## 3 Results and Discussion
Compared to P1, P2 has a more stretched contour and it is purposely designed to reproduce the elongated shape of the CO\({}_{2}\) molecule. Such a feature has indeed consequences on the selectivity of the CO\({}_{2}\)/CH\({}_{4}\) mixture: P2 selectivity to CO\({}_{2}\) is indeed higher (15) compared to the one of P1 (11) as see in Table 1. A lower overall permeance is reported both for CO\({}_{2}\) (0.17 \(\times\) 10\({}^{5}\) GPU) and CH\({}_{4}\) (0.01 \(\times\) 10\({}^{5}\) GPU) compared to pore P1 due to the smaller dimension of P2 along one pore axis that hinders the passage of both molecules. The energetic barriers calculated via NEB are accordingly higher in P2 than in P1 passing from 0.16 eV to 0.28 eV for CO\({}_{2}\) and from 0.27 eV to 0.69 eV for CH\({}_{4}\). As an additional proof of the shape selectivity of P2, we filtrated a CO\({}_{2}\)/H\({}_{2}\)S mixture. Despite having similar dimensions, CO\({}_{2}\) and H\({}_{2}\)S differ in shape. This simulation led to a permeance of 0.09 \(\times\) 10\({}^{5}\) GPU for CO\({}_{2}\) and 0.03 \(\times\) 10\({}^{5}\) GPU for H\({}_{2}\)S with a CO\({}_{2}\)/H\({}_{2}\)S selectivity of 2.7, proving that the filtration of molecules with comparable size is possible thanks to shape selective pores. In the framework of an \begin{table}
\begin{tabular}{c c c c} \hline
**CO\({}_{2}\)/CH\({}_{4}\)** & **CO\({}_{2}\) GPU [\(\times\)10\({}^{5}\)]** & **CH\({}_{4}\) GPU [\(\times\)10\({}^{5}\)]** & **S\({}_{\text{CO}_{2}\text{/CH}_{4}}\)** \\ \hline P1 & 0.25 & 0.02 & 11 \\ P2 & 0.17 & 0.01 & 15 \\ P3 & 2.39 & 2.4 & 1 \\ P3 OH & 3.25 & 0.55 & 5.9 \\ \hline
**H\({}_{2}\)S/CH\({}_{4}\)** & **H\({}_{2}\)S GPU [\(\times\)10\({}^{5}\)]** & **CH\({}_{4}\) GPU [\(\times\)10\({}^{5}\)]** & **S\({}_{\text{H}_{2}\text{S}\text{/CH}_{4}}\)** \\ \hline P1 & 0.23 & 0.01 & 20 \\ P3 & 5.79 & 2.04 & 2.8 \\ P3 OH & 11.91 & 2.07 & 5.7 \\ \hline
**CO\({}_{2}\)/H\({}_{2}\)S** & **CO\({}_{2}\) GPU [\(\times\)10\({}^{5}\)]** & **H\({}_{2}\)S GPU [\(\times\)10\({}^{5}\)]** & **S\({}_{\text{CO}_{2}\text{/H}_{2}\text{S}}\)** \\ \hline P2 & 0.09 & 0.03 & 2.7 \\ \hline \end{tabular}
\end{table} 
Table 1: Permeability and selectivity calculated for different gas mixtures and pore types. industrial process, graphene membranes with elliptic pores can be used as second stage separation membranes to severely reduce the concentration of the dangerous H\({}_{2}\)S gas in the CO\({}_{2}\)/H\({}_{2}\)S mixture remaining after the initial natural gas purification process. For the H\({}_{2}\)S to cross pore P2, an energy barrier comparable with CH\({}_{4}\) must be overcome (0.69 eV). For this reason, there is not a noticeable difference in gas permeance of the H\({}_{2}\)S/CH\({}_{4}\) mixture within the timespan of our simulation (51 ns). To test the effects of pore dimensions on the separation process, we run simulations with a larger pore, P3, having an area of 35.15 A\({}^{2}\) compared to 27.34 A\({}^{2}\) of P1. The increased area of the pore leads to a higher permeance (2.39 \(\times\) 10\({}^{5}\) GPU for CO\({}_{2}\) and 5.79 \(\times\) 10\({}^{5}\) GPU for H\({}_{2}\)S, respectively for the CO\({}_{2}\)/CH\({}_{4}\) and H\({}_{2}\)S/CH\({}_{4}\) mixtures) and less selectivity (1 for CO\({}_{2}\)/CH\({}_{4}\) and 2.8 H\({}_{2}\)S/CH\({}_{4}\)) than the previous pores. Therefore, the critical diameter that a circular pore must have in order to separate methane from CO\({}_{2}\) and H\({}_{2}\)S, is around 5.90 A. A similar diameter was also proven to filter salt ions from water during desalination [26]. In comparison, [PERSON] et al. [31] showed that CH\({}_{4}\) permeance is even larger than the one of CO\({}_{2}\) through a larger elliptic pore (10-12 A). Such a behavior indicates that, beyond a critical diameter, steric hindrance no longer contributes to selectivity and that for the natural gas purification to take place one must rely on other factors such as pore shape and functionalization.