Primary electroviscous effect in a dilute suspension of charged mercury drops

The standard theory of the primary electroviscous effect in a dilute suspension of charged spherical rigid particles in an electrolyte solution (Watterson, I. G.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1981, 77, 1115) is extended to cover the case of a dilute suspension of charged mercury drops of viscosity eta(d). A general expression for the effective viscosity or the electroviscous coefficient p of the suspension is derived. This expression tends to that for the case of rigid particles in the limit of eta(d) --> infinity. We also derive an approximate analytical viscosity expressions applicable to mercury drops carrying low zeta potentials at arbitrary kappaa (where kappa is the Debye-Hückel parameter and a is the drop radius) and to mercury drops as well as rigid spheres with arbitrary zeta potentials at large kappaa. It is shown that the large-kappaa expression of p for rigid particles predicts a maximum when plotted as a function of zeta potential. This result for rigid particles agrees with the exact numerical results of Watterson and White. It is also shown that in the limit of high zeta potential the effective viscosity of a suspension of mercury drops tends to that of uncharged rigid spheres given by Einstein's formula (Einstein, A. Ann. Phys. 1906, 19, 289), whereas in the opposite limit of low zeta potential the effective viscosity approaches that of a suspension of uncharged liquid drops derived by Taylor (Taylor, G. I. Proc. R. Soc. London, Ser. A 1932, 138, 41).     

Diffusiophoresis of concentrated suspensions of spherical particles with identical ionic diffusion velocities

Diffusiophoresis of concentrated suspensions of spherical particles subject to a small electrolyte gradient is analyzed theoretically at arbitrary levels of zeta potential and double-layer thickness. The Kuwabara unit cell model is adopted to describe the system under consideration. The effect of double-layer polarization is taken into account. It is found that the diffusiophoretic mobility exhibits a local maximum as well as a local minimum with varying zeta potential or double-layer thickness, similar to the corresponding dilute dispersion. The direction of the particle movement may even change back and forth. The previous low-zeta-potential approach is found to significantly overestimate the diffusiophoretic mobility as the zeta potential goes high. The deviation may be several fold sometimes. The effect of the volume fraction ratio of colloids is also examined. The higher the ratio, the lower the mobility.     

Sedimentation Velocity and Potential in Concentrated Suspensions of Charged Spheres with Arbitrary Double-Layer Thickness

The sedimentation in a homogeneous suspension of charged spherical particles 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. Overlap of the double layers of adjacent particles is allowed, and the polarization effect in the double layer surrounding each particle is considered. The electrokinetic equations that govern the ionic concentration distributions, the electric potential profile, and the fluid flow field in the electrolyte solution 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 for a symmetrically charged electrolyte with the surface charge density (or zeta potential) of the particle as the small perturbation parameter. An analytical expression for the settling velocity of the charged sphere in closed form is obtained from a balance among its gravitational, electrostatic, and hydrodynamic forces. A closed-form formula for the sedimentation potential in a suspension of identical charged spheres is also derived by using the requirement of zero net electric current. Our results demonstrate that the effects of overlapping double layers are quite significant, even for the case of thin double layers. Copyright 2000 Academic Press.     

Sedimentation of concentrated spherical particles with a charge-regulated surface

The sedimentation of a concentrated colloidal dispersion is examined for the case of an arbitrary double-layer thickness. Here, a general mixed-type condition on particle surface is assumed, and the classic models, which assume constant surface properties, can be recovered as the special cases of the present analysis. In particular, the behavior of biological cells, which carry dissociable functional groups on their surfaces, and particles, which are capable of exchanging ions with the surrounding medium, can be simulated by the present model. The mixed-type boundary condition leads to several interesting results in both sedimentation velocity and sedimentation potential as double-layer thickness and the concentration of particles vary.     

Electrophoresis of a concentrated aqueous dispersion of non-Newtonian drops

The electrophoresis of a concentrated dispersion of non-Newtonian drops in an aqueous medium, which has not been investigated theoretically in the literature, is analyzed under conditions of low zeta potential and weak applied electric field. The results obtained provide a theoretical basis for the characterization of the nature of an emulsion and a microemulsion system. A Carreau fluid, which has wide applications in practice, is chosen for the non-Newtonian drops, and the unit cell model of Kuwabara is adopted to simulate a dispersion. The effects of the key parameters of a dispersion, including its concentration, the shear-thinning nature of the drop fluid, and the thickness of the double layer, on the electrophoretic behavior of a drop are discussed. In general, the more significant the shear-thinning nature of the drop fluid is, the larger the mobility is, and this effect is pronounced as the thickness of the double layer decreases. However, if the double layer is sufficiently thick, this effect becomes negligible. In general, the higher the concentration of drops is, the smaller the mobility is; however, if the double layer is either sufficiently thin or sufficiently thick, this effect becomes unimportant.     

Electrophoresis of concentrated mercury drops

The electrophoretic behavior of concentrated monodispersed, positively charged mercury drops is investigated theoretically. The present study extends previous analyses by considering arbitrary surface potentials, double-layer polarization, and the interaction between adjacent double layers. The coupled equations describing the spatial variations in the flow field, the electric field, and the concentration field are solved by a pseudo-spectral method. For a low surface potential phi(r), the mobility increases monotonically with kappaalpha; kappa and alpha are respectively the reciprocal Debye length and the radius of a mercury drop. For medium and high phi(r), the mobility curve has a reflection point, which arises from the interaction of adjacent double layers, for kappaalpha. Also, if phi(r) is high, the mobility curve may exhibit a local minimum as kappaalpha varies. This phenomenon is pronounced if the concentration of the dispersed phase is high. If the double layer is thick, the mobility increases with phi(r), and the reverse is true if it is thin. We show that the higher the concentration of the dispersed phase the smaller the mobility, and as kappaalpha becomes large the mobility approaches a constant value, which is independent of the concentration of the dispersed phase. The mobility of mercury drops is larger than that of the corresponding rigid particles.     

Streaming potential generated by a long viscous drop in a capillary

The streaming potential generated by motion of a long drop of viscosity mu(d) = lambdamu in a uniform circular capillary filled with fluid of viscosity mu is investigated by means of a model previously used to study electrophoresis of a charged mercury drop in water. The capillary wall is at potential zeta c relative to the bulk fluid within it, and the surface of the drop is at potential zeta(d). Potentials are assumed to be sufficiently small so that the charge cloud is described by the linearized Poisson-Boltzmann equation, and the Debye length characterizing the thickness of the charge cloud is assumed to be thin compared with the gap h(0) between the drop and the capillary wall. Ions in the external fluid are not allowed to discharge at the surface of the drop, and the wall of the capillary has a nonzero surface conductivity sigma c. The drop is assumed to be sufficiently long so that end effects can be neglected. Recirculation of fluid within the drop gives rise to an enhanced streaming current when zeta(d) is nonzero, leading to an anomalously high streaming potential. This vanishes as the drop viscosity becomes large. If V is the velocity of the drop and gamma is the coefficient of interfacial tension between the two fluids, then the capillary number is Ca = mu V/gamma, and the gap varies as h(0)planck'sCa(2/3). When Ca is small, the gap h(0) is small and electrical conduction along the narrow gap is dominated by the surface conductivity sigma(c) of the capillary wall, which is constant. The electrical current convected by flowing fluid is proportional to Ca, as is the change in streaming potential caused by the presence of the drop. If sigma(c) = 0, then the electrical conductance of the gap depends on its width h(0) and on the bulk fluid conductivity sigma and becomes small as h(0) approximately equal to Ca(2/3) --> 0. The streaming potential required to cancel the O(Ca) convection current therefore varies as Ca(1/3). If sigma(c) = 0 and the drop is rigid (lambda --> infinity), then the change in streaming potential over and above that expected due to the change in pressure gradient is proportional to the difference in potentials zeta(c)-zeta(d).     

Electrophoresis of a concentrated dispersion of spherical particles covered by an ion-penetrable membrane layer

The electrophoretic behavior of a concentrated dispersion of soft spherical particles is investigated theoretically, taking the effects of double-layer overlapping and double-layer polarization into account. Here, a particle comprises a rigid core and an ion-penetrable layer containing fixed charge, which mimics biocolloids and particles covered by artificial membrane layers. A cell model is adopted to simulate the system under consideration, and a pseudo-spectral method based on Chebyshev polynomials is chosen for the resolution of the governing electrokinetic equations. The influence of the key parameters, including the thickness of the double layer, the concentration of particles, the surface potential of the rigid core of a particle, and the thickness, the amount of fixed charge, and the friction coefficient of the membrane layer of a particle on the electrophoretic behavior of the system under consideration is discussed. We show that while the result for the case of a dispersion containing rigid particles can be recovered as the limiting case of a dispersion containing soft particles, qualitative behaviors that are not present in the former are observed in the latter.     

Sedimentation of a concentrated dispersion of composite colloidal particles

The sedimentation of a concentrated spherical dispersion of composite particles, where a particle comprises a rigid core and a membrane layer containing fixed charge, is investigated theoretically. The dispersion is simulated by a unit cell model, and a pseudo-spectral method based on Chebyshev polynomials is adopted to solve the problem numerically. The influences of the thickness of double layer, the concentration of particles, the surface potential of the rigid core of a particle, and the amount of fixed charge in the membrane layer on both the sedimentation potential and the sedimentation velocity are discussed. Several interesting results are observed; for example, depending upon the charged conditions on the rigid core and in the membrane layer of a particle, the sedimentation potential might have both a local maximum and a local minimum and the sedimentation velocity can have a local minimum as the thickness of double layer varies. Also, the sedimentation velocity can have a local maximum as the surface potential varies. We show that the sedimentation potential increases with the concentration of particles. The relation between the sedimentation velocity and the concentration of particles, however, depends upon the thickness of double layer.     

Stability of dispersions of colloidal alumina particles in aqueous suspensions

The colloidal stability of suspensions of alumina particles has been investigated by measuring particle size distribution, sedimentation, viscosity, and zeta potential. Alumina particles were found to be optimally dispersed at pH around 3 to 7.8 without dispersant and at pH 8.5 and beyond with dispersant. The above results corroborate zeta potential and viscosity measurement data well. The surface charge of alumina powder changed significantly with anionic polyelectrolyte (ammonium polycarboxylate, APC) and the iep shifted toward more acidic range under different dispersant conditions. It was found that the essential role played by pH and dispersant (APC) on the charge generation and shift in the isoelectric point of alumina manifests two features: (i) the stability decreases on approaching the isoelectric point from either side of pH, and (ii) the maximum instability was found at pH 9.1 for alumina only and at pH 6.8 for alumina/APC, which is close to the isoelectric points for both the system, respectively. Using the model based on the electrical double-layer theory of surfactant adsorption through shift in isoelectric points, the authors could estimate the specific free energy of interaction (7.501 kcal/mol) between particles and dispersant. The interaction energy, zeta potential, sedimentation, and viscosity results, were used to explain the colloidal stability of the suspension.     

Electrophoresis of a colloidal sphere in a spherical cavity with arbitrary zeta potential distributions and arbitrary double-layer thickness

The electrophoretic motion of a dielectric sphere situated at the center of a spherical cavity with an arbitrary thickness of the electric double layers adjacent to the particle and cavity surfaces is analyzed at the quasisteady state when the zeta potentials associated with the solid surfaces are arbitrarily nonuniform. Through the use of the multipole expansions of the zeta potentials and the linearized Poisson-Boltzmann equation, the equilibrium double-layer potential distribution and its perturbation caused by the applied electric field are separately solved. The modified Stokes equations governing the fluid velocity field are dealt with using a generalized reciprocal theorem, and explicit formulas for the electrophoretic and angular velocities of the particle valid for all values of the particle-to-cavity size ratio are obtained. To apply these formulas, one only has to calculate the monopole, dipole, and quadrupole moments of the zeta potential distributions at the particle and cavity surfaces. In some limiting cases, our result reduces to the analytical solutions available in the literature. In general, the boundary effect on the electrophoretic motion of the particle is a qualitatively and quantitatively sensible function of the thickness of the electric double layers relative to the radius of the cavity.     

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.     

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.     

Diffusiophoresis in a suspension of spherical particles with arbitrary double-layer thickness

The diffusiophoresis in a homogeneous suspension of identical dielectric spheres with an arbitrary thickness of the electric double layers in a solution of a symmetrically charged electrolyte with a constant imposed concentration gradient is analytically studied. The effects of particle interactions (or particle volume fraction) 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 ionic concentration distributions, the electrostatic 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 diffusiophoretic velocity of the dielectric sphere in closed form correct to the second order of its surface charge density or zeta potential are obtained from a balance between its electrostatic and hydrodynamic forces. Comparisons of the results of the cell model with different conditions at the outer boundary of the cell are made.     

Primary electroviscous effect in a moderately concentrated suspension of charged spherical colloidal particles

On the basis of the standard theory of the primary electroviscous effect in a moderately concentrated suspension of charged spherical particles in an electrolyte solution presented by Ruiz-Reina et al. (Ruiz-Reina, E.; Carrique, F.; Rubio-Hernández, F. J.; Gómez-Merino, A. I.; García-Sánchez, P. J. Phys. Chem. B 2003, 107, 9528), which is applicable for the case where overlapping of the electrical double layers of adjacent particles can be neglected, the general expression for the effective viscosity or the primary electroviscous coefficient p of the suspension is derived. This expression is applicable for a suspension of spherical particles of radius a carrying arbitrary zeta potentials zeta at the particle volume fraction phi < or = 0.3 for the case of nonoverlapping double layers, that is, at kappaalpha > 10 (where kappa is the Debye-Hückel parameter). A simple approximate analytic expression for p applicable for particles with large kappaalpha and arbitrary zeta is presented. The obtained viscosity expression is a good approximation for moderately concentrated suspensions of the particle volume fraction phi < or = 0.3, where the relative error is negligible for kappaalpha > or =100 and even at kappaalpha = 50 the maximum error is approximately 20%. It is shown that a maximum of p, which appears when plotted as a function of the particle zeta potential, is due to the relaxation effect as in the case of the electrophoresis problem.     

Dynamic electrophoresis of a droplet in a spherical cavity

The electrophoretic behavior of a droplet in a spherical cavity subject to an alternating electric field is analyzed theoretically under the conditions of an arbitrary level of surface potential and double-layer thickness. The influences of the thickness of the double layer, the level of surface potential, the size of a droplet, the viscosity of the droplet fluid, and the frequency of the applied electric field on the electrophoretic behavior of a droplet are examined through numerical simulations. We show that, because of the effect of double-layer deformation, the magnitude of the electrophoretic mobility of a droplet could have a local maximum and the phase angle could have a negative (phase lead) local minimum as the frequency of the applied electric field varies. In general, the lower the surface potential, the thicker the double layer and the larger the viscosity of the droplet fluid, and the more significant the boundary effect, the smaller the magnitude of the electrophoretic mobility of a droplet.     

High-order field electrophoresis theory for a nonuniformly charged sphere

An electrophoresis theory is developed for a rigid sphere in a general nonuniform electric field. The zeta potential distribution and the double-layer thickness are both arbitrary. The zeta potential of the sphere is assumed to be small so that the deformation of the double layer can be neglected. Explicit expressions for the translational and rotational velocities of the sphere are derived in terms of the multipole moments of the zeta potential distribution and the tensor coefficients of the applied electric field. The presence of the kth-order component in the electrical potential field applied to the sphere results in a translation of the sphere only when the sphere possesses the (k-1)th- or (k+1)th-order multipole moments of the zeta potential distribution. In addition, the kth-order component in the electrical potential field causes a rotation of the sphere only when the sphere possesses the kth-order moment of the zeta potential distribution. As an illustrative example for the utility of our theory, we theoretically devise an electrophoresis analysis scheme for estimating the dipole moment of a dipolar sphere by observing the electrophoretic translation of the sphere in a quadratic potential field.     

Electrophoresis of a concentrated spherical dispersion at arbitrary electrical potentials

The electrophoretic behavior of a concentrated spherical dispersion is investigated theoretically. The present analysis extends those in the literature in that both the surface potential of a particle and the strength of the applied electric field are arbitrary and both the effects of double-layer polarization and the overlapping between neighboring double layers are taken into account. Results based on these conditions are highly desirable since they cover essentially all the possible experimental conditions in practice. We show that, for a fixed surface potential and strength of applied electric field, the higher the concentration of particle, the smaller the mobility. Counterions are found to accumulate at the downstream side of a particle. Double-layer polarization is inappreciable if either it is thick or the concentration of the particle is high.     

Electrophoretic motion of a slightly deformed sphere with a nonuniform zeta potential distribution

Electrophoretic motion is analyzed for a rigid, slightly deformed sphere with a nonuniform zeta potential distribution. Hydrodynamics and electrostatics solutions for the deformed sphere with an arbitrary double-layer thickness are determined by using the domain perturbation method. The surface shape and the zeta potential distribution for the deformed sphere are expressed by using the multipole expansion representation. In terms of monopole, dipole, and quadrupole moments of the surface shape and the zeta potential distribution, explicit expressions are obtained for the translational and rotational electrophoretic mobility tensors. The ensemble average for the mobility of the deformed sphere with a uniform orientation distribution is also derived. The utility of the general mobility expression is demonstrated by studying the electrophoretic motion of axisymmetric and ellipsoidal particles. The translational and rotational mobilities of axisymmetric particles are both affected by the monopole, dipole, and quadrupole moments of the zeta potential. For ellipsoidal particles, however, the dipole moment of the zeta potential does not affect the translational mobility, while the rotational mobility depends only on the dipole moment. The mobility of the deformed sphere with either a thick or a thin double layer is also derived.