Manning condensation in ion exchange membranes: A review on ion partitioning and diffusion models

Application of the Born Model to Describe Salt Partitioning in Hydrated Polymers

The classic Born model can be used to predict salt partitioning properties observed in hydrated polymers, but there are often significant quantitative discrepancies between these predictions and the experimental data. Here, we use an updated version of the Born model, reformulated to account for the local environment and mesh size of a hydrated polymer, to describe previously published NaCl, KCl, and LiCl partitioning properties of model cross-linked poly(ethylene glycol) diacrylate polymers. This reformulated Born model describes the influence of polymer structure (i.e., network mesh size and its relationship with water content) and external salt concentration on salt partitioning in the polymers with a significant improvement relative to the classic Born model. The updated model most effectively describes NaCl partitioning properties and provides an additional fundamental understanding of salt partitioning processes, for NaCl, KCl, and LiCl, in hydrated polymers that are of interest for a variety of environmental and biological applications.

A hard sphere model for single-file water transport across biological membranes

We use Gürsey's statistical mechanics of a one-dimensional fluid to find a formula for theP f / P d ratio in the transport of hard spheres across a membrane through a narrow channel that can accommodate molecular movement only in single file. P f is the membrane permeability for osmotic flow and P d the permeability for exchange across the membrane in the absence of osmotic flow. The deviation of the ratio from unity indicates the degree of cooperative transport relative to ordinary diffusion of independent molecules. In contrast to an early idea thatP f / P d must be equal to the number of molecules in the channel, regardless of the physical nature of the interactions among the molecules, we find a functional dependence on the fractional occupancy of the length of the channel by the hard spheres. We also attempt a random walk calculation for P d individually, which gives a result for P f as well when combined with the ratio.

Membrane Design Principles for Ion-Selective Electrodialysis: An Analysis for Li/Mg Separation

Selective electrodialysis (ED) is a promising membrane-based process to separate Li+ from Mg2+, which is the most critical step for Li extraction from brine lakes. This study theoretically compares the ED-based Li/Mg separation performance of different monovalent selective cation exchange membranes (CEMs) and nanofiltration (NF) membranes at the coupon scale using a unified mass transport model, i.e., a solution-friction model. We demonstrated that monovalent selective CEMs with a dense surface thin film like a polyamide film are more effective in enhancing the Li/Mg separation performance than those with a loose but highly charged thin film. Polyamide film-coated CEMs when used in ED have a performance similar to that of polyamide-based NF membranes when used in NF. NF membranes, when expected to replace monovalent selective CEMs in ED for Li/Mg separation, will require a thin support layer with low tortuosity and high porosity to reduce the internal concentration polarization. The coupon-scale performance analysis and comparison provide new insights into the design of composite membranes used for ED-based selective ion-ion separation.

Mobility of Condensed Counterions in Ion-Exchange Membranes: Application of Screening Length Scaling Relationship in Highly Charged Environments

Ion-exchange membranes (IEMs) are widely used in water, energy, and environmental applications, but transport models to accurately simulate ion permeation are currently lacking. This study presents a theoretical framework to predict ionic conductivity of IEMs by introducing an analytical model for condensed counterion mobility to the Donnan-Manning model. Modeling of condensed counterion mobility is enabled by the novel utilization of a scaling relationship to describe screening lengths in the densely charged IEM matrices, which overcame the obstacle of traditional electrolyte chemistry theories breaking down at very high ionic strength environments. Ionic conductivities of commercial IEMs were experimentally characterized in different electrolyte solutions containing a range of mono-, di-, and trivalent counterions. Because the current Donnan-Manning model neglects the mobility of condensed counterions, it is inadequate for modeling ion transport and significantly underestimated membrane conductivities (by up to ≈5× difference between observed and modeled values). Using the new model to account for condensed counterion mobilities substantially improved the accuracy of predicting IEM conductivities in monovalent counterions (to as small as within 7% of experimental values), without any adjustable parameters. Further adjusting the power law exponent of the screen length scaling relationship yielded reasonable precision for membrane conductivities in multivalent counterions. Analysis reveals that counterions are significantly more mobile in the condensed phase than in the uncondensed phase because electrostatic interactions accelerate condensed counterions but retard uncondensed counterions. Condensed counterions still have lower mobilities than ions in bulk solutions due to impedance from spatial effects. The transport framework presented here can model ion migration a priori with adequate accuracy. The findings provide insights into the underlying phenomena governing ion transport in IEMs to facilitate the rational development of more selective membranes.

Ion Partition in Polyelectrolyte Gels and Nanogels

Polyelectrolyte gels provide a load-bearing structural framework for many macroscopic biological tissues, along with the organelles within the cells composing tissues and the extracellular matrices linking the cells at a larger length scale than the cells. In addition, they also provide a medium for the selective transportation and sequestration of ions and molecules necessary for life. Motivated by these diverse problems, we focus on modeling ion partitioning in polyelectrolyte gels immersed in a solution with a single type of ionic valence, i.e., monovalent or divalent salts. Specifically, we investigate the distribution of ions inside the gel structure and compare it with the bulk, i.e., away from the gel structure. In this first exploratory study, we neglect solvation effects in our gel by modeling the gels without an explicit solvent description, with the understanding that such an approach may be inadequate for describing ion partitioning in real polyelectrolyte gels. We see that this type of model is nonetheless a natural reference point for considering gels with solvation. Based on our idealized polymer network model without explicit solvent, we find that the ion partition coefficients scale with the salt concentration, and the ion partition coefficient for divalent ions is higher than for monovalent ions over a wide range of Bjerrum length (lB) values. For gels having both monovalent and divalent salts, we find that divalent ions exhibit higher ion partition coefficients than monovalent salt for low divalent salt concentrations and low lB. However, we also find evidence that the neglect of an explicit solvent, and thus solvation, provides an inadequate description when compared to experimental observations. Thus, in future work, we must consider both ion and polymer solvation to obtain a more realistic description of ion partitioning in polyelectrolyte gels.

Chain Reduction of CHCl3 Photocatalyzed by SPEEK/PVA Films Swollen in Air-Saturated HCO2Na Solutions

Thin cross-linked films containing sulfonated poly(ether etherketone), SPEEK, and poly(vinyl alcohol), PVA, served as efficient photocatalysts for the reduction of CHCl3 when swollen in air-saturated solutions of formate buffers were photolyzed with 350 nm photons. The phototransformation generated CH2Cl2, CO2 and Cl- as products. The utilization of the continuous extraction method coupled with in situ potentiometry enabled kinetic determinations of the reaction progress. Quantum yields of halide ion formation, ϕ(Cl-), larger than 1 were obtained in the presence of air. These findings, together with the occurrence of a post-irradiation Cl- formation, indicated that the photoreduction took place via a chain process. Reductions photoinitiated by swollen films exhibited ϕ(Cl-) values between 3 and 20 times higher than the reactions induced in solutions containing the two polymers. Also, the dependencies of ϕ(Cl-) on CHCl3 or HCO2- concentration diverged significantly from the trends observed using solutions. Most findings are consistent with the occurrence of a reaction mechanism involving SPEEK radicals, •CO2- and •CHCl2 as chain carriers.

Sustainable Lithium Recovery from Hypersaline Salt-Lakes by Selective Electrodialysis: Transport and Thermodynamics

Evaporative technology for lithium mining from salt-lakes exacerbates freshwater scarcity and wetland destruction, and suffers from protracted production cycles. Electrodialysis (ED) offers an environmentally benign alternative for continuous lithium extraction and is amenable to renewable energy usage. Salt-lake brines, however, are hypersaline multicomponent mixtures, and the impact of the complex brine-membrane interactions remains poorly understood. Here, we quantify the influence of the solution composition, salinity, and acidity on the counterion selectivity and thermodynamic efficiency of electrodialysis, leveraging 1250 original measurements with salt-lake brines that span four feed salinities, three pH levels, and five current densities. Our experiments reveal that commonly used binary cation solutions, which neglect Na+ and K+ transport, may overestimate the Li+/Mg2+ selectivity by 250% and underpredict the specific energy consumption (SEC) by a factor of 54.8. As a result of the hypersaline conditions, exposure to salt-lake brine weakens the efficacy of Donnan exclusion, amplifying Mg2+ leakage. Higher current densities enhance the Donnan potential across the solution-membrane interface and ameliorate the selectivity degradation with hypersaline brines. However, a steep trade-off between counterion selectivity and thermodynamic efficiency governs ED's performance: a 6.25 times enhancement in Li+/Mg2+ selectivity is accompanied by a 71.6% increase in the SEC. Lastly, our analysis suggests that an industrial-scale ED module can meet existing salt-lake production capacities, while being powered by a photovoltaic farm that utilizes <1% of the salt-flat area.

Accounting for Ion Pairing Effects on Sulfate Salt Sorption in Cation Exchange Membranes

Ion exchange membranes (IEMs) are frequently used in water treatment and electrochemical applications, with their ion separation properties largely governed by equilibrium ion partitioning between a membrane and contiguous solution. Despite an expansive literature on IEMs, the influence of electrolyte association (i.e., ion pairing) on ion sorption remains relatively unexplored. In this study, salt sorption in two commercial cation exchange membranes equilibrated with 0.01-1.0 M MgSO₄ and Na₂SO₄ is investigated experimentally and theoretically. Association measurements of salt solutions using conductometric experiments and the Stokes-Einstein approximation show significant concentrations of ion pairs in MgSO₄ and Na₂SO₄ relative to those in simple electrolytes (i.e., NaCl), which is consistent with prior studies of sulfate salts. The Manning/Donnan model, developed and validated for halide salts in previous studies, substantially underpredicts sulfate sorption measurements, presumably due to ion pairing effects not accounted for in this established theory. These findings suggest that ion pairing can enhance salt sorption in IEMs due to partitioning of reduced valence species. By reformulating the Donnan and Manning models, a theoretical framework for predicting salt sorption in IEMs that explicitly considers electrolyte association is developed. Remarkably, theoretical predictions of sulfate sorption are improved by over an order of magnitude by accounting for ion speciation. In some cases, good quantitative agreement is observed between theoretical and experimental values for external salt concentrations between 0.1 and 1.0 M using no adjustable parameters.

Ion and Water Transport in Ion-Exchange Membranes for Power Generation Systems: Guidelines for Modeling

Artificial ion-exchange and other charged membranes, such as biomembranes, are self-organizing nanomaterials built from macromolecules. The interactions of fragments of macromolecules results in phase separation and the formation of ion-conducting channels. The properties conditioned by the structure of charged membranes determine their application in separation processes (water treatment, electrolyte concentration, food industry and others), energy (reverse electrodialysis, fuel cells and others), and chlore-alkali production and others. The purpose of this review is to provide guidelines for modeling the transport of ions and water in charged membranes, as well as to describe the latest advances in this field with a focus on power generation systems. We briefly describe the main structural elements of charged membranes which determine their ion and water transport characteristics. The main governing equations and the most commonly used theories and assumptions are presented and analyzed. The known models are classified and then described based on the information about the equations and the assumptions they are based on. Most attention is paid to the models which have the greatest impact and are most frequently used in the literature. Among them, we focus on recent models developed for proton-exchange membranes used in fuel cells and for membranes applied in reverse electrodialysis.

Effect of Dielectric Saturation on Ion Activity Coefficients in Ion Exchange Membranes

Polymeric ion exchange membranes are used in water purification processes to separate ions from water. The distribution and transport of ionic species through these membranes depend on a variety of factors, including membrane charge density, morphology, chemical structure, and the specific ionic species present in the fluid. The electrical potential distribution between membranes and solutions is typically described using models based on Donnan theory. An extension of the original theory is proposed to account for the nonideal behavior of ions both in the fluid and in the membrane as well to provide a more robust description of interactions of solutes with fixed charge groups on the polymer backbone. In this study, the variation in dielectric permittivity in the membrane medium with electric field strength is taken into account in a model based on Gouy-Chapman double-layer theory to provide a more accurate description of ion activity coefficients in an ion exchange membrane. A semianalytical model is presented that accounts for the variation in dielectric permittivity of water in a charged polymer membrane. A comparison of this model with Manning's counterion condensation model clearly demonstrates that by incorporating changes in water dielectric permittivity with electric field strength, much better agreement with experiments can be obtained over a range of salt concentrations for different ions.