Brownian motion near a partial-slip boundary: A local probe of the no-slip condition

Abstract

Motivated by experimental evidence of violations of the no-slip boundary condition for liquid flow in micrometer-scale geometries, we propose a simple, complementary experimental technique that has certain advantages over previous studies. Instead of relying on externally induced flow or probe motion, we suggest that colloidal diffusivity near solid surfaces contains signatures of the degree of fluid slip exhibited on those surfaces. To investigate, we calculate the image system for point forces ͑Stokeslets͒ oriented perpendicular and parallel to a surface with a finite slip length, analogous to Blake's solution for a Stokeslet near a no-slip wall. Notably, the image system for the point source and perpendicular Stokeslet contain the same singularities as Blake's solution; however, each is distributed along a line with a magnitude that decays exponentially over the slip length. The image system for the parallel Stokeslet involves a larger set of fundamental singularities, whose magnitude does not decay exponentially from the surface. Using these image systems, we determine the wall-induced correction to the diffusivity of a small spherical particle located "far" from the wall. We also calculate the coupled diffusivities between multiple particles near a partially slipping wall. Because, in general, the diffusivity depends on "local" wall conditions, patterned surfaces would allow differential measurements to be obtained within a single experimental cell, eliminating potential cell-to-cell variability encountered in previous experiments. In addition to motivating the proposed experiments, our solutions for point forces and sources near a partial-slip wall will be useful for boundary integral calculations in slip systems.

Effect of surface slip on Stokes flow past a spherical particle in infinite fluid and near a plane wall

“…Lauga and Squires [21] determined the force on a no-slip sphere translating normal or parallel to a slip wall, β p = ∞, at large distances. Their analysis for parallel motion furnishes the asymptotic prediction…”

“…Palaniappan and Daripa [18] derived a family of two-dimensional Stokes flows inside a circular cylinder with the slip boundary condition applied. More recently, Elasmi and Feuillebois [19,20] and Lauga and Squires [21] derived the fundamental singularity of Stokes flow in a semi-infinite domain bounded by a plane wall where the slip boundary condition applies, and developed integral and asymptotic solutions.…”

“…Our main goal is to compute resistance coefficients in the case of translation and rotation, and the translational and angular velocities in the case of free motion. The resistance coefficients for translation can be used to compute the diffusivity of small Brownian particles [21]. In the case of infinite flow, we seek a prediction for the effective viscosity of an infinite dilute suspension.…”

Using surface force apparatus, diffusion and velocimetry to measure slip lengths

“…Tracer dynamics is affected by confinement, and this dependence reflects the h.b.c. that apply on both solid substrates (Lauga & Squires 2005;Saugey et al 2005). The results for different wetting properties of the solid substrates lead to measurable differences in the diffusion coefficient, allowing us to deduce the corresponding surface slippage.…”

“…These two quantities allow two independent determinations of the slip length b with an accuracy of 10 nm in both cases. The speed profile straightforwardly provides a direct measurement of b and the variations in the diffusion coefficient are a function of the slip length value due to hydrodynamic coupling of the particle with the surface (Goldman et al 1967;Lauga & Squires 2005). It appears that on the hydrophilic surfaces no slippage was observed: bZ3G7 nm with the speed profile and bZK1G12 nm with the diffusion coefficient profile ( figure 8a,c).…”