Arrested states in persistent active matter: Gelation without attraction

We explore phase separation and kinetic arrest in a model active colloidal system consisting of self-propelled, hard-core particles with nonconvex shapes. The passive limit of the model, namely cross-shaped particles on a square lattice, exhibits a first-order transition from a fluid phase to a solid phase with increasing density. Quenches into the two-phase coexistence region exhibit an aging regime. The nonconvex shape of the particles eases jamming in the passive system and leads to strong inhibition of rotations of the active particles. Using numerical simulations and analytical modeling, we quantify the nonequilibrium phase behavior as a function of density and activity. If we view activity as the analog of attraction strength, the phase diagram exhibits strong similarities to that of attractive colloids, exhibiting both aging, glassy states and gel-like arrested states. The two types of dynamically arrested states, glasses and gels, are distinguished by the appearance of density heterogenities in the latter. In the infinitely persistent limit, we show that a coarse-grained model based on the asymmetric exclusion process quantitatively predicts the density profiles of the gel states. The predictions remain qualitatively valid for finite rotation rates. Using these results, we classify the activity-driven phases and identify the boundaries separating them.

Figure 1

[(a)–(f)] Snapshots of the system obtained at the end of simulations runs, $t_{\max }=2\times {10}^6$. The color bar represents the stationary time for each particle on log scale, ${log}_{10}({\tau }_i\left(t_{max}\right))$ (see Sec. 3). Unoccupied lattice sites are colored white. Particles that have not taken a single step through the duration of the simulation are colored darkest. The phases we find by varying the density $\rho$ and activity $\mathrm{\Delta }v$ are (a) steady-state passive fluid, (b) passive fluid-solid/aging glass states, (c) finite-activity states resembling the aging regime of the passive system, and [(d)–(f)] active phase with void-solid coexistence (see Sec. 5). There is a progression from majority nonarrested states (d) to majority arrested states [(e), (f)] with increasing density. Panel (g) displays a color map of the fraction of arrested states (see Sec. 4). Locations of the snapshots shown in panels (a)–(f) are marked on the phase diagram with pink circles. ${\rho }_{\mathrm{rcp}}$ in the figure denotes the random-close-packing density of hard crosses: the maximal density that can be reached via the RSAD process [31]. The morphology exhibited by the snapshots at high densities [(e), (f)] are reminiscent of high-density gels in attractive colloids, while those in panels (c) and (d) resemble the gel-bubble states at low densities [32].

Figure 2

(a) Nearest-neighbor labels and hopping rates for our active hard-cross model on a square lattice. The particles can perform thermal moves in any of the four directions with a rate $v_0$ (black arrows), and active moves with a rate $v_0+\mathrm{\Delta }v$ along the direction of their orientation (red arrow). (b) The cross at the central site (green) is prevented from rotating by the presence of neighboring crosses occupying the fourth and fifth nearest-neighbor sites.

Figure 3

Distributions of stationary times, $P\left({log}_{10}{\tau }_i\right)$. In the passive system $\left(\mathrm{\Delta }v=0\right)$, a signature of the crossover from the steady-state passive liquid to an aging glass regime is the appearance of a bimodal distribution, indicating two populations of particles with mobilities that differ by several orders of magnitude. This bimodal distribution appears at lower densities when persistent active motion is introduced. The different figures represent (a) passive crosses $\mathrm{\Delta }v=0$ and (b) the transition to an “active glass” with increasing density at a small activity $\mathrm{\Delta }v=0.025$. The transition at a fixed density (c) $\rho =0.14$ and (d) $\rho =0.16$ with increasing activity.

Figure 4

Time series of the ensemble averaged stationary time variance, $\langle Q\left(t\right)\rangle$, for the same data sets as in Fig. 3. (a) Passive system: $\langle Q\left(t\right)\rangle$ increases with time for $\rho \ge 0.1650$, indicating the emergence of aging states. (b) $\mathrm{\Delta }v=0.025$: $\langle Q\left(t\right)\rangle$ increases with time at densities $\rho \ge 0.1600$. (c) $\rho =0.1400$: $\langle Q\left(t\right)\rangle$ increases with time for $\mathrm{\Delta }v\ge 0.050$. (d) $\rho =0.1600$: $\langle Q\left(t\right)\rangle$ increases with time for $\mathrm{\Delta }v\ge 0.025$. The onset time of active dynamics, ${\tau }_{\mathrm{active}}$, is marked by black arrows in panels (b)–(d).

Figure 5

Mean-squared displacements (MSD) $R^2\left(t\right)$ (insets) and their logarithmic derivatives $\gamma \left(t\right)$ (main panels) at (a) $\rho =0.14,\mathrm{\Delta }v=0.2$, (b) $\rho =0.13,\mathrm{\Delta }v=0.07$, (c) $\rho =0.15,\mathrm{\Delta }v=0.07$, and (d) $\rho =0.12,\mathrm{\Delta }v=0.05$. Blue curves show “arrested” states $\left(\gamma <0.5\right)$, while red curves show nonarrested states. The ensemble average $\langle \gamma \left(t\right)\rangle$ is shown in black, and the mean-field prediction for a single active tracer is shown in green. The single tracer prediction crosses over from diffusive $\left(\gamma =1\right)$ to ballistic $\left(\gamma =2\right)$ at a characteristic time $t^{*}$ (black arrows), which we estimate from $\gamma \left(t^{*}\right)=3/2$.

Figure 6

Contour maps of the coarse-grained density profile of arrested states at the longest simulation time $t_{\max }=2\times {10}^6$. We used a coarse-graining box size with area $L^2/900$, so that each box encloses ${15}^2$ lattice sites. The color bar represents the local density which varies from $0\le \rho \le 0.2$. The arrows represent the gradient of the density field, and the lowest density point for each figure has been shifted to the center of the frame (making use of the periodic boundary conditions). Rows show fixed activity $\mathrm{\Delta }v=0.20,0.09,0.05$, and columns show densities increasing left to right $\rho =0.13,0.14,0.15,0.16$. Density fluctuations for a typical nonarrested active liquid state $(\rho =0.12,\mathrm{\Delta }v=0.040)$ are shown in the lower left corner. As the density is increased at a fixed activity, the width of the solid phase increases, and the active liquid fills the void region until it disappears. We observe that the width of the interface region is controlled solely by the activity $\mathrm{\Delta }v$.

Figure 7

One-dimensional linear density profiles illustrating the nonequilibrium phase classification based on the observed configurations and local density distributions. (a) A dense solid region along with an active liquid interface which fills the remaining space available in the system (see, for example, bottom row in Fig. 6). (b) Solid region, bordered by a narrow active liquid (AL) interface of total width $\xi$. The remaining area in the system is left empty of particles, creating a void (see, for example, top two rows in Fig. 6).

Figure 8

The divergence of the length scale of the “wetting active liquid” as the activity is decreased along with the theoretical prediction in Eq. (7). The data display good agreement with the $\mathrm{\Delta }{v}^{-1}$ decay. The dashed black line shows the best fit $\xi =\frac{10}{\mathrm{\Delta }v}$.

Figure 9

(a) Classifications of the activity-induced phases, based on linear density profiles. The solid lines indicate the theoretically predicted boundary between the active liquid and solid $+$ active liquid regions in Eq. (10), and the activity beyond which void regions open up [predicted by Eq. (13)]. We have used the observed value $\frac{D}{\alpha }=10, {\rho }_{\mathrm{solid}}=0.19$, and the simulated system size $L=450$. (b) Phase diagram obtained from the numerically sampled phase space of the active lattice gas, including the passive and the active states. Each point is classified as either pure active liquid, solid $+$ active liquid, or solid $+$ active liquid along with void regions. Open symbols denote the transition from the passive liquid to the aging glassy regime. These classifications are based on ensemble-average density profiles. The topology of this phase diagram is the same as that observed in attractive colloids [14, 32, 65]. The black region denotes densities larger than ${\rho }_{\mathrm{solid}}=0.19$. Note that the predicted phase diagram shown in panel (a) is valid only at finite activities and is expected to accurately capture the physics only at large enough activities where activity-induced correlations overwhelm the correlations intrinsic to the passive hard crosses at $\rho \approx {\rho }_{rcp}$. This is because, as discussed in the text, the coarse-grained model includes only simple exclusion, not the extended N3 exclusion.

Figure 10

Comparison of the local density distributions for persistent $\left(D_R=0\right)$ and finite rotation rate $\left(D_R=0.005\right)$ active hard crosses. At $\mathrm{\Delta }v=0.5$, the zero density peak corresponding to the empty void regions at $D_R=0$ (a) is replaced by a well-defined low density peak at $D_R=0.005$ (b). For the smaller activity value, the nonrotating crosses show phase separation into solid and void-like regions, with an intervening active liquid region (c). In contrast, at $D_R=0.005$, there is a simpler fluid-solid coexistence indicated by a clearly bimodal distribution (d).

Figure 11

Phase diagram at $D_R=0.005$ showing the binodal constructed from the peaks in ensemble density distributions with triangles marking the high-density peak and circles the low-density one. Crosses indicate phase space points $(\mathrm{\Delta }v,\rho )$ at which the system is in a homogeneous fluid state with a single peak at the global density in the density distribution. Colors indicate the global density $\rho$. Density distributions were averaged over five runs for each sampled phase space point.

Figure 12

Mean-squared displacements (MSD) of passive hard crosses starting from (a) the initial state produced by the RSAD protocol, $t_w=0$, and (b) $t_w=200000$, showing significant dependence on $t_w$, for densities $\rho \ge 0.1650$: Note the absence of the plateau observed in panel (a) at longer waiting times (b).

Figure 13

Self-intermediate scattering functions (ISF) of passive hard crosses starting from (a) the initial state produced by the RSAD, $t_w=0$, and (b) $t_w=200000$, showing significant dependence on $t_w$, for densities $\rho \ge 0.1650$: note the absence of the plateau observed in panel (a) at longer waiting times (b).

Figure 14

(a) Stretched exponential fits (black lines) to the ISFs at densities $\rho \le 0.1625$, showing that the fit fails at $\rho =0.1625$. The exponent, $\beta$, decreases from $0.8$ at $\rho =0.1$ to and $0.4$ at $\rho =0.16$. (b) ${\tau }_{\alpha }$ extracted from the stretched exponential fits.

Figure 15

Waiting time $\left(t_w\right)$ dependence of the long time decay of the ISF at $\rho =0.1650,0.1675,0.1715$ (bottom to top). The decay becomes slower with increasing $t_w$, and as shown in the inset, depends only on the ratio $\frac{t}{t_w}$.

Figure 16

Ensemble averaged local density distributions for smaller activity values showing the transition from the uniform liquid state to coexistence of solid regions with liquid regions. The liquid peaks shows large low-density fluctuations for the largest activity values. For the passive system $\mathrm{\Delta }v=0.000$, a coexistence between fluid and solid regions appears for densities $\rho >0.1625$. At nonzero but small values of the activity $\mathrm{\Delta }v>0$, this coexistence region expands to reach lower densities.

Figure 17

Ensemble average distributions of the local density (measured in square boxes with side length 45 lattice sites, for configurations at the simulation time $t_{\max }=2\times {10}^6)$ compared for different activities $\mathrm{\Delta }v$ and different overall global densities $\rho$. For sufficiently large activity $\mathrm{\Delta }v\ge 0.2$, there are only two clear peaks, a peak for the solid phase with density ${\rho }_{\mathrm{solid}}\approx 0.19$ and a peak for the empty void regions ${\rho }_{\mathrm{void}}=0$. As the activity is reduced, a liquid-like peak near $\rho \approx 0.14–0.15$ appears. At sufficiently small activity, the clear void peak disappears, and only large low density fluctuations of the liquid peak remain, suggesting that void regions have been filled in by the growing activity liquid interface. Note these distributions are ensemble averages over both arrested and nonarrested states.

Figure 18

Finite-size dependence of the local density distributions for several different activities at $\rho =0.15$ and linear system sizes $L=225,450,600$. The active liquid state (unimodal peak) present at $\mathrm{\Delta }v=0.03$ becomes a bimodal active liquid $+$ solid coexistence for $L=600$. Similarly, at larger activity, for instance, $\mathrm{\Delta }v=0.07$, increasing the system size opens up enough space for empty void regions to form, so that the active liquid $+$ solid coexistence at $L=225$ transforms into an active liquid $+$ solid $+$ void state at $L=600$. The results qualitatively support the predicted finite size dependence of the boundaries between the three nonequilibrium phase types, active liquid, solid-active liquid, and solid-active liquid-void.

Figure 19

Snapshots illustrating the finite-size dependence of the final nonequilibrium state reached, for $\rho =0.15$ and $L=225,450,600$. For small activity $\mathrm{\Delta }v=0.03$, increasing the system size transforms the final state from an active liquid in the smaller systems with $L=225,450$, to an active liquid $+$ solid state in the largest system $L=600$. For larger activity $\mathrm{\Delta }v=0.07$, increasing the system size allows room for voids to open up, so that the active liquid $+$ solid state at $L=225$ clearly becomes an active liquid $+$ solid $+$ void state at $L=600$. The black scale bars indicate a length of 100 lattice sites.

Figure 20

Comparison between the final gel-like configurations observed for the infinitely persistent $D_R=0$ active crosses [(a)–(d)] and for finite rotation rate $D_R=0.005$ [(e)–(h)]. Color bars denote the stationary times of each cross, ${log}_{10}\left({\tau }_i\right)$. For this large activity value $\mathrm{\Delta }v=0.5$, the percolating gel-like solid network can still form for active crosses with a finite rotation rate, suggesting that the rotational-locking mechanism of crosses greatly facilitates the global arrest of the system. For finite rotation rates, the rectangular empty void regions found for nonrotating crosses become filled with a finite density gas, and the boundaries of the low-density region become more rounded. The term “gas” here refers to very dilute regions and “liquid” is used to the describe higher densities (no significant physical difference is meant to be implied).

Figure 21

Comparison of long time configurations for nonrotating and finite rotation rate $D_R=0.005$ active crosses at a smaller activity value $\mathrm{\Delta }v=0.100$. Here, the nonrotating active crosses [(a)–(d)] separate into solid, void, and intervening active liquid regions. The probability of forming a percolating solid network and becoming arrested increase with increasing global density. For the finite rotation rate [(e)–(h)], a more standard two phase coexistence between liquid and solid is restored, and the structures resemble a two-phase coexistence between a fluid and a solid with little density heterogeneity in the fluid phase.