The Jammed Phase of Infinitely Persistent Active Matter

Abstract

We study an extreme active matter system, which is essentially a dense assembly of athermal, soft and infinitely persistent active particles. Using extensive numerical simulations we obtain jammed configurations of this system in two dimensions and probe the stability of such structures under increasing active forcing magnitude. We show that the critical active forcing magnitude for the jammed phase to yield scales with virial pressure as $f_c\sim p^α$, with $α=1.17$, describing the yielding line. Using a Laplacian framework, we redistribute the active forces into a modified contact force network. By analysing the statistics of these redistributed forces, we obtain a very robust scaling law consistent with the passive limit, not just near the unjamming line, but in the entire jammed active phase. The probability distribution of the magnitude of the contact force deviates from the power-law form found in passive systems for values smaller than the active force. Moreover, within the jammed phase, the system displays elastic, plastic, and yielding events with increasing active forcing. This active plasticity appears abruptly and can not be captured by the continuous softening of the Hessian spectrum. However, we demonstrate that the Hessian still retains the ability to predict relaxation times. These results clarify how activity modifies force distributions and leads to deformation, plasticity and yielding in dense, jammed, infinitely persistent active matter.

Figures (9)

• Figure 1: Yielding, force-balance and stability in infinitely persistent active matter. (a) Top: A jammed packing of soft bidisperse particles. Bottom: Under the effect of spatially uncorrelated active forces (red arrows), the contact forces get into mechanical equilibrium with the imposed active forces. (b) In the phase space spanned by pressure ($p$) and active forces ($f_0$), the circles are the critical active forces, $f_c$, that mark the boundary between the active solid and active liquid. At small pressures, $f_c$ scales as $f_c\sim p^{1.17}$ (dashed-line) for system size $N=8192$. See Sec. \ref{['unjam']}. Error bars are scaled up by a factor of five for visibility. c) Top: The contact forces $\bm{f}$ (blue-arrows) themselves are not in force-balance in active solids. Bottom: Using an auxiliary field $\phi$ defined on the particles, we redistribute the contact and active forces to construct a unique network of modified contact forces $\bm{f'}$ (purple-arrows) that are in force-balance. We detail this method in Sec. \ref{['refo']}. d) In the limit of infinite persistence time, an effective potential energy $U_\text{eff}$ governs the deformation dynamics in active solids. See Sec. \ref{['macro']}.
• Figure 2: Universal scaling of force distributions in jammed active solids. (a) active danglers ($z=2$) such as the black particle whose active force is balanced by elastic forces of the two gray particles are systematically removed prior to force analysis. (b) Inset: The contact forces $f$ for the passive jammed systems (blue) obey power-law distributions at $f/\langle f\rangle<1$ unlike the active systems with $f_0\rightarrow f_c^-$ (red) at small $f$. On redistributing the contact and active forces into a network of redistributed forces $f^{\prime}$, a decent collapse is obtained at small pressures $p$. (c) The scaling law (Eq. \ref{['ansatz']}) is applicable in the entirety of jammed active solid phase (Fig. \ref{['1panel']}) - at all pressures $p$ and active forces including $f_0<f_c$ (orange). (d) The scatter plot of $f^{\prime}$ and the angle $\theta$ it makes with the bond vector. (e) The normalized density distributions of $\theta$ has a cusp-like shape at $\theta\rightarrow0$ described by $\rho(\theta/\Delta \theta)\sim (\theta/\Delta \theta + c)^{-2.9}$ with $c=0.23$ (black line). The inset shows the standard deviation $\Delta \theta$ monotonically increasing with $p$. (f) The collapse of density distributions of $\text{tan}\,\theta=f_\perp/f_\parallel$, the ratio of the tangential and the normal components of $f^\prime$ has a plateau for $\text{tan}\,\theta/\text{tan}\,\theta_0\ll1$ and a power-law at large values. The inset is the normalization angle $\theta_0(p)$ monotonically increasing with $p$ akin to $\Delta \theta(p)$ in (e). System size, $N=8192\,(4096)$ for top (bottom) respectively.
• Figure 3: Elasticity, plasticity and yielding in jammed active solids. (a) The typical strain response during a quasistatic increase in imposed active force, using $\delta{f_0}=10^{-7}$, on a configuration of jammed particles displays elastic deformations (red) punctuated by plastic events (dark-green) and terminates by yielding. (b,c,d) Typical time evolution of the contact potential energy $\Delta U(r_{ij})$ (black) and effective potential energy $\Delta U_\text{eff}$ (blue) during elastic, plastic periods and yielding. (Insets of b,c,d) The change in active strain $\Delta \tilde{\gamma}$ scales with the change in active stress $\Delta \tilde{\sigma}$ for elastic events in contrast to plastic events. At yielding, strain increases monotonically with time. (e,f,g) Displacement fields (vectors scaled up by a factor of $3\times10^2$ for visibility) during a typical elastic, plastic and yielding event. (h) The relaxation dynamics of the effective potential energy for elastic (red) and plastic events (dark green) have exponential form at long times. (i) The inverse relationship between equilibration time $t_\text{eq}$ and the smallest non-trivial eigenvalue of the Hessian $\lambda_\text{min}$ can be seen for the elastic events (red). The plastic events lie above the line of $t_\text{eq}=\lambda_\text{min}^{-1}$ as the time to relax is typically longer in plastic events as seen in the dark green curves of (h). (j) The numerically calculated time constant at long times, $\tau$, match with $1/(2\lambda_\text{min})$. All figures use $N=1024$ and $p=10^{-4}$.
• Figure SI.1: (a) Each run at a fixed active force $f_0$ is a FIRE minimization procedure which terminates when the average speed of the particles is $\le 10^{-10}$. We remove the active danglers with $z=2$ and re-minimize $U_\text{eff}$ at the same $f_0$. (b) The probability distributions of the packing fractions before and after removal of rattlers displayed in translucent and solid line-points respectively for $N=1024$. For visual clarity, $P(\phi)$ of every pressure is offset by a value of 0.3 relative to $P(\phi$) of its immediately lower pressure.
• Figure SI.2: The yielding point $f_c$ at different system sizes. The dotted line is $f_c\sim p^{1.17}$. Note that $f_c$ found using Brownian dynamics is larger than the ones found using the FIRE algorithm.