# Theoretical and Experimental Insights into the Mechanism for Gas Separation through Nanochannels in 2D Laminar MXene Membranes

^{1}

^{2}

^{*}

## Abstract

**:**

_{2}transport in molecular sieving model was calculated to be 20.54 kJ mol

^{−1}. From the model, we can predict that the gas permeance of hydrogen (with smaller kinetic diameter) is contributed from both Knudsen diffusion and molecular sieving mechanism, but the permeance of larger molecular gases like nitrogen is sourced from Knudsen diffusion. The effects of the critical conditions such as temperature, the diffusion pore diameter of structural defects, and the thickness of the prepared MXene lamellar membrane on hydrogen and nitrogen permeance were also investigated to understand the hydrogen permeation difference from Knudsen diffusion and molecular sieving. At room temperature, the total hydrogen permeance was contributed 18% by Knudsen diffusion and 82% by molecular sieving. The modeling results indicate that molecular sieving plays a dominant role in controlling gas selectivity.

## 1. Introduction

_{2}) [12], molybdenum disulphide (MoS

_{2}) [13,14], and metal-organic frameworks (MOFs) [15,16] have been receiving particular interest.

_{n}

_{+1}X

_{n}T

_{x}, where M represents a transition metal, X is carbon and/or nitrogen, and T is referred to the surface termination [17,18]. Very recently, MXene-based 2D materials have been introduced to fabricate 2D lamellar membranes because of their tunable nanochannel width, excellent mechanical strength, and easy fabrication and integration [2,19,20]. It has been reported that the assembled MXene 2D membranes exhibit a range of attractive characters in separation, e.g., precise ion sieving [21], ultrafast water permeation [22], and gas separation [19,20].

_{2}and N

_{2}) in 2D MXene lamellar membranes. The simulation results are well consistent with the experimental results and their significance to the gas diffusion (e.g., permeance and selectivity) was discussed. The structural effects from MXene nanochannels formed during MXene nanosheets assembling on the transportation of different gas molecules (e.g., size and mass) were studied. Moreover, we provided the effects of temperature, pore diameter of structural defects, and the MXene thickness of the lamellar membrane on hydrogen and nitrogen permeances.

## 2. Experimental

_{3}C

_{2}T

_{x}was synthesized by etching Ti

_{3}AlC

_{2}powders using 50 wt% HF solution at a certain temperature followed with DMSO intercalation. The interlayer interaction became weak due to the removal of Al atom between layers, leading to a facile exfoliation to form MXene nanosheets under sonication. The suspended MXene nanosheets were deposited by filtration on the anodic aluminum oxide (AAO) support with a pore diameter of 200 nm (Whatman Co., Maidstone, UK) by a vacuum pump. The resultant MXene membrane was dried at 120 °C for 8 h in a vacuum oven to remove the water molecular between interlayers.

_{2}and N

_{2}gas permeances were derived from their gas mixture separation performance measurement through the MXene membrane supported on AAO [20]. The gas mixture permeance was carried out in a home-made device as reported previously [20]. The gas mixture containing 50 vol% H

_{2}and 50 vol% N

_{2}as the feed gas was supplied to the membrane feed side with a flow rate of 50 mL min

^{−1}, while the argon sweep gas with a flow rate of 40 mL min

^{−1}at a standard pressure was supplied to the sweep side. The exit gas from the membrane sweep side was transferred to an online GC (6890N, Agilent Technologies, Inc, Waldbronn, Germany) with TCD to measure the permeated H

_{2}or N

_{2}concentration. The permeance and mixture selectivity or separation factor were defined by:

_{i}is the permeance of the derived gas (i) (mol m

^{−2}s

^{−1}Pa

^{−1}), N

_{i}is the molar flux of the gas (i) (mol m

^{−2}s

^{−1}), P

_{i}

_{1}and P

_{i}

_{2}are the partial pressures of gas (i) at the feed side and sweep side.

_{i}

_{1}, y

_{i}

_{2}, y

_{j}

_{1}and y

_{j}

_{2}, the volumetric fraction of the gas component i or j in the feed or permeated side gas mixtures, respectively. Experimental results are summarised in Table 1.

## 3. Theoretical Models of Transport Mechanism

_{k}) and 2 nm. Accordingly, two mathematical models of the gas transport mechanism for the permeation through the MXene lamellar membrane can be proposed based on the different nanochannels width using the following assumptions:

- (1)
- One-dimensional transport model is used.
- (2)
- There are two kinds of transport channels as illustrated in Figure 1, which are the straight nanochannels (2 nm < d
_{p}< λ) and tortuous nanochannels (Φ_{k}< d_{p}< 2 nm). Their size remains unchanged with the temperature. - (3)
- The MXene/AAO lamellar membrane operates at a steady-state isothermal condition.
- (4)
- The linear adsorption isotherm (or non-adsorption) for all gases on the surface of MXene lamellar membranes is neglected.
- (5)
- (6)
- Gas mixture (hydrogen and nitrogen) transport through the membrane is under ideal conditions on which the actual separation factor is equal to the ideal selectivity and pure gas permeance is equal to the mixture permeance.

_{2}(kinetic diameter of 2.89 Å) and N

_{2}(3.64 Å) diffuse through the MXene lamellar membrane based on our experimental results [20], so the geometrical structure of nanochannels derived from the structural defects and interlayer spacing plays a significant role in the gas transportation. For gas permeation, two transport models were proposed corresponding to the nanochannels in order to explain the experimental results. The Knudsen diffusion is assumed to occur within the straight nanochannels with the lager pore diameter (2 nm < d

_{p}< λ). The tortuous nanochannels are mainly composed of randomly distributed nanoscale wrinkles, inter-galleries, and interlayer spacing between stacked MXene sheets. Therefore, the gas transportation within tortuous nanochannels containing the interlayer spacing (Φ

_{k}< d

_{p}< 2 nm) is mainly related to the molecular sieving.

_{total}, F

_{defects}, and F

_{int}

_{erlayer}are the total gas permeance, the gas permeance through straight nanochannels of structural defects, and tortuous nanochannels containing interlayer spacing, respectively. F

_{Kn}and F

_{g}are the permeances contributed by the Knudsen diffusion and the molecular sieving, respectively.

#### 3.1. Knudsen Diffusion (KD) Model

_{p}< λ), i.e., the average distance a molecule traversed by collisions, which is comparable or larger than transport channels, transport falls in Knudsen regime [27]. The mass transport of gas may be described by Fick’s first law as Equation (4).

_{K,i}) for gas can be obtained in terms of pressure gradient. In this case, D

_{K}is the Fick diffusion coefficient (Knudsen diffusivity), which may be expressed as the product of a geometric factor by diffusion pore diameter and the velocity of gas molecules by Equation (5):

_{p}is the diffusion pore diameter, the velocity of diffusing molecules is given by the kinetic theory of gases, and M is the molecular weight of the diffusing gas. The geometric factor is 1/3 since only these molecules moving in the considered direction will be taken into account [28]. The expression for Knudsen diffusion flux (J

_{K,i}) obtained by combining Equations (4) and (5) is expressed as Equation (6).

_{p}), Equation (7) was transformed to Equation (8) to reveal the gas permeance dependence on the temperature. Thus, the gas permeance in Knudsen regime is pressure-independent and decreases with temperature as indicated by Equation (8).

#### 3.2. Molecular Sieving (MS) Model

_{k}< d

_{p}< 2 nm), channel size changes into the molecular dimensions and molecules are no longer as free as these in Knudsen diffusion. For simplification purpose, the individual gas molecular adsorption difference is not considered. This is a reasonable assumption for these gases with less adsorption like He, H

_{2}, and N

_{2}than CO

_{2}. The molecular sieving flux can be expressed in terms of pressure gradient. As a result, the molecular sieving flux (J

_{s}

_{,i}) can be written as:

_{s,i}is the molecular sieving coefficient in the MXene laminates, which is given by Equation (10).

_{s}is the diffusion distance (the distance between two adjacent sites of the low energy regions), Z is the number of adjacent sites [28], and E

_{a,g}is the activation energy, which is required for molecules to surmount the attractive constrictions imposed by the nanochannels structure. However, the geometrical factor (1/Z) is the probability of a molecule moving in the direction under consideration. The expression of molecular sieving flux (J

_{s}

_{,i}) obtained by combining Equations (9) and (10) is shown as the following.

_{a,g}and the diffusion distance

_{.}

_{a,g}and the diffusion distance in the logarithmic plots can be regressed based on experimental data by Equation (13).

#### 3.3. Diffusion Contribution to Total Transport

## 4. Results and Discussion

#### 4.1. Morphology and Structure

_{3}C

_{2}T

_{x}), high magnification over the cross-section of MXene membrane, external surface of MXene membrane, and low magnification of the cross-section of MXene membrane are displayed as Figure 2a–d, respectively.

_{3}C

_{2}Tx) nanosheets were deposited as an outer layer on top of a porous AAO support with a pore diameter of 200 nm using the vacuum impregnation method to form the supported MXene membrane (Figure 2c,d).

#### 4.2. The Experimental Nitrogen Permeance and the Parameters Regression of Knudsen Diffusion (KD) Model

_{2}(3.64 Å), CH

_{4}(3.84 Å), C

_{3}H

_{6}(4.30 Å), and C

_{3}H

_{8}(4.50 Å) (Figure 4). Since we assume that the width of straight nanochannels are larger than the kinetic diameters of these gases, therefore, for the transport permeance of N

_{2}, CH

_{4}, C

_{3}H

_{6}, and C

_{3}H

_{8}, Knudsen diffusion is the main process in the gas transportation in the straight channels. Since straight nanochannels flow dominates gas permeation, the Knudsen diffusion modeling was performed by the ordinary least squares method using MATLAB 7.0 (The Math Works Inc., Natick, MA, USA) [29] to obtain the parameters in Equations (7) and (8) and the regression results were shown in Figure 5. We can observe that the calculations using the Knudsen diffusion model with the obtained parameters can fit the experimental data well with the resultant correlation coefficient up to 0.9966.

_{p}can be obtained from the regression. Since the tortuosity factor τ in Equation (17) can be estimated to ~1 in inner-sheet structural defects because of the straight diffusion channels. The geometrical effects (φ) of the porous structure derived from Equation (16) is 0.25, and the average diffusion pore diameter d

_{p}is 5.05 Å which is larger than the kinetic diameter of N

_{2}(3.64 Å), CH

_{4}(3.84 Å), C

_{3}H

_{6}(4.30 Å), and C

_{3}H

_{8}(4.50 Å). It can be concluded that the Knudsen diffusion through straight nanochannels for nitrogen is reasonable.

#### 4.3. The Experimental Hydrogen Permeance and the Parameters Regression of Molecular Sieving (MS) Model

_{2}(2.89 Å) (Figure 4), thus it is favorable to separate them by molecular sieving diffusion. Therefore, hydrogen transport mechanism models in the MXene membrane are the combined Knudsen diffusion and molecular sieving, which correspond to the diffusion through straight nanochannels of structural defects, and tortuous nanochannels contained interlayer spacing between stacked MXene sheets, respectively. Hence, H

_{2}permeance is higher than N

_{2}permeance in the temperature range of 295~593 K in experimental (Table 1 and Figure 6). For example, H

_{2}and N

_{2}permeances were 2.34 and 0.174 × 10

^{−7}mol m

^{−2}s

^{−1}Pa

^{−1}, respectively, at 295 K. The calculated separation factor or selectivity of H

_{2}/N

_{2}was ~13 based on the gas permeance, and it greatly exceeded the Knudsen selectivity of 3.74 for H

_{2}/N

_{2}pairs. It indicates the promising potential application of the 2D MXene membrane to separate H

_{2}from its gas mixture. The modeling results show that molecular sieving plays a dominant role in the selectivity of gas separation.

^{−1}in the feed side of the membrane, and an argon sweep gas with a flow rate of 40 mL (STP) min

^{−1}on the permeate side to remove the permeated hydrogen and nitrogen. The values for the model parameters φ, l

_{s}, and E

_{a,g}can be obtained by Equations (12) and (13).

_{a,g}) for H

_{2}permeance displayed in Figure 6 is 20.54 kJ mol

^{−1}. Although the geometrical effects of the porous structure (φ) and diffusion distance (l

_{s}) are difficult to be determined in Equation (12), by performing the ordinary least squares method using MATLAB 7.0 (The Math Works Inc., Natick, MA, USA) [29] for regression, the value of multiple (l

_{s}× φ) can be calculated to be 1.73 × 10

^{−12}. In addition, φ is the ratio of the membrane porosity ε to the tortuosity factor τ (see Equation (16)) and the tortuosity factor τ is difficult to be determined in Equation (17) as transport nanochannels from wrinkles and inter-galleries between stacked MXene sheets. The calculations using the model incorporating the obtained parameters fit the experimental data well with the resultant correlation coefficient of 0.9966.

#### 4.4. Temperature Dependent Permeance and Relative Contribution to Total Gas Transport from Knudsen Diffusion and Molecular Sieving

_{2}/N

_{2}) between 260 and 700 K. We can see that the hydrogen permeances are higher than nitrogen. For example, the hydrogen and nitrogen permeance through the MXene lamellar membrane is 2.11 and 0.11 × 10

^{−7}mol m

^{−2}s

^{−1}Pa

^{−1}at 363 K, respectively. Moreover, the corresponding selectivity of H

_{2}/N

_{2}is ~19 based on the gas permeance calculation. These results can be explained by the gas diffusion mechanism. The nitrogen transport is dominated by Knudsen diffusion through straight nanochannel of structural defects with the width of ~5.05 Å, while hydrogen transports based on Knudsen diffusion and molecular sieving through straight nanochannels of structural defects and tortuous nanochannels from interlayer spacing.

^{−7}mol m

^{−2}s

^{−1}Pa

^{−1}at 408 K. However, the selectivity of H

_{2}/N

_{2}is always enhanced. Such results can be interpreted by the joint effects of molecular sieving and Knudsen diffusion. From molecular sieving (Equation (12)), it reveals an exponential dependence of gas permeance on the temperature, which is obviously different from that of Knudsen diffusion. However, these trends are ascribed to the coverage of functional groups such as −OH and −O which will affect the adsorbed amount of hydrogen [31,32].

#### 4.5. The Effect of the Diffusion Pore Diameter of Straight Nanochannels on the Gas Permeance and Relative Contribution from Knudsen Diffusion and Molecular Sieving to Total Gas Transport

^{−7}mol m

^{−2}s

^{−1}Pa

^{−1}and nitrogen permeance 0.73 to 18.6 × 10

^{−8}mol m

^{−2}s

^{−1}Pa

^{−1}with the pore diameter of straight nanochannel alternation between 0.364 and 1.0 nm, respectively. The discrepancy between them becomes more pronounced with the increasing average pore diameter of straight nanochannels. This reflects that the larger nanochannel of structural defects will enhance Knudsen diffusion more significantly for permeance through the MXene lamellar membrane. Therefore, the selectivity of H

_{2}/N

_{2}decreases from 26 to 4.87 (close to 3.74 of the Knudsen selectivity). This indicates that it is important to reduce the average pore diameter of straight nanochannels resulting from structural defects to ensure the good gas selectivity.

_{c}) of 2.32 nm (23.2 Å) was also obtained from Figure 10, at which the Knudsen diffusion and molecular sieving equally share the transport. in addition, increasing the average diffusion pore diameter of defects more than such characteristic value (d

_{c}) will lead to the higher proportion of Knudsen diffusion than molecular sieving in the total permeance (Figure 10).

#### 4.6. The Effect of MXene Layer Thickness on the Gas Permeance and Fractional Diffusion of Knudsen Diffusion and Molecular Sieving

^{−7}mol m

^{−2}s

^{−1}Pa

^{−1}, respectively, when the MXene thickness is reduced from 1000 to 20 nm. On the other hand, the H

_{2}/N

_{2}selectivity maintains at 13.5. It indicates that the thinner MXene layer can lead to higher gas permeance for hydrogen or nitrogen (Equations (7) and (12)) but maintain the gas selectivity unaltered.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

a | Thickness of monolayer MXene lamellar, m |

c | Concentration of gas species, mol m^{−3} |

D_{K} | Knudsen diffusivity coefficient, m^{2} s^{−1} |

D_{g,i} | Molecular sieving coefficient, m^{2} s^{−1} |

d | d-spacing of MXene laminates, m |

d_{p} | Diffusion pore diameter, m |

d_{c} | Characteristic diffusion pore diameter, m |

E_{a,g} | Activation energy of gas diffusion, J mol^{−1} |

F | Permeance of the gas, mol m^{−2} s^{−1} Pa^{−1} |

F_{total} | Total gas permeance, mol m^{−2} s^{−1} Pa^{−1} |

F_{defects} | Permeance through structural defects, mol m^{−2} s^{−1} Pa^{−1} |

F_{interlayer} | Permeance through interlayer spacing, mol m^{−2} s^{−1} Pa^{−1} |

f_{Kn} | Fractional of Knudsen diffusion contribution |

f_{s} | Fractional of molecular sieving contribution |

l_{s} | Distance between two adjacent sites, m |

M | Molecular weight of the diffusing gas |

N | Molar flux of gas, mol m^{−2} s^{−1} |

P_{1} | Pressures at the feed side, Pa |

P_{2} | Pressures at permeate side, Pa |

R | Ideal gas constant, J mol^{−1} K^{−1} |

S | Selectivity of the gas pair |

T | Absolute temperature, K |

Z | Number of adjacent sites |

z | Distance coordinate, m |

Greek Letters | |

Φ_{k} | Kinetic diameter, m |

λ | Mean free path, m |

φ | Geometrical effects of the porous structure |

τ | Tortuosity factor |

δ | Membrane thickness, m |

ε | Membrane porosity |

Subscripts | |

i, j | Gas species i and j |

Kn | Knudsen diffusion |

s | Molecular sieving |

## References

- Sholl, D.S.; Lively, R.P. Seven chemical separations to change the world. Nature
**2016**, 532, 435–437. [Google Scholar] [CrossRef] [PubMed] - Ding, L.; Wei, Y.; Li, L.; Zhang, T.; Wang, H.; Xue, J.; Ding, L.X.; Wang, S.; Caro, J.; Gogotsi, Y. MXene molecular sieving membranes for highly efficient gas separation. Nat. Commun.
**2018**, 9, 155. [Google Scholar] [CrossRef] [PubMed] - Park, H.B.; Kamcev, J.; Robeson, L.M.; Elimelech, M.; Freeman, B.D. Fructose-driven glycolysis supports anoxia resistance in the naked mole-rat. Science
**2017**, 356, 1138–1148. [Google Scholar] [CrossRef] [PubMed] - Mi, B. Graphene oxide membranes for ionic and molecular sieving. Science
**2014**, 343, 740–742. [Google Scholar] [CrossRef] [PubMed] - Cheng, L.; Liu, G.; Jin, W. Recent progress in two-dimensional-material membranes for gas separation. Acta Phys. Chim. Sin.
**2019**, 35, 1090–1098. [Google Scholar] - Liu, Y.; Wang, N.; Cao, Z.; Caro, J. Molecular sieving through interlayer galleries. J. Mater. Chem. A
**2014**, 2, 1235–1238. [Google Scholar] [CrossRef] - Koenig, S.P.; Wang, L.; Pellegrino, J.; Bunch, J.S. Selective molecular sieving through porous grapheme. Nat. Nanotech.
**2012**, 7, 728–732. [Google Scholar] [CrossRef] [PubMed] - Celebi, K.; Buchheim, J.; Wyss, R.M.; Droudian, A.; Gasser, P.; Shorubalko, I.; Kye, C.L.J.I.; Park, H.G. Ultimate permeation across atomically thin porous graphene. Science
**2014**, 344, 289–292. [Google Scholar] [CrossRef] - Gao, X.; Li, Z.K.; Xue, J.; Qian, Y.; Zhang, L.Z.; Caro, J.; Wang, H. Titanium carbide Ti
_{3}C_{2}T_{x}(MXene) enhanced PAN nanofiber membrane for air purification. J. Membr. Sci.**2019**, 586, 162–169. [Google Scholar] [CrossRef] - Shen, J.; Liu, G.; Huang, K.; Chu, Z.; Jin, W.; Xu, N. Subnanometer two-dimensional graphene oxide channels for ultrafast gas sieving. ACS Nano
**2016**, 10, 3398–3409. [Google Scholar] [CrossRef] - Jin, Y.; Meng, X.; Yang, N.; Meng, B.; Sunarso, J.; Liu, S. Modeling of hydrogen separation through porous YSZ hollow fiber-supported graphene oxide membrane. AIChE J.
**2018**, 64, 2711–2720. [Google Scholar] [CrossRef] - Sun, L.; Ying, Y.; Huang, H.; Song, Z.; Mao, Y.; Xu, Z.; Peng, X. Ultrafast molecule separation through layered WS
_{2}nanosheet membranes. ACS Nano**2014**, 8, 6304–6311. [Google Scholar] [CrossRef] [PubMed] - Wang, D.; Wang, Z.; Wang, L.; Hua, L.; Jin, J. Ultrathin membranes of single-layered MoS
_{2}nanosheets for high-permeance hydrogen separation. Nanoscale**2015**, 7, 17649–17652. [Google Scholar] [CrossRef] [PubMed] - Achari, A.S.; Eswaramoorthy, S.M. High performance MoS
_{2}membranes: Effects of thermally driven phase transition on CO_{2}separation efficiency. Energy Environ. Sci.**2016**, 9, 1224–1228. [Google Scholar] [CrossRef] - Wang, X.; Chi, C.; Zhang, K.; Qian, Y.; Gupta, K.M.; Kang, Z.; Jiang, J.; Zhao, D. Reversed thermo-switchable molecular sieving membranes composed of two-dimensional metal-organic nanosheets for gas separation. Nat. Commun.
**2017**, 8, 14460. [Google Scholar] [CrossRef] - Peng, Y.; Li, Y.; Ban, Y.; Jin, H.; Jiao, W.; Liu, X.; Yang, W. Metal-organic framework nanosheets as building blocks for molecular sieving membranes. Science
**2014**, 346, 1356–1359. [Google Scholar] [CrossRef] - Naguib, M.; Kurtoglu, M.; Presser, V.; Lu, J.; Niu, J.; Heon, M.; Hultman, L.; Gogotsi, Y.; Barsoum, M.W. Two-dimensional nanocrystals produced by exfoliation of Ti
_{3}AlC_{2}. Adv. Mater.**2011**, 23, 4248–4253. [Google Scholar] [CrossRef] - Ghidiu, M.; Lukatskaya, M.R.; Zhao, M.Q.; Gogotsi, Y.; Barsoum, M.W. Conductive two-dimensional titanium carbide ‘clay’ with high volumetric capacitance. Nature
**2014**, 516, 78–82. [Google Scholar] [CrossRef] - Ding, L.; Wei, Y.; Wang, Y.; Chen, H.; Caro, J.; Wang, H. A two-dimensional lamellar membrane: MXene nanosheet stacks. Angew. Chem. Int. Ed.
**2017**, 56, 1825–1829. [Google Scholar] [CrossRef] - Fan, Y.; Wei, L.; Meng, X.; Zhang, W.; Yang, N.; Jin, Y.; Wang, X.; Zhao, M.; Liu, S. An unprecedented high-temperature-tolerance 2D laminar MXene membrane for ultrafast hydrogen sieving. J. Membr. Sci.
**2019**, 569, 117–123. [Google Scholar] [CrossRef] - Ren, C.E.; Hatzell, K.B.; Alhabeb, M.; Ling, Z.; Mahmoud, K.A.; Gogotsi, Y. Charge-and size-selective ion sieving through Ti
_{3}C_{2}T_{x}MXene membranes. J. Phys. Chem. Lett.**2015**, 6, 4026–4031. [Google Scholar] [CrossRef] [PubMed] - Wang, J.; Chen, P.; Shi, B.; Guo, W.; Jaroniec, M.; Qiao, S.Z. A regularly channeled lamellar membrane for unparalleled water and organics permeation. Angew. Chem. Int. Ed.
**2018**, 57, 6814–6818. [Google Scholar] [CrossRef] [PubMed] - Li, W.; Zheng, X.; Dong, Z.; Li, C.; Wang, W.; Yan, Y.; Zhang, J. Molecular dynamics simulations of CO
_{2}/N_{2}separation through two-dimensional graphene oxide membranes. J. Phys. Chem. C**2016**, 120, 26061–26066. [Google Scholar] [CrossRef] - Li, L.; Zhang, T.; Duan, Y.; Wei, Y.; Dong, C.; Ding, L.; Qiao, Z.; Wang, H. Selective gas diffusion in two-dimensional MXene lamellar membranes: Insights from molecular dynamics simulations. J. Mater. Chem. A
**2018**, 6, 11734–11742. [Google Scholar] [CrossRef] - Lito, P.F.; Cardoso, S.P.; Rodrigues, A.E.; Silva, C.M. Kinetic modeling of pure and multicomponent gas permeation through microporous membranes: Diffusion mechanisms and influence of isotherm type. Sep. Purif. Rev.
**2014**, 44, 283–307. [Google Scholar] [CrossRef] - Eliseev, A.A.; Poyarkov, A.A.; Chernova, E.A.; Eliseev, A.A.; Chumakov, A.P.; Konovalov, O.V.; Petukhov, D.I. Operando study of water vapor transport through ultra-thin graphene oxide membranes. 2D Mater.
**2019**, 6, 035039. [Google Scholar] [CrossRef] - Knudsen, M. Die Gesetze der Molekularströmung und der inneren Reibungsströmung der Gase durch Röhren. Ann. Phys.
**1909**, 333, 75–130. [Google Scholar] [CrossRef] - Xiao, J.; Wei, J. Diffusion mechanism of hydrocarbons in zeolites—I. Theory. Chem. Eng. Sci.
**1992**, 47, 1123–1141. [Google Scholar] [CrossRef] - The Math Works Inc. MATLAB 7.0.4 [Computer Program]; The Math Works Inc.: Natick, MA, USA, 2005. [Google Scholar]
- Ibrahim, A.; Lin, Y.S. Gas permeation and separation properties of large-sheet stacked graphene oxide membranes. J. Membr. Sci.
**2018**, 550, 238–245. [Google Scholar] [CrossRef] - Gao, G.; O’Mullane, A.P.; Du, A. 2D MXenes: A new family of promising catalysts for the hydrogen evolution reaction. ACS Catal.
**2016**, 7, 494–500. [Google Scholar] [CrossRef] - Seh, Z.W.; Fredrickson, K.D.; Anasori, B.; Kibsgaard, J.; Strickler, A.L.; Lukatskaya, M.R.; Gogotsi, Y.; Jaramillo, T.F.; Vojvodic, A. Two-dimensional molybdenum carbide (MXene) as an efficient electrocatalyst for hydrogen evolution. ACS Energy Lett.
**2016**, 1, 589–594. [Google Scholar] [CrossRef]

**Figure 1.**Schematic diagram of the two kinds of gas transport model through different nanochannels within the MXene lamellar membranes.

**Figure 2.**SEM images of the cross-section of MXene (Ti

_{3}C

_{2}T

_{x}) (

**a**), the cross-section of MXene membrane (

**b**), external surface of MXene membrane (

**c**), and the cross-section of MXene membrane (low magnification) (

**d**).

**Figure 5.**Comparison of the experimental data and the model predictions of the temperature dependent nitrogen permeance through the MXene lamellar membrane.

**Figure 6.**Comparison of the experimental data and the model predictions of temperature dependent hydrogen permeance through the MXene lamellar membrane.

**Figure 7.**The gas permeance and selectivity (H

_{2}/N

_{2}) through the MXene lamellar membrane as a function of temperature.

**Figure 8.**Temperature dependent fractional diffusion of H

_{2}to total transport from Knudsen diffusion and molecular sieving. Note: Data point represents experimental data while the continuous line indicates model predictions.

**Figure 9.**The effect of the average pore diameter of straight nanochannels on gas permeance at 295 K.

**Figure 10.**Average pore diameter of the straight nanochannels dependent fractional diffusion to total transport of Knudsen diffusion and molecular sieving for H

_{2}at 295 K.

**Figure 12.**Effects of the MXene membrane thickness on the transport fractions from Knudsen diffusion and molecular sieving to total gas transport operated at 295 K.

**Table 1.**Experimental results of the supported 800-nm-thick MXene lamellar membrane for temperature dependent nitrogen and hydrogen permeances measured from gas mixture separation performance tests [20].

Testing Temperature (K) | Permeance of N_{2} (10^{−8} mol m^{−2} s^{−1} Pa^{−1}) | Permeance of H_{2} (10^{−7} mol m^{−2} s^{−1} Pa^{−1}) | Separation Factor (H_{2}/N_{2}) |
---|---|---|---|

295 | 1.74 | 2.34 | 13.45 |

343 | 1.13 | 2.22 | 19.65 |

423 | 0.84 | 2.11 | 25.11 |

443 | 0.74 | 2.09 | 28.24 |

473 | 0.64 | 2.07 | 32.34 |

503 | 0.62 | 2.06 | 33.22 |

533 | 0.59 | 2.05 | 34.74 |

563 | 0.55 | 2.03 | 36.90 |

593 | 0.50 | 2.03 | 40.60 |

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Jin, Y.; Fan, Y.; Meng, X.; Zhang, W.; Meng, B.; Yang, N.; Liu, S.
Theoretical and Experimental Insights into the Mechanism for Gas Separation through Nanochannels in 2D Laminar MXene Membranes. *Processes* **2019**, *7*, 751.
https://doi.org/10.3390/pr7100751

**AMA Style**

Jin Y, Fan Y, Meng X, Zhang W, Meng B, Yang N, Liu S.
Theoretical and Experimental Insights into the Mechanism for Gas Separation through Nanochannels in 2D Laminar MXene Membranes. *Processes*. 2019; 7(10):751.
https://doi.org/10.3390/pr7100751

**Chicago/Turabian Style**

Jin, Yun, Yiyi Fan, Xiuxia Meng, Weimin Zhang, Bo Meng, Naitao Yang, and Shaomin Liu.
2019. "Theoretical and Experimental Insights into the Mechanism for Gas Separation through Nanochannels in 2D Laminar MXene Membranes" *Processes* 7, no. 10: 751.
https://doi.org/10.3390/pr7100751