Viscoelastic Properties of Transient Polymer Networks with Stimuli-Responsive Cross-Link Junctions

Chemical reactions in polymer networks have long been a focus of polymer science because of their strong coupling to the network rheology. Herein, a dynamic model is presented for the study of polymer networks in which reversible first-order chemical reactions (A ⇄ B) between two states of the functional group take place inside the cross-link junctions in response to the stimuli triggered by thermal, pH, light, salt, and mechanical changes. Because network chains are specified by the three states as AA, AB, and BB according to their end state, the proposed model treats the coupled three types of network chains. We focus on the reversible dynamic/static conversion from a physical binding (A) to a chemical one (B) and calculate the dynamic mechanical moduli as functions of the rate constants of the reaction with special attention to its degree of reversibility. The viscosity increases with the irreversibility of the reaction, and in the limit of perfect irreversibility, a conversion from a viscoelastic fluid to elastic rubber takes place. Divergence of the viscosity accompanied by the appearance of a finite equilibrium storage modulus at zero angular frequency is demonstrated by the calculation of them as functions of the backward rate constants of the reaction. For reversible reactions, there appear, in general, three fundamental relaxation modes, two of which characterize the speed of the reaction (slow and fast chemical relaxation times), and the other gives the lifetime of the cross-links (rheological relaxation time). The complex modulus of the network turns out to be a parallel connection of three Maxwell elements. Their relaxation times and intensities are found in terms of the rate constants and other molecular parameters such as the chain length and dissociation rate of the functional groups in the state A.