An improved Space-Charge model for flow through charged microporous membranes

The Space-Charge model is modified to better analyze the steady-state electrohydrodynamic behavior of aqueous monovalent electrolytes in charged microporous membranes. The effects of changes in solvent dielectric constant near the wall, ion hydration effects, finite ion sizes, and charge regulating surface effects, are incorporated into the governing electrohydrodynamic equations (i.e., Navier-Stokes (NSE), Nernst-Planck (NPE), and Poisson-Boltzmann (PBE) equations). Their effect on streaming potential, pore conductivity, excess conductivity, and maximum energy conversion efficiency for electro-osmosis is illustrated. It is shown that the dielectric saturation and ion hydration effects cause significant changes in the electric potential field and ion concentration inside the capillary tubes. Quantitative comparisons of model results with measured electrokinetic data reveal better agreement when compared with the existing model.     

Ionic exclusion phase transition in neutral and weakly charged cylindrical nanopores

A field theoretic variational approach is introduced to study ion penetration into water-filled cylindrical nanopores in equilibrium with a bulk reservoir [S. Buyukdagli, M. Manghi, and J. Palmeri, Phys. Rev. Lett. 105, 158103 (2010)]. It is shown that an ion located in a neutral pore undergoes two opposing mechanisms: (i) a deformation of its surrounding ionic cloud of opposite charge, with respect to the reservoir, which increases the surface tension and tends to exclude ions from the pore, and (ii) an attractive contribution to the ion self-energy due to the increased screening with ion penetration of the repulsive image forces associated with the dielectric jump between the solvent and the pore wall. For pore radii around 1 nm and bulk concentrations lower than 0.2 mol/l, this mechanism leads to a first-order phase transition, similar to capillary "evaporation," from an ionic-penetration state to an ionic-exclusion state. The discontinuous phase transition exists within the biological concentration range (～0.15 mol/l) for small enough membrane dielectric constants (ε(m) < 5). In the case of a weakly charged pore, counterion penetration exhibits a nonmonotonic behavior and is characterized by two regimes: at low reservoir concentrations or small pore radii, coions are excluded and counterions enter the pore to enforce electroneutrality; dielectric repulsion (image forces) remain strong and the counterion partition coefficient decreases with increasing reservoir concentration up to a characteristic value. For larger reservoir concentrations, image forces are screened and the partition coefficient of counterions increases with the reservoir concentration, as in the neutral pore case. Large surface charge densities (>2 × 10(-3) e/nm(2)) suppress the discontinuous transition by reducing the energy barrier for ion penetration and shifting the critical point toward very small pore sizes and reservoir concentrations. Our variational method is also compared to a previous self-consistent approach and yields important quantitative corrections. The role of the curvature of dielectric interfaces is highlighted by comparing ionic penetration into slit and cylindrical pores. Finally, a charge regulation model is introduced in order to explain the key effect of pH on ionic exclusion and explain the origin of observed time-dependent nanopore electric conductivity fluctuations and their correlation with those of the pore surface charge.     

Electrochemistry of capillary systems with narrow pores. I. Overview

In capillary systems with narrow pores the Helmholtz electrochemical double layer located at the pore wall extends over the entire cross section of the pores. It loses its character as the “charge on the wall”. It will be shown that not only the electrokinetic phenomena but also the electrical conductivity and the dialysis potential of membranes with narrow pores can be understood from the same point of view, namely: the electrolyte solution in the pores of a membrane with narrow pores is considered to be an approximately homogeneous solution in contact with immobilised charges located at the pore wall. In this case the electrochemical equations contain the fixed ion concentration as a parameter instead of the ζ potential. This makes it possible to describe quantitatively to a good approximation data on the electroosmosis, the electrical conductivity, the streaming potential and the dialysis potential taken from the literature, as well as results of our own measurements, by using a single membrane constant.     

Electrochemistry of capillary systems with narrow pores. I. Overview

In capillary systems with narrow pores the Helmholtz electrochemical double layer located at the pore wall extends over the entire cross section of the pores. It loses its character as the “charge on the wall”. It will be shown that not only the electrokinetic phenomena but also the electrical conductivity and the dialysis potential of membranes with narrow pores can be understood from the same point of view, namely: the electrolyte solution in the pores of a membrane with narrow pores is considered to be an approximately homogeneous solution in contact with immobilised charges located at the pore wall. In this case the electrochemical equations contain the fixed ion concentration as a parameter instead of the ζ potential. This makes it possible to describe quantitatively to a good approximation data on the electroosmosis, the electrical conductivity, the streaming potential and the dialysis potential taken from the literature, as well as results of our own measurements, by using a single membrane constant.     

Influence of steric, electric, and dielectric effects on membrane potential

The membrane potential arising through nanofiltration membranes separating two aqueous solutions of the same electrolyte at identical hydrostatic pressures but different concentrations is investigated within the scope of the steric, electric, and dielectric exclusion model. The influence of the ion size and the so-called dielectric exclusion on the membrane potential arising through both neutral and electrically charged membranes is investigated. Dielectric phenomena have no influence on the membrane potential through neutral membranes, unlike ion size effects which increase the membrane potential value. For charged membranes, both steric and dielectric effects increase the membrane potential at a given concentration but the diffusion potential (that is the high-concentration limit of the membrane potential) is affected only by steric effects. It is therefore proposed that membrane potential measurements carried out at high salt concentrations could be used to determine the mean pore size of nanofiltration membranes. In practical cases, the membrane volume charge density and the dielectric constant inside pores depend on the physicochemical properties of both the membrane and the surrounding solutions (pH, concentration, and chemical nature of ions). It is shown that the Donnan and dielectric exclusions affect the membrane potential of charged membranes similarly; namely, a higher salt concentration is needed to screen the membrane fixed charge. The membrane volume charge density and the pore dielectric constant cannot then be determined unambiguously by means of membrane potential experiments, and additional independent measurements are in need. It is suggested to carry out rejection rate measurements (together with membrane potential measurements).     

Dielectric exclusion of ions from membranes

Dielectric exclusion is caused by the interactions of ions with the bound electric charges induced by ions at interfaces between media of different dielectric constants. It is considered as one of mechanisms of nanofiltration. The transport properties of capillary model are expressed through ion distribution and diffusion coefficients. Due to local equilibrium the distribution coefficient is directly related to the excess solvation energy of ion. First, this energy is considered for single ions in single neutral pores in terms of pore size, ion charge, dielectric constants of solvent and membrane matrix and pore geometry. The dielectric exclusion from pores with closed geometry like circular cylinders is shown to be essentially stronger than that from pores with relatively open geometry like slits. Furthermore, the role of finite membrane porosity is analysed for the model of infinite slabs with alternating dielectric constants. The presence of other ions is accounted for within the scope of a mean-field approach, and the screening of dielectric exclusion is thus introduced and considered in some detail. A fixed electric charge is shown to cause additional screening. At the same time the dielectric exclusion makes the Donnan exclusion of ions stronger. Therefore the interaction between those two rejection mechanisms turns out to be non-trivial. Finally, the effect of solvent molecular structure is considered within the scope of non-local electrostatics. It is shown that the solvent non-locality typically results in somewhat stronger dielectric exclusion, however, its most important effect is slowing down the decline of dielectric exclusion with increasing bulk electrolyte concentration.     

Aqueous electrolytes confined within functionalized silica nanopores

Molecular dynamics simulations have been carried out to investigate structural and dynamical characteristics of NaCl aqueous solutions confined within silica nanopores in contact with a "bulk-like" reservoir. Two types of pores, with diameters intermediate between 20 A? and 37.5 A?, were investigated: The first one corresponded to hydrophobic cavities, in which the prevailing wall-solution interactions were of the Lennard-Jones type. In addition, we also examined the behavior of solutions trapped within hydrophilic cavities, in which a set of unsaturated O-sites at the wall were transformed in polar silanol Si-OH groups. In all cases, the overall concentrations of the trapped electrolytes exhibited important reductions that, in the case of the narrowest pores, attained 50% of the bulk value. Local concentrations within the pores also showed important fluctuations. In hydrophobic cavities, the close vicinity of the pore wall was coated exclusively by the solvent, whereas in hydrophilic pores, selective adsorption of Na(+) ions was also observed. Mass and charge transport were also investigated. Individual diffusion coefficients did not present large modifications from what is perceived in the bulk; contrasting, the electrical conductivity exhibited important reductions. The qualitative differences are rationalized in terms of simple geometrical considerations.     

Influence of atomistic physics on electro-osmotic flow: an analysis based on density functional theory

Molecular density profiles and charge distributions determined by density functional theory (DFT) are used in conjunction with the continuum Navier-Stokes equations to compute electro-osmotic flows in nanoscale channels. The ion species of the electrolyte are represented as centrally charged hard spheres, and the solvent is treated as a dense fluid of neutral hard spheres having a uniform dielectric constant. The model explicitly accounts for Lennard-Jones interactions among fluid and wall molecules, hard sphere repulsions, and short range electrical interactions, as well as long range Coulombic interactions. Only the last of these interactions is included in classical Poisson-Boltzmann (PB) modeling of the electric field. Although the proposed DFT approach is quite general, the sample calculations presented here are limited to symmetric monovalent electrolytes. For a prescribed surface charge, this DFT model predicts larger counterion concentrations near charged channel walls, relative to classical PB modeling, and hence smaller concentrations in the channel center. This shifting of counterions toward the walls reduces the effective thickness of the Debye layer and reduces electro-osmotic velocities as compared to classical PB modeling. Zeta potentials and fluid speeds computed by the DFT model are as much as two or three times smaller than corresponding PB results. This disparity generally increases with increasing electrolyte concentration, increasing surface charge density and decreasing channel width. The DFT results are found to be comparable to those obtained by molecular dynamics simulation, but require considerably less computing time.     

Colloido-chemical characteristics of porous glasses with different compositions in KNO3 solutions. 1. Structural and electrokinetic characteristics of membranes

The structural (specific surface area, liquid-filtration coefficient, average pore radius, volume porosity, and structural-resistance coefficient) and electrokinetic (counterion transport numbers, specific electrical conductivity, and electrokinetic potential) characteristics of porous glasses with different compositions have been determined in potassium nitrate solutions with concentrations of 10^?3–10^?1 M. All the membranes under investigation have been shown to exhibit the dependences of efficiency coefficients and counterion transport numbers on electrolyte concentration and pore size that are predicted by the theory of an electrical double layer. It has been established that, at a constant electrolyte concentration, the absolute values of electrokinetic potential increase with the average pore radius because of variations in the slipping-plane position.     

Determining the Zeta Potential of Porous Membranes Using Electrolyte Conductivity inside Pores

The zeta potential is an important and reliable indicator of the surface charge of membranes, and knowledge of it is essential for the design and operation of membrane processes. The zeta potential cannot be measured directly, but must be deduced from experiments by means of a model. The possibility of determining the zeta potential of porous membranes from measurements of the electrolyte conductivity inside pores (lambda(pore)) is investigated in the case of a ceramic microfiltration membrane. To this end, experimental measurements of the electrical resistance in pores are performed with the membrane filled with KCl solutions of various pHs and concentrations. lambda(pore) is deduced from these experiments. The farther the pH is from the isoelectric point and/or the lower the salt concentration is, the higher the ratio of the electrolyte conductivity inside pores to the bulk conductivity is, due to a more important contribution of the surface conduction. Zeta potentials are calculated from lambda(pore) values by means of a space charge model and compared to those calculated from streaming potential measurements. It is found that the isoelectric points are very close and that zeta potential values for both methods are in quite good agreement. The differences observed in zeta potentials could be due to the fact that the space charge model does not consider the surface conductivity in the inner part of the double layer. Measurements of the electrolyte conductivity within the membrane pores are proved to be a well-adapted procedure for the determination of the zeta potential in situations where the contribution of the surface conduction is significant, i.e., for small and charged pores. Copyright 2001 Academic Press.     

Structural and Electrosurface Properties of Porous Glasses of Various Composition in 1 : 1 Electrolyte Solutions

Complex studies of structural (the specific surface area, the volume porosity, the structural resistance coefficient, and the average pore radius), adsorption and electrokinetic (the electrical conductivity, the ion transport numbers, and the electrokinetic potential) characteristics as functions of pH and the concentration of KCl solution were carried out on porous glasses (PGs) with or without lead oxide and leached under various conditions. It was established that temperature of the leaching solution affects the colloidochemical parameters of PGs, while the addition of salt to the leaching solution exerts practically no influence on the PG behavior. It was shown that the addition of lead oxide results in the formation of membranes with thinner pores and higher surface charge.     

Electrokinetic Characteristics of Porous Glasses in Solutions of Sodium and Iron(III) Chlorides

The structural (structural resistance coefficient, bulk porosity, average pore radius, and specific surface area) and electrokinetic (surface conductivity and electrokinetic potential) characteristics of high-silica micro- and macroporous glasses produced from two-phase sodium borosilicate glass have been compared in solutions of an indifferent electrolyte (sodium chloride) and iron(III) chloride containing iron ions, which have a high specificity to silica surfaces. It has been shown that, in the presence of iron ions, the electrokinetic behavior of porous glasses is governed by two factors. First, the superequivalent adsorption of these ions in the Stern layer leads to positive values of the electrokinetic potential, and, second, their mobility in the pore space decreases, thereby resulting in the appearance of equilibrium solution concentration ranges, in which the specific conductivity of a pore solution becomes lower than that of a free solution.     

Colloid chemical characteristics of nanoporous glasses with different compositions in solutions of simple and organic electrolytes. 3. Calculation of electrochemical characteristics of membranes in terms of a homogeneous model

The electrosurface characteristics of nanoporous glass membranes–ion concentrations in pores with taking into account the specificity of counterions, electrokinetically mobile charge, the convective component of pore solution electrical conductivity, electroosmotic mobility of a liquid in the field of streaming potential and ion mobilities in pore space–were calculated within the homogeneous model. The effects of the type of counterion (sodium, potassium, ammonium, tetramethylammonium, and tetraethylammonium ions), solution concentration, glass composition, and pore size on the equilibrium and transport characteristics of membrane systems have been analyzed. A method for the determining of electrolyte activity coefficients in the membranes has been proposed.     

Electrokinetics of charged spherical colloidal particles taking into account the effect of ion size constraints

The electrokinetic properties of suspended spherical particles are examined using a modified standard electrokinetic model, which takes into account the finite ion size and considers that the minimum approach distance of ions to the particle surface need not be equal to their effective radius in the bulk solution. We calculate the conductivity increment and the electrophoretic mobility and present a detailed interpretation of the obtained results, based on the analysis of the equilibrium and field-induced ion concentrations, as well as the convective fluid flow in the neighborhood of the particle surface. We show that when charge reversal takes place, the sign of the concentration polarization remains unchanged while the sign of the electrophoretic mobility only changes under favorable circumstances.     

Concentration dependence of ionic transport in dilute organic electrolyte solutions

Ion transport is studied in dilute organic liquid electrolyte solutions in which close cation-anion interactions are minimized either through steric hindrance imposed by the bulky tetrabutylammonium cation or by strong solvation of alkali metal cations by DMSO or 1-propanol. In these solutions, the molar conductivity does not appear to depend on either the solvent viscosity or the size of the solvated charge carrier in a manner consistent with Walden's rule. The molar conductivities plotted as a function of the solvent dielectric constant from epsilon = 5.48 to 63.5 appear to lie on a smooth curve for a set of 0.0055 M solutions of tetrabutylammonium trifluoromethanesulfonate in a variety of aprotic solvents. The molar conductivity smoothly increases with increasing dielectric constant to a maximum at roughly epsilon = 33 and then decreases with further increase of the dielectric constant. The conductivity appears to depend only on the dielectric constant and not the specific functional group in this broad family of solvents. A similar plot for a series of linear alcohols as solvents also led to a smooth curve, although the values of the molar conductivity were lower than values in the aprotic solvents by almost an order of magnitude at corresponding values of the solvent dielectric constant.     

Electrokinetic characteristics of initial porous glasses and those modified with titanium- and aluminum-oxide particles

The structural (volume porosity, structural resistance coefficient, and average pore radius) and electrokinetic (specific electrical conductivity, ion-transport numbers, and electrokinetic potential) characteristics of macroporous glass membranes obtained from two-phase sodium-borosilicate glasses with different times of thermal treatment have been studied in solutions of hydrochloric acid and potassium chloride. The properties of the initial membranes have been compared with the characteristics of the same membranes modified by filtering through them suspensions of aluminum- and titanium-oxide nanoparticles with different weight concentrations. It has been shown that, at low degrees of pore channel surface coverage with nanoparticles (<0.1), the structural parameters of the membranes remain almost unchanged. In addition, it has been found that the presence of positively charged nanoparticles on the negatively charged surface increases the surface conductivity and the absolute value of the electrokinetic potential.     

Transition from dilute to concentrated electrokinetic behavior in the dielectric spectra of a colloidal suspension

Dielectric spectroscopy is used to measure the complex permittivity of 200 and 100 nm diameter polystyrene latex suspended in potassium chloride (KCl) solutions over the frequency range 10(4)-10(7) Hz as a function of particle volume fraction (?) and ionic strength. Dilute suspension dielectric spectra are in excellent agreement with electrokinetic theory. A volume fraction dependence of the dielectric increment is observed for low electrolyte concentrations (0.01, 0.05, and 0.1 mM) above ? ≈ 0.02. This deviation from the dilute theory occurs at a critical frequency ω* that is a function of volume fraction, particle size, and ionic strength. The dielectric increment of suspensions at the highest salt concentration (1 mM) shows no volume fraction dependence up to ? = 0.09. Values of ω* are collapsed onto a master curve that accounts for the length and time scales of ion migration between neighboring particles. The measured conductivity increment is independent of volume fraction and agrees with theory after accounting for added counterions and nonspecific adsorption.     

Polarizability and complex conductivity of dilute suspensions of spherical colloidal particles with charged (polyelectrolyte) coatings

The dielectric relaxation of polyelectrolyte-coated colloidal particles is examined via "exact" numerical solutions of the governing electrokinetic equations. The charged polymer coatings are characterized by a nominal charge density, thickness, and permeability. Brush-like segment density profiles are considered here, but more sophisticated segment and charge density profiles are accommodated by the model. The role of added counterions and nonspecific adsorption is considered briefly before examining how the experimentally measured conductivity and dielectric constant increments reflect the frequency of the applied electric field, the strength of the electrolyte, and characteristics of the polymer coatings, namely the charge, charge density, and permeability. Finally, a strategy is suggested by which dielectric spectroscopy and electrophoresis can be used to characterize polymer-coated particles. This approach complements experiments where electroviscous effects such as dynamic light scattering and sedimentation are weak.