Frequency-dependent electrical conductivity of concentrated dispersions of spherical colloidal particles

This paper outlines the application of a self-consistent cell-model theory of electrokinetics to the problem of determining the electrical conductivity of a dense suspension of spherical colloidal particles. Numerical solutions of the standard electrokinetic equations, subject to self-consistent boundary conditions, are implemented in formulas for the electrical conductivity appropriate to the particle-averaged cell model of the suspension. Results of calculations as a function of frequency, zeta potential, volume fraction, and electrolyte composition, are presented and discussed.     

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.     

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.     

How to impose stick boundary conditions in coarse-grained hydrodynamics of Brownian colloids and semi-flexible fiber rheology

Long-range hydrodynamics between colloidal particles or fibers is modelled by the fluid particle model. Two methods are considered to impose the fluid boundary conditions at colloidal surfaces. In the first method radial and transverse friction forces between particle and solvent are applied such that the correct friction and torque follows for moving or rotating particles. The force coefficients are calculated analytically and checked by numerical simulation. In the second method a collision rule is used between colloidal particle and solvent particle that imposes the stick boundary conditions exactly. The collision rule comprises a generalisation of the Lowe-Anderson thermostat to radial and transverse velocity differences.     

Critical concentration for colloidal crystallization determined with microliter centrifuged suspensions

We report an elegant method using centrifugal sedimentation for determining the critical particle concentration for colloidal crystallization. A small amount of a dilute suspension of monodispersed particles stored in a flat capillary cell was centrifuged to temporarily generate a nonequilibrium gradient of the particle concentration including a crystalline-noncrystalline phase boundary in the cell. In the recovering process after the centrifugation, the particle concentration of the crystalline phase at the boundary was found to always have the equilibrium value, although the global concentration distribution evolved with time. The critical concentration was determined based on spatially resolved spectrometry. The present method requires only one batch of a suspension of the order of microliters and is applicable up to high concentration regions near the closest packing without the effect of the particle aggregation.     

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.     

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.     

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.     

Effect of Charge,Size and Temperature on Stability of Charged Colloidal Nano Particles

Molecular simulation of charged colloidal suspension is performed in NVT canonical ensemble using Monte Carlo method and primitive model. The well-known Derjaguin-Landau-Verwey-Overbeek theory is applied to account for effective interactions between particles. Effect of temperature, valance of micro-ions and the size of colloidal particles on the phase stability of the solution is investigated. The results indicate that the suspension is more stable at higher temperatures. On the other hand, for a more stable suspension to exist, lower micro-ion valance is favorable. For micro-ions of higher charge the number of aggregates and the number of particle in each of aggregate on average is higher. However for the best of our results larger colloidal particle are less stable. Comparing the results with theoretical formula considering the influence of surface curvature shows qualitative consistency.     

Anisotropic diffusion in confined colloidal dispersions: the evanescent diffusivity

We employ an analogy to traditional dynamic light scattering to describe the inhomogeneous and anisotropic diffusion of colloid particles near a solid boundary measured via evanescent wave dynamic light scattering. Following this approach, we generate new expressions for the short-time self- and collective diffusivities of colloidal dispersions with arbitrary volume fraction. We use these expressions in combination with accelerated Stokesian dynamics simulations to calculate the diffusivities in the limit of large and small scattering wave numbers for evanescent penetration depths ranging from four particle radii to one-fifth of a particle radius and volume fractions from 10% to 40%. We show that at high volume fractions, and larger penetration depths, the boundaries have little effect on the dynamics of the suspension parallel to the wall since, to a first approximation, the boundary acts hydrodynamically much as another nearby particle. However, near and normal to the wall, the diffusivity shows a strong dependence on penetration depth for all volume fractions. This is due to the lubrication interactions between the particles and the boundary as the particle moves relative to the wall. These results are novel and comprehensive with respect to the range of penetration depth and volume fraction and provide a complete determination of the effect of hydrodynamic interactions on colloidal diffusion adjacent to a rigid boundary.     

Reversible coagulation of colloidal suspension in shallow potential wells: Direct numerical simulation

Brownian dynamics computer simulations of aggregation in 2D colloidal suspensions are discussed. The simulations are based on the Langevin equations, pairwise interaction between colloidal particles and take into account Brownian, hydrodynamic and colloidal forces. The chosen mathematical model enables to predict the correct values of diffusion coefficient of freely moving particle, the mean value of kinetic energy for each particle in ensemble of interacting colloidal particles and residence times of colloidal particles inside the potential wells of different depths. The simulations allow monitoring formation and breakage of clusters in a suspension as well as time dependence of the mean cluster size. The article is published in the original.     

Particle Interactions in Diffusiophoresis and Electrophoresis of Colloidal Spheres with Thin but Polarized Double Layers

The diffusiophoretic and electrophoretic motions of two colloidal spheres in the solution of a symmetrically charged electrolyte are analyzed using a method of reflections. The particles are oriented arbitrarily with respect to the electrolyte gradient or the electric field, and they are allowed to differ in radius and in zeta potential. The thickness of the electric double layers surrounding the particles is assumed to be small relative to the radius of each particle and to the gap width between the particles, but the effect of polarization of the mobile ions in the diffuse layer is taken into account. A slip velocity of fluid and normal fluxes of solute ions at the outer edge of the thin double layer are used as the boundary conditions for the fluid phase outside the double layers. The method of reflections is based on an analysis of the electrochemical potential and fluid velocity disturbances produced by a single dielectric sphere placed in an arbitrarily varying electrolyte gradient or electric field. The solution for two-sphere interactions is obtained in expansion form correct to O(r(12)(-7)), where r(12) is the distance between the particle centers. Our analytical results are found to be in good agreement with the available numerical solutions obtained using a boundary collocation method. On the basis of a model of statistical mechanics, the results of two-sphere interactions are used to analytically determine the first-order effect of the volume fraction of particles of each type on the mean diffusiophoretic and eletrophoretic velocities in a bounded suspension. For a suspension of identical spheres, the mean diffusiophoretic velocity can be decreased or increased as the volume fraction of the particles is increased, while the mean electrophoretic velocity is reduced with the increase in the particle concentration. Generally speaking, the particle interaction effects can be quite significant in typical situations. Copyright 2000 Academic Press.     

The migration of styrene butadiene latex during the drying of coating suspensions: when and how does migration of colloidal particles occur?

Surface elemental compositions of model latex clay coatings on an impervious substrate consolidated under various conditions were measured using the XPS technique, in order to clarify when and how colloidal latex particles migrate to the surface during drying. Under similar drying conditions, surface carbon content decreased with the addition of a water-soluble polymer to the coating colors, while remaining virtually unchanged for coatings of different coat weights made with a given color, indicating that surface carbon content variation is mainly caused by migration of latex rather than of water-soluble polymer. The results also showed that for coatings made with a given suspension, surface carbon content decreased with increasing delay time between coating and heating. For coatings frozen during consolidation and dried by sublimation, surface carbon content increased with increasing drying time before freezing. These results suggest that for the model coatings studied, latex migration mainly occurs after coating application before capillary formation during the initial drying stage when coatings are in the liquid phase, contradicting both the conventional capillary transport and boundary wall migration mechanisms. An alternative mechanism which attributes latex migration to surface trapping effect and to higher Brownian mobility of the smaller latex particles compared with pigment appears to provide a systematically consistent explanation to those phenomena. The new particle migration mechanism implies that segregation of colloidal particles is a ubiquitous phenomenon that would occur not only during the drying of paper coatings but also during consolidation of colloidal films containing particles of different sizes. This is of great importance in the control of surface compositions of nanocomposite coatings.     

Effective medium approximation and deposition of colloidal particles in fibrous and granular media

Laminar flow of fluids through fibrous and granular media and deposition of colloidal particles from a liquid suspension are two fundamental phenomena encountered in many industrial applications. An Effective Medium Approximation (EMA) is used to determine the fluid flow permeability and particle capture efficiency of random arrays of cylindrical and spherical collectors. The EMA assumes a model system in which a packing element (a single fiber in the fibrous medium and a single sphere in the granular medium) is surrounded by a fluid envelope and an effective-medium beyond the envelope. It integrates the important features of both the cell models and Brinkman's model. The Stokes equation and Brinkman equation are solved for the fluid envelope and effective medium regions, respectively, to obtain the permeability and close-to-surface velocity field around the collectors. The convective diffusion equation is then solved to determine the particle deposition rate. The analytical expressions for the permeability and particle deposition rate are derived for all possible cases of random packing of uniform and non-uniform cylinders and spheres. Effects of various system properties and operating conditions on deposition of colloidal particles are investigated. The physical or chemical conditions include the properties which affect the magnitude of double layer interaction: the electrolyte concentration and surface potentials, and the property which affects the van der Waals interaction: the Hamaker constant. It was found that the effects of the above properties is much more significant when the surface interactions play more important roles in the particle deposition process, or when the height of the total interaction energy barrier is higher than 5 kBT. Particle deposition becomes virtually impossible when the height of the repulsive energy barrier increases beyond 20 kBT.     

Equilibrium characteristic at ordered-disordered phase boundary in centrifuged nonequilibrium colloidal-crystal system

We show that a crystalline-noncrystalline boundary that temporarily appeared in a sediment centrifuged from a relatively dilute colloidal suspension can be regarded to be at a phase equilibrium. On the basis of this, we can determine the critical particle concentration for colloidal crystallization using an extremely small amount of specimen by spatially resolved spectroscopy.     

Advantages and limitations of the synchrotron based transmission X-ray microscopy in the study of the clay aggregate structure in aqueous suspensions

This paper reports new application of new transmission X-ray microscopy powered by a synchrotron source for the study of aqueous based clay suspensions. This paper delineates the advantages and limitations of this method. The tested transmission X-ray microscopy (TXM) technique has shown good agreement with the cryo-stage SEM technique. The spacial resolution of this TXM technique is 60 nm and clay particles with diameter below 500 nm are clearly visible and their pseudohexagonal symmetry is recognizable in detail. It is clearly demonstrated the methodology of implementing TXM to study aqueous based clay suspensions that are close to approximately 60 nm tomographic resolution. The technique enables us to study discrete structure of clay suspensions in water and within aggregates. This has never been previously possible. Larger crystals, more compact aggregates and less colloidal fraction present in kaolinite from Georgia has impact on faster settling and gelling in denser suspension than for Birdwood kaolinite in which colloidal particles create gel-like networking in less dense aqueous suspension.     

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.     

Electrical Conductivity of a Concentrated Suspension of Spherical Colloidal Particles

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

Use of a cell model for the evaluation of the dynamic mobility of spherical silica suspensions

In this paper we evaluate the validity of a cell model for the calculation of the dynamic mobility of concentrated suspensions of spheres. The key point is the consideration of the boundary conditions (electrical and hydrodynamic) at the boundary of the fluid cell surrounding a single probe particle. The model proposed is based on a universal criterion for the averages of fluid velocity, electric potential, pressure field or electrochemical properties in the cell. The calculations are checked against a wide set of experimental data on the dynamic mobility of silica suspensions with two different radii, several ionic strengths, and two particle concentrations. The comparison reveals an excellent agreement between theory and experiment, and the model appears to be extremely suitable for correctly predicting the behavior of the dynamic mobility, including the changes in the zeta potential, zeta, with ionic strength, the frequency and amplitude of the Maxwell-Wagner-O'Konski relaxation, and the inertial relaxation occurring at the top of the frequency range accessible to our experimental device.