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.