Micromechanics of polydisperse polymer networks with irregular topologies

Polymer network models connect the behavior of individual chains to the constitutive relationships of a broader polymer network. These network models have been used to characterize multiscale phenomena in a variety of contexts such as rubber elasticity, soft multifunctional materials, biological materials, and even the curing of polymers. However, more traditional network models are restricted to networks of monodisperse chains with constant degree crosslinks (i.e. uniform topology). While recent work has attempted to address polydisperse networks with varying degree cross-links, open questions persist on how to both describe the network structure statistically, and relate macroscopic state variables (such as deformation) to individual chains within the network. Here we argue cross-links can serve as a suitable fundamental unit of the network. To do so, we consider the elasticity of statistical distributions of cross-links with varying degrees, and consisting of chains with differing molecular weights. We derive relationships between statistical measures of the polydispersity of the network and its elasticity, then conclude with an outlook towards making elusive yet significant connections between polydispersity and toughness.