During the molecular permeation, the hBN surface is towards the feed-side while the graphene surface is towards the permeate-side. Thus, the CO₂/CH₄ mixtures are initially arranged in the hBN side. The graphene-hBN bilayer nanopore is constructed by stacking the graphene and hBN layers together and generating a nanopore with the same size (Figure 1A). To realize a selectivity of the separation of CO₂/CH₄ mixtures, the pores are functionalized by the N and H atoms (Figure 1B), as done in our early work (Sun and Bai, 2017). The N functionalization not only enlarges the size of the pore to achieve a high permeance of CO₂ molecules, but also enhances the adsorption intensities of CO₂ molecules on the surface which further improves the permeability and selectivity. The spacing distance of two layers is set as 3.4 Å in the simulation (Figure 1C).

To examine the sieving effect of nanopores, we perform MD simulations for the separation of CO₂/CH₄ mixtures in a non-equilibrium system based on the platform of LAMMPS. In the system, there are 1,000 mixture gas molecules, namely 500 CO₂ molecules and 500 CH₄ molecules. Initially, the molecules are arranged alternatively and uniformly in the feed side, while the permeate side is kept vacuum, as seen from Figure 1A. The nanopore is located at the center of the simulation box of height 140 nm. The heights of box in the permeate side and feed side are both 66.6 nm. Periodic boundary conditions are applied in the x- and y-directions (parallel to the surface), while reflective wall condition is applied in the z-direction (perpendicular to the surface). The spacing distance among the nanopores in the porous membranes is assigned as 4 nm. In this case, the number density of the pores is 6.25×10¹² cm⁻², which is comparable with the pore densities in the fabricated NPG membranes in laboratory (O’Hern et al., 2014). Owing to the application of periodic boundary conditions, the area of sheet is assigned as 4 × 4 nm² based on the spacing distance among nanopores. The simulation is run in a NVT ensemble with a temperature of 350 K, which is close to the running temperature in gas separation industry. A simulation period of 6.6×10⁷ timesteps (time step is 0.3 fs) is chosen such that the permeable gases transport through the nanopore with a relatively fast rate, ensuring a high accuracy in the calculation of permeance. Based on the ideal gas equation, the initial pressure of the feed side is estimated as 21.1 bar.

For carbon and hydrogen atoms in graphene sheet and CH₄ molecules, the atomic interactions are modeled by the AIREBO potential model; while for CO₂ molecules and hBN sheets, they are modeled by the Lennard–Jones (LJ) potential model. The detailed introduction and parameters of AIREBO potential can be found in the literature (Stuart et al., 2000). A three-site model with three partial charges is adopted for CO₂ molecules. In the model, the LJ potentials and Coulombic potentials are coupled together, as follows: ϕ ( r ij ) = { 4 ε [ ( σ r ij ) 12 − ( σ r ij ) 6 ] + C q i q j χ r ij ( r ij < r c u t ) 0                                                                                                                     ( r ij ≥ r c u t ) (1) where σ is length scale, ε is energy scale, q _i and q _j are the charges on atom i and atom j, C is an energy-conversion constant, and χ is the dielectric constant. The parameters for cross atoms are estimated by the Lorentz-Berthelot mixing rules based on the following parameters for C-C interactions: σ = 3.40 Å, ε = 2.413 × 10⁻³ eV (Du et al., 2011), H-H interactions: σ = 2.50 Å, ε = 1.300 × 10⁻³ eV (Liu et al., 2015), and N_graphene-N_graphene interactions: σ = 3.297 Å, ε = 3.126 × 10⁻³ eV (Chae and Violi, 2011), B-B interactions: σ = 3.453 Å, ε = 4.116 × 10⁻³ eV (Baowan and Hill, 2007), N_hBN-N_hBN interactions: σ = 3.365 Å, ε = 6.281 × 10⁻³ eV (Baowan and Hill, 2007). The bond and angle potentials in the gas molecules (CO₂) and the functionalized groups are all modeled by harmonic potential. The parameters in these potential models can refer to our previous work (Sun and Bai, 2017). The charges on the atoms near the pore are obtained through a density functional theory calculation using the DMol₃ module in Material Studio software.

As seen from Figure 2A, the number of permeated molecules increases over time but the increasing rate slows down gradually owing to the non-equilibrium permeation process. The molecular number in the permeate side (z < −0.6 nm) of permeating molecules sharply increases with the passage of time, while that of non-permeating molecules increases very slowly and only a few CH₄ molecules appear in the permeate-side. Namely, the graphene-hBN bilayer nanopore exhibits a phenomenon of selective molecular sieving for the separation of CO₂/CH₄ mixtures. Comparing to the single-layer graphene nanopore, the performance of selective molecular sieving of graphene-hBN bilayer nanopore is more significant.

In order to quantitatively analyze the molecular sieving performance of nanopores, the molecular permeance is calculated based on the relationship between the number of permeated molecules N and time t, as follows (Sun and Bai, 2017): N = ( 250 − N a l / 2 ) × ( 1 − e − 8.24 × 10 10 P t ) (2) where N _al is the average number of molecules adsorbed on both sides of graphene, P is molecular permeance, and the constant 250 is related to the molecular number 500 in the simulation box. On the basis of this function, the curves of molecular number N versus time t are fitted and the molecular permeance P is then obtained.

The molecular permeance (P) and selectivity for the separation of CO₂/CH₄ mixtures through the bilayer and single-layer nanopores are displayed in Figure 2B. For CO₂/CH₄, the CO₂ permeance of bilayer nanopore is higher than that of single-layer nanopore while the CH₄ permeance is lower, accordingly a higher selectivity appears for the bilayer nanopore. The bilayer pore presents a 3.7 times higher selectivity comparing to the single-layer graphene nanopore with the same size. Meanwhile, we compare the separation performance of the bilayer nanopore and graphene nanopore with available simulation data of NPG membranes in literature (Wen et al., 2015; Sun and Bai, 2017), as shown in Figure 2C. It can be found that both the single-layer graphene and bilayer nanopores show a high molecular permeance. The graphene-hBN bilayer nanopore exhibits an ultra-high permeance and high selectivity, such that its separation performance exceeds the upper bound of polymer membranes.

To understand why the graphene-hBN bilayer nanopore can present the quasi-unidirectional transport characteristic and finally exhibit a better molecular separation performance comparing to the single-layer nanopore, we firstly analyze the molecular adsorption characteristics on the two sides of the graphene-hBN bilayer surface and single-layer graphene surface. From the number density distributions of gas molecules along the perpendicular direction (Figure 3A,B), it can be found that the density of adsorbed molecules for CO₂ is higher than that of CH₄. This means that the interactions between CO₂ molecules and hBN or graphene surfaces are stronger than those of CH₄ molecules. Owing to the non-equilibrium permeation process, the density of adsorbed molecules in permeate side is weaker than that in feed side. Comparisons between the bilayer surface and single-layer surface in Figure 3C show that the bilayer nanopore has a stronger adsorption ability of CO₂ gas molecules because the interaction of the hBN surface is greater than that of graphene surface. Meanwhile, the competitive adsorption of CO₂ and CH₄ results in a reduction in the adsorption density of CH₄ on the feed-side surface of bilayer nanopore.

The distinctive molecular adsorption on the surface can partially explain why the graphene-hBN bilayer nanopore can improve selectivity while maintain ultra-high permeability. The adsorption of gas molecules on the surfaces is beneficial for the molecular permeation through the nanopore (Drahushuk and Strano, 2012; Sun et al., 2014), therefore the stronger molecular adsorption in the feed side can promote the molecular permeation from feed-side to permeate-side. Thus, the distinctive adsorption abilities of CO₂ and CH₄ molecules on the feed side of the graphene-hBN bilayer surface is the main reason for the high-efficiency molecular sieving.

The molecular adsorption on the graphene surface may have another positive contribution on the molecular permeance by inhibiting molecular backflow from permeate-side to feed-side. This inhibition effect can be well analyzed from the aspect of molecular trajectories during permeation. We obtain the molecular crossing number N _F−P and N _P−F, respectively; N _F−P is the molecular crossing number from feed-side to permeate-side, while N _P−F is the molecular crossing number from permeate-side to feed-side. Figure 4A shows the time variations of N _F−P, N _P−F for the CO₂ molecules. Over time, the two numbers all necessarily increase gradually with the permeation of molecules, but N _F−P increases more sharply and N _P−F increases very slowly. Interestingly, for the bilayer nanopore N _F−P increases while N _P−F decreases comparing to the single-layer nanopore, demonstrating that the bilayer nanopore can promote the molecular permeation from feed-side to permeate-side while inhibit the molecular backflow. This phenomenon exactly proves the quasi-unidirectional transport characteristics of the graphene-hBN bilayer nanopore.

Then, we further obtain the interacting energy distributions between gas molecules and graphene-hBN bilayer nanopore by calculating the potential of mean force (PMF). The PMF profiles of gas molecules passing through the nanopore are calculated based on the sampling method of Adaptive Biasing Force (Sun et al., 2019b). It can be clearly seen from Figure 4B that the PMF distribution is asymmetric, namely, CO₂ molecules have a relatively deeper attractive potential well in the feed side comparing to the permeate side. This further proves that there is a stronger interaction between gas molecules and graphene-hBN bilayer surface in feed side. PMF calculations give a strong evidence for the distinctive molecular adsorption abilities on the two sides and the resulted quasi-unidirectional molecular transport characteristics from the insight of thermodynamic physics.