Defect-Mediated Phase Transitions in Active Soft Matter

How do topological defects affect the degree of order in active matter? To answer this question we investigate an agent-based model of self-propelled particles, which accounts for polar alignment and short-ranged repulsive interactions. For strong alignment forces we find collectively moving polycrystalline states with fluctuating networks of grain boundaries. In the regime where repulsive forces dominate, the fluctuations generated by the active system give rise to quasi-long-range transitional order, but—unlike the thermal system—without creating topological defects.

Figure 1

Global polar order parameter $P$ (a) and hexatic order parameter ${\mathrm{\Psi }}_6$ (b) as a function of the control parameters $\rho$ and $\nu$. The dashed white lines indicate tentative boundaries between polar and unpolarized states, and crystalline states exhibiting crystalline order from fluidlike states. Snapshots of local hexatic order $|{\mathrm{\Psi }}_{6,i}|$ for a relative interaction value $\nu =0.25$ (c), $\nu =0.75$ (d), and $\nu =1.5$ (e); all three snapshots correspond to a high packing fraction of $\rho =0.85$. See also videos in the Supplemental Material [36]. Scale bars indicate a distance of $20R$.

Figure 2

(a) Defect ratio $d(t)$ as a function of time $t$ for $(\rho ,\nu )=(0.85,1.5)$. The black curve is an average over 50 realizations (gray lines) with error bars indicating the standard deviation. The steep decline in the time traces corresponds to the fast annihilation processes of grain boundaries [see (d), white errors]. (b) Probability distribution $P(d)$ [lin(d)-log] of the defect ratio for two stationary (flowing) polycrystalline states, i.e., $(\rho ,\nu )=(0.85,0.25)$ (green, triangles) and $(\rho ,\nu )=(0.85,0.75)$ (blue, circles). (c),(d) Snapshots illustrating the defect dynamics for an active crystal [$(\rho ,\nu )=(0.85,1.5)$], where the system shows a high degree of crystalline order. (c) Early phase with roughly homogeneous distributed defects and small hexagonal patches. (d) Formation of ringlike grain boundaries at larger times that contract (indicated by black arrows), leading to a sudden decrease of $d(t)$ [see (a)]. The left and right half of each figure depict the Voronoi triangulation and the local hexatic order parameter $|{\mathrm{\Psi }}_{6,i}|$, respectively. Disclinations are indicated by red and yellow dots, and green dots corresponds to particles with more than sevenfold or less than fivefold coordination. Scale bars: $20R$.

Figure 3

Static structure factor $S(\overrightarrow{q})$ for (a) the active crystal [$(\rho ,\nu )=(0.85,1.5)$] and (b) a polycrystal [$(\rho ,\nu )=(0.85,0.25)$], respectively, both in the stationary regime. Reciprocal lattice vectors $\overrightarrow{G}$ are indicated by white arrows. (c) Correlation function ${\mathrm{C}}_6(r)$ in lin(r)-log (inset: log-log) for the same parameters as in Figs. 1: $\rho =0.85$, and $\nu =1.5$ (red, square), $\nu =0.75$ (blue, circle), and $\nu =0.25$ (green, triangle). Correlation function $C_{\overrightarrow{G}}(r)$ for $(\rho ,\nu )=(0.85,1.5)$ (d) [log(r)-lin, dashed line is a power law with exponent 0.04] and $(\rho ,\nu )=(0.85,0.25)$ (e). All results correspond to a simulation box of size $L=400$ containing $N=172156$ particles; see Supplemental Material [36] for the data evaluation.