Series approach to modeling ion size effects for symmetric electrolytes in the diffuse double layer

An analytical model is developed for the potential drop and differential capacity across the diffuse layer which considers the effects of ion size on these properties. For symmetric electrolytes, this potential drop is expressed in terms of a cubic polynomial in the corresponding estimate in the Gouy-Chapman theory. Optimal polynomial coefficients and model validation for 1:1 and 2:2 electrolytes are provided by fits of Monte Carlo data obtained for a restricted electrolyte in a primitive solvent. Simple relationships between these coefficients and parameters commonly associated with the mean spherical approximation are obtained. It is shown that the series approach accurately describes potential drops and differential capacities of the diffuse layer for 1:1 and 2:2 electrolytes for the chosen assumptions.     

Application of the hypernetted chain approximation to the electrical double layer for 2:1 and 1:2 electrolytes

The equations needed to estimate the potential drop across the diffuse layer according to the hypernetted chain approximation (HNCA) are derived in this paper for 2:1 and 1:2 electrolytes at the restricted primitive level. It is shown that HNCA results can be expressed in the same format as the corresponding Gouy-Chapman equations with inclusion of two modifying functions. One function depends on the fraction of the solution volume occupied by the ions, and the other depends on the reciprocal thickness of the ionic atmosphere surrounding each ion. In addition, an expression for the potential profile in the diffuse layer for 2:1 and 1:2 electrolyte solutions is derived according to Gouy-Chapman theory. The modifying functions in the HNCA are then estimated using the Henderson-Blum approach for solutions containing ions with diameters of 300 and 400 pm for concentrations in the range from 0.1 to 2 M. It is shown that the Henderson-Blum approach is inadequate for systems with multivalent ions except for charge densities very close to the point of zero charge.     

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.     

Assessment of ion size effects in the diffuse double layer with use of an integral equation approach

An analytical expression is developed for the potential drop across the diffuse layer phi(d) in terms of a cubic polynomial in the corresponding estimate in the Gouy-Chapman approximation, phi(d)(GC). The coefficients of this polynomial are defined in terms of the MSA volume fraction eta and the reciprocal distance parameter Gamma. The resulting expression is shown to describe the Monte Carlo estimates of phi(d) obtained in a primitive level simulation of diffuse layer properties.     

Mechanisms behind concentration profiles illustrated by charge and concentration distributions around ions in double layers

The mechanisms behind the behaviour of concentration and charge density profiles in diffuse electric double layers are investigated quantitatively for 1:1 and 2:2 electrolytes. This is done by analysing various contributions to the mean force that acts on each ion. The forces are obtained from the calculated ionic charge and concentration distributions around individual ions at various positions in the double layer. These distributions are presented graphically which allows an immediate visual illustration of the mechanisms in action. Some features studied are charge inversion in double layers for divalent aqueous electrolytes, overcompensation of surface charge due to large amounts of physisorbed counterions, ion size effects in the double layer structure and various mechanisms that cause deviations from the predictions of the Poisson–Boltzmann approximation. A major objective of the paper is to present the results in a visual form and explain aspects of modern double layer theory in a simple manner.     

New developments in the theory of the diffuse double layer

The role of ion size effects in determining diffuse layer properties is considered. Monte Carlo data relevant to this question are reviewed. Then the integral equation approach to the problem is considered with an emphasis on attempts to derive an analytical equation for the potential drop across the diffuse layer. An empirical model for ion size effects is described that allows one to estimate not only the diffuse layer potential drop but also the diffuse layer capacity, the ionic surface excesses, and the potential distribution in the diffuse layer. It is demonstrated that the resulting model can easily be applied by experimentalists to the analysis of experimental data for double layer phenomena in electrochemistry and colloid science.     

Separation of aqueous electrolyte solutions with asymmetric membranes containing one charged layer

The Nernst-Planck equation and fine-pore membrane model are applied to describe the ultra- and nanofiltration of electrolyte solutions through a inhomogeneous membrane containing one charged layer. Concentration and electric potential distributions, as well as dependences of electrolyte rejection coefficient (selectivity) and streaming potential on system parameters are determined. Asymmetry effect is revealed with respect to the rejection coefficient and streaming potential at different orientations of the selective charged layer relative to the direction of the filtration flow. The cases of 1: 1 and 1: 2 electrolytes are investigated in detail. Theoretical calculations demonstrate that the rejection coefficient of a bi-layer membrane rises in the following series of binary electrolytes: 1: 2 < 1: 1 < 2: 1, when the first layer is positively charged, and in the opposite series of these electrolytes, when the first layer is negatively charged.     

Temperature and concentration effects on electrolyte transport across porous thin-film composite nanofiltration membranes: Pore transport mechanisms and energetics of permeation

The influence of temperature and concentration on nanofilter charge density and electrolyte pore transport mechanisms is reported. Crossflow filtration experiments were performed to measure transport of several electrolytes (NaCl, NaNO3, NaClO4, CaCl2, MgCl2, and MgSO4) across two commercially available thin-film composite nanofiltration membranes in the range 5-41 degrees C. Experiments were also performed with selected salts in the range 1-50 meq/L to quantify concentration effects. Three different approaches, irreversible thermodynamics, extended Nernst-Planck formulation, and theory of rate processes, were employed to interpret retentions of these symmetric and asymmetric electrolytes at varying temperature and concentration. Increasing feed water temperature slightly increased electrolyte reflection coefficients and only weakly increased permeability compared with neutral solutes. Electromigration and convection tended to counteract each other at high fluxes explaining the weak temperature dependence of the reflection coefficient. Changes in membrane surface charge density with temperature were attributed to increased adsorption of electrolytes on the polymer constituting the active layer. Activation energy of permeation for charged solutes was primarily determined by the Donnan potential at the membrane-feed water interface. Electrolyte permeation was shown to be an enthalpy-driven process that resulted in small entropy changes. Increasing sorption capacity with temperature and low sorption energies indicated that co-ion sorption on polymeric membranes was an endothermic physicosorption process, which appears to determine temperature dependence of electrolyte permeation at increased feed concentrations.     

Influence of surface conductivity on the apparent zeta potential of TiO2 nanoparticles

Zeta potential is a physico-chemical parameter of particular importance in describing ion adsorption and electrostatic interactions between charged particles. Nevertheless, this fundamental parameter is ill-constrained, because its experimental interpretation is complex, particularly for very small and charged TiO(2) nanoparticles. The excess of electrical charge at the interface is responsible for surface conductance, which can significantly lower the electrophoretic measurements, and hence the apparent zeta potential. Consequently, the intrinsic zeta potential can have a larger amplitude, even in the case of simple 1:1 electrolytes like NaCl and KCl. Surface conductance of TiO(2) nanoparticles immersed in a NaCl solution is estimated using a surface complexation model, and this parameter and particle size are incorporated into Henry's model in order to determine a constrained value of the zeta potential from electrophoresis. Interior conductivity of the agglomerates is calculated using a differential self-consistent model. The amplitude of estimated zeta potential is greater than that derived from the von Smoluchowski equation and corresponds to the electric potential at the outer Helmholtz plane calculated by our surface complexation model. Consequently, the shear plane may be located close to the OHP, contradicting the assumption of the presence of a stagnant diffuse layer at the TiO(2)/water interface.     

Behavior of mixtures of symmetric and asymmetric electrolytes near discretely charged planar surfaces: a Monte Carlo study

Canonical Monte Carlo (CMC) simulations are employed in this work in order to study the structure of the electrical double layer (EDL) near discretely charged planar surfaces in the presence of symmetric and asymmetric indifferent electrolytes within the framework of a primitive model. The effects of discreteness and strength of surface charge, charge asymmetry, and size asymmetry are specific focuses of this work. The CMC simulation protocol is initially tested against the classical theory, the modified Gouy-Chapman (GC) theory, in order to assess the reliability of the simulation results. The CMC simulation results and the predictions of the classical theory show good agreement for 1:1 electrolytes and low surface charge, at which conditions the GC theory is valid. Simulations with symmetric and asymmetric electrolytes and mixtures of the two demonstrate that size plays an important role in determining the species present in the EDL and how the surface charge is screened. A size-exclusion effect could be consistently detected. Although it is energetically favorable that higher-valence ions screen the surface charge, their larger size prevents them from getting close to the surface. Smaller ions with lower valences perform the screening of the charge, resulting in higher local concentrations of small ions close to the surface. The simulations also showed that the strength of the surface charge enhances the size-exclusion effect. This effect will definitely affect the magnitude of the forces between interacting charged surfaces.     

Examination of the role of ion size in determining double layer properties on the basis of a generalized mean spherical approximation

A new model for the diffuse double layer which accounts for the effects of ion size and solution permittivity is described. It is then used to estimate the potential drop across the diffuse layer at negative charge densities for the cases that Li⁺ and Cs⁺ are the electrolyte cations. The potential drop in the Li⁺ system is considerably smaller than that in the Cs⁺ system at 1 M, and both values are smaller than the value predicted by the Gouy–Chapman model. As the electrolyte concentration decreases these differences become smaller so that at 0.01 M, the present model predicts that the diffuse layer potential drop is approximately 90% of the Gouy–Chapman estimate. The results of the model are used to examine the differences in inner layer structure at mercury electrodes with Li⁺ and Cs⁺ ions at the outer Helmholtz plane, and to reconsider the question of the specific adsorption of Cs⁺ at negative-charge densities.     

Transport in polymer-gel composites: response to a bulk concentration gradient

This paper examines the response of electrolyte-saturated polymer gels, embedded with charged spherical inclusions, to a weak gradient of electrolyte concentration. An electrokinetic model was presented in an earlier publication, and the response of homogeneous composites to a weak electric field was calculated. In this work, the influence of the inclusions on bulk ion fluxes and the strength of an electric field (or membrane diffusion potential) induced by the bulk electrolyte concentration gradient are computed. Effective ion diffusion coefficients are significantly altered by the inclusions, so-depending on the inclusion surface charge or zeta potential-asymmetric electrolytes can behave as symmetrical electrolytes and vice versa. The theory also quantifies the strength of flow driven by concentration-gradient-induced perturbations to the equilibrium diffuse double layers. Similarly to diffusiophoresis, the flow may be either up or down the applied concentration gradient.     

A rapidly convergent series for calculation of the interaction between two similar plane double-layers for z+/z- = -2 asymmetric electrolytes at positive surface potential

A rapidly convergent series for calculation of the interaction energies between two similar plane double-layers for z+/z- = -2 asymmetric electrolytes at positive surface potential are obtained by introducing a parameter lambda in elliptical integral. When dimensionless surface potential is less or equal to 20, the number of the series terms required to obtain the interaction energies with six significant digits are not more than 4. The accurate numeral results are given and they can be used to check up the validity of approximate expressions people obtain. The present results are also fit for z+/z- = -1/2 asymmetric electrolytes at negative surface potential.     

Unidimensional Approximation of the Diffuse Electrical Layer in the Inner Volume of Solid Electrolyte Grains in the Absence of Background Ions

In this paper we continue working on our theory of electrical double layers resulting exclusively from dissociation of a solid electrolyte, which we previously proposed as a medium for catalytic interaction between solid cellulose and solid acid catalysts of hydrolysis. Two theoretical unidimensional models of the inner grain volume are considered: an infinitely long cylindrical pore, and a gel electrolyte near a grain outer surface. Despite the model simplicity, the predictions for the cylindrical pore case are in semi-quantitative agreement with literature data on electroosmotic experiments, adequately explaining high proton selectivity of sulfonic membranes, and decline of such selectivity at high background acid concentration. The gel model predicts less concentrated diffuse layer in comparison to electrolytes with impenetrable skeleton (e. g., sulfonated carbons). This suggests limited suitability of gel electrolytes as catalysts if a substrate cannot diffuse into the gel bulk and the reaction is thereby spatially limited to the near-surface region, for example if a substrate is solid like aforementioned cellulose.     

Diffusion of electrolytes of different natures through the cation-exchange membrane

Diffusion of different electrolytes through a negatively charged (cation-exchange) membrane into distilled water has been studied. It has been established theoretically (with no regard to the presence of diffusion layers) that the integral diffusion permeability coefficient of an electrolyte depends on the diffusion coefficients and the ratio between the charge numbers of a cation–anion pair, the ratio between the density of charges fixed in the membrane and electrolyte concentration, and the averaged coefficient of equilibrium distribution of cation?anion ion pairs in the membrane matrix. It has been found that, when co-ions have a higher mobility, the dependence of diffusion permeability on electrolyte concentration passes through a maximum. Derived equations have been compared with experimental dependences of the diffusion permeability of an MC-40 membrane with respect to different solutions of inorganic 1: 1 and 2: 1 electrolytes. The developed method has been shown to be applicable for describing diffusion of any electrolytes (including asymmetric ones) through arbitrary uniformly charged membranes.     

Analysis of experimental data on the EDL capacitance for different metals in asymmetrical electrolytes: Accounting for the solution nonideality

Potential dependences of differential capacitanceC are measured in diluted aqueous La₂(SO₄)₃ solutions on Hg, Tl-Ga, and Cd-Ga electrodes and in aqueous Na₂SO₄ solutions on the Cd-Ga electrode. The 1/C vs. 1/C_d dependences (where C_d is the differential capacitance of the diffuse layer) are linear and have a unit slope at the potential of zero charge, provided C_d is calculated using the Gonzalez-Sanz theory. Calculations based on the Gouy-Chapman-Grahame theory yield inadequate results.     

The double layer and ion adsorption at the interface between two non miscible solutions: Part I. Interfacial tension measurements for the water-nitrobenzene tetraalkylammonium bromide systems

The classical treatment of the double layer has been extended to the interface between two immiscible solutions. The model presented is composed of an inner compact layer, characterized by a dipolar potential drop, between two diffuse type layers. The systems studied are composed of C₂ to C₅ quaternary ammonium bromides at partition equilibrium between water and nitrobenzene for which the inner potential difference, for a given electrolyte, is independent, at least in the lower concentration range, of the concentration. Drop weight interfacial measurements and the use of the Gouy-Chapman approach show that the tetraethyl-, tetrapropyl- and tetrabutylammonium ions are not adsorbed within the inner compact layer, and the dipolar potential drop of this layer can then be determined. Tetrapentylammonium ions on the contrary are specifically adsorbed but the amount of adsorbed ions within the compact inner layer cannot be evaluated because of the impossibility, in this case, of determining the dipolar potential drop.     

Electrokinetic studies on testosterone/aqueous electrolyte interface : Part 1. Electro-osmosis,electrophoresis, streaming potential and streaming current

The electrical properties of testosterone interfaces were investigated. For this purpose, measurements of electro-osmosis, hydrodynamic permeation, streaming potential and streaming currents of metabolically important solutions of the electrolytes NaCl, KCl and MgCl₂ (in the concentration range 10^?4?10^?3 mol/l) across a testosterone plug were carried out. Electrophoretic mobility of testosterone particles suspended in these electrolyte solutions was also studied. The data were analysed from the viewpoint of nonequilibrium thermodynamics. Phenomenological coefficients were evaluated from the linear transport equations and Saxen's relationship was verified. Dependence of phenomenological coefficients on electrolyte concentration was examined. Electro-osmotic and electrophoretic transport coefficients were found to vary linearly with concentration, whereas hydrodynamic permeation and membrane conductance coefficients show non-linear variation. The results are explained on the basis of structural modifications occurring during the passage of the permeating species through the membrane. The nature of the electrical double layer formed at the testosterone/solution interface was ascertained on the basis of the direction of electro-osmotic permeation and electrophoretic migration of testosterone particles.Zeta potentials were estimated in order to obtain a plausible picture of the electrical double layer at the testosterone/solution interfaces. Dependence of zeta potentials on concentration was examined and membrane parameters calculated. The double layer thickness was estimated, which reveals that the diffuse double layer is more compact in the case of MgCl₂ than in that of KCl.     

Application of the method of images on electrostatic phenomena in aqueous Al2O3 and ZrO2 suspensions

A new solution for the Poisson equation for the diffuse part of the double layer around spherical particles will be presented. The numerical results are compared with the solution of the well-known DLVO theory. The range of the diffuse layer differs considerably in the two theories. Also, the inconsistent representation of the surface and diffuse layer charge in the DLVO theory do not occur in the new theory. Experimental zeta potential measurements were used to determine the charge of colloidal Al2O3 and ZrO2 particles. It is shown that the calculated charge can be interpreted as a superposition of independent H+ and OH- adsorption isotherms. The corresponding Langmuir adsorption isotherms are taken to model the zeta potential dependence on pH. In the vicinity of the isoelectric point the model fits well with the experimental data, but at higher ion concentrations considerable deviations occur. The deviations are discussed. Furthermore, the numerical results for the run of the potential in the diffuse part of the double layer were used to determine the electrostatic interaction potential between the particles in correlation with the zeta potential measurements. The corresponding total interaction potentials, including the van der Waals attraction, were taken to calculate the coagulation half-life for a suspension with a particle loading of 2 vol%. It is shown that stability against coagulation is maintained for Al2O3 particles in the pH region between 3.3 and 7 and for ZrO2 only around pH 5. Stability against flocculation can be achieved in the pH regime between 4.5 and 7 for Al2O3, while the examined ZrO2 particles are not stable against flocculation in aqueous suspensions.     

Description of <Emphasis Type="Italic">n</Emphasis>-butanol adsorption at the Hg/(H<Subscript>2</Subscript>O + NaF) interface in terms of the model of three parallel capacitors combined with the classical theory of diffuse layer

The curves of differential capacitance of the electrical double layer dense part in the Hg/(H₂O + NaF + n-C₄H₉OH) are calculated by using the equations of the model of three parallel capacitors, with the corresponding adsorption parameters fitted. The curves of full differential capacitance in the system are calculated, basing on the obtained results combined with the classical theory of the diffuse layer, for the following concentrations of the supporting electrolyte: 0.003, 0.01, 0.03, 0.1, 0.3, and 1 M. By using the curves’ regression analysis it is shown that they agree very well with the model of two parallel capacitors, when six effective adsorption parameters are appropriately selected (provided the linear potential dependence of the effective attraction constant is allowed for). The dependences of all model’s effective parameters on the NaF concentration are found; they correspond well to the analogous experimental dependence in the system under study.