An Asymptotic Derivation of the Thin-Debye-Layer Limit for Electrokinetic Phenomena

Yariv, Ehud

doi:10.1080/00986440903076590

Cited by 50 publications

(56 citation statements)

References 26 publications

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“…With transverse non-uniformities in the electric potential appearing only at O(1), the radial Debye-layer electric fields retain their O(δ −1 ) equilibrium scaling, just as in standard electrokinetic phenomena under moderate applied fields (Yariv 2009).…”

Section: Preliminary Resultsmentioning

confidence: 89%

The Taylor–Melcher leaky dielectric model as a macroscale electrokinetic description

Schnitzer

Yariv

2015

J. Fluid Mech.

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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.

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Section: Preliminary Resultsmentioning

confidence: 89%

The Taylor–Melcher leaky dielectric model as a macroscale electrokinetic description

Schnitzer

Yariv

2015

J. Fluid Mech.

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“…The leading-order boundary-layer solution of (1), in conjunction with (3), and attenuation at distances ≫ δ from the boundary, is well known [26][27][28]. It is given by…”

Section: Thin-double-layer Analysis a Single Particlementioning

confidence: 99%

Ray-theory approach to electrical-double-layer interactions

Schnitzer

2015

Phys. Rev. E

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A novel approach is presented for analyzing the double-layer interaction force between charged particles in electrolyte solution, in the limit where the Debye length is small compared with both inter-particle separation and particle size. The method, developed here for two planar convex particles of otherwise arbitrary geometry, yields a simple asymptotic approximation limited to neither small zeta potentials nor the "close-proximity" assumption underlying Derjaguin's approximation.Starting from the nonlinear Poisson-Boltzmann formulation, boundary-layer solutions describing the thin diffuse-charge layers are asymptotically matched to a WKBJ expansion valid in the bulk, where the potential is exponentially small. The latter expansion describes the bulk potential as superposed contributions conveyed by "rays" emanating normally from the boundary layers. On a special curve generated by the centres of all circles maximally inscribed between the two particles, the bulk stress -associated with the ray contributions interacting nonlinearly -decays exponentially with distance from the centre of the smallest of these circles. The force is then obtained by integrating the traction along this curve using Laplace's method. We illustrate the usefulness of our theory by comparing it, alongside Derjaguin's approximation, with numerical simulations in the case of two parallel cylinders at low potentials. By combining our result and Derjaguin's approximation, the interaction force is provided at arbitrary inter-particle separations. Our theory can be generalized to arbitrary three dimensional geometries, non-ideal electrolyte models, and other physical scenarios where exponentially decaying fields give rise to forces.

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“…Thus, the fluid domain is naturally decomposed into the electro-neutral 'bulk' and the Debye layer in quasi-equilibrium, surrounding the particle, where the (different) ionic concentrations are governed by Boltzmann distributions (Prieve et al 1984;Rubinstein & Zaltzman 2001). The standard practice in electrokinetic analyses (Yariv 2010a) is to extract an effective bulk description wherein effective boundary conditions represent asymptotic matching of the 'outer' bulk fields with the 'inner' Debye-scale variables (which satisfy the boundary conditions (2.12)-(2.15) on the literal particle boundary). An important quantity in the Debye-layer structure is the Debye-layer voltage-the 'zeta potential' z.…”

Section: Macroscale Descriptionmentioning

confidence: 99%

Electrokinetic self-propulsion by inhomogeneous surface kinetics

Yariv

2010

Proc. R. Soc. A.

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Electrokinetic self-propulsion was conceptually proven in recent experiments wherein bimetallic nano-rods were observed to migrate when placed in aqueous solutions. We present here a systematic theoretical model of the self-propulsion mechanism, analysing the steady-state transport occurring about an autonomously moving inhomogeneous particle. The non-uniform catalysis on the particle surface is modelled via positiondependent cation redox coefficients. The particle shape is axisymmetric but otherwise arbitrary, as are the distributions of the (possibly discontinuous) kinetic coefficients along its boundary. We formulate the mathematical problem governing this electrokinetic transport. In the thin-Debye-layer limit, the microscale description is transformed into a macroscale one, applying in the electro-neutral bulk. Effective boundary conditions represent asymptotic matching with the Debye-layer fields. A linearized model is derived for weak variation of the kinetic coefficients and is solved for a spherical-particle geometry. With a view towards understanding existing experiments, the macroscale model is used for analysing slender particles. Matched asymptotic expansions provide the particle velocity as a functional of its shape and kinetic-coefficient distributions. The predicted self-propulsion is in the direction observed in nanorod experiments.

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An Asymptotic Derivation of the Thin-Debye-Layer Limit for Electrokinetic Phenomena

Cited by 50 publications

References 26 publications

The Taylor–Melcher leaky dielectric model as a macroscale electrokinetic description

The Taylor–Melcher leaky dielectric model as a macroscale electrokinetic description

Ray-theory approach to electrical-double-layer interactions

Electrokinetic self-propulsion by inhomogeneous surface kinetics

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