Impact of smallest loops and composition fluctuations on the structure of end-linked polymer model networks

A self-consistent scheme of differential equations is developed for predicting the frequency of the two smallest loop defects within polymer model networks. Without any adjustable parameter, we obtain excellent agreement with Monte Carlo simulations that sample loop formation only up to the given maximum loop size. The formation of loops of second generation leads to correlations between connected junctions that cannot be treated exactly by considering statistical arguments alone, which is in contrast to reversible networks where equilibrium statistics are sufficient. These correlations and the statistics of the junctions are provided by our model. Comparison with more realistic simulation data in three dimensions indicates that composition fluctuations (CF) of cross-links and chains clearly impact network formation. The differences between the statistics of the network junctions and our mean field predictions provide insight into the size of the domains with a predominance of chains or junctions, and thus, regarding the quality of the mixture. Our results are highly relevant for an accurate modeling of network structure, improved estimates of the elastic properties of polymer networks, and for advanced analysis techniques of the network structure like network disassembly spectrometry or multiple quantum nuclear magnetic resonance.