p c r
ð Þ ¼
3
2πNb
2
3
2
exp À
3r
2
2Nb
2
,
ð53Þ
where p c (r) is the probability density function describing the likelihood of the chain
having an end-to-end vector r, N is the number of Kuhn segments in the chain, and
b is the Kuhn length. A schematic illustrating the function p c (r) in Eq. (53) is shown
in Fig. 7c assuming a 1D distribution space for r. Accordingly, the entropy of the
single chain can be obtained by the Boltzmann’s equation:
η c r
ð Þ ¼ k B ln p c r
ð Þ,
ð54Þ
where k B is the Boltzmann’s constant. The free energy for the purely entropic
response of the single chain can be defined as
Polymer chains
Dynamic crosslinks
Static crosslinks
(c)
)
b
(
)
a
(
Fig. 7 Statistical description of polymer network. (a) A schematic showing the molecular structure
of an amorphous polymer network with static and dynamic crosslinks. (b) Distribution space for the
chain end-to-end vector. (c) A schematic of the Gaussian probability density function for chain
conformations in a 1D distribution space
Mechanics of Polymer Networks with Dynamic Bonds
149