Elastomers are polymers able to undergo large, reversible deformations, and their mechanical properties depend on the chemistry of individual chains as well as the topology of the crosslinked network. In this work we analyze the connection between micro-scale network structure and the macroscopic mechanical properties by performing molecular dynamics (MD) simulations using the Kremer & Grest bead-spring model. The chain length and the density at which crosslinking is performed are varied in order to produce systems ranging from crosslink-dominated to highly entangled, and stress-stretch results are obtained via MD in the large deformation regime. In analogy with recent work on social, technological, and biological networks, we apply mathematical graph theory to describe elastomer networks in a multi-scale modeling framework. A matrix formulation of crosslinked polymers is presented and applied in order to identify the network structure resulting from both chemical crosslinks and physical crosslinks (entanglements). We show that spectral analysis of the crosslink and chain entanglement adjacency matrices along with the corresponding degree distributions can be used to identify and differentiate between the different materials. The spectrum of the crosslink adjacency matrix resembles a sparse regular graph, and spectrum of the intermolecular chain entanglement matrix for the highly entangled systems is shown to resemble a random graph; however, deviations are noted which require further study. A comparison of the network properties with the stress-stretch response demonstrates the influence of both crosslinks and entanglements on the large deformation mechanical behavior of an elastomer material.

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