What do blood cells, raisins, mountains, automobiles, and packing materials have in common? To physicist Thomas Witten, all are examples of things that crumple. Thanks to his research, other scientists may better understand how a blood cell's membrane contorts to pass through the tiniest capillaries, how a car's fender absorbs the energy of a crash, and how the Earth's tectonic plates collide and buckle to form a mountain range.
A ubiquitous but poorly understoodphenomenon, crumpling occurs on scales from the microscopic to the geologic. An analysis of any of these situations starts with the same fundamental problem. Imagine crumpling a sheet of paper into a ball, then squeezing it until it's ten times smaller. How much energy does that take? And how much harder is it--or how much energy is required--to squeeze the ball 100 times smaller?
Until now, physicists didn't know how to make such quantitative predictions short of actual measurement. And in that case, says Witten, "you would have to describe [the crumpling] separately for paper, sheet metal, et cetera, because the way each one crumples might depend crucially on the type of material."
One such specimen, a crumpled Mylar sheet, decorates Witten's office wall in the James Franck Institute. A specialist in theoretical condensed-matter physics, he studies new types of structures produced in complex fluids, like those containing polymers. Witten approached the problem of crumpling, too, from a theorist's perspective: What are the common elements among the seemingly unlike materials that crumple? In the December 1 issue of Science, he and his colleagues answered that question for the first time.
To their surprise, he says, "we found that a sheet of one material crumples like a sheet of any other material, provided the sheets are large and thin." These requirements, of course, are relative: A tiny cell membrane is "large" compared to its two-molecule-wide thickness; a tectonic plate is "thin" only in contrast to its continent-sized area.
"We had thought the problem was too complex to be able to say anything general about it," says graduate student Alex Lobkovsky, one of Witten's collaborators and, like him, a member of the University's NSF-funded Materials Research Science & Engineering Center. Joining them were former U of C research associate Hao Li, now at the NEC Research Institute in Princeton, New Jersey; David Morse of the University of California, Santa Barbara; and fourth-year Sharon Gentges.
For an undergraduate like Gentges, a byline in Science is a rare honor. It was her summer project, in fact, that first hinted at a way to crack the complex case of crumpling. Gentges was trying to verify a prediction by Witten and Li about the geometry of fullerenes--unusual, 60-atom carbon molecules whose flexible shapes vaguely resemble geodesic domes or soccer balls. To that end, she developed a computer model of an elastic membrane, representing it as a lattice of "virtual" springs. Using her model, Witten and Lobkovsky looked at how membranes respond to various distortions--including crumpling.
That application of Gentges' model, however, required a conceptual leap: that all crumpled structures consist of sharp points connected by networks of ridges. The concept allowed Witten's team to study the problem through a simple distortion: a boat-like shape made by removing two wedges from a membrane, then joining the edges of the remaining sheet. The computer model calculated the energy needed to force the edges together and the sharpness of the ridge that formed between the boat's two vertices.
"If you thought about it naively," explains Lobkovsky, "you might think that the energy would be uniformly distributed throughout the sheet. But we found that it's concentrated in the narrow ridge between these two points." Moreover, he adds, the energy distribution is independent of the type of material--"as long as the material is thin, relative to the distance between the two points."
The physicists were surprised by another finding, too. The size of the material has an unexpectedly small effect on the energy needed to crumple it. Crumple a small sheet of paper and a sheet eight times larger into proportionally identical balls--balls with the same arrangement of ridges and points--and the energy increases by a mere factor of two.
"This was something we didn't expect," says Witten. "You might have thought the energy would be 64 times greater"--or 8 times 8--"since the area and volume are 64 times greater. But because the energy is concentrated in the ridges, the area over which the energy is spread is a small fraction of the total area of the sheet."
This is the first time that anyone has quantified the relation between a crumpled sheet's size and energy, and it's of more than theoretical interest. Witten's work may aid, for example, in the design of better packing materials and of cars that can absorb the energy of a high-speed crash without crushing their occupants. If materials could be designed to control where ridges form, he suggests--perhaps by creating deliberate defects--engineers might more effectively manipulate how or where materials crumple.
"If we want to understand how crumpling works, we will have to understand how ridges form," says Witten. "Our work is an essential ingredient in understanding these phenomena."