Smoluchowski’s formula for thin-double-layer electrophoresis does not apply for highly charged particles, where surface conduction modifies the electrokinetic transport in the electro-neutral bulk. To date, systematic studies of this nonzero Dukhin-number effect have been limited to weak fields. Employing our recent macroscale model [O. Schnitzer and E. Yariv, “Macroscale description of electrokinetic flows at large zeta potentials: Nonlinear surface conduction,” Phys. Rev. E 86, 021503 (2012)], valid for arbitrary Dukhin numbers, we analyze herein particle electrophoresis at small (but finite) Dukhin numbers; valid for arbitrary fields, this asymptotic limit essentially captures the practical range of parameters quantifying typical colloidal systems. Perturbing about the irrotational zero-Dukhin-number flow, we derive a linear scheme for calculating the small-Dukhin-number correction to Smoluchowski’s velocity. This scheme essentially amounts to the solution of a linear diffusion–advection problem governing the salt distribution in the electro-neutral bulk. Using eigenfunction expansions, we obtain a semi-analytic solution for this problem. It is supplemented by asymptotic approximations in the respective limits of weak fields, small ions, and strong fields; in the latter singular limit, salt polarization is confined to a diffusive boundary layer. With the salt-transport problem solved, the velocity correction is readily obtained by evaluating three quadratures, corresponding to the contributions of (i) electro- and diffuso-osmotic slip due to polarization of both the Debye layer and the bulk; (ii) a net Maxwell force on the electrical double layer; and (iii) Coulomb body forces acting on the space charge in the “electro-neutral” bulk. The velocity correction calculated based upon the semi-analytic solution exhibits a transition from the familiar retardation at weak fields to velocity enhancement at moderate fields; this transition is analytically captured by the small-ion approximation. At stronger fields, the velocity correction approaches a closed-form asymptotic approximation which follows from an analytic solution of the diffusive boundary-layer problem. In this régime, the correction varies as the 3/2-power of the applied field. Our small-Dukhin-number scheme, valid at arbitrary field strengths, naturally lends itself to a tractable analysis of nonlinear surface-conduction effects in numerous electrokinetic problems.
The Dukhin number increases with the Debye thickness and the zeta potential, and decreases with particle size. In thin-double-layer systems, one routinely encounters zeta potentials of about 120 mV. In aqueous solutions, the thickest Debye width is about 100 nm; at this extreme, the thin-double-layer limit is still reasonably satisfied for particle size of about 5 μm. For these values, the Dukhin number is about 0.2.
In principle, a finite value of γ introduces through condition (16) a coupling between φ and φs. The latter is governed by Laplace’s equation within the solid particle and the continuity condition φs = φ at r = 1. This coupling disappears in the thin-double-layer limit, considered next.
The case of a moderately charged particle (Du = 0) and that of highly charged particle [Du = O(1)] were actually considered separately in Ref. 24, leading to two macroscale models; combining them leads to a uniformly valid model, which may be used for arbitrary values of Du: see section VIII in Ref. 24. In the present paper, we employ the uniformly valid model.
Strictly speaking, a small-Dukhin-number approximation corresponds to the domain δ ≪ Du ≪ 1. The reason for the lower bound has to do with the derivation of the macroscale SY model. When Du = O(δ), errors of the same order as the correction might originate from higher-order terms in the thin-double-layer approximation δ ≪ 1.24
The approximation presented in Ref. 24 is slightly different, as it was derived using the high-surface-charge model, where the brackets may be replaced by ln16.