Chain Stiffness

We can get a sense of how stiff a polymer chain is from the characteristic ratio, C_∞

$$C_{\infty }=\frac{{\langle r^2\rangle }_0}{n{l}^2}$$

Higher steric parameter corresponds to a higher characteristic ratio (all else equal) because a higher steric parameter will give a higher value of ${\langle r^2\rangle }_0$. So polymers that do not rotate as freely are more stiff and have longer end to end distances. The denominator, $n{l}^2$ is the mean square end to end distance based on the “random walk” with free rotation and no restrictions to bond angles (i.e., the denominator represents the most flexible polymer possible!). The numerator is the valence bond model including hindered rotation; this value will be higher for stiffer polymers. Think of the two extremes: an infinitely stiff polymer is just a rod. It can’t coil at all, and the end to end distance is the same as the contour length. Compare that to a very flexible polymer, which will have the ability to coil, and thus will have a much lower end to end distance.

A table of some characteristic ratios and steric parameters is given below (textbook table 10.3)

As expected, bulkier groups correspond to higher steric parameters and higher characteristic ratios. Temperature also plays a role; at higher temperatures there is more energy available to overcome those barriers to rotation, so it becomes easier for single bonds to rotate. If it’s easier for bonds to rotate, then the polymer is overall more flexible, and we therefore see a decrease in the steric parameters and characteristic ratio as we go to higher temperatures.