Nonlinear electrophoresis at arbitrary field strengths: small-Dukhin-number analysis

Abstract

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 Dukhinnumber 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 smallDukhin-number scheme, valid at arbitrary field strengths, naturally lends itself to a tractable analysis of nonlinear surface-conduction effects in numerous electrokinetic problems. C 2014 AIP Publishing LLC. [http://dx

Nonlinear Electrophoresis of Highly Charged Nonpolarizable Particles

“…3(b) were approximately −21.4(±2.2) and −9.9(±1.0) ×10 −20 m 4 s −1 V −3 , respectively. Based on the model proposed by Schnitzer et al [13,14] for small Dukhin and Péclet numbers, the nonlinear coefficients for these parameters are −37.6 and −7.9 ×10 −20 m 4 s −1 V −3 , as also shown in Fig. 3(b).…”

“…For the analytical approach in the thin double layer limit and small Péclet number, the first two allowed terms by symmetry are described as v ep = µ ep E+µ (3) ep E 3 , where both coefficients µ ep and µ (3) ep are independent of E [14,28]. The coefficients µ (3) ep obtained from our results (PS 620 nm and PMMA 520 nm) in Fig.…”

“…This nonlinear effect, which is different from the nonlinear elec-trokinetics found for polarizable particles [20], is not well understood for intermediate to large κa in contrast to the case of small κa in nonpolar electrolytes [21][22][23][24][25][26]. Theoretical analyses have predicted that the electrophoretic velocity becomes nonlinear to the applied field at a moderate to high field β = aE/φ th 1 [14,[27][28][29]. However, a well-controlled experiment in this regime has been lacking.…”

Effect of core charge density on the electrophoresis of a soft particle coated with polyelectrolyte layer

“…Equations 4 and 5 show that though the potential inside the core varies with the core-to-electrolyte dielectric permittivity ratio ( R ), the core surface potential is independent of the dielectric permittivity of the core. The asymptotic analysis under a thin double layer assumption by Schnitzer and Yariv [29] and the references there-in have demonstrated that the solid permittivity may influence the electrophoresis under a strong applied electric field.…”

Surface conduction and electroosmotic flow around charged dielectric pillar arrays in microchannels

“…On the dielectric micropillar surfaces, the surface conduction was described by the fluxmatching condition between normal and tangential flux referred to Schnitzer and Yariv's analysis 55 . Note that Schnitzer and Yariv analyzed the surface conduction in spherical coordinate system, so that their analysis should be transformed into local cylindrical coordinate system as denoted in Figure 3.…”