Separation selectivity and structural flexibility of graphene-like 2-dimensional membranes

Abstract

Single-layer membranes of porous graphene, graphyne derivatives (α/α2/β-graphyne), and porous boron nitride (BN) with similar pore sizes (approximately 8 × 6 Å) have been evaluated for separating pentane isomers using first-principles calculations. In spite of their slightly bigger pore sizes, graphyne derivatives only allow linear pentane molecules to go through their native pores, while in contrast porous graphene and BN membranes only block dibranched isomers, i.e., neopentane molecules. Analyses of the geometric changes reveal that the structural flexibility of the membrane determines the penetration barrier. During the penetration of molecules, rigid membranes like graphyne derivatives may exhibit similar distortion to flexible porous graphene and BN; however the energy increase for the former is twice that for the latter. More importantly, the passing molecules experience about two times geometry distortion and four times energy barrier increase when going through rigid membranes compared with flexible membranes. The more deformed passing molecule and the less deformed rigid membrane can result in a much higher penetration barrier. Our results show that the flexibility of porous BN is comparable to that of porous graphene while graphyne derivatives are much more rigid.

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1. Introduction

Single-layer two-dimensional (2D) membranes are ideal separation materials with maximum possible permeance.^1,2 Graphene as a representative 2D material can be readily made into few-layer or single-layer sheets.^3,4 Therefore, a lot of research proposes using porous graphene sheets as separation materials for all kinds of mixtures, including sea water desalination,^5–8 small molecule purification like H₂/N₂,³ H₂/CH₄,¹ and CO₂/N₂,⁹ alkane isomer separation,^10,11etc. However, making graphene membranes with a small pore size and uniform pore distribution is still a challenge,^12–14 especially when the pore is required to be smaller than 5 nm which will be suitable for separating small gas molecules.

Graphynes, as a carbon allotrope, can not only be made into few- or even single-layer sheets, but are also formed with uniformly distributed single-sized pores.^15–17 For examples, graphyne derivatives like graphdiyne sheets have been synthesized on copper substrates via cross-coupling reactions,¹⁸ whose pore is best characterized by the longest and the shortest edge-to-edge distances of 8.1 × 7.1 Å (Fig. 1a).¹⁶ A number of graphyne derivatives have been theoretically proposed and studied for their separation properties for small molecules. Jiao et al. explored the diffusion of H₂/CH₄/CO through graphdiyne and found that H₂ could be screened from the others.¹⁹ Additionally, graphdiyne has been proved to be an efficient membrane for He³/He⁴ isotope separation.¹⁴ Zhang et al. investigated the separation of H₂ from CO, N₂, and CH₄ by sheets of graphyne (5.5 × 4.8 Å, Fig. 1b), square-octagon graphyne (3.4 × 3.4 Å, Fig. 1c) and rhombic-graphyne (8.0 × 6.3 Å, Fig. 1d).¹⁶ They found that the former two with smaller pores are unsuitable for H₂ separation as they exhibit high diffusion barriers (nearly 2 eV) even for H₂. The latter one has high diffusion barriers for other gases rather than H₂, thus achieving high H₂ selectivity with respect to CH₄ (10¹⁶:1).

Fig. 1 Top view of selected graphyne derivatives.¹⁵ (a) Graphdiyne, (b) graphyne, (c) square-octagon, and (d) rhombic-graphyne. Gray balls represent carbon atoms. The orange color s and blue color l represent the shortest and the longest diameters of a pore, respectively.

However, Jiang et al. have shown that graphene sheets with smaller pore sizes (e.g. 3.0 × 3.8 Å) can efficiently separate H₂ from CH₄ and we have shown that holey graphene membranes (e.g. the pore16 model features an elliptic pore of 7.8 × 5.8 Å, Fig. 2a) with similar pore sizes to those of rhombic-graphyne (Fig. 1d) can efficiently separate bigger molecule mixtures like isomers of short alkanes (C5–C7), which is critical to producing high-octane gasoline and thus is important to the petrochemical industry.^9,10 Thus, the relationship between the separation properties and pore size in porous graphenes seems to differ from those in graphyne derivatives. Recently, the α/α2/β-graphyne family with similar pore sizes to those of graphdiyne (8 × 7 Å, Fig. 2b–d) has been proposed to be stable.^20–22 Their pore sizes are slightly bigger than those of rhombic-graphyne (Fig. 1d) and porous graphene (the Hpore16 model, Fig. 2a), which may make them less suitable for separating small gas molecule mixtures like H₂/N₂/CH₄,^1,23,24 but may make them suitable for alkane isomer separation. We would like to contrast the separation properties of the α/α2/β-graphyne family with those of porous graphenes and also would like to understand the difference in their separation behavior.

Fig. 2 Models of the membranes and the passing molecules. (a) Hpore16, the porous graphene model, (b) α-graphyne, (c) α2-graphyne, and (d) β-graphyne. Left column, the molecular structures of the models with their geometric pore diameters labeled. Middle column, the electron density plots for the models (isovalue 0.02 e Å⁻³) with their electron density pore diameters labeled. s and l represent the shortest and the longest diameters of a pore, respectively. Right column, pentane isomers labeled with their minimum geometric diameters and their kinetic diameters in terms of the electron density isosurface (isovalue 0.02 e Å⁻³): (e) pentane, (f) isopentane, and (g) neopentane. Gray balls: carbon; white balls: hydrogen; red dots: carbon atoms used to calculate the distance (adsorption height) between the molecule and the membrane.

Another well-studied graphene-like 2D material is boron nitride (BN). Porous BN nanosheets have been made using methods such as dynamic templating^25–27 and elemental substitution.^28,29 Similar to porous graphene, a high surface area and pore volume make BN highly suitable for gas storage and wastewater treatment.^25,30,31 Liu et al. found that holes in BN sheets possessed super-hydrophobicity, i.e., high selectivity and capacity to adsorb oils, organic solvents and dyes, and thus can be used to clean contaminated water.³¹ Whether porous BN exhibits similar separation properties to porous graphene is also worth some investigation.

To address the above concerns, we carried out a systematic first-principles study on the separation performance of single-layer porous graphene sheets, the α/α2/β-graphyne family, and porous BN. To facilitate a direct comparison, the pore sizes of porous graphene/BN models were set close to those of the graphyne family (approximately 8 × 6 Å) and they were tested for separating pentane isomers. To simplify our discussion, the unsaturated carbon atoms, boron and nitrogen atoms at the pore edges were H-passivated,^32,33 as H-ion bombardment is one common way to prepare porous graphene.³⁴ We designed one porous graphene model and one porous BN model, which were compared with three graphyne models. For each model, we calculated the diffusion potential energy surface for each of the three pentane isomers. Then, we evaluated the selectivity of the isomers for each model. Later, we quantitatively analyzed geometric deformation during penetration for both the isomers and the membrane models. Finally, we rationalized selectivity as a result of the rigidity (or flexibility) of the membrane, which can be expressed as the relation between the averaged geometric displacement of the atoms in the passing molecules, and the membranes and the corresponding energy change.

2. Methods

The first-principles-based density functional theory (DFT) calculations were performed using the software package Atomistix Tool Kit (ATK).^35,36 The generalized gradient approximation (GGA) in the form of the Perdew–Burke–Ernzerhof (PBE) functional was employed.³⁷ The Fritz-Haber-Institute (FHI) pseudopotential was chosen and the plane wave cutoff energy was set to 75 Hartree. The wave functions were expanded by using a double-ζ (DZ) basis for H atoms and double-ζ plus polarization (DZP) bases for B, C, and N atoms, respectively. The Brillouin zone was sampled with 4  ×  4  ×  1 Monkhorst–Pack k-point grids, which ensured a high numerical accuracy. Furthermore, along the perpendicular direction, a 20 Å vacuum was introduced to reduce the interactions among periodic images.^32,38 The structures were fully relaxed until the residual force on each atom became smaller than 0.05 eV Å⁻¹. The minimum energy paths for molecules diffusing through the pores were calculated using the Nudged Elastic Band (NEB) method.³⁹

This calculation setup was tested by reproducing literature values, where graphdiyne was used to separate small molecules like H₂ from CH₄ (Fig. S1a, ESI†).¹⁹ The diffusion activation barrier (distance from the bottom to the top of the potential energy surface) obtained by our NEB calculations was 0.70 eV, quantitatively close to the literature value (0.73 eV). The dispersion interaction-inclusive functionals (e.g. the DFT-D2 method) have been found to only shift the diffusion energy profile on the whole, the same calculation setup was also checked against in the case of CH₄ passing through a porous graphene sheet (Fig. S1b, ESI†).¹⁰ Again, the activation barrier was close to the results of the functional with van der Waals forces. Therefore, to save computational cost, we did not consider dispersion interactions in the following calculations.

3. Results and discussion

3.1 Model of membranes

The porous graphene model (Hpore16) is named after the number of deleted C atoms and is prefixed with letter H to represent hydrogen passivation (Fig. 2a). The pore size is best described by the longest and the shortest diameters (7.8 × 5.8 Å, the average pore diameter is 6.8 Å). The pore size according to the electron density isosurface is naturally smaller (5.5 × 3.9 Å, the average pore diameter is 4.7 Å, when electron density isosurface is 0.02 e Å⁻³).¹ α2- and β-graphyne have two types of pores,^21,40 but the smaller ones actually cannot allow any gas to penetrate under normal conditions, as revealed by the nearly complete blockage in the electron density plots (Fig. 2c and d).^41,42 Thus the graphyne derivatives (α/α2/β-graphyne) all have similar pore sizes with Hpore16 (Fig. 2b–d).^14,39,40 The adsorption height is defined as the distance between the passing molecule (red carbon atoms in Fig. 2e and f) and the membrane plane.

3.2 Diffusion barrier and selectivity

The transition states were confirmed by vibration normal mode analysis. For example, the transition states of isopentane and neopentane diffusing through α-graphyne and β-graphyne sheets each features a single imaginary frequency of 70–100 cm⁻¹ (Table S1, ESI†). The vibration normal modes are consistent with the penetration direction through the pore. The values of the imaginary frequencies indicate that the corresponding transition states involve only physical interactions, i.e., they are diffusion in nature.

The diffusion energetics of three pentane isomers passing through the four membranes models was calculated by using the NEB method and the results are shown in Fig. 3. The diffusion energy barrier (E_b) is defined as

E_b = E_TS − E_ref (1)

where E_TS and E_ref represent the energies of the transition state (TS) and the reference state (the binding energy of the molecule at 4 Å is set to zero), respectively.⁴³

Fig. 3 Diffusion energetics of pentane isomers penetrating the membranes. (a) Hpore16, (b) α-graphyne, (c) α2-graphyne, and (d) β-graphyne. Filled symbols are the values obtained from the actual NEB images.

For Hpore16, there are small and moderate barriers of 0.33 and 0.63 eV for isopentane and neopentane, respectively, while n-pentane can pass through freely. As the pore sizes of α/α2/β-graphyne are very close, their separation performance is essentially the same. All three can efficiently separate linear n-pentane from branched ones, as isopentane and neopentane have to overcome high energy barriers of 1.2–1.5 eV and 3.3–3.7 eV, respectively (Table 1).

Table 1 Energy barrier and selectivity of pentane isomers passing through porous graphene and graphyne derivative membranes

Membrane	Barrier/eV			Selectivity
	Pentane	Isopentane	Neopentane	Sp:Si:Sn
Hpore16	0	0.33	0.63	10^10:10^5:1
α-Graphyne	0.06	1.21	3.32	10^54:10^19:1
α2-Graphyne	0.10	1.47	3.63	10^59:10^21:1
β-Graphyne	0.08	1.36	3.70	10^60:10^23:1

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We calculated the relative selectivity for the mixture molecules through the Arrhenius equation for diffusion:

where D is the diffusion rate, A is the diffusion prefactor, and E is the diffusion energy barrier. Enthalpy and entropy corrections are assumed to be identical for all molecules in this study; in other words, all selectivities are calculated based on electronic energies only. We assumed that the prefactors of pentane isomers are identical A_P/A_I ≈ 1 and the temperature T = 300 K (room temperature).^22,41,42 This method mainly employs two approximations: (1) the diffusion barriers are electronic energy rather than enthalpy; (2) the entropy contribution is wrapped in the prefactors and the prefactors are assumed to be equal for different molecules. Therefore, the calculated selectivity is only estimation. The selectivity for pentane, isopentane, and neopentane passing through Hpore16 is 10¹⁰:10⁵:1 (Table 1), which is low compared with that of the graphyne membranes (e.g. for α-graphyne, 10⁵⁴:10¹⁹:1). In practice, these graphyne derivatives actually block monobranched and dibranched and cannot separate them from isomers.

For a pore model, pentane isomers exhibit diffusion barriers proportional to both their geometric diameter and kinetic diameter in terms of the electron density isosurface (Fig. 2). In fact, there is a threshold of the ratio of the molecule's diameter over the pore diameter, beyond which an appreciable barrier (>0.1 eV) increases. The ratio of the two should be larger than 0.5 in terms of geometric diameters (i.e., the pentane diameter of 2.9 Å over the shortest diameter of 5.8 Å of the smallest pore16) or 0.09 in terms of the kinetic diameter according to the electron density isosurface (i.e., the pentane diameter of 3.7 Å over the shortest diameter of 3.9 Å of the smallest pore16).

In spite of their bigger pore sizes (the shortest diameter is longer by 1 Å than that of Hpore16), these graphyne derivatives only allow linear pentane molecules to go through, while Hpore16 virtually only blocks dibranched isomers. This is evidently against our intuition and the rule that the larger the pore size, the lower the barrier reported in previous studies.^44,45 The electron density plots provide a clue. According to electron density, all the four membrane models show closer pore sizes (Fig. 2). The triple bonds (–C≡C–) in graphyne sheets have high electron density, thus the pore size is much smaller than the geometric pore size. This can be further illustrated by plotting electron density isosurfaces for both neopentane (placed at the center of the pores) and the membranes (Fig. S2, ESI†). For neopentane and the graphyne membranes, the density covers the entire pore area, while for neopentane and Hpore16 there is some gap in between, indicating weaker repulsion in the latter case.

3.3 Geometric deformation analysis

Analysis of the geometric change during molecule penetration may help us understand better the abnormally high barriers of the graphyne membranes, as the geometric relaxations of both the passing molecule and the membrane are crucial to the barrier and selectivity (Fig. S3, ESI†),^10,46 which were quite often assumed to be rigid.^46–49

We used α-graphyne as an example. Snapshots of pentane isomers passing through α-graphyne at the adsorption heights from −4 to 4 Å are shown in Fig. 4a and c. For each snapshot, careful analyses of geometric deformation of both the passing molecule and the membrane were carried out (Fig. 4d and f), as well as their corresponding energy changes (Fig. 4e and g).

Fig. 4 Geometric deformation analysis for pentane isomers penetrating α-graphyne. Snapshots of pentane isomers passing through α-graphyne: (a) pentane, (b) isopentane, and (c) neopentane. (d) Average z-direction deformation of pore rim atoms in α-graphyne. (e) Energy change (|dE|) of α-graphyne. (f) Average displacement of all atoms of pentane isomers . (g) Energy change (|dE|) of pentane isomers.

α-Graphyne presents substantial out-of-plane deformation when isopentane and neopentane diffuse through, especially in the range of −2 to 2 Å. In comparison, there is no noticeable change during pentane diffusion.

The deformation of the membrane is quantitatively expressed by the out-of-plane deformation , which is the average absolute z-direction shift of rim atoms around the pores (red atoms in Fig. S4, ESI†), whose has been shown to best represent the deformation.¹⁰ While for pentane isomers, the average absolute position change of all atoms, , is used for quantifying the deformation. Their energy change, compared to the energy of the isolated membrane, is represented by the absolute value d|E|.

The membrane deformation exhibits a surprising M-shape (Fig. 4d): as the passing molecule approaches the pore, the pore distortion first increases and then decreases. When the adsorption height is 0 Å (the penetration point), the membrane deformation features a local minimum. The deformation behavior is quite different from the porous graphene sheets,¹⁰ where a peak appears near or right at 0 Å (e.g. the deformation analysis for Hpore16 in Fig. S5, ESI†). The biggest distortions of α-graphyne membranes occur at 2 or −2 Å, with the values of 0.025, 0.022 and 0.005 Å for the penetrations of neopentane, isopentane and pentane, respectively. In comparison, when the adsorption height is 0 Å, the values are 0.007, 0.005 and 0.0002 Å for the same penetrations. The corresponding energy change of the membrane keeps increasing as the molecule approaches the pore and peaks at an adsorption height of 0 Å (Fig. 4e). For the penetration of pentane, the slight deformation of α-graphyne corresponds to only the maximum energy increase of 0.02 eV. For the penetration of isopentane and neopentane, moderate (0.16 eV) and large (0.55 eV) energy increases are observed for the membrane.

On the other hand, the distortions of pentane isomers are quite normal: they increase as the adsorption heights decrease and peak at 0 Å with the values of 0.009, 0.008 and 0.004 Å for neopentane, isopentane and pentane, respectively (Fig. 4g). Their energy changes synchronize with the distortions quite well (Fig. 4f). The maximum energy changes for pentane, isopentane and neopentane are about 0.07, 0.31, and 0.77 eV, respectively, even higher than the energy increase experienced by the membrane. The total energy change due to the deformation of both α-graphyne (0.55 eV) and neopentane (0.77 eV) is 1.32 eV, which accounts for almost 35% of the energy barrier (3.32 eV). Thus, to accurately calculate the barrier, the geometric change of both the membrane and the passing molecule must be taken into consideration.

The unconcerted change in geometry and energy of the α-graphyne membrane is really based on the out-of-plane deformation we have been focusing on. A closer look at the pore diameters reveals the unconventional deformation process, which is exemplified by passing neopentane. Originally, the pore size of α-graphyne is 8.00 × 7.00 Å (characterized by the longest and the shortest diameters of the pore). At an adsorption height of −2 Å, the pore size is slightly enlarged with the new longest and shortest diameters of 8.03 × 7.04 Å. With the ensuing reduction of the adsorption height to −1 Å, the shortest diameter is increased substantially while the longest diameter shortens evidently to 7.90 × 7.33 Å. At 0 Å, the longest diameter returns to its original value and the shortest diameter is still elongated to 8.00 × 7.30 Å. The diameter changes after the molecule goes to the other side of the pore are almost symmetric and are listed in Table S2 (ESI†). The subsequent pore size always increases as the molecule approaches the pore, during which the out-of-plane deformation reduces and the in-plane deformation increases. The membranes of these graphyne derivatives deforms differently from the graphene membrane. The deformation of the latter is always dominated by the out-of-plane component rather than the in-plane component at all adsorption heights.

The structural change when pentane isomers pass through the α-graphyne model turns out to be true for other graphyne models (Fig. 5). At an adsorption height of 0 Å, their out-of-plane structural deformation is far less than that of porous graphene Hpore16, although their energy change (|dE|) maximizes and is about twice as large as that of Hpore16. At an adsorption height of 0 Å, structural deformation is largely in-plane rather than out-of-plane (0.007, 0.0 and 0.005 Å for α-, α2- and β-graphyne, respectively, Fig. 5b) for graphynes.

Fig. 5 Deformation and energy analysis for the graphyne and graphene membrane models at an adsorption height of 0 Å. Energy change (|dE|) of the membranes (a) and pentane isomers (c) due to structural deformation. (b) Structural deformation of the membranes and pentane isomers . As the guide to the eye, lines of the same color link the data of the same diffusing molecule. Black: pentane; red: isopentane; blue: neopentane.

In contrast, both structural deformation and energy changes of pentane isomers are much higher when passing through graphynes than through porous graphenes (Fig. 5c and d). For example, the structural deformation and the energy change of neopentane passing through Hpore16 are 0.004 Å and 0.15 eV, respectively. When penetrating the graphyne derivatives, the geometry deformation and energy change of neopentane increase by one and four times, respectively. In particular, the energy change of neopentane is even higher than those of the graphyne derivatives, opposite to the porous graphene models. From the viewpoint of structural rigidity, we calculated the energy cost to bend the membranes into ideal convex surfaces of different degrees of curvature, as the penetrating molecule would likely deform the membrane by bulging it. We found that the “rigidities” of the graphyne membranes are greater than that of graphene (Fig. S2e, ESI†).

The origin of the higher degree of rigidity of the graphyne derivatives probably lies in the composition of their pores, which are made of more rigid carbon–carbon triple bonds (–C≡C–). In turn, their high electron density renders smaller accessible pores. For example, at an electron density isovalue of 0.02 e Å⁻³ (Fig. 2), α-, α2- and β-graphyne derivatives have pore-tangential circles with diameters of 4.5, 4.6 and 4.5 Å, respectively, which are all smaller than that of Hpore16 (4.7 Å). Meanwhile, the rigidity requires the graphyne derivatives to distort at more energy cost than Hpore16. Moreover, the rigidity does not allow too much out-of-plane distortion when the passing molecule is near the penetration point and the out-of-plane distortion is converted to in-plane pore expansion. Still, the passing molecule experiences higher distortion and a much higher energy increase. As a result, the barrier for pentane penetration is raised to a much higher value for the graphyne derivatives than for porous graphenes (Table 1 and Fig. 3).

It is natural to wonder what other graphene-like materials (i.e., porous boron nitride (BN) sheets) behave when used for separating alkane isomers. We built a BN Hpore16 model, similar to the porous graphene Hpore16. Eight pairs of boron and nitrogen atoms were removed from the single-layer BN sheet and the pore was passivated by hydrogen atoms. The BN bond length we used is 1.44 Å, which is close to literature values^30,42,50 and is slightly smaller than that of graphene (1.47 Å). The resulting pore size of BN Hpore16 (8.0 × 5.4 Å, the average pose size is 6.7 Å) as well as the pore size according to the electron density (0.02 e Å⁻³, 5.3 × 3.7 Å, the average pose size is 4.5 Å) is slightly smaller than that of graphene Hpore16 (the average pore sizes are 6.8 and 4.7 Å).

The diffusion barriers of the three pentane isomers passing through the BN Hpore16 model are 0.81, 0.16 and 0.0 eV for neopentane, isopentane and pentane, respectively (Fig. 6c). This means that the BN Hpore16 can only block dibranched neopentane while pentane and isopentane can pass easily. The energy barrier for neopentane is higher than that of the graphene Hpore16. The selectivity of pentane isomers is estimated to be 10¹³:10⁴:1 by using the Arrhenius equation, which is greater than that of the graphene Hpore16 (10¹⁰: 10⁵:1).

Fig. 6 The BNH pore16 membrane and the penetration energy barriers of pentane isomers. (a and b) are the molecular structure and electron density plots of the model (isovalue 0.02 e Å⁻³). (c) Diffusion energetics of pentane isomers passing through BN Hpore16. Energy barrier are obtained by NEB calculations. Blue balls: nitrogen; pink balls: boron; white balls: hydrogen.

To compare the behaviors of the porous graphene and BN sheets, we further analyzed the geometric distortion and the corresponding energy change during penetration (at an adsorption height of 0 Å) of the pentane isomers (Fig. 7). The structural deformation of the graphene Hpore16 is slightly larger than that of the BN membrane, by 0.01 and 0.007 Å for isopentane and neopentane passing through the membranes, which implies a higher rigidity of the BN Hpore16 in the z direction. The structural deformation of the pentane isomers are almost zero in both cases and their energy changes are very similar (Fig. 7 and Fig. S5, ESI†). A detailed deformation analysis of pentane isomers penetrating the BN Hpore16 is shown in Fig. S6 (ESI†). In general, graphene and BN membranes behave similarly during separation.

Fig. 7 Comparison of the BN Hpore16 and graphene Hpore16 membrane models for separating pentane isomers. Deformation and energy analysis are for membranes (a and b) and pentane isomers (c and d) at an adsorption height of 0 Å. As the guide to the eye, lines of the same color link the data of the same membrane.

As mentioned earlier, there are many factors (e.g. pore size, pore shape, passivating atoms/functional groups, etc.) that affect the energy barriers of molecules diffusing through pores.^1,3,8,15 Our data show once again that there is no simple linear relationship between the barrier and the pore size (Fig. 8). The pore rigidity has a significant effect on this relationship: as rigid membranes, the graphyne derivatives (α/α2/β-graphyne) have bigger geometric pore sizes than the flexible porous graphene sheets (Hpore16), but they have higher energy barriers for pentane isomers (Fig. 8a). This may be explained by using an electron density pore size: porous graphene sheets have the largest pores in terms of the electron density isosurface and therefore have the lowest barriers for pentane isomers (Fig. 8b). In fact, effective pore sizes are determined by the electronic structure and bonding of the pores, which also are reflected as different rigidities of the pores (Fig. S2, ESI†). However, the rigidity effect on the barrier can be more complex. For example, β-graphyne and α-graphyne have identical electron density pore size and the diffusion barriers of isopentane still differ by 0.15 eV, and the difference even increases to 0.38 eV for neopentane. In addition, the diffusion barrier of isopentane passing through α2-graphyne sheets is even higher than those for passing through α/β-graphyne sheets, although the former have bigger pores in both metrics than the latter. These abnormal barrier orders should be attributed to the complex interactions between the passing molecules and the pores, because static properties such as rigidity cannot completely determine the kinetic properties such as the diffusion barrier.

Fig. 8 Pore size and diffusion barrier of pentane isomers passing through graphyne derivatives, BN, and porous graphene sheets. (a) Geometric pore size. (b) Electron density pore size.

4. Conclusions

We have computationally evaluated the performance of single-layer porous graphene, graphyne derivatives (α/α2/β-graphyne), and porous boron nitride (BN) membranes for separating pentane isomers by first-principles calculations. Our results show that the membrane models of graphene (pore size of 7.8 × 5.8 Å) and BN (pore size of 8.0 × 5.4 Å) behave similarly and both block neopentane from other isomers, while the graphyne derivatives (models with pore sizes of about 8.0 × 7.0 Å) effectively block both the monobranched and dibranched pentane isomers.

The rigid –C≡C– triple bonds in the graphyne derivatives and their higher electron density make the effective space of the pores smaller and harder to distort. While the maximum out-of-plane deformations of all membranes are similar during separation, the maximum energy increases of the rigid graphyne membranes are about twice those of the flexible graphene and BN membranes. Meanwhile, the geometry distortion and energy rise of pentane isomers are correspondingly two and four times when passing through the rigid membranes compared with the flexible ones. As a result, the barriers for the rigid membranes are much higher than for the flexible membranes with similar pore sizes.

The graphyne derivatives also exhibit unusual geometry deformation during separation. At first, the out-of-plane deformation increases with the shrinking distances between the membrane and the passing molecule. Then at the passing point (the adsorption height is 0 Å), the out-of-plane deformation diminishes and transforms into the in-plane pore expansion. As the molecule passes through to the other side, the out-of-plane deformation increases again. While for the graphene and BN membranes, the out-of-plane deformation increases monotonically as the molecule approaches the pore.

Rigid membranes tend to cause larger geometry deformation of the passing molecules, which leads to much higher penetrating barriers and ultimately to different separation properties. Therefore, in the design of porous membranes, the flexibility of the membranes should be carefully considered.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 51320105014, 51621063, 21477096).

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c8cp00466h

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