A constitutive model for slidable cross-links mediated dual cross-linked polymers to understand coupling and hysteresis of dual cross-links

The mechanical properties of slidable cross-links mediated dual cross-linked polymers present an improvement over those of standard single cross-linked polymers. Nonetheless, the specific role played by the two cross-links within the same condensed polymer state remains ambiguous, and the distinct characteristics and interconnected impacts of these two cross-links necessitate further investigation. The introduction of different cross-links can result in polymers displaying wholly disparate mechanical behaviors. In this study, a constitutive model is developed by integrating self-avoiding walk and scaling theory to examine the rubber elastic behavior of dual cross-linked polymers undergoing slidable cross-links mediation. Following the principles of rubber elasticity, the mechanism underlying the formation of dual cross-linked polymers is elucidated, encompassing stable cross-linked networks, slidable cross-linked networks, and their coupling effects, which correspond to molecular mechanisms, such as the Gent model, self-avoiding walk chains, and scaling theory, respectively. Moreover, based on the Kardar-Parisi-Zhang scaling, the stable cross-linked network entails boundary conditions, while the sliding chain experiences constrained self-avoiding walking to relax stress and dissipate energy. The study further offers insights into the free energy of slidable cross-links mediated dual cross-linked polymers to analyze their rubber elasticity and hysteresis (loading-unloading cycle) effects. Finally, the efficacy of the proposed constitutive models is affirmed through comparison with experimental results documented in the literature, shedding light on the exceptional mechanical properties of dual cross-linked polymers.