Dynamic Mobility of Two Spherical Particles with Thick Double Layers.

The dynamic electrophoretic mobility of a pair of nearby spherical particles is analyzed in the case when the thickness of the electrical double layer around each particle is comparable to the particle radius. By means of an integral reciprocal relation, a formal expression is obtained for the force and torque on N spheres subject to an oscillating electric field which may be spatially varying. Upon linearizing in the surface potential, this expression is shown to depend upon a set of purely hydrodynamic problems involving N neutral spheres, the calculation of the electric field around N neutral spheres, and the equilibrium charge distribution around N charged spheres. In the case of a single particle, the known analytic formula for the dynamic mobility is recovered. For a pair of identical particles, the dynamic mobility is calculated numerically, using known solutions to the required subproblems. An analytical expression for the mobility of a pair of widely separated spheres is also obtained by a method of reflections, and this is in excellent agreement with the numerical results outside the range of double layer overlap. Copyright 2000 Academic Press.

Dynamic Mobility of Particles with Thick Double Layers in a Nondilute Suspension.

The dynamic mobility of a nondilute suspension of spherical particles is investigated in the case where the thickness of the electrical double layer around each particle is comparable to the particle radius. A formula is obtained for the O(φ) correction in a random suspension of particles with volume fraction φ, involving an integral over the dynamic mobility of a pair of spheres. This formula is then evaluated using both analytical approximations and numerical results previously obtained for the pair mobilities and valid for low surface potentials. The effect of double-layer thickness on the O(φ) coefficient is most pronounced at low frequencies, and lessens once the hydrodynamic penetration depth is smaller than the particle radius. Various approximations are considered that use the O(φ) result to predict the dynamic mobility in concentrated suspensions, and at high frequencies these approximations are shown to give results qualitatively different from those of recent cell models. Copyright 2000 Academic Press.

Interactions between Two Spherical Particles with Nonuniform Surface Potentials: The Linearized Poisson-Boltzmann Theory.

Using the linearized Poisson-Boltzmann theory, electrical double layer interactions are calculated between two nonuniform spherical colloidal particles with mean potential zero. Most results are for the case of surface potentials modeled by a single spherical harmonic and aligned relative to each other. As previously observed for flat surfaces, interactions decay more rapidly as a function of separation between spheres with such periodic, "single-mode" potentials than between spheres with uniform potentials. The Deryaguin approximation for single-mode spheres is tested and calculations are made of the force and torque that particles in a doublet exert on one another. Many of the concepts developed from models of flat plates with periodic aligned surface potentials are shown to be useful in this more general case. Attempts to explain recent differential electrophoresis experiments on the basis of nonuniform double layers fail in that the maximum restraining torques produced under plausible assumptions about the amplitude of nonuniformity are an order of magnitude smaller than those implied by the measurements. The main reason for this is that the torques are too small at the separations characteristic of a secondary minimum. The effect of misalignment of single-mode spheres is assessed by calculating the distribution of interaction energies over a set of relative orientations generated by quasi-random sampling. Finally, a method for generating spheres with "random" surface potentials is devised and potentials of mean force are calculated for pairs of spheres with such surface potentials. Comparison with the single-mode case is kept at a qualitative level, in the absence of detailed knowledge of how realistic such "random" surfaces are. Copyright 1999 Academic Press.

Electrophoretic Motion of a Spherical Particle with a Thick Double Layer in Bounded Flows.

The electrophoretic mobility of a spherical colloidal particle with low zeta potential near a solid charged boundary is calculated numerically for arbitrary values of the double layer thickness by a generalization of Teubner's method to the case of bounded flow. Three examples are considered: a sphere near a nonconducting planar wall with electric field parallel to the wall, near a perfectly conducting planar wall with electric field perpendicular to the wall, and on the axis of a cylindrical pore with electric field parallel to the axis. The results are compared with recent analytical calculations using the method of reflections. For the case of a charged sphere near a neutral surface, the reflection results are quite good, provided there is no double layer overlap, in which case there can be extra effects for constant potential particles that are entirely missed by the analytical expressions. For a neutral sphere near a charged surface, the reflection results are less successful. The main reason is that the particle feels the profile of the electroosmotic flow, an effect ignored by construction in the method of reflections. The general case is a combination of these, so that the reflections are more reliable when the electrophoretic motion dominates the electroosmotic flow. The effect on particle mobility of particle-wall interactions follows the trend expected on geometric grounds in that sphere-plane interactions are stronger than sphere-sphere interactions and the effect on a sphere in a cylindrical pore is stronger still. In the latter case, particle mobility can fall by more than 50% for thick double layers and a sphere half the diameter of the pore. The agreement between numerical results and analytical results follows the same trend, being worst for the sphere in a pore. Nevertheless, the reflections can be reliable for some geometries if there is no double layer overlap. This is demonstrated for a specific example where reflection results have previously been compared with experiments on protein mobility through a membrane (J. Ennis et al., 1996, J. Membrane Sci. 119, 47). Copyright 1999 Academic Press.

2-D and 3-D Interactions in Random Sequential Adsorption of Charged Particles

We explore the influence of electrolyte concentration on the adsorption of charged spheres using modeling techniques based on random sequential adsorption (RSA). We present a parametric study of the effects of double layer interactions between the charged particles and between the particle and the substrate on the jamming limit using a two-dimensional RSA simulation similar to that of Z. Adamczyk et al. (1990, J. Colloid Interface Sci. 140, 123) along with a simple method of estimating jamming limit coverages. In addition, we present a more realistic RSA algorithm that includes explicit energetic interaction in three dimensions, that is, particle-particle and particle-surface interactions during the approach of a particle to the substrate. The calculation of interaction energies in the 3-D RSA model is achieved with the aid of a three-body superposition approximation. The 3-D RSA model differs from the 2-D model in that the extent of coverage is controlled by kinetic rather than energetic considerations. Results of both models capture the experimentally observed trend of increased surface coverage with increased electrolyte concentration, and both models require the value of a key model parameter to be specified for a quantitative match to experimental data. However, the 3-D model more effectively captures the governing physics, and the parameter in this case takes on more meaningful values than for the 2-D model. Copyright 1997 Academic Press. Copyright 1997Academic Press

Electrophoretic Motion of Two Spherical Particles with Thick Double Layers

The electrophoretic mobilities of two interacting spheres are calculated numerically for arbitrary values of the double-layer thickness. A general formula for the electrophoretic translational and angular velocities of N interacting particles is derived for low-zeta-potential conditions. The present calculation complements the well-studied case of thin double layers. The results are compared with recent reflection calculations and are used to compute the O(phi) contribution to the electrophoretic mobility of a suspension. Particle interactions can be significant for values of the scaled particle radius kappaa