A viscoelastic model for multivalent cation-doped polyelectrolyte complexes: from doping levels to ion-responsive shape memory behaviour

Abstract – Polyelectrolyte complexes (PECs) are promising for diverse applications due to their ion sensitivity, enabling spontaneous, non-thermal actuation and ion-induced shape memory. However, due to their complex chemical kinetics, particularly in multivalent systems, few constitutive models have been reported to explore these thermo-mechano-chemical couplings. This work presents a phenomenological model to investigate working principles governing the ion-responsive thermodynamic and dynamic behaviours of PECs. Initially, the corrected doping constant equations are extended to incorporate the bridging and mixed modes of multivalent cations. Doping levels are solved by combining these equations with a swelling equilibrium condition derived from Flory–Huggins solution theory, the phantom network model and the tube model. Subsequently, the non-monotonic influence of multivalent cation concentration is explained through the competition between polymer internal energy and ionic strength within a superposition framework. Furthermore, we integrate the relaxation times across three characteristic scales in PECs into a generalized Maxwell model to reveal how ion specificity and concentration affect the glass transition temperature, storage modulus and capture their ion-induced shape memory behaviour. Finally, our analytical results are sufficiently validated against experimental data in previously reported literature. This work provides a new insight into the complex thermodynamics and constitutive relationships in ion-responsive PECs.