Determinative scrolling and folding of membranes through shrinking channels

Flat membranes ubiquitously transform into mysterious complex shapes in nature and artificial worlds. Behind the complexity, clear determinative deformation modes have been continuously found to serve as basic application rules but remain unfulfilled. Here, we decipher two elemental deformation modes of thin membranes, spontaneous scrolling and folding as passing through shrinking channels. We validate that these two modes rule the deformation of membranes of a wide thickness range from micrometer to atomic scale. Their occurrence and the determinative fold number quantitatively correlate with the Föppl-von Kármán number and shrinkage ratio. The unveiled determinative deformation modes can guide fabricating foldable designer microrobots and delicate structures of two-dimensional sheets and provide another mechanical principle beyond genetic determinism in biological morphogens.