Dynamic electrophoretic mobility of spherical colloidal particles in salt-free concentrated suspensions

In this contribution, the dynamic electrophoretic mobility of spherical colloidal particles in a salt-free concentrated suspension subjected to an oscillating electric field is studied theoretically using a cell model approach. Previous calculations focusing the analysis on cases of very low or very high particle surface charge are analyzed and extended to arbitrary conditions regarding particle surface charge, particle radius, volume fraction, counterion properties, and frequency of the applied electric field (sub-GHz range). Because no limit is imposed on the volume fractions of solids considered, the overlap of double layers of adjacent particles is accounted for. Our results display not only the so-called counterion condensation effect for high particle charge, previously described in the literature, but also its relative influence on the dynamic electrophoretic mobility throughout the whole frequency spectrum. Furthermore, we observe a competition between different relaxation processes related to the complex electric dipole moment induced on the particles by the field, as well as the influence of particle inertia at the high-frequency range. In addition, the influences of volume fraction, particle charge, particle radius, and ionic drag coefficient on the dynamic electrophoretic mobility as a function of frequency are extensively analyzed.     

Transient electrophoresis in a suspension of charged particles with arbitrary electric double layers

The startup of electrophoretic motion in a suspension of spherical colloidal particles, which may be charged with constant zeta potential or constant surface charge density, due to the sudden application of an electric field is analytically examined. The unsteady modified Stokes equation governing the fluid velocity field is solved with unit cell models. Explicit formulas for the transient electrophoretic velocity of the particle in a cell in the Laplace transforms are obtained as functions of relevant parameters. The transient electrophoretic mobility is a monotonic decreasing function of the particle-to-fluid density ratio and in general a decreasing function of the particle volume fraction, but it increases and decreases with a raise in the ratio of the particle radius to the Debye length for the particles with constant zeta potential and constant surface charge density, respectively. On the other hand, the relaxation time in the growth of the electrophoretic mobility increases substantially with an increase in the particle-to-fluid density ratio and with a decrease in the particle volume fraction but is not a sensitive function of the ratio of the particle radius to the Debye length. For specified values of the particle volume fraction and particle-to-fluid density ratio in a suspension, the relaxation times in the growth of the particle mobility in transient electrophoresis and transient sedimentation are equivalent.     

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

Approximate expressions are derived for the electrophoretic mobility of dilute cylindrical colloidal particles in a salt-free medium containing only counterions. The cylinder is assumed to be infinitely long. It is shown that as in the case of a spherical particle, there is a certain critical value of the particle surface charge separating two cases. When the particle surface charge is lower than the critical value (case 1), the electrophoretic mobility increases with increasing particle surface charge per unit length. When the particle surface charge is higher than the critical value (case 2), the mobility becomes constant (for a cylinder in a transverse field) or the increase in the electrophoretic mobility with the particle surface charge becomes suppressed (for a cylinder in a tangential field). These phenomena are caused by the effect of counterion condensation in the vicinity of the particle surface. The critical value of the particle charge is essentially independent of the particle volume fraction phi for the dilute case, unlike the case of a sphere, in which case the critical charge value is proportional to ln(1/phi).     

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.     

Electrophoresis and electric conduction in a salt-free suspension of charged particles

The electrophoresis and electric conduction of a suspension of charged spherical particles in a salt-free solution are analyzed by using a unit cell model. The linearized Poisson-Boltzmann equation (valid for the cases of relatively low surface charge density or high volume fraction of the particles) and Laplace equation are solved for the equilibrium electric potential profile and its perturbation caused by the imposed electric field, respectively, in the fluid containing the counterions only around the particle, and the ionic continuity equation and modified Stokes equations are solved for the electrochemical potential energy and fluid flow fields, respectively. Explicit analytical formulas for the electrophoretic mobility of the particles and effective electric conductivity of the suspension are obtained, and the particle interaction effects on these transport properties are significant and interesting. The scaled zeta potential, electrophoretic mobility, and effective electric conductivity increase monotonically with an increase in the scaled surface charge density of the particles and in general decrease with an increase in the particle volume fraction, keeping each other parameter unchanged. Under the Debye-Hückel approximation, the dependence of the electrophoretic mobility normalized with the surface charge density on the ratio of the particle radius to the Debye screening length and particle volume fraction in a salt-free suspension is same as that in a salt-containing suspension, but the variation of the effective electric conductivity with the particle volume fraction in a salt-free suspension is found to be quite different from that in a suspension containing added electrolyte.     

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.     

Excluded volume effect on the electrophoretic mobility of colloidal particles

In a recent work [J. Colloid Interface Sci. 316 (2007) 196] we studied the influence of the excluded volume effect on spatial distributions of ionic species and electrostatic potential in the neighborhood of a suspended spherical particle. It was shown that the excluded volume effect considerably increases the surface potential (for a given value of the particle charge) as compared to the case when ideal ion behavior is assumed. In the present work we extend our previous equilibrium results to the perturbed/nonequilibrium problem and analyze the effect of ion size constraints on the electrophoretic mobility of a rigid spherical particle immersed in a general electrolyte solution. We find that the electrophoretic mobility always increases with the excluded volume effect, which might broaden the range of experimental data that can be interpreted, including those cases where the measured mobility exceeded the theoretical maximum value predicted by the standard model.     

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.     

Re-entrant melting and freezing in a model system of charged colloids

We studied the phase behavior of charged and sterically stabilized colloids using confocal microscopy in a low polarity solvent (dielectric constant 5.4). Upon increasing the colloid volume fraction we found a transition from a fluid to a body centered cubic crystal at 0.0415+/-0.0005, followed by reentrant melting at 0.1165+/-0.0015. A second crystal of different symmetry, random hexagonal close packed, was formed at a volume fraction around 0.5, similar to that of hard spheres. We attribute the intriguing phase behavior to the particle interactions that depend strongly on volume fraction, mainly due to the changes in the colloid charge. In this low polarity system the colloids acquire charge through ion adsorption. The low ionic strength leads to fewer ions per colloid at elevated volume fractions and consequently a density-dependent colloid charge.     

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.     

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.     

Thermal control over the electrophoresis of soft colloidal particles

We study the electrophoresis of surface-charged thermosensitive microgel particles based on poly-N-isopropylacrylamide (PNIPAM); these deswell with increasing temperature T. Our results show that the electrophoretic mobility mu is affected by the temperature-induced volume phase transition. It increases with increasing temperature, as a result of the charge density increase induced by particle deswelling. Temperature thus allows control of mu, in contrast to the more conventional charged hard spheres for which mu is T independent. Salt also affects the mu behavior and gives rise to rich phenomenology, sharing common characteristics with charged hard spheres and polyelectrolyte-coated colloids depending on whether the microgels are swollen or deswollen. We interpret the effects of salt concentration n by considering that particle charges are located in an external shell, as confirmed by titrations, and that it is this shell-salt-induced compression that affects the resulting mu behavior.     

Effect of core charge density on the electrophoresis of a soft particle coated with polyelectrolyte layer

The electrophoresis of a charged soft particle with charged rigid core is considered under a weak imposed field condition. The rigid core of the soft particle is considered to have a finite dielectric permittivity and a fixed volume charge density. The electric potential distribution is determined by solving the Poisson-Boltzman equation out side the rigid core and a Poisson equation within the core along with continuity conditions on the core-shell interface. We have extended the analytic expression of Ohshima (Electrophoresis 27:526–533, 2006) for the electrophoretic mobility of a soft particle with a charged shell to include the effect of the volume charge density of the rigid core. Mobility based on the present expression matches exactly with the existing analytical solutions for a soft particle with an uncharged core. We have also made a comparison of our solution for mobility with an uncharged rigid core with the existing experimental results. The impact of the core charge density on the soft particle mobility is analyzed.     

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.     

Measuring the electrophoretic mobility of concentrated suspensions in nonaqueous media

We present a new method of measuring the electrophoretic mobility of a particle in a concentrated suspension. The method is used to measure the electrophoretic mobility of PMMA particles (diameter 10 microm) suspended in a mixture of liquid hydrocarbons. The particle volume fraction of the suspension is varied from 0 up to 0.30 and the resulting variation of the electrophoretic mobility is discussed. The suspending liquid is such that its refractive index is very close to that of the particles. Thus the suspension is almost transparent and it is possible to follow through a microscope the motion of one particle. The suspension is subjected to a low-frequency electric field (0.5 Hz). The cell containing the suspension is mounted on a piezoelectric crystal. The displacement that compensates for the particle motion (when the particle image is steady) is determined.     

Electrophoretic mobility of a particle covered with an ion-penetrable membrane

The electrophoretic behavior of a planar particle covered by an ion-penetrable membrane, which simulates a biological entity, is investigated. We show that, in general, a point charge model will overestimate the electrophoretic mobility of a particle and the deviation increases with the increase in the concentration of fixed charge and with the decrease in the thickness of membrane layer. As in the case of a point charge model, the present model also predicts a local maximum in the absolute mobility as the thickness of membrane layer varies. If the sizes of counterions of various valences are the same, then the lower the valence of counterions, the larger the mobility, and the larger the counterions, the greater the mobility. The latter is consistent with the experimental observations in the literature. For the level of the concentration of fixed charge examined, the effect of coions on the mobility is negligible.     

Protein Adsorption onto Zirconia Modified with Terminally Grafted Polyvinylpyrrolidone

Electroacoustics was used to study SDS-stabilized sunflower oil-in-water emulsions, with oil volume fractions between 2% and 50%. The dynamic mobility of the oil droplets was measured; the size and electric charge on the drops were calculated using formulas derived for dilute and concentrated systems and the results were compared. The relation derived for concentrated systems appears to be valid up to at least 50% provided the particles remain within the size range of the instrument, which shifts upward with rising concentration. Conductivity and pH had little effect on particle properties in the range studied; higher oil volume fraction (φ) had a substantial influence on the particle size produced in a homogenizer, but not on the zeta potential. Both median size and spread decreased with increases in φ. In contrast, both size and charge were hardly affected at volume fractions less than 10%. Dilution of the emulsion with a surfactant solution of the same composition as the water phase changed neither the particle size nor the zeta potential. The temperature of the emulsification process had a significant influence on the particle size but the zeta potential was hardly affected. Surfactant concentration had some effect on size at low volume fractions but not for φ>10%. The electroacoustic method hence could be applied to analyze both the dilute and the concentrated emulsions directly. Copyright 2001 Academic Press.     

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.     

Effect of pressure on electrophoretic mobility of polystyrene latex particles

A capillary electrophoresis system that can apply arbitrary helium gas pressures at both inlet and outlet reservoirs was constructed. The system was used to investigate the effect of pressure on electrophoretic behavior of polystyrene latex particles. The electrophoretic mobility of latex particles was increased with the application of pressure (< 3.0 kgf/cm2). The shrinkage of particle diameter under pressurization was observed using a microscope, however, the magnitude of shrinkage was not enough to explain the increase in electrophoretic mobility. Therefore, the application of pressure might increase the electric charge of the latex particle. Since methanol inhibited the enhancement in the electrophoretic mobility of the latex particles, water might play an important role in increasing mobility.     

Viscoelastic properties of polyacrylonitrile particles stabilised by a poly-2-vinylpyridine/polytert butylstyrene block copolymer

The viscoelastic properties of a dispersion of polyacrylonitrile particles stabilised by a block copolymer poly-2-vinylpyridine/polytert butylstyrene dispersed in solvesso have been measured as a function of particle concentration and frequency at ambient temperatures. At low volume fraction of particles it was found that the loss modulus of the dispersions was larger than the storage modulus, whilst at volume fractions > 0.40 the storage modulus dominates the rheology. This is attributable to there being a steric repulsion between the particles as a result of an increasing concentration of particles and the resultant reduction in interparticle separation in the dispersion. In addition the observed exponential increase of the storage modulus with increasing particle volume fraction mirrors the exponential increase in force with decreasing surface separation of the same type of polymers adsorbed to mica.