Computational Studies on Holey TMC6 (TM = Mo and W) Membranes for H2 Purification

The purification of hydrogen (H2) has been a vital step in H2 production processes such as steam–methane reforming. By first-principle calculations, we revealed the potential applications of holey TMC6 (TM = Mo and W) membranes in H2 purification. The adsorption and diffusion behaviors of five gas molecules (including H2, N2, CO, CO2, and CH4) were compared on TMC6 membranes with different phases. Though the studied gas molecules show weak physisorption on the TMC6 membranes, the smaller pore size makes the gas molecules much more difficult to permeate into h-TMC6 rather than into s-TMC6. With suitable pore sizes, the s-TMC6 structures not only show an extremely low diffusion barrier (around 0.1 eV) and acceptable permeance capability for the H2 but also exhibit considerably high selectivity for both H2/CH4 and H2/CO2 (>1015), especially under relatively low temperature (150–250 K). Moreover, classical molecular dynamics simulations on the permeation process of a H2, CO2, and CH4 mixture also validated that s-TMC6 could effectively separate H2 from the gas mixture. Hence, the s-MoC6 and s-WC6 are predicted to be qualified H2 purification membranes, especially below room temperature.


Introduction
With the increasing environmental pollution issues and global energy crisis, more and more attention has been paid to green energy resources, particularly hydrogen (H 2 ) energy [1,2]. Though lots of methods have been developed, steam-methane reforming is still the main approach for industrial production of H 2 [3]. During the steam-methane reforming process, however, the mixture composed of H 2 , CO 2 , and CH 4 is inevitable, which renders the purification of H 2 highly crucial in hydrogen production. The membrane separation technology is one of the most widely accepted methods for H 2 separation and purification [4]. Among the reported membranes, two-dimensional (2D) carbon-based membranes have been extensively studied due to their distinct advantages, e.g., low energy consumption and good cyclicity through physical interactions [5][6][7].
Material design by computational methods has been an effective tool to achieve novel 2D carbon-based ultrathin membranes for H 2 separation and purification, especially for the ones with intrinsic pores. As one of the most well-known representatives, the porous graphitic carbon nitride (g-C 3 N 4 ) monolayer has received a lot of attention for its potential as an effective gas separation membrane. Under room temperature, g-C 3 N 4 could exhibit extremely high theoretical selectivity for H 2 /CH 4 in the order of 10 46 [8]. Moreover, further theoretical simulations indicated that g-C 3 N 4 is also capable in helium (He) purification from both natural gas and noble gas molecules [9]. With the help of theoretical simulations, the porous C 2 N monolayer was reported to be suitable for He separation from other gases (Ne, CH 4 , CO 2 , etc.) [10]. Wang et al. investigated the diffusion properties of He, Ne, CO 2 , Ar, N 2 , CO, and CH 4 through a porous monolayer covalent triazine-based framework (CTF) membrane. Calculation results demonstrate that the selectivity for He and H 2 against common gas molecules (such as CO 2 , N 2 , CO, and CH 4 ) is highly promising for practical applications [11]. Meng et al. theoretically explored the structural and mechanical properties of metal-free fused-ring polyphthalocyanine (H 2 PPc) and halogenated H 2 PPc (F-H 2 PPc and Cl-H 2 PPc) membranes. It was found that fluorination and chlorination can effectively tune the permeable pores. Particularly, F-H 2 PPc is fascinating as a separation membrane for H 2 purification [12]. Recently, a series of 2D γ-C 4 X (X = O, S, or Se) membranes with intrinsic pores were theoretically designed, among which γ-C 4 O shows both low diffusion barriers (0.35 eV) and high permeance (5.0 × 10 −7 mol m −2 s −1 Pa −1 ) for H 2 . Moreover, γ-C 4 O is highly promising as a H 2 purification membrane from the H 2 /CH 4 mixture with a selectivity of about 10 19 [13]. The existing studies indicate that there are abundant possibilities for carbon-based ultrathin membranes with intrinsic pores. Therefore, carbon-based membranes with different pore sizes and termination on the pore edges are indispensable for H 2 purification.
Besides the ones entirely composed of non-metal elements, 2D membranes composed of carbon and metal atoms may also play a vital role as gas separation membranes due to the incorporation of metal atoms. In the pioneering work by Li et al., a novel 2D transition metal carbide (h-TMC 6 , TM = Mo, W) structure was theoretically designed [14]. It was found that the crystal structure of h-TMC 6 belongs to the hexagonal Kagome lattice. The stability of h-TMC 6 was confirmed by molecular dynamics simulations and phonon spectra calculations. Later, Liu et al. reported other transition-metal carbides with the same composition of TMC 6 (TM = Mo, W) but a tetragonal lattice [15], therefore being named s-TMC 6 . In general, the TMC 6 monolayers show triple atomic layer structures with Mo/W atomic layers sandwiched between two carbon atomic layers, with TM atoms coordinated with six nearest neighboring C atoms. More importantly, in both h-TMC 6 and s-TMC 6 , there are intrinsic pores surrounded by the TM atoms and carbon atoms, whose sizes are mainly determined by the lattice structures.
In this work, the capability of holey structures of TMC 6 (M = Mo, W) membranes for H 2 purification was theoretically explored, both square and hexagonal phases. By comparing the pore size and separation performance against H 2 and other gas molecules (N 2 , CO, CO 2 , CH 4 ), it was found that s-TMC 6 is more promising for H 2 purification, especially from H 2 /(CO 2 , CH 4 ) mixtures below room temperature (150-250 K). Our work not only predicts the potential applications of the TMC 6 membranes but also recommends the novel membrane materials for H 2 purification under low temperature.

Computational Methods
The Vienna ab initio simulation package (VASP) [16,17] was used for the first-principle calculations with plane-wave basis set and the projector augmented-wave (PAW) [18] method. The Perdew-Burke-Ernzerhof (PBE) [19] functional was adopted, with a cutoff energy of 500 eV. The structural relaxations were considered to be converged until the change in total energy reaches 10 −5 eV and the residual force per atom reaches 0.02 eV Å −1 . To avoid the interaction between the neighboring periodic images, a vacuum slab around 2 nm was applied for all the structures. The Brillouin zone was sampled with Monkhorst-Pack [20] k-point grids of 11 × 11 × 1 and 5 × 5 × 1 for unit cell of h-TMC 6 and s-TMC 6 , respectively. Furthermore, 4 × 4 × 1 k-point grids were adopted for the 3 × 3 supercell of h-TMC 6 , while 3 × 3 × 1 k-point grids were adopted for the 2 × 2 supercell of s-TMC 6 . For accurate description of weak van der Waals (vdW) interactions, Grimme's dispersion correction (DFT-D3) was included during the adsorption-related calculations [21]. The climbing image nudged elastic band (CI-NEB) method [22] was used to search the minimum energy pathway (MEP) and confirm the transition state (TS) during the diffusion of gas molecules. The diffusion barriers (E b ) were calculated to evaluate the capability of gas molecules passing through the intrinsic pores in the TMC 6 membranes by the following definition: where the E TS and E IS denote the energy of transition state (TS) and initial state (IS), respectively. Additionally, the adsorption energies (E ad ) of gas molecules on 2D TMC 6 were calculated by where E mol * , E mol , and E * stand for the energy of adsorption system, isolated molecule, and 2D TMC 6 , respectively. Herein, a negative E ad value means that the adsorption is favorable as exothermic process. The classical molecular dynamics (MD) simulations on permeation process of H 2 , CO 2 , and CH 4 mixture were implemented by Forcite module available in the Materials Studio software package. Gas molecules were interspersed between the TMC 6 membranes, and the initial condition was 250 K with a total simulation time of 5000 ps. The NVT ensemble and universal force field [23] were employed during the simulation with a time step of 1 fs.

Results and Discussion
As predicted in previous work, the single-layered TMC 6 (M = Mo, W) includes two different phases with holey structure, namely, the square phase (s-TMC 6 ) and the hexagonal phase (h-TMC 6 ). s-TMC 6 shows P4/mbm symmetry with a lattice constant of 8.541 and 8.543 Å for s-MoC 6 and s-WC 6 , respectively. There are 4 TM atoms and 24 carbon atoms in the unit cell of s-TMC 6 . Due to the extremely close lattice parameters, s-MoC 6 and s-WC 6 exhibit intrinsic pores with the same diameter of about 5.65 Å, which were obtained by directly measuring from the optimized atomic positions of carbon in the edge of pores. The intrinsic pores are composed of four TM atoms and eight C atoms, as displayed in Figure 1a. Differently, the h-TMC 6 exhibits P6m2 symmetry with the lattice constant of 4.381 and 4.383 Å for h-MoC 6 and h-WC 6 , respectively. The unit cell of h-TMC 6 contains one TM atom and six carbon atoms. Due to the more compact atomic configurations of h-TMC 6 , their intrinsic pores have smaller sizes when compared with s-TMC 6 . As indicated in Figure 1b, the pores in h-TMC 6 are composed of three TM atoms and six C atoms, whose diameters are about 4.48 Å.
After the confirmation of pore structures in the 2D TMC 6 , we then tested their performance in gas adsorption and diffusion. In this work, five different kinds of gas molecules (containing H 2 , N 2 , CO, CO 2 , and CH 4 ) were tested, which are the main components of the gas mixture in the steam-methane reforming process. All the adsorption configurations of gas molecules were fully relaxed. The adsorption energies (E ad ), equilibrium adsorption heights (h), and diffusion barriers (E b ) are systematically summarized in Table 1. In the equilibrium configurations, the adsorption heights between the gas molecules and substrate are mostly in the range of 2.0-2.5 Å, along with adsorption energies within −0.18 eV, as shown in Figure 2a,b. The large adsorption heights and weak adsorption strength evidently indicate physisorption through van der Waals interaction between the gas molecules and TMC 6 substrates. Generally speaking, the E ad of the same gas molecule on MoC 6 and WC 6 is almost the same, implying the negligible influence of TM atoms on the gas adsorption. Meanwhile, the E ad values of a gas molecule on s-TMC 6 will be slightly smaller than those on h-TMC 6 . However, it is interesting to see that the diffusion of the same molecule on h-TMC 6 will be much more difficult than that on s-TMC 6 due to the much greater E b values (see Figure 2c). Another finding is that for s-MoC 6 and s-WC 6 , due to the relatively small E b values, the difference between E b of the same gas molecule on them is also very small (<0.1 eV). For h-MoC 6 and h-WC 6 , a significant difference between E b of the same gas molecule on them arises (up to 0.33 eV). Therefore, though the gas molecules have close adsorption interaction on the TMC 6 membranes, their diffusion performance is highly distinguished. After the confirmation of pore structures in the 2D TMC6, we then tested their performance in gas adsorption and diffusion. In this work, five different kinds of gas molecules (containing H2, N2, CO, CO2, and CH4) were tested, which are the main components of the gas mixture in the steam-methane reforming process. All the adsorption configurations of gas molecules were fully relaxed. The adsorption energies (Ead), equilibrium adsorption heights (h), and diffusion barriers (Eb) are systematically summarized in Table 1. In the equilibrium configurations, the adsorption heights between the gas molecules and substrate are mostly in the range of 2.0-2.5 Å, along with adsorption energies within −0.18 eV, as shown in Figure 2a,b. The large adsorption heights and weak adsorption strength evidently indicate physisorption through van der Waals interaction between the gas molecules and TMC6 substrates. Generally speaking, the Ead of the same gas molecule on MoC6 and WC6 is almost the same, implying the negligible influence of TM atoms on the gas adsorption. Meanwhile, the Ead values of a gas molecule on s-TMC6 will be slightly smaller than those on h-TMC6. However, it is interesting to see that the diffusion of the same molecule on h-TMC6 will be much more difficult than that on s-TMC6 due to the much greater Eb values (see Figure 2c). Another finding is that for s-MoC6 and s-WC6, due to the relatively small Eb values, the difference between Eb of the same gas molecule on them is also very small (<0.1 eV). For h-MoC6 and h-WC6, a significant difference between Eb of the same gas molecule on them arises (up to 0.33 eV). Therefore, though the gas molecules have close adsorption interaction on the TMC6 membranes, their diffusion performance is highly distinguished.  Table 1. The adsorption energies (E ad , eV), equilibrium adsorption heights (h, Å), diffusion barriers for the studied gases (H 2 , N 2 , CO, CO 2 , and CH 4 ) passing through the intrinsic pore (E b , eV) of 2D TMC 6 , energy differences (∆E sub , eV) of the TMC 6 membranes between free-standing state and TS without adsorbates, the bond lengths (l 1 and l 2 , Å) of the gas molecules with free-standing state and TS. ble selectivity against H2, attributed to the suitable pore size. Notably, though the por size of s-WC6 is slightly larger than that of s-MoC6, it causes higher Eb for the same ga molecule. Hence, the metal species has relatively less influence on the diffusion of ga molecules, but it is not negligible. In the meantime, the unoccupied vdW diameters o pore in h-TMC6 are 0, which explains why h-TMC6 exhibits very high diffusion barrier for all the studied gas molecules. In addition to the comparison between pore sizes, the steric deformation of 2D TMC caused by electrostatic repulsion was also checked. We measured the main bond length of different molecules in isolated state (l1) and TS (l2), that is, H-H bond length for H2, N N bond length for N2, C-O bond length for CO and CO2, and C-H bond length for CH taking in consideration the non-polarized structures of CO2 and CH4. The comparison be tween l1 and l2 (seeing Table 1) indicates that the structural changes of the gas molecule under TS are negligible, except for CO and N2 on h-TMC6. Then, by removing the ga molecule in its TS structure and calculating the single-point energy (SPE) of the 2D TMC we can obtain the energy differences (ΔEsub) of the 2D TMC6 between free-standing stat and TS without adsorbates, as listed in Table 1. It can be summarized that the ΔEsub i directly proportional to the corresponding Eb. For H2, N2, and CO, the electrostatic repu sion takes the proportion of 10-18% in the Eb, and the proportion increases to 20-30% fo CO2 and CH4. It is revealed that CO2 and CH4 have more significant electrostatic repulsio After the comparison of adsorption and diffusion performance of different gas molecules on 2D TMC 6 , we then turned to the microscopic insights. As displayed in Figure 3, the electron density isosurfaces of H 2 , N 2 , CO, CO 2 , and CH 4 at their corresponding transition states are provided with the same isosurface value of 0.015 e Å −3 . For s-TMC 6 , it is evident that the smaller pores are fully occupied by the electron density isosurfaces, and the unoccupied parts only appear in larger pores with square shapes. Meanwhile, for h-TMC 6 , the triangle shapes appear in the pores as unoccupied parts, whose area is significantly smaller than that in s-TMC 6 . As we can see, under the transition states, there is significant overlapping between the electron density distribution of h-TMC 6 and gas molecules, while the overlapping between s-TMC 6 and gas molecules is lower, especially for H 2 , N 2 , CO, and CO 2 . The overlapping of electron density distribution will cause significant electrostatic interactions. The electrostatic interaction plays a leading part during the permeation process because higher overlapping of electron density distribution corresponds to larger E b values. In this work, the electrostatic interaction mainly originates from the different pore sizes of 2D TMC 6 . As a useful tool to quantitatively understand the selectivity of TMC 6 against different gases, a comparison between the measured diameters of the cross section in the van der Waals (vdW) surface (D c ) of different gas molecules and the pore size of unoccupied vdW surface in 2D TMC 6 was also performed. As measured, the unoccupied vdW diameters of pore in s-MoC 6 and s-WC 6 are 2.29 and 2.32 Å, significantly higher than those of g-C 3 N 4 (about 1.70 Å) [8] which results in significantly smaller E b values for gas molecules permeating g-C 3 N 4 when compared with s-TMC 6 . As reported, the D c of H 2 , N 2 , CO, CO 2 , and CH 4 is 2.44, 3.20, 3.46, 3.44, and 3.78 Å, respectively. Therefore, the pore size of s-TMC 6 is very close to the D c of H 2 but much smaller than those of N 2 , CO, CO 2 , and CH 4 . This finding means that s-TMC 6 will have considerable selectivity against H 2 , attributed to the suitable pore size. Notably, though the pore size of s-WC 6 is slightly larger than that of s-MoC 6 , it causes higher E b for the same gas molecule. Hence, the metal species has relatively less influence on the diffusion of gas molecules, but it is not negligible. In the meantime, the unoccupied vdW diameters of pore in h-TMC 6 are 0, which explains why h-TMC 6 exhibits very high diffusion barriers for all the studied gas molecules. with the pore edge of 2D-TMC6 when compared with other smaller gas molecules. Moreover, as indicated by Figure 2d, the ΔEsub in CO2 involved TS is obviously larger than that in the corresponding CH4 involved TS, implying stronger electrostatic repulsion of CO2 to the substrate. This finding explains the origin of the high Eb values of CO2 permeating into the pores of 2D TMC6. To evaluate the gas separation performance quantitatively, we calculated the permeability and selectivity based on diffusion energy barriers. For the permeability, gas kinetic theory was employed under the ideal gas approximation. Herein, the permeance (P, in mol m −2 s −1 Pa −1 ) of the penetrated gases is determined by [12] ∫ ∞ ( ) 1 ∞ Figure 3. Electron density isosurfaces of H 2 , N 2 , CO, CO 2 , and CH 4 on (a) s-TMC 6 and (b) h-TMC 6 at their corresponding transition states (TSs). The upper panels mark MoC 6 , while the lower panels denote WC 6 . The isosurface value is 0.015 e Å −3 .

Property
In addition to the comparison between pore sizes, the steric deformation of 2D TMC 6 caused by electrostatic repulsion was also checked. We measured the main bond lengths of different molecules in isolated state (l 1 ) and TS (l 2 ), that is, H-H bond length for H 2 , N-N Membranes 2022, 12, 709 7 of 12 bond length for N 2 , C-O bond length for CO and CO 2 , and C-H bond length for CH 4 , taking in consideration the non-polarized structures of CO 2 and CH 4 . The comparison between l 1 and l 2 (seeing Table 1) indicates that the structural changes of the gas molecules under TS are negligible, except for CO and N 2 on h-TMC 6 . Then, by removing the gas molecule in its TS structure and calculating the single-point energy (SPE) of the 2D TMC 6 , we can obtain the energy differences (∆E sub ) of the 2D TMC 6 between free-standing state and TS without adsorbates, as listed in Table 1. It can be summarized that the ∆E sub is directly proportional to the corresponding E b . For H 2 , N 2 , and CO, the electrostatic repulsion takes the proportion of 10-18% in the E b , and the proportion increases to 20-30% for CO 2 and CH 4 . It is revealed that CO 2 and CH 4 have more significant electrostatic repulsion with the pore edge of 2D-TMC 6 when compared with other smaller gas molecules. Moreover, as indicated by Figure 2d, the ∆E sub in CO 2 involved TS is obviously larger than that in the corresponding CH 4 involved TS, implying stronger electrostatic repulsion of CO 2 to the substrate. This finding explains the origin of the high E b values of CO 2 permeating into the pores of 2D TMC 6 .
To evaluate the gas separation performance quantitatively, we calculated the permeability and selectivity based on diffusion energy barriers. For the permeability, gas kinetic theory was employed under the ideal gas approximation. Herein, the permeance (P, in mol m −2 s −1 Pa −1 ) of the penetrated gases is determined by [12] In Equation (1), A p denotes the area of the pores, and the total number of collisions per unit time per area (N) is described as N = p 2πmk B T , where p, m, k B , and T stand for pressure, the mass of the molecule, the Boltzmann constant, and temperature, respectively. Hence, A p N could be viewed as the number of molecules that collide with the pore area per unit time. The portion of molecules with a speed large enough to overcome the diffusion barrier through the pore (i.e., v > v b = 2E b m ) is counted as the penetrant portion. A m , ∆p, and f (v) represent the total area of the membrane, the pressure difference (absolute value) between the two sides of the membrane, and the Maxwell velocity distribution, respectively. It is worth noting that A m is explicit for a 2D membrane, while A p is related to the pore shapes and the effective radii (R eff ) of the atoms at the pore rim. Here, R eff is calculated as R eff = R vdw / √ 2, where R vdW denotes the vdW radius. The feed pressure and the pressure difference are p = 3 × 10 5 Pa and ∆p = 10 5 Pa as provided in previous work [24].
The calculated permeance vs. temperature for H 2 , N 2 , CO, CO 2 , and CH 4 passing through the intrinsic pore of s-TMC 6 and h-TMC 6 are displayed in Figure 4. The green dashed line indicates the industrially acceptable permeance capability for gas separation (green dashed line, 6.7 × 10 −9 mol m −2 s −1 Pa −1 ). Over the temperature range of 100-500 K, the permeance values for each gas molecule through s-TMC 6 are evidently larger than those through h-TMC 6 . s-TMC 6 shows good permeance capability against hydrogen, while h-TMC 6 has low permeance capability for the different gas molecules. Specifically, in the temperature range of 100-300 K, the permeance values of H 2 through s-TMC 6 are always higher than the industrially acceptable one for gas separation. Meanwhile, the permeance values of CO 2 and CH 4 are always lower than the standard. For N 2 and CO, the permeance values will not be higher than the standard until the temperature is higher than 300 K. It is suggested that s-TMC 6 could be a potential H 2 purification membrane to separate H 2 from a mixture composed of N 2 , CO, CO 2 , or CH 4 below room temperature (100-300 K). temperature range of 100-300 K, the permeance values of H2 through s-TMC6 are always higher than the industrially acceptable one for gas separation. Meanwhile, the permeance values of CO2 and CH4 are always lower than the standard. For N2 and CO, the permeance values will not be higher than the standard until the temperature is higher than 300 K. I is suggested that s-TMC6 could be a potential H2 purification membrane to separate H from a mixture composed of N2, CO, CO2, or CH4 below room temperature (100−300 K).  It is well accepted that the performance of a separation membrane is characterized by both the permeance capability and selectivity. Herein, the selectivity between two gas species is defined as the ratio of the diffusion rates, S gas-1/gas-2 = A gas-1 /A gas-2 , which comes from the Arrhenius equation: In Equation (2), the A 0 means the diffusion prefactor that can be taken as the same value for all gases (10 11 s −1 ). Based on the definition, the selectivity gradually decreases with the increase in temperature. In Figure 5, the selectivity versus temperature for H 2 /N 2 , H 2 /CO, H 2 /CH 4 , and H 2 /CO 2 separation by s-MoC 6 and s-WC 6 are illustrated. It is easy to find that the s-TMC 6 membranes mainly exhibit excellent selectivity for H 2 /CH 4 and H 2 /CO 2 due to the great differences in E b between H 2 and CH 4 /CO 2 . Different from other carbon-based 2D membranes, under room temperature (300 K), the selectivity of s-TMC 6 for H 2 against other gas molecules is not ideal enough, with 10 11 and 10 13 for H 2 /CH 4 and H 2 /CO 2 , respectively. It is suggested that s-TMC 6 could be applied as H 2 purification membranes under low temperatures (100-250 K), with the selectivity of H 2 /(CO 2 , CH 4 ) larger than 10 15 . As indicated by the selectivity, s-WC 6 possesses better H 2 purification ability when compared with s-MoC 6 .
carbon-based 2D membranes, under room temperature (300 K), the selectivity of s-TMC6 for H2 against other gas molecules is not ideal enough, with 10 11 and 10 13 for H2/CH4 and H2/CO2, respectively. It is suggested that s-TMC6 could be applied as H2 purification membranes under low temperatures (100−250 K), with the selectivity of H2/(CO2, CH4) larger than 10 15 . As indicated by the selectivity, s-WC6 possesses better H2 purification ability when compared with s-MoC6. The convection-diffusion process is a problem in the field of fluid mechanics. Generally, the finite difference method (FDM) is a major method to treat with the convectiondiffusion equation [25] which only applies to macroscopic systems. However, for microscopic systems such as the TMC6 membranes in this work, it is extremely difficult to perform quantitative calculations on convection and diffusion of gas flow. As a result, the MD simulations were widely adopted to visualize the time-dependence diffusion process of the molecules, as well as to assess some parameters such as gas diffusion coefficient and the permeated number of gas molecules [26][27][28][29]. Due to the lack of quantitative results, MD simulations are often applied to confirm the results of DFT calculations.
To better understand the transmembrane processes during gas separation, we performed classical MD simulations (at 250 K) on the permeation of gas mixture through the TMC6 membranes. During the MD simulations, a box of about 4 nm × 4 nm × 12 nm was employed; then, 60 H2, 60 CO2, and 60 CH4 molecules were randomly put into a chamber composed of two TMC6 membranes (the distance between the membranes was set as 4 The convection-diffusion process is a problem in the field of fluid mechanics. Generally, the finite difference method (FDM) is a major method to treat with the convectiondiffusion equation [25] which only applies to macroscopic systems. However, for microscopic systems such as the TMC 6 membranes in this work, it is extremely difficult to perform quantitative calculations on convection and diffusion of gas flow. As a result, the MD simulations were widely adopted to visualize the time-dependence diffusion process of the molecules, as well as to assess some parameters such as gas diffusion coefficient and the permeated number of gas molecules [26][27][28][29]. Due to the lack of quantitative results, MD simulations are often applied to confirm the results of DFT calculations.
To better understand the transmembrane processes during gas separation, we performed classical MD simulations (at 250 K) on the permeation of gas mixture through the TMC 6 membranes. During the MD simulations, a box of about 4 nm × 4 nm × 12 nm was employed; then, 60 H 2 , 60 CO 2 , and 60 CH 4 molecules were randomly put into a chamber composed of two TMC 6 membranes (the distance between the membranes was set as 4 nm). In Figure 6a,b, the number of permeated H 2 molecules versus the simulation time for gas mixture in the chamber composed of s-TMC 6 are depicted. It is found that after MD simulations of 5000 ps, there are 35 and 40 H 2 molecules diffusing outside of the s-MoC 6 and s-WC 6 membranes, respectively. It is worth noting that the diffusion equilibrium was not achieved within 5000 ps; we can expect that after a long enough time, all the H 2 molecules will diffuse through the s-TMC 6 membranes into the product chamber. The snapshots of the gas mixture permeating through the s-TMC 6 membranes at 0 ps, 500 ps, 1000 ps, and 5000 ps are given in Figure 6c,d. During the permeation process, the H 2 molecules gradually migrate from the feed chamber to the product chamber. Moreover, none of CO 2 or CH 4 is found outside the membranes, clearly indicating that the s-TMC 6 membranes could efficiently separate H 2 molecules from the H 2 , CO 2 , and CH 4 mixture. Therefore, the MD simulations could well simulate the transmembrane processes of H 2 molecules, which greatly supports our first-principle calculation results.
In addition, it should be mentioned that we also performed the same MD simulations on h-TMC 6 . However, there are no gas molecules that run out from the chamber after 5000 ps at 250 K, indicating that h-TMC 6 could not work as effective gas separation membranes.

Conclusions
In this work, the potential applications of holey TMC6 membranes in H2 purification were uncovered by comparative first-principle calculations. The adsorption and diffusion behaviors of five gas molecules (including H2, N2, CO, CO2, and CH4) were investigated on h-TMC6 and s-TMC6. All the studied gas molecules showed weak physisorption on the

Conclusions
In this work, the potential applications of holey TMC 6 membranes in H 2 purification were uncovered by comparative first-principle calculations. The adsorption and diffusion behaviors of five gas molecules (including H 2 , N 2 , CO, CO 2 , and CH 4 ) were investigated on h-TMC 6 and s-TMC 6 . All the studied gas molecules showed weak physisorption on the TMC 6 membranes, but distinguishing diffusion barriers were obtained for different gas molecules across the pores of TMC 6 membranes. The smaller pore size makes the gas molecules much more difficult to permeate into h-TMC 6 rather than into s-TMC 6 . With suitable pore sizes, the s-TMC 6 structures not only show an extremely low diffusion barrier and acceptable permeance for the H 2 but also exhibit considerably high selectivity for H 2 /CH 4 and H 2 /CO 2 , under relatively low temperature (150-250 K). Moreover, classical MD simulations on the permeation process also validated that the s-TMC 6 could effectively separate H 2 from the gas mixture composed of H 2 , CO 2 , and CH 4 . Therefore, s-MoC 6 and s-WC 6 are qualified as separation membranes for H 2 purification from a gas mixture consisting of H 2 , CH 4 , and CO 2 below room temperature.