Activity-Induced Collapse and Arrest of Active Polymer Rings

We investigate, using numerical simulations, the conformations of isolated active ring polymers. We find that their behavior depends crucially on their size: Short rings ($N\lesssim 100$) swell, whereas longer rings ($N\gtrsim 200$) collapse, at sufficiently high activity. By investigating the nonequilibrium process leading to the steady state, we find a universal route driving both outcomes; we highlight the central role of steric interactions, at variance with linear chains, and of topology conservation. We further show that the collapsed rings are arrested by looking at different observables, all underlining the presence of an extremely long timescales at the steady state, associated with the internal dynamics of the collapsed section. Finally, we found that in some circumstances the collapsed state spins about its axis.

Figure 1

(a) Gyration radius as function of $N$, for active self-avoiding rings (different values of Pe) and active ghost rings ($\text{Pe}=10$) (b) Mean internal distance as function of the distance $s$ along the contour. (c) Bond-bond correlation function as function of $s$. In panel (b) and (c), full lines refer to active rings at $\text{Pe}=100$; orange dashed lines refer to the passive case in good solvent for $N=500$; dash-dotted lines refer to the passive case in bad solvent for $N=100$, 200, 500. Panels (b) and (c) share the legend. In panel (a) and (b) snapshots of short $N=70$ (inflated) and long $N=500$ (collapsed) rings, with colors referring to beads: (i) in the dangling sections or in the inflated state (green), (ii) on the surface of the collapsed structure (blue), (iii) in the interior of the collapsed core (red).

Figure 2

Route to the steady state: (a) position of the minimum of the bond correlation; (b) tangleness $T(t)/N$ as a function of time, normalized by ${\tau }_0$, for rings of different lengths at $\mathrm{Pe}=100$. The black symbol in (a) marks the time at which the snapshots shown are taken. Snapshots: $N=100$, green; $N=200$, blue; $N=500$, orange.

Figure 3

Fraction of survived neighbors as function of time; data refer to rings of size $N=500$ and several Pe at the steady state. The gray dashed line is a guide for the eye, highlighting the intermediate plateau.

Figure 4

(a) Monomer velocity distributions at the steady state for fixed $N=500$, several Pe. (b) Dynamic phase diagram for active rings. Symbols refer to state points sampled by means of numerical simulations: Collapsed rings are reported as blue squares, inflated rings as red circles, passivelike rings as green diamonds, and systems showing a more complex behavior as gray crosses.