A Phenomenological Coupled Model for Ion Transport and Deformation in Superabsorbent Polymers in Calcium-Containing Solutions

Understanding the absorption and desorption behavior of superabsorbent polymers (SAPs) in ionic environments is critical for their practical applications. Ion exchange between monovalent counterions within the SAP and multivalent cations (e.g., Ca2+) in solution not only induces macroscopic desorption but also generates non-uniform internal strain, creating a complex feedback loop with ion transport. This study establishes a phenomenological coupled model that integrates Fickian diffusion for ion transport with an elastic wave equation for SAP deformation. The coupling is realized through deformation-dependent diffusion coefficients and an ion-concentration-modulated elastic modulus, with the latter described by a first-order linear relationship over a limited range. Taking Ca2+ as a representative divalent cation, we systematically investigate the effects of solution concentration, SAP particle size, and ion dissociation degree. The model predicts several non-intuitive phenomena, including transient internal free Ca2+ concentrations exceeding the boundary concentration by up to ~15% and concentration gradient inversions for small SAP particles (radius 75 μm) at later times. Characteristic absorption time constants τa range from 98 s to 179 s depending on particle size and Ca2+ level. Simulated total Ca2+ uptake agrees with experimental data within an 8% mean relative error. The model is validated against macroscopic absorption/desorption curves and total Ca2+ uptake, while the predicted internal concentration and strain fields remain to be confirmed by spatially resolved experiments. These findings provide new mechanistic insights into the chemo-mechanical coupling in SAPs and offer guidance for their tailored design.