At large zeta potentials, surface conduction becomes appreciable in thin-double-layer electrokinetic transport. In the linear weak-field regime, where this effect is quantified by the Dukhin number, it is manifested in non-Smoluchowski electrophoretic mobilities. In this paper we go beyond linear response, employing the recently derived macroscale model of Schnitzer and Yariv [“Macroscale description of electrokinetic flows at large zeta potentials: Nonlinear surface conduction,” Phys. Rev. E 86, 021503 (2012)10.1103/PhysRevE.86.021503] as the infrastructure for a weakly nonlinear analysis of spherical-particle electrophoresis. A straightforward perturbation in the field strength is frustrated by the failure to satisfy the far-field conditions, representing a non-uniformity of the weak-field approximation at large distances away from the particle, where salt advection becomes comparable to diffusion. This is remedied using inner-outer asymptotic expansions in the spirit of Acrivos and Taylor [“Heat and mass transfer from single spheres in Stokes flow,” Phys. Fluids 5, 387 (1962)10.1063/1.1706630], with the inner region representing the particle neighborhood and the outer region corresponding to distances scaling inversely with the field magnitude. This singular scheme furnishes an asymptotic correction to the electrophoretic velocity, proportional to the applied field cubed, which embodies a host of nonlinear mechanisms unfamiliar from linear electrokinetic theories. These include the effect of induced zeta-potential inhomogeneity, animated by concentration polarization, on electro-osmosis and diffuso-osmosis; bulk advection of salt; nonuniform bulk conductivity; Coulomb body forces acting on bulk volumetric charge; and the nonzero electrostatic force exerted upon the otherwise screened particle-layer system. A numerical solution of the macroscale model validates our weakly nonlinear analysis.
For highly charged dielectric surfaces, the asymptotic structure underlying electrokinetic phenomena in the thin-double-layer limit reshuffles. The large counterion concentration near the surface, associated with the Boltzmann distribution in the diffuse layer, supports appreciable tangential fluxes appearing as effective surface currents in a macroscale description. Their inevitable nonuniformity gives rise in turn to comparable transverse currents, which, for logarithmically large zeta potentials, modify the electrokinetic transport in the electroneutral bulk. To date, this mechanism has been studied only using a weak-field linearization. We present here a generic thin-double-layer analysis of the electrokinetic transport about highly charged dielectric solids, which is not restricted to weak fields. We identify the counterion concentration amplification with the emergence of an internal boundary layer--within the diffuse part of the double layer--characterized by distinct scaling of ionic concentrations and electric field. In this multiscale description, surface conduction is conveniently localized within the internal layer. Our systematic scheme thus avoids the cumbersome procedure of retaining small asymptotic terms which change their magnitude at large zeta potentials. The electrokinetic transport predicted by the resulting macroscale model is inherently accompanied by bulk concentration polarization, which in turn results in nonlinear bulk transport. A novel fundamental subtlety associated with this intrinsic feature, overlooked in the weak-field approximation, has to do with the ambiguity of the "particle zeta potential" concept: In general, even uniformly charged surfaces are characterized by a nonuniform zeta-potential distribution. This impairs the need for a careful identification of the dimensionless number representing the transition to large zeta potentials.
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
While the Taylor-Melcher electrohydrodynamic model entails ionic charge carriers, it addresses neither ionic transport within the liquids nor the formation of diffuse spacecharge layers about their common interface. Moreover, as this model is hinged upon the presence of nonzero interfacial-charge density, it appears to be in contradiction with the aggregate electro-neutrality implied by ionic screening. Following a brief synopsis published by Baygents & Saville (Third International Colloquium on Drops & Bubbles, vol. 7, 1989, p. 7) we systematically derive here the macroscale description appropriate for leaky dielectric liquids, starting from the primitive electrokinetic equations and addressing the double limit of thin space-charge layers and strong fields. This derivation is accomplished through the use of matched asymptotic expansions between the narrow space-charge layers adjacent to the interface and the electro-neutral bulk domains, which are homogenized by the strong ionic advection. Electrokinetic transport within the electrical 'triple layer' comprising the genuine interface and the adjacent space-charge layers is embodied in effective boundary conditions; these conditions, together with the simplified transport within the bulk domains, constitute the requisite macroscale description. This description essentially coincides with the familiar equations of Taylor & Melcher (Annu. Rev. Fluid Mech., vol. 1, 1969, p. 111). A key quantity in our macroscale description is the 'apparent' surface-charge density, provided by the transversely-integrated triple-layer microscale charge. At leading order, this density vanishes due to the expected Debye-layer screening; its asymptotic correction provides the 'interfacial' surface-charge density appearing in the Taylor-Melcher model. Our unified electrohydrodynamic treatment provides a reinterpretation of both the Taylor-Melcher conductivity-ratio parameter and the electrical Reynolds number. The latter, expressed in terms of fundamental electrokinetic properties, becomes O(1) only for intense applied fields, comparable with the transverse field within the space-charge layers; at this limit, however, the asymptotic scheme collapses. Surface-charge advection is accordingly absent in the macroscale description. Because of the inevitable presence of (screened) net charge on the genuine interface, the drop also undergoes electrophoretic motion. The associated flow, however, is asymptotically smaller than that corresponding to the the Taylor-Melcher circulation. Our successful matching procedure contrasts the analysis of Baygents & Saville (1989), who considered more general electrolytes and were unable to directly match the inner and outer regions. We discuss this difference in detail.
We revisit the problem of extraordinary transmission of acoustic (electromagnetic) waves through a slit in a rigid (perfectly conducting) wall. We use matched asymptotic expansions to study the pertinent limit where the slit width is small compared to the wall thickness, the latter being commensurate with the wavelength. Our analysis focuses on near-resonance frequencies, furnishing elementary formulae for the field enhancement, transmission efficiency, and deviations of the resonances from the Fabry-Pérot frequencies of the slit. We find that the apertures' near fields play a dominant role, in contrast with the prevalent approximate theory of Takakura [Physical Review Letters, 86 5601 (2001)]. Our theory agrees remarkably well with numerical solutions and electromagnetic experiments [Suckling et al., Physical Review Letters, 92 147401 (2004)], thus providing a paradigm for analyzing a wide range of wave propagation problems involving small holes and slits.
Standard thin-double-layer modeling of electro-osmotic flows about metal objects typically predicts an induced zeta-potential distribution whose characteristic magnitude varies linearly with the applied voltage. At moderately large zeta potential, comparable with several thermal voltages, surface conduction enters the dominant electrokinetic transport, throttling that linear scaling. We derive here a macroscale model for induced-charge electro-osmosis accounting for that mechanism. Unlike classical analyses of surface conduction about dielectric surfaces, the present nonlinear problem cannot be linearized about a uniform-zeta-potential reference state. With the transition to moderately large zeta potentials taking place nonuniformly, the Dukhin number, representing the magnitude of surface conduction, is reinterpreted as a local dimensionless group, varying along the boundary. Debye-scale analysis provides effective boundary conditions about two types of generic boundary points, corresponding to small and moderate Dukhin numbers. The boundary decomposition into the respective asymptotic domains is unknown in advance and must be determined throughout the solution of the macroscale problem, itself hinging upon the proper formulation of effective boundary conditions. This conceptual obstacle is surmounted via introduction of a uniform approximation to these conditions.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.