Introducing interacting diffuse layers in TLM calculations: a reappraisal of the influence of the pore size on the swelling pressure and the osmotic efficiency of compacted bentonites

The truncation of the Gouy-Chapman diffuse part in compacted clay-rocks and bentonite is introduced into the electrical triple-layer model (TLM) recently developed by P. Leroy and A. Revil [J. Colloid Interface Sci. 270 (2004) 371]. The new model is used to explain the dependence of the osmotic efficiency and the swelling pressure as functions of the mean pore size of the medium, determined from the porosity and the specific surface. The truncation of the diffuse layer introduces a new variable in the system of equations to be solved, the electrical potential at the midplane between adjacent charged surfaces. This new variable is evaluated through a Taylor expansion of the electrical potential. The present model is able to capture the variation of the osmotic efficiency and the swelling pressure with the mean pore size. The partition of counterions between the Stern layer and the diffuse layer as a function of the pore size calculated by the TLM also shows a good consistency with the model. This implies that more than 90% of the counterions are located in the Stern layer.     

Complex conductivity of water-saturated packs of glass beads

The low-frequency conductivity response of water-saturated packs of glass beads reflects a combination of two processes. One process corresponds to the polarization of the mineral/water interface coating the surface of the grains. The other process corresponds to the Maxwell-Wagner polarization associated with accumulation of the electrical charges in the pore space of the composite medium. A model of low-frequency conductivity dispersion is proposed. This model is connected to a triple-layer model of electrochemical processes occurring at the surface of silica. This model accounts for the partition of the counterions between the Stern and the diffuse layers. The polarization of the mineral/water interface is modeled by the electrochemical polarization model of Schurr for a spherical grain. We take into account also the DC surface conductivity contribution of protons of the sorbed water and the contribution of the diffuse layer. At the scale of a macroscopic representative elementary volume of the porous material, the electrochemical polarization of a single grain is convoluted with the grain size distribution of the porous material. Finally, the Maxwell-Wagner polarization is modeled using the complex conductivity of a granular porous medium obtained from the differential effective medium theory. The predictions of this model agree well with experimental data of spectral induced polarization. Two peaks are observed at low frequencies in the spectrum of the phase. The first peak corresponds to the distribution of the size of the beads and the second peak is due to the roughness of the grains.     

The water-amorphous silica interface: analysis of the Stern layer and surface conduction

To explain why dynamical properties of an aqueous electrolyte near a charged surface seem to be governed by a surface charge less than the actual one, the canonical Stern model supposes an interfacial layer of ions and immobile fluid. However, large ion mobilities within the Stern layer are needed to reconcile the Stern model with surface conduction measurements. Modeling the aqueous electrolyte-amorphous silica interface at typical charge densities, a prototypical double layer system, the flow velocity does not vanish until right at the surface. The Stern model is a good effective model away from the surface, but cannot be taken literally near the surface. Indeed, simulations show no ion mobility where water is immobile, nor is such mobility necessary since the surface conductivity in the simulations is comparable to experimental values.     

Analysis of diffuse-layer effects on time-dependent interfacial kinetics

We investigate the subtle effects of the diffuse charged layer on interfacial kinetics by solving the governing equations for ion transport (Nernst–Planck) with realistic boundary conditions representing reaction kinetics (Butler–Volmer) and compact-layer capacitance (Stern) in the asymptotic limit =λ_D/L→0, where λ_D is the Debye screening length and L is the distance between the working and counter electrodes. Using the methods of singular perturbation theory, we derive the leading-order steady-state response to a nonzero applied current in the case of the oxidation of a neutral species into cations, without any supporting electrolyte. In certain parameter regimes, the theory predicts a reaction-limited current smaller than the classical diffusion-limited current; this over potential effect is not due to ohmic drop effects in the bulk of the cell but rather to antagonist processes involved in the surface charge transfer and diffuse layer charging respectively. We demonstrate that the charging of diffuse charge, since it is intimately coupled to the surface reaction and cannot be considered independently, plays a fundamental role in nonequilibrium surface reactions when the transport of one of the reacting species is coupled to the total interfacial response of the compact and diffuse layers.     

A discussion on ion conductivity at cation exchange membrane/solution interface

By Gouy–Chapman–Stern–Grahame (CGSG) model, the electric double layer at ion exchange membrane/solution interface consists of two parts: the Stern layer and the diffusion layer. The ions in Stern layer are compacted and considered to be immobile. The relation of diffusion layer mean conductivity K with outer Stern layer potential φ₀, the boundary potential φ_δ and the electrolyte concentration C₀ is educed for symmetric electrolyte system. The results show that K is higher than that of the bulk solution and is greatly influenced by φ₀, φ_δ and C₀.The examination of PE01 cation exchange membrane/solution interface resistance R_e measured by ac impedance technique, shows that R_e value decreases quickly as the KCl electrolyte concentration rises. The effect of electrolyte concentration on the resistance of EDL can be explained by the electrical interactions between ions and charged groups of the membrane. Since the membrane/solution interface resistance is much higher than that of bulk solution, therefore, a further analysis based on the theory developed in this study proves that the ion transfer resistance R_e of membrane–solution interface predominantly occurs at Stern layer as a result of static electrical interaction.     

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.     

Measurement and Modeling of Th Sorption onto TiO(2)

The sorption and desorption of Th onto TiO₂ (10 g/L) was studied as a function of pH (1–11), ionic strength (0.005–0.1 M NaCl, NaClO₄), thorium concentration (from 1 × 10⁻⁹ to 3 × 10⁻² M), and carbonate concentration (up to 10⁻² M) using ²³⁴Th tracer. The results indicate the reversible formation of an inner sphere complex with a strong pH dependence. No effect of the carbonate on sorption could be measured in the pH range investigated (5–10.5). The data were fitted with five different conceptual models (constant capacitance, diffuse layer, Stern, triple layer, and a nonelectrostatic model) using FITEQL. Different hydroxy complex surface reactions were tried, one at a time, varying the site density (1–12 sites/nm²). The reaction that provided the best fit depended strongly on the surface site density and the model used. To be able to fit a reaction to the pH-dependent data with the compact layer models, the effect of the electrostatic term had to be decreased by decreasing the site density below 5 sites/nm². None of the electrostatic models could be used to fit all of the isotherm data. The nonelectrostatic model could be used to reasonably fit both the pH-dependent and isotherm data.     

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.     

A double layer model of the gas bubble/water interface

Zeta potential is a physico-chemical parameter of particular importance to describe sorption of contaminants at the surface of gas bubbles. Nevertheless, the interpretation of electrophoretic mobilities of gas bubbles is complex. This is due to the specific behavior of the gas at interface and to the excess of electrical charge at interface, which is responsible for surface conductivity. We developed a surface complexation model based on the presence of negative surface sites because the balance of accepting and donating hydrogen bonds is broken at interface. By considering protons adsorbed on these sites followed by a diffuse layer, the electrical potential at the head-end of the diffuse layer is computed and considered to be equal to the zeta potential. The predicted zeta potential values are in very good agreement with the experimental data of H₂ bubbles for a broad range of pH and NaCl concentrations. This implies that the shear plane is located at the head-end of the diffuse layer, contradicting the assumption of the presence of a stagnant diffuse layer at the gas/water interface. Our model also successfully predicts the surface tension of air bubbles in a KCl solution.     

Electrokinetics of concentrated suspensions of spherical colloidal particles with surface conductance, arbitrary zeta potential, and double-layer thickness in static electric fields

In this paper the electrophoretic mobility and the electrical conductivity of concentrated suspensions of spherical colloidal particles have been numerically studied under arbitrary conditions including zeta potential, particle volume fraction, double-layer thickness (overlapping of double layers is allowed), surface conductance by a dynamic Stern layer model (DSL), and ionic properties of the solution. We present an extensive set of numerical data of both the electrophoretic mobility and the electrical conductivity versus zeta potential and particle volume fraction, for different electrolyte concentrations. The treatment is based on the use of a cell model to account for hydrodynamic and electrical interactions between particles. Other theoretical approaches have also been considered for comparison. Furthermore, the study includes the possibility of adsorption and lateral motion of ions in the inner region of the double layers (DSL model), according to the theory developed by C. S. Mangelsdorf and L. R. White (J. Chem. Soc. Faraday Trans.86, 2859 (1990)). The results show that the correct limiting cases of low zeta potentials and thin double layers for dilute suspensions are fulfilled by our conductivity formula. Moreover, the presence of a DSL causes very important changes, even dramatic, on the values of both the electrophoretic mobility and the electrical conductivity for a great range of volume fractions and zeta potentials, specially when double layers of adjacent cells overlap, in comparison with the standard case (no Stern layer present). It can be concluded that in general the presence of a dynamic Stern layer causes the electrophoretic mobility to decrease and the electrical conductivity to increase in comparison with the standard case for every volume fraction, zeta potential, and double-layer thickness.     

The modelling of solvent structure in the electrical double layer

A Civilized Model electrolyte is one in which the ions and solvent molecules are regarded as distinct molecular species and treated on an equal basis. Recent efforts to use a Civilized Model to study the effects of solvent structure in the properties of the electrical double layer are discussed. By modelling the electrolyte as a simple ion-dipole mixture, it is possible to gain valuable insight in areas such as: 1) the successes and limitations of the Gouy-Chapman-Stern picture; 2) the derivation as opposed to a postulation of the Stern layer; 3) the influence of the charged surface on the magnitudes of the apparent Stern capacities (e.g. the “low” capacitances of the mercury/solution interface vs. the “high” capacitance of inorganic oxides); 4) the effect of solvent structure on the potential profile in the diffuse layer; 5) the interpretation of the electrokinetic potential; and 6) the role of solvent orientation on the x potential.     

The conditions governing the specific adsorption of counter-ions in the interaction of two parallel plate-like colloidal particles

Frens and Overbeek have proposed that during the Brownian collision of two colloidal particles in a hydrophobic sol, the surface charge density due to potential-determining (p.d.) ions remains virtually unchanged. It is argued here that the cause of this behaviour is the low concentration of p.d. ions in the diffuse layer. However, equilibrium can be maintained with respect to counter-ions adsorbed into the Stern region from the supporting electrolyte, because the concentration of such electrolyte in the dispersion medium is considerably greater than that of p.d. ions.A general expression is quoted from earlier work for the electric double layer interaction between two parallel plate-like particles in the case where surface charge due to p.d. ions is fixed, but where counter-ions adsorbed into the Stern region can equilibrate with ions of the same species in the diffuse layer. Incorporating discreteness-of-charge and ion-size effects into the adsorption isotherm of the counter-ions, the double layer interaction energy of the two plates is calculated at contact of the two outer Helmholtz planes (o.h.p.'s). It is shown that although this energy exceeds the classical expression obtained by assuming the potential at the o.h.p. to be independent of plate separation, it remains finite.     

The surface conductivity at the polystyrene-latex particle/electrolyte interface

Summary The surface conductivity of polystyrene latex particles has been measured in a range of concentrations of potassium chloride, barium perchlorate, aluminium perchlorate, and hexol nitrate. There was an increase of surface conductivity with increasing concentration for the three simple electrolytes; the surface conductivity eventually attained the same constant value in each electrolyte. The measured surface conductivity was several orders of magnitude larger than that calculated for theGouy layer; the results have been interpreted as indicative of a large contribution to the total surface conductivity from the fixed layer and a negligible contribution from the diffuse layer.

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Zusammenfassung Die Oberflächenleitfähigkeit von Polystyrol-Latexpartikeln wurde in Abhängigkeit von der Konzentration in Kaliumchlorid, Bariumperchlorat, Aluminiumperchlorat und Hexolnitrat gemessen. Es besteht ein Anwachsen der Oberflächenleitfähigkeit mit wachsender Konzentration für die drei einfachen Elektrolyte. Dabei erreicht die Oberflächenleitfähigkeit anscheinend in jedem Elektrolyt denselben konstanten Wert. Die gemessene Oberflächenleitfähigkeit ist mehrere Größenordnungen höher als die für eineGouy-Schicht berechnete. Die Resultate wurden dahingehend interpretiert, daß sie auf einen großen Beitrag zur Gesamtleitfähigkeit durch eine fixierte Schicht und einen vernachlässigbaren Beitrag aus der diffusen Schicht hinweisen.

     

Cationic Interracial Polymerization of p-Methoxystyrene by Means of Triphenylmethylium Halide-Aerosil Initiation

Cationic polymerizations of p-methoxystyrene initiated by triphenylmethylium halides in conjunction with silica surfaces proceed via ion pair intermediates. The polymerization yields both aerosil-polymer composites and soluble polymers. The triphenylmethylium halides are active only on the surface of the solid, which is demonstrated by means of adsorption and zeta-potential measurements in 1,2-dichloroethane. The influence of the overall monomer concentration and initiator concentration on the MWD curves and composite formation is discussed in relation to the mechanism of cationic polymerization. Triphenylmethylium bromide-aerosil adsorbates yield soluble polymers with narrow MWD curves (M _w/M _n ? 1,2). This behavior approaches living conditions. The formation of composites mainly proceeds in the Stern layer of the ion pair layer which covers the aerosil particle while the generation of soluble polymers occurs in the diffuse layer. The behavior of interfacial polymerization is discussed in relation to surface chemistry and the mechanism of cationic polymerization. A general model for interfacial polymerization is proposed for triphenylmethylium halide-aerosil initiation.     

The impact of ionic strength on the adsorption of protons, Pb, Cd, and Sr onto the surfaces of Gram negative bacteria: testing non-electrostatic, diffuse, and triple-layer models

Bacterial surface adsorption reactions are influenced by electric field effects caused by changes in ionic strength; however, existing datasets are too sparse to definitively constrain these differences or to determine the best way to account for them using thermodynamic models. In this study, we examine the ionic strength dependence of proton and metal adsorption onto the surfaces of Pseudomonas mendocina and Pseudomonas putida by conducting proton, Cd(II), Pb(II), and Sr(II) adsorption experiments over the ionic strength range of 0.001 to 0.6 M. Chosen experimental results are thermodynamically modeled using a non-electrostatic approach, a diffuse layer model (DLM), and a triple-layer model (TLM). The results demonstrate that bacterial surface electric field effects are negligible for proton, Cd, and Pb adsorption onto P. putida and P. mendocina, and that the discrete site non-electrostatic model developed in this study is adequate for describing these reactions. The extent of Sr adsorption is influenced by changes in the bacterial surface electric field; however, the non-electrostatic model better describes Sr adsorption behavior than the DLM or TLM. The DLM and TLM greatly overpredict the effect of the electric field for all adsorption reactions at all ionic strengths tested.     

Conductivity of micellar solutions of ionic surfactants and surface conductivity of micelles

Within the effective medium model, a method is proposed for determining the surface conductivity of micelles. The known experimental data on the conductivity of aqueous sodium dodecyl sulfate micellar solutions are analyzed employing the developed approach. Specific surface conductivity λ_s of micelles is shown to be 10⁻⁸Ω⁻¹. The high values of λ_s are indicative of a noticeable contribution of the dense part of the electrical double layer, which is comparable with the contribution of its diffuse part, to the micellar solution conductivity. This estimate is strongly dependent on the initial information obtained in different studies. The marked influence of the micelle surface conductivity on the effective conductivity of micellar solutions allows one to put the question of the correctness of the method commonly used to determine the degree of counterion binding from the slopes of the dependences of the solution conductivity on the overall surfactant concentrations in the premicellar and micellar regions.     

Colloid stability of silver iodide in water-ethylene glycol mixtures

Summary This paper treats the stability of silver iodide sols in water-ethylene glycol (EG) mixtures. Rates of coagulation, obtained by a stopped-flow technique are analyzed in combination with data on surface charge, conductivity, and density. To some extent, bulk solution properties are reflected in the stability. For instance, indications of contraction, with a maximum at a molar ratio water/EG 2 recur in the rates of fast coagulation. It is impossible to describe the transition from water to EG through the variation of the bulk dielectric permittivityonly, because drastic alterations occur also in the Stern layer. Especially its change in thickness and the increasing specific counterion adsorption with increasing EG content are important stability-determining factors. Particularly at high EG concentrations, the outerHelmholtz plane may not be considered as a smeared-out equipotential plane. The mosaic-like potential pattern of this layer suggests the necessity to reconsider the dynamics of particle interactions.     

Non-equilibrium electric surface phenomena

Non-equilibrium aspects of traditional electrokinetic phenomena (electrophoresis, electroosmosis, streaming potential, sedimentation potential), electrostatic interaction of particles and new electrokinetic phenomena are considered. The significance of non-equilibrium electric surface phenomena for many major areas of modern colloid science (characterization of colloids, membrane science, transport phenomena and separation, particle interaction and coagulation) is established.The study of non-equilibrium electric surface phenomena is connected with the validation of the standard electrokinetic model (SEM), the development of a non-standard model and the development of an extensive programme of disperse system characterization based on integrated electrokinetic investigations. Experimental and theoretical studies of systems with a smooth, non-porous impermeable surface (mica in Anderson's experiments, and quartz microcapillaries with a molecule-smooth surface in Churaev's experiments) have shown that usually there are no significant difficulties in interpreting electrokinetic investigations despite the possible anomaly in the water structure near the surface and the possibility of maximum shear stress (yield stress), i.e. the anomalous viscosity and decreased dissolving power with respect to ions. However, systems which do not satisfy the conditions of the SEM are widely distributed, owing to the porosity, roughness or permeability of the boundary layer of the surface of the solid body which simultaneously belongs to the solid and liquid phases. In this layer, enclosed between the outer Helmholtz plane and the slipping plane, the motion of the liquid strongly slows down and the tangential flow of ions is characterized purely by the mobility which is close to the normal. Thus, a general property of a non-standard electrokinetic model is the presence of an anomalous (additional) surface conductivity in excess of the surface conductivity determined according to Bikerman's equation based on the ζ -potential alone.Confidence in modelling the electrokinetic phenomena has grown with the development of methods for modifying the surface such that its properties approach those of the SEM (Bijsterbosch and co-workers; Saville and co-workers).Extension of the particle characterization concept requires the measurement of both the mobile charge and the electrokinetic charge and from this an estimate of the thickness of the additional conductivity zone can be made. With the additional measurement of a titratable charge, it is possible to estimate the ion distribution between the dense and diffuse parts of the double layer (DL) and to estimate the decreased mobility of ions in the Stern layer or in the immobilized part of the DL.Quantitative laws governing the interaction of particles and corresponding to the non-standard model substantially differ from the traditional laws described by the DVLO theory as applied to the SEM. This is also true for adsorption properties which are characterized without sufficient reason by means of the ζ-potential. Therefore both the development of models of interaction and adsorption of ions, allowing for the non-standard electrokinetic model, and the extension of the particle characterization programme to integrated investigations of electric surface phenomena are required.Further generalization of the theory of electrokinetic phenomena is achieved. In addition to the surface charge another variety of surface force can be the origin of the electrokinetic phenomena.