Fig. 4: Regulatory mechanism of the enhanced co-cation selectivity of DP-M.

a Pore size distributions of P-M and DP-M membranes and the sizes of Li⁺ and Mg²⁺ cations. Schematic illustration of transmembrane transport of hydrated Li⁺ and Mg²⁺ through the subnanometer pores in DP-M. b Analysis of the binding configurations between P-M/DP-M molecular fragments and hydrated Li⁺ and Mg²⁺ using the independent gradient model based on Hirshfeld partition (IGMH). c, d Differential scanning calorimetry (DSC) thermogram derived crystallization and melting behavior of the DP-M membrane after being equilibrated with 1000 ppm c MgCl₂ and (d) LiCl aqueous solutions. The lower right inset shows the melting enthalpy (ΔH_m), crystallization enthalpy (ΔH_c), and their difference (ΔH_m−c). e The binding energies between the hydrated Li⁺/Mg²⁺ and the PA molecular fragments of P-M/DP-M. The numbers represent the calculated energy gaps between Li⁺ and Mg²⁺ in P-M and DP-M. f Interaction region indicator analysis of the PA fragments interacting with Mg²⁺. The effects of the hydration layer and TMC were shielded (the arb. units here represents energy, and 1 arb. units is approximately 27.21 eV). g Interaction region indicator analysis of the PA fragments derived from DAT-NH₂ and the visualized structure diagram. h van der Waals surface area ratio corresponding to various electrostatic potential intervals of the PA molecular fragments.