(402e) Steric Effects On Electrophoresis of a Colloidal Particle
For over a century, the Poisson-Nernst-Planck (PNP) equations have served as the de facto theoretical model of electrokinetic phenomena. However, the non-interacting, point-sized ions assumed by PNP can lead to impossibly large ion concentrations near even moderately charged surfaces. Here, we revist the classic problem of electrophoresis of a spherical colloidal particle, using modified PNP equations that account for steric repulsion between finite sized ions, through Bikerman's model . Steric effects are controlled by the bulk volume fraction of ions ν, and for ν= 0 the standard PNP equations are recovered. An asymptotic analysis in the thin-double-layer limit reveals at small zeta potentials ( ζ kBT/e ≈25 mV) the electrophoretic mobility Me to increase linearly with ζ for all ν, as expected from the Helmholtz-Smoluchowski (HS) formula. For larger ζ, however, it is well known that surface conduction of ions within the double layer reduces Me below the HS result. Crucially, in the PNP equations surface conduction becomes significant precisely because of the aphysically large and unbounded counter-ion densities predicted at large ζ. In contrast, steric effects impose a limit on the counter-ion density, thereby mitigating surface conduction. Hence, Me does not fall as far below HS for finite sized ions (ν≠0). Indeed, at sufficiently large ν, steric effects are so dramatic that a maximum in Me is not observed for physically reasonable values of ζ (< 10 kBT/e), in stark contrast to the PNP-based calculations of O'Brien and White . Finally, by calculating a Dukhin-Bikerman number characterising the relative importance of surface conduction, we collapse mobility versus zeta data for different volume fractions onto a single master curve.
 J. J. Bikerman, Philos. Mag. 33, 384 (1942)
 R. W. O'Brien and L. R. White, J. Chem. Soc. Faraday Trans. II 74, 1607 (1978).