Ory Schnitzer

Weakly nonlinear electrophoresis of a highly charged colloidal particle

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 ma...

Dielectric-solid polarization at strong fields: Breakdown of Smoluchowski's electrophoresis formula

We investigate the thin-double-layer electrophoretic drift of a uniformly charged dielectric particle, driven by an intense electric field comparable to the transverse Debye-layer field. Under these circumstances, solid polarization affects the leading-order electrokinetic transport in the fluid by inducing a nonuniform zeta-potential distribution. The resulting expression for the particle velocity is accordingly nonlinear in the applied field. The electrophoretic “mobility”—the ratio of this velocity and the applied field—depends upon two parameters, the first quantifying the surface-charge density, and the second constituting the product of the solid-to-liquid permittivity ratio and the scaled applied-field magnitude. At weak values of this product, solid polarization results in field-cubed deviations from Smoluchowskis velocity; at large values of it, the particle velocity is a slowly increasing function of the applied field, essentially varying with its logarithm. The transition between these two lim...

Osmotic self-propulsion of slender particles

We consider self-diffusiophoresis of axisymmetric particles using the continuum description of Golestanian et al. [“Designing phoretic micro-and nano-swimmers,” New J. Phys. 9, 126 (2007)], where the chemical reaction at the particle boundary is modelled by a prescribed distribution of solute absorption and the interaction of solute molecules with that boundary is represented by diffusio-osmotic slip. With a view towards modelling of needle-like particle shapes, commonly employed in experiments, the self-propulsion problem is analyzed using slender-body theory. For a particle of length 2L, whose boundary is specified by the axial distribution κ(z) of cross-sectional radius, we obtain the approximation −μ2DL∫−LLj(z)dκ(z)dz dz for the particle velocity, wherein j(z) is the solute-flux distribution, μ the diffusio-osmotic slip coefficient, and D the solute diffusivity. This approximation can accommodate discontinuous flux distributions, which are commonly used for describing bimetallic particles; it agrees s...

Asymptotics of surface-plasmon redshift saturation at subnanometric separations

Many promising nanophotonics endeavours hinge upon the unique plasmonic properties of nanometallic structures with narrow non-metallic gaps, which support super-concentrated bonding modes that singularly redshift with decreasing separations. In this letter, we present a descriptive physical picture, complemented by elementary asymptotic formulae, of a nonlocal mechanism for plasmon-redshift saturation at subnanometric gap widths. Thus, by considering the electron-charge and field distributions in the close vicinity of the metal-vacuum interface, we show that nonlocality is asymptotically manifested as an effective potential discontinuity. For bonding modes in the near-contact limit, the latter discontinuity is shown to be effectively equivalent to a widening of the gap. As a consequence, the resonance-frequency near-contact asymptotics are a renormalisation of the corresponding local ones. Specifically, the renormalisation furnishes an asymptotic plasmon-frequency lower bound that scales with the

Singular effective slip length for longitudinal flow over a dense bubble mattress

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

-power of the Fermi wavelength. We demonstrate these remarkable features in the prototypical cases of nanowire and nanosphere dimers, showing agreement between our elementary expressions and previously reported numerical computations.

Waves in Slowly Varying Band-Gap Media

The effective slip length for flow over a mattress of bubbles is considered in the singular limit of close separation of the bubbles. The slip length is shown to diverge inversely with the square root of the solid fraction and three terms in an asymptotic expansion are determined.

Singular perturbations approach to localized surface-plasmon resonance: Nearly touching metal nanospheres

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 Dukhin-number 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 regime, the correction varies as the 3/2-power of the applied field. Our small-Dukhin-number scheme, valid at arbitrary field strengths, naturally lends itself to a tractable analysis of nonlinear surface-conduction effects in numerous electrokinetic problems.

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

This paper is concerned with waves in locally periodic media, in the high-frequency limit where the wavelength is commensurate with the period. A key issue is that the Bloch-dispersion curves vary with the local microstructure, giving rise to hidden singularities associated with band-gap edges and branch crossings. We suggest an asymptotic approach for overcoming this difficulty, which we develop in detail in the case of time-harmonic waves in one dimension. The method entails matching adiabatically propagating Bloch waves, captured by a two-variable Wentzel--Kramers--Brillouin (WKB) approximation, with complementary multiple-scale solutions spatially localized about dispersion singularities. The latter solutions, obtained following the method of high-frequency homogenization (HFH), hold over dynamic length scales intermediate between the periodicity (wavelength) and the macro-scale. In particular, close to a spatial band-gap edge the solution is an Airy function modulated on the short scale by a standing...

Bloch Waves in an Arbitrary Two-Dimensional Lattice of Subwavelength Dirichlet Scatterers

Metallic nano-structures characterised by multiple geometric length scales support low-frequency surface-plasmon modes, which enable strong light localization and field enhancement. We suggest studying such configurations using singular perturbation methods, and demonstrate the efficacy of this approach by considering, in the quasi-static limit, a pair of nearly touching metallic nano-spheres subjected to an incident electromagnetic wave polarized with the electric field along the line of sphere centers. Rather than attempting an exact analytical solution, we construct the pertinent (longitudinal) eigen-modes by matching relatively simple asymptotic expansions valid in overlapping spatial domains. We thereby arrive at an effective boundary eigenvalue problem in a half-space representing the metal region in the vicinity of the gap. Coupling with the gap field gives rise to a mixed-type boundary condition with varying coefficients, whereas coupling with the particle-scale field enters through an integral eigenvalue selection rule involving the electrostatic capacitance of the configuration. By solving the reduced problem we obtain accurate closed-form expressions for the resonance values of the metal dielectric function. Furthermore, together with an energy-like integral relation, the latter eigen-solutions yield also closed-form approximations for the induced-dipole moment and gap-field enhancement under resonance. We demonstrate agreement between the asymptotic formulas and a semi-numerical computation. The analysis, underpinned by asymptotic scaling arguments, elucidates how metal polarization together with geometrical confinement enables a strong plasmon-frequency redshift and amplified near-field at resonance.