S Levine

The prediction of electrokinetic phenomena within multiparticle systems. I. Electrophoresis and electroosmosis

Abstract Henrys classic theory for the electrophoresis of a single isolated sphere is extended analytically to cover the more practical problem of the electrophoresis of a swarm of identical, dielectric, spherical particles. The effects of interaction of the individual particles and their associated electric fields are taken into explicit account by employing a fundamental cell-model representation which is known to provide good predictions for the motion of a swarm of spheres within a fluid in the absence of electrical effects. For thin double layers, the electrophoretic velocity of (or the electroosmotic velocity within) a swarm of identical dielectric spheres is shown to be invariant, practically speaking, with respect to the void fraction of the swarm. However, as the double layer thickness increases the electrophoretic (electroosmotic) velocity decreases sharply and becomes strongly dependent on the void fraction. The computed results provide theoretical justification for Smoluchowskis widely quoted approximate result for electroosmotic flow within porous media of arbitrary pore geometry.

Fine particle deposition in laminar flow through parallel-plate and cylindrical channels

Abstract The deposition of colloidal particles from a suspension in steady fully developed laminar flow onto the walls of a channel is rationalized as equivalent to mass transfer in the bulk with a first-order reaction at the walls. The resulting extended Graetz problem is solved for both parallel-plate and cylindrical channels. Through the use of confluent hypergeometric functions combined with asymptotic techniques, an evaluation of the resulting series solutions is made possible which is more accurate than all previous solutions, especially for the deposition of colloids and for cylindrical channels. Simple Leveque-type asymptotic solutions are also obtained for the case of large Peclet numbers, and when the reaction rate constant is infinite, these reduce to the corresponding well-established results for convective diffusion.

The prediction of electrokinetic phenomena within multiparticle systems: II. Sedimentation potential

Abstract A geometric cell model for a concentrated suspension in aqueous electrolyte of identical charged dielectric colloidal spheres was employed in an earlier paper in a theory of electrophoresis and electroosmosis. The same model is used here to study the related phenomenon of the sedimentation of such a suspension. The theory is restricted to small surface potentials (in the linear Debye-Huckel range) and to values of κa ≳ 10, where a is the particle radius and 1/κ is the double layer thickness. The distortion from spherical symmetry in the diffuse layer charge density within a cell (the relaxation effect) due to the sedimentation is described by a simple approximation. A steady state “streaming potential” condition of zero net current flow through a cell and also Henrys condition of zero normal component of current flow at a particle surface are used to calculate the induced dipole moment, the induced sedimentation electric field and the sedimentation velocity. For very dilute suspensions and thin double layers, these calculated quantities tend towards the classic results of Smoluchowski, as required. In the limiting case of a single particle the value obtained for the sedimentation velocity lies between those calculated by Smoluchowski and by Booth. The predictions for the sedimentation velocity in concentrated suspensions are consistent with the rather limited experimental data available in the literature.

Stabilization of emulsions by fine particles II. capillary and van der Waals forces between particles

Abstract The effective surface tension (free energy per unit area) of a planar oil/water interface completely covered with a closely-packed monolayer of identical spherical particles is determined as a function of contact angle. The surface tension diminishes as contact angle increases from 0° to 90° or decreases from 180° to 90°. Preliminary experimental results show this general trend. A theoretical study is made of the capillary forces acting between the interfacial spherical particles which stabilize oil/water emulsion droplets. Applying a two-dimensional cell model, the oil/water meniscus immediately surrounding a given particle of a closely-packed monolayer structure is assumed to have circular symmetry. Changes in interfacial areas between the oil, water and solid occur when an isolated surface particle becomes a member of the monolayer structure. For the simplified model used here the accompanying energy change due to interfacial tension yields a repulsion between the surface particles for all contact angles. By applying the well-known Derjaguin method of determining the interaction of particles at close separation, the van der Waals attraction between adjacent spheres in the monolayer is calculated as a function of the contact angle. The magnitudes of the capillary and van der Waals energy per particle are smaller by three or four orders of magnitude than the depth of the energy well in which an isolated solid sphere is trapped at the oil/water interface.

Capillary interaction of spherical particles adsorbed on the surface of an oil/water droplet stabilized by the particles. Part II

Abstract We develop a conical-cell model for determining the capillary interaction between identical spherical particles forming a monolayer adsorbed at variable packing on the surface of an oil-in-water (Pickering) emulsion droplet. Each adsorbed particle is assigned a cone-shaped section of the droplet volume. The vertex of the cone is located at the center of the droplet and the axis of the cone passes through the center of the adsorbed particle. A typical adsorbed particle is surrounded by an oil/water interfacial shell having circular symmetry about the axis of the cone. The shape of the oil/ water interface is obtained by solving the Young—Laplace equation. It is required that the volume of the dispersed phase in the droplet, which is contained in the cone, does not change on adsorption of the monolayer of particles. The interfacial energy assigned to a single adsorbed particle and its surrounding oil/water interface situated within its cone is determined. The capillary interaction is obtained by subtracting the corresponding interfacial energy when capillary interaction between the adsorbed particles is ignored. One method of obtaining interfacial energy without capillary interaction between the particles is based on the model of Menon, Nagarajan and Wasan, described in part I (S. Levine and B.D. Bowen, Colloids Surfaces, 59 (1991) 377). With this choice of interfacial energy without particle interactions, the capillary interaction is small and attractive. The leading term is identical with that obtained in Part I, by developing further the model of Menon, Nagarajan and Wasan. This term is proportional to the fourth power of the particle radius and diminishes as the inverse square of the separation between the particles. The use of the conical-cell model yields an additional term which is expressed in terms of the difference between the so-called effective (macroscopic) interfacial tension, due to the layer of adsorbed particles, and the conventional (microscopic) tension of the oil/water interface without particles. Although our result is consistent with the order of magnitude of the capillary interaction found with an earlier cylindrical-cell model, which was intended to apply to a very large droplet, the latter gave an incorrect capillary repulsion between the adsorbed particles.

Electrostatic interaction between two water-in-oil emulsion droplets in an electric field

Abstract The electrostatic interaction of two charged water-in-oil emulsion droplets at arbitrary separation in an applied uniform electric field is investigated theoretically. The problem is greatly simplified by assuming the droplets to be perfectly conducting spheres. After the potential distribution in the continuous-oil medium is expressed in terms of bipolar coordinates, the interaction energy of the two spheres is obtained in two steps. First, the energy of two uncharged spheres in the presence of the applied field is found by adapting a simple formula for the energy of an electric dipole in a uniform electric field. Then, the additional energy due to charges on the droplets is determined by means of a standard charging process. The force exerted by one sphere on the other is calculated from the gradient of the interaction energy. Valid for any nonzero separation, the result is shown to be equivalent to a previously published solution.

Double-layer interaction between parallel solid cylinders partially immersed in an oil/water interface

Abstract We consider two identical, parallel, infinitely long solid cylinders at a given separation, lying flat on a plane oil/water interface and both immersed to the same extent in the oil and water phases. The part of the surface of each cylinder in contact with the aqueous phase is charged, forming an electric double layer in a symmetrical aqueous binary electrolyte. The electrical potential in the overlapping electric double layers in the aqueous phase satisfies the Poisson-Boltzmann equation. The potentials within the uncharged interiors of the solid cylinders and in the oil phase satisfy Laplaces equation. The equations for the three potentials are solved simultaneously using the finite element method with Galerkin weighted residuals. The double-layer interaction per unit length of the cylinders is then calculated. Of the numerical results obtained, three deserve special mention. First, a short-range double-layer repulsion, decaying exponentially with separation between the two cylinders, acts through the aqueous electrolyte medium, whereas in the case of an uncharged oil/water interface a weaker, but much longer-ranging, repulsive interaction acts through the oil medium. Second, reasonable estimates of the short-range interaction between cylinders in a planar interface can be obtained from the Derjaguin approximation for thin double layers. Third, in addition to the repulsive force between the cylinders parallel to the oil/water interface, a force normal to the interface acts on the cylinders in the direction of the aqueous electrolyte phase.

A theory of electrophoresis of emulsion drops in aqueous two-phase polymer systems

An electrophoresis study has been carried out in an emulsion formed from an electrically neutral aqueous mixture of dextran and polyethylene glycol equilibrated at sufficient concentrations in the presence of electrolytes. Electrophoresis of a drop of one phase suspended in the other is observed, and the direction of the drops motion is reversed when the disperse phase and the continuous phase are interchanged. In the presence of sulfate, phosphate, or citrate ions, an electrostatic potential difference of the order of a few mV exists between the two phases. The potential implied by the direction of the electrophoretic motion is opposite to the Donnan potential observed between the two phases. The mobility of an emulsion drop increases with the drop radius and depends on ion concentration. These results are explained in terms of a model postulating an electric dipole layer associated with a mixture of oriented polymer molecules at the surface of a drop, with a potential difference between the interiors of the two phases resulting from the unequal ion distribution.