Article | Published:

Curvature-induced symmetry breaking determines elastic surface patterns

Nature Materials volume 14, pages 337342 (2015) | Download Citation

Subjects

Abstract

Symmetry-breaking transitions associated with the buckling and folding of curved multilayered surfaces—which are common to a wide range of systems and processes such as embryogenesis, tissue differentiation and structure formation in heterogeneous thin films or on planetary surfaces—have been characterized experimentally. Yet owing to the nonlinearity of the underlying stretching and bending forces, the transitions cannot be reliably predicted by current theoretical models. Here, we report a generalized Swift–Hohenberg theory that describes wrinkling morphology and pattern selection in curved elastic bilayer materials. By testing the theory against experiments on spherically shaped surfaces, we find quantitative agreement with analytical predictions for the critical curves separating labyrinth, hybrid and hexagonal phases. Furthermore, a comparison to earlier experiments suggests that the theory is universally applicable to macroscopic and microscopic systems. Our approach builds on general differential-geometry principles and can thus be extended to arbitrarily shaped surfaces.

Access optionsAccess options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

References

  1. 1.

    , & How filaments of galaxies are woven into the cosmic web. Nature 380, 603–606 (1996).

  2. 2.

    , , , & Dune formation under bimodal winds. Proc. Natl Acad. Sci. USA 106, 22085–22089 (2009).

  3. 3.

    Phase Transition Dynamics (Cambridge Univ. Press, 2002).

  4. 4.

    Symmetry breaking and the evolution of development. Science 306, 828–833 (2004).

  5. 5.

    , & Mechanical basis of morphogenesis and convergent evolution of spiny seashells. Proc. Natl Acad. Sci. USA 110, 6015–6020 (2013).

  6. 6.

    & Patterns and collective behavior in granular media: Theoretical concepts. Rev. Mod. Phys. 78, 641–692 (2006).

  7. 7.

    & On the theory of superconductivity. Zh. Eksp. Teor. Fiz. 20, 1064–1082 (1950).

  8. 8.

    The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. B 237, 37–72 (1952).

  9. 9.

    & Transition from a uniform state to hexagonal and striped Turing patterns. Nature 352, 610–612 (1991).

  10. 10.

    & Hydrodynamic fluctuations at the convective instability. Phys. Rev. A 15, 319–328 (1977).

  11. 11.

    , & Landau theory of the crumpling transition. Phys. Rev. Lett. 60, 2638–2640 (1988).

  12. 12.

    , , & Swarming ring patterns in bacterial colonies exposed to ultraviolet radiation. Phys. Rev. Lett. 87, 158102 (2001).

  13. 13.

    Chemotactic patterns without chemotaxis. Proc. Natl Acad. Sci. USA 107, 11653–11654 (2010).

  14. 14.

    & Introduction to Applied Nonlinear Dynamical Systems and Chaos Vol. 2 (Springer-Verlag, 1990).

  15. 15.

    & Geometry and physics of wrinkling. Phys. Rev. Lett. 90, 074302 (2003).

  16. 16.

    et al. Nested self-similar wrinkling patterns in skins. Nature Mater. 4, 293–297 (2005).

  17. 17.

    , , & Mechanical model of brain convolutional development. Science 189, 18–21 (1975).

  18. 18.

    , , , & Spontaneous formation of ordered structures in thin films of metals supported on an elastomeric polymer. Nature 393, 146–149 (1998).

  19. 19.

    & Fabricating microlens arrays by surface wrinkling. Adv. Mater. 18, 3238–3242 (2006).

  20. 20.

    , & Smart morphable surfaces for aerodynamic drag control. Adv. Mater. 26, 6608–6611 (2014).

  21. 21.

    et al. Finite deformation mechanics in buckled thin films on compliant supports. Proc. Natl Acad. Sci. USA 104, 15607–15612 (2007).

  22. 22.

    et al. Stress and fold localization in thin elastic membranes. Science 320, 912–916 (2008).

  23. 23.

    et al. Multiple-length-scale elastic instability mimics parametric resonance of nonlinear oscillators. Nature Phys. 7, 56–60 (2011).

  24. 24.

    , , & 3d finite element modeling for instabilities in thin films on soft substrates. Int. J. Solids Struct. 51, 3619–3632 (2014).

  25. 25.

    , , & Wrinkle to fold transition: Influence of the substrate response. Soft Matter 9, 8177–8186 (2013).

  26. 26.

    , , , & Self-assembled triangular and labyrinth buckling patterns of thin films on spherical substrates. Phys. Rev. Lett. 100, 036102 (2008).

  27. 27.

    , , , & Surface wrinkling patterns on a core-shell soft sphere. Phys. Rev. Lett. 106, 234301 (2011).

  28. 28.

    & Curvature-controlled wrinkle morphologies. Soft Matter 9, 3624–3630 (2013).

  29. 29.

    et al. Capillary deformations of bendable films. Phys. Rev. Lett. 111, 014301 (2013).

  30. 30.

    , , & Elastic sheet on a liquid drop reveals wrinkling and crumpling as distinct symmetry-breaking instabilities. Proc. Natl Acad. Sci. USA 109, 9716–9720 (2012).

  31. 31.

    & Buckling of a thin film bound to a compliant substrate, part i: Formulation, linear stability of cylindrical patterns, secondary bifurcations. J. Mech. Phys. Solids 56, 2401–2421 (2008).

  32. 32.

    , , , & Periodic patterns and energy states of buckled films on compliant substrates. J. Mech. Phys. Solids 59, 1094–1114 (2011).

  33. 33.

    Mathematical Elasticity Vol. 3 (North Holland, 2000).

  34. 34.

    , , & Surface wrinkling on polydimethylsiloxane microspheres via wet surface chemical oxidation. Sci. Rep. 4, 5710 (2014).

  35. 35.

    Riemannian Geometry and Geometric Analysis (Springer, 2008).

  36. 36.

    & Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112 (1993).

  37. 37.

    Analysis and Design of Structural Sandwich Panels (Pergamon, 1969).

  38. 38.

    & Self-Assembly, Pattern Formation and Growth Phenomena in Nano-Systems (Springer, 2006).

  39. 39.

    Regular Polytopes (Courier Dover Publications, 1973).

  40. 40.

    , & Bouncing localized structures in a liquid-crystal light-valve experiment. Phys. Rev. E 71, 015205(R) (2005).

  41. 41.

    & Nonvariational real Swift–Hohenberg equation for biological, chemical, and optical systems. Chaos 17, 037103 (2007).

  42. 42.

    , , & Localized hexagon patterns of the planar Swift–Hohenberg equation. SIAM J. Appl. Dyn. Syst. 7, 1049–1100 (2008).

  43. 43.

    & Localized states in the generalized Swift–Hohenberg equation. Phys. Rev. E 73, 056211 (2006).

  44. 44.

    , & Subdivision surfaces: A new paradigm for thin-shell finite-element analysis. Int. J. Numer. Methods Eng. 47, 2039–2072 (2000).

  45. 45.

    , , , & Self-contact and instabilities in the anisotropic growth of elastic membranes. Phys. Rev. Lett. 105, 068101 (2010).

Download references

Acknowledgements

This work was supported by the Swiss National Science Foundation grant No. 148743 (N.S.), by the National Science Foundation, CAREER CMMI-1351449 (P.M.R.) and by an MIT Solomon Buchsbaum Award (J.D.).

Author information

Author notes

    • Denis Terwagne

    Present address: Faculté des Sciences, Université Libre de Bruxelles (ULB), Bruxelles 1050, Belgium.

Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue Cambridge, Massachusetts 02139-4307, USA

    • Norbert Stoop
    • , Romain Lagrange
    •  & Jörn Dunkel
  2. Department of Civil & Environmental Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue Cambridge, Massachusetts 02139-4307, USA

    • Denis Terwagne
    •  & Pedro M. Reis
  3. Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue Cambridge, Massachusetts 02139-1713, USA

    • Pedro M. Reis

Authors

  1. Search for Norbert Stoop in:

  2. Search for Romain Lagrange in:

  3. Search for Denis Terwagne in:

  4. Search for Pedro M. Reis in:

  5. Search for Jörn Dunkel in:

Contributions

N.S., R.L. and J.D. developed the theory. N.S. and R.L. performed analytical calculations. N.S. implemented and performed the numerical simulations. D.T. and P.M.R. developed the experiments. N.S., R.L. and D.T. analysed data. All authors discussed the results and contributed to writing the paper.

Competing interests

The authors declare no competing financial interests.

Corresponding author

Correspondence to Jörn Dunkel.

Supplementary information

PDF files

  1. 1.

    Supplementary Information

    Supplementary Information

About this article

Publication history

Received

Accepted

Published

DOI

https://doi.org/10.1038/nmat4202

Further reading