Dispersion of Self-Propelled Rods Undergoing Fluctuation-Driven Flips

Synthetic microswimmers may someday perform medical and technological tasks, but predicting their motion and dispersion is challenging. Here we show that chemically propelled rods tend to move on a surface along large circles but curiously show stochastic changes in the sign of the orbit curvature. By accounting for fluctuation-driven flipping of slightly curved rods, we obtain analytical predictions for the ensemble behavior in good agreement with our experiments. This shows that minor defects in swimmer shape can yield major long-term effects on macroscopic dispersion.

Figure 1

Structure and trajectory patterns of Au-Pt rods. (a) Scanning electron microscopy image of rods (Zeiss Merlin scanning electron microscope). The inset shows the presence of Au and Pt segments confirmed by energy-dispersive x-ray spectroscopy. (b) The rods are not perfectly axisymmetric in shape. Representative trajectories of (c) slow rods (average speed $U\sim 8\mu \mathrm{m}{\mathrm{s}}^{-1}$) and (d) fast rods ($U\sim 39\mu \mathrm{m}{\mathrm{s}}^{-1}$) tracked over 15 seconds.

Figure 2

Turning and flipping rods. (a) Turning number, defined as the cumulative sum of the path curvature divided by $2\pi$, plotted for 20 fast rods (average speed $\sim 38\mu \mathrm{m}{\mathrm{s}}^{-1}$). Open circles represent flips, defined to have occurred when the sign of the path curvature switches and then stays the same for at least three frames. (b) Frequency of flips plotted against $|\kappa |$, estimated for each rod by averaging the absolute change in total curvature divided by the distance traveled between flips. The solid lines are given by Eq. (1) with $h_0=5$, 10, 15 nm from left to right, respectively. (c) Sketch of curved rods flipping between CW and CCW orbits. (d) A straight rod moves along a series of straight paths. (e) A slightly curved rod turns and flips. (f) A highly curved rod turns without flipping.

Figure 3

Measured and predicted changes in the direction of self-propelled rods. (a) Correlation in the direction of three sets of rods moving at different speeds $U$. Experimental data (symbols) have error bars representing lower and upper quartiles. Corresponding predictions (lines) are given by Eq. (6). (b) The persistence time and its error bars decrease with speed in our experiments (circles), simulations (crosses), and theory (solid line).

Figure 4

Measured and predicted diffusion of rods. (a) MSD of rods moving at various speeds in six different experiments (symbols). Predictions (lines) are given by Eq. (7). (b) The diffusivity saturates with increasing speed in our experiments (circles), simulations (crosses), and theory (solid line). Theoretical curves are given by Eq. (8) with ${\kappa }_0=0$ for no curvature and $f=0$ for no flipping.

Figure 5

Long-time swimmer concentrations $\overline{C}$ (symbols and solid lines) in chemical fuel gradients. Swimming speeds $U(x)$ (blue dashed lines) are prescribed and range from (a) 0 to 40 or (b) 40 to $80\mu \mathrm{m}{\mathrm{s}}^{-1}$. Symbols show simulations of straight and curved rods 30 minutes after initiating with uniform distribution ${\overline{C}}_0$. Solid lines are corresponding numerical solutions to the steady form of Eq. (5).