# 20.11 Elasticity of rubber

Before you read this, I suggest you read posts 20.2, 20.6 and 20.8.

In post 20.5, we saw that a material that maintained a constant volume, when it is stretched, has a Poisson’s ratio of 0.5. The Poisson’s ratio of rubber is approximately equal of 0.5, so any volume change, when it is stretched, are very small. We also saw, in post 20.5 that the value of Poisson’s ratio for metals was about 0.3 because their atoms move further apart when the metal is stretched. So, it appears that rubber can be stretched without its atoms moving much further apart. The picture above shows the stress(σ)-strain(ε) curve (post 20.3) for rubber when it is stretched and when it recoils. You can see that the loading (stretching) and unloading (recoil) curves are different – rubber follows different paths on the stress-strain curve during the loading process and when the process is reversed. When a material follows different paths on a graph for a process and for the reverse process, it is said to show hysteresis. The same type of behaviour can be observed in magnetising and demagnetising materials.

The area under the loading curve is the energy given to a unit volume of rubber when it is stretched (post 20.6); the area under the unloading curve is the energy used, by a unit volume, in recoil. So, rubber needs less energy to recoil than is needed to stretch it. In other words, it requires less energy to reverse the stretching process than was needed to make it happen.

We need to explain (1) how can rubber be stretched without its atoms moving further apart and (2) why it needs less energy to recoil than it does to be stretched? Let’s answer question 1 first. The picture above is a diagram showing the polymer molecules in a sample of rubber; they form a random, tangled network, as explained in post 20.7. I have shown the molecules in different colours, so that you can follow their individual paths, but in reality they have the same composition – although they could be different lengths. If we apply a tensile force to this network, the molecules will be pulled straighter and more parallel so the sample of rubber will get longer – just like pulling a heap of noodles or spaghetti. What is the evidence for what I have written in the previous paragraph? Remember that the distance between atoms in a solid is comparable to the wavelength of x-rays; so when a beam of x-rays passes through a solid we see diffraction effects (post 20.5). In the picture above, an x-ray beam passes perpendicular to the sheet of rubber (left) and its diffraction pattern (right) is recorded by a plane detector parallel to the rubber sheet. The pattern that is recorded has circular symmetry, because the rubber molecules have random orientations. When the rubber is stretched this circular symmetry is destroyed because the molecules become aligned. The black circle in the centre is formed by the x-ray beam that passes straight through the material.

Aligning the molecules creates a more ordered pattern and so increases the entropy of the system (post 16.38). To increase the entropy of a system, we need to do work – as in the examples of reverse osmosis (post 18.29) and the heat pump (post 18.30). So most of the work done in stretching rubber is used not to increase its energy but to increase its entropy. Recoil requires much less work (so it uses less energy, post 20.6) because the entropy of the system will spontaneously increase (post 16.38).

So, the recoil of rubber is driven largely by the tendency of the system to increase its entropy, according to the second law of thermodynamics (see posts 16.34, 16.35 and 16.38) and not by the tendency to minimise its potential energy.

Entropy explains many other phenomena that we observe in everyday life, like conduction of heat and diffusion. Diffusion is so familiar to us, for example smelling food cooking from a distance, that we tend not to think about it. If we saw a film of a broken object spontaneously reassembling itself, we would know that it was running backwards because otherwise the entropy of the system would be spontaneously decreasing (post 16.38). So we have an intuitive idea of entropy, even if we have never heard of it. Given the importance of entropy, it is surprising that it is neglected in most elementary science teaching – I spent seven years studying science at school without ever learning anything about it.

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