## 3 Results and Discussion  
Gas separation by porous graphene membranes is a process that involves the direct interaction between a single gas molecule and the pores. It is then important to understand what differentiates the gas molecules we are aiming at separating so as to determine the factors that influence their tendency to pass through the pore. A semi-quantitative estimation of the gas molecular size can be obtained considering the Van der Waals radii of the atoms that compose the molecules to be separated. In Figure 3, CO\({}_{2}\), H\({}_{2}\)S and CH\({}_{4}\) are represented schematically. In correspondence of the position of each atom, a sphere representing the Van der Waals surface has been drawn. Methane is the molecule that occupies the largest volume, having a rather spherical steric hindrance with a diameter of 8.5 A. Both carbon dioxide and hydrogen su  
Figure 3: CO\({}_{2}\) (**a**), CH\({}_{4}\) (**b**) and H\({}_{2}\)S (**c**) molecules with their relative sizes in Å. Carbon atoms are shown in black, oxygens in red, hydrogens in white and sulfur in yellow. The dimensions are calculated considering their Van der Waals volumes, whose surfaces are represented as shaded areas. Figure 2: Ball and stick representation of the porous graphene structures employed for gas separation (zoomed on the pore area). Carbon atoms are represented in black, oxygens in red and hydrogens in white. Distances are in Angstrom. dimension smaller than the diameter of CH\({}_{4}\), making them easier to filter by steric hindrance with respect to methane. Although CO\({}_{2}\) and H\({}_{2}\)S have similar sizes, they differ in shape: CO\({}_{2}\) has an elongated shape while H\({}_{2}\)S is more rounded. We will show that such a difference has particular consequences on their permeance across the pore. Unlike methane, CO\({}_{2}\) and H\({}_{2}\)S are characterized by polar bonds suggesting that polar terminations of the pores, such as hydroxyl groups, might influence the separation performances. In our work we study the factors that influence the separation process based on the characteristics of the molecules to be filtered and the pore size, shape and edge functionalization. We start analyzing pore P1 which has a circular shape and a diameter of 5.90 A. At the studied pressures, gas molecules do not aggregate and their interaction occurs only through elastic collisions which appear to be a quite rare event at our simulation scale. As one can see in Table 1, the pore efficiently separates both the CO\({}_{2}\)/CH\({}_{4}\) and the H\({}_{2}\)S/CH\({}_{4}\) mixtures. The CH\({}_{4}\) permeance in pore P1 is around 0.01-0.02\(\times 10^{5}\) GPU, whereas CO\({}_{2}\) and H\({}_{2}\)S achieve much larger values, 0.25 \(\times\) 10\({}^{5}\) GPU and 0.23 \(\times\) 10\({}^{5}\) GPU respectively, leading to a selectivity of 11 for the CO\({}_{2}\)/CH\({}_{4}\) mixture and 20 for the H\({}_{2}\)S/CH\({}_{4}\) mixture. For the case of pore P1, the filtration occurs mostly by steric effect: the methane, as highlighted before, has a larger cross section compared to CO\({}_{2}\) and H\({}_{2}\)S, consequently the barrier required to cross the pore is expected to be higher. We verified such a hypothesis performing a Nudged Elastic Band (NEB) simulation of the energy profile of a molecule traveling from the feed to the permeate. The NEB analysis shows that CH\({}_{4}\) has to overcome an energy barrier of 0.27 eV to cross the membrane while CO\({}_{2}\) and H\({}_{2}\)S have to overcome a lower barrier, respectively 0.16 eV and 0.17 eV. A pore of diameter of 5.90 A is therefore sufficient to effectively prevent CH\({}_{4}\) from passing through the membrane while at the same time it permits the passage of smaller molecules. The results agree with the work by [PERSON] et al. who, with the same pore, reported comparable permeances for the CO\({}_{2}\)/CH\({}_{4}\) and H\({}_{2}\)S/CH\({}_{4}\) mixtures [15]. We then tested the performances of pore P2 to understand how pore shape impacts the gas permeance.