monomers on a real polymer chain has a restricted bond angle. However, even if the
bond angle keeps fixed, the internal rotation of each bond around the previous bond
on the chain is still possible. Therefore, supposing no hindrance in the internal
rotation, we obtainthe freely-rotating-chain model. As illustrated in Fig.2.1, the
angle between the bond vector and its preceding is defined asy, and for the
backbone carbon chains,y¼ 180 –10928 ́. The mean-square end-to-end distance
of the freely-rotating-chain model can be derived by making a correction term for
the fixed bond angles, on the basis of the mean-square end-to-end distance of the
freely-jointed-chain model, giving by
<R^2 f:r:>¼nb^2
1 þcosy
1 cosy(2.8)
2.2.3 Hindered Rotating Chains
When real polymer chains perform the internal rotation along the backbone bonds,
the substituted side groups will interact with each other, causing a hindrance to the
internal rotation. Therefore,the hindered-rotating-chain modelmust be considered.
As illustrated in Fig.2.1, along the chain backbone, a bond can perform internal
rotation around the previous bond with a fixed bond angle. The trajectory made by
the end of this rotating bond forms a circle. On this circle, by making a reference to
the sectional line of the face formed by the previous two bonds along the chain, one
can define the angle of the projected line of the rotating bond asthe rotation anglef.
When two hydrogen substitutes on two separate carbon atoms of ethane CH 3 –CH 3
locate at the overlapping positions of internal rotation, their distance is 2.26 A ̊,
smaller than the sum of van der Waals radius of hydrogen atoms 2.40 A ̊. Thus, the
Fig. 2.1 Illustration of the
fixed bond angleyand the
internal rotating anglef
along the chain backbone
2.2 Semi-Flexibility of Polymer Chains 17