Valence Angle Model (freely rotating)

Figure 9.8: Valence Angle Model

Source: Lauren Zarzar

In the valence angle model, we try to account for the fact that the bonds along the backbone of the polymer will have a preferred bond angle. For a fully singled bonded carbon backbone in which all carbons are sp³ we expect those bonds angles to be 109.5°. In this model, although we fix the bond angle, we will still consider each single bond to freely rotate.

We can alter our formula for the mean square end to end distance with that fixed bond angle:

$${\langle r^2\rangle }_{fa}=n{l}^2\frac{1-\cos \theta }{1+\cos \theta }$$

Notice that the subscript “fa” here is to indicate that this is free rotation around a fixed angle. Also notice that this formula is for the mean square end to end distance, not the root mean square. You’d have to take the square root to find RMS end to end distance.