Fig. 2

Ion transport mechanism of surface-charged graphene oxide (GO) membranes. a MgCl₂ permeability and H₂O/MgCl₂ selectivity and b Na₂SO₄ permeability and H₂O/Na₂SO₄ selectivity of surface-charged GO membranes with various membrane zeta potentials obtained by streaming potential measurements and surface charge densities calculated from Gouy–Chapman equation. Orange squares are MgCl₂ permeability and orange circles are H₂O/MgCl₂ selectivity; Green squares are Na₂SO₄ permeability and green circles are H₂O/ Na₂SO₄ selectivity; Solid lines are best fits for the data. c, d Salt permeability and water/salt selectivity of c positively charged and d negatively charged GO membranes for MgCl₂, MgSO₄, and Na₂SO₄ salts with varied Z⁺/Z^– (ratio of the valence of cation and anion) values. e–g Surface element integration model predictions of Derjaguin–Landau–Verwey–Overbeek (DLVO) interaction energies between a charged ion and the charged membrane surface by adding Van der Waals attraction and electrostatic repulsion. e Orange: net DLVO interaction energy (solid line), electrostatic repulsion (dashed line), and Van der Waals attraction (dotted line) between Mg²⁺ and the GO-PDDA membrane; Yellow: net DLVO interaction energy (solid line), electrostatic repulsion (dashed line), and Van der Waals attraction (dotted line) between Na⁺ and the GO-PDDA membrane; f Navy: net DLVO interaction energy (solid line), electrostatic repulsion (dashed line), and Van der Waals attraction (dotted line) between SO₄²⁻ and the GO-PSS membrane; Blue: net DLVO interaction energy (solid line), electrostatic repulsion (dashed line), and Van der Waals attraction (dotted line) between Cl⁻ and the GO-PSS membrane; g Navy: net DLVO interaction energy (solid line), electrostatic repulsion (dashed line), and Van der Waals attraction (dotted line) between SO₄²⁻ and the GO-PSS membrane; Green: net DLVO interaction energy (solid line), electrostatic repulsion (dashed line), and Van der Waals attraction (dotted line) between SO₄²⁻ and the GO membrane. h Calculation formulas and schematic of the surface element integration (SEI) model in the calculation for DLVO interactions. Error bars represent standard deviations for three measurements