Predicting adsorption selectivities from pure gas isotherms for gas mixtures in metal–organic frameworks

We perform Grand Canonical Monte Carlo simulations on a lattice of Mg2+ sites (GCMC) for adsorption of four binary A/B mixtures, CH4/N2, CO/N2, CO2/N2, and CO2/CH4, in the metal–organic framework Mg2(2,5-dioxidobenzedicarboxylate), also known as CPO-27–Mg or Mg–MOF-74. We present a mean field co-adsorption isotherm model and show that its predictions agree with the GCMC results if the same quantum chemical ab initio data are used for Gibbs free energies of adsorption at the individual sites and for lateral interaction energies between the same, A⋯A and B⋯B, and unlike, A⋯B, adsorbed molecules. We use both approaches to test the assumption underlying Ideal Adsorbed Solution Theory (IAST), namely approximating A⋯B interaction energies as the arithmetic mean of A⋯A and B⋯B interaction energies. While IAST works well for mixtures with weak lateral interactions, CH4/N2 and CO/N2, the deviations are large for mixtures with stronger lateral interactions, CO2/N2 and CO2/CH4. Motivated by the theory of London dispersion forces, we propose use of the geometric mean instead of the arithmetic mean and achieve substantial improvements. For CO2/CH4, the lateral interactions become anisotropic. To include this in the geometric mean co-adsorption model, we introduce an anisotropy factor. We propose a protocol, named co-adsorption mean field theory (CAMT), for co-adsorption selectivity prediction from known (experiment or simulation) pure component isotherms which is similar to the IAST protocol but uses the geometric mean to approximate mixed pair interaction energies and yields improved results for non-ideal mixtures.


S1. Mean Field approximation for lateral interactions in a multicomponent mixture.
Let us consider a gas mixture with m number of components and let c, cʹ denote the indices of the gas components. At equilibrium, the Gibbs free energy of adsorption for c-th component of the gas mixture for a given coverage (θc) can be calculated from the partial pressure, Pc of that component: Here, is the ab initio interaction energy for a particular pair of adsorbed molecules of components c and cʹ, respectively.
The first term on the right-hand side of the Eq. (S2), ΔGc is the coverage independent Gibbs adsorption free energy and is related to the zero coverage adsorption equilibrium constant, . ΔGc is the binding strength of gas component c to the isolated adsorption sites. The second and third terms of the right-hand side of Eq. (S2) are the lateral interaction energies between the gas molecules of same (cc, pure gas term) and different components (ccʹ, cʹ ≠ c, mixing term), respectively. The last term is the configurational entropy (Langmuir entropy term). Here, it is assumed that molecules of different components are distributed randomly on the surface, which was also previously assumed by Fowler and Guggenheim 2 . This assumption makes the configurational entropy term, i.e., the last term in Eq. (S2), dependent only on the total coverage, which is the sum of coverages of all components, i.e., .
Comparing the right-hand sides of the Eq. (S1) and (S2), we get, The mean-field equilibrium constant for the c-th mixture component is, Replacing the first three terms of the right-hand side of Eq. (S4) using the expression from Eq. (S5) we obtain, The Eq. (S6) is a system of m equations. Summing over all components on both side of the equation we obtain,  Figure S1 for the results for a 10:90 CO2/N2 mixture. Figure S1, right panel, shows that selectivities of CO2/N2 are almost converged after the first iteration. In Figure S2, the CO2 selectivities obtained from converged ab initio CMF calculations (lines) are compared with converged ab initio GCMC simulations (red triangles). CMF results, regardless of fugacity corrected or not corrected, are in excellent agreement with the reference GCMC results. A small deviation between the selectivities obtained from MF model without fugacity correction and GCMC simulation is noticeable when total pressure is higher than 2 atm. The reason is that, Redlich-Kwong equation of state 3 predicts a smaller fugacity coefficient for CO2 than for N2. Moreover, at pressures higher than 2 atm, the ratio of Redlich-Kwong fugacity coefficients for CO2 and N2 decreases with increasing pressure, making CO2 selectivities almost constant with increasing pressure.

S2. Derivation of the geometric mean mixing rule
The Lennard-Jones potential between two molecules: A and B can be expressed as with RAB and σAB denoting the center of mass distance and collision diameter (i.e. distance when the inter-atomic potential is zero) between the molecule A and B , respectively. The depth of the potential well, εAB, is the interaction energy of the pair of the molecules at the potentialminimum, which appears at inter-molecular distance, 2 1/6 σAB. In molecular simulations, the σ and ε parameters for AB pairs are routinely estimated from the average of those parameters for the AA and BB pairs. The collision diameter for the AB pair is chosen as the arithmetic mean of the collision diameters of the AA and BB pairs. 5 (S10) This rule, which is known as Lorentz combination rule, is mathematically correct for hard sphere molecules.
The fundamental origin of the attractive part of the Lennard-Jones potential is the London dispersion interaction. 5 (S11) Comparing the coefficients of in Eqs. (S9) and (S11) and utilizing the well-known expression of C6AB from London approximation we get, where α and I represent the polarizabilities and the ionization potentials of the molecules, respectively. By eliminating α from the above equation, Hudson and McCoubrey obtained the following relation, 6 a a e s = where Lorentz combination rule, Eq.(S10), is used to approximate the collision diameter, σAB, for unlike gas molecules.
The b-parameter of the van der Waal's equation of state, with V̅ denoting the molar volume of the gas, can be obtained from critical temperature and pressure measurements of the considered gases and they are available in the literature 7 . If molecules are approximated as hard spheres then it can be shown that the b-parameters are proportional to the volume of a single molecule (VA), 8 (S15) with NAvo denoting the Avogadro's number. Inserting this result into Eq. (S13) we obtain, where the bÃB and ĨAB are the correction factors due to different size and different ionization energies of the considered molecules. They have the following expressions, (S17) which can be calculated by using the available experimental data for the van der Waal's bparameters 7 and the ionization potentials (I) 9 of the considered gas molecules.
Comparison of the van der Waal's b parameters for the different gases confirms the size of the molecules are very similar and consequently the correction factors b̃ are close to unity. All the molecules considered here have also very similar ionization energies as all of their highest occupied molecular orbitals (HOMO) are originating from 2p-levels. Consequently, the factors Ĩ are also close to unity.   selectivities of CO(10%) and CH4(90%) over N2 as a function of total pressure. Right: CO and CH4 selectivities as a function of gas phase composition. S11 Table S3 shows lateral interaction energies calculated for two different models: for the full periodic structures ("In MOF") and the isolated pairs of adsorbed molecules cut from these periodic structures ("Isolated"). For comparison of these two models we have approximated the periodic structures CCSD(T) results with the hybrid MP2:DFT+D + DCCSD(T) method as applied before for calculation of adsorption energies. 1, 10,11 For that finite size model systems (see Figure S5)  Details of the calculations are the same as used in our previous works. 1, 10, 11 Figure S5. Example of the Model system adopted for the hybrid calculations in case of adsorption pure N2.

S12
The last column in Table S3, |Hybrid -Isolated CCSD(T)|, shows the lateral interaction energy differences between isolated pair of molecules (in the gas phase) and the same pair in the MOF. The average of these energies is 0.12 kJ/mol. Also, the average lateral interaction energies of the short and long mixed pairs match very well (the average absolute difference is 0.11 kJ/mol) with the geometric mean of the interaction energies of the corresponding pure gas molecules (the "GM -S, L av." rows in Table S3). In this study, we use the isolated pair lateral interaction energies which is consistent with the idea of GCMC simulations on a lattice of sites and with the mean field approximation.
Let us assume that three calculations for the periodic systems are performed with all sites occupied with A-type molecules, with B-type molecules, or alternating with A and B type molecules yielding the energies E (A-A), E(B-B) and E(A-B). A two-body expansion yields (S19) (S20) Where, e.g.

The arithmetic mean of the total energies E(A-A)/A-A and E(B-B)/B-B is
(S21) and the mixing energy becomes There is an extra term D originating from the change of the molecule-surface interactions in Anisotropy factor. for isolated interaction pairs with two different "comput." methods, CCSD(T)/CBS and PBE+D2/QZVP, and the resulting mixed pair interaction terms and mixing energies .

S5. Ideal Adsorption Solution Theory (IAST)
Ideal adsorption solution theory (IAST) 12,13 is the most widely used model for prediction of co-adsorption from pure gas adsorption isotherms. It assumes that the adsorbed phase behaves like an ideal solution of the adsorbed components. Consequently, the equilibrium between the adsorbed phase and gas phase can be described analogously to Raoult's law, according to which the partial vapor pressure of a gas component c, Pc, in an ideal liquid mixture can be calculated from the composition of the liquid mixture, xc, where the vapor pressure of the pure liquid is P c°. The partial pressure can also be expressed as a product of gas phase composition (yc) and total pressure (P).
The spreading pressure of a gas component c can be calculated by the following equation, where A is the area of the surface. πc and are the spreading pressure and the pure gas adsorbed amount of component c, respectively. The equilibrium between the adsorbed components is attained when all the pure components have the same spreading pressure that is equal to the spreading pressure of the adsorbed mixture (π) itself. Considering a binary mixture (c = 1 or 2) this can be written as, The equilibrium pressures of the individual components, P c°, which are unknown, can be determined from the known composition of the gas (yc) and the total pressure (P) by applying Eq. (S25). (S28) To simplify the notation, let x and y represent the adsorbed and gas phase mole fraction of the first component (c = 1), respectively. Substituting the expressions of π and P° from Eqs.  ) and (20), for fitting our single component data points obtained from the ab initio lattice GCMC simulations.
After obtaining the adsorbed phase composition the total adsorbed amount in the mixture, q, is calculated according to the following equation, where q c° is the amount adsorbed in the pure gas isotherm at the same spreading pressure and temperature as that of the adsorbed mixture.