Structural Analysis of the Linear Elasticity of End-Cross-Linked Star Elastomers with a Number of Trapped Entanglements via Coarse-Grained Molecular Dynamics Simulations

To elucidate the fundamental structure–property relationship of inhomogeneous polymer networks with different numbers of trapped entanglements, we have analyzed the mechanical properties and network structures of end-cross-linked star elastomers in the cross-linking procedure by coarse-grained molecular dynamics simulations and a graph-based structural analysis. From graph structural analysis based on the Scanlan–Case criterion, we identified the "effective chains" after excluding mechanically redundant network strands (loops, bridge centers, and dangling chains). The elastic moduli of the networks without trapped entanglements were not directly proportional to the number of effective chains but were proportional to the number of closed cycles formed by effective chains, in agreement with the phantom network theory. For networks with trapped entanglements, it was clarified that the elastic moduli can be predicted by a linear combination of the number of effective cycles and the number of entanglements formed by the effective chains, as expected by Langley's formulation.