Nonlinear electrophoretic velocity of a spherical colloidal particle

Electrophoresis is the motion of a charged colloidal particle in an electrolyte under an applied electric field. The electrophoretic velocity of a spherical particle depends on the dimensionless electric field strength beta = a* e*E *(infinity) /k * (B) T *, defined as the ratio of the product of the applied electric field magnitude E *(infinity) and particle radius a*, to the thermal voltage k * (B) T * /e*, where k * B is Boltzmann's constant, T * is the absolute temperature, and e* is the charge on a proton. In this paper, we develop a spectral element algorithm to compute the electrophoretic velocity of a spherical, rigid, dielectric particle, of fixed dimensionless surface charge density sigma over a wide range of beta. Here, sigma = (e* a* /is an element of * k * (B) T *)sigma *, where sigma * is the dimensional surface charge density, and is an element of * is the permittivity of the electrolyte. For moderately charged particles ( sigma = O(1)), the electrophoretic velocity is linear in beta when beta << 1, and its dependence on the ratio of the Debye length (1/kappa*) to particle radius (denoted by delta = 1/(kappa * a*)) agrees with Henry's formula. As beta increases, the nonlinear contribution to the electrophoretic velocity becomes prominent, and the onset of this behaviour is delta-dependent. For ss >> 1, the electrophoretic velocity again becomes linear in field strength, approaching the Huckel limit of electrophoresis in a dielectric medium, for all delta. For highly charged particles (sigma >> 1) in the thin-Debye-layer limit (delta << 1), our computations are in good agreement with recent experimental and asymptotic results.