Knudsen diffusion differs from Fickian diffusion

We investigate the gas transport enhancement through nanotubes, relative to the prediction by the prevailing century-old Knudsen diffusion model. This enhancement is usually attributed to the partly specular molecular reflections at the smooth nanotube surface, which break the model assumption of completely diffusive reflections. However, an oversighted cause of the discrepancy between the measurement and theory that we found is that even for the gas transport with completely diffusive reflections, the Knudsen diffusion model based on Fick's first law is accurate only for long nanotubes. Additionally, for smooth nanotubes with partly specular reflections, the Knudsen diffusion model is also invalid even if the diffusion coefficient is corrected to account for the atomic-scale surface smoothness. On the other hand, the Knudsen diffusion model might be used for interpretations instead of predictions, and then the diffusion coefficient inferred from the measured mass flow rate could be completely different from the actual value. All those discrepancies and confusions stem from the implementation of Fick's first law can be avoided by using the molecular transmission probability obtained by the kinetic theory to quantify the flow rate of the Knudsen diffusion process. This work provides the correction to the Knudsen diffusion model for accurate predictions of gas diffusion through nanotubes and better interpretations of experimental measurements.