Numerical and analytical studies of the electrical conductivity of a concentrated colloidal suspension

In the past few years, different models and analytical approximations have been developed facing the problem of the electrical conductivity of a concentrated colloidal suspension, according to the cell-model concept. Most of them make use of the Kuwabara cell model to account for hydrodynamic particle-particle interactions, but they differ in the choice of electrostatic boundary conditions at the outer surface of the cell. Most analytical and numerical studies have been developed using two different sets of boundary conditions of the Neumann or Dirichlet type for the electrical potential, ionic concentrations or electrochemical potentials at that outer surface. In this contribution, we study and compare numerical conductivity predictions with results obtained using different analytical formulas valid for arbitrary zeta potentials and thin double layers for each of the two common sets of boundary conditions referred to above. The conductivity will be analyzed as a function of particle volume fraction, phi, zeta potential, zeta, and electrokinetic radius, kappaa (kappa(-1) is the double layer thickness, and a is the radius of the particle). A comparison with some experimental conductivity results in the literature is also given. We demonstrate in this work that the two analytical conductivity formulas, which are mainly based on Neumann- and Dirichlet-type boundary conditions for the electrochemical potential, predict values of the conductivity very close to their corresponding numerical results for the same boundary conditions, whatever the suspension or solution parameters, under the assumption of thin double layers where these approximations are valid. Furthermore, both analytical conductivity equations fulfill the Maxwell limit for uncharged nonconductive spheres, which coincides with the limit kappaa --> infinity. However, some experimental data will show that the Neumann, either numerical or analytical, approach is unable to make predictions in agreement with experiments, unlike the Dirichlet approach which correctly predicts the experimental conductivity results. In consequence, a deeper study has been performed with numerical and analytical predictions based on Dirichlet-type boundary conditions.     

Primary electroviscous effect in a dilute suspension of charged mercury drops

The standard theory of the primary electroviscous effect in a dilute suspension of charged spherical rigid particles in an electrolyte solution (Watterson, I. G.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1981, 77, 1115) is extended to cover the case of a dilute suspension of charged mercury drops of viscosity eta(d). A general expression for the effective viscosity or the electroviscous coefficient p of the suspension is derived. This expression tends to that for the case of rigid particles in the limit of eta(d) --> infinity. We also derive an approximate analytical viscosity expressions applicable to mercury drops carrying low zeta potentials at arbitrary kappaa (where kappa is the Debye-Hückel parameter and a is the drop radius) and to mercury drops as well as rigid spheres with arbitrary zeta potentials at large kappaa. It is shown that the large-kappaa expression of p for rigid particles predicts a maximum when plotted as a function of zeta potential. This result for rigid particles agrees with the exact numerical results of Watterson and White. It is also shown that in the limit of high zeta potential the effective viscosity of a suspension of mercury drops tends to that of uncharged rigid spheres given by Einstein's formula (Einstein, A. Ann. Phys. 1906, 19, 289), whereas in the opposite limit of low zeta potential the effective viscosity approaches that of a suspension of uncharged liquid drops derived by Taylor (Taylor, G. I. Proc. R. Soc. London, Ser. A 1932, 138, 41).     

Primary electroviscous effect in a moderately concentrated suspension of charged spherical colloidal particles

On the basis of the standard theory of the primary electroviscous effect in a moderately concentrated suspension of charged spherical particles in an electrolyte solution presented by Ruiz-Reina et al. (Ruiz-Reina, E.; Carrique, F.; Rubio-Hernández, F. J.; Gómez-Merino, A. I.; García-Sánchez, P. J. Phys. Chem. B 2003, 107, 9528), which is applicable for the case where overlapping of the electrical double layers of adjacent particles can be neglected, the general expression for the effective viscosity or the primary electroviscous coefficient p of the suspension is derived. This expression is applicable for a suspension of spherical particles of radius a carrying arbitrary zeta potentials zeta at the particle volume fraction phi < or = 0.3 for the case of nonoverlapping double layers, that is, at kappaalpha > 10 (where kappa is the Debye-Hückel parameter). A simple approximate analytic expression for p applicable for particles with large kappaalpha and arbitrary zeta is presented. The obtained viscosity expression is a good approximation for moderately concentrated suspensions of the particle volume fraction phi < or = 0.3, where the relative error is negligible for kappaalpha > or =100 and even at kappaalpha = 50 the maximum error is approximately 20%. It is shown that a maximum of p, which appears when plotted as a function of the particle zeta potential, is due to the relaxation effect as in the case of the electrophoresis problem.     

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.     

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.     

Electrical conductivity and stability of concentrated aqueous alumina suspensions

This work describes the effect of solids load and ionic strength on the electrical conductivity (K(S)) of concentrated aqueous suspensions of commercial alpha-alumina (1-35 vol% solids). The results obtained show that the dependency of the electrical conductivity of the suspending liquid (K(L)) on the volume fraction of solids is well described by Maxwell's model. The change in the conductivity of the suspensions relative to that of the suspending liquid (K(S)/K(L)) was found to be inversely proportional to the solids content, as predicted by Maxwell's model. The relative conductivity rate, DeltaK, could be interpreted in terms of the DLVO theory and the particles double layer parameter, kappaa, and used as a stability criterion. As kappaa changes, in response to the changes in ionic strength, so does the conducting to insulating character of the particles and, as such, their contribution to the overall suspension conductivity (expressed by DeltaK). When the particles become insulating, the suspension conductivity decreases when the solids load increases. The turning point in this particle behaviour corresponds to a critical concentration of ions in the solution that destabilises the suspension and is associated with the critical coagulation concentration (ccc). It is the electrical double layer that ultimately determines the conducting or insulating character of the particles, and that character can be made to change, as required for suspension stability, and accessed by the relative conductivity rate.     

Electrical Conductivity of a Concentrated Suspension of Soft Particles

A general expression for the electrical conductivity of a concentrated suspension of spherical soft particles (polyelectrolyte-coated particles) is obtained for the case where the overlapping of the electrical double layers of adjacent particles is negligible by using Kuwabara's cell model. It is shown that in the limit of very low potentials the obtained conductivity expression reduces to Maxwell's relation with respect to the volume fraction of the particle core and the contribution from the polyelectrolyte layer becomes negligible. An approximate conductivity expression is derived for the case of low potentials. Copyright 2000 Academic Press.     

On the influence of size, zeta potential, and state of motion of dispersed particles on the conductivity of a colloidal suspension

The dependence of the DC conductivity of diluted colloidal suspensions on the size, zeta potential, and state of motion of the dispersed particles is analyzed both theoretically and numerically. It is shown that the simple formula that represents the conductivity as a sum of products: charge times mobility, taken over all the carriers present in the suspension, is only valid for exceedingly low values of the product kappaa. In contrast, the formulation based on the value of the dipolar coefficient of the suspended particles seems to be valid for all the range of particle sizes. This assertion is only true if the dipolar coefficient is calculated taking into account the electrophoretic motion of the particles. For very low values of the product kappaa, the dipolar coefficient of particles free to move can be several orders of magnitude larger than that of immobile particles.     

Dynamic Mobility of Particles with Thick Double Layers in a Nondilute Suspension

The dynamic mobility of a nondilute suspension of spherical particles is investigated in the case where the thickness of the electrical double layer around each particle is comparable to the particle radius. A formula is obtained for the O(φ) correction in a random suspension of particles with volume fraction φ, involving an integral over the dynamic mobility of a pair of spheres. This formula is then evaluated using both analytical approximations and numerical results previously obtained for the pair mobilities and valid for low surface potentials. The effect of double-layer thickness on the O(φ) coefficient is most pronounced at low frequencies, and lessens once the hydrodynamic penetration depth is smaller than the particle radius. Various approximations are considered that use the O(φ) result to predict the dynamic mobility in concentrated suspensions, and at high frequencies these approximations are shown to give results qualitatively different from those of recent cell models. Copyright 2000 Academic Press.     

Particle interactions in electrophoresis: I. Motion of two spheres along their line of centers

The electrophoretic motion of two charged colloidal spheres with very thin electrical double layers in a constant applied electric field along their line of centers is considered. The particles may differ in radius and in zeta potential at the surface. The electrostatic and hydrodynamic governing equations are solved in the quasi-steady situation using bipolar coordinates and the electrophoretic velocities of particles are calculated for various cases. The interaction effect between particles can be very significant when the distance between particle surfaces gets close to zero. The particle with smaller zeta potential is speeded up by the motion of the other, which is retarded at the same time by the motion of the former one, if the two spheres have unequal zeta potentials of the same electrical sign. For two particles of different signs in zeta potential, motions of both are hindered by each other. The influence of the interaction between particles in general is stronger on the smaller one than on the larger one. For the special case of two electrophoretic spheres with identical zeta potentials, there is no particle interaction for all particle sizes and separations.     

On the limiting electrophoretic mobility of a highly charged colloidal particle in an electrolyte solution

The electrophoretic mobility of a spherical charged colloidal particle in an electrolyte solution with large kappaa (where kappa= Debye-Hückel parameter and a= particle radius) tends to a nonzero constant value in the limit of high zeta potential. It is demonstrated that this is caused by the fact that counterions condensed near the highly charged particle surface do not contribute to the electrophoretic mobility and only co-ions govern the mobility. A simple method to derive the limiting electrophoretic mobility expression is given. The present method is also applied to cylindrical particles, showing that the leading term of the limiting electrophoretic mobility of a cylindrical particle in a transverse field with large kappaa is the same as that of a spherical particle. The electrophoretic mobility of a cylindrical particle in a tangential field, on the other hand, is proportional to the particle zeta potential and does not exhibit a constant limiting value for high zeta potentials.     

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.     

Transient electrophoresis of spherical particles at low potential and arbitrary double-layer thickness

A theoretical study is presented for the dynamic electrophoretic response of a charged spherical particle in an unbounded electrolyte solution to a step change in the applied electric field. The electric double layer surrounding the particle may have an arbitrary thickness relative to the particle radius. The transient Stokes equations modified with the electrostatic effect which govern the fluid velocity field are linearized by assuming that the system is only slightly distorted from equilibrium. Semianalytical results for the transient electrophoretic mobility of the particle are obtained as a function of relevant parameters by using the Debye-Huckel approximation. The results demonstrate that the electrophoretic mobility of a particle with a constant relative mass density at a specified dimensionless time normalized by its steady-state quantity decreases monotonically with a decrease in the parameter kappaa, where kappa(-1) is the Debye screening length and a is the particle radius. For a given value of kappaa, a heavier particle lags behind a lighter one in the development of the electrophoretic mobility. In the limits of kappaa --> infinity and kappaa = 0, our results reduce to the corresponding analytical solutions available in the literature. The electrophoretic acceleration of the particle is a monotonic decreasing function of the time for any fixed value of kappaa. In practical applications, the effect of the relaxation time for the transient electrophoresis is negligible, regardless of the value of kappaa or the relative mass density of the particle.     

Sedimentation Velocity and Potential in Concentrated Suspensions of Charged Spheres with Arbitrary Double-Layer Thickness

The sedimentation in a homogeneous suspension of charged spherical particles 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. Overlap of the double layers of adjacent particles is allowed, and the polarization effect in the double layer surrounding each particle is considered. The electrokinetic equations that govern the ionic concentration distributions, the electric potential profile, and the fluid flow field in the electrolyte solution 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 for a symmetrically charged electrolyte with the surface charge density (or zeta potential) of the particle as the small perturbation parameter. An analytical expression for the settling velocity of the charged sphere in closed form is obtained from a balance among its gravitational, electrostatic, and hydrodynamic forces. A closed-form formula for the sedimentation potential in a suspension of identical charged spheres is also derived by using the requirement of zero net electric current. Our results demonstrate that the effects of overlapping double layers are quite significant, even for the case of thin double layers. Copyright 2000 Academic Press.     

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.     

The Effect of Protein Concentration on Electrophoretic Mobility

A theory of sedimentation in a concentrated suspension of spherical soft particles (i.e., polyelectrolyte-coated particles) is developed to obtain general expressions for sedimentation velocity of soft particles and sedimentation potential in the suspension. An Onsager relation between sedimentation potential and electrophoretic mobility of spherical soft particles in concentrated suspensions is derived for the case of low potentials and nonoverlapping electrical double layers of adjacent particles. Copyright 2000 Academic Press.     

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.     

Particle Interactions in Diffusiophoresis and Electrophoresis of Colloidal Spheres with Thin but Polarized Double Layers

The diffusiophoretic and electrophoretic motions of two colloidal spheres in the solution of a symmetrically charged electrolyte are analyzed using a method of reflections. The particles are oriented arbitrarily with respect to the electrolyte gradient or the electric field, and they are allowed to differ in radius and in zeta potential. The thickness of the electric double layers surrounding the particles is assumed to be small relative to the radius of each particle and to the gap width between the particles, but the effect of polarization of the mobile ions in the diffuse layer is taken into account. A slip velocity of fluid and normal fluxes of solute ions at the outer edge of the thin double layer are used as the boundary conditions for the fluid phase outside the double layers. The method of reflections is based on an analysis of the electrochemical potential and fluid velocity disturbances produced by a single dielectric sphere placed in an arbitrarily varying electrolyte gradient or electric field. The solution for two-sphere interactions is obtained in expansion form correct to O(r(12)(-7)), where r(12) is the distance between the particle centers. Our analytical results are found to be in good agreement with the available numerical solutions obtained using a boundary collocation method. On the basis of a model of statistical mechanics, the results of two-sphere interactions are used to analytically determine the first-order effect of the volume fraction of particles of each type on the mean diffusiophoretic and eletrophoretic velocities in a bounded suspension. For a suspension of identical spheres, the mean diffusiophoretic velocity can be decreased or increased as the volume fraction of the particles is increased, while the mean electrophoretic velocity is reduced with the increase in the particle concentration. Generally speaking, the particle interaction effects can be quite significant in typical situations. Copyright 2000 Academic Press.     

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.     

Transient electrophoresis of dielectric spheres

The dynamic electrophoretic response of a spherical dielectric particle suspended in an electrolyte solution to a step change in the applied electrics field is analytically studied. The electrical double layer surrounding the particle may have either a small but finite thickness or a very large thickness relative to the particle radius. For the case of electrophoresis of a particle with a thin double layer, the local electroosmotic velocity at the outer edge of the double layer evolving with time after the external field is imposed is used as an apparent slip boundary condition at the particle surface so that the unsteady equation of motion for the fluid flow outside the double layer is solved. Closed-form formulas for the transient electrophoretic mobility of the particle are derived as functions of relevant parameters. The results demonstrate that, when the double layer surrounding the particle is relatively thin, the normalized electrophoretic mobility at a given dimensionless time decreases monotonically with a decrease in the parameter kappaa, where kappa(-1) is the Debye screening length and a is the particle radius. When the double layer of the particle is relatively thick, the particle mobility can have magnitudes comparable to those for a particle with a thin double layer in the initial stage, but will become much smaller afterward. In general, the effect of the relaxation time for transient electrophoresis is negligible, regardless of the value of kappaa.