Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Selective and collective actuation in active solids

Nature Physics volume 18, pages 1234–1239 (2022)Cite this article

Subjects

Abstract

Active solids consist of elastically coupled out-of-equilibrium units performing work1,2,3,4,5,6,7,8,9,10,11,12,13. They are central to autonomous processes, such as locomotion, self-oscillations and rectification, in biological systems14,15,16,17,18,19,20,21,22,23,24,25, designer materials26 and robotics27,28,29,30,31. Yet, the feedback mechanism between elastic and active forces as well as the possible emergence of collective behaviours in a mechanically stable elastic solid remains elusive. Here we introduce a minimal realization of an active elastic solid in which we characterize the emergence of selective and collective actuation resulting from the interplay between activity and elasticity. Polar active agents exert forces on the nodes of a two-dimensional elastic lattice. The resulting displacement field nonlinearly reorients the active agents. For a large-enough coupling, a collective oscillation of the lattice nodes around their equilibrium position emerges. Only a few elastic modes are actuated and crucially, they are not necessarily the lowest energy ones. By combining experiments with the numerical and theoretical analyses of an agent’s model, we unveil the bifurcation scenario and selection mechanism by which the collective actuation takes place. Our findings may provide a new mechanism for oscillatory dynamics in biological systems14,19,21,24 and the opportunity for bona fide autonomy in metamaterials32,33.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Active-solids design principle.
Fig. 2: Selective and collective actuation in two-dimensional elastic lattices, pinned at their edges.
Fig. 3: Large-N lattices.
Fig. 4: Mean-field, single-particle, one-dimensional-lattice phase diagrams.

Similar content being viewed by others

References

  1. Koenderink, G. H. et al. An active biopolymer network controlled by molecular motors. Proc. Natl Acad. Sci. USA 106, 15192–15197 (2009).

    Article  ADS  Google Scholar 

  2. Henkes, S., Fily, Y. & Marchetti, M. C. Active jamming: self-propelled soft particles at high density. Phys. Rev. E 84, 040301 (2011).

    Article  ADS  Google Scholar 

  3. Menzel, A. M. & Löwen, H. Traveling and resting crystals in active systems. Phys. Rev. Lett. 110, 055702 (2013).

    Article  ADS  Google Scholar 

  4. Berthier, L. & Kurchan, J. Non-equilibrium glass transitions in driven and active matter. Nat. Phys. 9, 310–314 (2013).

    Article  Google Scholar 

  5. Ferrante, E., Turgut, A. E., Dorigo, M. & Huepe, C. Elasticity-based mechanism for the collective motion of self-propelled particles with springlike interactions: a model system for natural and artificial swarms. Phys. Rev. Lett. 111, 268302 (2013).

    Article  ADS  Google Scholar 

  6. Prost, J., Jülicher, F. & Joanny, J.-F. Active gel physics. Nat. Phys. 11, 111–117 (2015).

    Article  Google Scholar 

  7. Ronceray, P., Broedersz, C. P. & Lenz, M. Fiber networks amplify active stress. Proc. Natl Acad. Sci. USA 113, 2827–2832 (2016).

    Article  ADS  Google Scholar 

  8. Woodhouse, F. G., Ronellenfitsch, H. & Dunkel, J. Autonomous actuation of zero modes in mechanical networks far from equilibrium. Phys. Rev. Lett. 121, 178001 (2018).

    Article  ADS  Google Scholar 

  9. Briand, G., Schindler, M. & Dauchot, O. Spontaneously flowing crystal of self-propelled particles. Phys. Rev. Lett. 120, 208001 (2018).

    Article  ADS  Google Scholar 

  10. Giavazzi, F. et al. Flocking transitions in confluent tissues. Soft Matter 14, 3471–3477 (2018).

    Article  ADS  Google Scholar 

  11. Klongvessa, N., Ginot, F., Ybert, C., Cottin-Bizonne, C. & Leocmach, M. Active glass: ergodicity breaking dramatically affects response to self-propulsion. Phys. Rev. Lett. 123, 248004 (2019).

    Article  ADS  Google Scholar 

  12. Maitra, A. & Ramaswamy, S. Oriented active solids. Phys. Rev. Lett. 123, 238001 (2019).

    Article  ADS  Google Scholar 

  13. Scheibner, C. et al. Odd elasticity. Nat. Phys. 16, 475–480 (2020).

    Article  Google Scholar 

  14. Abercrombie, M. & Heaysman, J. E. Observations on the social behaviour of cells in tissue culture: II. ‘Monolayering’ of fibroblasts. Exp. Cell Res. 6, 293–306 (1954).

    Article  Google Scholar 

  15. Vilfan, A. & Frey, E. Oscillations in molecular motor assemblies. J. Phys.: Condens. Matter 17, S3901 (2005).

    ADS  Google Scholar 

  16. Szabo, B. et al. Phase transition in the collective migration of tissue cells: experiment and model. Phys. Rev. E 74, 061908 (2006).

    Article  ADS  Google Scholar 

  17. Mizuno, D., Tardin, C., Schmidt, C. F. & MacKintosh, F. C. Nonequilibrium mechanics of active cytoskeletal networks. Science 315, 370–373 (2007).

    Article  ADS  Google Scholar 

  18. Banerjee, S., Utuje, K. J. & Marchetti, M. C. Propagating stress waves during epithelial expansion. Phys. Rev. Lett. 114, 228101 (2015).

    Article  ADS  Google Scholar 

  19. Smeets, B. et al. Emergent structures and dynamics of cell colonies by contact inhibition of locomotion. Proc. Natl Acad. Sci. USA 113, 14621–14626 (2016).

    Article  ADS  Google Scholar 

  20. Bi, D., Yang, X., Marchetti, M. C. & Manning, M. L. Motility-driven glass and jamming transitions in biological tissues. Phys. Rev. X 6, 021011 (2016).

    Google Scholar 

  21. Chen, C., Liu, S., Shi, X.-q, Chaté, H. & Wu, Y. Weak synchronization and large-scale collective oscillation in dense bacterial suspensions. Nature 542, 210–214 (2017).

    Article  ADS  Google Scholar 

  22. Needleman, D. & Dogic, Z. Active matter at the interface between materials science and cell biology. Nat. Rev. Mater. 2, 17048 (2017).

    Article  ADS  Google Scholar 

  23. Holmes, D. F. et al. Synchronized mechanical oscillations at the cell–matrix interface in the formation of tensile tissue. Proc. Natl Acad. Sci. USA 115, E9288–E9297 (2018).

    Article  Google Scholar 

  24. Peyret, G. et al. Sustained oscillations of epithelial cell sheets. Biophys. J. 117, 464–478 (2019).

    Article  ADS  Google Scholar 

  25. Henkes, S., Kostanjevec, K., Collinson, J. M., Sknepnek, R. & Bertin, E. Dense active matter model of motion patterns in confluent cell monolayers. Nat. Commun. 11, 1405 (2020).

    Article  ADS  Google Scholar 

  26. Brandenbourger, M., Locsin, X., Lerner, E. & Coulais, C. Non-reciprocal robotic metamaterials. Nat. Commun. 10, 4608 (2019).

    Article  ADS  Google Scholar 

  27. Brambilla, M., Ferrante, E., Birattari, M. & Dorigo, M. Swarm robotics: a review from the swarm engineering perspective. Swarm Intell. 7, 1–41 (2013).

    Article  Google Scholar 

  28. Pratissoli, F., Reina, A., Lopes, Y. K., Sabattini, L. & Groß, R. A soft-bodied modular reconfigurable robotic system composed of interconnected kilobots. In 2019 International Symposium on Multi-Robot and Multi-Agent Systems (MRS) 50–52 (IEEE, 2019).

  29. Li, S. et al. Particle robotics based on statistical mechanics of loosely coupled components. Nature 567, 361–365 (2019).

    Article  ADS  Google Scholar 

  30. Dorigo, M., Theraulaz, G. & Trianni, V. Reflections on the future of swarm robotics. Sci. Robot. 5, eabe4385 (2020).

  31. Oliveri, G., van Laake, L. C., Carissimo, C., Miette, C. & Overvelde, J. T. Continuous learning of emergent behavior in robotic matter. Proc. Natl Acad. Sci. USA 118, e2017015118 (2021).

  32. Bertoldi, K., Vitelli, V., Christensen, J. & Van Hecke, M. Flexible mechanical metamaterials. Nat. Rev. Mater. 2, 17066 (2017).

    Article  ADS  Google Scholar 

  33. Pishvar, M. & Harne, R. L. Foundations for soft, smart matter by active mechanical metamaterials. Adv. Sci. 7, 2001384 (2020).

    Article  Google Scholar 

  34. Toner, J. & Tu, Y. Long-range order in a two-dimensional dynamical XY model: how birds fly together. Phys. Rev. Lett. 75, 4326–4329 (1995).

    Article  ADS  Google Scholar 

  35. Vicsek, T. & Zafeiris, A. Collective motion. Phys. Rep. 517, 71–140 (2012).

    Article  ADS  Google Scholar 

  36. Bricard, A., Caussin, J.-B., Desreumaux, N., Dauchot, O. & Bartolo, D. Emergence of macroscopic directed motion in populations of motile colloids. Nature 503, 95–98 (2013).

    Article  ADS  Google Scholar 

  37. Marchetti, M. C. et al. Hydrodynamics of soft active matter. Rev. Mod. Phys. 85, 1143 (2013).

    Article  ADS  Google Scholar 

  38. Wittkowski, R. et al. Scalar φ4 field theory for active-particle phase separation. Nat. Commun. 5, 4351 (2014).

  39. Peshkov, A., Bertin, E., Ginelli, F. & Chaté, H. Boltzmann-Ginzburg-Landau approach for continuous descriptions of generic Vicsek-like models. Eur. Phys. J. Spec. Top. 223, 1315–1344 (2014).

    Article  Google Scholar 

  40. Dauchot, O. & Démery, V. Dynamics of a self-propelled particle in a harmonic trap. Phys. Rev. Lett. 122, 068002 (2019).

    Article  ADS  Google Scholar 

  41. Fruchart, M., Hanai, R., Littlewood, P. B. & Vitelli, V. Non-reciprocal phase transitions. Nature 592, 363–369 (2021).

    Article  ADS  Google Scholar 

  42. Miskin, M. Z. et al. Electronically integrated, mass-manufactured, microscopic robots. Nature 584, 557–561 (2020).

    Article  ADS  Google Scholar 

Download references

Acknowledgements

P.B. was supported by a PhD grant from ED564 ‘Physique en Ile de France’. D.S. was supported by a Chateaubriand fellowship. G.D. acknowledges support from Fondecyt grant no. 1210656 and ANID–Millenium Science Initiative Program—Code NCN17_092. We are grateful to M. Fruchart and V. Vitelli for fruitful discussions regarding non-reciprocity in elastic materials.

Author information

Authors and Affiliations

  1. Gulliver UMR CNRS 7083, ESPCI Paris, Université PSL, Paris, France

    P. Baconnier, D. Shohat, V. Démery & O. Dauchot

  2. School of Physics and Astronomy, Tel Aviv University, Tel Aviv, Israel

    D. Shohat

  3. Instituto de Física, Pontificia Universidad Católica de Chile, Santiago, Chile

    C. Hernández López & G. Düring

  4. ANID—Millenium Nucleus of Soft Smart Mechanical Metamaterials, Santiago, Chile

    C. Hernández López & G. Düring

  5. Van der Waals-Zeeman Institute, Institute of Physics, Universiteit van Amsterdam, Amsterdam, The Netherlands

    C. Coulais

  6. Univ Lyon, ENSL, CNRS, Laboratoire de Physique, Lyon, France

    V. Démery

Authors
  1. P. Baconnier
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. Shohat
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. C. Hernández López
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. C. Coulais
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. V. Démery
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. G. Düring
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. O. Dauchot
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

O.D., C.C. and G.D. conceived the project. P.B. and D.S. performed the experiments. P.B., D.S. and O.D. analysed the experimental results. P.B., V.D., O.D., C.H.L. and G.D. worked out the theory. All the authors contributed to the writing of the manuscript.

Corresponding authors

Correspondence to P. Baconnier or O. Dauchot.

Ethics declarations

Competing interests

The authors declare no competing interests.

Peer review

Peer review information

Nature Physics thanks Anton Souslov and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Additional information

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data

Extended Data Fig. 1 Fixed points stability thresholds for the experimental’s structures.

(a/c) Stability thresholds upper-bound \({\pi }_{c,u}^{\{j,k\}}\) computed from Eq. (S45) for every pair of modes. The darker the pixel, the greater the upper-bound found. Remarkably, the best bound is always achieved for the pair of modes concerned by the condensation. (b/d) Fraction of stable fixed points as a function of π. The fraction of stable fixed points (•) is computed by integrating the histogram of the stability thresholds found by Eq. (S36) on one million configurations of the polarity field, drawn randomly and independently. Inset: sketch of the two most excited modes, which realized the best bound in (a/c). (a/b) For the triangular lattice pinned at the edges (α = 1.29). \({\pi }_{c}^{\min }=0.676\), \({\pi }_{c}^{\max }=1.20\), \({\pi }_{c}^{{{{\rm{upp}}}}}=1.49\) (c/d) For the kagome lattice pinned at the edges (α = 1.02). \({\pi }_{c}^{\min }=0.375\), \({\pi }_{c}^{\max }=0.751\), \({\pi }_{c}^{{{{\rm{upp}}}}}=0.883\).

Extended Data Fig. 2 Effect of the noise on the collective actuation regime.

(a) Averaged (over time) mean angular frequency (over particles) in absolute value as a function of D for π = 2.0. The gray area represents the 1-σ fluctuations. (b) Averaged (over time) mean angular frequency (over particles) in absolute value as a function of π for increasing angular noise. From top to bottom, D = 0, 10−3, 10−2, 2.10−2, 5.10−2. Fluctuations are not shown for sake of clarity. (c) Mean (over particles) angular frequency as a function of time for π = 2.0 and D = 10−1 in the triangular lattice pinned at the edges.

Supplementary information

Supplementary Information

Supplementary Sections 1–10 and Figs. 1–12.

Supplementary Video 1

Alignment experiment: we impose a square motion to an active building block and look at the response of the polarity. At each side of the square, the polarity aligns towards the new velocity vector. Acquired at 40 fps and displayed in real time.

Supplementary Video 2

Experiment: translation in a triangular lattice (N = 37) with free-boundary condition. Acquired at 30 fps and displayed in real time.

Supplementary Video 3

Experiment: rotation dynamics in a triangular lattice (N = 37) with free-boundary condition. Acquired at 30 fps and displayed in real time.

Supplementary Video 4

Experiment: frozen disordered dynamics in the triangular lattice (N = 19 active units). Acquired at 40 fps and displayed in real time.

Supplementary Video 5

Experiment: collective actuation regime in the triangular lattice (N = 19 active units). Acquired at 40 fps and displayed in real time.

Supplementary Video 6

Experiment: heterogeneous regime in the triangular lattice (N = 19 active units). Acquired at 40 fps and displayed in real time.

Supplementary Video 7

Experiment: frozen disordered dynamics in the kagome lattice (N = 12 active units). Acquired at 40 fps and displayed in real time.

Supplementary Video 8

Experiment: collective actuation dynamics in the kagome lattice (N = 12 active units). Acquired at 40 fps and displayed in real time.

Supplementary Video 9

Experiment: frozen dynamics in the single-particle system. Acquired at 40 fps and displayed in real time.

Supplementary Video 10

Experiment: spontaneous oscillations in the single-particle system. Acquired at 40 fps and displayed in real time.

Supplementary Video 11

Numerical simulation: annealing in π in the triangular lattice (N = 19 active units), with the same tension as in the experiment shown in Supplementary Video 6. The elasto-active coupling π is decreased from the collective actuation regime until the system finds a fixed point. Individual polarities are represented by a black arrow, springs are colour coded by the stress state; an elongated spring turns red, whereas a compressed one turns blue.

Supplementary Video 12

Numerical simulation of a large triangular lattice (N = 1,141 active units), sized such that its lower-energy modes have squared eigenfrequencies equal to unity. The system is initialized with zero displacement in every node and random initial condition for the polarities orientations, and π = 2.0. Polarities are shown as arrows coloured by their orientations. Springs are represented in grey. The system quickly finds a collective actuation regime.

Supplementary Video 13

Numerical simulation of a large kagome lattice (N = 930 active units), sized such that its lower-energy modes have squared eigenfrequencies equal to unity. The system is initialized with zero displacement in every node and random initial condition for the polarities orientations, and π = 10.0. Polarities are shown as arrows coloured by their orientations. Springs are represented in grey. The system quickly finds a collective actuation regime.

Source data

Source Data Figs. 2–4 and Extended Data Figs. 1 and 2

Experimental and numerical source data.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Baconnier, P., Shohat, D., López, C.H. et al. Selective and collective actuation in active solids. Nat. Phys. 18, 1234–1239 (2022). https://doi.org/10.1038/s41567-022-01704-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41567-022-01704-x

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing