Revisiting the Elasticity Theory for Real Gaussian Phantom Networks

In the classical phantom network theory, the shear modulus of a polymer network is derived assuming the underlying network has a treelike topology made up of identical strands. However, in real networks, defects such as dangling ends, cyclic defects, and polydispersity in strand sizes exist. Moreover, studies have shown that cyclic defects, or loops, are intrinsic to polymer networks. In this study, we illustrate a general framework for calculating the rubber elasticity of phantom networks with arbitrary defects. Closed form solutions for the elastic effectiveness of strands near isolated loops and dangling ends are obtained, and it was found that under classical assumptions of phantom network theory loops with order ≥3 have zero net impact on the overall elasticity. However, when a simple approximation for strand prestrain is considered, the modified network theory agrees well with experimentally measured moduli of PEG gels.