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.     

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.     

Influence of the counterion and co-ion diffusion coefficient values on some dielectric and electrokinetic properties of colloidal suspensions

The dependences of the conductivity increment, the electrophoretic mobility, and the permittivity increment on the counterion diffusion coefficient value were numerically determined. The use of the network simulation method made it possible to solve the governing equations for the whole range of counterion and co-ion diffusion coefficients and for very low frequencies, despite the far-reaching field-induced charge density outside the double layer. Calculations performed for different zeta potential and electrolyte concentration values show that increasing the counterion mobility, while keeping constant the electrolyte solution conductivity and the kappa a values, strongly increases the conductivity increment, barely affects the electrophoretic mobility, and strongly decreases the permittivity increment. The numerical results are discussed and compared to analytical predictions derived from the Shilov-Dukhin model, which generally leads to a good agreement, at least for high kappa a and moderate zeta.     

The interpretation of dielectric spectroscopy measurements on silica and hematite sols

Experimental data on the dielectric response of silica and hematite sols from the literature are interpreted using an analytical theory developed previously (Chassagne, C., Bedeaux, D., and Koper, G. J. M., J. Phys. Chem B105, 11,743 (2001), and Physica A, to be published). The theory is found to correctly predict both the relaxation frequency and the dielectric permittivity enhancement at low frequencies with only one free parameter. This parameter can be the zeta potential or the Stern layer conductance, in the case that the zeta potential is fixed to agree with the electrophoretic mobility measurements.     

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.     

An Improved Method for Calculating Zeta-Potentials from Measurements of the Electrokinetic Sonic Amplitude

The surface electric properties of the commercially available silica, Monospher 1000 (Fa. Merck), have been studied by conductivity and ESA (electrokinetic sonic amplitude) experiments. It could be shown that accounting for the contribution of the stagnant layer to surface conductivity is indispensable in the interpretation of electrokinetic data at low ionic strength. A general method has been put forward which allows to take into account the total, experimentally accessible surface conductivity in the evaluation of ESA data of moderately concentrated suspensions. This includes additional conductivity measurements which serve for the independent estimation of the total relative surface conductivity. The resulting zeta-potentials are clearly higher than those obtained after neglecting the contribution of the stagnant layer to surface conductivity. In addition, the ionic mobilities of potassium and magnesium in the hydrodynamically stagnant layer have been investigated in some detail. It has been found that the ionic mobility of potassium is of the same order of magnitude as in the bulk solution while the mobility of magnesium is significantly reduced. Copyright 2000 Academic Press.     

Cell model of the direct current electrokinetics in salt-free concentrated suspensions: the role of boundary conditions

In this paper, a general electrokinetic theory for concentrated suspensions in salt-free media is derived. Our model predicts the electrical conductivity and the electrophoretic mobility of spherical particles in salt-free suspensions for arbitrary conditions regarding particle charge, volume fraction, counterion properties, and overlapping of double layers of adjacent particles. For brevity, hydrolysis effects and parasitic effects from dissolved carbon dioxide, which are present to some extent in more "realistic" salt-free suspensions, will not be addressed in this paper. These issues will be analyzed in a forthcoming extension. However, previous models are revised, and different sets of boundary conditions, frequently found in the literature, are extensively analyzed. Our results confirm the so-called counterion condensation effect and clearly display its influence on electrokinetic properties such as electrical conductivity and electrophoretic mobility for different theoretical conditions. We show that the electrophoretic mobility increases as particle charge increases for a given particle volume fraction until the charge region where counterion condensation takes place is attained, for the above-mentioned sets of boundary conditions. However, it decreases as particle volume fraction increases for a given particle charge. Instead, the electrical conductivity always increases with either particle charge for fixed particle volume fraction or volume fraction for fixed particle charge, whatever the set of boundary conditions previously referred. In addition, the influence of the electric permittivity of the particles on their electrokinetic properties in salt-free media is examined for those frames of boundary conditions.     

The effect of obstacle conductivity and electric field on effective mobility and dispersion in electrophoretic transport: a volume averaging approach

The method of volume averaging has been used to determine the effective electrophoretic mobility and dispersion coefficients for molecular transport of point-like solutes in a two-phase porous medium where the electrical conductivity and the diffusion and mobility coefficients may vary in both phases. The formal theory, derived in previous work, is numerically evaluated for cases where the obstacle phase has a large or small conductivity relative to the fluid phase and where the diffusion coefficient of the solute in the obstacle phase can be large or small relative to that in the fluid phase. In agreement with previous Monte Carlo methods, the effective electrophoretic mobility is not a function of media conductivity or electric field when the obstacles are impermeable to solute transport or have small diffusion solute diffusion coefficients. However, the dispersion coefficient is a strong function of electric field and varies with obstacle conductivity when diffusive transport is small in the obstacles relative to the fluid. In contrast, the effective electrophoretic mobility is a function of electric field when conductivity of the obstacles is much larger than the fluid and when the obstacles are very permeable to solute but have low electrical conductivity.     

Analysis of the response of suspended colloidal soft particles to a constant electric field

A network model, originally designed for an electrokinetic study of soft particle suspensions, has been used for an in-depth analysis of the physical behavior of these systems under the action of an externally applied DC electric field. The versatility of the network simulation method used makes it possible to obtain information readily not only about the electrophoretic mobility, but also about any physical variable of interest at all points around the suspended particle: electric potential, ion concentrations, fluid velocity. The field-induced polarization of the double layer is described in terms of the dependence of these and other derived variables (volume charge density, electric field components, ion flux components) on the distance to the membrane-solution interface. In contrast to colloidal suspensions of hard particles, which basically depend on just two parameters (the reciprocal Debye length multiplied by the particle radius, kappaa, and the zeta potential, zeta), soft particle suspensions require a wider parameter set. First, there are two characteristic diffusion lengths in the system (one inside the membrane and the other in the solution) and two geometrical lengths (the core radius a and the membrane thickness (b-a)). Furthermore, there is the fixed charge density inside the membrane (and possibly a surface charge density over the core) that cannot be represented by a zeta potential. Finally, the parameter that characterizes the interaction between the fluid and the permeable membrane, gamma, strongly influences the behavior of the system. Dependences on all these parameters (except the geometrical ones) are included in this study.     

Streaming potentials and conductivities of latex plugs. Influence of the valency of the counterion

Plugs consisting of polystyrene particles with sulphonate surface groups have been investigated in the presence of mono-, di- and trivalent counterions. The streaming potentials and conductivities of the plug and bulk solution have been measured. From the latter, the overall surface conductivity has been estimated. It is shown that for this system, accounting for the surface conductivity in the hydrodynamically stagnant layer is crucial to convert the streaming potentials into zeta potentials. The overall surface conductivity clearly depends on the charge of the counterion, and so do the ionic mobilities in the stagnant layer. The higher the valency of the counterion, the lower is the ionic mobility in the stagnant layer, but for all counterions, the ionic mobility in the stagnant layer is of the same order of magnitude as in the bulk. For the divalent counterions, no indications of specific interactions with the surface groups have been found.     

The Dynamic Mobility of Colloidal Semiconducting Antimony-Doped Tin(IV) Oxide Particles

The dynamic mobility spectra of suspensions of semiconducting tin(IV) oxide particles doped with antimony have been measured with the technique of electroacoustics. The magnitude of the complex mobility decreases essentially monotonically with increasing frequency, just as for a nonconducting (dielectric) particle under the same conditions. Unlike the case for a dielectric particle, however, the magnitudes at low frequency increase with increasing conductivity. The phase angle behavior is also different from that of a normal dielectric particle. The change in the phase angle behavior is most obvious at low suspension conductivity and high frequency where the phase angles showed a much smaller phase lag than at high conductivities. Reasonable agreement was found between the experimental mobility and the theoretical dynamic mobility spectra obtained with O'Brien's theory for the enhanced permittivity of semiconductors. Copyright 2001 Academic Press.     

The Electrophoretic Mobility and Electric Conductivity of a Concentrated Suspension of Colloidal Spheres with Arbitrary Double-Layer Thickness

The electrophoresis in a monodisperse suspension of dielectric spheres with an arbitrary thickness of the electric double layers is analytically studied. The effects of particle interactions are taken into account by employing a unit cell model, and the overlap of the double layers of adjacent particles is allowed. The electrokinetic equations, which govern the ionic concentration distributions, the electric potential profile, and the fluid flow field in the electrolyte solution surrounding the charged sphere in a unit cell, are linearized assuming that the system is only slightly distorted from equilibrium. Using a perturbation method, these linearized equations are solved with the surface charge density (or zeta potential) of the particle as the small perturbation parameter. Analytical expressions for the electrophoretic mobility of the colloidal sphere in closed form correct to O(zeta) are obtained. Based on the solution of the electrokinetic equations in a cell, a closed-form formula for the electric conductivity of the suspension up to O(zeta(2)) is derived from the average electric current density. Comparisons of the results of the cell model with different conditions at the outer boundary of the cell are made for both the electrophoretic mobility and the electric conductivity. Copyright 2001 Academic Press.     

The electrokinetic behavior of charged non-spherical colloids

The response of charged colloids to electric fields is determined by combined phenomena occurring first in the electric double layer to then develop into long-range perturbations of ion concentration, local fields, and solvent flows. When particles are non-spherical, the loss of symmetry affects the short- and long-ranged processes modifying their behavior as observed through their electrophoretic mobility, dielectric permittivity, and electro-optical response. Recent measurements and theoretical developments have revealed phenomena characteristic for non-spherical particles, such as the doubling of the relaxations in the dielectric spectra, the appearance of torque-inducing hydrodynamic flows, and the anomalous perpendicular alignment. In this article we discuss in a unifying frame the recent experimental and theoretical progresses about the electrokinetic behavior of charged non-spherical colloids.     

Dielectric response of a concentrated colloidal suspension in a salt-free medium

In this paper the complex dielectric constant of a concentrated colloidal suspension in a salt-free medium is theoretically evaluated using a cell model approximation. To our knowledge this is the first cell model in the literature addressing the dielectric response of a salt-free concentrated suspension. For this reason, we extensively study the influence of all the parameters relevant for such a dielectric response: the particle surface charge, radius, and volume fraction, the counterion properties, and the frequency of the applied electric field (subgigahertz range). Our results display the so-called counterion condensation effect for high particle charge, previously described in the literature for the electrophoretic mobility, and also the relaxation processes occurring in a wide frequency range and their consequences on the complex electric dipole moment induced on the particles by the oscillating electric field. As we already pointed out in a recent paper regarding the dynamic electrophoretic mobility of a colloidal particle in a salt-free concentrated suspension, the competition between these relaxation processes is decisive for the dielectric response throughout the frequency range of interest. Finally, we examine the dielectric response of highly charged particles in more depth, because some singular electrokinetic behaviors of salt-free suspensions have been reported for such cases that have not been predicted for salt-containing suspensions.     

Electrophoretic mobility of a highly charged colloidal particle in a solution of general electrolytes.

For a highly charged particle in an electrolyte solution, counterions are condensed very near the particle surface. The electrochemical potential of counterions accumulated near the particle surface is thus not affected by the applied electric field, so that the condensed counterions do not contribute to the particle electrophoretic mobility. In the present paper we derive an expression for the electrophoretic mobility mu(infinity) of a highly charged spherical particle of radius a and zeta potential zeta in the limit of very high zeta in a solution of general electrolytes with large ka (where k is the Debye-Hückel parameter) on the basis of our previous theory for the case of symmetrical electrolytes (H. Ohshima, J. Colloid Interface Sci. 263 (2003) 337). It is shown that zeta can formally be expressed as the sum of two components: the co-ion component, zetaco-ion, and the counterion component, zetacounterion (where zeta = zetaco-ion + zetacounterion) and that the limiting electrophoretic mobility mu(infinity) is given by mu(infinity) = epsilonr epsilon0 zetaco-ion(infinity)/eta + 0(1/ka), where zetaco-ion(infinity) is the high zeta-limiting form of zetaco-ion, epsilonr and eta are, respectively, the relative permittivity and viscosity of the solution, and epsilon0 is the permittivity of a vacuum. That is, the particle behaves as if its zeta potential were zetaco-ion(infinity), independent of zeta. For the case of a positively charged particle in an aqueous electrolyte solution at 25 degrees C, the value of zetaco-ion(infinity) is 35.6 mV for 1-1 electrolytes, 46.0 mV for 2-1 electrolytes, and 12.2 mV for 1-2 electrolytes. It is also found that the magnitude of mu(infinity) increases as the valence of co-ions increases, whereas the magnitude of mu(infinity) decreases as the valence of counterions increases.     

非导电球型粒子悬浮液宽频介电谱——薄双电层理论的计算机模拟

以Shilov等提出的带有紧密层表面电导率的非导电球型粒子悬浮液宽频介电弛豫的薄双电层理论为基础, 从电动力学角度解释了粒子分散系两种典型介电弛豫(高频和低频弛豫)的机制. 在此基础上, 利用Mathcad程序将该理论定量程序化并建立了粒子/水相分散系介电谱参数与体系内部相参数的关系. 进而利用该程序模拟了溶液浓度、Zeta电位以及分散粒子半径等内相参数对两种弛豫的影响, 结合该理论阐述了不同环境下这两种弛豫的变化规律, 从而为今后更好地利用这两种弛豫表征纳米至毫米级球形粒子分散系的各相电及界面性质提供了有价值的参考.     

Cell model calculation for electrokinetic phenomena in concentrated suspensions: an Onsager relation between sedimentation potential and electrophoretic mobility

Cell model calculations for the electrophoretic mobility, electrical conductivity and sedimentation potential in concentrated suspensions of colloidal particles with low zeta potentials are reviewed with particular emphasis on an Onsager relation between sedimentation potential and electrophoretic mobility. A general Onsager relation is derived on the basis of the thermodynamics of irreversible processes. This relation, which involves the ratio of the electrical conductivity K* of the suspension to the conductivity Kinfinity in the absence of the particles, reproduces the Onsager relation derived from cell model calculations at low zeta potentials, where K*/Kinfinity becomes (1 - phi)/(1 + phi/2), phi being the particle volume fraction.     

Colloid Vibration Potential in a Concentrated Suspension of Spherical Colloidal Particles

A relation between the dynamic electrophoretic mobility of spherical colloidal particles in a concentrated suspension and the colloid vibration potential (CVP) generated in the suspension by a sound wave is obtained from the analogy with the corresponding Onsager relation between electrophoretic mobility and sedimentation potential in concentrated suspensions previously derived on the basis of Kuwabara's cell model. The obtained expression for CVP is applicable to the case where the particle zeta potential is low, the particle relative permittivity is very small, and the overlapping of the electrical double layers of adjacent particles is negligible. It is found that CVP shows much stronger dependence on the particle volume fraction φ than predicted from the φ dependence of the dynamic electrophoretic mobility. It is also suggested that the same relation holds between the electrokinetic sonic amplitude of a concentrated suspension of spherical colloidal particles and the dynamic electrophoretic mobility. Copyright 1999 Academic Press.     

Determination of the dynamic electrophoretic mobility of a spherical colloidal particle through a novel numerical solution of the electrokinetic equations

The standard equations developed to describe the electrophoretic motion of a charged particle immersed in an electrolyte subjected to an oscillating electric field are solved numerically with a new technique suitable for stiff systems. The focus of this work is to use this solution to determine the dynamic particle mobility, one of several quantities that can be extracted from these equations. This solution is valid from low frequencies to indefinitely high frequencies and has no restriction on zeta potential, double-layer thickness, or electrolyte composition. The solution has been used to calculate the dynamic electrophoretic mobility of a particle for a wide range of double-layer thicknesses and zeta potentials. The solution agrees with analytic approximations obtained previously by other authors under the conditions of a thin double layer and low zeta potential. The results are also consistent with calculations valid at frequencies where the ion diffusion length extends a significant distance beyond the double layer as obtained by another numerical technique.