Polymer Physics

potential energy of internal rotation will increase due to the strong volume exclusion
of two overlapping substitutes, which is unfavorable for the stability of the confor-
mation. When the hydrogen atoms locate at the interleaving positions with a rotation
angle of 60, the potential energy of internal rotation can be effectively lowered, as
illustrated in Fig.2.2.
On each middle carbon-carbon bond of the backbone polyethylene, the rest
chains at two ends can be regarded separately as two big groups, replacing the
two hydrogen atoms of an ethane discussed above. Thus, the strong interaction
between two rest chains will greatly raise the potential energy at the overlapping
positions, and make a big difference from their interleaving positions. As illustrated
in Fig.2.3, the overlapping position exhibits the highest potential energy in the
internal rotation, thetrans(denoted as t) conformation shows the lowest, and the
gaucheconformations (separated into the left g+and the right g-) are two metasta-
ble states. Therefore, three relatively stable states, i.e. onetransand twogauche
conformations, can be regarded as the representative states in the statistics of
polymer chain conformations. Such an ideal chain model is often calledthe
rotational-isomerism-state model(RISM, see Ref. (Volkenstein 1963 ; Birshtein
and Ptitsyn 1966 )). This model can characterize well the semi-flexibility of real
polymer chains. Several examples for semi-flexibility of real polymer chains can be
found in Flory’s specialized text (Flory 1969 ).
The semi-flexibility of polymer chains due to the hindered internal rotation is
revealed by the correction from the contribution of the internal rotation in the mean-
square end-to-end distance, as

<R^2 h:r:>¼nb^2 

1 þcosy

1 cosy



1 þ<cosf>

1 <cosf>

(2.9)

where¼

Ð^2 p

0

eEðfÞ=kTcosfdf

Ð^2 p

0

eEðfÞ=kTdf

andkis the Boltzmann’s

constant withTthe absolute temperature.
From the potential energy curve of the internal rotation of polyethylene shown in
Fig.2.3, one can recognize two potential energy differences with important

Fig. 2.2 (a) Illustration of the overlapping position (f¼ 0 ) and the interleaving positions
(f¼ 60 ) of the hydrogen atoms substituted on two carbon atoms of ethane. (b) The potential
energy curve of the internal rotation for ethane

18 2 Structure–Property Relationships