Adsorption-based membranes for air separation using transition metal oxides

In this work, we use computational modeling to examine the viability of adsorption-based pore-flow membranes for separating gases when a purely size-based separation strategy is ineffective. Using molecular dynamics simulations of O2 and N2, we model permeation through a nanoporous graphene membrane. Permeation is assumed to follow a five-step adsorption-based pathway, with desorption being the rate-limiting step. Using this model, we observe increased selectivity between O2 and N2, resulting from increased adsorption energy differences. We explore the limits of this strategy, providing an initial set of constraints that need to be satisfied to allow for selectivity. Finally, we provide a preliminary exploration of some transition metal oxides that appear to satisfy those conditions. Using density functional theory calculations, we confirm that these oxides possess adsorption energies needed to operate as adsorption-based pore-flow membranes. These adsorption energies provide a suitable motivation to examine adsorption-based pore-flow membranes as a viable option for air separation.


Lennard-Jones potential
In this work, all the interactions were modeled using the Lennard-Jones (LJ) potential with LAMMPS. 1 A description of this potential is found in [2]. The form of the LJ 12-6 potential used is given by

The pore in NPG membrane
Since the selectivity isn't based on molecular sieving, we employed an NPG membrane with a pore large enough to allow unhindered motion of both gases. In the energy profile, this would ensure that the translocation step in the adsorbed-phase pathway wouldn't have the highest energy barrier.

Obtaining adsorption energies from LJ parameters and resulting permeability and selectivity
A setup as shown in Fig S2a was used where a dimer of O 2 and N 2 were separately placed vertically at various distances from the membrane surface and the potential energy was tabulated. The potential energy plots are depicted in Fig S2b and the minimum energies obtained for each of the LJ parameter are indicated in Table S1. The selectivity as a function of the LJ parameters is plotted in Fig S3. Permeability was calculated using equation S2 Here, J is the flow rate, is the partial pressure, A is the surface area, l is the thickness, is the

Theoretical framework for permeability and selectivity
The theoretical framework used to obtain selectivity is by adopting the methodology used by Sun et al. 19 Flow rate expressed as a function of permeability is given by Equation S3 Where J is flow rate, A is the surface area, is the difference of partial pressure and is the ∆ permeability. , in turn, can be expressed as a function of the initial pressure , number of ∆ molecules in the permeate region N, number of molecules adsorbed , and total number of molecules and is given by Equation S4.
Combining equation S3 and S4 gives us equation S5, Integrating equation S5, we get the following: is in the following form and all our calculations are fitted to the equation S7.

Pressure calculation
Pressure is calculated using the ideal gas law as shown in equation S8.
Where p, N, k B , T, and V represent the pressure, number of molecules, the Boltzmann constant, temperature, and volume respectively. For obtaining the initial pressure in the feed side, we use N = 200 molecules (100 O 2 and 100 N 2 ), T = 500 K, and V = 31.5*36*175*10 -30 m 3 . This results in an initial pressure of around 68.6 atm in the feed side.

Density Functional theory calculations
Density Functional Theory (DFT) was employed using Vienna Ab initio Simulation Package (VASP) [20][21][22] to obtain the adsorption energies of oxygen and nitrogen molecules on two transition metal oxides: α-Fe 2 O 3 and Co 3 O 4 . The projector augmented wave (PAW) method 21 was used and the exchange-correlation effects are described by the generalized gradient approximation (GGA) as developed by Perdew, Burke and Ernzerhof (PBE). 24 Since these materials are magnetic, spin polarized calculations were performed. In order to capture the physisorption of N 2 molecules, van der Waals (vdW) forces were incorporated using the D3 correction method of Grimme et al. 25 We used an energy cutoff of 650 eV for both systems. Monkhorst-Pack k-points meshes of 4x4x2 (4x4x1) and 2x2x2 (2x3x1) were used for bulk (surface) Fe 2 O 3 and Co 3 O 4 systems, respectively. To account for correlations in the 3d orbitals in Fe 2 O 3 and Co 3 O 4 , we used Hubbard U-J=4 26 and U-J=3 27 parameters respectively in the Dudarev approach. 28 All the geometry relaxation was converged to within 1x10 -5 eV of the total energy. For surface relaxations, it was found to be advantageous to relax the structure in stages. A rough relaxation to 1x10 -4 eV of the total electronic energy and 0.03 eV/Å of total force was followed by a refined relaxation to 1x10 -6 eV of the total electronic energy and 0.001 eV/ Å of total force. For asymmetric surfaces, a dipole correction 29,30 was added at the end.

Fe 2 O 3
A lot of theoretical study have been done for the (0001) surface. 6,7 The bulk and surface properties as well as the parameters used in this study have been listed in  The bulk and surface properties as well as the parameters used in this study have been listed in Table S3. The lattice parameters match closely with other theoretical studies as well as experimental ones.