Spontaneously Flowing Crystal of Self-Propelled Particles

We experimentally and numerically study the structure and dynamics of a monodisperse packing of spontaneously aligning self-propelled hard disks. The packings are such that their equilibrium counterparts form perfectly ordered hexagonal structures. Experimentally, we first form a perfect crystal in a hexagonal arena which respects the same crystalline symmetry. Frustration of the hexagonal order, obtained by removing a few particles, leads to the formation of a rapidly diffusing “droplet.” Removing more particles, the whole system spontaneously forms a macroscopic sheared flow, while conserving an overall crystalline structure. This flowing crystalline structure, which we call a “rheocrystal,” is made possible by the condensation of shear along localized stacking faults. Numerical simulations very well reproduce the experimental observations and allow us to explore the parameter space. They demonstrate that the rheocrystal is induced neither by frustration nor by noise. They further show that larger systems flow faster while still remaining ordered.

Figure 1

Flowing crystal configurations in the experiment at a finite noise level with geometrical frustration ($N=1104$, $\varphi =0.86$), and in simulations at successive times, without noise and without geometrical frustration. The gray levels code for the orientational order parameter; the colors blue and red code for five and seven neighbors, respectively ($N=1141$, $\varphi =0.88$).

Figure 2

Experimental data, static aspects: The left panels show time snapshots of isotropic disks (ISO, top row) and of self-propelled polar disks (SPP, bottom row). The gray scale indicates individual values of the local orientational order parameter ${\psi }_6\in [0,1]$, except when there are more or less six Voronoi neighbors (red and blue, respectively). The rightmost panels show the corresponding pair correlation functions, obtained from time averages over trajectories.

Figure 3

Experimental data, SPP dynamics: The panels on the left show typical trajectories for different $N$. Plotted are the perpendicular and (unwrapped) parallel components of the particle position [33]. The panels on the right show the corresponding mean squared displacements.

Figure 4

Simulation data: Variation of $N$ (top row) versus variation of the arena area $A$ (bottom row). Shown are mean squared displacements obtained from pairs $(N,A)$ with the common (bulk) volume fractions $\varphi =0.846$, 0.878, and 0.890.

Figure 5

Simulation data from systems of different sizes. Left: Pair-correlation functions. Right: Average displacement per unit time vs distance from the center rescaled by the system size ($A=1027.23$, $\varphi =0.878$, $\mathrm{\Delta }t=256$).

Figure 6

Simulation data, exploring the parameter space in the $(\alpha ,D)$ and $(\varphi ,D)$ planes. The color codes for coincidence of individual orientation vectors with the orientations of global flow, first averaged over the particles, then over time. Noise levels are scaled by the free-flight time ${\tau }_{\mathrm{ff}}(\varphi )$, as measured at equilibrium. The curves in the left panel are the disorder–polar transition lines in the dilute limit, taken from Ref. [31], but scaled by 0.2 for reasons of visual presentation.