Calculation of the permeability of the stackings of multi-walled nanotubes

Carbon nanotubes (CNTs) with open tips are ideal tunnels for molecules and atoms to move at an average velocity exceeding velocity at the tunnel entrance. A membrane with unique physical properties can be obtained by stackable packing of CNTs. The performance of such membrane is essentially higher than that of the membranes whose mass transfer rate is limited by diffusion transfer. In this paper, a quasi-deterministic description of the molecular penetration through an ideal tunnel structure, in particular, through the regular stacking of carbon nanotubes with open tips, is presented. The mathematical model is based on the fundamental concepts of classical mechanics and is related to development of the barrier theory for the motion of a representative molecule of penetrating particles. Determining of the barrier energy is implemented using a modified Lennard–Jones potential that provides a convergence of the integrals over infinite surfaces, which are perpendicular to the main direction of molecular transport. It is revealed that there exists a minimum velocity limit for a representative molecule motion trough the energy barrier found. The values of velocities exceeding the stated limit provide a transparent barrier for molecules. Therefore, the fraction of passing molecules is determined as integral of the Maxwell distribution function with the lower limit equal to the minimum rate of penetration. It is also discovered that three-walled tubes provide higher degree of separation in contrast to the tubes with fewer layers.