A mathematical model describing elastic response of hybrid hydrogel from TEMPO-oxidized cellulose nanofibrils and graphene oxide

This study introduces a mathematical model to elucidate the long-term elastic behavior of hybrid hydrogels, combining TEMPO-oxidized cellulose nanofibrils (TOCNF) and graphene oxide (GO), both in the presence and absence of calcium ions. The model quantifies the influence of hydrogel composition, specifically the TOCNF/GO ratio and calcium ion concentration, on elasticity. The elasticity is precisely measured by the equilibrium shear modulus, derived by fitting experimental data to the generalized Maxwell model. The model's parameters effectively assess the diverse interactions contributing to the hydrogel's overall elasticity. It was found that the elasticity of individual TOCNF and GO hydrogels increases with their respective concentrations, attributable to various interparticle interactions. The incorporation of calcium chloride markedly enhances elasticity through the formation of ionic bridges, which emerge as the dominant factor governing elastic properties. Notably, the efficiency of these ionic crosslinks remains consistent across both TOCNF and GO hydrogels, although the overall elastic contribution is comparatively lower for GO systems. In hybrid TOCNF/GO hydrogels lacking Ca 2+ , negative deviations from standard mixing rules' predictions suggest inhibitory interactions, such as electrostatic repulsion. Conversely, the presence of Ca 2+ leads to significant positive deviations, indicative of a synergistic effect. Here, calcium ions facilitate crucial crosslinks not only within the individual TOCNF and GO networks but, more importantly, at their interfacial regions. This interfacial bridging substantially enhances the long-term elasticity of the ternary hydrogel system. The developed model accurately quantifies these intricate and complex interactions. • New model quantifies long-term elasticity of TOCNF/GO hybrid hydrogels. • Equilibrium shear modulus (Ge) used to assess long-term hydrogel elasticity. • Ca 2+ crosslinks at interfaces boost elasticity in hybrid hydrogels. • Ca 2+ addition diminishes electrostatic repulsion and increases elasticity.