Skip to main content
SpringerLink
Log in

Lattice model for self-folding at the microscale

  • Regular Article - Soft Matter
  • Published:
The European Physical Journal E Aims and scope Submit manuscript

Abstract

Three-dimensional shell-like structures can be obtained spontaneously at the microscale from the self-folding of 2D templates of rigid panels. At least for simple structures, the motion of each panel is consistent with a Brownian process and folding occurs through a sequence of binding events, where pairs of panels meet at a specific closing angle. Here, we propose a lattice model to describe the dynamics of self-folding. As an example, we study the folding of a pyramid of N lateral faces. We combine analytical and numerical Monte Carlo simulations to find how the folding time depends on the number of faces, closing angle, and initial configuration. Implications for the study of more complex structures are discussed.

Graphic abstract

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Data Availability Statement

This manuscript has no associated data, or the data will not be deposited. [Authors’ comment: The data that support the findings of this study are available from the corresponding author upon reasonable request.]

References

  1. T.C. Shyu, P.F. Damasceno, P.M. Dodd, A. Lamoureux, L. Xu, M. Shlian, M. Shtein, S.C. Glotzer, N.A. Kotov, Nat. Mater. 14, 785 (2015)

    Article  ADS  Google Scholar 

  2. M.K. Blees, A.W. Barnard, P.A. Rose, S.P. Roberts, K.L. McGill, P.Y. Huang, A.R. Ruyack, J.W. Kevek, B. Kobrin, D.A. Muller et al., Nature 524, 204 (2015)

    Article  ADS  Google Scholar 

  3. Z. Liu, H. Du, J. Li, L. Lu, Z.Y. Li, N.X. Fang, Sci. Adv. 4, eaat4436 (2018)

    Article  ADS  Google Scholar 

  4. Y. Zhang, Z. Yan, K. Nan, D. Xiao, Y. Liu, H. Luan, H. Fu, X. Wang, Q. Yang, J. Wang et al., Proc. Natl. Acad. Sci. U.S.A. 112, 11757 (2015)

    Article  ADS  Google Scholar 

  5. K. Miura, Inst. Space Astronaut. Sci. Rep. 618, 1 (1985)

    Google Scholar 

  6. L. Mahadevan, S. Rica, Science 307, 1740 (2005)

    Article  Google Scholar 

  7. M. Schenk, S.D. Guest, Proc. Natl. Acad. Sci. U.S.A. 110, 3276 (2013)

    Article  ADS  Google Scholar 

  8. R. Fernandes, D.H. Gracias, Adv. Drug Deliv. Rev. 64, 1579 (2012)

    Article  Google Scholar 

  9. J. Shim, C. Perdigou, E.R. Chen, K. Bertoldi, P.M. Reis, Proc. Natl. Acad. Sci. U.S.A. 109, 5978 (2012)

    Article  ADS  Google Scholar 

  10. M. Filippousi, T. Altantzis, G. Stefanou, M. Betsiou, D.N. Bikiaris, M. Angelakeris, E. Pavlidou, D. Zamboulis, G. van Tendeloo, RSC Adv. 3, 24367 (2013)

    Article  ADS  Google Scholar 

  11. S. Felton, M. Tolley, E. Demaine, D. Rus, R. Wood, Science 345, 644 (2014)

    Article  ADS  Google Scholar 

  12. S. Pandey, M. Ewing, A. Kunas, N. Nguyen, D.H. Gracias, G. Menon, Proc. Natl. Acad. Sci. U.S.A. 108, 19885 (2011)

    Article  ADS  Google Scholar 

  13. N.A.M. Araújo, R.A. da Costa, S.N. Dorogovtsev, J.F.F. Mendes, Phys. Rev. Lett. 120, 188001 (2018)

    Article  ADS  Google Scholar 

  14. P.M. Dodd, P.F. Damasceno, S.C. Glotzer, Proc. Natl. Acad. Sci. U.S.A. 115, E6690 (2018)

    Article  ADS  Google Scholar 

  15. A. Azam, T.G. Leong, A.M. Zarafshar, D.H. Gracias, PLOS ONE 4, e4451 (2009)

    Article  ADS  Google Scholar 

  16. H.P.M. Melo, C.S. Dias, N.A.M. Araújo, Commun. Phys. 3, 154 (2020)

    Article  Google Scholar 

  17. M. Andersen, C. Panosetti, K. Reuter, Front. Chem. 7, 202 (2019)

    Article  ADS  Google Scholar 

  18. M. Stamatakis, J. Condens, Matter Phys. 27, 013001 (2014)

    Article  Google Scholar 

  19. K.E. Sickafus, E.A. Kotomin, B.P. Uberuaga, Radiation effects in solids (Springer, New York, 2007)

    Book  Google Scholar 

  20. B. C. Arnold, N. Balakrishnan and H. N. Nagaraja, A first course in order statistics (John Wiley & Sons, New York, 1992)

  21. S.B. Yuste, K. Lindenberg, J. Stat. Phys. 85, 501 (1996)

    Article  ADS  Google Scholar 

  22. G.H. Weiss, K.E. Shuler, K. Lindenberg, J. Stat. Phys. 31, 255 (1983)

    Article  ADS  Google Scholar 

  23. D. Holcman, Z. Schuss, SIAM Rev. 56, 213 (2014)

    Article  MathSciNet  Google Scholar 

  24. A. Singer, Z. Schuss, D. Holcman, J. Stat. Phys. 122, 491 (2006)

    Article  ADS  MathSciNet  Google Scholar 

  25. S.B. Yuste, Phys. Rev. Lett. 79, 3565 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  26. K. Basnayake, Z. Schuss, D. Holcman, J. Nonlinear Sci. 29, 461 (2019)

    Article  ADS  MathSciNet  Google Scholar 

  27. S. Redner, A guide to first-passage processes (Cambridge University Press, Cambridge, 2001)

    Book  Google Scholar 

  28. S.D. Lawley, J.B. Madrid, J. Nonlinear Sci. 30, 1207 (2020)

    Article  ADS  MathSciNet  Google Scholar 

Download references

Acknowledgements

We acknowledge financial support from the Portuguese Foundation for Science and Technology (FCT) under Contracts No. PTDC/FIS-MAC/28146/2017 (LISBOA–01–0145–FEDER–028146), UIDB/00618/2020, and UIDP/00618/2020.

Author information

Authors and Affiliations

  1. Centro de Física Teórica e Computacional, Faculdade de Ciências, Universidade de Lisboa, 1749-016, Lisboa, Portugal

    T. S. A. N. Simões, H. P. M. Melo & N. A. M. Araújo

  2. Centro de Física das Universidades do Minho e do Porto, Campus de Gualtar, 4710-057, Braga, Portugal

    T. S. A. N. Simões

  3. Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, 1749-016, Lisboa, Portugal

    N. A. M. Araújo

Authors
  1. T. S. A. N. Simões
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. H. P. M. Melo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. N. A. M. Araújo
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

T.S.A.N.S., H.P.M.M. and N.A.M.A. contributed equally to this work and writing of all text.

Corresponding author

Correspondence to H. P. M. Melo.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Simões, T.S.A.N., Melo, H.P.M. & Araújo, N.A.M. Lattice model for self-folding at the microscale. Eur. Phys. J. E 44, 46 (2021). https://doi.org/10.1140/epje/s10189-021-00056-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epje/s10189-021-00056-3

Search

Navigation