Folding sheets are a form of soliton.

Connected pendulums at left are swinging in a "breather" mode, a form of soliton. It remains localized in space because of its high swings. Height of the surface under the pendulums shows their swing angles. Surface extends to the right to show the time-dependence of the swinging pattern. The diagonal edge is accentuated. Red curve is drawn to have its slope-angle at each point equal to the swing angle along the edge. This curve has the shape of a film floating on a liquid compressed so that it buckles and folds. Black tie-lines indicate corresponding points on the two curves. Below the red line is a photo of a thin plastic film 6 cm long floating on a trough of water, viewed edge-on, from L Pocivavsek et al, Science 16 May 2008: 320 912-916 .

An important part of understanding materials is to understand spontaneous localization of energy. When we bend a pencil until it cracks, the smooth forcing of our hands produces a sudden moment of energy release in a tiny subpart of the pencil. Some forms of localization, as in a separating drop or an imploding cylindrical bubble have been explained in general in the U of C MRSEC. Another form of localization under active study is the folding of a thin sheet, such as the lipid layers of Prof. Lee or the nanoparticle layers of Dr. Lin. Here a rectangular sheet floating on a liquid is compressed at the two opposite sides. The compression makes the sheet buckle into a wavy pattern that then collapses and localizes into a static fold, like that seen in profile at the bottom of the figure. We can reproduce this localization phenomenon accurately using computers.

Some forms of localization have been understood more deeply. These are a type of motion called solitons. Solitons are a type of wave in a strongly disturbed medium, such as a steep wave traveling through a canal or a collective swinging motion of a line of connected pendulums. The distinctive feature of a soliton is that the strong disturbance acts to confine the energy to a small region so that no energy escapes. Solitons are used in practical life \eg to transmit signals in optical fiber. The self-preserving aspect of the micron-long solitons maintains the integity of the signal as it propagates through kilometers of fiber.

The precise self-preservation that occurs in a soliton cannot happen without a reason. For some solitons like the line of coupled pendulums, the reason has been discovered. These are so-called integrable systems that can be completely characterized using symmetry. In this respect, the motion of a soliton is as robust and predictable as the motion of a planet, comet or satellite.

Now MRSEC scientists and collaborators have brought the same depth of understanding to the localization of a folding sheet. They did this by finding a mathematical link between the sheet and the line of pendulums. The result was published by Haim Diamant and Thomas A. Witten, Physical Review Letters, 107 164302 (2011).

The picture above shows how the link works. Along the left edge is a line of connected rod pendulums that were set in motion by swinging the middle pendulum to a large angle. The motion settles down to the "breather" pattern shown here. The high swing angles keep the motion confined to a finite region indefinitely. The colored surface shows how the swing angles vary in space and in time. They stay localized in space while oscillating indefinitely with time. The surface has been cut along a diagonal. The variation of swing angle along this diagonal tells the shape of the folded sheet. Specifically, distance along the diagonal is equal to distance along the sheet, the swing angle at a given point on the diagonal is the angle that the sheet makes with the horizontal at that point. The link allows one to describe the shape of the strongly folded sheet shown with formulas as simple as the equations of planetary motion found long ago by Isaac Newton. The link also clears up some mysteries regarding the folding sheet. For example, it explains why the pressure varies exactly as the square of the geometric compression length.

This link opens the door to deeper understanding. The discoverers are working to understand the symmetry of a sheet that allows it to be described as simple soliton. They are also confident that the link can be generalized to describe other forms of folding. They anticipate finding other soliton-like phenomena in the domain of folding sheets.

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