P. Ball
Pepper physics. The wavy cross-sections of bell peppers may be the result of buckling of a smooth tube.
H. Lee, et al., Phys. Rev. Lett. (2012)
Making waves. A tube that
swells from one end while being constrained at the other deforms into a
trumpet-bell shape (top). This structure may lower its elastic energy by
buckling instead (bottom).
Some of nature’s most delicate forms and patterns, such
as the fluted head of a daffodil or the convoluted labyrinths of
fingerprints, are created by buckling and wrinkling of soft tissue. In
hopes of mimicking such structures for future technology, a team
describes in Physical Review Letters
a technique for controlling the buckling shapes of small, soft tubes
and provides a theory to explain the results. The researchers hope their
work will lead to simple techniques for creating complex structures.
In recent years, researchers have become interested in
how buckling might generate patterns seen in the natural world, such as
the arrangements of leaves and florets on a flower stem or the crinkling
and folding of the leaves themselves. But Nicholas Fang of the
Massachusetts Institute of Technology in Cambridge and his colleagues
had a more practical, technological motivation for studying the regular
forms that buckling can generate in soft matter. Fang studies photonic
structures for controlling the flow of light, and he wondered whether
regular arrays of tiny buckles and wrinkles might scatter and reflect
light or sound, channeling and filtering it in useful ways.
To “grow” a deformable material and induce buckling, Fang
and his colleagues used a polymer gel that swells when it absorbs
water. They used a microfabrication technique to make short tubes with
diameters of several millimeters and walls of various thicknesses and
lengths. The tubes were fixed at one end to a solid surface. To induce
swelling starting 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 sometimes buckled by forming waves around their circumference,
leading to a many-pointed, star-shaped cross section. In general, the
shorter the tubes—compared with the diameter—the more wrinkles there
were. Surprisingly, the wall thickness had relatively little influence.
To understand the results, Fang and colleagues used a
simple model to calculate the shape that minimizes the total elastic
energy of a tube. Buckling costs elastic energy because it deforms the
structure, but it also reduces the energy from outward bending that a
smooth, trumpet-bell shape would require. For given conditions, the
minimum energy is a balance of the two contributions and results in a
particular number of buckles—which turns out to depend only on the
length-to-diameter ratio. The experimental results agreed well with
these theoretical predictions of the most stable mode of deformation.
“These patterns are lovely to look at,” says Michael
Marder, a specialist in nonlinear dynamics at the University of Texas at
Austin, “and [even] 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.”
Fang says the results may help explain some natural
systems—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 grows under constraints from the ends,” he says.
“Often we find a slice of slender pepper displays a triangle shape, and
that of short and squat peppers appears square or even starlike.”
Xi Chen of Columbia University in New York, who has
studied the buckling patterns on the surfaces of fruits and vegetables,
is not yet convinced by the connection with nature. “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.”
But mathematician Alan Newell of the University of
Arizona in Tucson, who has studied the role of buckling in the growth
patterns of leaves, says the results represent important progress.
“What’s good about this work is that they do a precise experiment, and
their results tend to agree with simple theories.”
Philip Ball is a freelance science writer in London and author of Curiosity: How Science Became Interested in Everything (2012).
