(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 [1]. 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 k_BT/e ≈25 mV) the electrophoretic mobility M_e 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 M_e 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, M_e 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 M_e is not observed for physically reasonable values of ζ (< 10 k_BT/e), in stark contrast to the PNP-based calculations of O'Brien and White [2]. 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.

[1] J. J. Bikerman, Philos. Mag. 33, 384 (1942)

[2] R. W. O'Brien and L. R. White, J. Chem. Soc. Faraday Trans. II 74, 1607 (1978).