Fig. 4: Ion diffusion mechanism revealed by DFT calculations.

a Electrostatic potential (ESP) surface (isosurface value: 0.01 e.Å⁻³) of fully (100%) deprotonated and partially (50%) deprotonated 2DPI. The blue and red colors denote less and more electron density in the ESP surface, respectively. For the fully deprotonated 2DPI, electrons are strongly localized on oxygen anions followed by imine nitrogen, while the electrons are more concentrated on the deprotonated oxygen in the partially deprotonated 2DPI. b Binding energy of K⁺ and Na⁺ ions with different hydration numbers on fully and partially deprotonated 2DPI surfaces. c Pathways for Na⁺ ion diffusion through fully deprotonated 2DPI. The analyzed pathway (denoted by green color) involves the diffusion from the most stable site (2O-1N: hydrated ions occupy the interlayer space) to an intermediate metastable site (1O-1N: hydrated ions bind at the edge of the layer), through which the hydrated ion passes to reach its stable neighboring identical site (black color). d, e Energy profile of diffusion path for K⁺ and Na⁺ ions through fully d and partially e deprotonated 2DPI. f Comparison of the diffusion barriers of K⁺ and Na⁺ ions through fully and partially deprotonated 2DPI. g Diffusion barriers of K⁺ and Na⁺ ions with different hydration number through fully deprotonated 2DPI.