Boundaries Control Collective Dynamics of Inertial Self-Propelled Robots

Simple ingredients, such as well-defined interactions and couplings for the velocity and orientation of self-propelled objects, are sufficient to produce complex collective behavior in assemblies of such entities. Here, we use assemblies of rodlike robots made motile through self-vibration. When confined in circular arenas, dilute assemblies of these rods act as a gas. Increasing the surface fraction leads to a collective behavior near the boundaries: polar clusters emerge while, in the bulk, gaslike behavior is retained. The coexistence between a gas and surface clusters is a direct consequence of inertial effects as shown by our simulations. A theoretical model, based on surface mediated transport accounts for this coexistence and illustrates the exact role of the boundaries. Our study paves the way towards the control of collective behavior: By using deformable but free to move arenas, we demonstrate that the surface induced clusters can lead to directed motion, while the topology of the surface states can be controlled by biasing the motility of the particles.

Figure 1

Clustering transition. Left-hand panel: Experiments. Photographs of the arena ($R=30\mathrm{cm}$ with 1 mm thick walls) with different numbers $N$ of robots. Note the gas phase for $N=60$, $\varphi =0.136$, and the presence of a large surface cluster for $N=120$, $\varphi =0.273$. $P(n)$ for the gas phase for which the PDF is monotonic and the denser phase with large clusters for which the PDF is bimodal. The solid line is a fit to the functional form given in the text for the gas phase. Right-hand panel: Simulations. Photographs of the arena. Along with a plot of $P(n)$ for surface fractions $\varphi$ of 0.11 and 0.22. The insets show $P(n)$ calculated for the central region only where it is monotonic and exponential. The Péclet number $\text{Pe}=31$ for the experiments and $\text{Pe}=37$ for the simulations. (The rendering of the images from the simulations is carried out using ovito [33]).

Figure 2

Phase diagram in $\varphi -\text{Pe}$ representation for the experiments and the simulations. For each Pe value, the surface fraction for which a transition from gas to coexistence of gas and clusters is found. The model is given as a continuous line whereby for each Pe the value of $\varphi$ for which a cluster is stable (i.e., $\varphi =(N^{*}A/\pi R^2)$) is found. Inset: The residence time in the cluster needed to fit the phase diagram using the model compared to estimates of this timescale from experiments. Additional points from the simulations are also shown.

Figure 3

Chirality effects: Left: Phase diagram from simulations summarizing the effects of an additional torque $T_0$, giving rise to circular trajectories, on the surface fraction for the transition to surface cluster formation. For these simulations, the value of the friction coefficient is 6. Middle: Photograph of an assembly of left and right turning rods. Left turning rods are marked with blue dots while right turning rods are marked with red dots. The trajectories of the rods are displayed with the direction of movement indicated by increasing color intensity. The cluster at the bottom of the image is segregated by chirality. Right: Spatiotemporal evolution from experiments where left and right turning robots segregate within the cluster as it grows in size.

Figure 4

Deformation of the arena and onset of locomotion of the whole cell. The deformation is small and the locomotion is limited for small $N$. For a higher $N$, the deformation is large and the mobility of the arena is strong with velocities near those of the rods. The deformation is measured as $2(a-b)/p$, where $a$ is the long axis, $b$ the small axis, and $p$ the perimeter of the arena. The velocity of the center of mass of the arena is normalized by the mean velocity of the individual rods. The correlation between deformation and locomotion is near 0.72 for the higher $N$. The arena is made of 0.1 mm thick paper with $R=10.5\mathrm{cm}$. The photographs to the right illustrate that large deformations are not strictly correlated to locomotion as the presence of two clusters on each end balance the arena in a static position.

Figure 5

Collective motion. For a flexible arena made of a boundary with 0.1 mm thick steel walls with $R=21\mathrm{cm}$, the rods reside longer at the surface and form a cluster relatively quickly, deform the arena, and set it in motion to go through an aperture (left series) or go around an obstacle (right series). For each case, a time series is shown.