Glassy swirls of active dumbbells

TL;DR

This study investigates how active propulsion influences glass formation in dense dumbbell mixtures, revealing distinct dynamical behaviors and heterogeneities compared to temperature-induced glasses, with implications for biological systems.

Contribution

It demonstrates that activity induces a unique glass transition characterized by large swirling vortices, differing fundamentally from thermal glasses.

Findings

Activity-induced glass shows large vortices.

Dynamical heterogeneity differs between thermal and active glasses.

Swirling vortex scale diverges near the glass transition.

Abstract

The dynamics of a dense binary mixture of soft dumbbells, each subject to an active propulsion force and thermal fluctuations, shows a sudden arrest, first to a translational then to a rotational glass, as one reduces temperature $T$ or the self-propulsion force $f$. Is the temperature-induced glass different from the activity-induced glass? To address this question, 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 characterise the onset of glass formation - the activity-induced glass exhibits large swirls or vortices, whose scale is set by activity, and appears to diverge as one approaches the glass transition. This large collective swirling movement should have implications for collective cell migration in epithelial layers.

Equations

$$\rho {\partial }_t\mathbf{v}=-(\Gamma -\eta {\nabla }^2)\mathbf{v}+\nabla \cdot \sigma$$

$$F[\mathbf{Q}]=\int d\vec{x}\left[\frac{A}{2}{\mathbf{Q}}^2+\frac{C}{4}{\mathbf{Q}}^4+\frac{K}{2}(\nabla \mathbf{Q}{)}^2\right]$$

$$D_t\mathbf{Q}=\lambda \mathbf{U}+\mathbf{Q}\cdot \mathbf{\Omega }-\mathbf{\Omega }\cdot \mathbf{Q}-{\gamma }^{-1}\frac{\delta F}{\delta \mathbf{Q}}$$

$$\dot{\mathbf{Q}}$$

$$\dot{\mathbf{v}}$$

Full text

Assemblies of self-propelled objects jam at high densities kurchan:13 ; berthier:14 ; ni:13 ; marchetti:11 and low temperatures mandal:16 . On approaching dynamical arrest from the fluid side, these dense active assemblies are seen to exhibit typical glassy dynamics, with activity manifesting simply as an effective temperature mandal:16 ; flenner:16 . Likewise, starting from the jammed state, activity is seen to prematurely fluidize the system at a reduced (enhanced) transition temperature (volume fraction) kurchan:13 ; berthier:14 ; ni:13 ; marchetti:11 ; mandal:16 . On the face of it, it might appear that an active glass behaves very similar to a conventional one, albeit with a different effective temperature or density kurchan:13 ; flenner:16 . In this paper, we provide evidence to the contrary - we show that an active glass exhibits distinctive dynamical features on account of their local driving.

To address this, we study dense assemblies of generic oriented nonspherical self-propelled objects suma:15 ; tung:16 ; wensink:PNAS12 ; wensink:12 ; siebert:16 ; hinz:15 , which are free to explore both translational and orientational degrees of freedom. Living realisations include reconstituted layer of confluent epithelial cells manning1:15 , where cell shape anisotropy plays a crucial role in the jamming-unjamming transition manning2:15 , and jammed biofilms formed by a dense collection of rod-shaped bacteria. Likewise, shaken non-spherical grains at high packing densities constitute non-living examples kumar:14 .

We perform Brownian dynamics simulations of a dense binary assembly of dumbbells kob:94 ; chong:05 ; moreno:05 ; chong:09 in 2-dimensions (see Fig. 1(a) and Supplementary Information supplement for details); the 50:50 mixture of A and B type dumbbells ensures amorphous steady state structures. This assembly, subject to a temperature bath $T$, is made active by driving each dumbbell with a body-fixed propulsion force $f$ along the long axis of each dumbbell, and is characterized by a Pećlet number $Pe\equiv f\sigma_{AA}/k_{B}T$, measuring the relative strength of activity with respect to temperature suma:15 ; hinz:15 . The phase diagram Fig. 1(b) shows a jammed state upon reducing either temperature or self-propulsion force. Our main result is that the dynamical signatures of an active glass are fundamentally different from a conventional glass - (i) activity makes the glass less fragile (Fig. 1(c)), and (ii) the nature of dynamical heterogeneity in an active glass is very unique and exhibits large scale swirls and vortices (Fig. 2(a)), whose size increases and appears to diverge as one approaches the arrested state (Fig. 3(b)) by reducing the self propulsion force $f$. We understand these length scales from a continuum hydrodynamic theory of active dumbbells. These large scale swirls are the sluggish imprints of the collective turbulent motion observed in an active fluid of anisotropic particles wensink:PNAS12 ; wensink:12 ; giomi:15 .

To identify the onset of translational and rotational glassy behaviour in the ($T-f$) plane, we monitor the mean-square-displacement ($MSD$), $\langle\Delta\mathbf{r}(t)^{2}\rangle=\langle\frac{1}{N}\sum_{i}\langle|{\bf r}_{i}(t_{0}+t)-{\bf r}_{i}(t_{0})|^{2}\rangle$, where $\bf{r}_{i}$ is the center-of-mass of the $i^{th}$-dumbbell; the orientation correlation function $C_{2}(t)=\frac{1}{N}\sum_{i}\langle P_{2}\left[{{\bf n}}_{i}(t_{0})\cdot{{\bf n}}_{i}(t_{0}+t)\right]\rangle$ chong:05 ; moreno:05 , where $P_{2}$ is the $2^{nd}$-order Legendre polynomial and ${\bf n}_{i}$ is the unit vector along the long axis of the $i^{th}$ dumbbell; and the overlap function $Q(t)=\langle\frac{1}{N}\sum_{i}w(|{\bf r}_{i}(t_{0})-{\bf r}_{i}(t_{0}+t)|)\rangle$, where $w(r)=1$ if r $\leq$ a and 0 if r $>$ a, with a=0.3 smarajit:09 . Here, $\langle\cdots\rangle$ denotes an average over time origins $t_{0}$ and trajectories and $N$ is the number of dumbbells.

Note the following - (i) the dumbbells are subject to thermal noise and active propulsion; (ii) there is no explicit activity decorrelation time, orientational decorrelation occurs because of collisions and thermal fluctuations; and (iii) there is no imposed orientational alignment, any kind of orientation correlation appears entirely due to packing and collisions.

Starting from the liquid phase at high temperature $T$ and activity $f$, we reduce either $T$ or $f$. The MSD (Supplementary Figure S1 supplement ) shows cage diffusion, with a distinct intermediate plateau and a reduced late time diffusion coefficient, characteristic of the approach to a glass, as the temperature or activity is reduced. The boundary to the glassy phase is associated with the vanishing of the late time diffusion coefficient. This is accompanied by a slowing down of both the translational and rotational structural relaxation process, captured by the two-step decay of the correlation functions, $Q(t)$ and $C_{2}(t)$ (Supplementary Figure S2 supplement ). Thus, the corresponding $\alpha$-relaxation times $\tau^{T}_{\alpha}$ and $\tau^{R}_{\alpha}$ increase, as $T$ or $f$ decreases (Supplementary Figure S3 supplement ). As is usual in studies of glass forming systems, we estimate the glass transition temperature, separately for both rotational and translational relaxation, by fitting the respective $\alpha$-relaxation time to a Vogel-Fulcher-Tammann (VFT) form, $\tau_{\alpha}=\tau_{\infty}\exp\left[\frac{1}{\kappa\left(\frac{T}{T_{\mbox{\tiny{VFT}}}}-1\right)}\right]$, where $\tau_{\infty}$ is the relaxation time at large temperatures, $\kappa$ is the coefficient of kinetic fragility, and $T_{\mbox{\tiny{VFT}}}(f)$ is the putative glass transition temperature for an applied active force $f$. The resultant phase diagram in the $T-f$ plane, obtained by marking the different $T_{\mbox{\tiny{VFT}}}(f)$ values, is shown in Fig. 1(b). We find that the rotational and translational degrees of freedom freeze out from the liquid at different $T_{\mbox{\tiny{VFT}}}(f)$, resulting in two distinct glass phases, the translational glass (TG) and translational-rotational glass (TRG).

The phase diagram itself does not reveal any difference between approaching the glass by lowering $T$ or $f$. To see differences in the active and passive systems, distinguished by their Pećlet number, one needs to probe their dynamical heterogeneity. Already, the VFT fits give some hint of this: the kinetic fragility ($\kappa$) of translational glass shows a decrease with increasing activity $f$; see Fig. 1(c) and inset. A similar behaviour is also observed for the rotational relaxation (see Supplementary Figure S4 supplement ). Thus, the active glass becomes stronger or less fragile than a passive glass, both during translational and rotational arrest. This allows for tuning of the glassy behaviour from being fragile to strong, by tuning the magnitude of active propulsion. The decrease in fragility in an active soft supercooled liquid is consistent with some earlier numerical mandal:16 and theoretical nandi:16 studies. We will see later how this result is also consistent with observations in reconstituted epithelial tissues that have been reported to show glassy behaviour zhou:09 .

To explore the difference in behaviour while approaching the arrested state by either decreasing $T$ or $f$, we probe the dynamics in finer detail at specific points along an iso-relaxation time ($\tau^{T}_{\alpha}=102.85$) line (see Fig. 1(b)), which lies in the supercooled liquid regime. We measure the displacement field vectors of the centre of mass of the dumbbells, ${\bf d}(\mathbf{r})$, over a time $\tau_{\alpha}^{T}$, and construct a spatial map of corresponding streamlines, which are shown in the left panel of Fig. 2. For the passive system ($f=0$), such a map is structureless with no large scale spatial correlation, Fig. 2(b)(left panel). On increasing $Pe$ along the iso-$\tau^{T}_{\alpha}$ line, one suddenly begins to observe distinct vortex like structures, whose size increases with increasing $Pe$, as seen in Fig. 2(a)(left panel). This pattern (or lack of it in the passive case) is also quantified via the coarse-grained vorticity $\omega(\mathbf{r})=\nabla\times{\bf v}_{\tau}(\mathbf{r})$, where ${\bf v}_{\tau}(\mathbf{r})\equiv{\bf d}(\mathbf{r})/\tau_{\alpha}^{T}$ note , which are shown as underlying colormaps in the left panels of Fig. 2. Further, in the right panel of Fig. 2, we show spatial maps of the magnitude of single particle displacements during $\tau_{\alpha}^{T}$, to gauge the extent of dynamical heterogeneity exhibited in either situation. The active system, Fig. 2(a)(right panel), has much more spatial heterogeneity in dynamics than the passive one, Fig. 2(b)(right panel). Also, notice the anti-correlation in the vorticity and the magnitude of particle displacement: regions with low (high) vorticity correspond to fast (slow) dumbbells, particles between a vortex-antivortex pair are faster, while particles in the vortex core are slower.

In order to extract correlation lengths from the spatial structures visible in such maps, we calculate the angle-averaged correlation functions of (i) the orientation of the displacement vectors of the dumbbell, $C(r)=\langle 2\cos^{2}\Delta\theta(\mathbf{r})-1\rangle$, where $\Delta\theta(\mathbf{r})$ is the angular separation between two displacement vectors separated by distance $\mathbf{r}$ (Supplementary Figure S5 supplement ), and (ii) the vorticity, $G(r)=\langle\omega(\mathbf{0})\omega(\mathbf{r})\rangle$, evaluated over $\tau^{T}_{\alpha}$ (Supplementary Figure S6 supplement ). The extracted correlation lengths, $\zeta$ and $\chi$, respectively , show a crossover as we move along the iso-$\tau^{T}_{\alpha}$ line, and distinguish the passive (low $Pe$) from the active (high $Pe$) supercooled liquids; see Fig. 3(a).

If we now move towards the dynamically arrested regime from either extreme ends, i.e. along the passive direction ($Pe=0$) where the active forcing is absent or the athermal active direction ($Pe=\infty$) where thermal fluctuations are suppressed, we clearly observe the stark differences in the way the above-mentioned spatial correlations grow. As one goes towards the glass transition at $Pe=0$, there is no significant change in the correlation lengths $\zeta$ and $\chi$ - which continue to remain at the scale of the dumbbell - (Supplementary Figure S7 supplement ). There is of course the usual dynamical heterogeneity associated with the emergence and growth of fast moving and slow moving domains berthier:11 and corresponding increase in relaxation times smarajit:09 ; smarajit:14 . However, on approaching the glass transition from the active side, along ($Pe=\infty$), the nature of the dynamical heterogeneity is very different and is associated with swirling or vortex patterns that grow in size. The corresponding growing length scales (see Fig. 3(b)), goes as $1/\sqrt{f-f^{*}}$, far away from the glassy regime ($f^{*}=4.5$), and crosses over to $1/(f-f^{*})$, as one nears dynamical arrest.

To understand the origin of these vorticity scales, we construct a hydrodynamic description of the self-propelled dumbbells in the isotropic phase. Our simulations show that the velocity $\mathbf{v}_{\tau}$ is correlated with the orientation of dumbbell ${\bf n}$, or more precisely, $\mathbf{Q}\cdot{\bf f}$, where $\mathbf{Q}$ is the nematic orientation tensor describing the apolar orientation of the dumbbell and ${\bf f}$ is the body-attached propulsion force. This correlation increases with Pećlet number along the iso-$\tau^{T}_{\alpha}$-line before saturating due to packing considerations (Supplementary Figure S8 supplement ). This suggests that the appropriate hydrodynamic fields are the conserved densities, orientation tensor $\mathbf{Q}$ and the dumbbell velocity $\mathbf{v}$. Note that the polarity of the dumbbells is an acquired property, a consequence of $\mathbf{Q}\cdot\mathbf{v}$. To see the generation of large swirls or vortices, it suffices to consider the dynamics of $\mathbf{Q}$ and ${\bf v}$ alone (Supplementary Information supplement ). Newtons second law provides the equation for the dumbbell velocity $\mathbf{v}$,

[TABLE]

Our use of Brownian dynamics implies that there is a frictional damping $\Gamma$, in addition to the usual viscous damping associated with the transfer of momentum arising from collisions. The last term corresponds to local forces due to a stress $\sigma=\sigma^{op}+\sigma^{act}$, written as a sum of the order parameter stress, $\mathbf{\sigma}^{op}=\Lambda\left({\delta F}/{\delta\mathbf{Q}}\right)-\left[\mathbf{Q}\cdot({\delta F}/{\delta\mathbf{Q}})-({\delta F}/{\delta\mathbf{Q}})\cdot\mathbf{Q}\right]$ giomi:15 , derived from a free-energy functional, $F[\mathbf{Q}]$ for the nematic (Supplementary Information supplement )

[TABLE]

and $\mathbf{\sigma}^{act}=fl\mathbf{Q}$ is the active stress marchetti:14 . The dynamics of $\mathbf{Q}$ is given by

[TABLE]

where $D_{t}$ represents the convective derivative in the comoving frame, $\lambda$ is the stable-flow alignment parameter, $\gamma$ is the rotational viscosity and $\mathbf{U}$ and $\mathbf{\Omega}$ are the symmetric and anti-symmetric strain-rates. Now linearising about the isotropic state with no flow, we obtain,

[TABLE]

Taking the curl of the above equations gives us the equation for the vorticity $\omega$. Away from the glass transition, patterning is generated by balancing the active stress against the orientational elastic stress; this gives a length scale $\sqrt{\Lambda K/f\,l}$, consistent with the scaling observed in the simulations at large $f$ (Fig. 3 and fielding:16 ). On approaching the glass transition $f=f^{*}$, the translational motion gets sluggish and finally arrested; it would seem that we can no longer use an active fluid description. We argue however that when $f>f^{*}$ we should still be able to use these equations, as long as we renormalize time by $t\to t/\tau^{T}_{\alpha}$. In this highly viscous fluid regime, the patterns are generated by balancing the active stress with the viscous stress; this gives rise to a length scale that goes as $1/(f-f^{*})$ (Supplementary Information supplement ). The crossover between these two different forms of the characteristic length with active force, occurs at propulsion force $\propto\eta^{2}l/\Lambda K$.

To summarise, we study a model Brownian assembly of self-propelling soft anisotropic particles, wherein there is neither any explicit activity de-correlation time nor imposed orientational alignment, unlike many other model active systems that have been studied. We demonstrate that the active forcing makes the corresponding supercooled fluid less fragile and more importantly the heterogeneous dynamics observed in the active fluid is very different from its passive counterpart, exhibiting large-scale swirls and vortices. We further show that the correlation lengths, associated with these spatial structures, grow with decreasing active forcing and appear to diverge as the dynamics gets arrested. We rationalise the occurrence of such length scales using a continuum hydrodynamic theory of active dumbbells.

Our study of dynamical heterogeneities in active glass composed of anisotropic particles has implications for the collective cell dynamics in epithelial layers. This can be seen, by constructing a Voronoi tessellation around each dumbbell; the resulting pattern then resembles an epithelial sheet (Supplementary Movie supplement ). One of the first studies of glassy dynamics close to the jamming transition in a reconstituted epithelial sheet zhou:09 , revealed two striking features : (i) the epithelial glass was significantly less fragile compared to a conventional hard-sphere glass and (ii) systematically reducing cell motility by treatment with titrated amounts of actin de-polymerization agents rendered the epithelial sheet more fragile. Our result on how active propulsion makes the jammed state stronger or less fragile, is entirely consistent with this. In addition, our work suggests the possibility of observing large scale swirls in situations where cellular propulsion is high and the cell-substrate adhesion is weak (Supplementary Movie supplement ) Angelini:11 .

The swirling patterns or vortex structures that we observe for dense assemblies of anisotropic particles, are reminiscent of the dynamical patterns seen in active turbulence exhibited by a collection of fluid of active rods wensink:PNAS12 ; wensink:12 ; giomi:15 . Is the dynamical heterogeneity exhibited by the active glass a frozen memory of its active turbulent past? Our work shows unambiguously that an active glass is fundamentally distinct and not just a conventional glass with another name.

We thank A. Das, J. P. Banerjee, S. S. Ray and S. Sastry for useful discussions. RM and PJB acknowledge financial support from CSIR, India, and CD from DST, India.