Using the force landscape of an active solid to predict plastic deformation

Abstract

Non-active disordered solids feature quasilocalized excitations that control plasticity, similar to crystal lattice defects, and these excitations can be identified via harmonic or anharmonic analyses of the potential energy landscape. Here we explore whether such ideas can be extended to active matter, focusing on dense packings of self-propelled rods. We generalize the definition of nonlinear excitations to force landscapes that incorporate active, non-conservative forces and find that force-based cubic excitations robustly predict future plastic events, enabling control of active solids.

Figures (6)

• Figure 1: A) Schematic of a potential energy landscape in a disordered packing, which can be represented as a force landscape B) with stable and unstable fixed points. C) A landscape of non-conservative, active forces. D) The total force landscape is the sum of (A) and (B) which will still have stable and unstable fixed points if active forces are not too large. Cubic modes (green) point from metastable states towards nearby instabilities. E) Plastic deformation, i.e. spikes in the strain per unit stress, $\Delta \gamma_{\text{act}}/ \Delta \sigma_{\text{act}}$ (red curve), occur when this landscape changes (via increasing non-conservative forces $\sigma_{\text{act}}$) until an unstable fixed point combines with the stable fixed point in a saddle-node bifurcation, causing the system relax into a nearby metastable state, i.e. a discontinuous jump in a strain-stress curve (blue).
• Figure 2: A) An example low-$\kappa$ harmonic mode measured in a non-Hamiltonian, active solid. B) An example low-$b$ cubic mode from the same configuration as A). Note that while these modes consist of a translational and rotational component for each rod, we visualize them by plotting the resulting translations of the spherical sub-particles that comprise the rods. Additional details in the Supplemental Information. C-E) Distributions of the asymmetry $\tau(\hat{z})$ (C), energy barrier $b(\hat{z})$ (D), and participation ratio $P_r(\hat{z})$ (E) of an ensemble of harmonic modes (blue) and cubic modes (green) as a function of stiffness $\kappa(\hat{z})$. Dashed line in (C) indicates a scaling of $\tau \propto \sqrt{\kappa}$.
• Figure 3: A,B) The two cubic modes with the lowest $b(\hat{\pi})$ observed $\Delta\sigma_{\text{act}} = 10^{-4}$ before the upcoming avalanche shown in C), with corresponding soft spots highlighted. C) A $D^2_{\text{min}}$ field showing the plastic deformation during the avalanche. D) Two clusters of particles obtained through persistent homology corresponding with localized plastic events that comprise the avalanche in C). Note that the $D^2_{\text{min}}$ field and clustering are calculated with the individual spherical sub-particles that comprise the rods. E,F) $\kappa$ values over a range of $\sigma_{\text{act}}$ for the 6 lowest harmonic modes (E) and cubic modes (F) colored by their overlaps with the unstable mode $\hat{\phi}_{\text{us}}$ at $\sigma_{\text{act}} \approx 0.00088$. G) $b(\hat{\pi})$ values for the 6 lowest cubic modes, colored by their proficiency with the rearrangement at $\sigma_{\text{act}} \approx 0.00088$. H) Distribution of cubic mode energy barriers. Dashed lines indicate power laws $P(b) \propto b^{0.6}$ and $P(b) \propto b$. I) Excess fraction of successful predictions (relative to random clusters) of upcoming rearrangements. Inset shows same data rescaled by the system size $N$ to demonstrate collapse.
• Figure S1: A) Illustration of the various quantities associated with a single rod and its 5 constituent particles. B) Distribution of strain intervals $\Delta\gamma_{\text{avalanche}}$ between the onset of subsequent instabilities for different system sizes (colors in all plots given by legend in D). The most likely strain for each system size is denoted by a dashed line. C) Distribution of the maximum value of $D^2_{\text{min}}$ between subsequent stable configurations. D) Average fraction of particles that experience a neighbor change $\Delta_{\text{neighbors}}$ at different amounts of non-affine displacement (shaded regions show one standard deviation from the mean). Black lines indicate upper and lower limits on avalanches that we consider: that is, only $D^2_{\text{min}}$ fields with maximum values between these lines is clustered for further analysis.
• Figure S2: A) Values of the scalar field $\Pi$. B) Cubic mode $\hat{\pi}$ corresponding to $\Pi$ in A), along with the associated cluster of points (same as Fig 3B in the main text). C-E) Energy barriers (C), convergence probabilities (D) and proficiencies (E) of the lowest 40 cubic modes (indexed by $m$) of the configuration shown in A-B. The proficiencies are calculated between the cubic modes and the two rearrangements shown in Fig 3D of the main text, with different colors corresponding to each rearrangement. The lowest two cubic modes each have a high proficiency (greater than the threshold $\chi_{\text{th}}=0.02$, shown with a dashed line) with one of the rearrangement clusters. (F) Average number of cubic modes obtained at different intervals of stress $\Delta\sigma_{\text{act}}$ before an instability for different system sizes (given by legend in (H)). (G) Average value of the smallest energy barrier $b(\hat{\pi}^0)$ at different $\Delta\sigma_{\text{act}}$. Dashed line indicates the scaling $b(\hat{\pi}^0)\propto (\Delta\sigma_{\text{act}})^{4/3}$. (H) Average probability of converging to a cubic mode as a function of the energy barrier $b(\hat{\pi})$. (I) Relative probability density of the proficiency $\chi$ between cubic modes and rearrangements. Dashed line indicates the threshold value $\chi_{\text{th}}$ used to separate "successes" from "failures".