Drift without flux: Brownian walker with a space-dependent diffusion coefficient

Abstract

Space dependent diffusion of micrometer sized particles has been directly observed using digital video microscopy. The particles were trapped between two nearly parallel walls making their confinement position dependent. Consequently, not only did we measure a diffusion coefficient which depended on the particles' position, but also report and explain a new effect: a drift of the particles' individual positions in the direction of the diffusion coefficient gradient, in the absence of any external force or concentration gradient. PACS number(s): 05.40. Jc, 82.70.Dd, 67.40.Hf (February 1, 2008) Brownian motion of spherical colloidal particles in the vicinity of a wall has been extensively studied, both theoretically and experimentally . It has been shown that the diffusion coefficients parallel or perpendicular to the wall were greatly reduced when the particles were close enough to the obstacle, i.e. within distances comparable to or less than their radius [1]. When the Brownian particles are trapped in a more confined geometry, such as a porous medium, the theory is far more complicated and few experimental studies have been reported in model geometries, where the particles are trapped between two parallel walls [2,3]. In this article, we report some new experimental results concerning the Brownian motion of particles trapped between two nearly parallel walls, so that the confinement, and thus the diffusion coefficient, become space dependent. As a result, we not only measure a diffusion coefficient which varies with the confinement, but also a drift of the particules' individual positions in the direction of the diffusion coefficient gradient, in the absence of any external force or concentration gradient. This drift was not accompanied by any net particle flux, i.e. statistically the same number of particles crossed any imaginary surface in both directions. We first discuss the general problem of a Brownian walker with a spatially dependent diffusion coefficient to explain the origin of the expected drift, and then present the experimental set-up and results.As in our experiment the diffusion coefficient varies in only one direction, say x, we briefly sketch a heuristic derivation of the 1D Brownian walker algorithm. The velocity of a 1D Brownian particle subjected to a random force and a viscous drag follows the Langevin equation,where γ −1 is the velocity relaxation time and Γ(t) the random force per unit mass defined by its mean value Γ(t) = 0 and correlation function Γ(t)Γ(t ′ ) = qδ(t − t ′ ). Using the equipartition theorem it can be shown that q is related to the temperature T and the particle's mass, m, by the standard relation q = 2γkT /m. Discretizing the random function Γ(t) over time intervals ∆t >> γ −1 allows us to drop in Eq.(1) the inertial term, dv/dt, and to replace the velocity v by ∆x/∆t. Choosing for Γ(t) the simplest random function, Γ(t) = ± q/∆t, leads to the well known Brownian walker algorithm,with D = kT /mγ. When the diffusion coefficient D, i.e. when the temperature T and/or t...

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