The validity of the Smoluchowski equation in electrophoretic studies of lipid membranes

The Helmholtz-Smoluchowski theory, widely used for the calculation of zeta potential from the measured electrophoretic mobility (EPM), is known to be invalid in the region where the mobility is affected by the surface conductivity and polarization of the electrical double layer. In this region, the zeta potential found according to the Smoluchowski equation (zeta sm) is not identical to the true electrostatic potential at the hydrodynamic plane of shear (zeta), which is considered in the Gouy-Chapman-Stern theory of the electrical double layer. As a result, zeta sm cannot be used for the subsequent calculation of surface potential and surface charge density of a membrane studied. here we suggest a simple way, allowing one to decide between the validity and nonvalidity of the Smoluchowski equation in various sets of experimental conditions used in electrophoretic measurements on lipid membranes. We calculated the dimensionless criterion Rel, accepted in the Dukhin theory of electrophoresis as a measure of the extent of the influence of surface conductivity and double layer polarization on EPM. The Rel changes, with membrane charge density, ionic strength and vesicle radius, were found to be helpful in finding the combinations of these three parameters, corresponding to the validity of the Smoluchowski equation.