A Dual–Mode Sorption Model for Selective Ion Partitioning in Ligand-Functionalized Polymer Membranes

Ion-selective ligand-functionalized polymer membranes (LFPMs) are promising materials for critical mineral (CM) industrial wastewater refining, but the effect of ligand identity on LFPM selectivity remains poorly described. We developed a dual–mode sorption (DMS) model for cation partitioning to quantitatively describe how ligand identity modulates cation-selective partitioning in LFPMs. In this study, we trained a machine learning model to interpolate a sparse aqueous ion–ligand affinity (ΔGIon–Ligand) literature dataset and extrapolate to unobserved ion–ligand pairs. We then parameterized LFPM-specific DMS models, modeling (1) hydrated free volume cation absorption a priori using the pore model with ion solvation and (2) cation adsorption at ligand sites with a competitive Langmuir model. We correlate Langmuir-type equilibrium coefficients with ΔGIon–Ligand using linear free energy relationships. Once parameterized, the DMS model accurately predicts experimental partitioning selectivity for most cation pairs. Ultimately, this study elucidates connections between ΔGIon–Ligand and LFPM selectivity to accelerate systematic LFPM design efforts for CM wastewater refining.