Index of /~tten/TW pulications and lectures/Benasque

 Name                    Last modified      Size  Description
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 Bimodal.html            2003-08-08 18:36  5.2K  
 Soda.html               2003-08-08 21:58  7.2K  
 apollo.GIF              2003-08-08 18:14  8.7K  
 kit2.nb.pdf             2003-08-08 16:42   24K  
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 ruffling.sxd            2003-08-23 04:41  103K  
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projects for Bensasque summer school, Summer/fall 2003

You might want to subscribe to this page. Then when I finally get something on here, you will know.

---T. Witten.

More amusements and background

Chicago Crumpling web site

1. Bimodal crumpled sheet

2. 3d soda constructor

3. spontaneous saddle shapes from nonuniform metric

Recently the hierarchically ruffled edges of leaves and of distorted plastic sheets have been explained using differential geometry [Theory of edges of leaves Marder M, Sharon E, Smith S, Roman B EUROPHYSICS LETTERS 62 (4): 498-504 MAY 2003]. The ruffled shape is said to arise because the metric of the material---the resting area of a given section of material, increases as one approaches the edge. This means that the edge cannot be strain free unless it is longer than a parallel line drawn within the material. This project explores a simple instance of this phenomenon involving the lowly potato chip.

Often solid foods are cooked by slicing them into thin disks and then frying them. The frying causes chemical reactions in the food that make it contract or expand. This expansion or contraction is generally not uniform. Often the result of the process is that the originally flat disk becomes distorted and bends out of plane. It acquires a gaussian curvature. Remarkably, this gaussian curvature is often negative. The disks become saddle-shaped rather than cup-shaped. One obvious possibility for understanding why this kind of distortion is so general is to attribute it to a nonuniform metric like the ruffled leaves mentioned above.

A simple test of this idea is to assume the simplest distortion of a disk metric consistent with its symmetry. That is, suppose the resting length grows quadratically with distance from the center, with some small amplitude. (A positive disclination in a two-dimensional crystal lattice has, by contrast, a resting length tha varies linearly with distance from the center.)

• What is the resting shape of such a disk? How strongly do radial lines curve into the third dimension?
• What physical process might give rise to a distortion of the metric that varies with radial distance? Nonuniform swelling of a gel?
• Can one find a generic quantitative feature of these disks that could be compared to the real objects?
• Is there a shape that removes all the internal strain----that "satisfies" the metric? (cf. Shankar Venkataramani, preprint).
• Why don't potato chips have waves within waves, as the leaves of Marder et al do?

4. twist-writhe correlations in elastic cylinders

The standard elastic energy of a thin solid rod consists of bending energy and twisting energy. The bending energy is quadratic in the local curvature; the twisting energy is quadratic in the local twisting rate. The local curvature and the local twist are free to vary independently. This means that there is nothing in the elastic energy to cause correlations or coupling between bending and twist: an imposed bending should not cause twist, nor should an imposed twist cause bending. [STATISTICAL-MECHANICS OF SUPERCOILED DNA MARKO JF, SIGGIA ED PHYSICAL REVIEW E 52 (3): 2912-2938 Part A SEP 1995]. This lack of correlation continues to hold for a flexible rod bent to form a loop, provided the loop is free to swivel. For such a loop one can define the total twist and the writhe, an integral involving the bending but not the twist. Since twist and bending are uncorrelated, twist and writhe should also be uncorrelated. The elasticity of DNA is expressed in the literature in terms of this model

We studied a primitive form of chain whose elasticity arises entirely from entropy. It has strong local fluctuations in configuration. The paper is here. In this chain, one can directly measure twist and writhe in any configuration. They are strongly anti- correlated! This must mean that the effective elastic energy for our chain is not the simple quadratic form listed above.

Since no other local and quadratic energy is consistent with the symmetry of an elastic rod, only the above form is possible if one considers only quadratic energies. But if one considers corrections due to cubic and higher-order contributions to the energy, then such terms could explain the mysterious correlations.

The goal of this project is to explore the possible cubic forms of the energy and identify which of these (if any) can cause twist-writhe correlations. The next goal is to demonstrate the existence of these effects by exhibiting twist-writhe correlations in a macroscopic object, such as a plastic tube. Since the correlations are believed to come from corrections to the energy beyond quadratic order, they should die out in a predictable way as one makes the rod thinner, keeping the shape constant.