Reentrant transition to collective actuation in active solids with a polarizing field

Abstract

Collective actuation in active solids - the spontaneous coherent excitation of a few vibrational modes - emerges from a feedback between structural deformations and the orientation of active forces. It is an excellent candidate as a basic mechanism for oscillatory dynamics and regulation in dense living systems, and a better control over its onset would open new avenues in the life sciences. Combining model experiments, simulations and theory, we study the dynamics of such an active solid in the presence of an external field that polarizes the active forces. The experiments reveal a novel oscillatory regime absent at zero field. The theoretical analysis of a single agent demonstrates that the small field oscillations and the large field ones can be mapped onto the bounded and unbounded phase dynamics of a nonlinear pendulum. In the many agents case, the transition to collective actuation is promoted at low field, leading to a reentrant transition.

Figures (5)

• Figure 1: Active solids in a polarizing field. (a) Schematic phase diagram as a function of the elasto-active coupling $\Pi$ and the field amplitude $h$. At $h = 0$, a transition between a disordered regime and collective actuation – either noise induced (NICA), or chiral oscillations (CO) – takes place baconnier2022selectivebaconnier2024noise. Adding a field, the disordered phase polarizes (FP) (color-coded from light to dark blue as polarization increases). In the green region, oscillating dynamics take place, and for large enough fields, a new dynamical regime emerges, taking the form of bounded oscillations around the field orientation, analogous to the motion of Windscreen Wipers (WW). (b-c) Schematic FP and WW regimes; red arrows: polarities $\boldsymbol{\hat{n}}_i$; trajectories of particle positions color-coded from blue to red.
• Figure 2: Single active unit experiments. (a) Geometry and notations; (b) Phase diagram (white square: FP, red circles: WW, green diamonds: CO, red circles inside green diamonds: coexistence between WW and CO, the black lines indicate $\Pi_c = \omega_{0}^{2} + h$ and $\Pi^{\star} = \omega_{0}^{2} + 3h$). (c) Polarization $m$ as a function of $h/\omega_{0}^{2}$ for $\Pi/\omega_{0}^{2} \in [0.74, 1.08, 1.24]$ (same markers as (b)); (d/f/h/j) Real space dynamics of the FP, WW, FD, CO regimes (red arrows: polarity $\boldsymbol{\hat{n}}$, trajectories of particle positions color-coded from blue to red, scale bars: $10$ cm). (e/g/i/k) Probability densities $\rho(\theta, \varphi)$, the white arrows indicate the direction of the dynamics when relevant. The parameter values are: (d/e) FP: $\Pi/\omega_{0}^{2} = 0.91$, $h/\omega_{0}^{2} = 0.08$ ($\beta = 10.7^{\circ}$); (f/g) WW: $\Pi/\omega_{0}^{2} = 1.24$, $h/\omega_{0}^{2} = 0.11$ ($\beta = 10.7^{\circ}$); (h/i) FD: $\Pi/\omega_{0}^{2} = 0.91$, $h/\omega_{0}^{2} = 0.0$ ($\beta = 0^{\circ}$); (j/k) CO: $\Pi/\omega_{0}^{2} = 1.24$, $h/\omega_{0}^{2} = 0.0$ ($\beta = 0^{\circ}$).
• Figure 3: Collective actuation in the presence of an external field. (a/c/e) Real space dynamics (same color code and scale bars as in Fig. \ref{['fig:fig2']}-d) and (b/d/f) polarizations in the transverse (red) and longitudinal (black) direction for a triangular lattice pinned at its edges, for increasing field; (bottom) $h = 0$ ($\beta = 0^{\circ}$), CO; (middle) $h = 0.11$ ($\beta = 10.7^{\circ}$), WW; (top) $h = 0.21$ ($\beta = 21.4^{\circ}$), FP; (g/k) Real space dynamics for square lattices pinned at both ends (top and bottom rows are pinned) for $\beta = 8.5^{\circ}$, with $W=4$ (g) and $W=2$ (k); and (h-j) and (l-n) same as (b/d/f) for respectively the stiff ($W = 4$) and soft ($W = 2$) lattices for $h = 0$ ($\beta = 0^{\circ}$) (bottom), $h = 8.7 \times 10^{-4}$ ($\beta = 8.5^{\circ}$) (middle), $h = 1.1 \times 10^{-3}$ ($\beta = 10.7^{\circ}$) (top). (o-p) Root mean squared transverse (red) and longitudinal (black) polarizations as a function of the tilt $\beta$ for $W = 4$ (o), and $W = 2$ (p). The gray areas cover the range of polarization accessible for a system of $N$ randomly oriented unit vectors ($[-1/\sqrt{N}, 1/\sqrt{N}]$ for $M_{\perp/\parallel}$ and $[0, 1/\sqrt{N}]$ for $\langle M_{\perp/\parallel}^2 \rangle^{1/2}$); the green area indicates the region where the reentrance transition to a WW regime occurs.
• Figure 4: Noiseless single active unit, theory and numerics. (a-c) Dynamics of the displacements in the dynamical regimes as named, as obtained from noiseless simulations of Eqs. (\ref{['eq:sp']}); the trajectory is shown over one period, and colored with time running from blue to red; the dark arrows are snapshots of the orientation of the active force $\boldsymbol{\hat{n}}$; for $\Pi/\omega_{0}^{2} = 3.95$; (a) $h/\omega_{0}^{2} = 0.62$: CO, (b) $h/\omega_{0}^{2} = 1.15$: WW$^2$, (c) $h/\omega_{0}^{2} = 2.23$: WW; the dashed circles indicate the radius $\Pi/\omega_{0}^{2}$. (d) Numerical phase diagram (white: FP, green: CO, red: WW, blue: WW$^2$); the top solid black (resp. dashed gray) line indicates $\Pi_c = \omega_{0}^{2} + h$ (resp. $\Pi^{\star} = \omega_{0}^{2} + 3h$). The black dot indicates the tip of the WW$^2$ domain of existence. (e) Rescaled frequency $\omega_{\varphi}$ as a function of $\tilde{H}$, for small $h/\omega_{0}^{2} = 10^{-4}$, colored markers indicate simulations of Eqs. (\ref{['eq:sp']}) (green: CO, red: WW), and the solid black (resp. gray) lines are the stable (resp. unstable) orbits of the pendulum equations, as obtained from the energy drift $\delta E$. (f-g) Energy drift $\delta E$ as a function of $E/\tilde{H}$ for three values of $\tilde{H}$, as indicated. The green marks indicate the stable orbits.
• Figure 5: Coarse-grained description, theory and numerics. (a-c) Dynamics of the displacements in the dynamical regimes as named, in the degenerate case, obtained by simulating Eqs. (\ref{['eq:cg']}); same conventions as Figs. \ref{['fig:fig4']}a-c. (d-e) Phase diagrams in the degenerate case (d), $\omega_{\parallel}^{2} = \omega_{\perp}^{2} = \omega_{0}^{2}$, and the non-degenerate case (e), $\omega_{\parallel}^{2} = 2\omega_{\perp}^{2} = 2\omega_{0}^{2}$; same color code as Fig. \ref{['fig:fig4']}d, dark blue markers at $h = 0$ indicate the NICA regime; the black lines represent $\Pi_c(h,D)$ as given by Eq. (\ref{['eq:threshold']}); areas where two colors appear intermingled indicate zones of coexistence between regimes. In all panels, $D=0.1$ and $\omega_{0}^{2} = 1$ are fixed.