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Smooth wrinkles and sharply crumpled regions are familiar
motifs in biological and synthetic sheets, such as plant leaves and crushed
foils, say physicists Benny Davidovitch, Narayanan Menon, and colleagues at the
University of Massachusetts Amherst, but how a featureless sheet develops a
complex shape has long remained elusive.
Now, in a study published in the Proceedings of the
National Academy of Sciences, the physicists report that they have
identified a fundamental mechanism by which such complex patterns emerge
spontaneously.
Davidovitch says they were inspired and moved toward a
solution by thinking about how a familiar birthday balloon, made of two
circular mylar foils, wrinkles and crumples (two separate processes). The two
foils start flat, but when glued together around their edges and injected with helium
gas to create higher-than-atmospheric pressure inside, they spontaneously
changes shape to accommodate the gas.
“This simple process leads to a fascinating pattern of
wrinkles and crumples that emerge spontaneously near the perimeter of each
foil,” Davidovitch points out. “What we discovered is an unusual
sequence of transitions that underlie this and possibly other types of morphological
complexity.”
In the laboratory, rather than balloons, the researchers
including doctoral student Hunter King, who conducted many of the experiments,
and postdoctoral researcher Robert Schroll, who carried out theoretical
calculations, used microscopically thin films and a drop of water to model the
effects they wished to study. They cut a circle of ultra-thin film from a sheet
10,000 times thinner than a piece of paper, only tens of nanometers thick, and
place it flat on the water drop nestled in a circular collar, where surface
tension holds it in place.
“We then very, very gently inject more and more water
into the bubble, very gradually, so it becomes more and more curved without
spilling over,” says Davidovitch. “When the radius of the drop gets
small enough, the thin film starts to develop fine radial wrinkles near its
outer perimeter as the water pressure increases If you keep adding pressure,
decreasing the radius further, a second transition takes place and the film
starts to crumple and to look more like a table cloth, draping with sharp
creases over the edge of a flattened top,” he adds.
Watching this process through incremental steps, the
researchers were able to observe and describe through mathematical formulas how
the drop imposes confinement on circles of latitude of the sheet. “The
degree of this confinement increases as the drop’s radius decreases, and an
unusual sequence of transitions can then be observed,” says Davidovitch.
With this work the investigators, who had earlier proposed
quantitative predictions of wrinkle patterns in ultra-thin sheets by following
the principle that such sheets must be free of compression, confirm their
theoretical predictions. The current experiments also suggest that the
wrinkle-to-crumple transition reflects a dramatic change called “symmetry
breaking” in the distribution of stresses in the sheet, rather than just a
further disruption of its symmetric shape, Davidovitch points out.
The researchers are now working on new puzzles regarding the
formation of crumpled features posed by the experiment.
