Electrokinetic and dielectric response of a concentrated salt-free colloid: Different approaches to counterion finite-size effects.

In the present work, a general model is developed for the electrokinetics and dielectric response of a concentrated salt-free colloid that takes into account the finite size of the counterions released by the particles to the solution. The effects associated with the counterion finite size have been addressed using a hard-sphere model approach elaborated by Carnahan and Starling [N. F. Carnahan and K. E. Starling, Equation of state for nonattracting rigid spheres, J. Chem. Phys. 51, 635 (1969)0021-960610.1063/1.1672048]. A more simple description of the finite size of the counterions based on that by Bikerman has also been considered for comparison. The studies carried out in this work include predictions on the effect of the finite counterion size on the equilibrium properties of the colloid and its electrokinetic and dielectric response when it is subjected to constant or alternating electric fields. The results show how important the counterion finite-size effects are for most of the electrokinetic and dielectric properties of highly charged and concentrated colloids, mainly for the static and dynamic electrophoretic mobilities. Furthermore, new insights are provided on the counterion condensation effect when counterions are allowed to have finite size. Focus is placed on the changes undergone by their concentration in the condensation layer for low-salt and highly charged colloids.

Electrokinetic detection of the salt-free condition in colloids. Application to polystyrene latexes.

Because of their singular phenomenology, the so-called salt-free colloids constitute a special family of dispersed systems. Their main characteristic is that the dispersion medium ideally contains only the solvent and the ions compensating exactly the surface charge of the particles. These ions (often called released counterions) come into the solution when the surface groups responsible for the particles charge get ionized. An increasing effort is nowadays dedicated to rigorously compare theoretical model predictions for ideal salt-free suspensions, where only the released counterions are supposed to be present in solution, with appropriately devised experiments dealing with colloids as close as possible to the ideal salt-free ones. Of course, if the supporting solution is aqueous, the presence of atmospheric contamination and any other charged species different from the released counterions in the solution must be avoided. Because this is not an easy task, the presence of dissolved atmospheric CO2 and of H+ and OH- from water dissociation cannot be fully discarded in aqueous salt-free solutions (often denominated realistic in such case). Ultimately, at some point, the role of the released counterions will be comparable or even larger in highly charged concentrated colloids than that of added salts. These topics are covered in the present contribution. The model results are compared with experimental data on the dynamic mobility and dielectric dispersion of polystyrene spheres of various charges and sizes. As a rule, it is found that the model correctly predicts the significance of alpha and Maxwell-Wagner-O'Konski relaxations. Positions and amplitudes of such relaxations are well predicted, although it is necessary to assume that the released counterions are potassium or sodium instead of protons, otherwise the frequency spectra of experimental mobility and permittivity differ very significantly from those theoretically calculated. The proposed electrokinetic evaluation is an ideal tool for detecting in situ the possible contamination (or incomplete ion exchange of the latexes). A satisfactory agreement is found when potassium counterions are assumed to be in solution, mostly if one considers that the comparison is carried out without using any adjustable parameters.

Influence of ion size effects on the electrokinetics of aqueous salt-free colloids in alternating electric fields.

Electrokinetics is the science of the physical phenomena appearing at the solid-liquid interface of dispersed particles subjected to external fields. Techniques based on electrokinetic phenomena constitute an important set of tools for the electrical characterization of colloids because of their sensitivity to the properties of particle-solution interfaces. Their rigorous description may require inclusion of the effects of finite size of chemical species in the theoretical models, and, particularly in the case of salt-free (no external salt added) aqueous colloids, also consideration of water dissociation and possible carbon dioxide contamination in the aqueous solution. A new ac electrokinetic model is presented for concentrated salt-free spherical colloids for arbitrary characteristics of the particles and aqueous solution, including finite-size effects of chemical species by appropriate modifications of the chemical reaction equations to include such non-ideal aspects. The numerical solution of the electrokinetic equations in an alternating electric field has also been carried out by using a realistic non-equilibrium scenario accounting for association-dissociation processes in the chemical reactions. The results demonstrate the importance of including finite-size effects in the electrokinetic response of the colloid, mainly at high frequencies of the electric field, and for highly charged colloids. Findings of previous models for pointlike ions or for ideal salt-free colloids including finite ion size effects are recovered with the present model, for the appropriate limiting conditions.

Surface conductivity of colloidal particles: experimental assessment of its contributions.

In this work we investigate how combined data on dielectric dispersion and electrophoretic mobility of colloidal suspensions at different temperatures can be used to evaluate the two main quantities characterizing the solid/liquid interface, namely, the zeta potential and the stagnant layer conductivity (SLC). This is possible because the electric permittivity depends on the total surface conductivity, while the electrophoretic mobility is governed by both the zeta potential and that conductivity. Based on a simple analytical theory, we can also estimate the diffusion coefficient of counterions in the stagnant layer, D(SL), for each temperature. The results lead to a good agreement between theory and experiment, although with somewhat high values of D(SL). With the aim of improving this description, we use a full theory of the electric permittivity of suspensions that accounts for the existence both of SLC and of a finite volume fraction of solids. An excellent description of the whole dielectric spectrum and of the electrophoretic mobility is possible in this case, although with still overestimated diffusion coefficients. This fact is discussed, and the importance of considering particle concentration effects even for suspensions that are often considered dilute is also stressed.

Effect of stagnant-layer conductivity on the electric permittivity of concentrated colloidal suspensions.

A long-lasting experience in the electrokinetics of suspensions has shown that the so-called standard model may be partly in error in explaining experimental data. In this model, the stagnant layer is considered nonconducting (Ksigmai=0), and only the diffuse layer contributes to the total surface conductivity (Ksigma=Ksigmad). In the present work, the authors analyze the consequences of assuming a nonzero stagnant layer conductivity on the permittivity of concentrated suspensions. Using a cell model to account for the particle-particle interactions, and a well established ion adsorption isotherm on the inner region of the double layer, the authors find the frequency-dependent electric permittivity of suspensions of spherical particles with volume fractions of solids up to above 40%. It is demonstrated that the addition of Ksigmai significantly increases the contributions of the double layer to the polarization of the suspension: the alpha or concentration polarization at low (kilohertz) frequencies, and the Maxwell-Wagner-O'Konski (associated with conductivity mismatch between particle and medium) one at intermediate (megahertz) frequencies. While checking for the possibility that the results obtained in conditions of Ksigmai not equal 0 could be reproduced assuming Ksigmai=0 and raising Ksigmad to reach identical total Ksigma, it is found that this is approximately possible in the calculation of the permittivity. Interestingly, this does not occur in the case of electrophoretic mobility, where the situations Ksigma=Ksigmad and Ksigma=Ksigmad+Ksigmai (for equal Ksigma) can be distinguished for all frequencies. This points to the importance of using more than one electrokinetic technique to properly evaluate not only the zeta potential but other transport properties of concentrated suspensions, particularly Ksigmai.

Influence of cell-model boundary conditions on the conductivity and electrophoretic mobility of concentrated suspensions.

In the last few years, different theoretical models and analytical approximations have been developed addressing the problem of the electrical conductivity of a concentrated colloidal suspension. Most of them are based on the cell model concept, and coincide in using Kuwabara's hydrodynamic boundary conditions, but there are different possible approaches to the electrostatic boundary conditions. We will call them Levine-Neale's (LN, they are Neumann type, that is they specify the gradient of the electrical potential at the boundary), and Shilov-Zharkikh's (SZ, Dirichlet type). The important point in our paper is that we show by direct numerical calculation that both approaches lead to identical evaluations of the conductivity of the suspensions if each of them is associated to its corresponding evaluation of the macroscopic electric field. The same agreement between the two calculations is reached for the case of electrophoretic mobility. Interestingly, there is no way to reach such identity if two possible choices are considered for the boundary conditions imposed to the field-induced perturbations in ionic concentrations on the cell boundary (r = b), deltan(i) (r = b). It is demonstrated that the conditions deltan(i)(b) = 0 lead to consistently larger conductivities and mobilities. A qualitative explanation is offered to this fact, based on the plausibility of counter-ion diffusion fluxes favoring both the electrical conduction and the motion of the particles.

Determination of stagnant layer conductivity in polystyrene suspensions: temperature effects.

In the classical theory of electrokinetic phenomena, it is admitted that the whole electrokinetic behavior of any colloidal system is fully determined by the zeta potential, zeta, of the interface. However, both experimental data and theoretical models have shown that this is an incomplete picture, as ions in the stagnant layer (the region between the solid surface and the slip plane--the plane where the equilibrium potential equals zeta) may respond to the field. In this paper, we aim at the evaluation of this contribution by the estimation of both K(SL)(sigma) (the surface conductivity of the stagnant layer) and K(d)(sigma) (the conductivity associated with the diffuse layer). This will be done by measuring the high-frequency dielectric dispersion (HFDD) in polystyrene suspensions; here "high-frequency" means the frequency interval where Maxwell-Wagner-O'Konski relaxation takes place (typically at MHz frequencies). Prior to any conclusions, a treatment of electrode polarization effects in the measurements was needed: we used two methods, and both led to similar results. Simulating the existence of surface conductivity by bulk conductivity, we reached the conclusion that no consistent explanation can be given for our HFDD and additional electrophoresis data based on the existence of diffuse-layer conductivity alone. We thus show how K(SL)(sigma) can be estimated and demonstrate that it can be explained by an ionic mobility very close to that characteristic of ions in the bulk solution. Such mobility, and hence also the values of K(SL)(sigma), increases with temperature as expected on simple physical grounds.

The primary electroviscous effect in colloidal suspensions.

Although a well-defined electrokinetic phenomenon, the primary electroviscous effect in dilute colloidal suspensions is still an unsolved problem. Most of the experimental tests of the different theories that we have studied have shown a lack of agreement. We have developed, during the last years, new theoretical approaches obtaining, finally, a much better agreement with the experimental results. The corrections are defined in two lines: first, it is accepted that ions present into the Stern layer, in which the fluid is stagnant, can tangentially move; second, it is accepted that the hydrodynamic interaction between colloidal particles exists although the suspensions are extremely diluted. The remarkable conclusion of our work is that the combination of both corrections should give correct theoretical results.

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.

Rheological and Electrokinetic Properties of Sodium Montmorillonite Suspensions.

Because of their particular electric surface properties and crystal structure, most clay minerals possess a very high ion exchange capacity. Furthermore, the surface charge distribution is anisotropic: while faces of the laminar clay particles have a negative, pH-independent charge, edges may be positive or negative, depending on pH. In this work, we propose to contribute new data on particle-particle interaction and charge distribution, by means of measurements of the low-frequency dielectric dispersion (LFDD) of the clay suspensions. Because of the nonspherical shape of clay particles, there are no theoretical models capable of explaining the experimental relaxation spectra. Hence, we limit ourselves to obtaining indirect information by comparing LFDD spectra in different experimental conditions. The quantities of interest in LFDD are the value of the low-frequency dielectric constant, epsilon'(r)(0), and the characteristic or relaxation frequency, omega(cr). These two parameters were measured varying the weight fraction, straight phi, of clay (0.5, 1, and 1.5% w/v) and the pH of the dispersion medium (5, 7, and 9), while maintaining the ionic strength constant ([NaCl]=10(-4) M). It was found that the characteristic relaxation frequency of the dielectric constant was pH-dependent, with a significant minimum at pH 7 in all cases. The results are interpreted as the superposition of two independent relaxation phenomena, associated with edges and faces. With respect to the weight fraction influence, we have found a linear behavior of epsilon'(r)(0) with straight phi at pH 9, indicating the existence of no significant interaction between particles. However, at pH 7 a slight deviation of linearity is observed, and at pH 5 we observe a clearly nonlinear behavior, indicating a stronger degree of interaction between particles. This is in good agreement with the initial assumption that at acid pH values, the electric surface charge of faces is negative, whereas the edges possess a positive charge, thus favoring attractive face-to-edge interaction. Copyright 2000 Academic Press.

Effect of a Dynamic Stern Layer on the Sedimentation Velocity and Potential in a Dilute Suspension of Colloidal Particles.

In this paper the theory of the sedimentation velocity and potential (gradient) in a dilute suspension of charged spherical colloidal particles developed by Ohshima et al. (H. Ohshima, T. W. Healy, L. R. White, and R. W. O'Brien, J. Chem. Soc., Faraday Trans. 2, 80, 1299 (1984)) has been modified to include the presence of a dynamic Stern layer on the particle surfaces. The starting point has been the theory that Mangelsdorf and White (C. S. Mangelsdorf, and L. R. White, J. Chem. Soc., Faraday Trans. 86, 2859 (1990)) developed to calculate the electrophoretic mobility of a colloidal particle allowing for the lateral motion of ions in the inner region of the double layer (dynamic Stern layer). The effects of varying the different Stern layer parameters on the sedimentation velocity and potential are discussed and compared to the case when a Stern layer is absent. The influence of electrolyte concentration and zeta potential of the particles is also analyzed. The results show that regardless of the chosen set of Stern layer and solution parameters, the presence of a dynamic Stern layer causes the sedimentation velocity to increase and the sedimentation potential to decrease, in comparison with the standard case (no Stern layer present). These changes are almost negligible when sedimentation velocity is concerned, but they are very important when it comes to the sedimentation potential. A justification for this fact can be given in terms of an Onsager reciprocal relation, connecting the magnitudes of the sedimentation potential and the electrophoretic mobility. As previously reported, the presence of a dynamic Stern layer exerts a great influence on the electrophoretic mobility of a colloidal particle, and by means of the Onsager relation, the same is confirmed to occur when the sedimentation potential is concerned. Copyright 2000 Academic Press.

Effects of Temperature and Polydispersity on the Dielectric Relaxation of Dilute Ethylcellulose Suspensions.

The role of temperature on the low-frequency dielectric dispersion is analyzed for moderately polydisperse suspensions of spherical ethylcellulose latex particles. The study is carried out in the 10-50 degrees C temperature range for two different electrolyte concentrations, namely, 5 x 10(-5) and 10(-4) M NaCl. It is found that the relaxation frequency increases with temperature, whereas the amplitude of the dielectric dispersion decreases when temperature is raised. This agrees qualitatively with predictions based on the classical electrokinetic theory (DeLacey, E. H. B., and White, L. R., J. Chem. Soc., Faraday Trans. 2 77, 2007 (1983)). However, the quantitative agreement is very far from being satisfactory. To try to overcome these differences, we have applied a more complete model in which tangential motions of ions in the inner part of the electric double layer is allowed for (DSL model, Mangelsdorf, C. S., and White, L. R., J. Chem. Soc., Faraday Trans. 86, 2859 (1990)). Although in most situations DSL models considerably improve the agreement between theory and experiment, in our case the dynamic Stern layer correction does not seem to be enough to bring much closer experimental data and predictions. It is for this reason that we also consider the fact that our suspensions are not strictly monodisperse. Keeping polydispersity in mind (this can be done by simply taking the volume average particle radius as a representative size parameter) and introducing it in the DSL model, it is shown that a much better description of the main features of the dielectric dispersion, that is, the amplitude of the dielectric increment, and the characteristic relaxation frequency of the suspensions can be reached. Copyright 1999 Academic Press.

Dielectric Dispersion of Colloidal Suspensions in the Presence of Stern Layer Conductance: Particle Size Effects.

In this work, we discuss the role that particle size plays in the manifestations of surface conduction on the dielectric response of colloidal dispersions. To that aim, experimental data on the dielectric constant of polystyrene suspensions of two different particle diameters (23 and 530 nm) are first compared to the predictions of a classical or standard model (E. H. B. DeLacey and L. R. White, J. Chem. Soc. Faraday Trans. 2 77, 2007 (1983)), and it is found that, while the latter explains reasonably the dielectric behavior of the smallest particles, it considerably underestimates the phenomenon in the case of large particles. To explain these results in terms of contributions of ion motions in the inner region of the double layer of the particles, the approach followed by C. S. Mangelsdorf and L. R. White (J. Chem. Soc. Faraday Trans. 86, 2859 (1990)) is used to incorporate surface conductance in the theory of dielectric response of suspensions. In ac fields it is found that the model considerably improves the comparison between theory and experiment, whereas its use seems unnecessary for the smallest particles, where, whatever the combination used for the parameters of the theory, its predictions do not differ from the standard theory. Only in the case of the larger particles studied does the introduction of surface conductance play any role. A comparison between both types of theoretical results in a wide range of particle sizes demonstrates that Stern layer conductance always increases the magnitude of the low-frequency dielectric constant of suspensions, but its effect is less important the smaller the particle size and the larger the zeta potential for fixed ionic conditions in the dispersion medium. Copyright 1999 Academic Press.

Effect of Size Polydispersity on the Dielectric Relaxation of Colloidal Suspensions: A Numerical Study in the Frequency and Time Domains.

In this article, a systematic numerical study is described of the effect of the polydispersity of suspensions of spherical particles on their dielectric behavior, in both the frequency and time domains, starting from the model proposed by DeLacey and White (J. Chem. Soc., Faraday Trans. 2 77, 2007 (1981)) for monodisperse suspensions. The distribution function of relaxation times, characterizing the dielectric response of the systems, is also calculated. It is found that in both the frequency and time domains the predicted behavior does not differ in any essential way from the one obtained for a monodisperse suspension with particle radius close to the volume-averaged mean radius of the polydisperse system. Hence, no arguments related to polydispersity seem to be useful for explaining the discrepancies frequently found between measured and calculated dielectric increments in suspensions, namely, those concerning the magnitude of the dielectric constant of the suspension (its low-frequency value), the value of the characteristic or relaxation frequency, or the overall shape of the relaxation pattern. Copyright 1998 Academic Press.