Mathematical modeling of swelling and shrinking dynamics in pH-sensitive hydrogels composed of alginate, anionic cellulose, and chitosan

This study develops a mathematical model to predict the swelling and shrinkage dynamics of pH-sensitive hydrogels composed of alginate, TEMPO-oxidized cellulose nanofibers (TOCNF), or chitosan. We investigated the effects of biopolymer type, cross-linking density, and/or pH conditions on hydrogel behavior. Swelling and shrinkage experiments revealed that cross-link density primarily governs the final equilibrium state but may also modulate the swelling and shrinkage rate under specific environmental conditions. To quantify this behavior, we introduce a novel parameter, the swelling/shrinkage affinity ( a sw , a sh ), which enables predictive evaluation of hydrogel responsiveness. The model integrates experimental data with rheological measurements, linking cross-link density to mechanical properties such as shear modulus. Rheological measurements validated the findings, correlating mechanical properties with cross-link density and aligning with swelling and shrinkage data. This work enhances the fundamental understanding of hydrogel behavior and offers a predictive tool for optimizing formulations in biomedical, environmental, and industrial applications. This framework may be extended in future work to incorporate external stimuli such as temperature or electromagnetic fields, potentially broadening the application of smart hydrogel systems. • The swelling and shrinkage ratios are governed by the biopolymer and crosslinker concentrations. • The swelling and shrinkage rates are independent of biopolymer and crosslinker concentrations. • Cross-link density controls rigidity and swelling but not swelling rate. • Mathematical models predict hydrogel swelling and shrinkage behavior.