Polymer Conformation

How Do Rubber Bands Stretch So Much!?

Until now, we've focused on metals and ceramics - neither of which we'd describe as "stretchy" (or "compliant"). Indeed, these materials can only stretch a tiny bit before they start to permanently deform and break. This small amount of stretch is because bonded atoms can be pulled a little ways apart without breaking the bond (see @ref(CB7235) NOT FOUND if you need a refresher). However, this amount is quite small. Rubber bands are able to stretch many times their equilibrium length without any permanent deformation! How can this be explained?

Polymer Chain Conformation (or Shape)

Polymers are long molecules made of many repeating units. In most cases, the bonds between the repeat units are not at an angle 180° and they are free to rotate. Figure 1.11.1 illustrates how this allows polymers to change shape without breaking bonds. The rotation is possible because there is really no preferred orientation of the bonds between the mers - in the case of the polymer chain below, each bond can essentially occupy a position anywhere along the cone indicated with the dotted circle. This implies that there are many, many possible conformations that a chain can take!

Modeling Polymer Conformation

Because there's no energetic preference for the position of the atom about the cones in Section 1.11.2, it may, at any time and with a bit of thermal energy, position itself in a "random" position in the cone. With the same logic we used to justify the random walk model of diffusion (@ref(LP1155) NOT FOUND), we can justify modeling the these possible orientations within the polymer with a random walk behavior. However, this time we are a bit more limited... we need to make sure that we do not break the bonds along a mers, and must thereby constrain the possible positions that the atoms can occupy about the cone. NetLogo model 1.11.1 does just that.

In NetLogo model 1.11.1, we've started our initial condition with a polymer chain that is stretched from one side of the world to the other. Then, the simulation follows the rules below for each tick (time step):

1. Each mer selects a random neighboring position (which we call "patches" or "squares").
2. The mer then checks to make sure that moving there won't break its bonds with its neighbors. If it does, it doesn't move.
3. If the self-excluding? setting is one, it will also check to make sure this is not already a mer on that patch. If there is, it doesn't move on, that the patch is empty
4. If conditions 2. and 3. are met, it moves to the square it identified in 1.

If we think about a real polymer, self-excluding? should always be on, because mers (i.e., atoms) cannot occupy the same space. However, it is possible to do simple analytical calculations if we assume mers can move on the same patches, so we have allowed this model have overlapping mers if we desire. This non-self-excluding model is an useful, accessible, and efficient model that gives us intuition about polymer conformation.

Let's try exploring the NetLogo model 1.11.1 below.

Describing Molecular Conformation Using Random Walk Models

Our simple NetLogo model 1.11.1 tells us a few important things about polymer chains that we can start to relate to polymer behaviors. First, due to the energetic equivalence of rotations of bonds in space along a polymer chain, polymer chains often have coiled conformations. This leads to an amazing property of some polymers in that we can physically pull on the polymer, straighting the chains (with relatively little energy), and then when we release the chains they return to their original coiled shape. Because the chains in the preferred coiled conformation are so small compared to the stretched conformation, this gives us tremendous stretch in these polymers (which we call elastomers), a property that few other materials possess.

We can estimate the end-to-end distance $r$ in these polymer chains using the random walk model. In the exercise above, we used a simple square lattice to discover this behavior. However, in actual polymer chains (i.e., polyethylene) we can be a bit more precise in computing the actual length of the "straight" molecule by understanding the geometry of the mer itself. For example, we can calculate the entire length of a polyethylene chain by looking at the number of bonds in the molecule $n$, the angle between the bonds $\theta$, and the length of the bond $l$:

Using Figure 1.11.2 and a bit of trigonometry, the length of a "straight" polymer chain with bonds angled at $\theta$ is then $L = l n \sin{\theta/2}$. Using the random walk approximation to understand the coiled length vs the length of the bonds $l$ and their number, $n$, allows us to approximate the actual extents of the coiled ene-to-end length $r$ by $r = \sqrt{n}l$.

One Step Further...

Polymers are, of course, not made up of a single chain of atoms, but many coiled up chains. Also, you can't take rubber and extend it as far as would be predicted by the random walk model. You can only extend it a few times its original size. There's a few possible reasons for that, one of which has to do with architecture, and we'll discuss later. The other, which might be more obvious at this point in the conversation is the fact that if you have lots and lots of coiled up polymers, you'd expect some to become entangled. This will limit our ability to full stretch out the polymer length - just like when you try to untangle your wired headphones (which, with Bluetooth headphones, might start to be a dated references) or a length of string. Personally, I think of untangling Christmas lights, but that's not an experience we all have.

The degree of entanglement can limit how much we can stretch the polymers, but also gives them more stiffness (make them harder to stretch). We can control this degree of entanglement during processing to tune the mechanical properties of the polymer.