Figure 4. (a) Permeance, (b) ideal He/CH4selectivity as a function of pore diameter. (c) Effect of different pore diameters on He/CH4 separation performance for Ti4C3. The shaded area refers to the Ti4C3 nanopores with high separation performance. Note: The He/CH4 selectivity of the MXene nanopores with diameter of 2.13 Å is estimated to be greater than 550, because there was still no one CH4 molecule can be passed through the MXene nanopores with simulation time of 500 ns.
At small given d (~2 Å for He, and ~3 Å for CH4), the gas permeance difference among four kinds of MXene nanopores is quite significant, but vanish as d increases. This indicates that the nanopore’s structure details affect the gas transportation significantly for molecular sieving mechanism, but not for Knudsen diffusion. As for all studied MXene nanopores, the permeance of He and CH4both increase with d . They both increase sharply at some certaind value (d -threshold), then increase slowly (Figure 4a and 4b ). The d- threshold is 2~3 Å for He, and 3~4 Å for CH4, well agreeing with their kinetic diameters (He = 2.6 Å, CH4 = 3.8 Å). It is because of the sharp increasement of CH4 permeance at d~ 4 Å that, the He/CH4 selectivity (S He/CH4) drop dramatically to the Knudsen selectivity. For instance, when the size of Ti4C3 nanopores (2.92 Å) are smaller than the CH4 d -threshold region, they exhibit excellent He/CH4 selectivity (S He/CH4) of 62.5 and remarkably high He permeance of 1.34×106 GPU. However, when Ti4C3 nanopore’s d increase to 4.2 Å, the separation factor of He/CH4 significantly decreases to Knudsen selectivity (He/CH4 Knudsen selectivity = 2) due to CH4 permeance significantly rising. Such phenomenon is also observed with the other three kinds of MXene nanopores (Figures S3-S5 ), indicating the transport mechanism of CH4 through MXene nanopores changes whend increase above d- threshold of CH4.
The sharp increasement of gas permeance when d increase to above gas molecule’s kinetic diameter could be called the cut-off effect47, 59. This usually indicates the transport mechanism through the nanopore may change, and would be illustrated with Ti4C3 nanopore as an example. The PMF curves of gas passing through the nanopore were calculated based on the density of gas molecule along z direction.
\(PMF=\ -k_{B}T\times ln\frac{\rho_{z}}{\rho_{0}}\) (1)
where kBis the Boltzmann constant,T is temperature of simulations, ρ z and ρ 0refer to the density of gas molecules along z direction and bulk phase, respectively. Figure 5a and Figure S6 show that, the free energy barrier (ΔF ) decreases with d increasing, indicating gas molecules should pass through large nanopores easily. Furthermore, ΔF shows two trends as d varying: first, whend is smaller than the gas kinetic diameter (d gas), ΔF changes drastically. Second, when d is larger thand gas, ΔF changes slowly, with He/CH4 selectivity close to the Knudsen selectivity. In fact, the energy barrier calculated by Equation 1 for d> d gas is mainly contributed by the steric confinement of the nanopores, that is, in the membrane region (15.0 < z < 15.8 nm for Ti4C3), the gas molecules only exist in the nanopores (not in the Ti4C3material), leading to a substantial reduction of gas molecules’ translational entropy.
The above free energy barriers were further employed to calculateS CH4/He of MXene nanopores with the transition state theory (TST), with\(S_{\text{TST}}=\ \sqrt{M_{j}}/\sqrt{M_{i}}exp(E_{i}-E_{j}/k_{B}T)\),55, 60 where Mi , Mj are the molar masses of gas i and j , ΔEiand ΔEj refer to free energy barriers for gas molecules i and j passing through the nanopore, iand j refer to He and CH4 in this study, respectively. As for the four kinds of MXene nanopores with d~ 3.0 Å, the selectivity calculated with the TST (STST ) well agree with those calculated from gas permeance (SP ). For instance, the energy barrier for CH4 passing through the Ti4C3 nanopore (d = 2.92 Å) is 8.33 kJ/mol higher than He (Figure 5b ), yieldingSTST of 57.6, very close toSP of 62.5. Interestingly, density (ρ ) profiles of EMD could also be employed to predict the ratio of permeance for the same gas molecule passing through different nanopores, that is,\(\frac{P_{i}}{P_{j}=}\frac{\rho_{i}}{\rho_{j}}\), where P andρ are the permeance and the minimum density (along the density profile) for an gas, i andj refer to two different nanopores, while for the sake of simplicity, j usually refers to the smallest nanopore that the gas molecule could pass through (as shown in Figure S7 ). The\(\frac{P_{i}}{P_{j}}\) predicted with the above equation (usingρi &ρj ) agrees quite well with that calculated with the permeance from NEMD (as shown inFigure 4a ), no matter the gas is He or CH4(Figure S7 ). Therefore, while EMD cannot be employed to calculate He/CH4 selectivity, it can still be utilized to predict the gas permeance.