Significance of Co-ion Partitioning in Salt Transport through Polyamide Reverse Osmosis Membranes.

Salt permeability of polyamide reverse osmosis (RO) membranes has been shown to increase with increasing feed salt concentration. The dependence of salt permeability on salt concentration has been attributed to the variation of salt partitioning with feed salt concentration. However, studies using various analytical techniques revealed that the salt (total ion) partitioning coefficient decreases with increasing salt concentration, in marked contrast to the observed increase in salt permeability. Herein, we thoroughly investigate the dependence of total ion and co-ion partitioning coefficients on salt concentration and solution pH. The salt partitioning is measured using a quartz crystal microbalance (QCM), while the co-ion partitioning is calculated from the measured salt partitioning using a modified Donnan theory. Our results demonstrate that the co-ion and total ion partitioning behave entirely differently with increasing salt concentrations. Specifically, the co-ion partitioning increased fourfold, while total ion partitioning decreased by 60% as the salt (NaCl) concentration increased from 100 to 800 mM. The increase in co-ion partitioning with increasing salt concentration is in accordance with the increasing trend of salt permeability in RO experiments. We further show that the dependence of salt and co-ion partitioning on salt concentration is much more pronounced at a higher solution pH. The good co-ion exclusion (GCE) model─derived from the solution-friction model─is used to calculate the salt permeability based on the co-ion partitioning coefficients. Our results show that the GCE model predicts the salt permeabilities in RO experiments relatively well, indicating that co-ion partitioning, not salt partitioning, governs salt transport through RO membranes. Our study provides an in-depth understanding of ion partitioning in polyamide RO membranes and its relationship with salt transport.

Exceptional Water Desalination Performance with Anion-Selective Electrodes.

Capacitive deionization (CDI) typically uses one porous carbon electrode that is cation adsorbing and one that is anion adsorbing. In 2016, Smith and Dmello proposed an innovative CDI cell design based on two cation-selective electrodes and a single anion-selective membrane, and thereafter this design was experimentally validated by various authors. In this design, anions pass through the membrane once, and desalinated water is continuously produced. In the present work, this idea is extended, and it is experimentally shown that also a choice for anion-selective electrodes, in combination with a cation-selective membrane, leads to a functional cell design that continuously desalinates water. Anion-selective electrodes are obtained by chemical modification of the carbon electrode with (3-aminopropyl)triethoxysilane. After chemical modification, the activated carbon electrode shows a substantial reduction of the total pore volume and Brunauer-Emmett-Teller (BET) surface area, but nevertheless maintains excellent CDI performance, which is for the first time that a low-porosity carbon electrode is demonstrated as a promising material for CDI.

Adsorption of anionic surfactants in a nonionic polymer brush: experiments, comparison with mean-field theory, and implications for brush-particle interaction.

The adsorption of the anionic surfactants sodium dodecyl sulfate (SDS) and sodium dodecyl benzene sulfonate (SDBS) in poly(ethylene oxide) (PEO) brushes was studied using a fixed-angle optical flow-cell reflectometer. We show that, just as in solution, there is a critical association concentration (CAC) for the surfactants at which adsorption in the PEO brush starts. Above the critical micelle concentration (CMC) the adsorption is found to be completely reversible. At low brush density the adsorption per PEO monomer is equal to the adsorption of these surfactants in bulk solution. However, with increasing brush density, the number of adsorbed surfactant molecules per PEO monomer decreases rapidly. This decrease is explained in terms of excluded volume interactions plus electrostatic repulsion between the negatively charged surfactant micelles. Experimentally, a plateau value in the total adsorption is observed as a function of grafting density. The experimental results were compared to the results of an analytical self-consistent field (aSCF) model, and we found quantitative agreement. Additionally, the model predicts that the plateau value found is in fact a maximum. Both experiments and model calculations show that the adsorption scales directly with the polymerization degree of the polymers in the brush. They also show that an increase in the ionic strength leads to an increase in the adsorbed amount, which is explained as being due to a decrease in the electrostatic penalty for the adsorption of the SDS micelles. The adsorption of SDS micelles changes the interactions of the PEO brush with a silica particle. This is illustrated by atomic force microscopy (AFM) measurements of the pull-off force of a silica particle from a PEO brush: at high enough PEO densities, the addition of SDS leads to a very strong reduction in the force necessary to detach the colloidal silica particle from the PEO brush. We attribute this effect to the large amount of negative charge incorporated in the PEO brush due to SDS adsorption.

Non-Gaussian curvature distribution of actin-propelled biomimetic colloid trajectories.

We analyze the motion of colloids propelled by a comet-like tail of polymerizing actin filaments. The curvature of the particle trajectories deviates strongly from a Gaussian distribution, implying that the underlying microscopic processes are fluctuating in a non-independent manner. Trajectories for beads of different size all showed the same non-Gaussian behavior, while the mean curvature decreased weakly with size. A stochastic simulation that includes nucleation, force-dependent dissociation, growth, and capping of filaments, shows that the non-Gaussian curvature distribution can be explained by a positive feedback mechanism in which attached chains under higher tension are more likely to snap.

Adsorption of the protein bovine serum albumin in a planar poly(acrylic acid) brush layer as measured by optical reflectometry.

The adsorption of bovine serum albumin (BSA) in a planar poly(acrylic acid) (PAA) brush layer has been studied by fixed-angle optical reflectometry. The influence of polymer length, grafting density, and salt concentration is studied as a function of pH. The results are compared with predictions of an analytical polyelectrolyte brush model, which incorporates charge regulation and excluded volume interactions. A maximum in adsorption is found near the point of zero charge (pzc) of the protein. At the maximum, BSA accumulates in a PAA brush to at least 30 vol %. Substantial adsorption continues above the pzc, that is, in the pH range where a net negatively charged protein adsorbs into a negatively charged brush layer, up to a critical pH value. This critical pH value decreases with increasing ionic strength. The adsorbed amount increases strongly with both increasing PAA chain length and increasing grafting density. Experimental data compare well with the analytical model without having to include a nonhomogeneous charge distribution on the protein surface. Instead, charge regulation, which implies that the protein adjusts its charge due to the negative electrostatic potential in the brush, plays an important role in the interpretation of the adsorbed amounts. Together with nonelectrostatic interactions, it explains the significant protein adsorption above the pzc.

Polyelectrolyte brush amplified electroactuation of microcantilevers.

This paper describes the electroactuation of microcantilevers coated on one side with cationic polyelectrolyte brushes. We observed very strong cantilever deflection by alternating the potential on the cantilever between +0.5 and -0.5 V at frequencies up to 0.25 Hz. The actuation resulted from significant increases in the expansive stresses in the polymer brush layer at both negative and positive potentials. However, the deflection at negative bias was significantly larger. We have developed a theoretical framework that correlates conformational changes of the polymer chains in the brush layer with the reorganization of ions due to the potential bias. The model predicts a strong increase in the polymer volume fraction, close to the interface, which results in large expansive stresses that bend the cantilever at negative potentials. The model also predicts that the actuation responds much stronger to negative potentials than positive potentials, as observed in the experiments.

Salt-induced swelling-to-shrinking transition in polyelectrolyte multilayer capsules.

We study the size of polyelectrolyte multilayer capsules as a function of ionic strength, temperature, and time. A dynamic micromechanical model is developed which successfully describes the experiments. The model includes the polymer-solvent surface tension, an electrostatic force which is strongly ionic strength dependent, and a temperature-dependent mobility parameter. The activation of >50 kT suggests that multiple ion pairs must be broken simultaneously in the process of chain rearrangement. In support of our physical model capsules can repeatedly swell and shrink by varying ionic strength.

Phase behavior of mixtures of oppositely charged protein nanoparticles at asymmetric charge ratios.

We present experimental and theoretical results for the phase behavior of mixtures of oppositely charged globular protein molecules in aqueous solutions containing monovalent salt. These colloidal mixtures are interesting model systems, on the one hand for electrolyte solutions ("colloidal ionic liquids"), and on the other for mixtures of oppositely charged (bio)macromolecules, colloids, micelles, etc., with the range of the electrostatic interactions (Debye length) easily tunable from much smaller to much larger than the particle size, simply by adding different amounts of monovalent salt. In this paper we investigate the phase behavior of such mixtures in the case that equally sized colloids have a large difference in charge magnitude. This is possible at any mixing ratio because small ions compensate any colloidal charge asymmetry. Our experimental system is based on lysozyme, a positively charged "hard" globular protein molecule, and succinylated lysozyme, a chemical modification of lysozyme which is negatively charged. By changing the solution pH we can adjust the ratio of charge between the two molecules. To describe phase separation into a dilute phase and a dense "complex" phase, a thermodynamic model is set up in which we combine the Carnahan-Starling-van der Waals equation of state with a heterogeneous Poisson-Boltzmann cell model and include the possibility that protein molecules adjust their charge when they move from one phase to the other (charge regulation). The theory uses the nonelectrostatic attraction strength as the only adjustable parameter and reasonably well reproduces the data in that complexation is only possible at intermediate , not too asymmetric mixing ratios, and low enough ionic strength and temperature.

Self-consistent field theory of protein adsorption in a non-Gaussian polyelectrolyte brush.

To describe adsorption of globular protein molecules in a polyelectrolyte brush we use the strong-stretching approximation of the Edwards self-consistent field equation, combined with corrections for a non-Gaussian brush. To describe chemical potentials in this mixture of (globular) species of widely varying sizes (ions, brush polyelectrolyte segments, globular protein molecules), we use the Boublik-Mansoori-Carnahan-Starling-Leland equation of state derived for polydisperse mixtures of spherical particles. The polyelectrolyte chain is described in this approach as a string of beads with the beads of a size related to the chain diameter. We use the one-dimensional Poisson equation to describe the electrostatic field and include the ionizable character of both the brush polyions and the protein molecules. This model explains the experimental observation of high amounts of protein adsorption in a polyacid brush for values above the isoelectric point of the protein as being due to charge reversal of the protein molecules upon entry in the brush. We find a distinct minimum in protein concentration near the edge of the brush. With increasing this barrier to protein transfer becomes larger, but much less so when we increase the ionic strength, a difference that might relate to an experimentally observed difference in the protein release rate in these two cases. A free energy analysis shows that the release of small ions from the brush and the increase of brush ionization are the two driving forces for protein adsorption in a like-charged brush.

Phase behavior of mixtures of oppositely charged nanoparticles: heterogeneous Poisson-Boltzmann cell model applied to lysozyme and succinylated lysozyme.

We study the phase behavior of mixtures of oppositely charged nanoparticles, both theoretically and experimentally. As an experimental model system we consider mixtures of lysozyme and lysozyme that has been chemically modified in such a way that its charge is nearly equal in magnitude but opposite in sign to that of unmodified lysozyme. We observe reversible macroscopic phase separation that is sensitive not only to protein concentration and ionic strength, but also to temperature. We introduce a heterogeneous Poisson-Boltzmann cell model that generally applies to mixtures of oppositely charged nanoparticles. To account for the phase behavior of our experimental model system, in addition to steric and electrostatic interactions, we need to include a temperature-dependent short-ranged interaction between the lysozyme molecules, the exact origin of which is unknown. The strength and temperature dependence of the short-ranged attraction is found to be of the same order of magnitude as that between unmodified lysozyme molecules. The presence of a rather strong short-ranged attraction in our model system precludes the formation of colloidal liquid phases (or complex coacervates) such as those typically found in mixtures of globular protein molecules and oppositely charged polyelectrolytes.

A modified Poisson-Boltzmann model including charge regulation for the adsorption of ionizable polyelectrolytes to charged interfaces, applied to lysozyme adsorption on silica.

The equilibrium adsorption of polyelectrolytes with multiple types of ionizable groups is described using a modified Poisson-Boltzmann equation including charge regulation of both the polymer and the interface. A one-dimensional mean-field model is used in which the electrostatic potential is assumed constant in the lateral direction parallel to the surface. The electrostatic potential and ionization degrees of the different ionizable groups are calculated as function of the distance from the surface after which the electric and chemical contributions to the free energy are obtained. The various interactions between small ions, surface and polyelectrolyte are self-consistently considered in the model, such as the increase in charge of polyelectrolyte and surface upon adsorption as well as the displacement of small ions and the decrease of permittivity. These interactions may lead to complex dependencies of the adsorbed amount of polyelectrolyte on pH, ionic strength, and properties of the polymer (volume, permittivity, number, and type of ionizable groups) and of the surface (number of ionizable groups, pK, Stern capacity). For the adsorption of lysozyme on silica, the model qualitatively describes the gradual increase of adsorbed amount with pH up to a maximum value at pHc, which is below the iso-electric point, as well as the sharp decrease of adsorbed amount beyond pHc. With increasing ionic strength the adsorbed amount decreases (for pH > pHc), and pHc shifts to lower values.

A modified box model including charge regulation for protein adsorption in a spherical polyelectrolyte brush.

Recent experiments showed significant adsorption of bovine serum albumin (BSA) in spherical polyelectrolyte brushes (SPB) consisting of polyacrylic acid, even for pH values above the isoelectric point of the protein, when both protein and polyion are negatively charged. To describe these experimental findings theoretically, we have constructed a spherical box model for an annealed brush consisting of a weak polyelectrolyte that includes the adsorption of BSA. At equilibrium the chemical potential of BSA in solution equals that at each location in the brush, while the net force on the polyions (including osmotic, stretching, and excluded volume terms) is zero at each location. Protein adsorption is predicted above the isoelectric point and--in agreement with experimental data--is a strong function of ionic strength and pH. Adsorption of protein in the brush is possible because the pH in the brush is below the isoelectric point and protein reverses its charge from negative to positive when it adsorbs.

Electrostatic free energy of interacting ionizable double layers.

The electrostatic contribution to the interaction free energy of charge-regulating materials, similar as well as dissimilar, contains electric work as well as chemical work and can be obtained from an integration over the diffuse part of the double layer together with a summation of the surface contribution to the free energy over the two surfaces. Examples for the surface contribution are given for acid, base, zwitterionic, and amphoteric (1-pK and 2-pK) materials for a diffuse double layer and for the Stern-Gouy-Chapman model, with and without ion adsorption. For charge-regulating materials, the electrostatic contribution to the interaction free energy at contact (adhesion force of curved surfaces, or particles) is always finite and can be obtained from a simple calculation.

Ionizable polyelectrolyte brushes: brush height and electrosteric interaction.

Semi-analytical scaling theory is used to describe quenched and annealed (weakly charged, ionizable, charge-regulating) polyelectrolyte brushes in electrolyte solutions of arbitrary salt concentration. An Alexander-De Gennes box model with homogeneous distribution of polymer segments and the free ends located at the edge of the brush is assumed, as is local electroneutrality in the brush. For annealed polyelectrolyte and in the low-salt regime, the theory predicts that for sufficiently dense brushes, the salt concentration has a small influence on brush height, while the brush expands with increasing grafting density, in agreement with experiment. Expressions are presented for the interaction free energy of compressed ionizable and quenched polyelectrolyte brushes (proportional to the force between particles or curved surfaces). In all cases, the required prefactors are explicitly stated. The theory is compared directly with published experiments on the influence of salt concentration, pH, and grafting density on the thickness and interaction force of polystyrene sulfonate (quenched) and poly(meth)acrylic acid (annealed) brushes. In general, trends are well reproduced but significant deviations remain.

Electrostatic free energy of weakly charged macromolecules in solution and intermacromolecular complexes consisting of oppositely charged polymers.

When oppositely charged polyelectrolytes are mixed in water, attraction between oppositely charged groups may lead to the formation of polyelectrolyte complexes (associative phase separation, complex coacervation, interpolymer complexes). Theory is presented to describe the electrostatic free energy change when ionizable (annealed) (macro-)molecules form a macroscopic polyelectrolyte complex. The electrostatic free energy includes an electric term as well as a chemical term that is related to the dissociation of the ionic groups in the polymer. An example calculation for complexation of polyacid with polybase uses a cylindrical diffuse double layer model for free polymer in solution and electroneutrality within the complex and calculates the free energy of the system when the polymer is in solution or in a polyelectrolyte complex. Combined with a term for the nonelectrostatic free energy change upon complexation, a theoretical stability diagram is constructed that relates pH, salt concentration, and mixing ratio, which is in qualitative agreement with an experimental diagram obtained by Bungenberg de Jong (1949) for complex coacervation of arabic gum and gelatin. The theory furthermore explains the increased tendency toward phase separation when the polymer becomes more strongly charged and suggests that complexation of polyacid or polybase with zwitterionic polymer (e.g., protein) of the same charge sign (at the "wrong side" of the iso-electric point) may be due (in part) to an induced charge reversal of the protein.

Cylindrical cell model for the electrostatic free energy of polyelectrolyte complexes.

Associative phase separation (complex coacervation) in a mixture of oppositely charged polyelectrolytes can lead to different types of (inter-)polyelectrolyte complexes (soluble micelles, macroscopic precipitation). In a previous report [Langmuir 2004, 20, 2785-2791], we presented a model for the electrostatic free energy change when (weakly charged) polyelectrolyte forms a homogeneous complex phase. The influence of ionization of the polymer on the electrostatic free energy of the complex was incorporated but the influence of complex density neglected. In the present effort, cylindrical cells are assumed around each polyelectrolyte chain in the complex, and on the basis of the Poisson-Boltzmann equation, the electrostatic free energy is calculated as a function of the complex density. After combination with Flory-Huggins mixing free energy terms and minimization of the total free energy, the equilibrium complex density is obtained, for a given ratio of polycations to polyanions in the complex. The analysis is used in an example calculation ofpolyelectrolyte film formation by alternatingly applying a polycation and a polyanion solution. The calculation suggests that the often observed exponential growth of a polyelectrolyte film when the polymer is weakly charged has a thermodynamic origin: the polyelectrolyte complex shifts repeatedly between two equilibrium states of different densities and compositions. However, when the polyelectrolytes are strongly charged the difference in the compositions between the two equilibrium states is very small, and exponential growth by an absorption mechanism is no longer possible.

Application of the charge regulation model to transport of ions through hydrophilic membranes: one-dimensional transport model for narrow pores (nanofiltration).

The charge regulation concept is combined with the Navier-Stokes and Nernst-Planck equations to describe the ion retention of nanofiltration membranes consisting of narrow cylindrical pores. The charge regulation approach replaces the assumption of a constant charge or a constant potential at the membrane pore surface, and accounts for the influence of pH, salt concentration, and type of electrolyte on ion retention. In the current model, radial concentration and potential gradients are considered to be negligibly small (valid for narrow enough pores), resulting in a one-dimensional transport description. The model describes typical experimental data for nanofiltration membranes, such as the change of ion retention with pore radius, ion concentration, pH, and pressure both for monovalent and multivalent ions. For a constant solvent velocity, the model in some cases predicts an optimum pore size for retention. Nonequal retentions for anions and cations are predicted at low and high pH values, as well as a minimum solvent velocity for very low salt concentrations. For higher salt concentrations, and at a fixed pressure difference, an increase in solvent velocity with increasing ion concentrations is predicted, in agreement with other one-dimensional transport descriptions found in the literature, but in contrast to some experimental data.

Simplifications of the Poisson-Boltzmann Equation for the Electrostatic Interaction of Close Hydrophilic Surfaces in Water.

Simple solutions of the Poisson-Boltzmann (PB) equation for the electrostatic double-layer interaction of close, planar hydrophilic surfaces in water are evaluated. Four routes, being the weak overlap approximation, the Debye-Hückel linearization based on low electrostatic potentials, the Ettelaie-Buscall linearization based on small variations in the potential, and a new approach based on the fact that concentrations are virtually constant in the gap between close surfaces, are discussed. The Ettelaie-Buscall and constant-concentration approach become increasingly accurate for closer surfaces and are exact for touching surfaces, while the weak overlap approximation is exact for an isolated surface. The Debye-Hückel linearization is valid as long as potentials remain low, independent of separation. In contrast to the Ettelaie-Buscall approach and the weak overlap approximation, the Debye-Hückel linearization and constant-concentration approach can also be used for systems containing multivalent ions. Simulations in which the four approaches are compared with the PB equation for the constant-charge model, the constant-potential model, as being used in the DLVO theory, and the charge-regulation model are presented. Copyright 2001 Academic Press.