Diffusioosmosis and electroosmosis in a capillary slit with surface charge layers

This study analytically examines the steady diffusioosmotic and electroosmotic flows of an electrolyte solution in a fine capillary slit with each of its inside walls covered by a layer of adsorbed polyelectrolytes. In this solvent-permeable and ion-penetrable surface charge layer, idealized polyelectrolyte segments are assumed to distribute at a uniform density. The electric double layer and the surface charge layer may have arbitrary thicknesses relative to the gap width between the slit walls. The electrostatic potential distribution on a cross section of the slit is obtained by solving the linearized Poisson–Boltzmann equation, which applies to the case of low potentials or low fixed-charge densities. Explicit formulas for the fluid velocity profile due to the imposed electrolyte concentration gradient or electric field through the slit are derived as the solution of a modified Navier–Stokes/Brinkman equation. The results demonstrate that the structure of the surface charge layer can lead to an augmented or a diminished electrokinetic flow (even a reversal in direction of the flow) relative to that in a capillary with bare walls, depending on the characteristics of the capillary, of the surface charge layer, and of the electrolyte solution. For the diffusioosmotic flow with an induced electric field, competition between electroosmosis and chemiosmosis can result in more than one reversal in direction of the flow over a range of the Donnan potential of the adsorbed polyelectrolyte in the capillary.     

非均匀电场的电堆积理论研究

基于样品区非均匀电场电势分布的测试结果并利用异性双点电荷电场的理论模型,研究了非均匀电场电堆积效应,提出了非均匀电场电堆积作用的理论模型.采用一种改进的非均匀电场电堆积系统,以矿泉水中痕量Cr(Ⅵ)和Pb(Ⅱ)为考察对象,同时对其进行分离富集和测定,为理论模型提供了实验依据.     

Mathematical model of electromigration allowing the deviation from electroneutrality

The structure of the double layer on the boundary between solid and liquid phases is described by various models, of which the Stern–Gouy–Chapman model is still commonly accepted. Generally, the solid phase is charged, which also causes the distribution of the electric charge in the adjacent diffuse layer in the liquid phase. We propose a new mathematical model of electromigration considering the high deviation from electroneutrality in the diffuse layer of the double layer when the liquid phase is composed of solution of weak multivalent electrolytes of any valence and of any complexity. The mathematical model joins together the Poisson equation, the continuity equation for electric charge, the mass continuity equations, and the modified G-function. The model is able to calculate the volume charge density, electric potential, and concentration profiles of all ionic forms of all electrolytes in the diffuse part of the double layer, which consequently enables to calculate conductivity, pH, and deviation from electroneutrality. The model can easily be implemented into the numerical simulation software such as Comsol. Its outcome is demonstrated by the numerical simulation of the double layer composed of a charged silica surface and an adjacent liquid solution composed of weak multivalent electrolytes. The validity of the model is not limited only to the diffuse part of the double layer but is valid for electromigration of electrolytes in general.     

X-ray study of the electric double layer at the n-hexane/nanocolloidal silica interface

The spatial structure of the transition region between an insulator and an electrolyte solution was studied with x-ray scattering. The electron-density profile across the n-hexane/silica sol interface (solutions with 5, 7, and 12 nm colloidal particles) agrees with the theory of the electrical double layer and shows separation of positive and negative charges. The interface consists of three layers, i.e., a compact layer of Na(+), a loose monolayer of nanocolloidal particles as part of a thick diffuse layer, and a low-density layer sandwiched between them. Its structure is described by a model in which the potential gradient at the interface reflects the difference in the potentials of "image forces" between the cationic Na(+) and anionic nanoparticles and the specific adsorption of surface charge. The density of water in the large electric field (approximately 10(9)-10(10) Vm) of the transition region and the layering of silica in the diffuse layer is discussed.     

Diffusioosmosis of electrolyte solutions in a fine capillary tube

A theoretical study is presented for the steady diffusioosmotic flow of an electrolyte solution in a fine capillary tube generated by a constant concentration gradient imposed in the axial direction. The capillary wall may have either a constant surface potential or a constant surface charge density of an arbitrary quantity. The electric double layer adjacent to the charged wall may have an arbitrary thickness, and its electrostatic potential distribution is determined by an analytical approximation to the solution of the Poisson-Boltzmann equation. Solving a modified Navier-Stokes equation with the constraint of no net electric current arising from the cocurrent diffusion, electric migration, and diffusioosmotic convection of the electrolyte ions, the macroscopic electric field and the fluid velocity along the axial direction induced by the imposed electrolyte concentration gradient are obtained semianalytically as a function of the radial position in a self-consistent way. The direction of the diffusioosmotic flow relative to the concentration gradient is determined by the combination of the zeta potential (or surface charge density) of the wall, the properties of the electrolyte solution, and other relevant factors. For a prescribed concentration gradient of an electrolyte, the magnitude of fluid velocity at a position in general increases with an increase in its distance from the capillary wall, but there are exceptions. The effect of the radial distribution of the induced tangential electric field and the relaxation effect due to ionic convection in the double layer on the diffusioosmotic flow are found to be very significant.     

Effect of Inorganic Ions on Oleate Adsorption at a Nigerian Hematite–Water Interface

Using “pure” natural hematite selected from a high silica Nigerian hematitic ore, oleate adsorption densities at the hematite–water interface were determined in the presence of various inorganic ions (anions and cations) of different charges and at varying concentrations. Adsorption density was determined using electrical conductivity measurements. The specific surface area of the hematite particles was determined using the method of adsorption of paranitrophenol in aqueous solution. Inorganic ions in solution depressed oleate adsorption at the aqueous hematite surface. The charge of the ion proved to be the dominant factor determining the depression of oleate adsorption. Ionic strength also was an influence, up to a limiting value at which monolayer oleate coverage of the hematite surface occurred. The inorganic ions in solution are considered to function through nonspecific adsorption in the diffuse region of the electric double layer.     

Diffusioosmosis of electrolyte solutions along a charged plane wall

The steady diffusioosmotic flow of an electrolyte solution along a dielectric plane wall caused by an imposed tangential concentration gradient is analytically examined. The plane wall may have either a constant surface potential or a constant surface charge density of an arbitrary quantity. The electric double layer adjacent to the charged wall may have an arbitrary thickness, and its electrostatic potential distribution is determined by the Poisson-Boltzmann equation. The macroscopic electric field along the tangential direction induced by the imposed electrolyte concentration gradient is obtained as a function of the lateral position. A closed-form formula for the fluid velocity profile is derived as the solution of a modified Navier-Stokes equation. The direction of the diffusioosmotic flow relative to the concentration gradient is determined by the combination of the zeta potential of the wall and the properties of the electrolyte solution. For a given concentration gradient of an electrolyte along a plane wall, the magnitude of fluid velocity at a position in general increases with an increase in its electrokinetic distance from the wall, but there are exceptions. The effect of the lateral distribution of the induced tangential electric field in the double layer on the diffusioosmotic flow is found to be very significant and cannot be ignored.     

Chloride ion migration in natural bentonite

The disposal of nuclear wastes in geological formations demands the construction of engineering barriers. Bentonite clay rock is frequently used both as natural and engineering barrier. The natural bentonite rock in its original form is considered as compacted bentonite if the density is higher than 1.2 g/cm³. In this paper, the risk of the extrapolation of the laboratory experiments to field conditions and the high differences of the natural samples are emphasized: as much as 52 % standard deviation was obtained in the migration coefficients characterizing bentonite samples collected from the same site with a very small extent of sampling. Moreover, the bulk densities (1.18–1.87 g/cm³) and montmorillonite content are also rather different (45–71 m/m %).The contradictions of the effects of the swelling clay mineral (montmorillonite) content and the bulk density of bentonite are illustrated: it is shown that the migration rate of chloride anion is determined by the ratio of the different water types (interlayer water of montmorillonite to free pore water of bentonite, including the electric double layer water). The apparent migration coefficients of bentonite clay and concrete (natural and artificial engineering barrier) are compared.     

Electrophoretic motion of a spherical particle in a converging-diverging nanotube

A charged spherical particle is concentrically positioned in a converging-diverging nanotube filled with an electrolyte solution, resulting in an electric double layer (EDL) forming around the particle's surface. In the presence of an axially applied electric field, the particle electrophoretically migrates along the axis of the nanotube due to the electrostatic and hydrodynamic forces acting on the particle. In contrast to a cylindrical nanotube with a constant cross-sectional area in which the electric field is almost uniform, the presence of a converging-diverging section in a nanotube alters the electric field, perturbs the charge distribution, and induces a pressure gradient and a recirculating flow that affect the electrostatic and hydrodynamic forces acting on both the particle and the fluid. Depending on the magnitude of the surface charge density along the nanotube's wall, the particle's electrophoretic motion may be significantly accelerated as the particle transverses the converging-diverging section. A continuum model consisting of the Nernst-Planck, Poisson, and Navier-Stokes equations for the ionic concentrations, electric potential, and flow field is implemented to compute the particle's velocity as a function of the particle's size, the nanotube's geometry, surface charges, electric field intensity, bulk electrolyte concentration, and the particle's location. When the particle is negatively charged and the wall of the nanotube is uncharged, the particle migrates in the direction opposite to that of the applied electric field and the presence of the converging-diverging section significantly accelerates the particle's motion. This, however, is not always true when the nanotube's wall is charged with the same sign as that of the particle. Once the ratio of the surface charge density of the nanotube's wall to that of the particle exceeds a certain value, the negatively charged particle will not translocate through the tube toward the anode and does not attain the maximum velocity at the throat of the converging-diverging section. One can envision such a device to be a nanofilter that allows molecules with surface charge densities much higher than that of the wall to go through the nanofilter, while preventing molecules with surface charge densities lower than that of the wall from passing through the nanofilter. The induced recirculating flow may be used to enhance mixing and to stretch, fold, and trap molecules in nanofluidic detectors and reactors.     

Electrokinetic flow in a capillary with a charge-regulating surface polymer layer

An analytical study of the steady electrokinetic flow in a long uniform capillary tube or slit is presented. The inside wall of the capillary is covered by a layer of adsorbed or covalently bound charge-regulating polymer in equilibrium with the ambient electrolyte solution. In this solvent-permeable and ion-penetrable surface polyelectrolyte layer, ionogenic functional groups and frictional segments are assumed to distribute at uniform densities. The electrical potential and space charge density distributions in the cross section of the capillary are obtained by solving the linearized Poisson-Boltzmann equation. The fluid velocity profile due to the application of an electric field and a pressure gradient through the capillary is obtained from the analytical solution of a modified Navier-Stokes/Brinkman equation. Explicit formulas for the electroosmotic velocity, the average fluid velocity and electric current density on the cross section, and the streaming potential in the capillary are also derived. The results demonstrate that the direction of the electroosmotic flow and the magnitudes of the fluid velocity and electric current density are dominated by the fixed charge density inside the surface polymer layer, which is determined by the regulation characteristics such as the dissociation equilibrium constants of the ionogenic functional groups in the surface layer and the concentration of the potential-determining ions in the bulk solution.     

Simulation of the electrical double layer of a macroion with different counterion charges

An electrical double layer of a spherical macroion with single-, double-, and triple-charged counterions in aqueous solution of 1: 1 background electrolyte at different concentrations are studied by the molecular dynamics method for models with discrete and continuous surface charge distribution. Radial profiles of ion partial densities and the electric potential distribution in the double layer are calculated. The degree of counterion binding with a macroion is determined. The effect of water permittivity on the structure of electrical double layer is studied.     

Vertical motion of a charged colloidal particle near an AC polarized electrode with a nonuniform potential distribution: theory and experimental evidence

Electroosmotic flow in the vicinity of a colloidal particle suspended over an electrode accounts for observed changes in the average height of the particle when the electrode passes alternating current at 100 Hz. The main findings are (1) electroosmotic flow provides sufficient force to move the particle and (2) a phase shift between the purely electrical force on the particle and the particle's motion provides evidence of an E2 force acting on the particle. The electroosmotic force in this case arises from the boundary condition applied when faradaic reactions occur on the electrode. The presence of a potential-dependent electrode reaction moves the likely distribution of electrical current at the electrode surface toward uniform current density around the particle. In the presence of a particle the uniform current density is associated with a nonuniform potential; thus, the electric field around the particle has a nonzero radial component along the electrode surface, which interacts with unbalanced charge in the diffuse double layer on the electrode to create a flow pattern and impose an electroosmotic-flow-based force on the particle. Numerical solutions are presented for these additional height-dependent forces on the particle as a function of the current distribution on the electrode and for the time-dependent probability density of a charged colloidal particle near a planar electrode with a nonuniform electrical potential boundary condition. The electrical potential distribution on the electrode, combined with a phase difference between the electric field in solution and the electrode potential, can account for the experimentally observed motion of particles in ac electric fields in the frequency range from approximately 10 to 200 Hz.     

Diffusioosmosis of electrolyte solutions in a fine capillary slit

The steady diffusioosmotic flows of an electrolyte solution along a charged plane wall and in a capillary channel between two identical parallel charged plates generated by an imposed tangential concentration gradient are theoretically investigated. The plane walls may have either a constant surface potential or a constant surface charge density. The electrical double layers adjacent to the charged walls may have an arbitrary thickness and their electrostatic potential distributions are determined by the Poisson-Boltzmann equation. Solving a modified Navier-Stokes equation with the constraint of no net electric current arising from the cocurrent diffusion, electric migration, and diffusioosmotic convection of the electrolyte ions, the macroscopic electric field and the fluid velocity along the tangential direction induced by the imposed electrolyte concentration gradient are obtained semianalytically as a function of the lateral position in a self-consistent way. The direction of the diffusioosmotic flow relative to the concentration gradient is determined by the combination of the zeta potential (or surface charge density) of the wall, the properties of the electrolyte solution, and other relevant factors. For a given concentration gradient of an electrolyte along a plane wall, the magnitude of fluid velocity at a position in general increases with an increase in its electrokinetic distance from the wall, but there are exceptions. The effect of the lateral distribution of the induced tangential electric field and the relaxation effect in the double layer on the diffusioosmotic flow are found to be very significant.     

Diffusion of ionic species in bentonite

We present a macroscopic model of ionic diffusion in bentonites including the effect of the hydraulic-electrical-chemical couplings expected in such charged porous medium. The anomalous concentrations of the ions in the pore water of the bentonite are modeled with a modified Donnan model in which we account for the partition of the counterions between the Stern and Gouy-Chapman layers. This is accomplished using an electric triple layer (TLM) model combined with an explicit complexation model at the mineral/water interface. The porosity entering into the determination of the formation factor of the medium is an effective porosity obtained by removing the fraction of hydration water covering the surface of the clay minerals. We investigate two different cases of diffusion. In the first case, we consider a salinity gradient between two reservoirs in contact with a cylindrical sample of bentonite. The model predicts an increase of the diffusivity of the salt with the salinity of the solution in contact with the bentonite in agreement with experimental data. In the second case, we analyze a self-diffusion experiment of an ionic tracer. The model predicts an increase of the diffusivity of anions with the effective porosity and with the ionic strength. This is also in good agreement with experimental data.     

Structural change of ion-exchange membrane surfaces under high electric fields and its effects on membrane properties

Structural change of an ion-exchange membrane under a high electric field was investigated by comparing water dissociation and the FTIR spectra between the virgin membrane and that used at an overlimiting current density. From a series of water dissociation experiments at overlimiting current densities, it was observed that water dissociation in an anion-exchange membrane used at an overlimiting current density was higher than that in a virgin membrane at the same current density. The FTIR study revealed that the tertiary amine groups are formed from the quaternary ammonium groups on the anion-exchange membrane surface where ion depletion occurs under the influence of the applied strong electric field. The occurrence of increased water dissociation is considered to be caused by the protonation and deprotonation of the tertiary amine groups in the anion-exchange membrane. On the other hand, there was no structural change for the cation-exchange membrane under the electric field investigated in this study, which is coincident with the results of water dissociation experiments for the CMX membrane. In addition, we found that membrane resistance, permselectivity, and plateau length of the current-voltage curve were affected by the converted tertiary amine groups depending on the solution pH.     

Discrete charge effects on the Donnan potential and surface potential of a soft particle

The Donnan potential and surface potential of soft particles (i.e., polyelectrolyte-coated hard particles) in an electrolyte solution play an essential role in their electric behaviors. These potentials are usually derived via a continuum model in which fixed charges inside the surface layer are distributed with a continuous charge density. In this paper, for a plate-like soft particle consisting of a cubic lattice of fixed point charges, on the basis of the linearized Poisson–Boltzmann equation, we derive expressions for the electric potential distribution in the regions inside and outside the surface layer. This expression is given in terms of a sum of the screened Coulomb potentials produced by the point charges within the surface layer. We show that the deviation of the results of the discrete charge model from those of the continuous charge model becomes significant as the ratio of the lattice spacing to the Debye length becomes large.     

Transfer of electrolyte ions and water dissociation in anion-exchange membranes under intense current conditions

Polarization characteristics of electromembrane systems (EMS) based on the Russian commercial heterogeneous membranes MA-40 and MA-41, the anion-exchange heterogeneous membrane AMH (Mega, Czech Republic), and the modified membrane MA-40M are studied by the method of rotating membrane disk in dilute sodium chloride solutions. The effective transport numbers of ions are found; the partial voltammetric characteristics (VAC) with respect to chloride and hydroxyl ions are measured; the limiting current densities are calculated as a function of the membrane disk rotation rate. In terms of the theory of the overlimiting state of EMS, based on experimental VAC and the dependences of the effective transport numbers on the current density, the following internal parameters of systems under study are calculated: the space charge and electric field strength distribution over the diffusion layer and the membrane. It is shown that water dissociation can be virtually completely eliminated by substituting chemically stable quaternary ammonium groups inert with respect to water dissociation in the surface layer of a heterogeneous anion-exchange membrane MA-40 for the active ternary and secondary functional amino groups. The maximum electric field strength values at the membrane/solution interface, which were found in the framework of the theory of over-limiting state, turned out to be close for all anion-exchange membranes studied, namely, (7?C9) × 10⁶ V/cm. This suggests that it is the nature of ionogenic groups in the surface layer rather than the field effect that plays the decisive role in the membrane ability to accelerate the water dissociation reaction. It is proved experimentally that in highly intense current modes of the electrodialysis process, the thermal hydrolysis of quaternary ammonium bases occurs in strongly basic MA-41 and AMH membranes by the Hofmann reaction to form ternary amino groups catalytically active in water dissociation reaction. Based on the concept on the catalytic mechanism of water dissociation, the fraction of ternary amino groups formed by thermal hydrolysis in the surface layer (the space charge region) of monopolar anion-exchange membranes MA-41 and AMH is assessed quantitatively as 0.7 and 6.5%, respectively.     

Planar electric double layer for a restricted primitive model electrolyte at low temperatures

Monte Carlo simulation and the modified Poisson-Boltzmann theory are used to investigate the planar electric double layer for a restricted primitive model electrolyte at low temperatures. Capacitance as a function of temperature at low surface charge is determined for 1:1, 2:2, 2:1, and 3:1 electrolytes. Negative adsorption can occur for 1:1 electrolytes at low surface charge with low electrolyte concentration. The 1:1 electrolyte diffuse layer potential as a function of surface charge displays a maximum at low densities. At high densities, the diffuse layer potential is negative with a negative slope. The Gouy-Chapman-Stern theory fails in this low-temperature regime, whereas the modified Poisson-Boltzmann theory is fairly successful in this regard.     

Mathematical model for the overlimiting state of an ion-exchange membrane system

A three-layered mathematical model is proposed for describing the overlimiting state in an ion-exchange membrane system. The model’s prominent feature is the allowance for the space-charge region; the water dissociation reaction, which is catalyzed by active ionogenic groups; and the coupled gravitational and electroosmotic convection, which leads to the emergence of dependence of the effective diffusion layer thickness on the electric current density. The model is used for calculating, on the basis of known initial current-voltage curves and dependences of effective transport numbers on the current density, such internal characteristics of the system as the diffusion layer thickness, distribution of concentration of ions, space charge, and electric-field strength at various current densities.     

Molecular dynamics simulations of sorption of organic compounds at the clay mineral/aqueous solution interface

The adsorption of trichloroethene, C₂HCl₃, on clay mineral surfaces in the presence of water has been modeled as an example describing a general program that uses molecular dynamics simulations to study the sorption of organic materials at the clay mineral/aqueous solution interface. Surfaces of the clay minerals kaolinite and pyrophyllite were hydrated at different water levels corresponding to partial and complete monolayers of water. In agreement with experimental trends, water was found to outcompete C₂HCl₃ for clay surface sites. The simulations suggest that at least three distinct mechanisms coexist for C₂HCl₃ on clay minerals in the environment. The most stable interaction of C₂HCl₃ with clay surfaces is by full molecular contact, coplanar with the basal surface. This kind of interaction is suppressed by increasing water loads. A second less stable and more reversible interaction involves adsorption through single-atom contact between one Cl atom and the surface. In a third mechanism, adsorbed C₂HCl₃ never contacts the clay directly but sorbs onto the first water layer. To test the efficacy of existing force field parameters of organic compounds in solid state simulations, molecular dynamics simulations of several representative organic crystals were also performed and compared with the experimental crystal structures. These investigations show that, in general, in condensed-phase studies, parameter evaluations are realistic only when thermal motion effects are included in the simulations. For chlorohydrocarbons in particular, further explorations are needed of atomic point charge assignments. © 1998 John Wiley & Sons, Inc. J Comput Chem 19: 144–153, 1998