Glassy swirls of active dumbbells

Is an active glass different from a conventional passive glass? To address this, we study the dynamics of a dense binary mixture of soft dumbbells, each subject to an active propulsion force and thermal fluctuations. This dense assembly shows dynamical arrest, first to a translational and then to a rotational glass, as one reduces temperature $T$ or the self-propulsion force $f$. We monitor the dynamics along an iso-relaxation-time contour in the $(T-f)$ plane. We find dramatic differences both in the fragility and in the nature of dynamical heterogeneity, which characterize the onset of glass formation—the activity-induced glass exhibits large swirls or vortices, whose scale is set by activity, and it appears to diverge as one approaches the glass transition. This large collective swirling movement should have implications for collective cell migration in epithelial layers. We construct continuum hydrodynamic equations for the simulated system, and we show that the observed behavior of this growing dynamic length scale can be understood from these equations.

Figure 1

(a) Binary assembly of self-propelled dumbbells of $A$ (blue) or $B$ (red) type in the dilute limit, built from two identical soft Lennard-Jones (LJ) spheres of diameter $\sigma$ connected through a harmonic spring of stiffness $k$ and rest length $l$ (inset). The body-fixed self-propulsion force $f$ (arrow) is along the long axis of each dumbbell. (b) Stochastic trajectory of a single active dumbbell in the dilute regime ($\rho =0.1$) at $T=0.5$, in the presence of an active force ($f=5$), showing center-of-mass position and dumbbell orientation. (c) Rotational correlation time ${\tau }_{\alpha }^R$ vs density $\rho$ for the assembly of active dumbbells for $f=10.0$ at two temperatures, $T=0.2$ (red) and $T=0.6$ (blue). Orientational decorrelation happens because of both thermal fluctuations and collisions as a consequence of active and thermal driving. We see that at $T=0.6$, the rotational persistence time increases monotonically with density; this is because rotational fluctuations are hindered due to crowding. At $T=0.2$, however, ${\tau }_{\alpha }^R$ decreases at first from a high value before increasing again; this initial decrease with density arises from the increasing number of collisions encountered by persistent dumbbells.

Figure 2

The translational and rotational mean-square displacement (MSD) of a single active dumbbell in the dilute regime ($\rho =0.1$) at $T=0.5$, with and without an active force, $f=0$ and 5, respectively.

Figure 3

(a) A dense binary assembly of self-propelled dumbbells of $A$ (blue) or $B$ (red) type. (b),(c) Mean-square displacement $\left[\langle \mathrm{\Delta }r{\left(t\right)}^2\rangle \right]$ as function of time $t$: (b) for propulsion force $f=2.0$ for different temperatures $T=3$ (blue), 3.5 (orange), 4 (green), 5 (cyan), 6 (magenta), and 8 (red); and (c) for temperature $T=1.5$ for different propulsion forces $f=3$ (blue), 3.5 (orange), 4 (green), 5 (cyan), 6 (magenta), and 8 (red).

Figure 4

(a) Overlap function, $Q\left(t\right)$, measured for propulsion force $f=2$ at different temperatures $T=3$ (blue), 3.5 (orange), 4 (green), 5 (cyan), 6 (magenta), and 8 (red); and (b) for temperature $T=1.5$ at different propulsion forces $f=3$ (blue), 3.5 (orange), 4 (green), 5 (cyan), 6 (magenta), and 8 (red). The inset shows the rotational time correlation function, $C_2\left(t\right)$, and its behavior for the same set of parameters in both cases.

Figure 5

(a) Translational $\alpha$-relaxation time scale (in filled circles) as a function of temperature ($T$) for different values of self-propulsion force $[f=0$ (red), 2 (green), 3 (magenta), 4 (cyan), 5 (orange), 6 (dark gray), and 8 (blue)]. The black dotted lines represent the fitting of the data sets for each $f$ to the VFT form. (b) Rotational $\alpha$-relaxation time scale (in filled circles) ${\tau }_{\alpha }^R$ vs $T$ for different values of $f$, with fits (black dotted lines) to the VFT form.

Figure 6

Phase diagram in the ($T-f$) plane shows a liquid (L, brown), translational glass (TG, blue), and translational-rotational glass (TRG, green) with phase boundaries determined from VFT fits. The dashed line represents the iso-relaxation-time (${\tau }_{\alpha }^T=102.85$) line, along which the dynamics is probed at points marked in green.

Figure 7

Angell plots: (a) Variation of translational relaxation times ${\tau }_{\alpha }^T$ vs scaled temperature $(T/T_g)$ at different values of the active force. As in [27], $T_g$ is defined as the temperature at which ${\tau }_{\alpha }^T={10}^6$. The inset shows how the corresponding kinetic fragility $\kappa$ decreases with increased self-propulsion force. (b) Variation of rotational relaxation times ${\tau }_{\alpha }^R$ vs scaled temperature ($T_g/T$) with changing active force: $f=0$ (red), 1 (green), 2 (cyan), 3 (violet), and 3.5 (blue). $T_g$ is defined as the temperature at which ${\tau }_{\alpha }^R={10}^6$. The inset shows how the corresponding kinetic fragility $\kappa$ decreases with increased self-propulsion force.

Figure 8

(a) Left: Streamlines of displacement field, $\mathbf{d}\left(\mathbf{r}\right)$, for the active dumbbells $(f=10.3,T=0)$ computed over $t={\tau }_{\alpha }^T$ showing large vortexlike structures with the underlying color map reflecting the corresponding values of the vorticity $\omega \left(\mathbf{r}\right)$. Right: Map showing the magnitude of displacement during $t={\tau }_{\alpha }^T$, illustrating the extent of heterogeneous dynamics in the active glass, with blue particles being the fastest and black particles being the slowest. Note that the vorticity and particle velocities are anticorrelated (see the text). (b) Left: Streamlines of displacement for passive dumbbells $(f=0,T=5.5)$ along with the corresponding vorticity color map. Right: Displacement map for the passive liquid. Note the absence of any large-scale structures and the resultant low vorticity.

Figure 9

Displacement field for an athermal assembly of a binary LJ mixture of disks (Kob-Andersen model), computed over time scales of corresponding ${\tau }_{\alpha }\approx 5028$, for two different persistence time scales: left, ${\tau }_p=10$; and right, ${\tau }_p={10}^4$, for applied active forcing of $f=3.5$ and 1.28, respectively, illustrating that the swirls become large only when persistence time scales are large, as indicated by the measured correlation lengths ($\zeta$).

Figure 10

(a) Spatial correlation function of the orientation of the displacement vectors of the center of mass of the dumbbells, $C\left(r\right)=\frac{1}{2\pi }\int \langle 2{cos}^2\mathrm{\Delta }\theta \left(\mathbf{r}\right)-1\rangle d\varphi$, where $\mathrm{\Delta }\theta \left(\mathbf{r}\right)$ is the angular separation between two displacement vectors separated by distance $\mathbf{r}$. The simulation points are chosen along an iso-${\tau }_{\alpha }^T$ line on the ($T-f$) phase diagram. The correlation length $\zeta$ is extracted using the condition $C\left(\zeta \right)=1/e$. (b) Spatial correlation of the vorticity field of the coarse-grained velocity vectors $G\left(r\right)=\frac{1}{2\pi }\int \langle \omega \left(\mathbf{0}\right)\omega \left(\mathbf{r}\right)\rangle d\varphi$ over a time scale of ${\tau }_{\alpha }^T$. The simulation points are chosen along an iso-${\tau }_{\alpha }^T$ line on the ($T-f$) phase diagram. The correlation length $\chi$ is extracted using the condition $G\left(r\right)=0$.

Figure 11

(a) Correlation lengths $\zeta$, characterizing the degree of alignment of the displacement vectors, and $\chi$, characterizing the spatial correlation of vorticity, as a function of Pe, along the iso-${\tau }_{\alpha }^T$ line, show a crossover from a passive to an active supercooled liquid. To highlight the change, we subtract from the measured correlation lengths a microscopic length associated with the size of the dumbbell. The choice of plotting against $exp\left[-1/\sqrt{\text{Pe}}\right]$ ensures uniform spacing for the simulation data points over the range of Pe. (b) $\zeta$ and $\chi$ as functions of activity $f$ at $T=0$ show an increase and crossover from $1/\sqrt{f-f^{*}}$ dependence at large $f$ to $1/(f-f^{*})$ dependence close to the glass transition at $f^{*}=4.5$ (dashed line shows fit). (c) Length scale ($\zeta$) associated with spatial correlation of the direction of the displacement vectors for the dumbbell system at $f=0$ for different $T$ (blue) and at $T=0$ for different $f$ (red), i.e., when we approach the glass boundary along two different axes. The correlation length scale does not show any significant change if the glass transition is approached by reducing temperature, but it shows an increase as the system approaches a glass transition by reducing the activity. The inset shows the relaxation time scale (${\tau }_{\alpha }^T$) for a similar set of simulation points at $f=0$ for different $T$ and at $T=0$ for different $f$.

Figure 12

Average of the normalized dot product of the coarse-grained velocity (${\mathbf{v}}_{\tau }$) of each dumbbell and its average orientation vector ($\mathbf{n}=\frac{\mathbf{n}\mathbf{1}+\mathbf{n}\mathbf{2}}{2}$) over the $\alpha$-relaxation time (${\tau }_{\alpha }^T$) as we move through different points on the iso-${\tau }_{\alpha }^T$ line from passive to active supercooled liquid region. For a few initial points the correlation between displacement and orientation increases and then it saturates as maximum coupling between these two is reached, which is limited by the packing of the dumbbells.