Statistics of rupture in phantom chain network simulations

Phantom chain simulations have shown that the mean rupture properties of star polymer networks collapse onto master curves against the cycle rank density $\xi$. This study revisits this universality with a much larger ensemble than in earlier studies to discuss the statistics. Phantom Gaussian networks were made by end-linking star prepolymers, and 1,000 realizations were collected for each of 30 conditions with functionality $f=3$--$8$ and conversion $p=0.60$--$0.95$, giving 30,000 networks in total. For each realization, the breaking stretch $\lambda_b$, the breaking stress $\sigma_b$, the breaking energy $W_b$, and the cycle rank $\xi$ were recorded. The master curves are unchanged by the larger sample, demonstrating that the earlier conclusions reported for the averages of smaller ensembles hold. However, the individual realizations are inherently random, and their statistical properties, rather than the individual values, are examined. At fixed $f,p$, the fluctuation of $\xi$ is small, varying by less than 0.01, whereas $\lambda_b$, $\sigma_b$, and $W_b$ scatter by 0.05--0.3. The fluctuation of $\xi$ is almost uncorrelated with that of the breaking properties. In addition, the scatter has a definite structure; its magnitude decreases with the mean cycle rank density $\xi$, the $\lambda_b$--$\sigma_b$ correlation grows with $\xi$, and the distributions deviate from Gaussian. The $\lambda_b$ distribution is skewed to the right at small $\xi$, whereas $\sigma_b$ is skewed to the left at large $\xi$. These rupture statistics were discussed in the framework of extreme-value statistics to demonstrate that the observed trends are opposite to those of the random fuse model, in which strength decreases with size and weakest-link statistics appear for weak disorder. The difference may reflect the source of fluctuation, i.e., the cross-linking in the present networks.