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.     

Approximate analytic expression for the dynamic electrophoretic mobility of a spherical colloidal particle in an oscillating electric field

An approximate analytic expression is derived for the dynamic electrophoretic mobility of a spherical charged colloidal particle in an electrolyte solution in an applied oscillating electric field. This expression, which takes into account the relaxation effects, is applicable for all values of zeta potential at large kappa a (kappa a > or = ca. 30) and omega/2pi < or = ca. 10 MHz, where kappa is the Debye-Hückel parameter, a is the particle radius, and omega is the frequency of the electric field. It is shown that the obtained mobility expression is in excellent agreement with the exact numerical results of Mangelsdorf and White (J. Chem. Soc., Faraday Trans. 1992, 88, 3567).     

Low-Zeta-Potential Analytic Solution for the Electrophoretic Mobility of a Spherical Colloidal Particle in an Oscillating Electric Field

The equations developed by C. S. Mangelsdorf and L. R. White (J. Chem. Soc. Faraday Trans. 88, 3567 (1992)) to calculate the electrophoretic mobility of a solid, spherical colloidal particle subjected to an oscillating electric field are solved analytically for low zeta potential, ζ, to obtain the electrophoretic mobility correct to (eζ/k_BT). Due to severe numerical cancellation of the exponential integrals, two forms of the analytic solution are presented which are numerically stable for different regions of κa (where a is the particle radius and κ^-1 is the Debye screening length). This low-ζ analytic solution is valid for all frequencies, particle sizes, and electrolyte concentrations, and agrees to at least two significant figures with the "exact" results obtained by Mangelsdorf and White at eζ/k_BT = 1 (ζ ≈ 25 mV). A program implementing this low-zeta analytic formula for the electrophoretic mobility is available from the authors.     

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.     

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.     

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.     

Electrophoresis of soft particles at high electrolyte concentrations: an interpretation by the Henry theory

The existence of electrophoretic mobility at high electrolyte concentrations defines a remarkable peculiarity in the electrosurface characteristics of soft particles. According to Ohshima [H. Ohshima, Colloids Surf. 103 (1995) 249], this effect is caused by the electroosmotic flow within the soft particle shell. An explanation supporting Ohshima's conclusion can be derived from classic electrokinetic theories. Based on the Henry theory [D.C. Henry, Proc. R. Soc. London Ser. A 133 (1931) 106], we demonstrate that the electrophoretic mobility of soft particles does not disappear at decinormal concentration.     

Dielectric relaxation pattern of dilute colloidal suspensions

An immediate method of analysis of the relaxation characteristics of a colloidal suspension, like of any dielectric, is based on the so-called Cole-Cole representation (imaginary part versus real part) of its complex dielectric constant in a wide frequency range. In this work, we show theoretical plots calculated according to the models developed by DeLacey and White (J Chem Soc Faraday Trans 2 77:2007–2039), and by Rosen et al. (J Chem Phys 98: 4183–4194; this model uses the dynamic Stern layer theory). Both theoretical approaches to the dielectric relaxation pattern of a colloidal suspension are compared to each other, and to experimental data obtained on polystyrene suspensions. Although no significant differences are found between the theoretical predictions of the relaxation patterns (except for the values of the dielectric constant; the DSL model yields higher polarizabilities of the suspensions), none of the models can exactly reproduce the frequency dependence of the dielectric constant of a colloidal system. We propose a modification of DeLacey and White's model to include the possibility that the ionic drag coefficients depend on the ion position in the double layer. The final results show that the general trends of the frequency dependence of the quantities involved are not modified, irregardless of the changes in ionic drag coefficients.     

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.     

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).     

Stability in Colloidal Mixtures Containing Particles with a Large Disparity in Size

An experimental approach, based on turbidity measurements, is proposed for studies of the stability in colloidal mixtures containing particles with large disparity in size. The main advantage of this approach is that it permits investigations even under conditions of comparable particle number concentrations of the two colloidal populations. Binary mixtures containing a poly(vinyl acetate) (PVAc) latex and a Ludox AS-40 silica sol were investigated. The silica particles were much smaller than the latex ones. The experimental stability factors were compared with the theoretical values computed on the basis of the Kihira-Ryde-Matijevic model (J. Chem. Soc., Faraday Trans. 88(16), 2379 (1992)) for interaction between spherical particles with unevenly distributed surface charges. All the experimental results support the idea that, even when both sols are negatively charged, the small silica particles are adsorbed onto the latex surface. Under these conditions, the heteroaggregates, which are composed of PVAc cores surrounded with silica particles, can be modeled as PVAc particles having "modified" surface characteristics (i.e., average Stern potential and varying extents of the surface charge segregation). Copyright 2001 Academic Press.     

Electroviscous effect of moderately concentrated colloidal suspensions: Stern-layer influence

A previous model for the viscosity of moderately concentrated suspensions has been extended. The influence of a dynamic Stern layer (DSL), which produces an additional surface conductance at the electrolyte-particle interface, is included. The theoretical treatment is based on Happel's cell model with Simha's boundary conditions for the interparticle hydrodynamic interactions and on a dynamic Stern-layer model for ionic conduction on the particle surface according to Mangelsdorf and White (ref 39). The results are valid for arbitrary zeta potentials and double-layer thickness. Extensive theoretical predictions are shown and interesting new behaviors are found. The comparison with the results in the absence of additional surface conductance shows a great influence of this mechanism in the energy dissipation during the laminar flow of these suspensions. We conclude that the inclusion of a dynamic Stern layer will be required to match the predictions with the experimental results.     

Effects of structural properties of the Stern layer on the electrophoretic migration of a highly charged spherical macroion

The electrophoretic migration of a highly charged spherical macroion suspended in an aqueous solution of NaCl is studied using the molecular dynamic method. The objective is to examine the effects of the colloidal surface charge density on the electrophoretic mobility (μ) of the spherical macroion. The bare charge and the size of the macroion are varied separately to induce changes in the colloidal surface charge density. Our results indicate that μ depends on colloidal surface charge density in a nonmonotonic manner, but that this relationship is independent of the way the surface charge density is varied. It is found that an increase in colloidal surface charge density may lead to the formation of new sublayers in the Stern layer. The μ profile is also found to have a local maximum for a bare charge at which a new sublayer is formed in the Stern layer, and a local minimum for a bare charge at which the outer sublayer becomes relatively dense. Finally, the electrophoretic flow caused by the migration of the spherical macroion is studied to find that one decisive factor causing the electrophoretic flow is the ability of the macroion to carry anions in the electrolyte solution.     

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.     

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.     

Electrokinetics of soft particles

Theories of electrokinetics of soft particles, which are particles covered with an ion-penetrable surface layer of polyelectrolytes, are reviewed. Approximate analytic expressions are given, which describe various electrokinetics of soft particles both in dilute and concentrated suspensions, that is, electrophoretic mobility, electrical conductivity, sedimentation velocity and potential, dynamic electrophoretic mobility, colloid vibration potential, and electrophoretic mobility under salt-free condition.     

Numerical calculation of the electrophoretic mobility of a spherical particle in a salt-free medium

By extending an approximate theory of the electrophoretic mobility of dilute spherical colloidal particles in a salt-free medium containing only counterions (H. Ohshima, J. Colloid Interface Sci. 248 (2002) 499--503), a systematic numerical method is given for the calculation of the electrophoretic mobility, which is based on an iteration method. We assume that each sphere is surrounded by a spherical free volume, within which counterions are distributed so that electro-neutrality is satisfied. The electrophoretic mobility is found to be determined mainly by the pressure due to the counterions at the outer surface of the free volume. It is shown how the mobility values deviate from those expected from Hückel's formula for high particle charges or zeta potentials because of the counterion condensation effect.     

Heteroflocculation of binary latex dispersions of similar chemistry but varying size

The stability of dilute bimodal (diameter:100 and 200 nm) model latex dispersions is studied as a function of electrolyte concentration and particle number fraction by measuring perikinetic aggregation with dynamic light scattering. A formally correct expression for the effective, doublet stability ratio of a bimodal system is derived that accounts for the difference in the particle size and hence, extends the derivation by Hogg and co-workers [Trans. Faraday Soc. 62 (1966) 1638]. Including the particle size ratio predicts slightly lower stability ratios for polydisperse but chemically similar systems. Stability ratios for binary mixtures of model colloidal latices are extracted from aggregation measurements in the fractal aggregation regime and are compared to predictions based on DLVO calculations of the potential. The results suggest that the composition of the aggregates is dependent on the relative stability of the two components (and consequently, on electrolyte concentration) and is richer in the least stable component.     

Electrophoresis of soft particles: analytic approximations

An approximate analytic expression is derived for the electrophoretic mobility of a weakly charged spherical soft particle (i.e., a hard particle covered with a weakly charged polyelectrolyte layer) on the basis of the general mobility expression for soft particles (Ohshima, H., J. Colloid Interface Sci. 2000, 228, 190-193). The obtained mobility expression, which reproduces various approximate results so far derived and gives some new mobility formulas, covers all types of weakly charged soft particles with arbitrary values of the thickness of polymer layer, the radius of the particle core, the electrophoretic softness, and the Debye length, including spherical polyelectrolytes with no particle core as well as spherical hard particles with no polyelectrolyte layer.     

Electrophoretic mobility of colloidal particles coated with a layer of adsorbed polymers

A general expression is given for the electrophoretic mobility of a large charged colloidal particle coated with a layer of adsorbed charged polymers. A liquid flow within the polymer layer is taken into account. The potential distribution is calculated on the basis of the non-linear Poisson Boltzmann equation. Simple approximate analytic expressions for the electrophoretic mobility are derived for various cases.