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.     

On the General Expression for the Electrophoretic Mobility of a Soft Particle

Electrokinetic equations for electrophoresis of a soft particle (that is, a hard particle covered with a layer of polyelectrolytes) have been solved previously under the conditions that the net force acting on the soft particle as a whole (the particle core plus the polyelectrolyte layer) must be zero and that the electrical force acting on the polymer segment is balanced with a frictional force exerted by the liquid flow (J. Colloid Interface Sci. 163, 474 (1994)). In the present work we replaced the latter condition by the alternative and more appropriate condition that pressure is continuous at the boundary between the surface layer and the surrounding electrolyte solution to solve the electrokinetic equations and obtained the general mobility expression for the electrophoretic mobility of a spherical soft particle. It is found that the general mobility expression thus obtained reproduces all of the approximate mobility expressions derived previously and, in addition, that the continuous pressure condition leads to the correct limiting behavior of the electrophoretic mobility in the case where the frictional coefficient tends to zero (this behavior cannot be derived from the force balance condition for the polyelectrolyte layer). Copyright 2000 Academic Press.     

Ion size effects on the electrokinetics of salt-free concentrated suspensions in ac fields

We analyze the influence of finite ion size effects in the response of a salt-free concentrated suspension of spherical particles to an oscillating electric field. Salt-free suspensions are just composed of charged colloidal particles and the added counterions released by the particles to the solution that counterbalance their surface charge. In the frequency domain, we study the dynamic electrophoretic mobility of the particles and the dielectric response of the suspension. We find that the Maxwell-Wagner-O’Konski process associated with the counterions condensation layer is enhanced for moderate to high particle charges, yielding an increment of the mobility for such frequencies. We also find that the increment of the mobility grows with ion size and particle charge. All these facts show the importance of including ion size effects in any extension attempting to improve standard electrokinetic models.     

Influence of membrane layer properties on the electrophoretic behavior of a soft particle

The influence of the physical properties of the membrane layer of a soft particle, which comprises a rigid core and a porous membrane layer, on its electrophoretic behavior, is investigated. Because that influence was almost always neglected in the previous studies, the corresponding results can be unrealistic. The applicability of the model proposed is verified by the available theoretical and experimental results. The electrophoretic mobility of the particle under various conditions is simulated through varying the dielectric constant, the thickness, and the drag coefficient of the membrane layer, and the bulk ionic concentration. We show that under typical conditions, the deviation in the electrophoretic mobility arising from assuming that the dielectric constant of the membrane layer is the same as that of the bulk liquid phase can be in the order of 50%. In addition, the thicker the membrane layer and/or the higher the bulk ionic concentration, the larger the deviation. If the surface of the core of the particle is charged, as in the case of inorganic particles covered by an artificial membrane layer, the deviation at constant core surface potential is larger than that under other types of charged conditions. However, if the surface of the core is uncharged, as in the case of biocolloids, then that deviation becomes negligible. These findings are of fundamental significance to theoreticians in their analysis on the electrokinetic behaviors of soft particles, and to experimentalists in the interpretation of their data.     

Startup of electrophoresis in a suspension of colloidal spheres

The transient electrophoretic response of a homogeneous suspension of spherical particles to the step application of an electric field is analyzed. The electric double layer encompassing each particle is assumed to be thin but finite, and the effect of dynamic electroosmosis within it is incorporated. The momentum equation for the fluid outside the double layers is solved through the use of a unit cell model. Closed‐form formulas for the time‐evolving electrophoretic and settling velocities of the particles in the Laplace transform are obtained in terms of the electrokinetic radius, relative mass density, and volume fraction of the particles. The time scale for the development of electrophoresis and sedimentation is significantly smaller for a suspension with a higher particle volume fraction or a smaller particle‐to‐fluid density ratio, and the electrophoretic mobility at any instant increases with an increase in the electrokinetic particle radius. The transient electrophoretic mobility is a decreasing function of the particle volume fraction if the particle‐to‐fluid density ratio is relatively small, but it may increase with an increase in the particle volume fraction if this density ratio is relatively large. The particle interaction effect in a suspension on the transient electrophoresis is much weaker than that on the transient sedimentation of the particles.     

Counterion condensation in hydrosols of single-crystalline detonation nanodiamond particles obtained by air annealing of their agglomerates

The pattern of previously recorded dependences of the specific surface charge and electrophoretic mobility of monodisperse detonation nanodiamond particles on pH of aqueous KCl solutions suggests that counterions are condensed on the particle surface. Counterion condensation is considered in terms of the Levin model, and the experimental ratios between the densities of the electrokinetic and surface charges of dispersed particles, as well as the fractions of condensed counterions, are calculated as depending on pH and KCl concentration in nanodiamond hydrosols. The obtained dependences lead to the conclusion that counterion condensation on the surface of detonation nanodiamond particles does indeed take place.     

Anomalous low-frequency electro-optic behavior of ferric oxide particles in the presence of poly(ethylene oxide)

Electro-optic techniques were used to investigate the influence of poly(ethylene oxide) (PEO) on the surface electric state of positively charged oxide particles. The variations in particle electrophoretic mobility of beta-FeOOH particles in the presence of PEO indicate significant changes in the surface electric state of the particles in the concentration interval of PEO 10(-2)-10(-1) g dm(-3). The electro-optic results for the same conditions were unexpected: no significant difference is observed in the value and the relaxation frequency of particle electric polarizability in the frequency domain of the alpha-relaxation (detected in the kilohertz range); particle rotational relaxation time also remains unchanged; considerable changes are detected only in the relaxation interval of particle rotation (detected in the hertz range). The obtained results reject the possibility of the formation on the particle surface of a thick polymer layer. A thin adsorption layer cannot explain the significant decrease in particle electrophoretic mobility. The variations in electrophoretic mobility are well correlated with the effects in the domain of particle rotation. A possible explanation of the observed effects is proposed, based on our previous investigations of the effects in the low-frequency domain. The presented results demonstrate that the important information on the electrokinetic charge distribution is not found in the domain of the alpha-dispersion, but in the domain of particle rotation.     

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.     

The electrokinetic properties of latex particles: comparison of electrophoresis and dielectrophoresis

A comprehensive study of the AC and DC electrokinetic properties of submicrometre latex particles as a function of particle size and suspending medium conductivity and viscosity is presented. Electrophoretic mobility and dielectrophoretic cross-over results were measured for particle diameters ranging from 44 to 2000 nm. The zeta potentials of the particles were calculated from the electrophoretic mobility data for different suspending medium conductivities, using various models, with and without the inclusion of surface conduction. The dielectrophoretic data was analysed to derive values for the Stern layer conductance and zeta potentials.     

RHEOLOCY OF AQUEOUS DISPERSIONS OF ALUMINA,TITANIA AND MIXTURE OF ALUMINA AND TITANIA POWDERS

The electrophoretic mobility and rheology of alumina, titania and alumina / titania powder mixtures ( ratio of alumina : titania 1 : 2 and 2 : 1 ) were investigated in aqueous suspensions in the pH range 3 - 10. The results suggest that the electrophoretic mobility of the individual powders is independent of the particle concentration, but the electrokinetic behavior of the powder mixture depends upon the concentration of the individual constituents. The rheology of all the suspensions is charge related and while alumina and alumina / titania ( ratio 2 : 1 ) flocculate at pH 9.0 both titania and alumina / titania ( ratio 1 : 2 ) form low viscous dispersions. Similarly while later dispersions flocculate at pH 5.6, the former dispersions behave as Newtonian type fluids. Better correlation between the slip rheology and the electrokinetic behavior of suspensions containing both alumina and titania powders can be obtained provided the experimental results are evaluated based upon the total surface area of the individual powders but not the particle concentration.     

Mobility of protein through a porous membrane

The electrophoretic mobility of proteins in membrane pores has been investigated experimentally. When the size of the protein is small relative to the pore size, the protein mobility is identical to the free protein mobility. As the pore radius approaches the protein radius the mobility of the protein is significantly reduced. This phenomenon has been explained in terms of electrokinetic theory. Using a method of reflections, and taking into account the effect of the back-flow, an approximation has been developed for the average mobility in a closed system of a spherical particle moving under electrophoresis parallel to the axis of a cylindrical pore. This approximation assumes that the surface potential of the particle is low, and is valid for arbitrary double layer thickness relative to particle size, provided that there is minimal overlap between the double layers at the pore surface and around the particle. It is also predicted that when the protein and the membrane have surface potentials of the same sign, there can be a significant increase in protein mobility for medium-sized pores.     

Dynamic electrophoretic mobility of spherical colloidal particles in a salt-free medium

A theory is proposed for the dynamic electrophoretic mobility mu(omega) of spherical colloidal particles in a salt-free medium containing only counterions in an oscillating electric field of frequency omega. The dynamic mobility depends on the frequency omega of the applied electric field and on the particle volume fraction as well as on the particle surface charge. It is found that as in the case of the static electrophoretic mobility mu(0) in salt-free media, there is a certain critical value of the particle surface charge separating two cases, that is, the low-surface-charge case and the high-surface-charge case (in the latter case the counterion condensation takes place near the particle surface). For the low-surface-charge case, the dynamic mobility agrees with that of a sphere in an electrolyte solution in the limit of very low electrolyte concentrations kappaa-->0 (Hückel's limit), where kappa is the Debye-Hückel parameter and a is the particle radius. For the high-surface-charge case, however, the dynamic mobility becomes constant independent of the particle surface charge, because of the counterion condensation effects. A simple expression for the ratio mu(omega)/mu(0) applicable for all cases is given.     

A new generalization of the standard electrokinetic model

We present a new generalization of the standard electrokinetic model based on the assumption that there is a thin layer surrounding the suspended particle where the equilibrium ion density is not determined by the Gouy-Chapman distribution, while the standard model applies outside this layer. Our approach differs from existing models in that we consider that the surface layer is made both of free ions (mostly counterions) and of the fixed ions that constitute the charge of the particle. Furthermore, the free ion density is determined by appropriate boundary conditions without considering any adsorption isotherms. Finally, the fluid is allowed to freely flow inside the layer, only hindered by the presence of the fixed charges and the adhesion condition on the surface of the particle. We show that this generalization leads to results that qualitatively differ from those obtained using existing models: instead of always decreasing, the electrophoretic mobility can actually increase with the anomalous surface conductivity. This could make it possible to use our model for the interpretation of a broader set of experimental data, including those cases when the measured mobility is higher than predicted by the standard model.     

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 Effect of Mechanical Treatment on Colloidal and Chemical Properties and Reactivity of Powdered Cellulose

The effect of mechanical activation on colloidal and chemical properties of aqueous dispersions of powdered cellulose (particle size, electrokinetic parameters, surface electrical properties, adsorption, etc.) and its solubility in a methylmorpholine N-oxide monohydrate–dimethyl sulfoxide mixture is studied. A rise in the number of mechanical treatment cycles increases the surface charge density, adsorbability, and solubility of cellulose, but decreases particle size and electrophoretic mobility, as well as the viscosity of cellulose solutions due to a reduction in the degree of cellulose polymerization, development of microporosity, and a rise in the internal surface area of polymer particles as a result of their disintegration.     

Cationic and anionic poly(N-isopropylacrylamide) based submicron gel particles: electrokinetic properties and colloidal stability

A cationic and an anionic poly(N-isopropylacrylamide) (poly(NIPAM)) microgel latex were synthesized via batch radical polymerization under emulsifier-free conditions. The hydrodynamic properties, colloidal stability, and electrokinetic characteristics of these two samples were studied. The hydrodynamic particle size variation was discussed by considering the effect of salinity and temperature on the shrinkage of the thermally sensitive polymer domains. The colloidal stability also depended on temperature and electrolyte concentration. A stability diagram with two well-defined domains (stable and unstable) was obtained. The flow from one domain to the other was fully reversible due to the peculiar (de)hydration properties of the polymer. The electrokinetic behavior, which depends on electrical and frictional properties of the particles, was analyzed via electrophoretic mobility measurements. Results were discussed by considering both the particle structure dependence on temperature and salinity, and the electric double layer compression. In addition, the electrophoretic mobility data were analyzed using Ohshima's equations for particles covered by an ion-penetrable surface charged layer, as well as using another simpler equation for charges located on a hydrodynamic equivalent hard sphere. Differences between the properties of both latexes were justified by the presence of a hydrophilic comonomer, aminoethyl methacrylate hydrochloride (AEMH), in the cationic microgel.     

Translational motion of a spherical particle near a planar liquid-fluid interface

The translational electrophoretic motion of a colloidal spherical particle parallel to a planar liquid-fluid interface is analyzed by using the reciprocal theorem developed by Yariv and Brenner [E. Yariv, H. Brenner, J. Fluid Mech. 484 (2003) 85]. Based on the thin electric double layers assumption, analytical solutions of the forces acting on the particle are obtained, and the influence of the liquid-fluid interface on the electrophoretic velocity of the particle is studied. It is found that the speed of the particle's electrokinetic motion will increase as the separation distance between the particle and the interface decreases. This enhancement of electrophoretic mobility becomes more significant when the viscosity of the fluid phase becomes larger.     

Electrophoretic mobility of a colloidal particle with constant surface charge density

When the electrophoretic mobility of a particle in an electrolyte solution is measured, the obtained electrophoretic mobility values are usually converted to the particle zeta potential with the help of a proper relationship between the electrophoretic mobility and the zeta potential. For a particle with constant surface charge density, however, the surface charge density should be a more characteristic quantity than the zeta potential because for such particles the zeta potential is not a constant quantity but depends on the electrolyte concentration. In this article, a systematic method that does not require numerical computer calculation is proposed to determine the surface charge density of a spherical colloidal particle on the basis of the particle electrophoretic mobility data. This method is based on two analytical equations, that is, the relationship between the electrophoretic mobility and zeta potential of the particle and the relationship between the zeta potential and surface charge density of the particle. The measured mobility values are analyzed with these two equations. As an example, the present method is applied to electrophoretic mobility data on gold nanoparticles (Agnihotri, S. M.; Ohshima, H.; Terada, H.; Tomoda, K.; Makino, K. Langmuir 2009, 25, 4804).     

Electrophoretic mobility of a spherical colloidal particle in a salt-free medium

A general expression as well as approximate expressions are derived for the electrophoretic mobility of dilute spherical colloidal particles in a salt-free medium containing only counter ions. It is shown that there is a certain critical value of the particle surface charge. When the particle surface charge is lower than the critical value, the electrophoretic mobility is proportional to the particle surface charge or the particle zeta potential, following Hückel's formula. When the particle surface charge is higher than the critical value, the electrophoretic mobility becomes independent of the particle surface charge. This is due to the effect of counter ion condensation in the vicinity of the particle surface.