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Shell buckling for programmable metafluids

Nature volume 628, pages 545–550 (2024)Cite this article

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Abstract

The pursuit of materials with enhanced functionality has led to the emergence of metamaterials—artificially engineered materials whose properties are determined by their structure rather than composition. Traditionally, the building blocks of metamaterials are arranged in fixed positions within a lattice structure1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19. However, recent research has revealed the potential of mixing disconnected building blocks in a fluidic medium20,21,22,23,24,25,26,27. Inspired by these recent advances, here we show that by mixing highly deformable spherical capsules into an incompressible fluid, we can realize a ‘metafluid’ with programmable compressibility, optical behaviour and viscosity. First, we experimentally and numerically demonstrate that the buckling of the shells endows the fluid with a highly nonlinear behaviour. Subsequently, we harness this behaviour to develop smart robotic systems, highly tunable logic gates and optical elements with switchable characteristics. Finally, we demonstrate that the collapse of the shells upon buckling leads to a large increase in the suspension viscosity in the laminar regime. As such, the proposed metafluid provides a promising platform for enhancing the functionality of existing fluidic devices by expanding the capabilities of the fluid itself.

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Fig. 1: Metafluid comprising highly deformable capsules.
Fig. 2: Modelling metafluids.
Fig. 3: Programming metafluids for functionality.
Fig. 4: Harnessing nonlinearity for tunable rheology.

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Data availability

The data that support the findings of this study are openly available in GitHub at https://github.com/BertVanRaemdonck/Buckling-Capsule-Metafluids.

References

  1. Engheta, N. & Ziolkowski, R. W. Metamaterials: Physics and Engineering Explorations (Wiley, 2006).

  2. Craster, R. V. & Guenneau, S. Acoustic Metamaterials: Negative Refraction, Imaging, Lensing and Cloaking Vol. 166 (Springer Science & Business Media, 2012).

  3. Deymier, P. A. Acoustic Metamaterials and Phononic Crystals Vol. 173 (Springer Science & Business Media, 2013).

  4. Maldovan, M. Narrow low-frequency spectrum and heat management by thermocrystals. Phys. Rev. Lett. 110, 025902 (2013).

    Article  ADS  PubMed  MATH  Google Scholar 

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

  6. Kadic, M., Bückmann, T., Schittny, R. & Wegener, M. Metamaterials beyond electromagnetism. Rep. Prog. Phys. 76, 126501 (2013). ISSN 0034-4885.

    Article  ADS  PubMed  Google Scholar 

  7. Christensen, J., Kadic, M., Kraft, O. & Wegener, M. Vibrant times for mechanical metamaterials (book review). MRS Commun. 5, 453–462 (2015).

    Article  CAS  MATH  Google Scholar 

  8. Lee, G. et al. Piezoelectric energy harvesting using mechanical metamaterials and phononic crystals. Commun. Phys. 5, 94 (2022).

    Article  CAS  MATH  Google Scholar 

  9. Xu, X. et al. Multifunctional metamaterials for energy harvesting and vibration control. Adv. Funct. Mater. 32, 2107896 (2022).

    Article  CAS  Google Scholar 

  10. Hu, G., Tang, L., Liang, J., Lan, C. & Das, R. Acoustic-elastic metamaterials and phononic crystals for energy harvesting: a review. Smart Mater. Struct. 30, 085025 (2021).

    Article  ADS  CAS  Google Scholar 

  11. Chen, Z., Guo, B., Yang, Y. & Cheng, C. Metamaterials-based enhanced energy harvesting: a review. Physica B 438, 1–8 (2014).

    Article  ADS  CAS  MATH  Google Scholar 

  12. Fowler, C., Silva, S., Thapa, G. & Zhou, J. High efficiency ambient RF energy harvesting by a metamaterial perfect absorber. Opt. Mater. Express 12, 1242–1250 (2022).

    Article  ADS  CAS  Google Scholar 

  13. Ramahi, O. M., Almoneef, T. S., AlShareef, M. & Boybay, M. S. Metamaterial particles for electromagnetic energy harvesting. Appl. Phys. Lett. 101, 173903 (2012).

    Article  ADS  Google Scholar 

  14. Lin, Keng-Te., Lin, H., Yang, T. & Jia, B. Structured graphene metamaterial selective absorbers for high efficiency and omnidirectional solar thermal energy conversion. Nat. Commun. 11, 1389 (2020).

    Article  ADS  CAS  PubMed  PubMed Central  MATH  Google Scholar 

  15. Cortés, E. et al. Optical metasurfaces for energy conversion. Chem. Rev. 122, 15082–15176 (2022).

    Article  PubMed  PubMed Central  MATH  Google Scholar 

  16. Patel, S. K., Surve, J., Katkar, V. & Parmar, J. Optimization of metamaterial-based solar energy absorber for enhancing solar thermal energy conversion using artificial intelligence. Adva. Theory Simul. 5, 2200139 (2022).

    Article  CAS  Google Scholar 

  17. Chen, T., Li, S. & Sun, H. Metamaterials application in sensing. Sensors 12, 2742–2765 (2012).

    Article  ADS  PubMed  PubMed Central  MATH  Google Scholar 

  18. Molerón, M. & Daraio, C. Acoustic metamaterial for subwavelength edge detection. Nat. Commun. 6, 8037 (2015).

    Article  ADS  PubMed  MATH  Google Scholar 

  19. Fan, W., Yan, B., Wang, Z. & Wu, L. Three-dimensional all-dielectric metamaterial solid immersion lens for subwavelength imaging at visible frequencies. Sci. Adv. 2, e1600901 (2016).

    Article  ADS  PubMed  PubMed Central  Google Scholar 

  20. Urzhumov, Y. A. et al. Plasmonic nanoclusters: a path towards negative-index metafluids. Opt. Express 15, 14129–14145 (2007).

    Article  ADS  PubMed  Google Scholar 

  21. Sheikholeslami, S. N., Alaeian, H., Koh, Ai. Leen. & Dionne, J. A. A metafluid exhibiting strong optical magnetism. Nano Lett. 13, 4137–4141 (2013).

    Article  ADS  CAS  PubMed  Google Scholar 

  22. Yang, J. et al. Broadband absorbing exciton-plasmon metafluids with narrow transparency windows. Nano Lett. 16, 1472–1477 (2016).

    Article  ADS  CAS  PubMed  MATH  Google Scholar 

  23. Hinamoto, T., Hotta, S., Sugimoto, H. & Fujii, M. Colloidal solutions of silicon nanospheres toward all-dielectric optical metafluids. Nano Lett. 20, 7737–7743 (2020).

    Article  ADS  CAS  PubMed  Google Scholar 

  24. Kim, K., Yoo, SeokJae, Huh, Ji-Hyeok, Park, Q.-Han & Lee, S. Limitations and opportunities for optical metafluids to achieve an unnatural refractive index. ACS Photon. 4, 2298–2311 (2017).

    Article  CAS  MATH  Google Scholar 

  25. Cho, Y. et al. Using highly uniform and smooth selenium colloids as low-loss magnetodielectric building blocks of optical metafluids. Opt. Express 25, 13822–13833 (2017).

    Article  ADS  CAS  PubMed  Google Scholar 

  26. Brunet, T. et al. Soft 3D acoustic metamaterial with negative index. Nat. Mater. 14, 384–388 (2015).

    Article  ADS  CAS  PubMed  MATH  Google Scholar 

  27. Peretz, O., Ben Abu, E., Zigelman, A., Givli, S. & Gat, A. D. A metafluid with multistable density and internal energy states. Nat. Commun. 13, 1810 (2022).

    Article  ADS  CAS  PubMed  PubMed Central  MATH  Google Scholar 

  28. Djellouli, A., Marmottant, P., Djeridi, H., Quilliet, C. & Coupier, G. Buckling instability causes inertial thrust for spherical swimmers at all scales. Phys. Rev. Lett. 119, 224501 (2017).

    Article  ADS  PubMed  Google Scholar 

  29. Jambon-Puillet, E., Jones, T. J. & Brun, P.-T. Deformation and bursting of elastic capsules impacting a rigid wall. Nat. Phys. 16, 585–589 (2020).

    Article  CAS  MATH  Google Scholar 

  30. Utada, A. S. Monodisperse double emulsions generated from a microcapillary device. Science 308, 537–541 (2005).

    Article  ADS  CAS  PubMed  MATH  Google Scholar 

  31. Chen, Q. Robust fabrication of ultra-soft tunable PDMS microcapsules as a biomimetic model for red blood cells. Soft Matter 19, 5249–5261 (2023).

  32. Puglisi, G. & Truskinovsky, L. Mechanics of a discrete chain with bi-stable elements. J. Mech. Phys. Solids 48, 1–27 (2000).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Benichou, I. & Givli, S. Structures undergoing discrete phase transformation. J. Mech. Phys. Solids 61, 94–113 (2013).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  34. Nagelberg, S. et al. Reconfigurable and responsive droplet-based compound micro-lenses. Nat. Commun. 8, 14673 (2017).

    Article  ADS  PubMed  PubMed Central  Google Scholar 

  35. Guazzelli, É., Morris, J. F. & Pic, S. A Physical Introduction to Suspension Dynamics Cambridge Texts in Applied Mathematics (Cambridge Univ. Press, 2011).

  36. Shewan, H. M. & Stokes, J. R. Viscosity of soft spherical micro-hydrogel suspensions. J. Colloid Interface Sci. 442, 75–81 (2015).

    Article  ADS  CAS  PubMed  MATH  Google Scholar 

  37. Shewan, H. M. & Stokes, J. R. Analytically predicting the viscosity of hard sphere suspensions from the particle size distribution. J. Nonnewton. Fluid Mech. 222, 72–81 (2015).

    Article  MathSciNet  CAS  MATH  Google Scholar 

  38. Dressaire, E. & Sauret, A. Clogging of microfluidic systems. Soft Matter 13, 37–48 (2017).

    Article  ADS  CAS  Google Scholar 

  39. Chien, S., Usami, S. & Bertles, J. F. Abnormal rheology of oxygenated blood in sickle cell anemia. J. Clin. Invest. 49, 623–634 (1970).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

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Acknowledgements

This research was supported by NSF through the Harvard University Materials Research Science and Engineering Center grant number DMR-2011754, the Fund for Scientific Research-Flanders (FWO) and the European Research Council (ERC starting grant ILUMIS). We thank G. Mckinley and B. Keshavarz and S. Sun for their help with the rheology tests; G. Coupier for the idea of a pressure-driven flow; A. Meeussen, N. Rubin, S. Wei, D. Lim and A. Dorrah for their help with optical experiments; and C. McCann and A. Watkins for comments on the paper.

Author information

Author notes

  1. Shmuel Rubinstein

    Present address: The Racah Institute of Physics, The Hebrew University, Jerusalem, Israel

  2. Benjamin Gorissen

    Present address: Department of Mechanical Engineering, KU Leuven and Flanders Make, Heverlee, Belgium

  3. These authors contributed equally: Adel Djellouli, Bert Van Raemdonck, Yang Wang

Authors and Affiliations

  1. J.A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA

    Adel Djellouli, Yang Wang, Yi Yang, Anthony Caillaud, David Weitz, Shmuel Rubinstein, Benjamin Gorissen & Katia Bertoldi

  2. Department of Mechanical Engineering, KU Leuven and Flanders Make, Heverlee, Belgium

    Bert Van Raemdonck

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Contributions

A.D., B.G., B.V.R. and K.B. proposed and developed the research idea. A.D., Y.Y. and A.C. designed and fabricated the centimetre-scale capsules. Y.W. fabricated and characterized the micrometre-scale capsules. A.D. designed and conducted the experiments and optical simulations. B.V.R. conducted the numerical calculations. A.D., B.V.R. and K.B. wrote the paper. K.B., B.G., S.R. and D.W. supervised the research.

Corresponding authors

Correspondence to Benjamin Gorissen or Katia Bertoldi.

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The authors declare no competing interests.

Peer review

Peer review information

Nature thanks Thomas Brunet, Corentin Coulais and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

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Supplementary information

Supplementary Information

This file contains Supplementary text, Figs. 1–28, Tables 1 and 2, and References.

Peer Review File

Supplementary Video 1

Pressure–volume curve of a single capsule.

Supplementary Video 2

Pressure–volume curve of the metafluid.

Supplementary Video 3

Smart gripper.

Supplementary Video 4

Interactions with flexible structures.

Supplementary Video 5

Tunable optical properties.

Supplementary Video 6

Pressure-driven flow.

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Djellouli, A., Van Raemdonck, B., Wang, Y. et al. Shell buckling for programmable metafluids. Nature 628, 545–550 (2024). https://doi.org/10.1038/s41586-024-07163-z

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