A spatiotemporally-nonlocal continuum field theory of polymer networks

A physically-based continuum theory that captures the microstructure-dependent and temporal effects of both permanent and transient polymer networks is still lacking, despite the fact that it is greatly needed for the analysis of polymeric microstructures. To fill in this gap, this work proposes a physically-based spatiotemporally nonlocal continuum field theory. A general framework is established that quantitatively connects microscopic descriptions of polymer networks (chain energetics, chain-length distribution, assembly structure of the interpenetrating network, and rate of bond exchange reactions) to key components in the spatiotemporally nonlocal constitutive relations (explicit form of the nonlocal kernel function, magnitude of nonlocal characteristic length, two-phase weighting factors, and explicit form of the relaxation function), based on three hypotheses on the continuum viewpoint of the underlying discrete network structure: the existence of a finite bottom bound of volume to define intensive quantities, uniformity of energy density field inside the representative volume of a polymer network, and the condition for initiation of chain stretch. Applying the general framework to a permanent 8-chain concentric network yields a concrete two-phase nonlocal elasticity constitutive relation, where the explicit form of the kernel function can be derived by simply assuming an implicit form. Application to a transient network with bond exchange reactions yields a spatiotemporally nonlocal constitutive relation. The spatiotemporally nonlocal continuum theory can be helpful for exploring transformative and subversive high-performance materials involving the specific spatial stacking and arrangement of functional units through artificial design.