Activity coefficients and Donnan coion exclusion in charged membranes with weak-acid fixed charge groups

We have studied theoretically the effects that the dissociation equilibrium of weak-acid fixed charge groups (e.g. carboxyl groups) exerts on the mean activity coefficients in charged membranes using a Donnan formalism. The model calculations indicate that unless carbon dioxide is excluded from the external aqueous solution, the pH of the membrane solution can be low enough to affect significantly the effective fixed charge concentration and the coion exclusion when the membrane fixed charge concentration is high compared with the external solution salt concentration. Although this problem was already pointed out in previous studies, the possibility that the pH and salt concentration effects appeared in the mean activity coefficient was not considered. We have analyzed quantitatively these effects here and shown that they will eventually be reflected in the effective mean activity coefficients obtained experimentally unless an appropriate theoretical model incorporating the dissociation equilibrium equation explicitly is used to analyze the experimental data.     

Activity coefficients and coion exclusion in charged polymeric membranes with macroscopic inhomogeneities

 We consider a simple model to evaluate mean activity coefficients in charged polymeric membranes with an inhomogeneous distribution of the fixed charge concentration on the macroscopic spatial scale, and apply the results obtained to the coion exclusion characteristic of the Donnan equilibrium. Model calcula-tions with two fixed charge distributions of experimental interest (the asymmetric and the skin-core distributions) show that the inhomogeneity effects can be important when M28.8n the external salt solution concentration is lower than the membrane-fixed charge concen-tration, especially if spatial regions of low charge concentration exist within the membrane. The results obtained appear to be in qualitative agreement with previous experiments. Received 1 October 1996 Accepted 5 February 1997     

Membrane Potential of Composite Bipolar Membrane in Ethanol-Water Solutions: The Role of the Membrane Interface

The membrane potential across a composite bipolar membrane (CBM) composed of a cation-exchange membrane with an anion-exchange membrane is theoretically and experimentally analyzed for LiCl ethanol-water solutions. The theoretical approach is based on an extension of the Donnan equilibrium and the Nernst-Planck equation of monopolar charged membranes for the case of two ion-exchange layers by considering the effect of electrolyte ion pairing in the external solution. The experimental results show that the effective membrane charge densities of the two ion-exchange layers will become smaller than those which are separately estimated for each layer. We have introduced a contact factor, zeta, into the theoretical approach to clarify this phenomenon in this study, and the theoretical predictions were in good agreement with the experimental data. The membrane potential measurements show that CBM has the characteristics of a bipolar membrane and can significantly contribute to a better electrochemical characterization of the CBMs. Copyright 1999 Academic Press.     

Tangential streaming potential as a tool in modeling of ion transport through nanoporous membranes

Tangential streaming potential (TSP) measurements have been carried out so as to assess the electrokinetic properties of the active layer of organic nanofiltration (NF) membranes. Due to the porous structure of NF membranes, cares must be taken to convert the experimental data into zeta potential. Indeed, an assumption that is implicitly made in Smoluchowski's theory (or in related approaches accounting for the surface conduction phenomenon) is that both streaming and conduction currents involved in the streaming potential process flow through an identical path. Such an assumption does not hold with porous membranes since the conduction current is expected to flow wherever the electric conductivity differs from zero. Consequently, a non-negligible share of the conduction current is likely to flow through the membrane body filled with the electrolyte solution. This phenomenon has been taken into account by carrying out a series of TSP measurements at various channel heights. Experiments have been conducted with various electrolyte solutions. The inferred zeta potentials have been further converted into membrane volume charge densities which have been used to predict the membrane performances in terms of rejection rates. The conventional NF theory, i.e. based on a steric/Donnan exclusion mechanism, has been found to be unable to describe the experimental rejection rates. Using the volume charge density of the membrane as an adjustable parameter, it has been shown that the conventional theory even predicts the opposite sign for the membrane charge. On the other hand, the experimental rejection rates have been well described by including dielectric effects in the exclusion mechanism. In this case, a noticeable lowering of the effective dielectric constant of the electrolyte solution inside pores has been predicted (with respect to the bulk value).     

Dielectric relaxation on the intermediate layer in a bipolar membrane under the water splitting phenomenon. I. Shift of the dielectric properties by means of time-dependent impedance measurements

In this study, we examined the dielectric properties of an intermediate layer in a bipolar membrane, which is composed of a negatively charged layer and a positively charged layer joined in series. As a result of the time-dependent impedance measurements of charged membranes, the negative increment in electric conductivity and the positive increment in electric capacity were observed only in the case of a bipolar membrane under the application of reverse-biased voltages, which were quite different from the behavior of both monopolar membranes and of a bipolar membrane under forward-biased voltages. Further, the observed shifts showed a nearly constant value against the reverse-biased voltage. It is concluded that these characteristics coincide with the process of ion exclusion in the intermediate layer and are attributed to the water splitting mechanism.     

Equilibrium distribution of permeants in polyelectrolyte microcapsules filled with negatively charged polyelectrolyte: the influence of ionic strength and solvent polarity

The effects of ionic strength and solvent polarity on the equilibrium distribution of fluorescein (FL) and FITC-dextran between the interior of polyelectrolyte multilayer microcapsules filled with negatively charged strong polyelectrolyte and the bulk solution were systematically investigated. A negatively charged strong polyelectrolyte, poly(styrene sulfonate) (PSS), used for CaCO3 core fabrication, was entrapped inside the capsules. Due to the semipermeability of the capsule wall, a Donnan equilibrium between the inner solution within the capsules and the bulk solution was created. The equilibrium distribution of the negatively charged permeants was investigated by means of confocal laser scanning microscopy as a function of ionic strength and solvent polarity. The equilibrium distribution of the negatively charged permeants could be tuned by increasing the bulk ionic strength to decrease the Donnan potential. Decreasing the solvent polarity also could enhance the permeation of FL, which induces a sudden increase of permeation when the ethanol volume fraction was higher than 0.7. This is mainly attributed to the precipitation of PSS. A theoretical model combining the Donnan equilibrium and Manning counterion condensation was employed to discuss the results.     

A novel measurement method of Donnan potential at an interface between a charged membrane and mixed salt solution

We developed a novel measurement method of the Donnan potential difference at a charged membrane/salt solution interface. The method can measure the potential under the condition that the membrane charge density is much lower than the KCl concentration of the salt bridge. This method is very useful for obtaining the effective charge density of each layer of a bipolar membrane. The present experiments in a system of a negatively charged poly(vinyl alcohol) membrane and a single salt solution of KCl, NaCl, LiCl, CaCl₂ and LaC₃ revealed that the membrane effective charged density has the same value for all the ions. The experiments in mixed KCl and CaCl₂ solution revealed that the potential in the system is governed mainly by the concentration of the counterion having the highest valence in the system.     

Influence of the convective term in the Nernst-Planck equation on properties of ion transport through a layer of solution or membrane

The one-dimensional boundary-value problem of steady-state ion transport, which takes into account the convective component, is formulated and solved in terms of the Nernst-Planck model. This problem is investigated in connection with the diffusion layer, which is understood in a broad sense. This can be the diffusion layer as it is usually understood, i.e., located adjacent to a hydraulically permeable membrane. In another context it can be regarded as a capillary connecting two reservoirs filled with solutions of different concentration or as an uncharged macropore permeating the membrane and separating two solutions. Finally, the solution to the problem is applied to the membrane itself, which is represented as a quasi-homogeneous gel. In the latter case, a virtual electroneutral solution in local equilibrium with a small volume of membrane is considered. The problem is investigated in dimensionless form as a function of the Peclet number. It is shown that the Peclet number is numerically equal to the absolute value of the dimensionless convection velocity. The limiting current, concentration profiles, distributions of the field strength and potential, and effective transport numbers are analyzed as functions of the convective component.     

Streaming potential measurements as a characterization method for nanofiltration membranes

The streaming potentials of two different nanofiltration membranes were studied with several electrolyte solutions to investigate the influence of salt type and concentration on the zeta potential and kinetic surface charge density of the membranes. The zeta potentials decreased with increasing salt concentration, whereas the kinetic surface charge densities increased. The kinetic surface charge densities could be described by Freundlich isotherms, except in one case, indicating that the membranes had a negligible surface charge. The kinetic surface charge density observed was caused by adsorbed anions. Salt retention measurements showed different mechanisms for salt separation for the two investigated membranes. One membrane showed a salt retention that could be explained by a Donnan exclusion type of separation mechanism, whereas for the other membrane the salt rejection seemed to be a combination of size and Donnan excluion. Comparing the results obtained by the streaming potential measurements with those of the retention measurements, it could be concluded that the membrane with the highest kinetic surface charge density showed the Donnan exclusion type of separation, whereas the membrane with the lower surface charge density showed a separation mechanism that was not totally determined by Donnan exclusion, size effects seemed to play a role as well.     

Some properties of electrolyte solutions in nanoconfinement revealed by the measurement of transient filtration potential after pressure switch off

We have demonstrated that with a composite nanoporous ceramic membrane in a batch membrane cell it is technically feasible to switch off the trans-membrane hydrostatic pressure difference within tens of milliseconds. That enabled us to resolve practically the whole time evolution of transient filtration potential. Measurements of the latter have been complemented by measurements of steady-state salt rejection by the composite membrane and by measurements of the streaming potential and hydraulic permeability of membrane supports available separately. A theory has been developed in terms of network thermodynamics for the electrical response of a bilayer membrane to a pressure perturbation. In combination with the results of salt rejection measurements, from the time transients of filtration potential we could determine the ion transport numbers within the nanoporous layer. Besides that, from the dependence of steady-state salt rejection on the trans-membrane volume flow, we have determined the diffusion permeability of and the salt reflection coefficient in the nanoporous layer. This has enabled us to estimate the contributions of Donnan and non-Donnan mechanisms to the rejection of ions by the nanoporous membrane used in this study. It has been unexpectedly found that the Donnan exclusion played only a secondary role. Our hypothesis is that the non-Donnan exclusion of ions from the nanopores might be caused by changes in water properties in nanoconfinement. Proceeding from the results of steady-state filtration experiments with the membrane and the support, we also concluded that the nanoporous layer was imperfection-free and had a quite narrow pore size distribution, which made it a suitable object for fundamental studies of ion transfer mechanisms in nanopores.     

Validation of carrier-mediated transport of H(+) and Na(+) through mobile-site membranes

The response of membranes containing neutral ion carriers of either H(+) or Na(+) to an externally-applied potential step was investigated using previously developed techniques for the analysis of charge and mass transport in ion-selective membranes. The results from constant-resistance membranes, e.g. membranes with unperturbed negative site concentration profiles, showed that tridodecylamine behaved as a carrier for H(+), and there was no evidence for proton hopping from stationary carriers. In addition, the experimental outcome supported the assumption of a failure of the Donnan exclusion principle at very low pH levels in these membranes. The results from membranes containing the Na(+) carrier illustrated the significant concentration polarization of ionic species, which was related to significant changes in bulk membrane resistance.     

Reverse permeation of sodium ions driven by pH difference across a liquid membrane: time dependencies of membrane resistance and membrane potential in a reverse permeation system

The diffusional flux of sodium ions across a liquid membrane was observed as a reverse permeation phenomenon: sodium ions were transported across the membrane against their own concentration difference. A supported liquid membrane having stearic acid as an ionic carrier was used. The internal aqueous phase contained NaCl and HCl and the external aqueous phase contained NaOH of the same initial concentration as NaCl in the internal aqueous phase. The reverse permeation occurred with a long time delay. During the delay, sodium ions flowed from the acidic to alkaline solution. The diffusion coefficient of sodium ion estimated from the flux equation taking into account the Donnan equilibrium at the interface was found to be much greater than that in the membrane solvent, 1-octanol. In the same membrane system as for the flux measurement, the membrane conductance and the membrane potential were measured as a function of time. The time dependence of the membrane potential in the presteady state showed a biphasic behavior. The initial rapid phase could be attributed to the change in the phase boundary potential and the subsequent slow step to the change in the diffusion potential within the membrane. Before the steady membrane potential had been reached, the reverse permeation of sodium ions against their own concentration difference was not observed. During the slow relaxation process of the membrane potential, the membrane resistance decreased to approach the steady state. Moreover, the oscillation of membrane potential abruptly started at a time in the slow step of the potential change and continued during the steady state. It was suggested that, at the presteady state, the increase in the amount of water in the membrane would drive a drastic change in the state of the liquid membrane in the filter pore, e.g. an inverted micellar structure making.     

Dielectric analysis of nanofiltration membrane in electrolyte solutions: influences of electrolyte concentration and species on membrane permeation

Dielectric properties of a nanofiltration membrane immersed in dilute aqueous electrolyte solutions were measured, and frequency dependence of capacitance and conductance of the systems was analyzed, based on the interfacial polarization theory, giving values of permittivity and conductivity of the membrane and the solutions. Permittivity, epsilon m, of the membrane slightly decreased whereas conductivity, km, of the membrane increased with increasing electrolyte concentration, as a result of entrance of ions into the membrane. The ratio of membrane/solution conductivity, km/kw, also depended on the electrolyte concentration, showing that distribution of ions in the membrane and in solutions follow Donnan equilibrium, due to the presence of negative fixed charges in the membrane. New expressions were derived from Donnan equilibrium principle to explain this phenomenon, and negative fixed charge concentration ce of the membrane was obtained; thus the Donnan potential, DeltaPhi Don, of the membrane in solutions at various concentrations could be calculated. The new expressions could be expected to be usable to analyze ion permeation property through membrane.     

Donnan permselectivity in layer-by-layer self-assembled redox polyelectrolye thin films

Redox polyelectrolyte multilayers have been assembled with use of the layer-by-layer (LBL) deposition technique with cationic poly(allylamine) modified with Os(bpy)(2)ClPyCHO (PAH-Os) and anionic poly(styrene)sulfonate (PSS) or poly(vinyl)sulfonate (PVS). Different behavior has been observed in the formal redox potential of the Os(II)/Os(III) couple in the polymer film with cyclic voltammetry depending on the charge of the outermost layer and the electrolyte concentration and pH. The electrochemical quartz crystal microbalance (EQCM) has been used to monitor the exchange of ions and solvent with the external electrolyte during redox switching. At low ionic strength Donnan permselectivity of anions or cations is apparent and the nature of the ion exclusion from the film is determined by the charge of the topmost layer and solution pH. At high electrolyte concentration Donnan breakdown is observed and the osmium redox potential approaches the value for the redox couple in solution. Exchange of anions and water with the external electrolyte under permselective conditions and salt and water under Donnan breakdown have been observed upon oxidation of the film at low pH for the PAH-Os terminating layer. Moreover, at high pH values and with PVS as the terminating layer EQCM mass measurements have shown that cation release was masked by water exchange.     

Effect of proton concentration on membrane potential across a weak amphoteric polymer membrane

An amphoteric membrane consists of both positively and negatively fixed charge groups chemically bound to the polymer chains. If the external solution is changed from alkali to acid, it is possible to obtain an experimental result in which the membrane potential changes from positive to negative through the isoelectric point. It was characterized by examining the relationship between membrane potential and proton concentration (pH) obtained from both experimental and theoretical considerations. The Nernst-Planck flux equation and the Donnan equilibrium theory were also solved for a four-component system combined with the dissociation constant, in order to discuss the pH dependence of membrane potential in a weak amphoteric membrane by comparing the experimental results with the calculated results. It was proven that the calculated results substantially deviated from the theoretical results despite a similar tendency. Such a deviation was caused by the fact that the original theory disregarded the activity coefficient and the ionic mobility, which were dependent on the fixed charge concentration in a membrane. The original theoretical model was modified by adding the effect of a fixed charge group to the activity coefficient and ionic mobility. The calculated results using the modified model explained well the experimental results if the parameter called charge effectiveness, phi, was introduced into the equations. Introduction of phi into the prediction of membrane potential was already done by Kobatake et al. in a system of a strong polyelectrolyte monopolar membrane/salt aqueous solution. In this study, it was proved that phi can also be introduced into a weak amphoteric polymer membrane/salt aqueous solution system. Finally it was also concluded that the Donnan equilibrium and the Nernst-Planck flux equation were still applicable for examining the transport phenomena for the system of a weak amphoteric charged membrane and electrolyte solutions at various pH.     

Analysis of the pressure-induced potential arising through composite membranes with selective surface layers

When a pressure gradient is applied through a charged selective membrane, the transmembrane electrical potential difference, called the filtration potential, results from both the applied pressure and induced concentration difference across the membrane. In this work we investigate the electrokinetic properties relative to both active and support layers of a composite ceramic membrane close to the nanofiltration range. First, the volume charge density of the active layer is obtained by fitting a transport model to experimental rejection rates (which are controlled by the active layer only). Next, the value of the volume charge density is used to compute the theoretical filtration potential through the active layer. For sufficiently high permeate volume fluxes, the concentration difference across the active layer becomes constant, which allows assessing the membrane potential of the active layer. Experimental measurements of the overall filtration potential arising through the whole membrane are performed. The contribution of the support layer to this overall filtration potential is put in evidence. That implies that the membrane potential of the active layer cannot be deduced directly from the overall filtration potential measurements. Finally, the contribution of the support layer is singled out by subtracting the theoretical filtration potential of the active layer from the experimental filtration potential measured across the whole membrane (i.e., support + active layers). The amphoteric behavior of both layers is put in evidence, which is confirmed by electrophoretic measurements carried out with the powdered support layer and by recently reported tangential streaming potential measurements.     

Donnan potential and surface potential of a spherical soft particle in an electrolyte solution

A simple numerical method, which does not involve numerical integration of the Poisson-Boltzmann equations, is presented for obtaining the relationship between the Donnan potential and surface potential of a spherical soft particle (i.e., a polyelectrolyte-coated particle) in a symmetrical electrolyte solution. We assume that a soft particle consists of the particle core of radius a covered with an ion-penetrable surface layer of polyelectrolytes of thickness d and that ionized groups of valence Z are distributed at a uniform density of N in the polyelectrolyte layer and the relative permittivity takes the same value in the regions outside and inside the polyelectrolyte layer. The Donnan potential and surface potential are determined by the values of a, d, Z, N, and the Debye-Hückel parameter kappa of the electrolyte solution. Numerical results obtained by the present method are in excellent agreement with exact results obtained by solving the nonlinear spherical Poisson-Boltzmann equations for the both regions inside and outside the polyelectrolyte layer.     

Structure formation in films of weakly charged block polyelectrolyte solutions

A mean-field dynamic density functional theory is used to describe a phase diagram of concentrated solutions of weakly charged flexible block polyelectrolytes in a film. Electrostatics is taken into account by applying the local electroneutrality constraint (the Donnan membrane equilibrium approach). In the Donnan limit it is assumed that a salt added to the solution perfectly screens long-range electrostatic interactions. The phase diagram of a solution of a triblock polyelectrolyte in a film as a function of the solvent concentration and the charge of the polyelectrolyte (solvophilic) block is calculated for a given film thickness. The phase behavior of the block polyelectrolyte film arises from the interplay between surface-induced alignment and the electrostatically-driven structure formation. The observed mesoscopic structures (lamellar, perforated lamellar, cylindrical, micellar, and mixed phases) are oriented parallel to the surfaces for the considered case of morphologies unfrustrated by the film thickness. Structures with connections between parallel layers (bicontinuous, etc.) are not formed. As a result of surface-induced ordering, the region of ordered phases in a film is wider than in bulk and the phase boundary between ordered and disordered phases is more diffuse. As in the case of unconfined block polyelectrolyte solution, the solution in a film does not follow the lyotropic sequence of phases of such a block copolymer upon increase in the charge of the polyelectrolyte block. Upon changing the charge of the solvophilic copolymer block, transformations of copolymer morphology take place via change in curvature of polymeric domains. Due to confinement of a polyelectrolyte film, no swelling of solvophilic domains is observed.     

Rates of transport through a capsule membrane to attain Donnan equilibrium

The swelling of a capsule consisting of salt solution and polyelectrolyte, surrounded by a membrane, is studied. The membrane allows salt and water to pass, but is impermeable to polyelectrolyte molecules. Equilibrium swelling of the capsule is governed by Donnan equilibrium. Transport rates of a salt and water through the membrane are expressed in terms of a Darcy permeability and a salt diffusivity. The governing equations predict that the rate at which equilibrium is attained as the external salt concentration varies is controlled by the timescale for diffusion of salt, rather than by that for Darcy flow. Experiments were performed using capsules with membranes made of covalently linked HSA and alginate. The capsule volume varied with a single relaxation rate when the external salt concentration was changed, as predicted by theory. This constitutes the first step toward a simple method for determining the membrane properties of capsules by measuring rates of change of capsule volume.