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From
http://philipball.blogspot.com/2012_05_01_archive.html
On Friday, May 25, 2012 Philip Ball writes:
Buckled up
I have written a story for Physical Review Focus, of which the pre-edited version is below. There’s more on this topic in my book Shapes, and out of sheer Friday-night generosity I reproduce some of it below too.
Some of nature’s most delicate forms and patterns, such as the fluted head of a daffodil and the convoluted labyrinths of fingerprints, are created by buckling and wrinkling. These deformations are a response to internal stress, for example as a sheet of soft tissue grows while being constrained at its boundaries. Old paint wrinkles as the paint film swells in some places but stays pinned to the surface below in others.
Because buckling and wrinkling patterns can be highly regular and predictable, they could provide a way of creating complex structures by spontaneous self-organization. In a paper in Physical Review Letters [1], Nicholas Fang at the Massachusetts Institute of Technology in Cambridge and coworkers describe a way of controlling the buckling shapes in small tubes of soft material, and show that they can explain theoretically how the pattern evolves as the tube dimensions are altered.
“These patterns are lovely to look at”, says Michael Marder, a specialist on nonlinear dynamics at the University of Texas at Austin, “and if the ability to control patterns is not yet at the level of control that is likely to interest engineers, it’s a promising step forward.”
“Mechanical buckling has long been suggested as a means of pattern formation in biological tissues”, says mathematician Alan Newell of the University of Arizona, who has previously advanced this as an explanation for the spiral phyllotaxis patterns of leaves and florets. “What’s good about this work is that they do a precise experiment and their results tend to agree with simple theories.”
To ‘grow’ a deformable material so as to induce buckling, Fang’s team use a polymer gel that swells when it absorbs water. They explored tubular geometries not only because these are conceptually simple but because they are relevant to some natural buckling structures, such as the bronchial passage, which may become swollen and wrinkled in asthmatics.
The researchers used a microfabrication technique to make short tubes with diameters (D) of several mm, and walls of various thickness (t) and length (h). The tubes are fixed at one end to a solid substrate, creating the constraint that drives buckling. To induce swelling that begins at the free end of the tube, the researchers inverted the tubes in oil and let the ends poke into a layer of water below.
Swelling deformed the tubes into truncated cone shapes, which might then buckle into a many-pointed star-shaped cross-section. In general, the shorter the tubes (the smaller the ratio h/D), the more wrinkles there were around the tube circumference. Surprisingly, the wall thickness had relatively little influence: tubes with the same h/D tended to have a similar buckled shape regardless of the wall thickness.
To understand that, Fang and colleagues used a simple model to calculate the shape that minimizes the total elastic energy of a tube. Buckling costs elastic energy around the circumference, but it can also reduce the elastic energy due to outward bending of the tube as it swells. For a given set of parameters, the two contributions balance to minimize the total energy for a particular number of buckles – which turns out to depend only on h/D and not wall thickness. The experimental results mapped well onto these theoretical predictions of the most stable mode of deformation.
Fang, who usually works on photonic structures for controlling the flow of light, hopes that these regular buckles and wrinkles might offer ways of channeling and directing light and sound waves by scattering. “The basic idea is that the wrinkled structure could absorb or scatter the light field or acoustic waves in a directional way”, he says. “We’re currently testing such structures for applications in ultrasound-mediated drug delivery.”
The results may have implications for natural systems too. Fang says it’s no coincidence that the buckled gel rings resemble slices of bell pepper, for example. “Bell peppers can be considered as a tubular structure that grow under constraints from the ends”, he says. “Often we find a slice of slender peppers display a triangle shape and that of short and squat peppers appear in square or even star-like. Our model suggests that these patterns are determined by the ratio of length to diameter.” The team also thinks that the results might elucidate the buckling patterns of corals and brain tissue.
Xi Chen of Columbia University in New York, who has studied the buckling pattern on the surfaces of fruits and vegetables, is not yet convinced by this leap. “It’s not yet clear where the rather strict constraint on swelling – the key for obtaining the shapes described in their paper – comes from in nature. It’s interesting work but there’s still a large gap before it could be applied directly to natural systems.”
Newell raises a more general note of caution: similarities of form and pattern between an experiment and a theory are just suggestive and not conclusive proof that one explains the other. “To say that the pattern you observe is indeed the one produced by the mechanism you suggest requires one to test the dependence of the pattern on the parameters relevant to the model”, he says. “In this case, the authors test the h/D ratio dependence but it would also have been good to see the dependence of the outcomes on various of the elastic parameters.”
Reference
1. Howon Lee, Jiaping Zhang, Hanqing Jiang, and Nicholas X. Fang, "Prescribed Pattern Transformation in Swelling Gel Tubes by Elastic Instability", Phys. Rev. Lett. 108, 214304 (2012).
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