Phantom Force Balance Procedure for Predicting the Modulus of Entangled Polymer Networks

A computational phantom Force Balance, Maximum Entropy Homogenization Procedure is presented to predict the equilibrium shear modulus of entangled polymer networks. A Monte Carlo method is used to generate periodic bead–spring microstructures of polymer networks. Entanglements are introduced by merging two internal beads of adjacent network strands, yielding additional tetrafunctional cross-links. The microstructures are optimized to their minimum free energy state, for which the modulus is readily available. The procedure is validated by comparing its modulus predictions with those from both stress–relaxation molecular dynamics (MD) simulations and the Miller–Macosko theory (MMT). Near-perfect agreement with both the MD and MMT results is obtained, with the required computational resources being about four and more orders of magnitude smaller than those in the MD simulations. Finally, good agreement with experimental results over a variety of different polymer networks is demonstrated, including those with bottlebrush and comb-like polymer strands and also near-critical gels, suggesting that the presented procedure can be practically used to predict the modulus of arbitrary polymer networks.