Dynamically generated patterns in dense suspensions of active filaments

We use Langevin dynamics simulations to study dynamical behavior of a dense planar layer of active semiflexible filaments. Using the strength of active force and the thermal persistence length as parameters, we map a detailed phase diagram and identify several nonequilibrium phases in this system. In addition to a slowly flowing melt phase, we observe that, for sufficiently high activity, collective flow accompanied by signatures of local polar and nematic order appears in the system. This state is also characterized by strong density fluctuations. Furthermore, we identify an activity-driven crossover from this state of coherently flowing bundles of filaments to a phase with no global flow, formed by individual filaments coiled into rotating spirals. This suggests a mechanism where the system responds to activity by changing the shape of active agents, an effect with no analog in systems of active particles without internal degrees of freedom.

Figure 1

Schematic representation of the model for self-propelled semiflexible filaments. Spherical beads of diameter $\sigma$ are connected through harmonic springs of spring constant $k_b$. The filament flexibility is modeled as a bending penalty for angle $\theta$ with stiffness constant $\kappa$. Beads belonging to different chains, as well as beads on the same chain that are more than two beads apart, interact via a short-range, repulsive interaction, modeled with the Weeks-Chandler-Andersen potential. Finally, each bead is subject to an active force ${\mathbf{f}}_i^a$ pointing along the tangent to the filament contour at $i$. Note that, for clarity, distances between beads along the filament have been exaggerated.

Figure 2

Nonequilibrium phase diagram for ${\xi }_p/L$ vs Pe at packing fraction $\varphi =0.64905$. Results for different filament lengths are plotted on the same graph since they show the same scaling. Symbols mark individual simulations for different filament lengths: $N=5$ ($\blacksquare$), $N=10$ ($◯$), $N=25$ ($*$), and $N=50$ ($▴$), with phases coded by color. Different phases were identified by visually inspecting each individual run. Crossovers between phases are indicated by shading. Dashed lines indicate rough boundaries between phases and serve as a guide to the eye. The gray-shaded bar indicates the threshold to spiralling, set by the flexure number $\mathfrak{F}\approx {10}^3$ [42].

Figure 3

Snapshots of simulations for $N=50$ in four nonequilibrium phases: (a) flowing melt (${\xi }_p/L=0.2,\text{Pe}=35.5$), (b) swirl (${\xi }_p/L=0.04,\text{Pe}=177.58$), (c) segregated phase (${\xi }_p/L=0.82,\text{Pe}=355.16$), and (d) spirals (${\xi }_p/L=0.04,\text{Pe}=17757.8$). Individual filaments are marked by a different shade. Arrows in (a) point to examples of $+1/2$ (red) and $-1/2$ (yellow) topological defects. White areas in (c) are regions devoid of filaments, indicating substantial density fluctuations. Inset in panel (d) is a zoom-in on several spirals.

Figure 4

(a) Mean-square displacement (MSD) as a function of time for ${\xi }_p/L\approx 0.83$ for six values of $\text{Pe}=0,0.852,8.52,85.2,852,4260$. (b) Log-log plot of the diffusion coefficient as a function of Pe in the regime where the system is diffusive. Inset: linear scaling at low activity for a soft (${\xi }_p/L=0.08$) filament. (c) Ratio of active (${\xi }_p^a$) to passive (${\xi }_p=2\kappa b/k_{B}T$) persistence length for a range of values of bare bending stiffnesses. Note that due to steric effects the measured persistence length at low activity is always larger than ${\xi }_p$. Inset: ratio of active, $R_g^a$, to passive, $R_g$, radius of gyration as a function of Pe. (d) Average angular velocity of rotation $\omega$ defined in Eq. (9) around centers of mass of filaments as a function of Pe. Legend in (d) also applies to panels (b) and (c). $N=25$ in all plots.

Figure 5

Transition time from superdiffusive to diffusive behavior in the segregated phase as a function of Péclet number for ${\xi }_p/L=0.08,0.83$, and 1.67.

Figure 6

Local polar (a) and nematic (b) order parameter defined in Eq. (10) as a function of activity for a range of bare filament stiffnesses for $N=25$. Spatial correlation function $g_{m,2}\left(r\right)$ defined in Eq. (11) for the polar (c) and nematic (d) order parameter. Note that the upturn in (d) is an artifact of averaging for large $r$.

Figure 7

Comparison of the MSD curves for two values of the Péclet number using Langevin (solid lines) and Brownian (dashed lines) dynamics simulations. For large activity, at long times, the MSD curves are indistinguishable. For smaller values of Pe, there is a small difference between the two, but we attribute it to very slow dynamics in this regime: it is likely that the steady state has not been reached during the duration of the simulation.

Figure 8

Identification of topological defects. (a) Headless unit-length vectors pointing along the local tangent to the filament contour are placed at the vertices of a Delaunay triangulation (white lines). We then loop counterclockwise (red arrow) over the vertices of each triangle (marked in red) and compute a signed sum of the angles the headless vectors make with the $x$ axis. If this sum exceeds $\pm \pi$ a defect of appropriate topological charge is placed inside that triangle. Defects (yellow ball) are located at the vertices of the Voronoi diagram (black lines) dual to the Delaunay triangulation. (b) Application of this algorithm to a snapshot of the simulation. For visualization purposes, we zoomed in on only a couple of defects.