Angstrom-scale ionic streaming when the electrical double-layer concept fails

A knowledge gap exists for flows and transport phenomena at the angstrom scale when the Poisson–Nernst–Planck equation based on the concept of the electrical double layer fails. We discovered that streaming conductance becomes pressure-dependent in angstrom channels using latent-track membranes. The streaming current emerges only when the applied pressure exceeds a threshold value, which is inconsistent with the existing knowledge as a constant. With increasing channel size, we found that the pressure-dependent streaming conductance phenomenon weakens and vanishes into a constant streaming conductance regime when the mean channel radius exceeds $\sim$ 2 nm. The effective surface potential derived from the streaming conductance that divides conduction anomalously increases as the channel narrows. We suspect that the pressure-dependent streaming current is due to the reinforced Coulomb interaction between counterions and deprotonated carboxyl groups at the surface, which is close to the ion channel but different from that of electrified two-dimensional materials. The streaming current emerged due to hydrodynamic friction when the counterions were released from the surface. We approximated the stochastic process of counterion dissociation by a one-dimensional Kramer escape theory framework and defined the Damk $\ddot {\mathrm{o}}$ hler number to describe the transition from the nonlinear streaming conductance regime to the linear regime as functions of applied pressure and channel radius and well explained the enhanced effective surface potential in confinement.