


Elastomers and entropic springs
Let us begin by doing a Gedanken experiment. Or you could literally do it if you have the ingredients. Imagine a ring made of some thin wire or may be a regular piece of string. And now imagine stretching it between you fingers. What would happen? It would just become stiff and you cannot do much more. Now imagine doing the same experiment with a rubber band. You will be able to extend the rubber band to at least a few times its original size. And if you let it go, it goes right back to its original shape and size. So, the experiment shows me that rubbery materials can be "deformed reversibly" to many times their original size. This note tries to tell you how and why. Also, in the concluding paragraph, the context of such considerations as applicable to biological systems is mentioned (feel free to skip the section named theory, the rest is sufficient story by itself). 
Microscopics of a regular solid
As our starting point, let us consider the elasticity of a regular material, the string or the metal wire in our earlier experiment.What is elasticity anyway? It is the theory that tells you how much force you need to exert in order to deform/extend a material by a given amount. Let us simplify even further. Let us consider a spring attached to a wall and pulling on it. How much force do I need to exert to do this? This is given by the "Hooke's law" that we all know and love. Let F be the force and x be the extension of the spring from its rest length. Then F=kx is the statement of Hooke's law, i.e., the amount of force I need to exert grows linearly with the extension. And the force law is characterized by a single constant, the spring constant of the particular spring I am using.
Now, why did we start there? The reason is that you can get a reasonable theory for the elasticity of a solid if you considered it to be made up of atoms connected by springs (see picture along side), whose spring constants are determined by the electromagnetic interaction among the atoms [1]. What is the typical energy scale of this spring like interaction, i.e., how stiff are these springs? It is about a few electron volts typically, which makes them really stiff springs (but for us to realize that the spring is really stiff, I need to compare this energy scale to something else right? We will come back to this later). 


Microscopics of rubber
Note that the picture above is how a string or a wire looks like in the microscopic scale. Next let us ask what does a rubber band look like?[2] It looks like the picture along side. A rubbery material is made of coiled up flexible polymers that are "crosslinked" by some chemical agents (the big black dots in my picture) [3]. For clarity I have caricatured the three polymers shown in the picture with different colors. The message here is that the polymers are in some coiled state. When I stretch such a material, what I am doing is pulling the black dots apart. What will this do to a polymer? It will uncoil it some. You already know that uncoiling a string costs much less energy than trying to pull at a fully extended string in an attempt to lengthen it. This is primarily the difference between rubbery materials and regular crystalline solid. These kinds of solids, to which class the rubber band belongs, have a special name, they are called "elastomers". Note that I can say that I understand the elasticity of rubberlike materials, if I can get the equivalent of "Hooke's law" for this system. And for this purpose I need to understand what happens when I pull on a polymer. In the rest of this note, we will try to explain how to describe this theoretically. 

The theory
Before we proceed with the theory, let us pause for a moment.How can what I said above be right? If I had a string coiled up on my table and I pull it open to its full length, it does not cost me any energy at all. It comes apart nice and easy. If I were to take the analogy above seriously, the rubber band should not offer me any resistance at all. Pulling at a rubber band must be like pulling on water, it should just come apart. But this is clearly not true. So what did I miss? What I missed is called “entropy”. Suppose my rubber band is at zero temperature (no no, not 0C or 0F but 0K). Then the analogy with the macroscopic string holds and the rubber band should indeed flow like water till the polymers are completely extended. But at all finite temperature, the polymers in rubber are jiggling around with some kinetic energy. And that makes all the difference as we will try to show below. 

In order to quantify this notion we need to ask what makes physical systems happy. Let us consider a regular spring again. If we just let the spring be, it has a characteristic length, let us call this the rest length of the spring. This is the length in which the spring is happiest. Now I pull on the ends of the string. I have to do some work to pull it because I am moving the spring away from the state it is happiest in. This work gets stored in the spring as potential energy. Alright, now let us ask at finite temperatures what is the state in which the polymer is happiest? It is happiest when it has the largest entropy.
What is this entropy? For a polymer, we can understand it as follows. Now suppose you have a coiled up string. It looks like a disc right? The size of this disk is called the "radius of gyration" of the polymer (a measure of the lateral extension of the polymer). Suppose the length of the polymer is L. I ask you, "how many ways can you make an object of length L with it?", you will tell me, "Exactly one way, stretch the polymer out to its full length". Similarly, if I wanted to make an object of some length A which is much much smaller than L, again we can do this in exactly one way, namely make a tight coil out of the polymer with each turn of the coil having a radius A. But, if I wanted some intermediate sized object, then I can make it in many many ways, in each of these ways, the polymer will be coiled in a slightly different way. The entropy of a polymer of radius of gyration R is the number of different configurations the polymer can have given this radius as its lateral dimensions. As stated earlier, the polymer wants to have maximum entropy. Hence, from the arguments above it does not want to be fully extended or tightly coiled, but rather be coiled up in some intermediate state. This intermediate state has a size equal to the square root of its length L [4].
So in my unstretched rubber band, I have polymer coils that are happy, i.e., in the state of maximum entropy. Now I pull on the rubber band. What happens? I stretch the polymers. Their radius of gyration increases above the optimal value, their entropy goes down and they are unhappy. So, just like you pay an energy cost to stretch a normal spring, you pay an entropy cost to stretch a polymer. If you calculate this cost, you can derive the equivalent of Hooke’s law for these polymers [5]. Then you find that if I stretch a polymer by an amount x, then the force I need to apply is F=(CT/L)x, where C is just a constant, and T is the temperature of the system. So, a polymer behaves like a spring with a spring constant determined by the temperature of the system! And the reason it behaves like a spring is because of a loss in entropy rather than a gain in energy. This is what people call an "entropic spring". Now, suppose I compare this spring constant with the spring constant associated with atomic solids we considered earlier, I find that it is 0.00001 times smaller! Thus polymers form very loose springs. And rubber is exactly like a regular solid but with a really ittybitty spring constant that scales with the temperature of the solid [6]. 
Conclusion
In summary, rubbers are solids made by crosslinking polymers. Polymers form entropic springs whose stiffness increase as the temperature increases. And this explains why rubber bands become brittle and break when you try to stretch them on hot summer days! But at the start of the post, I said that such considerations are biologically relevant. How is that? The cell wall is a rubber! It is a crosslinked polymer mesh made of polymers that are called actins and microtubules. You will now say to me, why would I want to know about the elasticity of a cell wall? The reason is that it is the elasticity of the cell wall that allows a cell (those that are not swimmers that is) to crawl. And all questions associated with the motility of such cells boils down to understanding the elasticity of the cell wall. And you need to start by understanding the elasticity of the plain old rubber band first! 
Jargon, Caveats and Disclaimers
[1] Coulumb, screened coulomb, LennardJones, you take your pick.
[2] I am fudging scales here, mapping Angstroms to a fraction of a micrometer, but for simplicity we ignore this difference here.
[3] This process is called vulcanization, that turns a complex fluid into a solid, gives it a finite zero frequency shear modulus.
[4] You can see this easily if you think of the polymer as a 1D random walk of length L. Then the RMS distance the walker would travel is Square root of L right?
[5] For the experts, note that you can derive this readily. Take a Boltzmann definition for the entropy as S=k_BLogW, where W is the number of configurations of a polymer of radius of gyration R. Using the random walk analogy earlier, this is W=exp(R^2/L). So the loss in entropy due to stretching must be S(R)S(R+x). And force F=T(dS/dx)x (just a standard response relation).
[6] You should ask me now how come I ignored entropy when considering elasticity of an atomic solid. The fact is that entropy strain independent for harmonic solids and plays no role in the elasticity. 
