Nonlinear and nonequilibrium electrophoresis of spherical particles of radius a is shown to be possible when the solid surface allows field or current penetration. At low particle Peclet numbers, transient capacitative charging occurs until the surface polarization completely screens the external field. For a DC applied field E-infinity the resulting electrokinetic velocity reaches Dukhin's maximum value of (ε) over capE(infinity)(2)a/mu, where (ε) over cap and mu are the liquid permittivity and viscosity. At high Peclet numbers, electroosmotic convection of the electroneutral bulk stops the transient charging before complete field-line exclusion. For an ion-selective and conducting spherical granule, the polarization is then determined by the steady-state Ohmic current driven by the penetrated external field. The high-Peclet electrokinetic velocity is lower, diffusivity-dependent and scales as E(infinity)(2/3)a(1/3). (C) 2004 Elsevier Inc. All rights reserved.