Electrical Conductivity of a Concentrated Suspension of Spherical Colloidal Particles

A general expression for the electrical conductivity of a concentrated suspension of spherical colloidal particles is obtained for the case where the particle zeta potential is low and the overlapping of the electrical double layers of adjacent particles is negligible by using Kuwabara's cell model. It is shown how the conductivity of a concentrated suspension depends on the particle volume fraction, the zeta potential zeta, and the reduced particle radius kappaa (kappa = Debye-Hückel parameter and a = particle radius). It is also found that the obtained conductivity formula tends to Maxwell's formula for two different extreme cases: (i) when the particles are uncharged (zeta = 0) and (ii) when the electrical double layers around the particles are infinitesimally thin (kappaa --> infinity). That is, in the latter limiting case (kappaa --> infinity), the conductivity becomes independent of the zeta potential, just as in the case of dilute suspensions. Copyright 1999 Academic Press.     

An experimental test of Booth's primary electroviscous effect theory

The primary electroviscous effect has been investigated in dilute suspensions of titanium oxide (anatase), the viscosities of which were measured by means of a capillary viscometer with automatic timing. The linear relation between viscosity and solids volume fraction was first determined at the isoelectric point of the particles when the particles are uncharged, and the electroviscous contribution to the intrinsic viscosity was then determined at other values of pH. Booth's theory (Proc. R. Soc. London Ser. A203, 533 (1950)) agrees well with the experimental results when the particle zeta potential is small and the double layer is thin (kappa alpha approximately 7.3), but agreement is poor when the double layer is thick (kappa alpha approximately 0.6).     

Sedimentation velocity and potential in a concentrated suspension of charged liquid drops

The sedimentation behavior of a concentrated suspension of charged liquid drops is analyzed theoretically at arbitrary surface potential and arbitrary double-layer thickness; that is, the effects of double-layer polarization and double-layer overlapping are taken into account. Kuwabara's unit cell model is employed to model the suspension system, and a pseudospectral method based on the Chebyshev polynomial is adopted to solve the governing electrokinetic equations numerically. Several interesting phenomena, which are of significant influence if the internal flow inside a liquid drop is taken into account, are observed. Key factors are examined such as the thickness of the electric double layer, the magnitude of the surface potential, the volume fraction of liquid drops, and the viscosity of the internal fluid. The results presented here add another dimension to the previous studies, which include concentrated suspensions of rigid particles and mercury drops under low zeta potential, with the consideration of the internal flow of liquid drops and double-layer polarization, characterized by its viscosity and the zeta potential respectively. It is found, among other things, that the smaller the viscosity of the internal fluid is, the higher the sedimentation velocity of liquid drops. The higher the zeta potential is, the larger the decrease in sedimentation velocity. In particular, the sedimentation velocity of an inviscid drop (gas bubble) is about three times higher than that of a rigid one. The decrease in sedimentation velocity resulting from the effect of double-layer polarization achieves about 50% if the zeta potential is sufficiently high.     

The electroviscous effect in ethylcellulose latex suspensions. Effect of ionic strength and correlation between theory and experiments

 This article describes an experimental and theoretical investigation of the so-called primary electroviscous effect, i.e., the increase in suspension viscosity due to the existence of an electrical double layer around the particles. By measuring the viscosity of ethylcellulose latex suspensions, the electroviscous coefficient, the quantity measuring the effect, was estimated for different concentrations of 1-1 electrolyte in the dispersion medium. These data were compared with the predictions of Watterson and White's model, using the zeta potential of the particles deduced from electrophoretic mobility measurements. It was found that the theory considerably underestimates the effect. In an attempt to improve the agreement between data and predictions, the model was generalized to include the possibility (dynamic Stern layer) that ions in the inner part of the double layer have nonzero mobility. The general theory, however, predicts even lower values of the electroviscous coefficients, thus increasing the separation between calculated and measured electroviscous coefficients. A careful analysis of the ionic concentrations and velocity profiles with and without dynamic Stern layer corrections can account for this fact, but leaves unsolved the problem of the large discrepancies found in the theoretical explanation of the strength of the electroviscous effect. Received: 19 October 1999/Accepted: 17 December 1999     

Primary electroviscous effect in a suspension of charged porous spheres

Primary electroviscous effect for a dilute suspension of porous spheres with fixed volumetric charge density is investigated theoretically. In the absence of flow, the electrical potential and solution charge density are assumed to satisfy the linearized Poisson-Boltzmann equation. With incorporation of the electrical body force, the Brinkman equation and the Stokes equation are used to govern the fluid flow inside and outside a sphere. The theory is formulated by assuming weak deviation of the charge cloud from its equilibrium state. However, the electrical body force is not restricted to be small compared to the viscous force in the fluid momentum equation. The results show that the double layer distortion is increased with increasing particle permeability, thereby enhancing the relative importance of its stress contribution. Nonetheless, the intrinsic viscosity remains a decreasing function of permeability, similar to the case of uncharged particles.     

Primary electroviscous effect in a dilute suspension of soft particles

A theory for the primary electroviscous effect in a dilute suspension of soft particles (i.e., particles coated with an ion-penetrable surface layer of polyelectrolytes) in an electrolyte solution is presented. The general expression for the effective viscosity eta s of the suspension and the primary electroviscous coefficient p, which is further expressed in terms of a function L, is given. On the basis of the general expressions, we derive approximate analytic expressions for eta s and p, which are applicable when the density of the fixed charges distributed within the surface layer is low. Further we obtain a simple approximate analytic expression (without involving numerical integrations) for p applicable for most practical cases. It is found that the function L exhibits a minimum when plotted as a function of kappa a (kappa is the Debye-Hückel parameter and a is the particle core radius), unlike the case of a suspension of hard particles, in which case L decreases as kappa a increases, exhibiting no minimum. The presence of a minimum for the case of a suspension of soft particles is due to the fact that L is proportional to 1/kappa 2 at small kappa a and to kappa 2 at large kappa a. Because of the presence of this minimum, the difference in L between soft and hard particles becomes very large for large kappa a.     

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.     

Electroviscous Effect in Dilute Suspensions of Alumina

A study on the electroviscous effect of alumina suspensions has been made. At the low volume fraction of the particles studied here only a first-order effect was detected. Ubbelohde-type capillary viscometers have been used. A simple method to determine the hydrodynamic constant k(1) has been proposed. The experimental primary electroviscous coefficients corresponding to different electrolyte concentrations have been compared with two different theoretical approachs (I. G. Watterson, and L. R. White, J. Chem. Soc. Faraday Trans. 2 77, 1115 (1981); F. J. Rubio-Hernández, E. Ruiz-Reina, and A. I. Gómez-Merino, J. Colloid Interface Sci. 206, 334 (1998)) and the results suggest that the presence of a dynamic Stern layer plays a certain role in this effect. Copyright 2000 Academic Press.     

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.     

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.     

Electrophoresis of spheres with uniform zeta potential in a gel modeled as an effective medium

The effective medium model [H.C. Brinkman, Appl. Sci. Res. A 1 (1947) 27] is used to calculate the electrophoretic mobility of spheres in a gel with uniform zeta potential on their surface. In the absence of a gel support medium or ion relaxation (the distortion of the ion atmosphere from equilibrium due to the presence of an external flow or electric field), our results reduce to those of Henry [D.C. Henry, Proc. R. Soc. London Ser. A 133 (1931) 106]. The relaxation effect can be ignored for weakly charged particles, or for particles with low absolute zeta potential. Using a procedure similar to that employed by O'Brien and White [R.W. O'Brien, L.R. White, J. Chem. Soc. Faraday Trans. 2 74 (1978) 1607], the relaxation effect is accounted for in the present work and results are presented over a wide range of particle sizes, gel concentrations, and zeta potentials in KCl salt solutions. In the limit of no gel, our results reduce to those of earlier investigations. The procedure is then applied to the mobility of Au nanoparticles in agarose gels and model results are compared to recent experiments [D. Zanchet, C.M. Micheel, W.J. Parak, D. Gerion, S.C. Williams, A.P. Alivisatos, J. Phys. Chem. B 106 (2002) 11758; T. Pons, H.T. Uyeda, I.L. Medintz, H. Mattoussi, J. Phys. Chem. B 110 (2006) 20308]. Good agreement with experiment is found for reasonable choices of the model input parameters.     

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.     

The primary electroviscous effect of prolate silica sols

The intrinsic viscosity and the dynamic mobility of four silica sols have been measured as a function of the ionic strength. It was found that intrinsic viscosity decreased with increasing ionic strength, which we attribute to the primary electroviscous effect. The geometry and the charge of the particles were fitted using experimental viscosity, light scattering, and dynamic mobility data, where the intrinsic viscosity measured at the highest ionic strength for a given sol was used as input data in our analysis. Further, the boundary element (BE) method was used to calculate the primary electroviscous effect and electrophoretic mobility of charged prolate ellipsoids. These calculations were then compared with experimental data, and the primary electroviscous effect was subtracted from the intrinsic viscosity at a given ionic strength, which led to a slightly altered geometry of the particles. This revised geometry was used as input data using the BE method, and the procedure was repeated iteratively until agreement was obtained at high ionic strength. In general, good agreement between theory and experiment was found.     

Viscous solvent colloidal system for direct visualization of suspension structure, dynamics and rheology

We introduce a model colloid system comprised of particles dispersed in a viscous solvent that can be applied to 3D direct visualization studies of suspension structure, dynamics and rheology. The colloids are poly(methyl methacrylate) (PMMA) spheres sterically stabilized by a copolymer of poly(diphenyl-dimethyl) (DPDM) siloxane that matches the refractive index of PMMA. The monodisperse particles, synthesized with mean diameter varying from 0.7 to 1.1 microm, are stably dispersed in a DPDM siloxane solvent, with viscosity varying from 2.2 to 4.3 Pa s at 20 degrees C. As opposed to other classes of PMMA colloids dispersed in organic solvents, this system displays minimal charge interactions. At room temperature, pair potential interactions (measured by extrapolation of pair correlation functions to infinite dilution) are well modeled by a generalized Lennard-Jones alpha-2alpha potential (alpha=10) with dimensionless interaction energy, epsilon/k(B)T=0.6. We use the DPDM-PMMA colloidal system in conjunction with confocal microscopy studies to measure: (i) the radial distribution function in 3D at dilute concentrations and (ii) the colloid self-diffusivity in 3D at dilute concentrations. Both measurements, neither previously reported in uncharged systems, are facilitated by the slow, viscous dynamics of the system. We also show that the viscosity and particle size of the system are such that the high-volume fraction shear thickening transition can be accessed at shear rates amenable to direct visualization.     

Cell models for the primary electroviscous effect

The primary electroviscous effect in a nondilute suspension of charged spherical particles is studied by means of cell models. The governing equations are derived, and then analytic results are obtained by restricting attention to the limit of thin double layers, small Hartmann and Peclet numbers, and small potentials. Previous work has assumed that the velocity at the outer boundary of the cell is identical to the imposed flow, as proposed by Simha (J. Appl. Phys. 1952, 23, 1020). Results with this boundary condition are compared against those predicted when the tangential shear stress on the outer boundary is assumed to be unperturbed, as proposed by Happel (J. Appl. Phys. 1957, 28, 1288). Both the hydrodynamic and electroviscous contributions to the effective viscosity are smaller with the Happel boundary condition, showing that such cell models offer a range of predictions and should be used with caution.     

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.     

Effect of small particles on the near-wall dynamics of a large particle in a highly bidisperse colloidal solution

We consider the hydrodynamic effect of small particles on the dynamics of a much larger particle moving normal to a planar wall in a highly bidisperse dilute colloidal suspension of spheres. The gap h(0) between the large particle and the wall is assumed to be comparable to the diameter 2a of the smaller particles so there is a length-scale separation between the gap width h(0) and the radius of the large particle b>h(0). We use this length-scale separation to develop a new lubrication theory which takes into account the presence of the smaller particles in the space between the larger particle and the wall. The hydrodynamic effect of the small particles on the motion of the large particle is characterized by the short time (or high frequency) resistance coefficient. We find that for small particle-wall separations h(0), the resistance coefficient tends to the asymptotic value corresponding to the large particle moving in a clear suspending fluid. For h(0)>a, the resistance coefficient approaches the lubrication value corresponding to a particle moving in a fluid with the effective viscosity given by the Einstein formula.     

Electroviscous effect of moderately concentrated colloidal suspensions under overlapping conditions

In this work we present a theoretical model for the calculation of the electroviscous coefficient of a colloidal suspension. The treatment is not limited for dilute suspensions and includes the contribution of the overlapping between adjacent ionic layers. The development here used is based on a cell model, which is applicable to Newtonian suspensions under low shear conditions and without crystalline ordering. Also presented are a complete study of the new numerical results and comparisons with previous results. We find new behaviors for the case of moderate volume fractions that do not appear in the dilute limit.     

Effect of silica colloids on the rheology of viscoelastic gels formed by the surfactant cetyl trimethylammonium tosylate

The effects of the addition of submicrometer-sized colloidal silica spheres on the linear and nonlinear rheology of semidilute solutions of a viscoelastic gel are studied. For a 1.4 wt% solution of the surfactant CTAT, a peak in the zero-shear rate viscosity eta(0) is observed at approximately equal weight percents of silica and CTAT. This peak shifts to lower silica concentrations on increasing either the CTAT concentration or the surface charge on silica and disappears when the CTAT concentration is increased to 2.6 wt%. The increases in eta(0) and the high frequency plateau modulus G(0) on the introduction of SiO(2) are explained by considering the increasingly entangled wormlike micelles that are formed due to the enhanced screening of the electrostatic interactions. The observed decrease in the values of G(0) and eta(0) at higher concentrations of silica particles is explained in terms of the formation of surfactant bilayers due to the adsorption of the positively charged cetyl trimethylammonium to the negatively charged silica.     

Adsorption behavior of oxyethylated surfactants at the air/water interface

The low-shear viscosity eta(0) of colloidal suspensions of acrylic latex or silica in aqueous gelatin has been measured at a temperature above the sol-gel transition. Measurements were made on dilution of a concentrated suspension with water or a gelatin solution. Thus, either the gelatin : colloid ratio was maintained or it was varied at constant aqueous gelatin concentration. Systems were studied with four lime-processed gelatins of different molecular weights at two concentrations of added salt. In addition, the latex particle size and the thickness of the adsorbed gelatin layer were measured by photon correlation spectroscopy (PCS) under dilute conditions. The dependence of the low-shear viscosity eta(0) on particle concentration was exponential and did not follow the well-established Krieger-Dougherty model for simple hard-sphere suspensions over the concentration range studied. A simple phenomenological model, eta(0)=eta(o)10(phi(e)/phi(s)), was found to predict the behavior well. Here, eta(o) is the viscosity of a gelatin solution of the corresponding solution concentration, phi(e) is proportional to the volume fraction of the particles, and phi(s) is a scaling factor, which was determined to have a value of 0.85. With this value of phi(s), the dimensions determined from PCS could be used to predict the viscosity values.