The Euler Characteristic of Phantom Elastomeric Polymer Networks

It is shown that through a relationship between the Euler characteristic and the cycle rank of polymer networks one has access to a quantitative description of functionality heterogeneity in these fascinating structures. This relationship is applied and extended to study heterogeneous network structures resulting from the random removal of chains from a perfect 2D tetrafunctional network. It is shown that the distribution of crosslink functionality is correlated strongly with the mechanical properties of the networks as those are manifested in the cycle rank.