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Curvature-induced symmetry breaking determines elastic surface patterns

Nature Materials volume 14, pages 337–342 (2015)Cite this article

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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.

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Figure 1: Macroscopic and microscopic wrinkling morphologies of stiff thin films on spherically curved soft substrates.
Figure 2: Notation and experimental system.
Figure 3: Phase diagram of wrinkling morphologies.
Figure 4: Bifurcation diagram of wrinkling patterns.

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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.).

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Author notes

  1. Denis Terwagne

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

Authors and 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

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  1. Norbert Stoop
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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.

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Correspondence to Jörn Dunkel.

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Stoop, N., Lagrange, R., Terwagne, D. et al. Curvature-induced symmetry breaking determines elastic surface patterns. Nature Mater 14, 337–342 (2015). https://doi.org/10.1038/nmat4202

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