Gas Separation in Nanoporous Graphene from First Principle

Article pubs.acs.org/JPCC

Gas Separation in Nanoporous Graphene from First Principle Calculations Alberto Ambrosetti* and Pier Luigi Silvestrelli Dipartimento di Fisica e Astronomia, Università di Padova, via Marzolo 8, I-35131 Padova, Italy DEMOCRITOS National Simulation Center, 34136 Trieste, Italy ABSTRACT: Thanks to its single atom thickness and its mechanical strength, nanoporous graphene is currently being regarded as a promising candidate for efficient and reliable gas separation applications. Clearly, the accurate energetic characterization of the penetration processes involving relevant gas-phase molecules is a fundamental prerequisite for any possible application. Here we evaluate permeation barriers and adsorption energies of the H2O, CH4, CO, CO2, O2, and H2 molecules and of the Ar atom on two types of hydrogen saturated pores by means of ab initio simulations, based on the density functional theory (DFT), able to include dispersion corrections too. We find that, although the qualitative trend followed by the values of the permeation barriers of the considered molecules is independent of the adopted DFT functional, at a quantitative level the results are noticeably affected by the dispersion corrections and the chosen exchange contribution characterizing the different functionals, as well as by the allowed graphene sheet distortions. Interestingly, we observe that, due to the occurrence of nontrivial H-bond interactions with the pore-saturating H atoms, the permeation barrier of water remains low even considering a small-size pore. The barrier is further diminished when considering the interaction with a second water molecule on the opposite side of the pore. These observations, combined with the relatively strong binding of the water molecule with the defected surface, suggests that porous graphene could also represent a promising membrane for water filtration.

I. INTRODUCTION Gas separation is a subject of direct chemical and industrial relevance, whose applications encompass processes such as oil refinery, waste gas purification, or production of controlled gas phase moieties. Conventional separation mechanisms, relying, for instance, on phase change or adsorption, are still widely used in industrial production. However, the important practical and economical interests involved in gas separation processes are determining a steady search for more reliable and energetically efficient techniques.1 In this regard, pressuredriven membrane gas separation techniques are increasingly applied and appear as ideal candidates for reducing the environmental impact and economical cost of the related industrial processes: in fact, membranes do not require inefficient thermally driven separation procedures, and the absence of moving parts facilitates applications in remote or hardly accessible locations. Several types of nanoporous membranes have been proposed in the last years, based, for instance, on organic polymeric aggregates, glass, metals, or carbon structures. Among these, in particular, single graphene sheets2 combine chemical inertness and high-temperature stability, typical of carbon-based materials, with an improved mechanical strength. Moreover, since the permeability of a membrane is inversely proportional to its thickness,3 the peculiar one-atom thickness of graphene makes it a suitable candidate for the design and production of highly permeable membranes. Due to the delocalized π electrons of the constituent carbon atoms, responsible for Pauli repulsion in the hollow sites, © 2014 American Chemical Society

pristine graphene is impermeable even to the very small He atoms.4 The creation of nanopores patterns is thus essential to construct a functioning membrane. Although at present the practical realization of controlled-shape nanopores might be somewhat challenging, these can be, for instance, produced in two-dimensional sheets making use of a focused electron beam generated by a transmission electron microscope,5 by a focused ion beam,6 or through self-organized growth.7 Various types of pores, characterized by different size, shape, and carbon bond saturation have been recently considered in a number of theoretical studies.8−15 Depending on the geometrical and chemical properties of the pores, different energy barriers for the permeation of gas molecules through the membrane are found, thus opening the way to the engineering of nanoporous membranes featuring specific selectivity and permeation rates.8,16−18 Ab initio theoretical methods provide in this context an invaluable tool for analyzing in detail the interaction between gaseous molecules and pores, supporting the experimental search for optimal porous patterns. Among available ab initio theoretical methods, semilocal density functional theory (DFT) currently represents the method of choice in surface physics,19 as it allows for a correlated quantum mechanical description of rather extended systems at an affordable computational cost. However, given the short-range character of the commonly used approximate semilocal Received: May 19, 2014 Revised: July 9, 2014 Published: August 1, 2014 19172

dx.doi.org/10.1021/jp504914u | J. Phys. Chem. C 2014, 118, 19172−19179

The Journal of Physical Chemistry C

Article

The two types of defects, referred to as A- and B-type pores, are shown in Figures 1 and 2, respectively. While the A-type

correlation functionals, they do not properly describe dispersion or van der Waals (vdW) effects, that are therefore included through a suitable nonlocal term in the correlation functional,20,21 or through the (postprocessing) addition of long-range correlation energy contributions.22−28 Despite the intense development and wide use of dispersion-corrected DFT methods, at present little is known regarding the effect of the vdW interactions on the permeation energy barriers. While the vdW interaction is expected to enhance the attraction between surface and noncovalently bonded molecules,24,29,30 the scenario appears more complex in the case of compressed systems; in fact, most of the popular dispersion-corrected DFT methods rely on damping functions or range-separation schemes that are typically parametrized in such a way to reproduce the correct energetics corresponding only to equilibrium or long-distance configurations. The overall vdW effect in compressed configurations is thus difficult to predict in advance, since indeed short intermolecular distances might represent a challenge for dispersion-corrected DFT schemes.31 Here we evaluate the permeation barriers of a set of molecules, namely, H2O, CH4, CO, CO2, O2, H2, and Ar, interacting with two types of possible graphene nanopores. The first six molecules are found in the atmosphere and have particular relevance in gas filtration processes, while the Ar monomer is considered due to its very low reactiveness and the consequent predominant role played by the dispersion interactions in characterizing its binding energy with a substrate. By comparing the results obtained using semilocal and dispersion-corrected DFT schemes, we assess the role of different possible exchange-correlation functionals and of the vdW effects on those configurations that are mostly relevant to the gas-separation processes. Moreover, we investigate the limitations of estimating the permeation barriers on the basis of simple constrained-geometry calculations instead of adopting the more accurate (but also more expensive) nudged elastic band (NEB) approach.

Figure 1. A-Type pore obtained by single C ring removal from pristine graphene sheet.

II. METHOD We performed ab initio DFT calculations in the framework of the Quantum ESPRESSO suite.32 Electron−ion interactions were described using norm-conserving pseudopotentials, and wave functions were expanded in plane waves with a kinetic energy cutoff of 70 Ry. A periodically repeated hexagonal supercell was adopted, consisting of 6 × 6 replicas of the pristine graphene unit cell in the x−y plane. A spacing of 36 Å was introduced along the z direction among periodically repeated graphene layers, to minimize the interaction with replicas. In this study, two different, periodically repeated, defects (pores) formed in the graphene structure are considered: these are obtained by removing either one or two adjacent carbon-atom rings from pristine graphene (corresponding to 6 or 10 C atoms) and saturating the resulting dangling bonds by H atoms. Controlled pores of analogous subnanometer size and shape were created in graphene, for instance, by O’Hern et al.33 using an experimental technique based on chemical oxidation of nucleated defects. As the hydrogenation procedure is concerned, the H saturation of C dangling bonds is a common practice in quantum chemistry simulations, and its application to graphene defects was previously considered by other authors.8,16,34 Hydrogen, on the other hand, was shown to stabilize vacancies in graphene,35 while H ion bombardment36 can indeed lead, according to Lehtinen et al.,37 to the formation of H-saturated defects.

Figure 2. B-Type pore obtained by two C ring removal from pristine graphene sheet.

pores are isotropic and are characterized by a maximum H−H distance of 3.53 Å, the B-type pores (also considered in previous investigations8,10,16) are elongated with a maximum H−H distance of 5.86 Å. Since our periodic simulation supercell is relatively large (corresponding to 72 C atoms in the case of pristine graphene), this makes the distance, 14.77 Å, between the centers of two adjacent defects sufficiently large to assume that the interactions among periodic replicas of the defects are weak (see Figure 3). The sampling of the Brillouin zone was carried out through a 4 × 4 × 1 k-point mesh. Two semilocal, Generalized Gradient Approximation (GGA) exchange-correlation (xc) functionals were used, namely, PBE38 and revPBE.39 Dispersion-corrected calculations were performed by adopting the well established vdW-DF21 and vdW-DF240 functionals, and also the newly implemented rVV1041 approach. Both the rVV10 and the vdWDF2 functional describe short-range interactions by means of the refitted PW86 GGA functional (rPW86),42 while vdW-DF adopts the revPBE one. We applied these dispersion-corrected schemes using the fully self-consistent variant, as currently implemented within the Quantum-ESPRESSO package. In fact, one expects that a self-consistent approach is better suited for the present study than a more standard method where 19173



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By looking at the results reported in Table 1 one can see that the trend of the barrier values is roughly the same both for the Table 1. Permeation Energy Barriers (ΔE) for A- and BType Nanoporous Graphene (eV) A-type H2O CH4 CO2 CO H2 O2 Ar

PBE

rVV10

NEB(PBE)

0.54 2.82 1.57 1.21 0.35 0.50 2.84

0.48 2.78 1.24 0.91 0.30 0.44 2.87

0.95 3.17

PBE

rVV10

NEB(PBE)

0.55 0.79 0.59 0.71 0.15 0.07 0.79

0.46 0.60 0.35 0.54 0.09 0.00 0.71

0.69 1.22

0.41 0.71 3.01

B-type

Figure 3. Periodically repeated B-type pores. Green lines indicate x−y plane cell vectors. The same supercell was employed also for the isotropic A-type pores, where the defect orientation is unimportant.

H2O CH4 CO2 CO H2 O2 Ar

dispersion effects are evaluated as a postprocessing correction, since we simulated compressed configurations where substantial density changes could be induced by the intermolecular interactions. Finally, to get accurate estimates of the energy barriers, nudged elastic band43 (NEB) calculations were also performed (only in combination with the PBE functional), typically adopting chains made of seven beads, where the initial and final configurations were kept fixed, as obtained by preliminary geometry optimization procedures.

0.25 0.13 0.83

A- and the B-type pore and both for the approach based on the PBE functional and that based on the rVV10 one. In line with previous calculations,10 the ΔE values tend to increase according to the following ordering: H2, O2, H2O, CO2, CO, CH4, and Ar. Deviations from this trend are represented by small energy differences, and, due to the uncertainties of different vdW-corrected schemes (see below) in describing compressed configurations, should not be overemphasized. In agreement with Jiang et al.8 and Hauser and Schwerdtfeger,16 for H2 we find very low barriers in all considered cases (up to a minimum value of 0.09 eV in B-type pores using rVV10), thus making these nanoporous systems promising for efficient hydrogen separation applications. On the other hand, our results indicate that, while an adequate separation of H2 from, for example, CH4 can be probably achieved, the same is not true for O2 and H2, since in both the smaller A-type and the larger B-type pore the permeation barriers relative to these two molecules turn out to be small and very similar. Interestingly, a substantial increase of ΔE is observed for all the molecules, moving from B- to the smaller A-type pores (up to ∼2 eV for Ar and CH4 using both PBE and rVV10), but for the case of water, in spite of its larger size than that of O2 and H2; in fact, the permeation barriers of H2O through the B- and A-type pores are almost the same, which clearly deserves a deeper investigation. A structural analysis of the in-pore configurations, largely responsible for the determination of ΔE, shows that a substrate saturating H atom, Hs, is pointing toward the water O atom in the A-type pore (see Figures 4 and 5). This observation suggests a tendency to form an hydrogen bond with the water molecule, which clearly could lead to a lowering of the permeation energy barrier. A detailed geometrical analysis shows that both the HO−Hs and CHs− O angles (99.7° and 170.7°, respectively) are compatible with the assignment of a H-bonding character. The same holds for the distance between the water O and the Hs atom, 2.04 Å, in line with typical hydrogen-bonded systems, where it ranges from ∼1.5 to 3 Å.44 In this respect, a calculation based on the

III. RESULTS AND DISCUSSION We considered the interaction of each type of pore with seven different molecules, namely H2O, CH4, CO, CO2, O2, H2, and Ar. First we carried out calculations using the PBE semilocal functional, which misses long-range correlation effects, and the rVV10 dispersion-corrected functional where instead longrange interactions are accurately included, as reported in recent successful applications.24,27,41 We stress again that dispersioncorrected DFT methods by construction tend to be mostly reliable for describing configurations where the intermolecular distances are close to the equilibrium values or/and are relatively large,26,31 while the performances in compressed configurations characterized by reduced distances are less tested and, probably, more dependent on the specific underlying functional. Permeation energy barriers are first estimated by following a simple approach based on the comparison of the total energy, Eads, of every molecule when it is located in its most stable adsorption configuration, well above the pore, to that, Ez0, of the molecule when the z coordinate of its center of mass is constrained to remain at the same (z0) level of the graphene plane. The energy barriers are then defined as ΔE = −Eads + Ez 0 (1) Obviously, a rigorous evaluation of the actual permeation rates would require information relative to pressure and entropy contributions, which could, for instance, be extracted from ab initio molecular dynamics simulations. The present permeation barriers, however, already provide a clear indication concerning the energetics relative to the filtration process, allowing at the same time a direct comparison among different xc functionals. 19174



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the water molecules on the other side of the pore, although the effect is less pronounced than in graphdyine. On the other hand, in the larger B-type pore, a striking ΔE decrease of 0.43 eV with PBE and 0.44 eV with rVV10 (78% and 96%, respectively) is found. The barrier hence is here almost completely removed by the water−water interaction, similarly to the case of graphdyine.46 The dramatic enhancement of the effect with respect to A-type pores can be attributed to a weaker interaction of water with the saturating H atoms, that allows for the formation of a stronger water−water H-bond. To further assess the importance of hydrophilic pore terminations in water filtration, we considered an additional type of defect saturation, as proposed by Cohen-Tanugi and Grossman.34 This consists of replacing half of the hydrogen atoms with OH groups, creating a H, OH alternating pattern. Also in this case, we found low permeation barriers (0.77 and 0.67 eV for A and B-type pores, respectively, using rVV10), confirming the relevance of electrostatic and H-bond type interactions. On the other hand, despite the hydrophilic nature of hydroxyl groups, a slight increase of ΔE with respect to the considered H-saturated pores (more evident in the A-type pore) could be explained in terms of steric effects, due to the reduced size of the defects and the relatively large size of the OH complexes. Coming back to the overall analysis, a further inspection of Table 1 shows that, although the PBE and rVV10 results do not differ substantially, the rVV10 vdW-corrected ΔE estimated barriers are generally lower (between and 0.04 and 0.33 eV) than those obtained by PBE. To explain the origin of these differences, we explicitly report in Table 2 the binding energies

Figure 4. Water molecule placed inside the pore (in-pore configuration), upper view.

Figure 5. Water molecule in in-pore configuration, side view.

generation of maximally localized Wannier functions (MLWFs)45 reveals a substantial enhancement (from 1.82 to 2.33 D, using the PBE functional) of the dipole moment of the water molecule, indicating that electrostatic effects play a role. In particular, while the largest contribution to the change of the dipole moment is oriented along the z axis, the remaining component (corresponding to ∼30% of the total variation) is oriented along the direction connecting O to Hs. Hydrogen bonding can indeed give rise to nontrivial effects in filtration processes, as reported, for instance, by Bartolomei et al.46 who have recently showed how the formation of a Hbond across the pore, between two water molecules located on the opposite faces of graphdyine can strongly reduce the permeation barrier of water. Such mechanism could possibly be effective also in the narrow A-type pore considered in the present study, although here it could interfere with the observed tendency to form H-bonds with the saturating H atoms. We investigated this aspect computing the permeation barrier of water in A- and B-type pores, keeping a second water molecule fixed at 2.00 Å on the opposite side of the pore, with one H atom pointing toward the pore center (see Figure 6). Our results confirm that even in the small A-type pore the additional water−water interaction further lowers the barrier: in fact, with respect to the results of Table 1, PBE and rVV10 predict a ΔE decrease of 0.06 and 0.11 eV, respectively, corresponding to a reduction by 11% and 23%. Hence, also in our case the permeation barrier is decreased by the presence of

Table 2. Binding Energies for Molecules Interacting with Aand B-Type Nanoporous Graphene (meV) A-type H2O CH4 CO2 CO H2 O2 Ar

B-type

PBE

rVV10

PBE

rVV10

160 31 55 32 1 218 24

340 167 218 141 49 444 142

138 27 56 27 4 261 39

253 132 162 98 39 522 123

of the molecules, when they are adsorbed in the equilibrium configuration well above the pores, defined as E bind = Emol + Esur − Eads

(2)

where Eads (see eq 1) is the total energy of the molecule adsorbed on the defect, Emol is the energy of the isolated molecule, and Esur that of the defected surface. Our results confirm the expectation29,47,48 that including dispersion effects (as in the rVV10 approach) leads to a considerable stabilization of the adsorbed molecules due to the long-range interaction with the substrate. The observed overall reduction of the adsorption height is a direct consequence of this interaction (see Table 3). On the other hand, in-pore configurations are characterized by shorter molecule−surface distances, leading, in principle, to even stronger dispersion effects, so that the permeation barriers could be lowered by vdW interactions. The slight reduction of the ΔE values, in Table 1, going from PBE to rVV10, seems to support this conjecture. However, one must point out that at short distances

Figure 6. Two water molecules sitting on opposite sides of the pore. The bottom molecule is kept at fixed distance with a hydrogen pointing toward the pore center. 19175



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Table 3. PBE and rVV10 Equilibrium Adsorption Distances for Molecules Adsorbed on A- and B-Type Nanoporous Graphene, Computed from the Adsorbate Center of Mass (Å) A-type H2O CH4 CO2 CO H2 O2 Ar

Table 4. Permeation Energy Barriers (ΔE) for A- and BType Nanoporous Graphene (eV) Obtained Using Semilocal (PBE and revPBE) and vdW-Corrected Functionals for H2O and Ar Adsorptiona

B-type

A-type

PBE

rVV10

PBE

rVV10

2.07 3.59 3.40 3.22 2.73 2.30 2.96

1.99 2.91 2.96 3.20 2.72 1.99 2.57

2.15 3.37 3.26 3.23 2.74 1.50 2.89

2.00 2.87 2.93 3.21 2.71 1.23 2.31

H2O Ar

H2O Ar a

the dispersion corrections are strongly damped13 to avoid double-counting effects, because the semilocal functionals are expected to describe well short-range interactions. Moreover, at short-range, the rVV10 method does not rely on PBE but instead on the rPW86 exchange and LDA correlation functionals,42,49 so that inferring dispersion effects on the permeation barriers just directly comparing rVV10 and PBE results might be questionable since the different estimates could be also related to the different underlying semilocal functionals. This comment actually applies to most of the current selfconsistent and postprocessing dispersion-corrected DFT methods; actually, an accurate estimate of genuine dispersion effects at all distances could be only achieved by adopting rigorous range-separation approaches.50 To further investigate the dependence of the energy barriers on the chosen functional, in the interesting case of water interacting with the pore, we performed additional calculations using the revPBE semilocal functional, and the vdW-DF and vdW-DF2 schemes (see Figure 7 and Tables 4 and 5). We applied these methods also to the case of Ar which is characterized by the highest permeation barriers among the molecules studied. Looking at Table 4 one can see that revPBE consistently predicts higher permeation barriers than PBE (up to ∼0.7 eV higher for Ar on the A-type pore), thus indicating that exchange contributions, differently evaluated by the two schemes, play indeed a significant role in compressed configurations, possibly more important than long-range correlation effects. This result can be clearly explained in terms of the significant overlap between the wave functions of neighboring fragments when the intermolecular distances are short, in line with previous findings.26,51 In this regard, we stress that an analogous exchange contribution analysis could

PBE

revPBE

0.54 2.84

0.70 3.52

rVV10

PBE

revPBE

rVV10

0.55 0.79

0.75 1.16

0.46 0.71

vdW-DF

vdW-DF2

NEB(PBE)

0.58 3.23

0.95 3.01

vdW-DF

vdW-DF2

NEB(PBE)

0.67 0.97

0.55 0.84

0.69 0.83

0.48 0.67 2.87 3.25 B-type

NEB(PBE) indicates barrier estimated using the NEB approach.

Table 5. Binding Energies for Molecules Interacting with Aand B-Type Nanoporous Graphene (meV): Comparison among Semilocal (PBE and revPBE) and vdW-Corrected Functionals for H2O and Ar Adsorption A-type pores H2O Ar

H2O Ar

PBE

revPBE

160 24

71 2

PBE

revPBE

138 39

73 3

rVV10

vdW-DF

vdW-DF2

322 178

316 149

rVV10

vdW-DF

vdW-DF2

253 123

259 156

237 113

340 142 B-type pores

be carried out also between the PBE and PBE052 xc functionals. However, the presence of fractional exact exchange limits the PBE0 applicability to rather small systems due to the strongly nonlinear computational cost scaling. With regard to the vdW-DF and vdW-DF2 dispersioncorrected functionals, these give energy barrier values intermediate between the revPBE and the PBE ones. Since the vdW-DF functional describes short-range exchange effects just at the revPBE level, the observed reduction of the permeation barriers with respect to the pure-revPBE approach can therefore genuinely be ascribed to correlation effects. By comparing the results of the vdW-DF2 and rVV10 schemes, both based on the rPW86 functional, one can see that vdW-DF2 gives slightly higher energy barriers than rVV10, an effect probably related to the smaller attraction typically predicted by vdW-DF2 (see also the binding energies reported in Table 5), which affects the relative stability of in-pore and

Figure 7. Comparison between barrier (ΔE) and adsorption energy (Eads) for Ar and H2O for the considered exchange-correlation functionals. 19176



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adsorbed configurations. We can add that a detailed analysis48 of the vdW-DF class of methods showed a nontrivial dependence of the binding energy of chemisorbed systems on the underlying semilocal functionals; in particular, vdW-DF and vdW-DF2 were found to be too repulsive at short distances and improved performances were obtained by replacing the original semilocal exchange and correlation contributions with alternative functionals.53,54 Clearly similar considerations could apply also to the present case, although a partial cancellation of pronounced functional dependence is expected because the underlying semilocal functional will influence the binding energy of a given molecule both when it is well above the pore and when is constrained in the in-pore configuration. As a general remark, we stress that while the chosen dispersion methods provide a self-consistent treatment of dispersion, all of these effectively rely on second-order perturbative expressions for the long-range correlation energy. On the other hand, a fully many-body description51,55,56 of the long-range correlation, beyond second-order, is expected not to alter the present qualitative conclusions. In fact, many body contributions typically represent only a fraction of the total vdW energy,51 and will likely be hidden by the discrepancies observed among the different xc functionals. Moreover, while a reduction of Ebind is expected in the adsorbed configuration due to screening effects,56−58 improved stabilization will likely take place for in-pore configurations, due to the large in-plane polarizability of graphene.59 As a consequence, a full manybody description of dispersion is expected to slightly enhance the permeation barrier reduction, further supporting the above conclusions. Finally, we investigated how the values of the estimated permeation barriers can be affected by our simplified approach based on the evaluation (see above) of the difference between the binding energy of the adsorbed molecule well above the pore and that obtained by constraining the molecule center of mass to remain at the same level of the graphene plane. In fact, particularly for the largest considered molecules, the selected inpore configurations might not be well representative of the actual potential-energy saddle point. For instance, due to the relatively strong repulsion interaction between the constrained molecule and the defective substrate, the latter could be significantly distorted (see Figure 5) in such a way to slightly shift the pore away from the molecule, thus leading to somewhat underestimated barriers. This effect can be indeed observed in Figure 8, where the permeation barrier, ΔE, is

plotted as a function of the distance between the center of mass of the H2 molecule and the graphene reference plane (negative distance values mean that the molecule is below the graphene plane). As can be seen, due to the pore deformation, actually the maximum of the barrier is not achieved precisely at z0 but at a value corresponding to a negative adsorption height. Although a rigorous computation of the barrier could in principle be accomplished through a set of geometry optimizations (as shown), this approach is somewhat impractical due to the need of reiterating the relaxation procedure upon gradual penetration of the molecule in the pore. To improve the reliability of the permeation-barrier estimates we therefore carried out more expensive NEB calculations, adopting the PBE semilocal functional and considering the following set of molecules (representative of the more interesting cases discussed above): H2O, CH4, O2, H2, and Ar. The permeation barriers evaluated by this alternative approach, reported in Table 1, are consistently larger than those obtained by the previous method, this effect being more pronounced, as expected, for molecules interacting with the smaller A-type pore. Our results for H2 and CH4, compare favorably with those of Jiang 8 (∼0.15 and ∼1.3 eV, respectively). The minor differences found can be attributed to technical details of the simulations such as the different adopted supercell and pore orientation. Since the changes in the permeation-barrier values found by using NEB in place of the preliminary approach range from a few tens of meV to hundreds of meV, this suggests that deformation effects of defected graphene are nontrivial and depend considerably on the selected molecule. Interestingly, however, the computed NEB energy barriers follow roughly the same trend of the previous results, which therefore turn out to be qualitatively preserved despite the inaccuracies in the description of the graphene deformation effects. As the water molecules are concerned, although the permeation barriers predicted for the A- and B-pore by the NEB scheme are no more within a few tens of meV from each other, nonetheless the increase in their values going from the Bto A-pore is still much smaller than that observed for the other considered large molecules (CH4 and CO2) and is instead similar to that relative to the small H2 molecule, thus confirming, even using this more sophisticated approach, that the tendency to form hydrogen bonding plays an important role. Moreover, one can observe that H2O has a relatively large adsorption energy on the A-type pore accompanied by a reduced adsorption height (i.e., 340 meV and 1.99 Å, respectively, according to rVV10). Therefore, one could expect in realistic conditions an accumulation of water molecules around the defect. As shown by Du et al.,9 a sizable availability of molecules close to the defect, a condition strictly related to the surface wetting and thus to graphene functionalization, could enhance the final permeation rate due to the shorter permeation trajectories. As a final remark, we also stress that NEB calculations, although in principle preferable, are computationally much more expensive than single geometry optimizations, due to the simulataneous optimization of a number of interconnected beads accounting for the whole reaction path. NEB calculations can thus become soon unaffordable in larger systems (for instance larger porus defects to be investigated in future studies), where a careful geometry optimization would therefore represent the only viable approach to estimate ΔE.

Figure 8. ΔE (see definition in the text) as a function of the distance (z − z0) between the H2 center of mass and the graphene reference plane (z0, placed at the dashed line) in the A-type pore. 19177



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In addition, geometry optimization approaches allow a more direct comparison among different xc functionals and a more transparent analysis of the different energy components, as shown above.

IV. CONCLUSIONS We investigated the interaction of seven molecules of interest with porous graphene by particularly focusing on the estimates of the permeation barriers as obtained by different DFT-based methods, including schemes where self-consistent dispersion corrections are included. We found that dispersion effects tend to consistently decrease the permeation barrier energies (up to a few hundreds meV) with respect to semilocal DFT functionals where these effects are neglected. This result can be qualitatively explained by the increased interactions characterizing the in-pore configurations where the molecules are very close to the surface, although a quantitative assessment of the dispersion corrections is not trivial because of spurious short-range effects which might be induced by the adopted damping functions and by the approximated range-separations of the correlation energy involved in current dispersioncorrected methods. Besides dispersion effects, semilocal exchange energy contributions play an equally relevant role, as seen through a comparison between the results obtained by the PBE and revPBE functional, and deserve further investigations. Contextually, due to the complexity of the permeation process, an accurate evaluation of permeation barriers certainly requires calculations that go beyond that based on separate, constrained geometry optimization calculations, as shown by a comparison with the results obtained by the more sophisticated NEB approach. In fact, due to the pore deformation, finding the correct in-pore saddle point within a single geometry optimization is a challenging task, whereas NEB naturally accounts for distortion effects. However, despite all the theoretical challenges mentioned above and the influence of the xc functional of choice on the relevant energetics, the general trend of the predicted permeation barriers is preserved, regardless of the chosen approach, which supports the basic reliability of DFT-based methods for the evaluation of permeation barriers in defected graphene. Finally, our results point out that water represents indeed a particularly interesting case for filtration by graphene-based membranes: in fact, the tendency to form hydrogen bonds with saturating hydrogens and other water molecules on the opposite side of the pore can have a significant effect on the reduction of the permeation barriers.

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AUTHOR INFORMATION

Notes

The authors declare no competing financial interest.

ACKNOWLEDGMENTS We acknowledge financial support by MIUR within the PRIN 2010-2011 GRAF project. We also acknowledge allocation of computer resources at CINECA within the Iscra C ABBILGRA project.

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REFERENCES

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Gas Separation in Nanoporous Graphene from First Principle

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