Rubber elasticity

Why does a stretched rubber band pull back, and why does it warm when you stretch it?

A steel spring pulls back because you have stretched its bonds and they want their old length. It is tempting to say a rubber band does the same thing, only more so.

But check the numbers before accepting that. A chemical bond breaks if you lengthen it by a few percent. A rubber band can be stretched to five or six times its resting length and snap back unharmed. Whatever pulls it back, then, cannot be a stretched bond — nothing in it has been stretched at all. So what is doing the pulling?

Here is a second clue, and it is the one that gives the game away. Stretch a rubber band quickly and press it to your lip: it is warm. Let it snap back and press it again: it is cool. Do that to a steel spring and nothing of the sort happens.

Look at what rubber is made of. Long chain molecules, thousands of carbon atoms end to end, tied to each other here and there by cross-links — the sulfur bridges vulcanisation introduces, which stop the whole thing flowing like treacle. Between cross-links, the chain segments rotate freely about their bonds, and at room temperature they are doing so constantly, thermal motion reshuffling each chain's shape.

Now ask about one such chain. If the segments point in random directions, where do its two ends sit relative to each other? Overwhelmingly, close together — not because anything attracts them, but for the same reason a dropped necklace lands in a heap: there are enormously more ways to arrange the links that leave the ends near each other than ways that leave them far apart. Fully extended is one arrangement out of an astronomical number.

So a relaxed rubber band is not at rest in the mechanical sense. It is at rest in the statistical sense: its chains are wandering through the huge population of coiled shapes because that is where nearly all the shapes are.

Now pull. You are forcing the chain ends apart, and the further apart they go the fewer shapes remain available to the chain between them. The count of accessible conformations collapses. That is a fall in entropy, and it is being imposed by you against the constant reshuffling, which is trying at every instant to put the chain back among the many. Release the band and it does exactly that. The retractive force is the macroscopic sum of that statistical pressure — and this is the surprise: it is entropic, not energetic. Nothing stored the energy in a bond.

Follow that and two predictions fall out, both of which you can test in a kitchen.

First, the warming. You did work on the band and its internal energy barely changed, because no bonds were strained; the energy has to go somewhere, and it leaves as heat. Reversed on release, the band takes heat back and cools. That is the Gough-Joule effect, noticed by John Gough in 1805 and put on a quantitative footing by Joule half a century later.

Second, and stranger: because the retractive force comes from thermal agitation reshuffling the chains, more agitation means more force — roughly in proportion to absolute temperature. So hang a weight from a rubber band and warm it with a hairdryer, and the band contracts and lifts the weight, where a metal wire would lengthen. Rubber gets stiffer when heated. Almost everything else gets softer.

Three qualifications, because the ideal picture above is cleaner than the material.

Real rubber is not purely entropic. A modest share of the retractive force in natural rubber — commonly put at something like a tenth to a fifth at moderate extension — comes from genuine changes in internal energy as segments shift between conformations of unequal energy. The entropic account is the dominant term and the right one to reason with, but it is leading order, not the whole of it.

The ideal model also fails at large extension. Stretch natural rubber far enough and the chains align sufficiently to crystallise under strain, which stiffens it sharply; the simple counting argument does not predict that.

And the mechanism has an off switch. Cool rubber below its glass transition — liquid nitrogen does it in seconds — and the segments can no longer rotate. The reshuffling stops, the statistics have nothing to work with, and what is left is an ordinary brittle solid that shatters when struck. The elasticity was never a property of the molecules alone, but of molecules plus enough thermal motion to keep exploring their shapes.

• How the same entropic reasoning explains the elasticity of DNA and of the proteins that give tissue its give.
• Why a rubber band left stretched for months slowly loses its pull, and what creep tells you about the cross-links.

Every term the collection defines is gathered in the glossary.