Lifetime of evaporating two-dimensional sessile droplets.

Prevailing diffusion-limited analyses of evaporating sessile droplets are facilitated by a quasisteady model for the evolution of vapor concentration in space and time. When attempting to employ that model in two dimensions, however, one encounters an impasse: the logarithmic growth of concentration at large distances, associated with the Green's function of Laplace's equation, is incompatible with the need to approach an equilibrium concentration at infinity. Observing that the quasisteady description breaks down at large distances, the diffusion problem is resolved using matched asymptotic expansions. Thus the vapor domain is conceptually decomposed into two asymptotic regions: one at the scale of the drop, where vapor transport is indeed quasisteady, and one at a remote scale, where the drop appears as a point singularity and transport is genuinely unsteady. The requirement of asymptotic matching between the respective regions furnishes a self-consistent description of the time-evolving evaporation process. Its solution provides the droplet lifetime as a universal function of a single physical parameter. Our scheme avoids the use of a remote artificial boundary, which introduces a nonremovable dependence upon a nonphysical parameter.

Self-Diffusiophoresis of Slender Catalytic Colloids.

We consider the self-diffusiophoresis of axisymmetric particles using a continuum description where the interfacial chemical reaction is modeled by first-order kinetics with a prescribed axisymmetric distribution of rate-constant magnitude. We employ the standard macroscale framework where the interaction of solute molecules with the particle boundary is represented by diffusio-osmotic slip. The dimensionless problem governing the solute transport involves two parameters (the particle slenderness ϵ and the Damköhler number Da) as well as two arbitrary functions which describe the axial distributions of the particle shape and rate-constant magnitude. The resulting particle speed is determined throughout the solution of the accompanying problem governing the flow about the force-free particle. Motivated by experimental configurations, we employ slender-body theory to investigate the asymptotic limit ϵ ≪ 1. In doing so, we seek algebraically accurate approximations where the asymptotic error is smaller than a positive power of ϵ. The resulting approximations are thus significantly more useful than those obtained in the conventional manner, where the asymptotic expansion is carried out in inverse powers of ln ϵ. The price for that utility is that two linear integral equations need to be solved: one governing the axial solute-sink distribution and the other governing the axial distribution of Stokeslets. When restricting the analysis to spheroidal particles, no need arises to solve for the Stokeslet distribution. The integral equation governing the solute-sink distribution is then solved using a numerical finite-difference scheme. This solution is supplemented by a large-Da asymptotic analysis, wherein a subtle nonuniformity necessitates a careful treatment of the regions near the particle ends. The simple approximations thereby obtained are in excellent agreement with the numerical solution.

Velocity amplification in pressure-driven flows between superhydrophobic gratings of small solid fraction.

With diminishing fraction of their solid portion, compound gas-solid superhydrophobic surfaces exhibit a large amount of slip which allows for appreciable velocity amplification in pressure-driven microchannel flows. We address this small solid-fraction limit in the context of a grating-like configuration, where superhydrophobicity is provided by a periodic array of flat-meniscus bubbles which are trapped in a Cassie state within the grooved channel walls. Asymptotic analysis for both longitudinal and transverse flows reveals a logarithmic scaling of the effective slip length in the solid fraction of the compound boundaries, thus refuting earlier claims of an algebraic singularity. The logarithmic scaling in the longitudinal problem is explained using an analogy between the unidirectional velocity and the velocity potential in two-dimensional irrotational flows. In the transverse problem it has to do with the Stokes paradox. The mechanisms identified herein explain the absence of slip-length singularity in the comparable asymmetric configuration, where only one of the channel walls is superhydrophobic.

Dielectrophoretic sphere-wall repulsion due to a uniform electric field.

When a zero-net-charge particle is placed under a uniform electric field, the decay of the Maxwell stress with the third power of distance ensures a nil electric force. A nonzero force may nonetheless be generated in the presence of a planar wall due to a mechanism which resembles conventional dielectrophoresis under nonuniform fields. In the prototypical case of a spherical particle this force acts perpendicular to the wall; its magnitude depends upon the pertinent boundary conditions governing the electric potential. When a particle is suspended in an electrolyte solution, where the double-layer structure ensures zero net charge, these conditions are electrokinetic in nature; they involve a balance between bulk conduction and diffusion, represented by normal derivatives, and an effective surface-conduction mechanism, represented by surface-Laplacian terms whose magnitude is quantified by appropriate Dukhin numbers. The dimensionless force depends upon the particle and wall Dukhin numbers as well as the ratio λ of the size of the particle to its distance from the wall. The remote-particle limit λ ≪ 1 is addressed using successive reflections. Calculation of the first few terms in the asymptotic expansion of the force only requires the evaluation of a single reflection from the wall. The leading-order term, scaling as λ(4), is repulsive, with a magnitude that varies non-monotonically with the particle Dukhin number and is independent of the wall Dukhin number. Surface conditions on the wall enter only at the O(λ(5)) leading-order correction.

Ratcheting of Brownian swimmers in periodically corrugated channels: a reduced Fokker-Planck approach.

We consider the motion of self-propelling Brownian particles in two-dimensional periodically corrugated channels. The point-size swimmers propel themselves in a direction which fluctuates by Brownian rotation; in addition, they undergo Brownian motion. The impermeability of the channel boundaries in conjunction with an asymmetry of the unit-cell geometry enables ratcheting, where a nonzero particle current is animated along the channel. This effect is studied here in the continuum limit using a diffusion-advection description of the probability density in a four-dimensional position-orientation space. Specifically, the mean particle velocity is calculated using macrotransport (generalized Taylor-dispersion) theory. This description reveals that the ratcheting mechanism is indirect: swimming gives rise to a biased spatial particle distribution which in turn results in a purely diffusive net current. For a slowly varying channel geometry, the dependence of this current upon the channel geometry and fluid-particle parameters is studied via a long-wave approximation over a reduced two-dimensional space. This allows for a straightforward seminumerical solution. In the limit where both rotational diffusion and swimming are strong, we find an asymptotic approximation to the particle current, scaling inversely with the square of the swimming Péclet number. For a given swimmer-fluid system, this limit is physically realized with increasing unit-cell size.

Assessing corrections to the Fick-Jacobs equation.

We utilize macrotransport theory to compute the effective diffusion coefficient of a point-sized particle in a periodic channel of slowly varying cross-section to the second order in the long-wavelength limit. This asymptotic result serves as a benchmark test for the respective modifications of the Fick-Jacobs equation proposed by Zwanzig [J. Phys. Chem. 96, 3926 (1992)], Reguera and Rubi [Phys. Rev. E 64, 061106 (2001)], and Kalinay and Percus [Phys. Rev. E 74, 041203 (2006)]. While all three modifications result in an identical effective diffusivity at first order, only the model proposed by Kalinay and Percus agrees at second order with our asymptotic result.

Strong electro-osmotic flows about dielectric surfaces of zero surface charge.

We analyze electro-osmotic flow about a dielectric solid of zero surface charge, using the prototypic configurations of a spherical particle and an infinite circular cylinder. We assume that the ratio δ of Debye width to particle size is asymptotically small, and consider the flow engendered by the application of a uniform electric field; the control parameter is E-the voltage drop on the particle (normalized by the thermal scale) associated with this field. For moderate fields, E=O(1), the induced ζ potential scales as the product of the applied-field magnitude and the Debye width; being small compared with the thermal voltage, its resolution requires addressing one higher asymptotic order than that resolved in the comparable analysis of electrophoresis of charged particles. For strong fields, E=O(δ-1), the ζ potential becomes comparable to the thermal voltage, depending nonlinearly on δ and E. We obtain a uniform approximation for the ζ-potential distribution, valid for both moderate and strong fields; it holds even under intense fields, E≫δ-1, where it scales as log|E|. The induced-flow magnitude therefore undergoes a transition from an E2 dependence at moderate fields to an essentially linear variation with |E| at intense fields. Remarkably, surface conduction is negligible as long as E≪δ-2: the ζ potential, albeit induced, remains mild even under intense fields. Thus, unlike the related problem of induced-charge flow about a perfect conductor, the theoretical velocity predictions in the present problem may actually be experimentally realized.

Nonlinear oscillations in an electrolyte solution under ac voltage.

The response of an electrolyte solution bounded between two blocking electrodes subjected to an ac voltage is considered. We focus on the pertinent thin-double-layer limit, where this response is governed by a reduced dynamic model [L. Højgaard Olesen, M. Z. Bazant, and H. Bruus, Phys. Rev. E 82, 011501 (2010)]. During a transient stage, the system is nonlinearly entrained towards periodic oscillations of the same frequency as that of the applied voltage. Employing a strained-coordinate perturbation scheme, valid for moderately large values of the applied voltage amplitude V, we obtain a closed-form asymptotic approximation for the periodic orbit which is in remarkable agreement with numerical computations. The analysis elucidates the nonlinear characteristics of the system, including a slow (logarithmic) growth of the zeta-potential amplitude with V and a phase straining scaling as V-1lnV. In addition, an asymptotic current-voltage relation is provided, capturing the numerically observed rapid temporal variations in the electric current.

Electric conductance of highly selective nanochannels.

We consider electric conductance through a narrow nanochannel in the thick-double-layer limit, where the space-charge Debye layers adjacent to the channel walls overlap. At moderate surface-charge densities the electrolyte solution filling the channel comprises mainly of counterions. This allows to derive an analytic closed-form approximation for the channel conductance, independent of the salt concentration in the channel reservoirs. The derived expression consists of two terms. The first, representing electromigratory transport, is independent of the channel depth. The second, representing convective transport, depends upon it weakly.

Nonlinear electrokinetic flow about a polarized conducting drop.

In the thin-double-layer limit κa>>1, electrokinetic flows about free surfaces are driven by a combination of an electro-osmotic slip and effective shear-stress jump. An intriguing case is that of a highly conducting liquid drop of radius a, where the inability to balance the viscous shear by Maxwell stresses results in an O(κa) velocity amplification relative to the familiar electro-osmotic scale. To illuminate the inherent nonlinearity we consider uncharged drops, where the induced surface-charge distribution results in a fore-aft symmetric electrokinetic flow profile with no attendant drop translation. This problem is analyzed using a macroscale model, where the double layer is represented by effective boundary conditions. Because of the intense flow, ionic convection within the O(1/κ)-wide diffuse-charge layer is manifested by a moderate-zeta-potential surface-conduction effect. The drop deforms to a prolate shape in response to the combination of hydrodynamic forces and the effective electrocapillary reduction of the surface-tension coefficient, both mechanisms being asymptotically comparable. The flow field and the concomitant drop deformation are calculated using both a weak-field approximation and numerical simulations of the nonlinear macroscale model.

Electrokinetic particle-electrode interactions at high frequencies.

We provide a macroscale description of electrokinetic particle-electrode interactions at high frequencies, where chemical reactions at the electrodes are negligible. Using a thin-double-layer approximation, our starting point is the set of macroscale equations governing the "bounded" configuration comprising of a particle suspended between two electrodes, wherein the electrodes are governed by a capacitive charging condition and the imposed voltage is expressed as an integral constraint. In the large-cell limit the bounded model is transformed into an effectively equivalent "unbounded" model describing the interaction between the particle and a single electrode, where the imposed-voltage condition is manifested in a uniform field at infinity together with a Robin-type condition applying at the electrode. This condition, together with the standard no-flux condition applying at the particle surface, leads to a linear problem governing the electric potential in the fluid domain in which the dimensionless frequency ω of the applied voltage appears as a governing parameter. In the high-frequency limit ω>>1 the flow is dominated by electro-osmotic slip at the particle surface, the contribution of electrode electro-osmosis being O(ω(-2)) small. That simplification allows for a convenient analytical investigation of the prevailing case where the clearance between the particle and the adjacent electrode is small. Use of tangent-sphere coordinates allows to calculate the electric and flows fields as integral Hankel transforms. At large distances from the particle, along the electrode, both fields decay with the fourth power of distance.

Macroscale description of electrokinetic flows at large zeta potentials: nonlinear surface conduction.

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.

Induced-charge electro-osmosis beyond weak fields.

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.

One-dimensional conduction through supporting electrolytes: two-scale cathodic Debye layer.

Supporting-electrolyte solutions comprise chemically inert cations and anions, produced by salt dissolution, together with a reactive ionic species that may be consumed and generated on bounding ion-selective surfaces (e.g., electrodes or membranes). Upon application of an external voltage, a Faraday current is thereby established. It is natural to analyze this ternary-system process through a one-dimensional transport problem, employing the thin Debye-layer limit. Using a simple model of ideal ion-selective membranes, we have recently addressed this problem for moderate voltages [Yariv and Almog, Phys. Rev. Lett. 105, 176101 (2010)], predicting currents that scale as a fractional power of Debye thickness. We address herein the complementary problem of moderate currents. We employ matched asymptotic expansions, separately analyzing the two inner thin Debye layers adjacent to the ion-selective surfaces and the outer electroneutral region outside them. A straightforward calculation following comparable singular-perturbation analyses of binary systems is frustrated by the prediction of negative ionic concentrations near the cathode. Accompanying numerical simulations, performed for small values of Debye thickness, indicate a number unconventional features occurring at that region, such as inert-cation concentration amplification and electric-field intensification. The current-voltage correlation data of the electrochemical cell, obtained from compilation of these simulations, does not approach a limit as the Debye thickness vanishes. Resolution of these puzzles reveals a transformation of the asymptotic structure of the cathodic Debye layer. This reflects the emergence of an internal boundary layer, adjacent to the cathode, wherein field and concentration scaling differs from those of the Gouy-Chapman theory. The two-scale feature of the cathodic Debye layer is manifested through a logarithmic voltage scaling with Debye thickness. Accounting for this scaling, the complied current-voltage data collapses upon a single curve. This curve practically coincides with an asymptotically calculated universal current-voltage relation.

Irreversible electrokinetic repulsion at zero-Reynolds-number sedimentation.

Stokes-flow reversibility is violated in electrolyte solutions by a streaming-potential mechanism, where nonuniform convective currents within Debye layers surrounding charged particles induce electric fields in the electroneutral Ohmic bulk. We demonstrate the irreversibility consequences of this phenomenon for the problem of particle-pair sedimentation, where the two particles experience a repulsive force driven by bulk Maxwell stresses. At small separations the force scales inversely with the third power of separation distance. This singular behavior is associated with the counterrotation of the two torque-free particles, which leads through a lubrication mechanism to an intense electric field in the narrow gap between them. At large separations the force follows an inverse dependence upon the fourth power of separation, now associated with rectilinear particle motion.

Communication: The phoretic drift of a charged particle animated by a direct ionic current.

A charged colloidal particle which is suspended in an electrolyte solution drifts due to an external voltage application. For direct currents, particle motion is affected by two separate mechanisms: electro-osmotic slip associated with the electric field and chemi-osmotic slip associated with the inherent salt concentration gradient in the solution. These two mechanisms are interrelated and are of comparable magnitude. Their combined effect is demonstrated for cation-exchange electrodes using a weak-current approximation. The linkage between the two mechanisms results in an effectively modified mobility, whose dependence on the particle zeta potential is nonlinear. At small potentials, the electro-osmotic mechanism dominates and the particle migrates according to the familiar Smoluchowski mobility, linear in the electric field. At large zeta potentials, chemiosmosis becomes dominant: for positively charged particles, it tends to arrest motion, leading to mobility saturation; for negatively charged particles, it enhances the drift, effectively leading to a shifted linear dependence of the mobility on the zeta potential, with twice the Smoluchowski slope.

Ionic currents in the presence of supporting electrolytes.

We analyze one-dimensional charge conduction within an ionic solution in the presence of supporting electrolytes that do not discharge on the electrodes. For thin Debye layers, numerical simulations predict current abatement, in agreement with experimental knowledge; in addition, they reveal unconventional features absent from classical analyses of binary solutions, such as high cation concentration near the electrodes. We derive a companion asymptotic description of the problem in the singular thin-Debye-layer limit, reproducing these attributes. The asymptotic analysis reveals a nested boundary-layer structure about the reactive electrodes and furnishes a universal current-voltage relation.

Asymptotic current-voltage relations for currents exceeding the diffusion limit.

We consider the one-dimensional transport of ions into a perm-selective solid. Direct attempts to evaluate the current-voltage characteristics for currents exceeding the diffusion limit are frustrated by the appearance of nonconverging integrals. We describe how to overcome this obstacle using a regularization scheme.