes of individual ions and therefore influences the retention of ions.

Surfactant-dependent critical interfacial tension in silicon carbide membranes for produced water treatment

During fossil oil extraction, a complex water stream known as produced water (PW), is co-extracted. Membrane treatment makes PW re-use possible, but fouling and oil permeation remain major challenges. In this work, membrane fouling and oil retention of Synthetic PW stabilized with a cationic, anionic, zwitterionic or nonionic surfactant, were studied at various surfactant and salt concentrations. We discuss our results in the framework of the Young-Laplace (YL) equation, which predicts for a given membrane, pressure and oil-membrane contact angle, a critical interfacial tension (IFT) below which oil permeation should occur. We observe such a transition from high to low oil retention with decreasing IFT for the anionic (SDS), cationic (CTAB) and non-ionic (TX) surfactant, but at significantly higher critical IFTs than predicted by YL. On the other side, for the zwitterionic DDAPS we do not observe a drop in oil retention, even at the lowest IFT. The discrepancy between our findings and the critical IFT predicted by YL can be explained by the difference between the measured contact angle and the effective contact angle at the wall of the membrane pores. This leads to a surfactant-dependent critical IFT. Additionally, our results point out that zwitterionic surfactants even at the lowest IFT did not present a critical IFT and exhibited low fouling and low oil permeation.

Layering of bidisperse charged nanoparticles in sedimentation

Bidisperse mixtures of charged nanoparticles form separate layers upon centrifugation as a result of minimization of the system's free energy in sedimentation-diffusion equilibrium. Different factors were investigated experimentally for their effects on the layering, and are supported by theoretical calculations of the full sedimentation profiles. Surprisingly, lighter/smaller nanoparticles can even sink below heavier/larger ones when the particle surface charge is carefully tuned. This study provides deeper insights into the control of layering in polydisperse particle mixtures during sedimentation.

Selective adsorption of nitrate over chloride in microporous carbons

Activated carbon is the most common electrode material used in electrosorption processes such as water desalination with capacitive deionization (CDI). CDI is a cyclic process to remove ions from aqueous solutions by transferring charge from one electrode to another. When multiple salts are present in a solution, the removal of each ionic species can be different, resulting in selective ion separations. This ion selectivity is the result of combined effects, such as differences in the hydrated size and valence of the ions. In the present work, we study ion selectivity from salt mixtures with two different monovalent ions, chloride and nitrate. We run adsorption experiment in microporous carbons (i.e., without applying a voltage), as well as electrosorption experiments (i.e., based on applying a voltage between two carbon electrodes). Our results show that i) during adsorption and electrosorption, activated carbon removes much more nitrate than chloride; ii) at equilibrium, ion selectivity does not depend strongly on the composition of the water, but does depend on charging voltage in CDI; and iii) during electrosorption, ion selectivity is time-dependent. We modify the amphoteric Donnan model by including an additional affinity of nitrate to carbon. We find good agreement between our experimental results and the theory. Both show very high selectivity towards nitrate over chloride, [Formula: see text] ∼10, when no voltage is applied, or when the voltage is low. The selectivity gradually decreases with increasing charging voltage to [Formula: see text] ∼6 at V_ch = 1.2 V. Despite this decrease, the affinity-effect for nitrate continues to play an important role also at a high voltage. In general, we can conclude that our work provides new insights in the importance of carbon-ion interactions for electrochemical water desalination.

Performance metrics for the objective assessment of capacitive deionization systems

In the growing field of capacitive deionization (CDI), a number of performance metrics have emerged to describe the desalination process. Unfortunately, the separation conditions under which these metrics are measured are often not specified, resulting in optimal performance at minimal removal. Here we outline a system of performance metrics and reporting conditions that resolves this issue. Our proposed system is based on volumetric energy consumption (Wh/m³) and throughput productivity (L/h/m²) reported for a specific average concentration reduction, water recovery, and feed salinity. To facilitate and rationalize comparisons between devices, materials, and operation modes, we propose a nominal standard separation of removing 5 mM from a 20 mM NaCl feed solution at 50% water recovery. We propose this particular separation as a standard, but emphasize that the rationale presented here applies irrespective of separation details. Using our proposed separation, we compare the desalination performance of a flow-through electrode (fte-CDI) cell and a flow between membrane (fb-MCDI) device, showing how significantly different systems can be compared in terms of generally desirable desalination characteristics. In general, we find that performance analysis must be considered carefully so to not allow for ambiguous separation conditions or the maximization of one metric at the expense of another. Additionally, for context and clarity, we discuss a number of important underlying performance indicators and cell characteristics that are not performance measures in and of themselves but can be examined to better understand differences in performance.

Wettability of Amphoteric Surfaces: The Effect of pH and Ionic Strength on Surface Ionization and Wetting

We present a novel theory to predict the contact angle of water on amphoteric surfaces, as a function of pH and ionic strength. To validate our theory, experiments were performed on two commonly used amphoteric materials, alumina (Al₂O₃) and titania (TiO₂). We find good agreement at all pH values, and at different salt concentrations. With increasing salt concentration, the theory predicts the contact angle-pH curve to get steeper, while keeping the same contact angle at pH = PZC (point of zero charge), in agreement with data. Our model is based on the amphoteric 1-p K model and includes the electrostatic free energy of an aqueous system as well as the surface energy of a droplet in contact with the surface. In addition, we show how our theory suggests the possibility of a novel responsive membrane design, based on amphoteric groups. At pH ∼ PZC, this membrane resists flow of water but at slightly more acidic or basic conditions the wettability of the membrane pores may change sufficiently to allow passage of water and solutes. Moreover, these membranes could act as active sensors that only allow solutions of high ionic strength to flow through in wastewater treatment.

Energy consumption in capacitive deionization - Constant current versus constant voltage operation

In the field of Capacitive Deionization (CDI), it has become a common notion that constant current (CC) operation consumes significantly less energy than constant voltage operation (CV). Arguments in support of this claim are that in CC operation the endpoint voltage is reached only at the end of the charging step, and thus the average cell voltage during charging is lower than the endpoint voltage, and that in CC operation we can recover part of the invested energy during discharge. Though these arguments are correct, in the present work based on experiments and theory, we conclude that in operation of a well-defined CDI cycle, this does not lead, for the case we analyze, to the general conclusion that CC operation is more energy efficient. Instead, we find that without energy recovery there is no difference in energy consumption between CC and CV operation. Including 50% energy recovery, we find that indeed CC is more energy efficient, but also in CV much energy can be recovered. Important in the analysis is to precisely define the desalination objective function, such as that per unit total operational time -including both the charge and discharge steps- a certain desalination quantity and water recovery must be achieved. Another point is that also in CV operation energy recovery is possible by discharge at a non-zero cell voltage. To aid the analysis we present a new method of data representation where energy consumption is plotted against desalination. In addition, we propose that one must analyze the full range of combinations of cycle times, voltages and currents, and only compare the best cycles, to be able to conclude which operational mode is optimal for a given desalination objective. We discuss three methods to make this analysis in a rigorous way, two experimental and one combining experiments and theory. We use the last method and present results of this analysis.

Theory of water treatment by capacitive deionization with redox active porous electrodes

Capacitive deionization (CDI) for water treatment, which relies on the capture of charged species to sustain the electrical double layers (EDLs) established within porous electrodes under an applied electrical potential, can be enhanced by the chemical attachment of fixed charged groups to the porous electrode electrodes (ECDI). It has recently been demonstrated that further improvements in capacity and energy storage can be gained by functionalization of the electrode surfaces with redox polymers in which the charge on the electrodes can be modulated through Faradaic reactions under different cell voltages in a capacitive process that can be called "Faradaic CDI" (FaCDI). Here, we extend recent mathematical models developed for the characterization of CDI and ECDI systems to incorporate the redox mediated contributions by allowing for the variable chemical charges generated by reactions in FaCDI. The lumped model developed here assumes the spacer channel is well-mixed with uniform electrosorption in each electrode. We demonstrate that the salt adsorption performance characterization of the fixed chemical charge ECDI and variable chemical charge FaCDI materials can be unified within a common theoretical framework based on the point of zero charge (PZC) of the electrode material. In the latter case the PZC is determined by the equilibrium potentials of the redox couples immobilized on the porous electrodes. The new model is able to predict the experimentally observed enhanced and inverted performance of CDI cells, and illuminates the benefit of choosing redox active materials for water treatment applications. The deionization performance of FaCDI cells is shown to be superior to that of CDI and ECDI systems with equilibrium adsorption capacities 50-100% higher than attained with CDI systems, and at smaller cell voltages, depending on the redox potentials of the Faradaic moieties.

Coulometry and Calorimetry of Electric Double Layer Formation in Porous Electrodes

Coulometric measurements on salt-water-immersed nanoporous carbon electrodes reveal, at a fixed voltage, a charge decrease with increasing temperature. During far-out-of-equilibrium charging of these electrodes, calorimetry indicates the production of both irreversible Joule heat and reversible heat, the latter being associated with entropy changes during electric double layer (EDL) formation in the nanopores. These measurements grant experimental access-for the first time-to the entropic contribution of the grand potential; for our electrodes, this amounts to roughly 25% of the total grand potential energy cost of EDL formation at large applied potentials, in contrast with point-charge model calculations that predict 100%. The coulometric and calorimetric experiments show a consistent picture of the role of heat and temperature in EDL formation and provide hitherto unused information to test against EDL models.

Theory of pH changes in water desalination by capacitive deionization

In electrochemical water desalination, a large difference in pH can develop between feed and effluent water. These pH changes can affect the long-term stability of membranes and electrodes. Often Faradaic reactions are implicated to explain these pH changes. However, quantitative theory has not been developed yet to underpin these considerations. We develop a theory for electrochemical water desalination which includes not only Faradaic reactions but also the fact that all ions in the water have different mobilities (diffusion coefficients). We quantify the latter effect by microscopic physics-based modeling of pH changes in Membrane Capacitive Deionization (MCDI), a water desalination technology employing porous carbon electrodes and ion-exchange membranes. We derive a dynamic model and include the following phenomena: I) different mobilities of various ions, combined with acid-base equilibrium reactions; II) chemical surface charge groups in the micropores of the porous carbon electrodes, where electrical double layers are formed; and III) Faradaic reactions in the micropores. The theory predicts small pH changes during desalination cycles in MCDI if we only consider phenomena I) and II), but predicts that these pH changes can be much stronger if we consider phenomenon III) as well, which is in line with earlier statements in the literature on the relevance of Faradaic reactions to explain pH fluctuations.

Theory of linear sweep voltammetry with diffuse charge: Unsupported electrolytes, thin films, and leaky membranes

Linear sweep and cyclic voltammetry techniques are important tools for electrochemists and have a variety of applications in engineering. Voltammetry has classically been treated with the Randles-Sevcik equation, which assumes an electroneutral supported electrolyte. In this paper, we provide a comprehensive mathematical theory of voltammetry in electrochemical cells with unsupported electrolytes and for other situations where diffuse charge effects play a role, and present analytical and simulated solutions of the time-dependent Poisson-Nernst-Planck equations with generalized Frumkin-Butler-Volmer boundary conditions for a 1:1 electrolyte and a simple reaction. Using these solutions, we construct theoretical and simulated current-voltage curves for liquid and solid thin films, membranes with fixed background charge, and cells with blocking electrodes. The full range of dimensionless parameters is considered, including the dimensionless Debye screening length (scaled to the electrode separation), Damkohler number (ratio of characteristic diffusion and reaction times), and dimensionless sweep rate (scaled to the thermal voltage per diffusion time). The analysis focuses on the coupling of Faradaic reactions and diffuse charge dynamics, although capacitive charging of the electrical double layers is also studied, for early time transients at reactive electrodes and for nonreactive blocking electrodes. Our work highlights cases where diffuse charge effects are important in the context of voltammetry, and illustrates which regimes can be approximated using simple analytical expressions and which require more careful consideration.

Induced-Charge Capacitive Deionization: The Electrokinetic Response of a Porous Particle to an External Electric Field

We demonstrate the phenomenon of induced-charge capacitive deionization that occurs around a porous and conducting particle immersed in an electrolyte, under the action of an external electric field. The external electric field induces an electric dipole in the porous particle, leading to its capacitive charging by both cations and anions at opposite poles. This regime is characterized by a long charging time, which results in significant changes in salt concentration in the electrically neutral bulk, on the scale of the particle. We qualitatively demonstrate the effect of advection on the spatiotemporal concentration field, which, through diffusiophoresis, may introduce corrections to the electrophoretic mobility of such particles.

Analysis of ionic conductance of carbon nanotubes

We use space-charge (SC) theory (also called the capillary pore model) to describe the ionic conductance, G, of charged carbon nanotubes (CNTs). Based on the reversible adsorption of hydroxyl ions to CNT pore walls, we use a Langmuir isotherm for surface ionization and make calculations as a function of pore size, salt concentration c, and pH. Using realistic values for surface site density and pK, SC theory well describes published experimental data on the conductance of CNTs. At extremely low salt concentration, when the electric potential becomes uniform across the pore, and surface ionization is low, we derive the scaling G∝sqrt[c], while for realistic salt concentrations, SC theory does not lead to a simple power law for G

Revisiting Morrison and Osterle 1965: the efficiency of membrane-based electrokinetic energy conversion

We revisit Morrison and Osterle (1965) who derived a phenomenological expression for the 'figure-of-merit' [Formula: see text] of the electrokinetic energy conversion (EKEC) of a pressure difference into electric energy (and vice versa) using charged nanotubes, nanopores or ion-exchange membranes. We show the equivalence with Morrison and Osterle of a novel expression of [Formula: see text] derived by Bentien et al (2013). We analyze two physical models for ionic and solvent flow which directly relate [Formula: see text] to nanopore characteristics such as pore size and wall charge density. For the uniform potential model, we derive an analytical expression as a function of pore size, viscosity, ion diffusion coefficients and membrane charge density, and compare results with the full space-charge model by Osterle and co-workers as a function of pore size and ion diffusion coefficient. We present a novel expression for [Formula: see text] for salt solutions with ions with unequal diffusion coefficients (mobilities) and show that to increase [Formula: see text] the counterion mobility must be low and the coion mobility high.

Analysis of electrolyte transport through charged nanopores

We revisit the classical problem of flow of electrolyte solutions through charged capillary nanopores or nanotubes as described by the capillary pore model (also called "space charge" theory). This theory assumes very long and thin pores and uses a one-dimensional flux-force formalism which relates fluxes (electrical current, salt flux, and fluid velocity) and driving forces (difference in electric potential, salt concentration, and pressure). We analyze the general case with overlapping electric double layers in the pore and a nonzero axial salt concentration gradient. The 3×3 matrix relating these quantities exhibits Onsager symmetry and we report a significant new simplification for the diagonal element relating axial salt flux to the gradient in chemical potential. We prove that Onsager symmetry is preserved under changes of variables, which we illustrate by transformation to a different flux-force matrix given by Gross and Osterle [J. Chem. Phys. 49, 228 (1968)JCPSA60021-960610.1063/1.1669814]. The capillary pore model is well suited to describe the nonlinear response of charged membranes or nanofluidic devices for electrokinetic energy conversion and water desalination, as long as the transverse ion profiles remain in local quasiequilibrium. As an example, we evaluate electrical power production from a salt concentration difference by reverse electrodialysis, using an efficiency versus power diagram. We show that since the capillary pore model allows for axial gradients in salt concentration, partial loops in current, salt flux, or fluid flow can develop in the pore. Predictions for macroscopic transport properties using a reduced model, where the potential and concentration are assumed to be invariant with radial coordinate ("uniform potential" or "fine capillary pore" model), are close to results of the full model.

Complementary surface charge for enhanced capacitive deionization

Commercially available activated carbon cloth electrodes are treated using nitric acid and ethylenediamine solutions, resulting in chemical surface charge enhanced carbon electrodes for capacitive deionization (CDI) applications. Surface charge enhanced electrodes are then configured in a CDI cell to examine their salt removal at a fixed charging voltage and both reduced and opposite polarity discharge voltages, and subsequently compared to the salt removal of untreated electrodes. Substantially improved salt removal due to chemical surface charge and the use of a discharge voltage of opposite sign to the charging voltage is clearly demonstrated in these CDI cycling tests, an observation which for the first time validates both enhanced CDI and extended-voltage CDI effects predicted by the Donnan model [Biesheuvel et al., Colloids Interf. Sci. Comm., 10.1016/j.colcom.2015.12.001 (2016)]. Our experimental and theoretical results demonstrate that the use of carbon electrodes with optimized chemical surface charge can extend the CDI working voltage window through discharge voltages of opposite sign to the charging voltage, which can significantly enhance the salt adsorption capacity of CDI electrodes. Thus, in addition to carbon pore size distribution, chemical surface charge in carbon micropores is considered foundational for salt removal in CDI cells.

Resistance identification and rational process design in Capacitive Deionization

Capacitive Deionization (CDI) is an electrochemical method for water desalination employing porous carbon electrodes. To enhance the performance of CDI, identification of electronic and ionic resistances in the CDI cell is important. In this work, we outline a method to identify these resistances. We illustrate our method by calculating the resistances in a CDI cell with membranes (MCDI) and by using this knowledge to improve the cell design. To identify the resistances, we derive a full-scale MCDI model. This model is validated against experimental data and used to calculate the ionic resistances across the MCDI cell. We present a novel way to measure the electronic resistances in a CDI cell, as well as the spacer channel thickness and porosity after assembly of the MCDI cell. We identify that for inflow salt concentrations of 20 mM the resistance is mainly located in the spacer channel and the external electrical circuit, not in the electrodes. Based on these findings, we show that the carbon electrode thickness can be increased without significantly increasing the energy consumption per mol salt removed, which has the advantage that the desalination time can be lengthened significantly.

Adsorption of charged and neutral polymer chains on silica surfaces: the role of electrostatics, volume exclusion, and hydrogen bonding

We develop an off-lattice (continuum) model to describe the adsorption of neutral polymer chains and polyelectrolytes to surfaces. Our continuum description allows taking excluded volume interactions between polymer chains and ions directly into account. To implement those interactions, we use a modified hard-sphere equation of state, adapted for mixtures of connected beads. Our model is applicable to neutral, charged, and ionizable surfaces and polymer chains alike and accounts for polarizability effects of the adsorbed layer and chemical interactions between polymer chains and the surface. We compare our model predictions to data of a classical system for polymer adsorption: neutral poly(N-vinylpyrrolidone) (PVP) on silica surfaces. The model shows that PVP adsorption on silica is driven by surface hydrogen bonding with an effective maximum binding energy of about 1.3k(B)T per PVP segment at low pH. As the pH increases, the Si-OH groups become increasingly dissociated, leading to a lower capacity for H bonding and simultaneous counterion accumulation and volume exclusion close to the surface. Together these effects result in a characteristic adsorption isotherm, with the adsorbed amount dropping sharply at a critical pH. Using this model for adsorption data on silica surfaces cleaned by either a piranha solution or an O(2) plasma, we find that the former have a significantly higher density of silanol groups.

Energy from CO2 using capacitive electrodes - a model for energy extraction cycles

A model is presented for the process of harvesting electrical energy from CO2 emissions using capacitive cells. The principle consists of controlling the mixing process of a concentrated CO2 gas stream with a dilute CO2 gas stream (as, for example, exhaust gas and air), thereby converting part of the released mixing energy into electrical energy. The model describes the transient reactive transport of CO2 gas absorbed in water or in monoethanolamine (MEA) solutions, under the assumption of local chemical equilibrium. The model combines the selective transport of ions through ion-exchange membranes, the accumulation of charge in the porous carbon electrodes and the coupling between the ionic current and the produced electrical current and power. We demonstrate that the model can be used to calculate the energy that can be extracted by mixing concentrated and dilute CO2 containing gas streams. Our calculation results for the process using MEA solutions have various counterintuitive features, including: 1. When dynamic equilibrium is reached in the cyclical process, the electrical charge in the anode is negative both during charging and discharging; 2. Placing an anion-exchange membrane (AEM) in the system is not required, the energy per cycle is just as large with or without an AEM.

Theory of ion transport with fast acid-base equilibrations in bioelectrochemical systems

Bioelectrochemical systems recover valuable components and energy in the form of hydrogen or electricity from aqueous organic streams. We derive a one-dimensional steady-state model for ion transport in a bioelectrochemical system, with the ions subject to diffusional and electrical forces. Since most of the ionic species can undergo acid-base reactions, ion transport is combined in our model with infinitely fast ion acid-base equilibrations. The model describes the current-induced ammonia evaporation and recovery at the cathode side of a bioelectrochemical system that runs on an organic stream containing ammonium ions. We identify that the rate of ammonia evaporation depends not only on the current but also on the flow rate of gas in the cathode chamber, the diffusion of ammonia from the cathode back into the anode chamber, through the ion exchange membrane placed in between, and the membrane charge density.

Sedimentation dynamics and equilibrium profiles in multicomponent mixtures of colloidal particles

In this paper we give a general theoretical framework that describes the sedimentation of multicomponent mixtures of particles with sizes ranging from molecules to macroscopic bodies. Both equilibrium sedimentation profiles and the dynamic process of settling, or its converse, creaming, are modeled. Equilibrium profiles are found to be in perfect agreement with experiments. Our model reconciles two apparently contradicting points of view about buoyancy, thereby resolving a long-lived paradox about the correct choice of the buoyant density. On the one hand, the buoyancy force follows necessarily from the suspension density, as it relates to the hydrostatic pressure gradient. On the other hand, sedimentation profiles of colloidal suspensions can be calculated directly using the fluid density as apparent buoyant density in colloidal systems in sedimentation-diffusion equilibrium (SDE) as a result of balancing gravitational and thermodynamic forces. Surprisingly, this balance also holds in multicomponent mixtures. This analysis resolves the ongoing debate of the correct choice of buoyant density (fluid or suspension): both approaches can be used in their own domain. We present calculations of equilibrium sedimentation profiles and dynamic sedimentation that show the consequences of these insights. In bidisperse mixtures of colloids, particles with a lower mass density than the homogeneous suspension will first cream and then settle, whereas particles with a suspension-matched mass density form transient, bimodal particle distributions during sedimentation, which disappear when equilibrium is reached. In all these cases, the centers of the distributions of the particles with the lowest mass density of the two, regardless of their actual mass, will be located in equilibrium above the so-called isopycnic point, a natural consequence of their hard-sphere interactions. We include these interactions using the Boublik-Mansoori-Carnahan-Starling-Leland (BMCSL) equation of state. Finally, we demonstrate that our model is not limited to hard spheres, by extending it to charged spherical particles, and to dumbbells, trimers and short chains of connected beads.

In situ spatially and temporally resolved measurements of salt concentration between charging porous electrodes for desalination by capacitive deionization

Capacitive deionization (CDI) is an emerging water desalination technique. In CDI, pairs of porous electrode capacitors are electrically charged to remove salt from brackish water present between the electrodes. We here present a novel experimental technique allowing measurement of spatially and temporally resolved salt concentration between the CDI electrodes. Our technique measures the local fluorescence intensity of a neutrally charged fluorescent probe which is collisionally quenched by chloride ions. To our knowledge, our system is the first to measure in situ and spatially resolved chloride concentration in a laboratory CDI cell. We here demonstrate good agreement between our dynamic measurements of salt concentration in a charging, millimeter-scale CDI system to the results of a modified Donnan porous electrode transport model. Further, we utilize our dynamic measurements to demonstrate that salt removal between our charging CDI electrodes occurs on a longer time scale than the capacitive charging time scales of our CDI cell. Compared to typical measurements of CDI system performance (namely, measurements of outflow ionic conductivity), our technique can enable more advanced and better-controlled studies of ion transport in CDI systems, which can potentially catalyze future performance improvements.

Optimization of salt adsorption rate in membrane capacitive deionization

Membrane capacitive deionization (MCDI) is a water desalination technique based on applying a cell voltage between two oppositely placed porous electrodes sandwiching a spacer channel that transports the water to be desalinated. In MCDI, ion-exchange membranes are positioned in front of each porous electrode to prevent co-ions from leaving the electrode region during ion adsorption, thereby enhancing the salt adsorption capacity. MCDI can be operated at constant cell voltage (CV), or at a constant electrical current (CC). In this paper, we present both experimental and theoretical results for desalination capacity and rate in MCDI (both in the CV- and the CC-mode) as function of adsorption/desorption time, salt feed concentration, electrical current, and cell voltage. We demonstrate how by varying each parameter individually, it is possible to systematically optimize the parameter settings of a given system to achieve the highest average salt adsorption rate and water recovery.

Current-induced membrane discharge

Possible mechanisms for overlimiting current (OLC) through aqueous ion-exchange membranes (exceeding diffusion limitation) have been debated for half a century. Flows consistent with electro-osmotic instability have recently been observed in microfluidic experiments, but the existing theory neglects chemical effects and remains to be quantitatively tested. Here, we show that charge regulation and water self-ionization can lead to OLC by "current-induced membrane discharge" (CIMD), even in the absence of fluid flow, in ion-exchange membranes much thicker than the local Debye screening length. Salt depletion leads to a large electric field resulting in a local pH shift within the membrane with the effect that the membrane discharges and loses its ion selectivity. Since salt co-ions, H(+) ions, and OH(-) ions contribute to OLC, CIMD interferes with electrodialysis (salt counterion removal) but could be exploited for current-assisted ion exchange and pH control. CIMD also suppresses the extended space charge that leads to electro-osmotic instability, so it should be reconsidered in both models and experiments on OLC.

Water Desalination with Wires

We show the significant potential of water desalination using a novel capacitive wire-based technology in which anode/cathode wire pairs are constructed from coating a thin porous carbon electrode layer on top of electrically conducting rods (or wires). By alternately dipping an array of electrode pairs in freshwater with and in brine without an applied cell voltage, we create an ion adsorption/desorption cycle. We show experimentally how in six subsequent cycles we can reduce the salinity of 20 mM feed (brackish) water by a factor of 3, while application of a cation exchange membrane on the cathode wires makes the desalination factor increase to 4. Theoretical modeling rationalizes the experimental findings, and predicts that system performance can be significantly enhanced by material modifications. To treat large volumes of water, multiple stacks of wire pairs can be used simultaneously in a "merry-go-round" operational mode.

Water desalination using capacitive deionization with microporous carbon electrodes

Capacitive deionization (CDI) is a water desalination technology in which salt ions are removed from brackish water by flowing through a spacer channel with porous electrodes on each side. Upon applying a voltage difference between the two electrodes, cations move to and are accumulated in electrostatic double layers inside the negatively charged cathode and the anions are removed by the positively charged anode. One of the key parameters for commercial realization of CDI is the salt adsorption capacity of the electrodes. State-of-the-art electrode materials are based on porous activated carbon particles or carbon aerogels. Here we report the use for CDI of carbide-derived carbon (CDC), a porous material with well-defined and tunable pore sizes in the sub-nanometer range. When comparing electrodes made with CDC with electrodes based on activated carbon, we find a significantly higher salt adsorption capacity in the relevant cell voltage window of 1.2-1.4 V. The measured adsorption capacity for four materials tested negatively correlates with known metrics for pore structure of the carbon powders such as total pore volume and BET-area, but is positively correlated with the volume of pores of sizes <1 nm, suggesting the relevance of these sub-nanometer pores for ion adsorption. The charge efficiency, being the ratio of equilibrium salt adsorption over charge, does not depend much on the type of material, indicating that materials that have been identified for high charge storage capacity can also be highly suitable for CDI. This work shows the potential of materials with well-defined sub-nanometer pore sizes for energy-efficient water desalination.

Diffuse charge and Faradaic reactions in porous electrodes

Porous electrodes instead of flat electrodes are widely used in electrochemical systems to boost storage capacities for ions and electrons, to improve the transport of mass and charge, and to enhance reaction rates. Existing porous electrode theories make a number of simplifying assumptions: (i) The charge-transfer rate is assumed to depend only on the local electrostatic potential difference between the electrode matrix and the pore solution, without considering the structure of the double layer (DL) formed in between; (ii) the charge-transfer rate is generally equated with the salt-transfer rate not only at the nanoscale of the matrix-pore interface, but also at the macroscopic scale of transport through the electrode pores. In this paper, we extend porous electrode theory by including the generalized Frumkin-Butler-Volmer model of Faradaic reaction kinetics, which postulates charge transfer across the molecular Stern layer located in between the electron-conducting matrix phase and the plane of closest approach for the ions in the diffuse part of the DL. This is an elegant and purely local description of the charge-transfer rate, which self-consistently determines the surface charge and does not require consideration of reference electrodes or comparison with a global equilibrium. For the description of the DLs, we consider the two natural limits: (i) the classical Gouy-Chapman-Stern model for thin DLs compared to the macroscopic pore dimensions, e.g., for high-porosity metallic foams (macropores >50 nm) and (ii) a modified Donnan model for strongly overlapping DLs, e.g., for porous activated carbon particles (micropores <2 nm). Our theory is valid for electrolytes where both ions are mobile, and it accounts for voltage and concentration differences not only on the macroscopic scale of the full electrode, but also on the local scale of the DL. The model is simple enough to allow us to derive analytical approximations for the steady-state and early transients. We also present numerical solutions to validate the analysis and to illustrate the evolution of ion densities, pore potential, surface charge, and reaction rates in response to an applied voltage.

Direct power production from a water salinity difference in a membrane-modified supercapacitor flow cell

The entropy increase of mixing two solutions of different salt concentrations can be harnessed to generate electrical energy. Worldwide, the potential of this resource, the controlled mixing of river and seawater, is enormous, but existing conversion technologies are still complex and expensive. Here we present a small-scale device that directly generates electrical power from the sequential flow of fresh and saline water, without the need for auxiliary processes or converters. The device consists of a sandwich of porous "supercapacitor" electrodes, ion-exchange membranes, and a spacer and can be further miniaturized or scaled-out. Our results demonstrate that alternating the flow of saline and fresh water through a capacitive cell allows direct autogeneration of voltage and current and consequently leads to power generation. Theoretical calculations aid in providing directions for further optimization of the properties of membranes and electrodes.

Nonlinear dynamics of capacitive charging and desalination by porous electrodes

The rapid and efficient exchange of ions between porous electrodes and aqueous solutions is important in many applications, such as electrical energy storage by supercapacitors, water desalination and purification by capacitive deionization, and capacitive extraction of renewable energy from a salinity difference. Here, we present a unified mean-field theory for capacitive charging and desalination by ideally polarizable porous electrodes (without Faradaic reactions or specific adsorption of ions) valid in the limit of thin double layers (compared to typical pore dimensions). We illustrate the theory for the case of a dilute, symmetric, binary electrolyte using the Gouy-Chapman-Stern (GCS) model of the double layer, for which simple formulae are available for salt adsorption and capacitive charging of the diffuse part of the double layer. We solve the full GCS mean-field theory numerically for realistic parameters in capacitive deionization, and we derive reduced models for two limiting regimes with different time scales: (i) in the "supercapacitor regime" of small voltages and/or early times, the porous electrode acts like a transmission line, governed by a linear diffusion equation for the electrostatic potential, scaled to the RC time of a single pore, and (ii) in the "desalination regime" of large voltages and long times, the porous electrode slowly absorbs counterions, governed by coupled, nonlinear diffusion equations for the pore-averaged potential and salt concentration.

Diffuse-charge effects on the transient response of electrochemical cells

We present theoretical models for the time-dependent voltage of an electrochemical cell in response to a current step, including effects of diffuse charge (or "space charge") near the electrodes on Faradaic reaction kinetics. The full model is based on the classical Poisson-Nernst-Planck equations with generalized Frumkin-Butler-Volmer boundary conditions to describe electron-transfer reactions across the Stern layer at the electrode surface. In practical situations, diffuse charge is confined to thin diffuse layers (DLs), which poses numerical difficulties for the full model but allows simplification by asymptotic analysis. For a thin quasi-equilibrium DL, we derive effective boundary conditions on the quasi-neutral bulk electrolyte at the diffusion time scale, valid up to the transition time, where the bulk concentration vanishes due to diffusion limitation. We integrate the thin-DL problem analytically to obtain a set of algebraic equations, whose (numerical) solution compares favorably to the full model. In the Gouy-Chapman and Helmholtz limits, where the Stern layer is thin or thick compared to the DL, respectively, we derive simple analytical formulas for the cell voltage versus time. The full model also describes the fast initial capacitive charging of the DLs and superlimiting currents beyond the transition time, where the DL expands to a transient non-equilibrium structure. We extend the well-known Sand equation for the transition time to include all values of the superlimiting current beyond the diffusion-limiting current.

Counterion volume effects in mixed electrical double layers

When a monolayer of negatively charged surfactant molecules is brought in contact with an aqueous solution containing mixtures of counterions of different size and valency, very large deviations from Poisson-Boltzmann theory (PBT) develop at a high surface charge, with the smaller counterion outcompeting the larger one (even if divalent) near the interface, leading to counterion segregation [V.L. Shapovalov, G. Brezesinski, J. Phys. Chem. B 110 (2006) 10032]. We use a modified PBT that empirically includes an extended Carnahan-Starling equation-of-state to describe hard-sphere interactions in electrical double layers containing ions of different size and charge. Model calculations are made for ion concentration profiles, free energies, surface pressures, and differential capacities. At high surface charge, volume interactions become important, leading to significant deviations from PBT. In contrast to PBT, at high surface charge, contributions to energy and pressure are no longer mainly entropic, but instead volume and electrostatic field effects now dominate. When the hydrated size of the divalent ion is used as an adjustable parameter, the theory is in good agreement with the experimental data.

Volume exclusion effects in the ground-state dominance approximation for polyelectrolyte adsorption on charged interfaces

We consider the problem of polyelectrolyte molecules adsorbing on oppositely charged interfaces. For sufficiently long chains, the ground-state dominance approximation can be used which results in a (semi-)analytical solution of the self-consistent field equations (aSCF). Whereas existing aSCF theory assumes a low polyelectrolyte density, here the required electrostatic corrections for a high polymer density are implemented. Adsorbed polymer excludes volume for the solvent and small ions, a volume effect that also leads to a reduced dielectric permittivity and a resulting polarization term in the exchange potential. Calculations show the influence of volume exclusion on the polymer density profile.