Polymer Physics

XX

j>i

bibj¼ 0 (2.4)

Accordingly, we obtain

<R^2 f:j:>¼nb^2 (2.5)

where<...>means the ensemble average over many polymer chains. This
quantity is referred asthe mean-square end-to-end distanceof polymer chains.
As an organic chemist, Staudinger regarded macromolecules initially as rigid
rods (Staudinger and Nodzu 1930 ). Later on, two excellent physical chemists, Guth
and Mark, recognized the free internal rotation of polymer chains (Guth and Mark
1934 ). Almost at the same time, Kuhn proposed more explicitly the random coil
model and made an analogy for the conformation of a freely jointed chain in the
trajectory of a random-walking particle in Brownian motion, provided that
the chain length corresponds to the walking time (Kuhn 1934 ). One can see that
the formula (2–5) is actually consistent with Einstein’s calculation on Brownian
motion in 1905 (Einstein 1905 ). The consistency implies that the random
conformations of polymer chains actually result from an integration of random
Brownian motions of monomers in the chain.
Another quantity to characterize the coil size, probably a more widely used one,
isthe mean-square radius of gyration. This size appears to be more practical than
the mean-square end-to-end distance, since it can be measured directly by using
light scattering. The mean-square radius of gyration is defined as the summation of
mean-square distances of all the monomers relative to the mass center of the
polymer coil. Thus, if we define the square vectorsr^2 as the distances radiating
from the mass center of the whole polymer chain, their average overnþ1 chain
monomers (each with the massm) gives

<R^2 g>

Xn

i¼ 0

miri^2

Xn

i¼ 0

mi

,

(2.6)

When the polymer chain is long enough, we can derive that

<R^2 g>¼

1

6

<R^2 f:j:> (2.7)

2.2.2 Freely Rotating Chains

The carbon-carbon bonds constitute the backbone of a polyolefin chain. In princi-
ple, each carbon atom contains four bonds aligning along the tetrahedronsp^3 hybrid
orbits with the bond angles fixed at 10928 ́. In other words, the connection of

16 2 Structure–Property Relationships