Self-Oscillation and Synchronization Transitions in Elastoactive Structures

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Self-Oscillation and Synchronization Transitions in Elastoactive Structures

Ellen Zheng, Martin Brandenbourger, Louis Robinet, Peter Schall, Edan Lerner, and Corentin Coulais

Phys. Rev. Lett. 130, 178202 – Published 25 April 2023

Abstract

The interplay between activity and elasticity often found in active and living systems triggers a plethora of autonomous behaviors ranging from self-assembly and collective motion to actuation. Among these, spontaneous self-oscillations of mechanical structures is perhaps the simplest and most widespread type of nonequilibrium phenomenon. Yet, we lack experimental model systems to investigate the various dynamical phenomena that may appear. Here, we introduce a centimeter-sized model system for one-dimensional elastoactive structures. We show that such structures exhibit flagellar motion when pinned at one end, self-snapping when pinned at two ends, and synchronization when coupled together with a sufficiently stiff link. We further demonstrate that these transitions can be described quantitatively by simple models of coupled pendula with follower forces. Beyond the canonical case considered here, we anticipate our work to open avenues for the understanding and design of the self-organization and response of active biological and synthetic solids, e.g., in higher dimensions and for more intricate geometries.

Received 12 July 2021
Revised 23 February 2023
Accepted 5 April 2023

DOI:https://doi.org/10.1103/PhysRevLett.130.178202

Physics Subject Headings (PhySH)

Biological self-organization Biomimetic & bio-inspired materials Emergence of patterns

Living matter & active matter

Physics of Living SystemsPolymers & Soft Matter

Authors & Affiliations

Ellen Zheng, Martin Brandenbourger , Louis Robinet, Peter Schall, Edan Lerner, and Corentin Coulais

Institute of Physics, Universiteit van Amsterdam, Science Park 904, 1098 XH Amsterdam, Netherlands

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Issue

Vol. 130, Iss. 17 — 28 April 2023

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Images

Figure 1
Emergence of self-oscillations in elastoactive chains. (a) Configurations of seven active particles connected by a flexible rubber chain. (b) Close-up details of two active particles unveiling the design of the microbot. (c) A close-up of the linkage between each particle with width ( $W$ ), thickness ( $H$ ) and length ( $L$ ). (d) Histogram of the active force measurements conducted at $0.05 mm / \min$ . (e),(f) Snapshots of the trajectories of the active particles showing the oscillations changed from self-amplified to overdamped with $W = 5 mm$ , 4.4 mm, and 2 mm corresponding to elastoactive number $σ = 0.17$ , 0.21, and 0.80, respectively. See also Videos in Supplemental Material [47].
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Figure 2
Characterization of the dynamics of elastoactive chains. (a) Snapshot of the elastoactive chain with $W = 17.7 mm$ during its self-oscillation. (b) Time series of the angle between the first and last particle (mean polarization) was one of the parameters we chose to characterize the system, blue and orange represent the $σ = 0.695$ and $σ = 0.166$ chain, respectively. With the average of tangential angles of the particles (mean curvature) being the other parameter, plotting mean polarization ( $Θ$ ) against mean curvature ( $Ω$ ) gives (c) limit cycle showing that active forces balance with dissipation toward stable self-oscillation. (d) Schematics of an elastoactive chain of seven pendulums with active forces. (e),(f) Simulation results at $σ = 0.8$ showing good agreement with the experimental results.
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Figure 3
An active system featuring a Hopf bifurcation at $σ = 0.15$ . Simulation and experimental results showing the evolution of (a) amplitude and (b) frequency with increasing $σ$ , respectively, for the chain with $N = 7$ . The inset in (a) is a log-log plot demonstrating the power law between $Θ$ and $σ$ . (c) Real and imaginary (d) part of the eigenvalues of the Jacobian of Eqs. (1) and (2) vs elastoactive number $σ$ for a minimal chain with $N = 2$ . In all panels, the gray area represents the stable region.
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Figure 4
Elastoactive chains pinned at both ends. Stills of an active chain where both ends are pinned with overlaid trajectories of the Hexbugs. The end-to-end distance between the pinning points is $ℓ = 500 mm$ (left), $ℓ = 475 mm$ (middle), and $ℓ = 400 mm$ (right). The length of the undeformed chain is $L = 500 mm$ . (b) Model: sketch of the chain pinned at both ends. (c) Amplitude and (d) frequency of the self-snapping oscillations vs the compressive strain $(L - ℓ) / L$ . The insets are the corresponding data from the numerical model defined in Supplemental Material [47]. The elastoactive number of the chain is $σ = 0.8$ . See also Videos in Supplemental Material [47].
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Figure 5
Synchronization of two elastoactive chains with different elasticity coupled by the first particles only. (a) Snapshot of a pair of elastoactive chains with different elasticity coupled by another stiff silicon rubber chain. (b) Evolution of the instantaneous phase differences [instantaneous phases $Φ_{1}$ (dashed lines) and $Φ_{2}$ (solid lines) in inset] of two elastoactive chains with coupling strength $K = 0.8$ (yellow lines) and $K = 1.2$ (brown lines). (c) Instantaneous frequency difference extracted from the instantaneous phase difference $Ψ$ vs rescaled coupling stiffness $κ$ , with error bars corresponding to deviations on the linear regression [dashed blue line and black lines in panel (b)]. (d) Schematics of the numerical model adding a coupling spring ( $K$ ) to two previously established elastoactive chains. (e),(f) Same data as (b),(c) for the numerical simulations. An Arnold tongue is observed when the frequency mismatch is plotted against the ratio between both elastoactive numbers $ρ$ and the stiffness $κ$ [inset of panel (f)].
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