Addressing Challenges in Ion-Selectivity Characterization in Nanopores.

Ion selectivity is the basis for designing smart nanopore/channel-based devices, e.g., ion separators and biosensors. Quantitative characterization of ion selectivities in nanopores often employs the Nernst or Goldman-Hodgkin-Katz (GHK) equation to interpret transmembrane potentials. However, the direction of the measured transmembrane potential drop is not specified in these equations, and selectivity values calculated using absolute values of transmembrane potentials do not directly reveal the ion for which the membrane is selective. Moreover, researchers arbitrarily choose whether to use the Nernst or GHK equation and overlook the significant differences between them, leading to ineffective quantitative comparisons between studies. This work addresses these challenges through (a) specifying the transmembrane potential (sign) and salt concentrations in terms of working and reference electrodes and the solutions in which they reside when using the Nernst and GHK equations, (b) reporting of both Nernst-selectivity and GHK-selectivity along with solution compositions and transmembrane potentials when comparing different nanopores/channels, and (c) performing simulations to define an ideal selectivity for nanochannels. Experimental and modeling studies provide significant insight into these fundamental equations and guidelines for the development of nanopore/channel-based devices.

Computational Design of an Electro-Membrane Microfluidic-Diode System.

This study uses computational design to explore the performance of a novel electro-membrane microfluidic diode consisting of physically conjugated nanoporous and micro-perforated ion-exchange layers. Previously, such structures have been demonstrated to exhibit asymmetric electroosmosis, but the model was unrealistic in several important respects. This numerical study investigates two quantitative measures of performance (linear velocity of net flow and efficiency) as functions of such principal system parameters as perforation size and spacing, the thickness of the nanoporous layer and the zeta potential of the pore surface. All of these dependencies exhibit pronounced maxima, which is of interest for future practical applications. The calculated linear velocities of net flows are in the range of several tens of liters per square meter per hour at realistically applied voltages. The system performance somewhat declines when the perforation size is increased from 2 µm to 128 µm (with a parallel increase of the inter-perforation spacing) but remains quite decent even for the largest perforation size. Such perforations should be relatively easy to generate using inexpensive equipment.

Kinematic and volumetric analysis of coupled transmembrane fluxes of binary electrolyte solution components.

The paper deals with relationships between the individual transmembrane fluxes of binary electrolyte solution components and the experimentally measurable quantities describing rates of transfer processes, namely, the electric current, the transmembrane volume flow and the rates of concentration changes in the solutions adjacent to the membrane. Also, we collected and rigorously defined the kinetic coefficients describing the membrane selective and electrokinetic properties. A set of useful relationships between these coefficients is derived. An important specificity of the proposed analysis is that it does not use the Irreversible Thermodynamic approach by analyzing no thermodynamic forces that generate the fluxes under consideration. Instead, all the regularities are derived on the basis of conservation and linearity reasons. The terminology "Kinematics of Fluxes" is proposed for such an analysis on the basis of the analogy with Mechanics where Kinematics deals with regularities of motion by considering no mechanic forces. The only thermodynamic steps of the analysis relate to the discussion on the partial molar volumes of electrolyte and ions that are the equilibrium thermodynamic parameters of the adjacent solutions. These parameters are important for interrelating the transmembrane fluxes of the solution components and the transmembrane volume flow. The paper contains short literature reviews concerned with the partial molar volumes of electrolyte and ions: the methods of measurement, the obtained results and their theoretical interpretations. It is concluded from the reviews that the classical theories should be corrected to make them applicable for sufficiently concentrated solutions, 1 M or higher. The proposed correction is taken into account in the kinematic analysis.

Streaming Potential with Ideally Polarizable Electron-Conducting Substrates.

With nonconducting substrates, streaming potential in sufficiently broad (vs Debye screening length) capillaries is well known to be a linear function of applied pressure (and coordinate along the capillary). This study for the first time explores streaming potential with ideally polarizable electron-conducting substrates and shows it to be a nonlinear function of both coordinate and applied pressure. Experimental manifestations can be primarily expected for streaming potentials arising along thin porous electron-conducting films experiencing solvent evaporation from the film side surface. Model predictions are in good qualitative agreement with literature experimental data.

Ion Separations Based on Spontaneously Arising Streaming Potentials in Rotating Isoporous Membranes.

Highly selective ion separations are vital for producing pure salts, and membrane-based separations are promising alternatives to conventional ion-separation techniques. Our previous work demonstrated that simple pressure-driven flow through negatively charged isoporous membranes can separate Li+ and K+ with selectivities as high as 70 in dilute solutions. The separation mechanism relies on spontaneously arising streaming potentials that induce electromigration, which opposes advection and separates cations based on differences in their electrophoretic mobilities. Although the separation technique is simple, this work shows that high selectivities are possible only with careful consideration of experimental conditions including transmembrane pressure, solution ionic strength, the K+/Li+ ratio in the feed, and the extent of concentration polarization. Separations conducted with a rotating membrane show Li+/K+ selectivities as high as 150 with a 1000 rpm membrane rotation rate, but the selectivity decreases to 1.3 at 95 rpm. These results demonstrate the benefits and necessity of quantitative control of concentration polarization in highly selective separations. Increases in solution ionic strength or the K+/Li+ feed ratio can also decrease selectivities more than an order of magnitude.

Evaporation-driven electrokinetic energy conversion: Critical review, parametric analysis and perspectives.

Energy harvesting from evaporation has become a "hot" topic in the last couple of years. Researchers have speculated on several possible mechanisms. Electrokinetic energy conversion is the least hypothetical one. The basics of pressure-driven electrokinetic phenomena of streaming current and streaming potential have long been established. The regularities of evaporation from porous media are also well known. However, "coupling" of these two classes of phenomena has not, yet, been seriously explored. In this critical review, we will recapitalize and combine the available knowledge from these two fields to produce a coherent picture of electrokinetic electricity generation during evaporation from (nano)porous materials. For illustration, we will consider several configurations, namely, single nanopores, arrays of nanopores, systems with reduced area of electrokinetic-conversion elements and devices with side evaporation from thin nanoporous films. For the latter (practically the only one studied experimentally), we will formulate a simple model describing correlations of system performance with such principal parameters as the nanoporous-layer length, width and thickness as well as the pore size, pore-surface hydrophilicity, effective zeta-potential and electric conductivity in nanopores. These correlations will be qualitatively compared with experimental data available in the literature. We will see that experimental data not always are in agreement with the model predictions, which may be due to simplifying model assumptions but also because the mechanisms are different from the classical electrokinetic energy conversion. In particular, this concerns the mechanisms of conversion of evaporation-driven ion streaming currents into electron currents in external circuits. We will also formulate directions of future experimental and theoretical studies that could help clarify these issues.

Nanofiltration of Multi-Ion Solutions: Quantitative Control of Concentration Polarization and Interpretation by Solution-Diffusion-Electro-Migration Model.

For effective use of advanced engineering models of nanofiltration quality of experimental input is crucial, especially in electrolyte mixtures where simultaneous rejections of various ions may be very different. In particular, this concerns the quantitative control of concentration polarization (CP). This work used a rotating disklike membrane test cell with equally accessible membrane surface, so the CP extent was the same over the membrane surface. This condition, which is not satisfied in the conventional membrane test cell, made possible correcting for CP easily even in multi-ion systems. Ion rejections were studied experimentally for several dominant salts (NaCl, MgCl2, Na2SO4 and MgSO4) and trace ions (Na+, NH4+, Cl- and NO3-) using NF270 membrane. The solution-diffusion-electro-migration model was used to obtain ion permeances from the experimental measurements. The model could well fit the experimental data except in the case of NH4+. The correlations between the ion permeances and type of dominant salt are discussed in the context of the established mechanisms of NF such as Donnan and dielectric exclusion. The obtained information contributes to the systematic transport characterization of NF membranes and may be ultimately useful for computational fluid dynamics simulations of the performance of the membranes in various applications.

Analysis of Membrane Transport Equations for Reverse Electrodialysis (RED) Using Irreversible Thermodynamics.

Reverse electrodialysis (RED) is an electro-membrane process for the conversion of mixing energy into electricity. One important problem researchers' face when modeling the RED process is the choice of the proper membrane transport equations. In this study, using experimental data that describe the membrane Nafion 120 in contact with NaCl aqueous solutions, the linear transport equation of irreversible thermodynamics was applied to calculate the power density of the RED system. Various simplifying assumptions about transport equation (i.e., four-, three-, and two-coefficients approaches) are proposed and discussed. We found that the two-coefficients approach, using the membrane conductivity and the apparent transport number of ions, describes the power density with good accuracy. In addition, the influence of the membrane thickness and the concentration polarization on the power density is also demonstrated.

Flow through negatively charged, nanoporous membranes separates Li+ and K+ due to induced electromigration.

Flow through negatively charged nanopores separates Li+ and K+ with selectivities of up to 70 and Li+ passages from 20% to above 100%. Remarkably, both the Li+/K+ selectivity and Li+ passage initially increase with flow rate, breaking the permeability/selectivity trade-off. Modelling demonstrates that flow through the membranes creates electric fields that retard transport of cations. Selectivity increases with flow rate because the K+ electromigration velocity exceeds its convective velocity, but for Li+ electromigration is weaker than convection. Modelling also shows the importance of controlling concentration polarization. With further work, related separations might provide highly pure Li salts for battery manufacturing.

A Limiting Case of Constant Counterion Electrochemical Potentials in the Membrane for Examining Ion Transfer at Ion-Exchange Membranes and Patches.

Ion passage through ion-exchange membranes is vital in electrodialysis desalination, batteries and fuel cells, and water splitting. Simplified models of ion transport through such membranes frequently assume complete exclusion of co-ions (ions with the same sign of charge as the fixed charge in the membrane) from the membrane. However, a second assumption of constant counterion electrochemical potentials across the membrane leads to simple analytical expressions for ion fluxes and transmembrane potentials. Moreover, linear corrections to account for a small membrane electrical resistance yield analytical expressions with a wider applicability. For bi-ionic potential measurements and current-induced concentration polarization at low salt concentrations, these analytical solutions match the fluxes and potentials obtained numerically without the limiting assumptions. This gives confidence in both the limiting assumptions (under appropriate conditions) and the numerical solutions. At low ion concentrations, the analytical solutions may enable rapid characterization of membrane coatings or boundary layers in solution, and such boundary layers are important in many applications of ion-exchange membranes. In fact, the assumption of complete co-ion exclusion is sometimes more limiting than the constraint of constant electrochemical potentials of counterions across the membrane. Remarkably, this limiting case readily yields the ion accumulation and depletion regions above "ion-exchange patches" that reside beneath a solution with an applied electric field. Such regions are important for sample preconcentration in microfluidic devices.

Modelling nanofiltration of electrolyte solutions.

This review critically examines current models for nanofiltration (NF) of electrolyte solutions. We start from linear irreversible thermodynamics, we derive a basic equation set for ion transfer in terms of gradients of ion electrochemical potentials and transmembrane volume flux. These equations are extended to the case of significant differences of thermodynamic forces across the membrane (continuous version of irreversible thermodynamics) and solved in quadratures for single salts and trace ions added to single salts in the case of macroscopically-homogeneous membranes. These solutions reduce to (quasi)analytical expressions in the popular Spiegler-Kedem approximation (composition-independent phenomenological coefficients), which we extend to the case of trace ions. This enables us to identify membrane properties (e.g. ion permeances, ion reflection coefficients, electrokinetic charge density) that control its performance in NF of multi-ion solutions. Further, we specify the phenomenological coefficients of irreversible thermodynamics in terms of ion partitioning, hindrance and diffusion coefficients for the model of straight cylindrical capillaries. The corresponding expressions enable assessment of the applicability of the popular nanopore model of NF. This model (based on the use of macroscopic approaches at nanoscale) leads to a number of trends that have never been observed experimentally. We also show that the use of the Born formula (frequently employed for the description of dielectric exclusion) hardly leads to meaningful values of solvent dielectric constant in membrane pores because this formula disregards the very solvent structure whose changes are supposed to bring about the reduction of dielectric permittivity in nanopores. We conclude that the effect should better be quantified in terms of ion excess solvation energies in the membrane phase. As an alternative to the nanopore description of NF, we review recent work on the development of an advanced engineering model for NF of multi-ion solutions in terms of a solution-diffusion-electromigration mechanism. This model (taking into account spontaneously arising transmembrane electric fields) captures several trends observed experimentally, and the use of trace ions can provide model parameters (ion permeances in the membrane) from experiment. We also consider a recent model (ultrathin barrier layers with deviations from local electroneutrality) that may reproduce observed feed-salt concentration dependences of membrane performance in terms of concentration-independent properties like excess ion solvation energies. Due to its complexity, practical modelling of nanofiltration will probably be performed with advanced engineering models for the foreseeable future. Although mechanistic studies are vital for understanding transport and developing membranes, future simulations in this area will likely need to depart from typical continuum models to provide physical insight. For enhancing the quality of modelling input, it is essential to improve the control of concentration polarization in membrane test cells.

High Selectivities among Monovalent Cations in Dialysis through Cation-Exchange Membranes Coated with Polyelectrolyte Multilayers.

Cation-exchange membranes allow preferential passage of cations over anions, but they show minimal selectivity among cations, which limits their use in ion separations. Recent studies show that modification of cation-exchange membranes with polyelectrolyte multilayers leads to exceptional monovalent/divalent cation electrodialysis selectivities, but no studies report high selectivity among monovalent ions. This work demonstrates that adsorption of protonated poly(allylamine) (PAH)/poly(4-styrenesulfonate) (PSS) multilayers on Nafion membranes leads to high K+/Li+ selectivities in Donnan dialysis, where K+ and Li+ ions in a source phase pass through the membrane and exchange with Na+ ions in a receiving phase. Addition of 0.01 M HNO3 to a source phase containing 0.01 M KNO3 and 0.01 M LiNO3 increases the K+/Li+ selectivity from 8 to ∼60 through (PAH/PSS)5PAH-coated Nafion membranes, primarily because of a ≥fivefold increase in K+ flux. These selectivities are much larger than the ratio of 1.9 for the aqueous diffusion coefficients of K+ and Li+, and uncoated Nafion membranes give a K+/Li+ selectivity <3. Bi-ionic transmembrane potential measurements at neutral pH confirm that the membrane is more permeable to K+ than Li+, but this selectivity is less than in Donnan dialysis with acidic solutions. In situ ellipsometry data indicate that PAH/PSS multilayers (assembled at pH 2.3, 7.5, or 9.3) swell at pH 2.0, and this swelling may open cation-exchange sites that preferentially bind K+ to enable highly selective transport. The coated membranes also exhibit modest selectivity for K+ over H+, suggesting selective transport based on preferential partitioning of K+ into the coatings. Selectivity declines when increasing the source-phase KNO3 concentration to 0.1 M, perhaps because the discriminating transport pathway saturates. Moreover, selectivities are lower in electrodialysis than in Donnan dialysis, presumably because electrodialysis engages other transport mechanisms, such as electroosmosis and strong electromigration.

Current-Induced Concentration Polarization of Nanoporous Media: Role of Electroosmosis.

Current-induced concentration polarization of nanoporous media is explored theoretically by using approach of local thermodynamic equilibrium within nanopore cross-sections. The problem is solved in quadratures in terms of irreversible thermodynamics. The phenomenological coefficients are further specified by using capillary space-charge model for straight slit-like and cylindrical capillaries. This analysis reveals several novel features of current-induced concentration polarization related to the (electro)osmotic volume transfer. It confirms the previous finding that volume transfer can suppress the limiting-current phenomena but obtains more accurate criteria for this. In particular, it shows that the critical nanopore size and/or electrolyte concentration depend on the nanoporous-medium relative thickness. Under no-limiting-current conditions, the salt concentration at the interface between nanoporous medium and unstirred layer is a nonmonotone function of current density. This gives rise to unconventional current-voltage characteristics. Moreover, under certain conditions, the analysis predicts the existence of ranges of "prohibited" current densities where the problem does not have 1D stationary solution, which could give rise to a kind of "phase separation" with coexisting zones of different local current densities corresponding to the same voltage. Besides the advanced understanding of current-induced concentration polarization of nanoporous media, this analysis provides guidelines for the optimization of sample preconcentration systems in (bio)chemical microanalysis.

"Breakthrough" osmosis and unusually high power densities in Pressure-Retarded Osmosis in non-ideally semi-permeable supported membranes.

Osmosis is the movement of solvent across a membrane induced by a solute-concentration gradient. It is very important for cell biology. Recently, it has started finding technological applications in the emerging processes of Forward Osmosis and Pressure-Retarded Osmosis. They use ultrathin and dense membranes supported mechanically by much thicker porous layers. Until now, these processes have been modelled by assuming the membrane to be ideally-semipermeable. We show theoretically that allowing for even minor deviations from ideal semipermeability to solvent can give rise to a previously overlooked mode of "breakthrough" osmosis. Here the rate of osmosis is very large (compared to the conventional mode) and practically unaffected by the so-called Internal Concentration Polarization. In Pressure-Retarded Osmosis, the power densities can easily exceed the conventional mode by one order of magnitude. Much more robust support layers can be used, which is an important technical advantage (reduced membrane damage) in Pressure-Retarded Osmosis.

Deviations from Electroneutrality in Membrane Barrier Layers: A Possible Mechanism Underlying High Salt Rejections.

Reverse osmosis and nanofiltration (NF) employ composite membranes whose ultrathin barrier layers are significantly more permeable to water than to salts. Although solution-diffusion models of salt transport through barrier layers typically assume ubiquitous electroneutrality, in the case of ultrathin selective skins and low ion partition coefficients, space-charge regions may occupy a significant fraction of the membrane barrier layer. This work investigates the implications of these deviations from electroneutrality on salt transport. Both immobile external surface charge and unequal cation and anion solvation energies in the barrier layer lead to regions with excess mobile charge, and the size of these regions increases with decreasing values of either feed concentrations or ion partition coefficients. Moreover, the low concentration of the more excluded ion in the space-charge region can greatly increase resistance to salt transport to enhance salt rejection during NF. These effects are especially pronounced for membranes with a fixed external surface charge density whose sign is the same as that of the more excluded ion in a salt. Because of the space-charge regions, the barrier-layer resistance to salt transport initially rises rapidly with increasing barrier thickness and then plateaus or even declines within a certain thickness range. This trend in resistance implies that thin, defect-free barrier layers will exhibit higher salt rejections than thicker layers during NF at a fixed transmembrane pressure. Deviations from electroneutrality are consistent with both changes in NF salt rejections that occur upon changing the sign of the membrane fixed external surface charge, and CaCl2 rejections that in some cases may first decrease, then increase and then decrease again with increasing CaCl2 concentrations in NF feed solutions.

Coating of Nafion membranes with polyelectrolyte multilayers to achieve high monovalent/divalent cation electrodialysis selectivities.

Electrodialysis (ED) membranes typically exhibit modest selectivities between monovalent and divalent ions. This paper reports a dramatic enhancement of the monovalent/divalent cation selectivities of Nafion 115 membranes through coating with multilayer poly(4-styrenesulfonate) (PSS)/protonated poly(allylamine) (PAH) films. Remarkably, K(+)/Mg(2+) ED selectivities reach values >1000, and similar monovalent/divalent cation selectivities occur with feed solutions containing K(+) and Ca(2+). For comparison, the corresponding K(+)/Mg(2+) selectivity of bare Nafion 115 is only 1.8 ± 0.1. However, with 0.01 M KNO3 and 0.01 M Mg(NO3)2 in the source phase, as the applied current density increases from 1.27 to 2.54 mA cm(-2), the K(+)/Mg(2+) selectivities of coated membranes decrease from >1000 to 22. Water-splitting at strongly overlimiting current densities may lead to a local pH increase close to the membrane surface and alter film permeability or allow passage of Mg(OH)x species to decrease selectivity. When the source phase contains 0.1 M KNO3 and 0.1 M Mg(NO3)2, the K(+) transference number approaches unity and the K(+)/Mg(2+) selectivity is >20,000, presumably because the applied current is below the limiting value for K(+) and H(+) transport is negligible at this high K(+) concentration. The high selectivities of these membranes may enable electrodialysis applications such as purification of salts that contain divalent or trivalent ions.