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Numerical Estimates of the Topological Effects in the Elasticity of Gaussian Polymer Networks and Their Exact Theoretical Description

Gusev, Andrei A.
Macromolecules 2019 v.52 no.9 pp. 3244-3251
algorithms, computers, entropy, homogenization, mechanics, microstructure, models, modulus of elasticity, network theory, polymers, prediction, topology
A general elastic-thread homogenization procedure is introduced and then used to obtain direct numerical estimates of the shear modulus of Gaussian polymer networks on the basis of their representative 3D periodic microstructures. A force relaxation algorithm is employed to find equilibrium positions of the network junctions. It is shown that from a statistical mechanics perspective, this numerical algorithm can simply be regarded as a convenient method for finding the mean (averaged out over thermal fluctuations) positions of the junctions that maximize the entropy of the polymer network and therefore define the observable, equilibrium thermodynamic state of the network. As an illustration, the procedure is used to obtain estimates of the shear modulus of various model 3D periodic end-linked and vulcanized microstructures incorporating up to a million strands. The same computer microstructures are used to extract several specific topological factors operative in the considered theoretical and model descriptions. It is shown that the numerical stiffness estimates always agree exactly with the predictions of the affine network theory. It is found that while the phantom network model is consistently more accurate than the affine network model, neither of these two classical models are really suitable for general purpose quantitative predictions. It is argued that for such reliable predictions, representative 3D network microstructures are generally needed in order to estimate the single key topological factor Γ advocated by the exact affine network theory.