Numerical simulations

The numerical simulation was performed based on coupled Poisson and Nernst–Planck equations by setting appropriate boundary parameters. The Nernst–Planck Eq. (5) defines the flux of each ion species, which describes the transport character of charged nanochannels. The ion concentration induced electrical potential can be described by Poisson Eq. (6). When the system reaches a stationary regime, the ion flux should conform to the steady-state continuity Eq. (7).

where Ji→, ci, zi, and Di are the ionic flux, ion concentration, charge number of each ionic speciesi, and diffusion coefficient. φ and ε represent the electrical potential and the dielectric constant of medium. T, F, and R are the absolute temperature, Faraday constant and universal gas constant, respectively. The diffusion coefficients for K⁺ and Cl⁻ are 1.9 × 10⁻⁵ and 2.0 × 10⁻⁵ cm² s⁻¹, respectively.

For simplification, we use a cylindrical channel of 100 nm in length and 0.76 nm in diameter to simulate the channels inside HB_x@COF-301/PAN. To decrease the effect of entrance/exit mass transfer resistances on the overall ionic transport, two electrolyte reservoirs are introduced. The external potential is applied on the boundary w1, and w2 offered the reference potential. The boundary conditions for the electrical potential and ion flux are shown as below:

The physical quantity σ represents the surface charge density of the channel walls. The ionic current can be calculated by