Non-Gaussian statistics for the motion of self-propelled Janus particles: Experiment versus theory

Spherical Janus particles are one of the most prominent examples for active Brownian objects. Here, we study the diffusiophoretic motion of such microswimmers in experiment and in theory. Three stages are found: simple Brownian motion at short times, superdiffusion at intermediate times, and finally diffusive behavior again at long times. These three regimes observed in the experiments are compared with a theoretical model for the Langevin dynamics of self-propelled particles with coupled translational and rotational motion. Besides the mean square displacement also higher displacement moments are addressed. In particular, theoretical predictions regarding the non-Gaussian behavior of self-propelled particles are verified in the experiments. Furthermore, the full displacement probability distribution is analyzed, where in agreement with Brownian dynamics simulations either an extremely broadened peak or a pronounced double-peak structure is found, depending on the experimental conditions.

Figure 1

Schematic of the particle motion for two subsequent time steps and definition of several parameters used for its characterization. The translational motion is determined by the displacements $\Delta x$ and $\Delta y$ of the center-of-mass postion of the particle. The orientation vector $\widehat{\mathbf{u}}=(\sin \theta \cos \phi ,\sin \theta \sin \phi ,\cos \theta )$ coincides with the direction of self-propulsion. Note that $\theta ={90}^{\circ }$ in the figure for the sake of clarity. While $\theta$ and $\phi$ define the particle orientation, $\vartheta$ is the angle between the directions of subsequent displacements. Due to the combination of Brownian motion and self-propulsion, $\widehat{\mathbf{u}}$ is not necessarily collinear with the displacement direction.

Figure 2

Double logarithmic plot of the experimental results for the MSD of Janus particles with diameter $d_1=2.08\mu \mathrm{m}$ in water and in H${}_2$O${}_2$ solutions with different concentrations as a function of the scaled time $\tau =D_{\mathrm{r}}t$. Various regimes of motion are identified. Dashed and solid curves refer to different measurements for the same H${}_2$O${}_2$ concentrations. Left inset: visualization of the data in a linear plot. Right inset: experimental results for smaller particles with diameter $d_2=0.96\mu \mathrm{m}$ in water and in H${}_2$O${}_2$ solutions with concentrations of 2.5% and 5%.

Figure 3

Comparison of the measured MSD (symbols) with the theoretical prediction (solid curves). The fitting parameter $a$ is given in Table . Dashed lines indicate the transition times ${\tau }_1\left(a\right)=18/a^2$ and ${\tau }_2=1/2$ between the different regimes of motion.

Figure 4

(a) Experimental and (b) theoretical results for the excess kurtosis $\gamma$. The theoretical curves are calculated for the values of the self-propulsion force extracted from the MSD fits in Fig. 3 (see Table ). Inset in (b): Visualization of the theoretical data in a linear plot.

Figure 5

Time evolution of $\Psi (\Delta x,t)$: (a) experimental results for $5\%$ H${}_2$O${}_2$ concentration; (b) corresponding simulation for $a=62$.

Figure 6

Time evolution of $\Psi (\vartheta ,t)$: (a)–(c) experimental data for (a) water, (b) 2.5%, and (c) 5% H${}_2$O${}_2$ concentration. (d) Simulation results for $a=62$.

Figure 7

(a) Time evolution of the measured probability distribution $\Psi (\Delta x,t)$ for Janus particles in a $10\%$ H${}_2$O${}_2$ solution. The occurrence of the double peak indicates that the particle orientation does not diffuse freely on a unit sphere for high H${}_2$O${}_2$ concentrations. (b) Reference simulation for particles whose orientation is restricted to the $x$-$y$ plane.

Figure 8

Experimental results for $\Psi (\Delta x,t)$ after $2\mathrm{s}$ for different H${}_2$O${}_2$ concentrations. The inset shows the Gaussian distribution measured in water.

Figure 9

SEM images of the silica particles with diameters (a) $d_1=2.08\pm 0.05\mu \mathrm{m}$ and (b) $d_2=0.96\pm 0.03\mu \mathrm{m}$.

Figure 10

Image preprocessing with the program imagej: (a) the original image directly obtained by video microscopy from the experiments, (b) image after using the find edge function, and (c) image after using the Gaussian blur function.