§ 1. The theory of cataphoresis, and of the complementary phenomenon of electrosmosis, is based on the conception of an “ electrical double layer ” at the interface between the two phases whose relative motion is under consideration.* In the original theory, as propounded by Quincke and Helmholtz, this electrical double layer was regarded as a kind of parallel plate condenser made up of two laminar distributions of electrification, of which one—the so-called “ inner sheet ”—was firmly attached to the rigid phase, while the other—the “ outer sheet ”—resided in the mobile phase ; the separation between the two was considered to be a distance of the order of molecular dimensions. The currently accepted view, initiated by Gouy, differs from that of Helmholtz chiefly in that the outer sheet of the double layer is considered to be a diffuse distribution of electrification—an “ ionic atmosphere ” of the type investigated by Debye and his collaborators in connection with the theory of strong electrolytes. The net electric density in the ionic atmosphere varies continuously from a maximum in the immediate neighbourhood of the fixed inner sheet, to a negligibly small value in the bulk of the liquid, over a distance which is a function of the ionic concentration, and which lies as a rule between molecular dimensions and some thousand micromillimetres. In a deduction which appears to be completely consistent with this more modern view of the double layer, Smoluchowski deduced the expression U = DXζ/4π
η
(1) for the cataphoretic velocity U ; X is the applied field strength, ζ the potential difference across the double layer, D the dielectric constant and
η
the viscosity of the medium. The equation is identical with that developed by Helmholtz except for the inclusion of the dielectric constant, but was deduced on a much more general basis, and is claimed by Smoluchowski to be valid for rigid electrically insulating particles of any shape, subject only to the following four restrictions :— 1) That the usual hydrodynamical equations for the motion of a viscous fluid may be assumed to hold both in the bulk of the liquid and within the double layer; (2) That the motion is “stream line motion,” and slow enough for the “inertia terms” in the hydrodynamic equations to be neglected ; (3) That the applied field may be taken as simply superimposed on the field due to the electrical double layer ; and (4) That the thickness of the double layer (
i. e,
the distance normal to the interface over which the potential differs appreciably from that in the bulk of the liquid) is small compared with the radius of curvature at any point of the surface.
Effect of Droplet Size and Counterions on the Spatial Distribution of Ions