Ion steric effects on electrophoresis of a colloidal particle

We calculate the electrophoretic mobility M-e of a spherical colloidal particle, using modified Poisson-Nernst-Planck (PNP) equations that account for steric repulsion between finite sized ions, through Bikerman's mean-field model (Bikerman, Phil. Mag., vol. 33, 1942, p. 384). Ion steric effects are controlled by the bulk volume fraction of ions nu, and for nu = 0 the standard PNP equations are recovered. An asymptotic analysis in the thin-double-layer limit reveals at small zeta potentials (zeta < k(B)T/e approximate to 25 mV) M-e to increase linearly with zeta for all nu, as expected from the Helmholtz-Smoluchowski (HS) formula. For larger zeta, however, it is well known that Surface conduction of ions within the double layer reduces M-e below the HS result. Crucially, however, in the PNP equations surface conduction becomes significant precisely because of the aphysically large and unbounded counter-ion densities predicted at large zeta. In contrast, ion 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 (nu not equal 0). Indeed, at sufficiently large nu, ion steric effects are so dramatic that a maximum in M-e is not observed for physically reasonable values of zeta(<= 10k(B)T/e), in stark contrast to the PNP-based calculations of O'Brien & White (J. Chem. Soc. Faraday Trans. 11, vol. 74, 1978, p. 1607) and O'Brien (J. Colloid Interface Sci., vol. 92, 1983, p. 204). Finally, by calculating a Dukhin-Bikerman number characterizing the relative importance of surface conduction, we collapse M-e versus zeta data for different nu onto a single master curve.