Biased Brownian motion in narrow channels with asymmetry and anisotropy

We study Brownian motion of a single millimeter size bead confined in a quasi-two-dimensional horizontal channel with built-in anisotropy and asymmetry. Channel asymmetry is implemented by ratchet walls while anisotropy is introduced using a channel base that is grooved along the channel axis so that a bead can acquire a horizontal impulse perpendicular to the longitudinal direction when it collides with the base. When energy is injected to the channel by vertical vibration, the combination of asymmetric walls and anisotropic base induces an effective force which drives the bead into biased diffusive motion along the channel axis with diffusivity and drift velocity increase with vibration strength. The magnitude of this driving force, which can be measured in experiments on a tilted channel, is found to be consistent with those obtained from dynamic mobility and position probability distribution measurements. These results are explained by a simple collision model that suggests the random kinetic energy transfer between different translational degrees of freedom may be turned into useful work in the presence of asymmetry and anisotropy.

Figure 1

(a) Schematic diagram of experimental setup. (b) Schematic of a branch. (c) The shape and dimension of the grooves on the channel base. (d) Position distribution of the beads recorded at $f=60$ Hz and $A=1$ mm. (e) Positions of bead along the channel in 160 s duration every 0.2 s. The vertical color bar at the right shows the time in seconds.

Figure 2

Probability density of the bead $P_b\left(x\right)$ in the four branches at $A=1$ mm, $f=60$ Hz. Inset shows the probability density $P_c\left(x\right)$ of the cells in branch 1. The dash line in the inset is the expected $P_c\left(x\right)$ if the bead positions are uniformly distributed within the cell.

Figure 3

Velocity distribution of $u_x$ and $u_y$ at vibration condition $A=1$ mm, $f=60$ Hz. The solid line is a Gaussian fit for $P\left(u_y\right)$. Inset shows that thermal speed $u_{T_y}$ increases with increasing vibration strength $V_s$.

Figure 4

(a) The time dependent of the displacement $x$ (thick black line) and the fluctuation $\delta x$ (thin red line). (b) Variation of the drift velocity with thermal speed $u_{T_y}$. (c) MSD of $\delta x$. (d) Diffusion constant versus $u_{T_y}$.

Figure 5

Reduced probability density $P_b\left(x\right)$ in different branches measured at $A=1$ mm, $f=40$ Hz, and inclined angle $\varphi =1.5^{\circ }$. Inset shows that $\mathrm{\Delta }logP$ (defined in text) increases almost linearly with $sin\varphi$; the solid line is a linear fit with horizontal intercept = $-2.27\times {10}^{-2}\pm 14\%$.

Figure 6

(a) Schematic diagram of a bead colliding with a wall making an angle $\theta$ with the channel axis. (b) Ratio between drift velocity and diffusivity $v_d/D$ versus thermal speed $u_{T_y}$. Values of $\mathrm{\Delta }logP/L$ are obtained from position density measurements under different vibration conditions, and $F_d/T_g$ is that calculated using Eq. (5).