Thermal and athermal three-dimensional swarms of self-propelled particles

Swarms of self-propelled particles exhibit complex behavior that can arise from simple models, with large changes in swarm behavior resulting from small changes in model parameters. We investigate the steady-state swarms formed by self-propelled Morse particles in three dimensions using molecular dynamics simulations optimized for graphics processing units. We find a variety of swarms of different overall shape assemble spontaneously and that for certain Morse potential parameters at most two competing structures are observed. We report a rich “phase diagram” of athermal swarm structures observed across a broad range of interaction parameters. Unlike the structures formed in equilibrium self-assembly, we find that the probability of forming a self-propelled swarm can be biased by the choice of initial conditions. We investigate how thermal noise influences swarm formation and demonstrate ways it can be exploited to reconfigure one swarm into another. Our findings validate and extend previous observations of self-propelled Morse swarms and highlight open questions for predictive theories of nonequilibrium self-assembly.

Figure 1

Three-dimensional self-propelled swarms of $N=600$ Morse particles at $T^{*}=0$, $\alpha =2$, $\gamma =1$, and $\beta =0.5$. Steady-state swarms are observed in MD simulations after $2\times {10}^6$ time steps have evolved for particles within a $20\sigma$ simulation box with periodic boundary conditions. Particles are initialized with random velocities and are randomly distributed within a $5\sigma$ cubic region inside the simulation box. Blue arrows indicate particle velocities. Orange spheres represent particle positions, but the sphere diameters to not correspond to particle size. Scale bars are shown in the upper right and alignment order $A$ is shown in the lower left of each swarm. Swarm radii range from $0.06\sigma$ to $0.3\sigma$ (ball). (a) Ball ($C=1.5$, $l=0.5$) composed of concentric spherical layers of particles and identical velocities. (b) Stationary hollow shell ($C=0.5$, $l=0.5$) composed of particles that travel in circular orbits. (c) Stationary ring ($C=0.5$, $l=0.5$) with all particles traveling in the same circular orbit. (d) Stationary spherical swarm ($C=0.6$, $l=0.5$) with clumps of particles that travel in circular orbits. (e) Stationary cylindrical swarm ($C=0.6$, $l=0.5$) composed of clumps of particles traveling about the same circular orbit. The particles within the clumps move in a cylindrical fashion, but follow a complex path such that the cross-sectional shape of the cylinder appears to be time dependent. (f) Torus ($C=0.6$, $l=0.3$) composed of particles that travel in circular orbits of different radii, but share an axis of rotation. Particle paths within the torus span the entire volume: A particle that is near the hole eventually moves to the outer part and back again. See Supplemental Material videos 1–6 [32] for movies of these swarms.

Figure 2

“Phase diagram” for 3D stable structures assembled at values of $0.1⩽l⩽2.0$ and $0.1⩽C⩽2.0$, $T^{*}=0$, $\alpha =2$, $\gamma =1$, and $\beta =0.5$. The ratio of repulsion to attraction strength $C=\frac{C_r}{C_a}$ and the ratio of repulsion to attraction length scales $l=\frac{l_r}{l_a}$ determine the shape of the interaction potential and the swarms observed at steady state. In the upper right quadrant ($l>1$, $C>1$) repulsion dominates the potential, preventing organized clusters from assembling. In the lower right quadrant ($l<1$, $C>1$) the repulsion is short-ranged and the attraction is long-ranged, resulting in translating spherical balls. In the lower left quadrant ($l<1$, $C<1$) attraction dominates the potential and the phase diagram is most complex, with shells, rings, tori, balls, and clumps all stabilized within narrow ranges of $l$ and $C$. In the top left quadrant ($l>1$, $C<1$) particles are attractive, but the repulsive length scale dominates, giving rise to hollow shells.

Figure 3

Four initial conditions and their probabilities of transitioning to a ring. The shell and cylinder initial conditions have radii of 0.062. The ball has a radius of 1.35, and the random configuration is distributed randomly in cubic box with side length of 5. Ring transition probabilities are calculated from 100 independent runs performed for each of the indicated initial condition with $T^{*}=0$, $\alpha =1$, $\gamma =1$, $\beta =0.5$, $C_r=0.5$, $l_r=0.5$, $C_a=1$, $l_a=1$, and $N=600$ with random initial velocities.

Figure 4

Probability of a random shell transforming into a ring as a function of alignment order $A$ [Eq. (7)] at $T^{*}=0$, $\alpha =1$, $\gamma =1$, $\beta =0.5$, $C_r=0.5$, $l_r=0.5$, $C_a=1$, $l_a=1$, and $N=600$. The blue line is drawn as a guide to the eye. Inset are representative alignment order trajectories for swarms that form a shell (red triangles) and ring (blue circles).

Figure 5

Initial time evolution of the alignment order $A$ averaged over 10 independent runs for six different combinations of initial arrangements and initial velocities. The particles are spatially initialized either randomly within a $5\sigma$ cube, within a $2.7\sigma$ ball, or on a $0.062\sigma$ shell within a $20\sigma$ cubic simulation box with periodic boundary conditions. Velocities are initialized from a Gaussian distribution with mean 2 and standard deviation 1 (random $\vec{v}$) or drawn from the ring velocity distribution with mean 2, standard deviation 0.15, and $z$ component $=$ 0 (parallel $\vec{v}$). The number of observed rings after $1\times {10}^6$ time steps are shown in parentheses. Error bars are plotted every 2000 steps and denote one standard deviation.

Figure 6

Probability of shell [red (Low curve at small values of noise)] and ring [blue (Upper curve at small values of noise)] formation as a function of $T^{*}$ at $\alpha =2$, $\gamma =1$, $\beta =0.5$, $C_r=0.5$, $l_r=0.5$, $C_a=1$, $l_a=1$, and $N=600$. 100 independent runs are performed at each noise level, with particles initially randomized in a $5\sigma$ cube within a $20\sigma$ cubic simulation box and are evolved for $1\times {10}^6$ time steps. Error bars denote one standard deviation.

Figure 7

Normalized cylinder height $\Lambda$ as a function of dimensionless temperature $T^{*}$ and number of particles $N$. Particles are initialized as a ring with random velocities. $\alpha =2$, $\gamma =1$, $\beta =0.5$, $C_r=0.5$, $l_r=0.5$, $C_a=1$, and $l_a=1$. $\Lambda$ are averaged over 10 simulation snapshots and over 10 independent simulation runs, where the snapshots are the last 10 at $1\times {10}^4$ time step increments. The standard deviations of $\Lambda$ are less than $3\%$ of the mean values and are not shown.

Figure 8

Mechanisms of structural transitions among shells, rings, and cylinders. Increasing the temperature of a ring causes it to transition into a cylinder of increasing height. At large enough $T^{*}$, cylinders transition into hollow shells. The transformation of a hollow shell into a cylinder or ring requires not only that the temperature be decreased below $T^{*}=0.007$, but also that an instantaneous fluctuation in alignment order is sufficiently large to initiate the transition. The transition involves the shell polarizing into cylindrical bands, which align eventually turning into a cylinder with noise-dependent height or a ring if the noise is zero.