# Entropic force in polymers

According to my textbook, the elastic force in a rubber is caused to the tendency of the polymers to return to their initial disordered state of higher entropy.

But isn't this looking at entropy on a larger level (I'm hesitant to say macroscopic) than it is usually looked at? Generally, we consider the entropy of individual atoms in an ordered structure (a crystal lattice, for instance). But here, we are looking at the entropy of entire polymer chains.

Also, it just doesn't seem to make sense with the non-statistical view of entropy. The book explains it in the following way. The first law of thermodynamics,

$$dE = dQ - dW$$

Making some substitutions,

$$dE = TdS + Fdx$$

where $F$ is the elastic force, $dx$ is the displacement, $T$ is the temperature and $dS$ is the entropy change. Next, approximating $dE$ as zero, and solving for $F$,

$$F = -T\frac{dS}{dx}$$

which is supposed to tell us the that the force is proportional to the rate $dS/dx$ at which the rubber band's entropy changes. But how is there any chage in entropy when no heat is being added?

And, for that matter, if I take a string, lying all entangled, and try to straighten it, why doesn't it exert an entropic force and try to go back to its higher entropic state?

• Entropy is a molar quantity i.e. the total entropy is proportional to the number of molecules present. In a piece of rubber there are an enormous number of molecules, but with a piece of string you have just one molecule of string. The entropy change when you straighten the string is too small to produce a detectable force - if you had $6.023 \times 10^{23}$ pieces of string it might be a different matter. Mar 8, 2014 at 8:03
• @JohnRennie: But the string itself is composed of a large number of molecules, isn't it? Mar 8, 2014 at 8:23
• Yes, but you aren't changing the entropy of the molecules in the string. You're only changing the entropy of the string itself. Mar 8, 2014 at 8:39
• So, in the rubber, we are changing the entropy of the polymer chains. The entropic force is only significant because of their large number, am I right? Mar 8, 2014 at 14:03
• Yes. The point is that you are stretching the rubber and therefore changing the conformation of all the polymers in it. You aren't stretching the string, you are just rearranging it. Mar 8, 2014 at 14:57

Thermodynamically polymers can be discussed in two different ways:

• as molecules within a solution, containing zillions of polymer molecules
• as a single molecule that is a collection of many atoms: indeed, the degree of polymerization, i.e., the number of segments in a polymer, may reach $$10^9$$; typical chains contain millions of links - e.g., living DNA typiclaly comes in segments of $$10^6-10^7$$ nucleotides.

Statistical mechanics viewpoint
When a single polymer molecule is treated as a thermodynamical system, its entropy comes from the fact that the chain can be folded in many different configurations. The stretched configurations, with the extent of $$\sim Nb$$ ($$N$$ - degree of polymerization, $$b$$ - size of the link) are far less probable than the ball-like configuartions. In fact, the radius of gyration of a polymer is typically given by $$R_g\approx N^\nu b,$$ where $$\nu$$ is about $$3/5$$. Thus, A stretched polymer, when released, returns will behave as an isolated system: evolve towards the state of maximum entropy, i.e., sample different configurations, most of which are not the stretched ones.

Thermodynamics viewpoint
One has to perform work in order to stretch a polymer, i.e., increase its internal energy. When the polymer is releazed, it converts this energy to entropy. Note that supplying system with heat is not the only way to change its entropy, since entropy is a state function, while the heat is not - i.e., the same entropic state can be reach by different paths.

References

The process of extension of a rubber bands encompasses two phenomena

1. increase of disorderliness when the elastic strain is released, and

2. in repeated extension and de-extension the orientation of chain in parallel also goes on. Thus it is increase in entropy initially and its decreased ultimately culminating to the strain induced crystallisation.

Thus initially entropy increases and then decreases.