Driven Anisotropic Diffusion at Boundaries: Noise Rectification and Particle Sorting

We study the diffusive dynamics of a Brownian particle in the proximity of a flat surface under nonequilibrium conditions, which are created by an anisotropic thermal environment with different temperatures being active along distinct spatial directions. By presenting the exact time-dependent solution of the Fokker-Planck equation for this problem, we demonstrate that the interplay between anisotropic diffusion and hard-core interaction with the plain wall rectifies the thermal fluctuations and induces directed particle transport parallel to the surface, without any deterministic forces being applied in that direction. Based on current micromanipulation technologies, we suggest a concrete experimental setup to observe this novel noise-induced transport mechanism. We furthermore show that it is sensitive to particle characteristics, such that this setup can be used for sorting particles of different sizes.

Figure 1

(a1) and (a2) Illustration of the setup. A Brownian colloid diffuses in proximity to a two-dimensional surface, the $x_1\text{−}{x}_2$ plane at $x_3=0$ (illustrated by the gray areas). Its average, long-term velocity is indicated by the blue arrow. The synthetic noise is applied along the ${\mathit{e}}_{\sigma }$ direction in the $x_2\text{−}{x}_3$ plane (red bar). The external force $\mathit{f}$ points towards the surface (gray arrow). In (a1), the full three-dimensional setup is sketched; (a2) shows the $x_2\text{−}{x}_3$ plane in which the synthetic noise is applied. (b) Two different spherical particles move along opposite directions on the surface; solid orange curve: Brownian sphere with radius $0.25\mu \mathrm{m}$; dashed green curve: Brownian sphere with radius $0.4\mu \mathrm{m}$. Shown are the probability densities $p(x_2)={\int }_{-\infty }^{+\infty }d{x}_1{\int }_0^{+\infty }d{x}_{3}p(x_1,x_2,x_3)$ for the two particles, obtained from the exact solution (4) at time $t=1200\mathrm{s}$. (c) Net particle velocity $⟨{\dot{x}}_2⟩$, (12b), as a function of the particle radius $a$. The two particle sizes from (b) are marked by corresponding solid orange and dashed green lines. Parameters used in (b) and (c): $T=300\mathrm{K}$ (room temperature), $T_{\mathrm{kin}}=T+{\sigma }^2/(2k_B{\gamma }_0)=1000\mathrm{K}$ (synthetic kinetic temperature [13], specified for a reference spherical particle with radius $0.5\mu \mathrm{m}$ and Stokes friction coefficient ${\gamma }_0$), $\theta =0.9\pi$, $\mathit{f}=(0\text{fN},7.5\text{fN},-9\text{fN})$. The friction coefficients of the spherical particles have been calculated—as an approximation—for an unbounded fluid, i.e., $\tilde{\gamma }=6\pi \nu a$, where $\nu$ is the viscosity of water at room temperature.