Statistical Field Theory for Nonlinear Elasticity of Polymer Networks with Nonlocal Excluded Volume Interactions

Existing phenomenological and micromechanical elastic models for rubbery polymers are unable to account for polymer molecular structure and inter-segment interactions. To address these limitations, we have developed a statistical mechanics-based field-theoretic model for polymer networks, which provides an efficient mesoscale approach that enables the accounting of excluded volume effects without the expense of large-scale molecular modeling. A mesoscale representative volume element is populated with multiple interacting chains, and the macroscale nonlinear elastic deformation is imposed by mapping the chain end-to-end vectors.

In the absence of excluded volume interactions, it recovers the closed-form results of the classical theory of rubber elasticity. With excluded volume interactions, the model is solved numerically in 3D using a finite element method to obtain the energy, stresses, and linearized moduli under imposed macroscale deformation. Highlights of the numerical study include: (1) the linearized Poisson’s ratio is very close to the incompressible limit without a phenomenological imposition of incompressibility; (2) despite the harmonic Gaussian chain as a starting point, there is an emergent strain-softening and strain-stiffening response that is characteristic of real elastomers, driven by the interplay between the entropy and the excluded volume interactions; and (3) the emergence of a deformation sensitive localization instability at large excluded volumes.