Brownian Motion of Boomerang Colloidal Particles

We investigate the Brownian motion of boomerang colloidal particles confined between two glass plates. Our experimental observations show that the mean displacements are biased towards the center of hydrodynamic stress (CoH), and that the mean-square displacements exhibit a crossover from short-time faster to long-time slower diffusion with the short-time diffusion coefficients dependent on the points used for tracking. A model based on Langevin theory elucidates that these behaviors are ascribed to the superposition of two diffusive modes: the ellipsoidal motion of the CoH and the rotational motion of the tracking point with respect to the CoH.

Figure 1

(a) SEM image of the boomerang particles fabricated on a silicon wafer. (b) Optical microscopic image and schematics of the coordinate systems. ($x_1-x_2$) is the lab frame and ($X_1-X_2$) is the body frame. (c) Representative trajectories of 5 different total lengths of time, where the red spots represent the positions of the CoH, and the boomerang is colored-coded in time. (d) An exemplary 300 s trajectory for the CoH (green) and the CoB (blue).

Figure 2

(a) MSDs of the CoB in the lab frame vs. $t$. Red line: the best linear fit for $t<10\mathrm{s}$; dark brown line: theoretical fit using Eq. (2). Inset: linear plot of MSDs vs. $t$. (b) MSDs of $\theta$ vs. $t$ with the best linear fitting (red line). (c) MSDs for the CoB in the lab frame with ${\theta }_0=0$. Red lines: theory curves with Eq. (1b). (d) MDs in the lab frame with ${\theta }_0=0$. Red line: theory curve of Eq. (1a) using $D_{\theta }$ and $D_{2\theta }$ obtained from Fig. 2b, 3e.

Figure 3

(a)–(b) MDs vs $t$ in the CBF (a) and DBF (b). The red line in (b) is the theory curve of Eq. (4a). (c)–(d) MSDs vs t in the CBF (c) and DBF (d). In (c), the red lines are the best linear fittings. In (d), the dark brown curve is Eq. (4b) with $D_{\theta }$ and $D_{2\theta }$ obtained from the data in Fig. 2 and the red straight lines have the same slopes as those in (c). (e)–(f) Translation-rotation correlations vs $t$ in the CBF (e) and DBF (f). The red lines are the best linear fitting.

Figure 4

(a) $D_{11}$ and $D_{22}$ vs $d$. Red and green lines are theory curves based on Eqs. (3b, 3c). (b) $D_{1\theta }$ and $D_{2\theta }$ vs $d$. Red and green lines are the theory curves of $D_{2\theta }=r{D}_{\theta }$ and $D_{1\theta }=0$, respectively. The dashed lines indicate the CoH at $d_0=1.16\mu \mathrm{m}$.