Hereris the density of the polymer, andntis an empirical parameter that reflects
how many interpenetrated chains on average are required for displaying entangle-
ment effects,nt¼21.37.5 %,Nais the Avogadro constant, andpis thepacking
length, which is defined as
pVðnÞ
R^2 gðnÞ(6.2)
Clearly,pis the ratio of the occupied volume of each chain to its radius of gyration.
Here,nis the number of chain units in each polymer. Using the simple freely-jointed-
chain model with a chain unit holding the lengthland the widthw,wecanobtain
VðnÞnlw^2 (6.3)R^2 gðnÞnl^2 (6.4)Therefore,pw^2
l(6.5)
which is actually the ratio of the cross-sectional area of a chain unit to its length. Here
lcan be regarded as Kuhn segment length, reflecting the semi-flexibility of polymer
chains;w^2 can be regarded as the packing density of polymer chains, corresponding
to the interaction strength between polymer chains. Therefore, the value ofpis an
integrated result of two intrinsic features for the basic chemical structures of polymer
chains. Before the chain length reaches the entanglement lengthne,boththeKuhn
segment length and the packing density of short chains increase with the increase of
chain lengths, while the value ofpwill gradually decay to a constant. After the chain
length arrives atne, the limit value ofpreaches between 2 and 10 A ̊. The entangle-
ment lengthMe~p^3 , indicating that the entanglement effect is raised by the fixed
amount of interpenetrated chainsnt(about 21 chains) in a characteristic volumep^3 for
the molten chains. Therefore, such a scaling relationship reveals the topological
nature of chain entanglement (Lin 1987 ). In concentrated solutions, when the volume
Fig. 6.4 Illustration of
physical entanglement and
topological entanglement of
polymer chains
96 6 Polymer Deformation