Boundary-layer effects on electromagnetic and acoustic extraordinary transmission through narrow slits.

We study the problem of resonant extraordinary transmission of electromagnetic and acoustic waves through subwavelength slits in an infinite plate, whose thickness is close to a half-multiple of the wavelength. We build on the matched-asymptotics analysis of Holley & Schnitzer (2019 Wave Motion 91, 102381 (doi:10.1016/j.wavemoti.2019.102381)), who considered a single-slit system assuming an idealized formulation where dissipation is neglected and the electromagnetic and acoustic problems are analogous. We here extend that theory to include thin dissipative boundary layers associated with finite conductivity of the plate in the electromagnetic problem and viscous and thermal effects in the acoustic problem, considering both single-slit and slit-array configurations. By considering a distinguished boundary-layer scaling where dissipative and diffractive effects are comparable, we develop accurate analytical approximations that are generally valid near resonance; the electromagnetic-acoustic analogy is preserved up to a single parameter that is provided explicitly for both scenarios. The theory is shown to be in excellent agreement with GHz-microwave and kHz-acoustic experiments in the literature.

Slender-body theory for plasmonic resonance.

We develop a slender-body theory for plasmonic resonance of slender metallic nanoparticles, focusing on a general class of axisymmetric geometries with locally paraboloidal tips. We adopt a modal approach where one first solves the plasmonic eigenvalue problem, a geometric spectral problem which governs the surface-plasmon modes of the particle; then, the latter modes are used, in conjunction with spectral-decomposition, to analyse localized-surface-plasmon resonance in the quasi-static limit. We show that the permittivity eigenvalues of the axisymmetric modes are strongly singular in the slenderness parameter, implying widely tunable, high-quality-factor, resonances in the near-infrared regime. For that family of modes, we use matched asymptotics to derive an effective eigenvalue problem, a singular non-local Sturm-Liouville problem, where the lumped one-dimensional eigenfunctions represent axial voltage profiles (or charge line densities). We solve the effective eigenvalue problem in closed form for a prolate spheroid and numerically, by expanding the eigenfunctions in Legendre polynomials, for arbitrarily shaped particles. We apply the theory to plane-wave illumination in order to elucidate the excitation of multiple resonances in the case of non-spheroidal particles.

Surface plasmon resonances of arbitrarily shaped nanometallic structures in the small-screening-length limit.

According to the hydrodynamic Drude model, surface plasmon resonances of metallic nanostructures blueshift owing to the non-local response of the metal's electron gas. The screening length characterizing the non-local effect is often small relative to the overall dimensions of the metallic structure, which enables us to derive a coarse-grained non-local description using matched asymptotic expansions; a perturbation theory for the blueshifts of arbitrary-shaped nanometallic structures is then developed. The effect of non-locality is not always a perturbation and we present a detailed analysis of the 'bonding' modes of a dimer of nearly touching nanowires where the leading-order eigenfrequencies and eigenmode distributions are shown to be a renormalization of those predicted assuming a local metal permittivity.

A generalized Derjaguin approximation for electrical-double-layer interactions at arbitrary separations.

Derjaguin's approximation provides the electrical-double-layer interaction force between two arbitrary convex surfaces as the product of the corresponding one-dimensional parallel-plate interaction potential and an effective radius R (function of the radii of curvature and relative orientation of the two surfaces at minimum separation). The approximation holds when both the Debye length 1/κ and minimum separation h are small compared to R. We show here that a simple transformation, R⇒[R]√[[K1][K2]/K1K2], yields an approximation uniformly valid for arbitrary separations h; here, Ki is the Gaussian curvature of particle i at minimum separation, and [ ⋅ ] is an operator which adds h/2 to all radii of curvature present in the expression on which it acts. We derive this result in two steps. First, we extend the two-dimensional ray-theory analysis of Schnitzer [Phys. Rev. E 91, 022307 (2015)], valid for κh, κR ≫ 1, to three dimensions. We thereby obtain a general closed form expression for the force by matching nonlinear diffuse-charge boundary layers with a WKBJ-type expansion describing the bulk potential, and subsequent integration via Laplace's method of the traction over the medial surface generated by all spheres maximally inscribed between the two surfaces. Second, we exploit the existence of an overlap domain, 1 ≪ κh ≪ κR, where both the ray-theory and the Derjaguin approximations hold, to systematically form the generalized mapping. The validity of the result is demonstrated by comparison with numerical computations.

Ray-theory approach to electrical-double-layer interactions.

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 interparticle 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 centers 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 center 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 interparticle separations. Our theory can be generalized to arbitrary three-dimensional geometries, nonideal electrolyte models, and other physical scenarios where exponentially decaying fields give rise to forces.

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.

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.

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.