Self-propelled rods exhibit a phase-separated state characterized by the presence of active stresses and the ejection of polar clusters

We study collections of self-propelled rods (SPR) moving in two dimensions for packing fractions less than or equal to 0.3. We find that in the thermodynamical limit the SPR undergo a phase transition between a disordered gas and a novel phase-separated system state. Interestingly, (global) orientational order patterns—contrary to what has been suggested—vanish in this limit. In the found novel state, the SPR self-organize into a highly dynamical, high-density, compact region—which we call aggregate—which is surrounded by a disordered gas. Active stresses build inside aggregates as a result of the combined effect of local orientational order and active forces. This leads to the most distinctive feature of these aggregates: constant ejection of polar clusters of SPR. This novel phase-separated state represents a novel state of matter characterized by large fluctuations in volume and shape, related to mass ejection, and exhibits positional as well as orientational local order. SPR systems display new physics unseen in other active matter systems.

Figure 1

Transition from disorder to order and phase separation at small system sizes. Top, from left to right: polar-order parameter $S_1$, nematic-order parameter $S_2$, and average cluster size $\langle m\rangle$ over system size $N$, respectively, as function of the particle aspect ratio $\kappa$ for various packing fractions $\eta$. $N=10000$. Bottom: A simulation snapshot for $\kappa =10$ and $\eta =0.3$, with $S_1=0.5,S_2=0.26,\langle m\rangle /N=0.14$. Notice that clusters are polar. The orientation of rods is color coded as indicated in the bottom left panel. This convention is also used in Figs. 4 and 5.

Figure 2

Interactions in the SPR model. Top: Sketch of two interacting rods in the SPR model. Forces and torques resulting from the interaction are shown in the top-right panel. See text for explanations. Bottom: Chronological snapshots of a collision between two rods. Notice that even though the interaction is exclusively repulsive, it leads to an effective velocity alignment (an effective attraction). A movie of this interaction is provided in Ref. [43].

Figure 3

Finite-size scaling of the polar-order parameter $S_1$ and average cluster size with respect to system size $\langle m\rangle /N$ for aspect ratio $\kappa =10$ and several packing fractions $\eta$. Notice that, for $\eta \ge 0.15$, the system becomes disordered as the system size $N$ is increased, while remaining phase separated.

Figure 4

The orientationally ordered phase-separated state observed for $\kappa >{\kappa }_c$ in small systems $\left(N\ll N_{*}\right)$ becomes unstable when $N$ is increased. In very big systems $\left(N\gg N_{*}\right)$ we observe only aggregates (which correspond to an orientationally disordered phase-separated state). Top left: Evolution of $T_{\text{agg}}/T_{\text{tot}}$ with $N$, where $T_{\text{agg}}$ is the time the system spent in the aggregate phase and $T_{\text{tot}}$—here, $T_{\text{tot}}={10}^7$—is the total simulation time. Second line, left: the polar order $S_1$ as function of time for $N\sim N_{*}$—here, $N_{*}\sim {10}^4$. Large values of $S_1$ correspond to the system being highly ordered, typically due to the formation of a highly ordered band (panel i), while low values of $S_1$ correspond to the formation of an aggregate (panel ii). Third line: Histograms of the polar order parameter $p\left(S_1\right)$ for $N=5000,N=10000\sim N_{*}$ and $N=20000$. Bottom panel: Total elastic energy $U_{\text{tot}}={\sum }_{i=1}^N{U}_i$ of the system as function of time $t$, for $N\sim N_{*}$. High values of $U_{\text{tot}}$ correspond to the presence of an aggregate (inset on the left panel), while low values of $U_{\text{tot}}$ are related to the presence of polar structures such as bands. On both snapshots, the color of each rod indicates its interaction potential $U_i$: blue (red) color indicates small (large) $U_i$ values. The dashed black circle in the snapshot corresponding to an aggregate provides an idea of the boundary shell of the aggregate. For movies, see Supplemental Material [43]. Simulations correspond to $\kappa =4$ and $\eta =0.3$.

Figure 5

Dynamics of an aggregate. Top panel: aggregate size and gas density as function of time. The aggregate boundary exhibits large fluctuations due to the emergence of orientational defects that lead to the detachment of large polar clusters from the aggregate. Second line: The three panels display in chronological order one of these events. The corresponding time window is indicated by the vertical gray area in the top left panel. Dashed circles indicate the detachment of polar clusters. The inset shows that topological defect. Bottom panel: Orientational and positional order within in the aggregates. Left: $p\left(d\right)$ is the probability density that the distance between the centers of two rods is $d$ (in units of $\ell$). We observe peaks at multiples of the rod length $\ell$ (the smaller peaks between 0 and 1 correspond to $w$). This is the fingerprint of positional (smectic-like) order: the rods are arranged as on a lattice. Inset: Zoom on a subdomain of the aggregate. Colors encode rod orientation to show the domains of similarly orientated rods. The arrows indicate the local orientation of the rods. Simulations of the aggregate dynamics correspond to $N=80000,\eta =0.15$, and $\kappa =7$ ($N=10000,\eta =0.3$, and $\kappa =4$ for the aggregate in the bottom panel). For movies, see Supplemental Material [43].

Figure 6

Exploration of the parameter space (phase diagram). Each squared point in the plots corresponds to a parameter set $\{N,\eta ,\kappa \}$ for which we have performed simulations. The color of a point encodes the observed state of active matter. Black: disordered, not phase-separated state (gas). Red: ordered phase-separated state (polar clusters or polar band). Green: disordered phase-separated state (aggregate). Blue: unstable situation, where the system oscillates between the ordered phase-separated state and the disordered phase-separated state. The color code for the regions of the phase diagram is the same as for the simulation points.

Figure 7

Left: cluster-size distributions $p\left(m\right)$ and the corresponding polar-order-within-clusters distributions $S_{1,m}$ (i.e., the polar order within clusters as function of their size $m$). Right: Global polar order parameter effectively observed $S_1$ and global polar order parameter obtained under the hypothesis that cluster orientations are fully decorrelated $S_{1,\text{uncorrelated}}$. Simulations correspond to $\eta =0.15,\kappa =10$ (and $N=20000$ for the $S_{1,m}$ distribution, but this distribution is very similar for the other values of $N$).