A unified theory of dipolophoresis for nanoparticles

General mobility relations are derived for the translation and rotation of submicrometer-size freely suspended conducting (ideally polarizable) particles of arbitrary shape under dc or ac spatially nonuniform electric forcing. Both linear electrophoretic effects for an initially charged colloid and nonlinear induced-charge electrophoresis of an uncharged particle are considered within the same framework. A concise derivation of the total loads (electrostatic and hydrodynamic) exerted on a single colloid are obtained by integrating the Maxwell, shear and normal (relating to the unsteady Stokes equation) stresses over the particle. These newly derived expressions for the force and torque exerted on a nanoparticle, which is subject to any electric field, are given for an arbitrary Debye thickness and thus open the road to studying nonlinear phoretic motions on the nanoscale.