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#### Abstract

Active solids such as cell collectives, colloidal clusters, and active metamaterials exhibit diverse collective phenomena, ranging from rigid body motion to shape-changing mechanisms. The nonlinear dynamics of such active materials remains, however, poorly understood when they host zero-energy deformation modes and when noise is present. Here, we show that stress propagation in a model of active solids induces the spontaneous actuation of multiple soft floppy modes, even without exciting vibrational modes. By introducing an adiabatic approximation, we map the dynamics onto an effective Landau free energy, predicting mode selection and the onset of collective dynamics. These results open new ways to study and design living and robotic materials with multiple modes of locomotion and shape change.

- Received 21 October 2023
- Revised 26 January 2024
- Accepted 3 April 2024

DOI:https://doi.org/10.1103/PhysRevLett.132.238303

© 2024 American Physical Society

#### Physics Subject Headings (PhySH)

- Research Areas

Collective behavior

- Physical Systems

Auxetic materialsCollective dynamics

Condensed Matter, Materials & Applied Physics

#### Focus

#### Theory Predicts Collective States of Mobile Particles

###### Published 7 June 2024

Collections of interacting self-propelled objects held rigidly together show patterns of organized behavior that can be predicted.

See more in *Physics*

#### Authors & Affiliations

Claudio Hernández-López^{1,4,*}, Paul Baconnier^{2}, Corentin Coulais^{3}, Olivier Dauchot^{2}, and Gustavo Düring^{4,†}

^{1}Laboratoire de Physique de l’École Normale Supérieure, UMR CNRS 8023, Université PSL, Sorbonne Université, 75005 Paris, France^{2}Gulliver UMR CNRS 7083, ESPCI Paris, Université PSL, Paris, France^{3}Institute of Physics, Universiteit van Amsterdam, 1098 XH Amsterdam, The Netherlands^{4}Instituto de Física, Pontificia Universidad Católica de Chile, 8331150 Santiago, Chile

^{*}claudio.hernandez.lopez@bio.ens.psl.eu^{†}gduring@uc.cl

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##### Issue

Vol. 132, Iss. 23 — 7 June 2024

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#### Images

###### Figure 1

Active rigid body motion and active mechanisms. (a)Enlargement of the experimental active elastic lattice introduced in [22], with self-propelling units—hexbugs—trapped in 3D-printed annuli, connected by springs in a triangular lattice. (b)Experimental rotational dynamics observed under central pinning. Scale bar, 10cm. (c),(d) Experimental translational and rotational dynamics observed for a free structure. Scale bar, 20cm. (e)Alternating translational and rotation dynamics obtained numerically for the same free structure. (f)A rotational-auxetic regime observed numerically for a nonpinned auxetic square system (see the text, Fig.3, and Movie S1 [30]). Trajectories are color coded from blue to red by increasing time.

###### Figure 2

Second-order transitions to solid body motion and mechanisms for one-mode systems. Simulations consider ${\pi}_{r}=0.001$ unless specified. (a)Periodic boundary condition (translation-only) network. (b)–(d)consider a $100\times 100$ translating system. (b)Global polarization magnitude $\parallel \mathit{P}\parallel $ time series; ${\pi}_{\theta}=0.4$. (c)Phase diagram ${\mu}_{P}$ vs ${\pi}_{\theta}$ as a function of noise. ($S$) for simulations, ($T$) for theoretical predictions. (d)Landau free energy as a function of ${\mu}_{x}$ and ${\mu}_{y}$; left: ${\pi}_{\theta}=0.1$; right: ${\pi}_{\theta}=0.6$. (e)A two-layer hexagonal ring with the definition of the rotational angle $\varphi $. (f)–(h)consider a nine-layer hexagonal ring system. (f)Angular velocity $\dot{\varphi}$ time series; ${\pi}_{\theta}=0.4$. (g)Phase diagram ${\mu}_{\varphi}$ vs ${\pi}_{\theta}$ (left) and ${\mu}_{\varphi}$ vs ${\pi}_{r}$, when ${\pi}_{\theta}=0$ (right). (h)Landau free energy as a function of ${\mu}_{\varphi}$; blue: ${\pi}_{\theta}=0.25$; orange: ${\pi}_{\theta}=0.6$. (i)A one-layer auxetic system with the definition of the auxetic angle $\gamma $. (j)–(l)consider an eight-layer auxetic system. (j)Auxetic angular velocity $\dot{\gamma}$ time series; ${\pi}_{\theta}=0.4$. (k)Phase diagram ${\mu}_{\gamma}$ vs ${\pi}_{\theta}$ (left) and ${\mu}_{\gamma}$ vs ${\pi}_{r}$, when ${\pi}_{\theta}=0$ (right). (l)Landau free energy as a function of ${\mu}_{\gamma}$. Blue: ${\pi}_{\theta}=0.25$; orange: ${\pi}_{\theta}=0.6$.

###### Figure 3

Two-mode system phenomenology: switching between modes of actuation. (a)Time series of the magnitude of the global polarization $\mathit{P}$ (blue) and of the angular velocity $\dot{\varphi}$ (red) for a two-ring nonpinned triangular lattice; left to right: ${\pi}_{\theta}=0.10$, 0.35; ${\pi}_{r}=0.1$. (b)Phase diagram ${\mu}_{P}$ and ${\mu}_{\varphi}$ vs ${\pi}_{\theta}$ of a 30-ring nonpinned triangular lattice. (c)Landau free energy for a nonpinned two-ring triangular lattice as a function of ${\mu}_{\varphi}$ and ${\mu}_{x}$; from left to right: ${\pi}_{\theta}=0.10$, 0.35. (d)–(f)consider an eight-layer rotational-auxetic network. (d)Time series of the auxetic angular velocity $\dot{\gamma}$ (blue) and rotational angular velocity $\dot{\varphi}$ (red); from top to bottom: ${\pi}_{\theta}=0.001$, 0.006, 0.014. (e)Phase diagram $\dot{\gamma}$ (top) and $\dot{\varphi}$ (bottom) vs ${\pi}_{\theta}$. The theoretical values are the time average of each solution obtained from following the free energy minima as $\gamma $ varies. Inset: enlargement of the $\dot{\varphi}\ne 0$ region. (f)Landau free energy as it varies with $\gamma $ ($\gamma =0.2+n\pi /2$, with $n=0$, 1, 2, 3); ${\pi}_{\theta}=0.001$, and the evolution time step $\mathrm{\Delta}t=0.01$.