Fig. 2. Energy barriers to ion transport in subnanometer pores.

(A) Relating the measured energy barrier for transport of individual ions (blue columns, left vertical axis) to ion hydration energy (orange columns, right vertical axis). (B) Schematic diagram showing the partition of ions into the pores. To partition into the pores, hydrated ions have to first undergo partial dehydration at the pore mouth (orange arrows) and then be stabilized inside the pores (pink arrows). (C) Schematic diagram showing the diffusion of cations and anions inside the pores. For cations (green spheres), diffusion inside the pores follows a hopping mechanism, i.e., breaking and reforming electrostatic interactions with the negatively charged carboxyl groups in the pore wall. For anions (orange spheres), intrapore diffusion bypasses the negatively charged functional groups because of electrostatic repulsion. (D) Energy barriers for decoupled (individual) Na⁺ and F⁻ transport at pH 9.5, 5.7, and 4.5. Solution pH was adjusted using aliquots of 0.1 M H₂SO₄ and 0.1 M NaOH. Error bars represent SDs from duplicate measurements. (E) Energy barriers for Na⁺ and F⁻ to enter a subnanometer pore (orange columns) and diffuse inside the pore (green columns) from DFT simulations. Details of the DFT simulations can be found in the Supplementary Materials.