Thermodynamics of a Rubber Band

Preface

What makes rubber bands stretchy?

It seems like a question with an obvious answer, but perhaps obvious in the vein of answers to questions like “What makes water wet?” or “Why is the sky blue?” There’s no quick and easy explanations, and in many cases parents who find their children asking them will discover it’s easier to say that there is no reason; they just are.

However, rubber bands actually have a very good reason for being stretchy, and the full explanation relies on thermodynamics. So worry not, future physicist parents: studying thermodynamics will let you properly answer at least this tricky question, and you will never in your life have to let another child down by explaining to them, “That’s just how rubber bands work, sweetie.” Unless you’re a lazy parent—that’s on you.

The modern rubber band is made out of natural rubber, a polymer derived from the latex of a rubber tree (synthetic rubber is generally not as stretchy). Polymers are essentially long, chainlike molecules composed of many identical or nearly identical subunits. Many polymers can sort of combine through a process called “cross-linking,” and they end up behaving in many ways more like a single molecule than a large group of them.

The cross-linked polymers of a rubber band begin in a chaotic, low energy, tangled state. When the bands are stretched, the energy is increased and the polymers untangle until they reach a local maximum of their length, and a local minimum of entropy. When released, the entropy rapidly increases until they tangle again, compressing the rubber band back to its original state.

The “cross-linking” property of polymers is vital to the band’s elasticity. Without this property, the rubber band would have no reason to tend towards a tangled state, since the entropy would be about the same in both states. If you released a rubber band with properties like that, it would just stay in the outstretched position until you force into another state.

So that’s it, right? Case closed? Rubber bands compress because of entropy; it’s beautiful, it’s elegant, and it’s eye-opening, yes, but aren’t we done?

No, silly. We have maths to do.

Rubber Band Math

An ideal “band-like” model can be constructed using a finite series of $\widetilde{N}$ linked segments of length a, each having two possible states of “up” or “down.” For fun, let’s say one end is attached to the ceiling, and the other end is attached to an object of mass m. The segments themselves are weightless.

The entropy can be found by computing the microstates using combinatorics:

$\Omega(N_{up},N_{dn}) = \frac{N!}{N_{up}!(N-N_{up})!} = \frac{N!}{N_{up}!N_{dn}!}$

Given that any segment can be found either parallel or antiparallel to the vertical direction, the amount of segments $N_{up}$ pointing up and $N_{dn}$ pointing down can be determined from N using:

$N_{up} + N_{dn} = \widetilde{N}$

and

$L = a(N_{dn} - N_{up})$

which gives

$N_{dn} = \frac{1}{2}(\widetilde{N}-\frac{L}{a})$

$N_{up} = \frac{1}{2}(\widetilde{N}-\frac{L}{a})$

The Boltzmann entropy is thus given by:

$S = k_bln(\Omega) = k_b[ln(\widetilde{N}!)-ln(N_{up}!)-ln(N_{dn})!]$

Applying the earlier equations:

$S = k_bln(\Omega) = k_b[ln(\widetilde{N}!)-ln(\frac{1}{2}(\widetilde{N}-\frac{L}{a})!)-ln(\frac{1}{2}(\widetilde{N}+\frac{L}{a}))!]$

And we get something useful!

Some follow up questions:

Given the internal energy equation U = TS + W (where W is work), find U.
Find the chain length L as a function of T, U, and $\widetilde{N}$ given $dU = TdS + \tau dL$ where $\tau$ is the tension (in this case equivalent to the gravitaitonal force mg)
Does the chain obey Hooke’s law? If so, what is the value of the stiffness constant?