Abstract. The present paper deals with a numerical solution of the two-dimensional problem of helium and methane molecules motion through an ultra-thin layer of a porous material composed of spherical nanoparticles of the same size. The interaction potential "nanoparticlemolecule" is obtained by integrating paired molecular interactions over the nanoparticle volume. Using the method of classical molecular dynamics, permeability of a layer having the size of about 10 -8 m is studied.

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The problem of helium isotope separation via bi-layer membranes of graphitic carbon nitride g-C3N4 has been studied. The probability of passing isotopes through the membrane is derived from solving the Schrödinger integral equation using Hermite polynomials. The potential energy of the membrane is calculated based on modified Lennard-Johnes potential. The separation degree of the 3He/4He reaches the value of 1045 due to the resonant effect.

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Nanoporous membranes are widely used in various fields, such as gas separation, water purification, catalytic processes, and the use of batteries in electrodes. Nowadays, hollow carbon spheres or nanowires are attracting attention of researchers and experimenters due to high adjustability of their mechanical and chemical properties. This makes it possible, among other things, to more accurately adjust permeability of membranes created from this material for various atoms and molecules, which ensures a good degree of gas separation. The mathematical simulation of gas separation via nanocapsule and hollow nanowire porous membrane is performed. Research has shown that such membranes are able to separate He/$$\text{CH}_4$$ CH 4 /$$\text{H}_2$$ H 2 /$$\text{N}_2$$ N 2 gas mixtures.

The separation of isotopes of one substance is possible within the framework of the quantum mechanical model. The tunneling effect allows atoms and molecules to overcome the potential barrier with a nonzero probability. The membranes of two monoatomic layers enhance the differences in the components’ passage through the membrane, thereby providing a high separation degree of mixtures. The probability of overcoming the potential barrier by particles is found from the solving of the Schrödinger integral equation. Hermite polynomials are used to expand all the terms of the Schrödinger integral equation in a series to get a wave function. A two-layer graphdiyne membrane is used to separate the mixture.

The use of hydrogen and helium as a fuel requires well-developed technologies for the separation of gas mixtures, in particular, separation of isotopes of an element. The presence of quantum effects makes such processes possible, and the proposed system of two potential barriers enhances the differences in the passage of isotopes through two layers of matter providing high separation efficiency. The probability of atoms passing through composite barriers is derived from the solving of stationary Schrodinger equation.

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