Mechanochemical feedback drives complex inertial dynamics in active solids

Abstract

Active solids combine internal active driving with elasticity to realize states with nonequilibrium mechanics and autonomous motion. They are often studied in overdamped settings, e.g., in soft materials, and the role of inertia is less explored. We construct a model of a chemically active solid that incorporates mechanochemical feedback and show that, when feedback overwhelms mechanical damping, autonomous inertial dynamics can spontaneously emerge through sustained consumption of chemical fuel. By combining numerical simulations, analysis and dynamical systems approaches, we show how active feedback drives complex nonlinear dynamics on multiple time-scales, including limit cycles and chaos. Our results suggest design principles for creating ultrafast actuators and autonomous machines from soft, chemically-powered solids.

Figures (4)

• Figure 1: Mechanochemical coupling. (a,b) Feedback loops among reaction ($\chi$), strain ($\partial_x u$), strain rate ($\partial_x v$) in passive (a) and active (b) solids. In the passive limit (a), detailed balance enforces vanishing strain-rate coupling, and a reciprocal strain-reaction coupling ($k_BTC$). When active (b), the feedback drives nonequilibrium fluxes on independent time scales $\tau_{1,2}$. (c) Contour plot of reaction rate $\partial_t \chi$ (in A.U.) using the 0D model in Eq. \ref{['eq:0D']}, with the dashed line marking $\partial_t \chi=0$ so $\chi$ grows (decays) above (below) this curve. An example trajectory is shown where a perturbation from the origin leads to a limit cycle, with four representative states illustrated as insets, where the background shading refers to concentration $\chi$. Starting with the bottom left corner, increased strain breaks stress-sensitive crosslinkers, which drives $\chi$ to increase, followed by decrease in strain, and recovery back to the first state.
• Figure 2: Dynamics of the 1D mechanochemical system in Eq. \ref{['eq:1Deqs']}. We set $L = 2\pi$, $\eta = 0.16$, and $\tau_c = 100$ along with (a) $\alpha_u = 50, \alpha_v = 0$, (b) $\alpha_u = -10, \alpha_v = 0$, (c) $\alpha_u = -50, \alpha_v = 0$, (d) $\alpha_u = 50, \alpha_v = 77$, (e) $\alpha_u = 50, \alpha_v = 80$, (f) $\alpha_u = 50, \alpha_v = 82$. We include a small diffusion constant $D = 0.05$ to regularize chemical gradients. The initial conditions are $u(x,0) = 0.01\sin(x)$, $v(x,0) = 0 = \chi(x,0)$ The first row shows the displacement field $u(x,t)$. The second row shows the phase space $({\epsilon=\partial_x u},v)$ trajectory at the point $x = 9\pi/25$.
• Figure 3: Dynamical phases due to nonreciprocal strain-chemistry feedback ($A_v=0, A_u<0$) in the 0D model (Eq. \ref{['eq:0D']}). (a) Phase diagram ($\mathsf{Ch} = 100$). (b-d) Typical trajectories in the $(U,V)$ plane for (b) the outer (larger) LC with $\left( \mathsf{Vi}, A_u \right)=\left(0.70, -130.0\right)$ (underdamped regime) and the inner (smaller) LC with $\left( \mathsf{Vi},A_u \right)=\left(1.2,-140.0 \right)$ (overdamped regime), (c) one FP at $U_0 = X_0 = 0, V_0=0$ with $\left( \mathsf{Vi}, A_u \right)=\left(0.70, -20.0\right)$, and (d) FPs with the trajectory converging to one of them $U_+ = -X_{+} = \sqrt{1+A_u}$ with $\left( \mathsf{Vi}, A_u \right)=\left(0.70, -0.5\right)$. The FPs are shown as red (unstable) and green (stable) dots in (b-d), with the arbitrarily chosen initial conditions denoted by the black star. In panels (b-d), different symbols adjacent to the panel index represent the corresponding points in the phase diagram in (a). (e) Dependence of the time period $T$ on $A_u$ for different $\mathsf{Ch}$ in the LC region (in $\tau_{\rm{el}}=1$ units).
• Figure 4: Dynamical phases due to nonreciprocal strain-strain rate-chemistry feedback ($A_v>0, A_u>0$) in the 0D model (Eq. \ref{['eq:0D']}). (a) Phase diagram ($\mathrm{Ch} = 100$, $A_u = 50$). (b-f) Typical trajectories in the $(U,V)$ plane for the FPs with the trajectory converging to (b) one FP $U_+ = -X_+ = \sqrt{1+A_u}$ for $A_v=60.5$, (c) an LC around one unstable FP for $A_v=68.0$, (d) period doubling for $A_v=78.5$, (e) chaos for $A_v=84.0$, and (f) the large LC enclosing all FPs for $A_v=106.0$. In all cases, we set $\mathsf{Vi}=0.16$ and $A_u=50.0$. The black stars in (b-f) denote the arbitrarily chosen initial condition. (g) Bifurcation diagram showing the route from LC around the FP $U_{\pm} = -X_{\pm} = \sqrt{1+A_u}$ to chaos as a function of the parameter $A_v$. Here, $u_{\rm m}$ represents the local maximum value of $U$SM2024. (h) Dependence of the time period $T$ on $A_v$ at $\mathsf{Vi}=0.16$ in the large LC region, approaching $2\pi$ as $A_v$ increases (in $\tau_{\rm{el}}=1$ units).