Richard W. O'Brien

Electrophoretic mobility of a spherical colloidal particle

The equations which govern the ion distributions and velocities, the electrostatic potential and the hydrodynamic flow field around a solid colloidal particle in an applied electric field are reexamined. By using the linearity of the equations which determine the electrophoretic mobility, we show that for a colloidal particle of any shape the mobility is independent of the dielectric properties of the particle and the electrostatic boundary conditions on the particle surface. The mobility depends only on the particle size and shape, the properties of the electrolyte solution in which it is suspended, and the charge inside, or electrostatic potential on, the hydrodynamic shear plane in the absence of an applied field or any macroscopic motion.New expressions for the forces acting in the particle are derived and a novel substitution is developed which leads to a significant decoupling of the governing equations. These analytic developments allow for the construction of a rapid, robust numerical scheme for the solution of the governing equations which we have applied to the case of a spherical colloidal particle in a general electrolyte solution. We describe a computer program for the conversion of mobility measurements to zeta potential for a spherical colloidal particle which is far more flexible than the Wiersema graphs which have traditionally been used for the interpretation of mobility data. Furthermore it is free of the high zeta potential convergence difficulties which limited Wiersemas calculations to moderate values of ζ. Some sample computations in typical 1:1 and 2:1 electrolytes are exhibited which illustrate the existence of a maximum in the mobility at high zeta potentials. The physical explanation of this effect is given. The importance of the mobility maximum in testing the validity of the governing equations of electrophoresis and its implications for the colloid chemists picture of the Stern layer are briefly discussed.

The solution of the electrokinetic equations for colloidal particles with thin double layers

Abstract The calculation of the transport properties of a colloid usually involves the solution of a set of coupled differential equations for the electrical potential, ion densities, and fluid velocities in the suspending electrolyte. This paper is concerned with the solution of these equations for colloidal particles whose dimensions are much greater than the double-layer thickness. It is shown that the methods devised by Dukhin [outlined in “Surface and Colloid Science” (E. Matijevic, Ed.), Vol. 7, Chap. 3. Wiley, New York, 1974] for solving the equations for a symmetric two species electrolyte can be simplified, and extended to the case of a general electrolyte. To illustrate the procedure the electrophoretic velocity and the static conductivity of a dilute suspension of spherical particles in a general electrolyte are calculated.

The high-frequency dielectric dispersion of a colloid

Abstract The dielectric response of a colloid in which the particle radius a is much greater than the double-layer thickness κ −1 is investigated. Such colloids exhibit two types of dielectric dispersion: one at frequencies of order D / a 2 , and the other at higher frequencies of order κ 2 D where D is the ion diffusivity. It is the high-frequency dispersion that is of interest here. The calculation of this dispersion is greatly simplified by the fact that the applied field has a negligible effect on the ion densities in the electrolyte. Thus there is no diffusive component to the electric current, and the double layer and surrounding electrolyte can be treated as simple conductors. By combining the solution for the electrical potential around an isolated sphere with the corresponding result obtained by S. S. Dukhin and V. N. Sbilov (“Dielectric Phenomena and the Double Layer in Disperse Systems and Polyelectrolytes.” Wiley, New York, 1974) for the low-frequency regime we construct a complete picture of the dielectric response of a dilute suspension of spheres. To illustrate the effect of particle concentration on the form of the dielectric dispersion, computed values are presented for a random close-packed array of spheres.

The electrophoresis of a spheroid with a thin double layer

Abstract A formula is obtained for the electrophoretic mobility of a spheroidal particle in the thin double-layer limit. Unlike the Smoluchowski expression, this formula takes into account the change in the ion densities induced by the applied field, an effect which becomes significant as the particle ζ potential increases. In the course of deriving the electrophoresis formula, we also obtaine an expression for the static conductivity of a dilute suspension of randomly oriented spheroids.

The electrical conductivity of a dilute suspension of charged particles

This paper deals with the problem of calculating the electrical conductivity K∗ of a suspension of nonconducting charged particles, assuming that the particles and their double layers occupy only a small fraction of the total volume. For such a suspension it is shown that the contribution from each particle to the conductivity may be obtained from the forms of the electrical potential and ion densities at large distances from the particle. To obtain these far-field forms it is necessary to solve the same set of equations as in the electrophoresis problem; namely a set of coupled differential equations for the flow field and for quantities related to the ion densities and electric field outside the particle. An approximate solution to this problem is presented for the case of a spherical particle with low ζ potential immersed in a symmetric electrolyte. From the solution we obtain a formula for the effective conductivity of a dilute suspension of spheres, correct to 0(ζ2). Comparison of this approximate formula for K∗ with values obtained from numerical solutions of the exact equations indicates that the formula for K∗ is accurate to within a few percent provided ζ is less than 50 mV. With the aid of this formula it should be possible for the experimentalist to determine the ζ potential for a suspension from measurements of the electrical conductivity. In this way it may be possible to determine the ζ-potential of particles which are too small to be observed in the electrophoresis apparatus.

The electrical conductivity of a porous plug

Abstract A formula is derived for the electrical conductivity of a porous plug composed of closely packed dielectric spheres in an electrolyte. The radius of the spheres is assumed to be much larger than the double-layer thickness, and the spheres are taken to have a uniform ζ-potential. By combining this formula with measured values of conductivity it should be possible for experimentalists to estimate the ζ-potential for granular plugs.

Electroosmosis in porous materials

Abstract Electroosmosis is the phenomenon of fluid flow induced by an applied electric field. This paper is concerned with electroosmosis in a porous material composed of closely-packed spheres immersed in a general electrolyte. A formula is obtained for the electroosmotic flow rate in the case when the double layer is much thinner than the particle radius. By combining this formula with electroosmosis measurements it is possible to determine the particle ζ potential. To test the validity of the model which underlies this, and most other electrokinetic calculations, ζ potentials obtained from Van der Put and Bijsterboschs (J. Colloid Interface Sci. 92, 499, 1983) electroosmosis measurements are compared with potentials obtained from their conductivity and electrophoresis measurements.

The dielectric response of concentrated latices

Abstract Dielectric response measurements can, in principle, provide important information on the concentration of colloidal particles, their shape and charge, and the composition of the background electrolyte. In this paper we present measurements of the effect of particle volume fraction on the dielectric response of concentrated latices in the frequency range 1–10 MHz, a range where the theory predicts large dielectric dispersion effects. It is found that the measurements on suspensions with a volume fraction of 0.3 or more are in accordance with a result obtained using a cell model. As the volume fraction decreases the departures from the formula increase, but these may be due to errors in our measurement of the particle contribution to the complex conductivity at these low volume fractions.

Electroacoustic characterization of colloids with unusual particle properties

The AcoustoSizer (Matec Applied Sciences) has been used to follow the electroacoustic behaviour of oil-in-water emulsions, stabilised by an anionic surfactant, under various treatments. The property which is measured in an electroacoustic experiment is the particle dynamic mobility, which is a complex quantity having both a magnitude and a phase angle. For many systems it is possible to use the dynamic mobility to estimate the zeta potential and the size of the particles. In the emulsion system, at low surfactant concentrations, the estimated zeta potentials were very high (up to almost 200 mV (negative)). A comparison is made between the zeta values obtained on the emulsion system and the anticipated measurements using d.c. methods. Whilst the d.c. measurement is ambiguous, the electroacoustic measurement is unequivocal in ascribing a high zeta potential to the measured data. The dynamic mobility in this case shows a phase lead superimposed on the more usual phase lag caused by particle inertia. Unusual electroacoustic phase behaviour is also displayed by semiconductor particles, but in that case the origin of the effect is in the very large values of the effective particle dielectric permittivity, which is proportional to the particle size. For the 4 μm semiconducting silicon particles used in this study, the phase lag is large (50°) at low frequencies and becomes smaller as the frequency rises.