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A polymer network fractures by breaking covalent bonds, but theexperimentally measured strength of the polymer network is orders of magnitudelower than the strength of covalent bonds. We investigate the effect ofstatistical variation of the number of links in polymer chains on strengthusing a parallel chain model. Each polymer chain is represented by afreely-jointed chain, with a characteristic J-shaped force-extension curve. Thechain carries entropic forces for most of the extension and carries covalentforces only for a narrow range of extension. The entropic forces are orders ofmagnitude lower than the covalent forces. Chains with a statisticaldistribution of the number of links per chain are pulled between two rigidparallel plates. Chains with fewer links attain covalent forces and rupture atsmaller extensions, while chains with more links still carry entropic forces.We compute the applied force on the rigid plates as a function of extension anddefine the strength of the parallel chain model by the maximum force divided bythe total number of chains. With the J-shaped force-extension curve of eachchain, even a small scatter in the number of links per chain greatly reducesthe strength of the parallel chain model. We further show that the strength ofthe parallel chain model relates to the scatter in the number of links perchain according to a power law.

The effect of scatter of polymer chain length on strength