Freely Jointed Chain

Freely Jointed Chain jls164 Wed, 02/26/2014 - 13:07

One way to think of polymer conformations is to treat the polymer like a freely jointed chain where every bond is free to rotate to any angle, but the bond distances are fixed (“l”) and the number of bonds in the chain is fixed (“n”). We can then think of the polymer conformation as that of a “random walk”. What do we mean by that? Imagine you stand in one place, representing the end of the polymer, and take a step of length “l” in any direction. This corresponds to one bond distance and the position of the second atom in the polymer backbone. Take another step of length “l”, in any direction. You now stand in the position of the third atom in the polymer backbone. Continue for “n” steps, however many bonds are in the backbone of the polymer. Your path may look something like this:

an example a freely jointed chain polymer conformation

Figure 9.7: Example of a freely jointed chain polymer conformation

Source: Lauren Zarzar

A good way of characterizing this “walk” is the distance from the start to the finish, which we labeled as “r” and call the “end to end distance”. There are many different possibilities for r given values of l and n, just depending on the specific path you take. Similarly, the polymer is constantly fluctuating in conformation and changing end to end distance. While we cannot specify the exact conformation at any given instance, we can know something about the distribution for r values and can use an “average” value of r reflective of that distribution: root mean square (RMS) end to end distance, which is given by the equation 9.1:

R M S e n d t o e n d d i s t a n c e = {〈 r^{2} 〉}^{\frac{1}{2}} = n^{\frac{1}{2}} l

Eq 9.1

Note:

" $n$ " is the number of "steps" (i.e., number of bonds in the polymer backbone) and " $l$ " is the length of each "step" (i.e., bond).

Although RMS end to end distance of a freely jointed chain is a better representation of the polymer conformation than either the contour length or fully extended chain length, it’s still not great – after all, we made an assumption that the bond angles and positions could be anywhere, which is definitely not representative of actual polymer molecules. We can try to alter our model to take that into account.