The Electrophoretic Mobility and Electric Conductivity of a Concentrated Suspension of Colloidal Spheres with Arbitrary Double-Layer Thickness

Electrophoretic Mobility and Electric Conductivity of Salt-Free Suspensions of Charged Soft Particles

A unit cell model is employed to analyze the electrophoresis and electric conduction in a concentrated suspension of spherical charged soft particles (each is a hard core coated with a porous polyelectrolyte layer) in a salt-free medium. The linearized Poisson–Boltzmann equation applicable to a unit cell is solved for the equilibrium electrostatic potential distribution in the liquid solution containing the counterions only surrounding a soft particle. The counterionic continuity equation and modified Stokes/Brinkman equations are solved for the ionic electrochemical potential energy and fluid velocity distributions, respectively. Closed-form formulas for the electrophoretic mobility of the soft particles and effective electric conductivity of the suspension are derived, and the effect of particle interactions on these transport characteristics is interesting and significant. Same as the case in a suspension containing added electrolytes under the Debye–Hückel approximation, the scaled electrophoretic mobility in a salt-free suspension is an increasing function of the fixed charge density of the soft particles and decreases with increases in the core-to-particle radius ratio, ratio of the particle radius to the permeation length in the porous layer, and particle volume fraction, keeping the other parameters unchanged. The normalized effective electric conductivity of the salt-free suspension also increases with an increase in the fixed charge density and with a decrease in the core-to-particle radius ratio, but is not a monotonic function of the particle volume fraction.

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.

Electroosmosis and Electric Conduction of Electrolyte Solutions in Charge-Regulating Fibrous Media

An analytical study of the electroosmosis and electric conduction of electrolyte solutions in a fibrous medium composed of parallel charge-regulating cylinders with arbitrary electric double layer thickness is presented. A linearized charge regulation model was adopted for the association and dissociation reactions occurring at the amphoteric functional groups over the surfaces of the cylinders, and a unit cell model was employed to accommodate interactions among the cylinders. The electrokinetic equations governing the ionic concentration, electric potential, and liquid flow fields were solved at low zeta potential for the cylinders. Explicit formulas for the electroosmotic mobility and effective electric conductivity in the fiber matrix were obtained. The results indicate that the charge regulation characteristics, such as the equilibrium constants of the reactions occurring at the cylinders’ surfaces and the bulk concentration of the charge-determining ions, influence the surface charge density and potential, electroosmotic mobility, and effective electric conductivity substantially.

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.

Influence of ion size effects on the electrokinetics of aqueous salt-free colloids in alternating electric fields

Electrokinetics is the science of the physical phenomena appearing at the solid-liquid interface of dispersed particles subjected to external fields. Techniques based on electrokinetic phenomena constitute an important set of tools for the electrical characterization of colloids because of their sensitivity to the properties of particle-solution interfaces. Their rigorous description may require inclusion of the effects of finite size of chemical species in the theoretical models, and, particularly in the case of salt-free (no external salt added) aqueous colloids, also consideration of water dissociation and possible carbon dioxide contamination in the aqueous solution. A new ac electrokinetic model is presented for concentrated salt-free spherical colloids for arbitrary characteristics of the particles and aqueous solution, including finite-size effects of chemical species by appropriate modifications of the chemical reaction equations to include such non-ideal aspects. The numerical solution of the electrokinetic equations in an alternating electric field has also been carried out by using a realistic non-equilibrium scenario accounting for association-dissociation processes in the chemical reactions. The results demonstrate the importance of including finite-size effects in the electrokinetic response of the colloid, mainly at high frequencies of the electric field, and for highly charged colloids. Findings of previous models for pointlike ions or for ideal salt-free colloids including finite ion size effects are recovered with the present model, for the appropriate limiting conditions.

Electrokinetic Flow of Salt-Free Solutions in a Fibrous Porous Medium

The electrokinetic flow of a salt-free solution in the fibrous porous medium constituted by an array of parallel charged circular cylinders subject to a pressure gradient and an electric field imposed in the axial direction is analytically studied via the use of a unit cell model. The Poisson-Boltzmann equation and modified Navier-Stokes equation applicable to a unit cell accommodating the salt-free solution around an individual cylinder are solved to determine the electric potential profile and fluid velocity distribution. Results of the electroosmotic velocity and effective electric conductivity in the fiber matrix are obtained as functions of the surface charge density of the dielectric cylinders and the porosity of the fiber matrix. The effects of the porosity or interactions among the cylinders on the electric potential distribution, electroosmotic velocity, and effective electric conductivity are significant and interesting under practical conditions. The apparent zeta potential, electroosmotic velocity, and effective electric conductivity increase monotonically with an increase in the surface charge density of the cylinders. When the porosity of the fiber matrix and surface charge density of the cylinders are high, the increases of the apparent zeta potential and electroosmotic velocity with the surface charge density are substantially suppressed due to the counterion condensation effect. However, this effect becomes weak when the porosity is low.

Characterization of electroosmotic flow through nanoporous self-assembled arrays

Characterization of EOF mobility for Tris and TBE buffer solutions is performed in nanoporous arrays using the fluorescent marker method to examine the magnitude of EOFs through nanopores with mean diameters close to electric double layer thickness (Debye length). Structures made from solid silica nanospheres with effective pore sizes from 104 nm down to 8 nm are produced within the microchannel using an evaporation self-assembly method. EOF results in nanoporous matrices show higher EOF mobilities for stronger electrolyte solutions, which are drastically different compared to microchannel EOF. The effects of scaling are also examined by comparing the EOF mobility for varying ratios of pore diameters to the Debye length, which shows a surprising consistency across all particle sizes examined. This work demonstrates various factors which must be considered when designing nanofluidic devices, and discusses the causes of these small scale effects.

Electrophoresis and dielectric dispersion of spherical polyelectrolyte brushes

Spherical polyelectrolyte brushes (SPBs) consist of a rigid core on which polyelectrolyte chains are grafted in such a way that in certain conditions (low ionic strength and high charge of the chains) the polymer chains extend radially toward the liquid medium. Because of the hairy-like structure of the polymer brushes, the typical soft-particle approach used for explaining the behavior of polyelectrolyte-coated particles must be modified, using the assumptions that the density of charged segments in the polymer chains decreases with the squared distance to the rigid core surface and that the same happens to the friction between the brushes and the surrounding fluid. Interest in clarifying the electrokinetics of these systems is not just academic. It has recently been found experimentally (Jiménez et al., Soft Matter 2011, 7, 3758-3762) that the response of concentrated suspensions of spherical polyelectrolyte brushes in the presence of alternating electric fields shows a number of unexpected features. Both dielectric and dynamic electrophoretic mobility spectra (respectively, dependences of the electric permittivity and the AC electrophoretic mobility on the frequency of the applied field) showed very special aspects, with giant values of the mobility and an unusually strong dielectric relaxation in the kHz region. In the present paper we give a full account of the electrodynamics of such systems, based on a cell model for describing the hydrodynamic and electrical interactions between the particles. It is found that the low-frequency dynamic mobility of SPBs is much higher than that of rigid particles of comparable size and charge, making any interpretation based on zeta potential estimations of very limited applicability. The very characteristic feature of SPBs in concentrated suspensions, namely, the enhanced alpha relaxation, can be explained by considering an adequate description of the field-induced perturbations in the counterion and co-ion concentrations, well developed both outside and inside the soft layer in the case of brush-coated particles. It can be also pointed out that the dynamic electrophoretic mobility of SPBs increases with the volume fraction of particles, as a consequence of the large thickness of the brush. Predictions are also shown for the effects of friction coefficient and charge of the polyelectrolyte layer. The results compare well with experimental spectra of the dynamic mobility and electric permittivity of moderately concentrated suspensions of SPBs consisting of a 50 nm polystyrene core with grafted poly(styrene sulfonate) chains some 140 nm in length.

Electroosmotic velocity and electric conductivity in a fibrous porous medium in the transverse direction

The steady electroosmosis and electric conduction in a fibrous medium constructed by a homogeneous array of parallel, identical, charged, circular cylinders filled with an electrolyte solution is analytically examined. The imposed electric field is constant and normal to the axes of the cylinders. The electric double layer surrounding each dielectric cylinder may have an arbitrary thickness relative to the radius of the cylinder. A unit cell model that allows for the overlap of the double layers of adjacent cylinders is employed. The electrokinetic equations that govern the ionic concentration distributions, the electrostatic potential profile, and the fluid flow field in the electrolyte solution surrounding the charged cylinder in a cylindrical cell are linearized assuming that the system is only slightly distorted from equilibrium. Through the use of a regular perturbation method, these linearized equations are solved with the surface charge density (or zeta potential) of the cylinder as the small perturbation parameter. Analytical expressions for the electroosmotic velocity of the fluid solution and the effective electric conductivity in the array of cylinders are obtained in closed forms as functions of the porosity of the fiber matrix and other characteristics of the porous system. Comparisons of the results of the cell model with different conditions at the outer boundary of the cell are made. The cell model predicts that, under otherwise identical conditions, the electric conductivity in a porous medium composed of an array of parallel cylinders in the transverse direction in general is smaller than that of a suspension of spheres, but there are some exceptions. The effect of interactions among the cylinders or spheres on the effective conductivity can be significant under appropriate conditions.

AC electrokinetics of concentrated suspensions of soft particles

In this work, we show how the cell model traditionally used for the evaluation of the electrokinetic properties of concentrated suspensions can be modified to include the case of soft particles, that is, particles consisting of a rigid core and a polyelectrolyte membrane. The Navier-Stokes and Poisson's equations have been modified to account for the presence of extra friction and a volume-distributed charge in the membrane. In addition to the boundary conditions on the particle and the cell boundary, it is necessary to define conditions on the polymer-electrolyte solution interface. The frequency dependence of the dynamic mobility and electric permittivity of suspensions of soft particles with arbitrary solids concentration is computed. It is shown that the dynamic mobility of these systems is larger than that corresponding to hard particles with the same charge. For the permittivity, the same trends are observed: the R-relaxation amplitude increases upon coating. It is found that friction plays an important role in determining the mobility, while the permittivity is more affected by the concentration of solids. The model also predicts that the charges on the core and in the membrane are very important parameters, although their effects differ on the mobility and the permittivity. While the former depends mainly on the membrane charge, the latter is responsive to both charges at comparable extents.

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.

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.

Electrokinetic Phenomena in concentrated disperse systems: general problem formulation and Spherical Cell Approach

Electrokinetic Phenomena in concentrated disperse and colloid systems have been studied employing Spherical Cell Approach for over three decades. The critical review of the advances in this area, which is conducted in the present paper, demonstrates a number of contradictions between the results reported by different authors. These contradictions are largely associated with imposition of boundary conditions at the outer boundary of the representative Spherical Cell. In order to establish a correct version of the Spherical Cell Approach, in the present paper, the theory of Electrokinetic Phenomena in concentrated suspensions is revisited by primarily focusing on the boundary conditions employed at the Spherical Cell outer boundary. To this end, a general mathematical problem is formulated for addressing the behavior of a planar layer of a macroscopically homogeneous disperse system under simultaneous influence of the pressure difference, gravitation and applied electric fields. On the basis of the general problem formulation, we present strict definitions of the kinetic coefficients which describe the system behavior. Making use of such definitions, some general relationships are rederived for the kinetic coefficients, namely, the Smoluchowski asymptotic expressions and the Onsager irreversible thermodynamic relationships. The general problem is reformulated for describing the electric, hydrodynamic and ion concentration fields inside the representative Spherical Cell. Using an original approach, a complete set of the boundary conditions is derived by employing the only assumption: the average over the disperse system volume is equal to the average over a representative Spherical Cell volume. A general method for predicting the kinetic coefficients is developed by employing the solution of the formulated problem. The developed method is combined with the method of small perturbation parameter using the normalized zeta potential. Final expressions for the kinetic coefficients are obtained while accounting for the terms proportional to zeta potential. The predictions are compared with results of other publications. On this basis, conclusions are made about the validity of different models proposed in the literature.

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.

Salt concentration and particle density dependence of electrophoretic mobilities of spherical colloids in aqueous suspension

Using laser Doppler velocimetry in the superheterodyne mode, we conducted a systematic study of the electrophoretic mobility of dispersions of small silica spheres (a=18 nm) suspended in water at different salinities and particle concentrations. The concentration of NaCl was varied from 40 microM up to 16 mM, while the particle concentrations were varied between 4.2x10(18) and 2.1x10(20) m-3. We find a decrease of mobility with increasing salt concentrations and an increase with increased particle number densities. The latter observation is not backed by the standard cell model of electrophoresis with Shilov-Zharkikh boundary conditions. Rather, if the experimental data are interpreted within that model, an unexpected change of the zeta potential at constant added salt concentration results. Interestingly, all experimental data collapse onto a single master curve, if plotted versus the ratio C* of particle counterions to added salt ions. We obtain a logarithmic increase of mobility for C*<1 and a plateau for C*>1. This may indicate a change of the Stern layer structure not yet included in the theoretical model.

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.

Diffusiophoresis in a suspension of charge-regulating colloidal spheres

An analytical study of diffusiophoresis in a homogeneous suspension of identical spherical charge-regulating particles with an arbitrary thickness of the electric double layers in a solution of a symmetrically charged electrolyte with a uniform prescribed concentration gradient is presented. The charge regulation due to association/dissociation reactions of ionogenic functional groups on the particle surface is approximated by a linearized regulation model, which specifies a linear relationship between the surface charge density and the surface potential. The effects of particle-particle electrohydrodynamic 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 that govern the electric potential profile, the ionic concentration distributions, and the fluid flow field in the electrolyte solution surrounding the particle in a unit cell are linearized assuming that the system is only slightly distorted from equilibrium. Using a regular perturbation method, these linearized equations are solved with the equilibrium surface charge density (or zeta potential) of the particle as the small perturbation parameter. Closed-form formulas for the diffusiophoretic velocity of the charge-regulating sphere correct to the second order of its surface charge density or zeta potential are derived. Our results indicate that the charge regulation effect on the diffusiophoretic mobility is quite sensitive to the boundary condition for the electric potential specified at the outer surface of the unit cell. For the limiting cases of a very dilute suspension and a very thin or very thick electric double layer, the particle velocity is independent of the charge regulation parameter.

Spherical Cell Approach for the Effective Viscosity of Suspensions

In the present paper, the spherical cell approach is employed for addressing the effective viscosity of suspensions of spherical particles. The proposed derivation is based on the only assumption which constitutes the essence of the spherical cell approach: a representative part of the suspension is a spherical cell which contains a particle surrounded by the continuous phase. In contrast with the previous studies on this topic, no additional assumptions are used in the present analysis. The general method of derivation and the final result, which represents the effective viscosity as a function of the solid-phase volume fraction, are compared with earlier studies where the spherical cell approach was applied for describing the effective viscosity.

Numerical calculation of the electrophoretic mobility of concentrated suspensions of soft particles

The electrophoretic mobility of spherical soft particles in concentrated colloidal suspensions is numerically calculated. The particle is modeled as a hard core coated with an ion-penetrable membrane bearing a uniform distribution of fixed charges, while the high particle concentration is taken into account by means of a cell model. The network simulation method used makes it possible to solve the problem without any restrictions on the values of the parameters such as particle concentration, membrane thickness, fixed charge density in the membrane, viscous drag in the membrane, number and valence of ionic species, electrolyte concentration, etc. The theoretical model used is similar to the one presented by Ohshima [H. Ohshima, J. Colloid Interface Sci. 225 (2000) 233], except for the use of the Shilov-Zharkikh, rather than the Levine-Neale, boundary condition for the electric potential, and the inclusion in the force balance equation of an additional term corresponding to the force exerted by the liquid on the core of the moving particle [J.J. López-García, C. Grosse, J. Horno, J. Colloid Interface Sci. 265 (2003) 327]. The obtained results only coincide with existing analytical expressions for low particle concentrations, low particle charge, and when the electrolyte concentration is high, the membrane is thick, and its resistance to the fluid flow is high. This suggests that most interpretations of the electrophoretic mobility of soft particles in concentrated suspensions require numerical calculations.

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.

Dynamic electrophoretic mobility of concentrated dispersions of spherical colloidal particles. On the consistent use of the cell model

This paper outlines a complete and self-consistent cell model theory of the electrokinetics of dense spherical colloidal suspensions for general electrolyte composition, frequency of applied field, zeta potential, and particle size. The standard electrokinetic equations, first introduced for any given particle configuration, are made tractable to computation by averaging over particle configurations. The focus of this paper is on the systematic development of suitable boundary conditions at the outer cell boundary obtained from global constraints on the suspension. The approach is discussed in relation to previously published boundary conditions that have often been introduced in an ad hoc manner. Results of a robust numerical calculation of high-frequency colloidal transport properties, such as dynamic mobility, using the present model are presented and compared with some existing dense suspension models.

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.

Influence of cell-model boundary conditions on the conductivity and electrophoretic mobility of concentrated suspensions

In the last few years, different theoretical models and analytical approximations have been developed addressing the problem of the electrical conductivity of a concentrated colloidal suspension. Most of them are based on the cell model concept, and coincide in using Kuwabara's hydrodynamic boundary conditions, but there are different possible approaches to the electrostatic boundary conditions. We will call them Levine-Neale's (LN, they are Neumann type, that is they specify the gradient of the electrical potential at the boundary), and Shilov-Zharkikh's (SZ, Dirichlet type). The important point in our paper is that we show by direct numerical calculation that both approaches lead to identical evaluations of the conductivity of the suspensions if each of them is associated to its corresponding evaluation of the macroscopic electric field. The same agreement between the two calculations is reached for the case of electrophoretic mobility. Interestingly, there is no way to reach such identity if two possible choices are considered for the boundary conditions imposed to the field-induced perturbations in ionic concentrations on the cell boundary (r = b), deltan(i) (r = b). It is demonstrated that the conditions deltan(i)(b) = 0 lead to consistently larger conductivities and mobilities. A qualitative explanation is offered to this fact, based on the plausibility of counter-ion diffusion fluxes favoring both the electrical conduction and the motion of the particles.

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

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

Drude-type conductivity of charged sphere colloidal crystals: Density and temperature dependence

We report on extensive measurements in the low-frequency limit of the ac conductivity of colloidal fluids and crystals formed from charged colloidal spheres suspended in de-ionized water. Temperature was varied in a range of 5 degrees C < Theta < 35 degrees C and the particle number density n between 0.2 and 25 microm(-3) for the larger, respectively, 2.75 and 210 microm(-3) for the smaller of two investigated species. At fixed Theta the conductivity increased linearly with increasing n without any significant change at the fluid-solid phase boundary. At fixed n it increased with increasing Theta and the increase was more pronounced for larger n. Lacking a rigorous electrohydrodynamic treatment for counterion-dominated systems we describe our data with a simple model relating to Drude's theory of metal conductivity. The key parameter is an effectively transported particle charge or valence Z(*). All temperature dependencies other than that of Z(*) were taken from literature. Within experimental resolution Z(*) was found to be independent of n irrespective of the suspension structure. Interestingly, Z(*) decreases with temperature in near quantitative agreement with numerical calculations.

Diffusioosmosis and Electroosmosis of Electrolyte Solutions in Fibrous Porous Media

The steady diffusioosmotic and electroosmotic flows of an electrolyte solution in the fibrous porous medium constructed by a homogeneous array of parallel charged circular cylinders are analyzed under conditions of small Peclet and Reynolds numbers. The imposed electrolyte concentration gradient or electric field is constant and can be oriented arbitrarily with respect to the axes of the cylinders. The thickness of the electric double layers surrounding the cylinders is assumed to be small relative to the radius of the cylinders and to the gap width between two neighboring cylinders, but the polarization effect of the diffuse ions in the double layers is incorporated. Through the use of a unit cell model, the appropriate equations of conservation of the electrochemical potential energies of ionic species and the fluid momentum are solved for each cell, in which a cylinder is envisaged to be surrounded by a coaxial shell of the fluid. Analytical expressions for the diffusioosmotic and electroosmotic velocities of the bulk electrolyte solution as functions of the porosity of the ordered array of cylinders are obtained in closed form for various cases. Comparisons of the results of the cell model with different conditions at the outer boundary of the cell are made. In the limit of maximum porosity, these results can be interpreted as the diffusiophoretic and electrophoretic velocities of an isolated circular cylinder caused by the imposed electrolyte concentration gradient or electric field.

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.