On the flow field about an electrophoretic particle

The flow field about an electrophoretic body is theoretically investigated by analytical methods. An effective boundary condition for the electric potential at particle surface is derived. This condition, which generalizes the one obtained by Levich (Physicochemical Hydrodynamics, 475 (Prentice Hall, Englewood Cliffs, 1962). Ch. 9), captures the effect of (convective and electromigratory) surface current in the Debye layer and is valid as far as it is legitimate to neglect ion-concentration gradient in the bulk liquid. Conditions for negligible concentration gradients are also presented and discussed. The effect of surface current determines a deviation from Morrison’s “classical” theory, which predicts irrotational flow-field for any particle shape with electrophoretic velocity given by the well-known Smoluchowski formula and always directed along the applied electric field. It is shown here that in the presence of the above effect the irrotationality of the flow field is not preserved if the particle surface has non-uniform curvature. However, irrotational flow-field still subsists for a sphere and a cylinder and is analytically determined in terms of a new non-dimensional parameter, referred to as the electrophoretic number. The case of spheroidal objects is also examined in detail. In this case the flow field, though not strictly irrotational, is shown to be nearly approximated by an irrotational flow-field, which is also determined over wide ranges of electrophoretic number and spheroid aspect ratio. The quality of this approximation is expressed as a relative error on the Helmholtz-Smoluchowski condition and numerically evaluated both in longitudinal and transverse configuration. The limiting cases of spheroid degenerating into a needle and a disk are also addressed. In all above cases the respective mobilities deviate from Smoluchowski’s formula and depend on the electrophoretic number. An important effect of surface ion-transport in the double layer is anisotropy of electrophoretic mobility for non-spherical objects. That always bears a bias of the electrophoretic velocity with respect to the applied electric field when the latter is not collinear with a symmetry axis of the body. For a cylinder, bias is always towards the axis. For a spheroid, it is generally towards the polar axis; however, bias towards the equatorial axis is predicted for moderately oblate spheroids. In general, the bias angle is remarkable, which is of potential consequence in technical applications of electrophoresis. Comparison of results of the present theory with experimental work from the literature is presented and discussed.