Dipolophoresis of nanoparticles

A general methodology is presented for evaluating the dielectrophoretic velocity of a freely suspended spherical colloid in an electrolyte solutes under the action of a nonuniform electric field and for a Debye layer of arbitrary thickness. The nonlinear induced charge electrophoretic problem is first considered. General analytic expressions are derived for the mobility of an uncharged particle in the form of products between adjacent modes of the ambient electric field demonstrating a symmetry-breaking-type phenomena. It is shown that the mobility of a conducting (i.e., ideally polarizable) spherical particle vanishes for a quadratic electric field in the limiting case of a thin Debye layer. For an infinitely thick Debye layer it attains asymptotically a positive finite value. Yet, there is another value of a finite Debye length for which the mobility changes sign. This interesting nonintuitive effect may have implications to separation of particles by size. The linear case of a uniformly charged colloid is obtained as a special limit and the classical mobility expressions of Henry [Proc. R. Soc. (London) 4, 106 (1931)], Smulokowski [Handbook of Electricity and Magnetism, edited by L. Graetz (Barth, Leipzig, 1921), p. 2], and Huckel [Physik 25, 204 (1924)] for a spherical colloid are readily recovered. The formulation is based on an extension of Teubner's integral approach for nonuniform electric fields and on utilizing a variant of the Lorentz reciprocal theorem for Stokes flows. As an example demonstrating the effect of nearby boundaries, the method is finally applied for the radially symmetric case involving a freely suspended colloid within a hollow spherical capsule filled with electrolyte. It can be considered as an extension of Henry's unbounded solution for a confined three dimensional embodiment.