  A polymer network fractures by breaking covalent bonds, but the
experimentally measured strength of the polymer network is orders of magnitude
lower than the strength of covalent bonds. We investigate the effect of
statistical variation of the number of links in polymer chains on strength
using a parallel chain model. Each polymer chain is represented by a
freely-jointed chain, with a characteristic J-shaped force-extension curve. The
chain carries entropic forces for most of the extension and carries covalent
forces only for a narrow range of extension. The entropic forces are orders of
magnitude lower than the covalent forces. Chains with a statistical
distribution of the number of links per chain are pulled between two rigid
parallel plates. Chains with fewer links attain covalent forces and rupture at
smaller extensions, while chains with more links still carry entropic forces.
We compute the applied force on the rigid plates as a function of extension and
define the strength of the parallel chain model by the maximum force divided by
the total number of chains. With the J-shaped force-extension curve of each
chain, even a small scatter in the number of links per chain greatly reduces
the strength of the parallel chain model. We further show that the strength of
the parallel chain model relates to the scatter in the number of links per
chain according to a power law.
