Arpan
Kundu‡
^{a},
Kaido
Sillar‡
^{ab} and
Joachim
Sauer
*^{a}
^{a}Humboldt Universität zu Berlin, Institut für Chemie, Unter den Linden 6, 10099 Berlin, Germany. E-mail: js@chemie.hu-berlin.de
^{b}University of Tartu, Institute of Chemistry, Ravila 14a, 50411, Tartu, Estonia
First published on 6th December 2019
We perform Grand Canonical Monte Carlo simulations on a lattice of Mg^{2+} sites (GCMC) for adsorption of four binary A/B mixtures, CH_{4}/N_{2}, CO/N_{2}, CO_{2}/N_{2}, and CO_{2}/CH_{4}, in the metal–organic framework Mg_{2}(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, CH_{4}/N_{2} and CO/N_{2}, the deviations are large for mixtures with stronger lateral interactions, CO_{2}/N_{2} and CO_{2}/CH_{4}. 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 CO_{2}/CH_{4}, 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.
The selection and rational design of improved materials with optimized properties for a specific separation target requires reliable predictions of co-adsorption isotherms and adsorption selectivities. Because isotherm measurements are much more demanding for mixtures than for single components and require specific equipment,^{13,14} more than 50 years after its invention, in the vast majority of cases, Ideal Adsorbed Solution Theory (IAST)^{15} is still used to predict co-adsorption isotherms for gas mixtures from pure gas data.^{16,17} Also when simulation methods are used, the availability of methods for predicting mixture isotherms from pure components will speed up computational screening for optimal materials in separations, e.g., ref. 18.
IAST assumes that mixture components behave like an ideal solution in the adsorbed phase – an approximation that is not always valid, for example when one component adsorbs more strongly than the other^{19} or is of very different size than the other.^{20} The ideal behavior of the adsorbed phase implies that the mixing energy is zero, which also defines the underlying approximation for the lateral interactions – the mixing energy can be zero only if the intermolecular interactions between the molecules of unlike components, E_{AB}, are the average (arithmetic mean, AM) of the interactions between molecules of the individual mixture components A and B,^{21,22}
(1) |
Here, we propose an improved method for predicting mixture isotherms from pure gas data that approximate the interaction energy between unlike molecules as the geometric mean (GM) of the interactions between molecules of the individual components,
(2) |
There are other ways of dealing with non-ideal mixing behavior, e.g., Real Adsorbed Solution Theory (RAST)^{26–28} makes use of activity coefficients to take lateral interactions into account, but requires experimental or simulated co-adsorption data, whereas we focus here on predictions based on pure gas components only. RAST is typically applied to mixtures with strong adsorbate–adsorbate interactions, comparable in strength with adsorbate–surface interactions, e.g., water–alcohol mixtures on microporous silica.^{28,29} Systems with strong lateral interactions are beyond the scope of our study.
We consider four binary gas mixtures: CO_{2}/CH_{4} and N_{2}/CH_{4} relevant for the natural gas upgrade, CO_{2}/N_{2} relevant for flue gas separation for CCS, and CO/N_{2} for removal of toxic CO from gas mixtures (e.g., burnt air) which might be relevant for gas mask applications and syngas production and purification. As an adsorbent, we consider Mg_{2}(dobdc)^{30–32} (dobdc^{4−} = 2,5-dioxidobenzendicarboxylate), a MOF, also known as CPO-27–Mg and Mg–MOF-74, that is considered especially promising for CO_{2} adsorption because of its high concentration of accessible strong adsorption sites, “under-coordinated” (five-fold coordinated) Mg^{2+} ions. To these sites, CH_{4}, N_{2}, CO, and CO_{2} bind with 26, 29, 39, and 46 kJ mol^{−1}, respectively (Table 1). Comparatively, the average lateral adsorbate–adsorbate energies are small, −0.55, −0.35, −0.34, and −2.81 kJ mol^{−1}, respectively; only for CO_{2}, they exceed the thermal energy at 298 K (−2.5 kJ mol^{−1}).
Ab initio ^{ } | MF fit of ads. constant | MF fit of free energy | ||||||
---|---|---|---|---|---|---|---|---|
A | ΔE_{A} | ΔG_{A} | E _{AA} = E^{av}_{AA} | ^{ } | ^{ } | ΔG_{A} | −L_{AA}RT^{f} | |
a Zero point vibrational energy contributions are included. b Used for IAST, Section 4.1. c Used for CMFfit, Section 4.2. d Ref. 34–36. e Calculated using eqn (21) from . f See eqn (22), –L_{AA}RT is comparable with E_{AA} because N = 2 and E^{av}_{AA} = E_{AA} as there is no anisotropy in pure gas lateral interactions. g After fitting with continued fraction representation, −2.63 kJ mol^{−1}, see Section S6 in the ESI. | ||||||||
CO | −39.0 | −1.11 | −0.34 | −1.11 | 1.5682 | −0.34 | −1.11 | −0.33 |
N_{2} | −29.2 | 4.16 | −0.35 | 4.15 | 0.1872 | −0.33 | 4.16 | −0.32 |
CH_{4} | −25.8 | 3.90 | −0.55 | 3.87 | 0.2097 | −0.46 | 3.89 | −0.46 |
CO_{2} | −45.9 | −9.22 | −2.81 | −9.38 | 44.14 | −3.24^{g} | −9.25 | −2.70 |
Although with increasing pressure also the adsorption sites at the dobdc linker molecules will become populated (our previous study^{33} on pure CO_{2} adsorption has shown that this will be the case for pressures higher than 0.05 atm) the present study considers a homogeneous lattice of Mg^{2+} adsorption sites as a model that captures essential features of MOFs with open metal ion sites.
For the CH_{4}/N_{2}, CO/N_{2}, CO_{2}/N_{2}, and CO_{2}/CH_{4} mixtures, we first predict co-adsorption isotherms and adsorption selectivities from quantum chemical ab initio calculations employing two methods:
(i) Grand Canonical Monte Carlo (GCMC) simulations on a lattice of adsorption sites which use Gibbs free energies of adsorption as input for gas molecules at isolated individual sites and calculate the interaction energies for each configuration explicitly,^{33} both generated from quantum chemical ab initio calculations with chemical accuracy (4 kJ mol^{−1} or better).^{34–36} These results will serve as a benchmark for any other method used in this study.
(ii) Competitive mean-field (CMF) isotherm model that has clearly separated and physically meaningful parameters for molecule-surface and molecule–molecule (lateral) interactions.^{34–36} Before, we have used the mean field (Bragg–Williams) model for pure gases;^{34–36} here we extend it to mixture co-adsorption by including the interaction energy term between molecules of different gases in addition to the lateral interaction energies between the same gas molecules.^{37}
For pure gases, we have shown before that MF theory which assumes an average value for lateral interaction energies not only yields isotherms in close agreement with experiments,^{34–36} but also in close agreement with the results of GCMC simulations on a lattice of adsorption sites.^{33} Here, for mixtures, we find the same level of agreement between the results of the analytical CMF equations and the benchmark GCMC simulations with the same ab initio data as the input.
Next, we use both GCMC and CMF co-adsorption isotherms and selectivities to test the AM approximation and the CMF results to test both the AM and GM approximations. For the adsorbed mixtures with very weak lateral interactions, CH_{4}/N_{2} and CO/N_{2}, we find that mixing energies are indeed negligible (less than 0.2 kJ mol^{−1}) and, consequently, AM mixing is a good approximation and GM mixing has no advantage. For mixtures with stronger lateral interactions, CO_{2}/N_{2} and CO_{2}/CH_{4}, isotherms obtained with AM mixing deviate substantially from the GCMC benchmarks and GM mixing is always an improvement, whereas for CO_{2}/N_{2} with GM close agreement with the GCMC benchmark is reached, and for CO_{2}/CH_{4}, an “anisotropy” factor is needed to account for different mixed pair interactions in different directions of the adsorbate layer.
Based on the insight gained, we propose an improved protocol for predicting co-adsorption selectivities from measured (or calculated) pure gas adsorption isotherms. As with the classical IAST protocol, the starting point is fitting the pure component adsorption data with an isotherm model. For this we use the mean-field (Bragg–Williams) isotherm model. Co-adsorption data are then obtained with the CMF equations, using different mixing rules for unlike pair interactions. When AM mixing is used, it reproduces IAST,^{38} whereas improved results are obtained when GM mixing is applied. The final step is inclusion of an anisotropy factor which needs an atomistic model of adsorption structures. This is not an obstacle because structural optimization using force fields or density functional theory (DFT) has become routine.
This article is organized as follows. Section 2 describes ab initio calculations of Gibbs free energies for adsorption of N_{2}, CO, CO_{2}, and CH_{4} on Mg^{2+} sites and of molecule–molecule interaction energies in the adsorbate layers. Section 3 presents ab initio predictions of co-adsorption isotherms and selectivities using GCMC simulations and CMF equations including tests of the AM and GM approximations. Section 4 presents our geometric mean model for predicting co-adsorption isotherms and selectivities from fitted pure gas data as an alternative to IAST.
For three of the four binary mixtures, Fig. 1 shows the adsorption structures with full occupation of the Mg^{2+} sites taken from previous PBE+D2 structural optimizations under periodic boundary conditions, see ref. 33–36. The figure shows the relevant interactions in the a,b-plane of the hexagonal pore. The distances in the z-direction, a unit cell length of 689 pm, are much larger and the corresponding interactions can be neglected. For pairs of adsorbed molecules taken from these periodic structures the lateral interaction energies are calculated using Coupled Cluster (CC) theory with complete basis set (CBS) extrapolation (CCSD(T)/CBS(D,T)). The pairs are isolated, i.e., the framework is not present in these calculations.
Fig. 1 Adsorbate structures of CH_{4}/N_{2} (A) CO_{2}/N_{2} (B), and CO_{2}/CH4 (C) mixtures in Mg_{2}(dobdc). Distances between the centers of mass of the adsorbed molecules are given in pm. |
For the CO_{2}/CH_{4} mixture the “mixed” term for the lateral interaction energies between unlike molecules is taken from our previous work,^{33} whereas for the other mixtures the CO⋯N_{2}, CH_{4}⋯N_{2}, and CO_{2}⋯N_{2} interaction energies are calculated in this work. Table 2 shows the results. All Mg^{2+} adsorption sites are equivalent and for adsorbed pure gases all interactions between adsorbed molecules are also equivalent, but with mixed adsorbates “symmetry breaking” occurs. The distances and the interaction energies between neighboring unlike molecules depend on the direction of the interactions, i.e. they are anisotropic. When a pair of unlike molecules gets closer in one direction, the lateral interaction between them gets stronger, whereas the interactions with a molecule in opposite direction gets more distant with a weaker interaction, see Fig. 1 and Table 2.
A/B | CH_{4}/N_{2} | CO/N_{2} | CO_{2}/N_{2} | CO_{2}/CH_{4} | |
---|---|---|---|---|---|
a Corresponds to IAST. b See eqn (9). c Effective lateral interaction energy, E^{GM}_{AB}/2, see Section 3.5, eqn (16) and (17). d Effective energy of mixing. e Corresponds to IAST; mixing energy is 0 kJ mol^{−1}. f Effective L_{AB}-parameter, L^{GM}_{AB}/2, see Section 4.4, eqn (30). g L parameter for mixing, defined as ΔL_{mix} = L_{AB}−(L_{AA} + L_{BB})/2; see Table 1 for L_{AA} and L_{BB} parameters obtained from linear MF fitting. h In parenthesis, effective energy of mixing after considering f_{AB} = 0.5. | |||||
E _{AB} | Short | −0.54 | −0.81 | −1.68 | −1.07 |
Long | −0.40 | −0.14 | −0.48 | −0.22 | |
E ^{av}_{AB}(ab initio) | Average | −0.47 | −0.48 | −1.08 | −0.65 |
E ^{AM}_{AB}^{a} | −0.45 | −0.35 | −1.58 | −1.68 | |
ΔE_{mix}^{b} | −0.02 | −0.13 | 0.50 | 1.04 | |
E ^{GM}_{AB} | −0.44 | −0.34 | −0.99 | −1.24 (−0.62)^{c} | |
ΔE^{GM}_{mix} | 0.01 | 0.00 | 0.59 | 0.44 (1.06)^{d} | |
−RT L^{AM}_{AB}^{e} | Fit | −0.39 | −0.33 | −1.51 | −1.58 |
−RT L^{GM}_{AB} | Fit | −0.38 | −0.32 | −0.93 | −1.11 (−0.56)^{f} |
−RT ΔL^{GM}_{mix}^{g} | Fit | 0.01 | 0.00 | 0.58 | 0.47 (1.02)^{h} |
We consider a lattice that contains M Mg^{2+} adsorption sites (neglecting adsorption on the weaker linker sites). Each site can adsorb a gas molecule, either A or B from a binary gas mixture. For a particular lattice gas configuration i containing M_{A} and M_{B} adsorbed molecules of components A and B, respectively, there are M_{AA}, M_{BB}, and M_{AB} interacting A⋯A, B⋯B, and A⋯B pairs, respectively, and the lattice gas Hamiltonian, H_{i}, which represents the total free energy of the configuration i is
H_{i} = ΔG_{A}M_{A} + ΔG_{B}M_{B} + E_{AA}M_{AA} + E_{BB}M_{BB} + E_{AB}M_{AB}; i = 1,2,…,3^{M} | (3) |
As mentioned above, the lateral interaction energies are calculated for isolated pairs, i.e., the framework is not present in these calculations. Taking these energies from total energies for the full periodic structures would not be consistent with the use of constant ΔG_{A} and ΔG_{B} values in eqn (3), see also the ESI, Section S4.†
For a lattice gas model of moderate size with, e.g., M = 6 × 100 sites as used here, the number of possible configurations becomes enormously large – 2^{M} and 3^{M} for a pure gas and a binary mixture, respectively. Hence, adsorption isotherms cannot be calculated analytically from the partition function. A GCMC simulation on this lattice gas model samples only the important configurations at a constant chemical potential and temperature. From these configurations, adsorption isotherms are calculated as the ensemble average of the number of molecules adsorbed.^{33} Unlike the mean field approach introduced in Section 3.2, our GCMC simulations on a lattice of adsorption sites treat all the lateral interactions exactly, and we will refer to them as “GCMC” in the following, see Scheme 1.
Commonly, GCMC simulations are not performed on a lattice of sites but directly in configuration space which requires several orders of magnitude more energy evaluations. This is not affordable with ab initio, not even with DFT potential energy surfaces, and typically force fields are used. GCMC calculations and similar simulation techniques have been used with force fields before to test the accuracy of IAST.^{16,18,29,47–50}
Here, we extend the mean-field approximation to binary gas mixtures with components A and B. For the general case with many (m) components, see the ESI, Section S1.† The coverage dependent Gibbs free energy for component A in the mixture is
(4) |
(5) |
From eqn (4) we obtain the single component adsorption equilibrium constants for the mixture (see also Section S1 in the ESI†)
(6) |
(7) |
The surface coverage is obtained as
(8) |
Under the limiting conditions of no lateral interactions, i.e., E^{av}_{AA}(ab initio) = E^{av}_{AB}(ab initio) = 0, eqn (8) reduces to the competitive Langmuir model of co-adsorption (K^{MF,mix}_{c} = K_{c} = exp[−ΔG_{c}/RT], c = A, B). When lateral interactions are non-negligible, K^{MF,mix}_{A} and K^{MF,mix}_{B} become a function of the coverages and, hence, eqn (8) turns into a self-consistent equation where the coverage of each gas component, θ_{A or B} = f(θ_{A},θ_{B}), is also a function of itself. We solve this CMF model using an iterative process with coverages from the competitive Langmuir model as an initial guess. Usually, a few iterations are sufficient to yield converged surface coverages (see also Fig. S1 in the ESI†).
(9) |
(10) |
(11) |
Our lattice GCMC method offers the unique possibility of examining the AM approximation without any additional assumptions. If in every step of the GCMC simulation, the A⋯B interaction energies are approximated as the AM of A⋯A and B⋯B pair interaction energies, we refer to it as “GCMC-AM”, see Scheme 1. In addition, we will use the analytical and computationally much more efficient competitive MF (CMF) model for testing the AM mixing rule and for comparing it with the GM mixing rule. We will insert E^{av}_{AB} values in the CMF equations, eqn (8), that are approximated according to eqn (1) and (2). We will refer to the results as “CMF-AM” and “CMF-GM”, respectively, see Scheme 1.
(12) |
The selectivity of 1.1 calculated for a wide range of CH_{4}/N_{2} mixture compositions and gas phase pressures shows that it will not be possible to separate nitrogen impurities from natural gas. For CO/N_{2}, our ab initio lattice GCMC simulations yield a selectivity value of 8.4 which agrees well with the IAST selectivities of around 10 calculated for different CO/N_{2} mixtures based on measured pure gas data.^{17}
For the non-ideal CO_{2}/N_{2} and CO_{2}/CH_{4} mixtures, GCMC-AM substantially underestimates the selectivity for CO_{2} – the major component on the surface – by 16% and 32% for 10:90 mixtures of CO_{2}/N_{2} and CO_{2}/CH_{4}, respectively, in the pressure range between 0.5 and 5 atm, see Fig. 2, top. Correspondingly, the adsorbed amounts of the minor components, i.e., N_{2} and CH_{4} for CO_{2}/N_{2} and CO_{2}/CH_{4}, respectively, are overestimated with the AM approximation. The reason is that the CO_{2}⋯CO_{2} lateral interactions are stronger than the N_{2}⋯N_{2} and CH_{4}⋯CH_{4} ones, which makes the AM for the CO_{2}⋯N_{2} and CO_{2}⋯CH_{4} interactions, E^{AM}_{AB}, 46% and 158%, respectively, larger than the explicitly calculated E^{av}_{AB}(ab initio) values, see Table 2. Consequently, the stabilities of the minor components on the surface are overestimated which leads to the overestimation of their adsorbed amounts with the AM approximation. The dependence of the selectivity on the gas phase composition (bottom panels of Fig. 2) also shows that the AM approximation underestimates the GCMC CO_{2} selectivity for a high CO_{2} content, by 18% and 34% for CO_{2}⋯N_{2} and CO_{2}⋯CH_{4}, respectively.
The effect of the AM mixing rule seen with lattice GCMC simulations (GCMC-AM in Fig. 2) is reproduced by the CMF model if the AM approximation is applied to calculate the average lateral interactions in eqn (8), CMF-AM. Because of this and because the CMF model is computationally much more efficient, in the following we will use the CMF to test the geometric mean approximation.
Fig. 2 shows the comparison of the AM and GM approximations for CMF selectivities, CMF-AM and CMF-GM, respectively. For CO_{2}/N_{2} mixtures, CO_{2} selectivities obtained with GM are in close agreement with the CMF results. The reason is that the GM of the E^{av} parameters of CO_{2} and N_{2}, E^{GM}_{AB} = −1.0 kJ mol^{−1} (see Table 2), is very close to the average of the directly calculated lateral interaction energies between CO_{2} and N_{2} molecules in the adsorbed phase, E^{av}_{AB}(ab initio) = −1.1 kJ mol^{−1}.
For CO_{2}/CH_{4}, Fig. 2 shows an improvement of about 13% for the calculated selectivities when GM instead of AM mixing is used, but the deviations from the CMF results are still substantial, about 20%. The reason is that E^{GM}_{AB} = −1.24 kJ mol^{−1} gets closer to E^{av}_{AB}(ab initio) = −0.65 kJ mol^{−1} than E^{AM}_{AB} = −1.68 kJ mol^{−1}, but the deviation is still 0.59 kJ mol^{−1}. The origin of the remaining deviations observed with the CMF-GM model is the anisotropy of the lateral interaction energies (E_{AB}) for “mixed” pairs as our ab initio calculations show. This anisotropy will be addressed in the next section.
Co-adsorption selectivity, nevertheless, is independent of this anisotropy if adsorbed molecules do not interact with each other on the surface, e.g., at a very low surface coverage. In this case, the co-adsorption selectivity becomes the ratio of zero coverage equilibrium constants for both components. Using the ab initio values for the latter, the calculated zero coverage CO_{2}/N_{2} selectivity is 167 at 313 K and the CO_{2}/CH_{4} selectivity is 199 at 298 K, which are in good agreement with the experimental IAST selectivities of ca. 175 and 210–220, respectively.^{52–54}
We may take this into account by defining effective pair interaction energies regarding the weakly interacting CO_{2}⋯CH_{4} pairs as non-interacting. Assuming that there are n_{AA}, n_{BB} and n_{AB} numbers of A⋯A, B⋯B and A⋯B pairs, respectively, with non-negligible interaction energies we get:
(13) |
(14) |
If we apply the geometric mean approximation for the average interaction energy of a mixed pair, i.e.,
(15) |
(16) |
(17) |
(18) |
For the CO_{2}/CH_{4}, the long, weakly interacting (−0.22 kJ mol^{−1}) CO_{2}⋯CH_{4} pairs are regarded as non-interacting which means that there is only one interacting “mixed” pair, n_{AB} = 1, whereas there are two interacting “same” pairs, n_{AA} = n_{BB} = 2, hence
After inclusion of the factor 1/2 for the CO_{2}⋯CH_{4} interaction, we get E^{GM}_{AB}/2 = −0.62 kJ mol^{−1} which nicely agrees with the ab initio energy for the same interaction, E^{av}_{AB}(ab initio) = −0.65 kJ mol^{−1}. Moreover, the effective mixing energy, ΔE^{GM/2}_{mix} = 1.06, calculated using the former, matches well with that for the “real mixture”, ΔE_{mix} = 1.04 kJ mol^{−1}, see Table 2. As Fig. 2 shows, the CMF calculations that account for the anisotropy factor f_{AB} = 1/2, CMF-fGM, reproduce the target CMF and GCMC results.
For fitting the single component data points, we make use of the non-linear mean-field isotherm equation,^{36}
(19) |
(20) |
(21) |
(22) |
Table 1 shows the comparison of the fitted parameters with the ab initio data used for the GCMC simulations. For adsorbed gases with very weak lateral interactions (CO, N_{2}, and CH_{4}) there is agreement within 0.1 kJ mol^{−1}; for CO_{2} with stronger lateral interactions (−2.8 kJ mol^{−1}) the fit yields 0.4 kJ mol^{−1} (15%) stronger binding. The agreement can be improved by further expanding in continued fraction representation, see the ESI, Section S6.†
IAST predictions for mixture isotherms according to the standard protocol described here will be labelled “IAST” in the following, see Scheme 1. Any kind of pure component isotherms can be used, originating either from experiments^{16,38} or from simulations. The simulation method employed for the pure component isotherms is irrelevant, and it may range from ab initio GCMC on a lattice of sites (as we use in this work)^{34–36} to GCMC simulations in full configuration space using a force field, see, e.g., ref. 55.
Since IAST shares the assumption of ideal mixtures with GCMC-AM and CMF-AM (Fig. 3), eqn (1), we should expect to get the same results. This is shown in Fig. 3 which will be discussed in more detail below.
(23) |
(24) |
Table 1 shows that the linear, free energy fitting procedure, eqn (24), reproduces the pure component lateral interaction energies for adsorbed gases with very weak lateral interactions, i.e., for CO, N_{2}, and CH_{4}, within 0.1 kJ mol^{−1} of the directly calculated ab initio values, and within 0.2 kJ mol^{−1} for adsorbed gases with stronger lateral interactions, e.g. CO_{2}.
In the second step, the fitted L_{AA} and L_{BB} interaction parameters are used to approximate L_{AB} as the arithmetic mean, cf.eqn (1),
L^{AM}_{AB} = (L_{AA} + L_{BB})/2 | (25) |
With this, we have two different protocols for generating co-adsorption isotherms from pure component ones, both based on the assumption of ideal mixtures, IAST and CMFfit-AM. Since we use the same ab initio data as the input, they should yield the same results. While the IAST protocol does not require any explicit specification of the interactions between different adsorbed molecules, but involves an integration step, the CMFfit-AM protocol relies on the AM approximation for the interaction between pairs of unlike adsorbate molecules.
(26) |
The anisotropy factor f_{AB}, eqn (17), that has further improved the CMF-GM results for CO_{2}/CH_{4} cannot be derived from pure gas adsorption data, neither from experimental nor from simulated ones. It requires input from adsorption structures for the mixture. The latter is only available from atomistic simulations, but high-level quantum chemical calculations as we perform in this study are not required. Computational methods that are easily available also for non-specialists like DFT(+dispersion) or even simple force fields, e.g., ref. 17, 24, and 25 are sufficient as long as they provide qualitatively correct adsorption structures. We will refer to such methods as “computational” in the following.
After identifying the location of the adsorption sites, one needs to carry out three DFT (or force field) optimization runs for the MOF structures with all adsorption sites filled with: (i) A, (ii) B, and (iii) half A and half B. In structure (iii), A and B molecules should alternate at the adsorption sites (see Fig. 1). Then lateral interaction energies for each “same” pair (AA, BB) and “mixed” (AB) pair must be calculated explicitly.
In Section 3.5 the numbers of interacting neighbors were determined by inspection of the structure of the adsorbate layer for the mixture and from these numbers, f_{AB} was calculated using eqn (17). For a generally applicable protocol we recommend a different approach that is easier to implement into a computer code. Here, the anisotropy factor is calculated directly from the computed average lateral interaction energies. The GM approximation
(27) |
(28) |
Not much is gained so far, because we need to know E^{av}_{AB} to calculate f_{AB}. However, we may calculate f_{AB} from the results of a simple computational approach mentioned above,
(29) |
(30) |
The f_{AB} values calculated according to eqn (29) for CO_{2}/N_{2} and CO_{2}/CH_{4} are 1.09 and 0.52, respectively. The latter is in agreement with f_{AB} = 0.5 obtained in Section 3.5 for CO_{2}/CH_{4} from the numbers of non-interacting pairs. For the isotherms reported in Fig. 2 and 3 for CO_{2}/CH_{4}, we used f_{AB} = 0.5.
Table 2 shows the GM mixed pair interaction energies, −RTL^{GM}_{AB}, as well as the GM mixing energies, −RTΔL^{GM}_{mix}. They have been obtained from single component L_{AA} and L_{BB} parameters that resulted from linear MF fitting of the GCMC isotherms for pure components. Table 2 shows that they are very close to the directly calculated E^{GM}_{AB} and ΔE^{GM}_{mix} values. We therefore expect that our “CMFfit-GM” and “CMFfit-fGM” co-adsorption isotherms show the same improvements over “CMFfit-AM” (Fig. 3) as Fig. 2 shows for the “CMF-GM” and CMF-fGM” compared to CMF-AM” – and this is indeed the case.
For CO_{2}/N_{2} mixtures the “CMFfit-GM” predicted selectivities deviate only 4% from the GCMC reference values. This excellent agreement is reached because the GM of the L-parameters for CO_{2} and N_{2} (−0.9 kJ mol^{−1}) is very close to the average ab initio lateral interaction energy, E^{av}_{AB}(ab initio) = −1.1 kJ mol^{−1}, for a CO_{2}⋯N_{2} pair, see Table 2. Moreover, the mixing energy calculated using these L-parameters is also within 0.1 kJ mol^{−1} of the ab initio mixing energy, ΔE_{mix}.
For CO_{2}/CH_{4}, CMFfit-GM also improves the CO_{2} selectivities compared to CMFfit-AM, though the deviations from the reference GCMC results can be as large as 20%. The reason is that the geometric mean (−1.1 kJ mol^{−1}) improves −RTL_{AB} by 0.5 kJ mol^{−1} compared to the arithmetic mean (−1.6 kJ mol^{−1}), yet it is almost double the ab initio CO_{2}⋯CH_{4} lateral interaction energy, E^{av}_{AB}(ab initio) = −0.65 kJ mol^{−1}. After including an anisotropy factor of 0.5, the −RTL_{AB}-parameter (−0.56 kJ mol^{−1}) is within 0.1 kJ mol^{−1} of the E^{av}_{AB}(ab initio) for CO_{2}⋯CH_{4} interaction, and the mixing energy calculated using these L-parameters (−RTΔL_{mix}) is in perfect agreement with the ab initio energy of mixing, ΔE_{mix}. Fig. 3 shows that for CO_{2}/CH_{4} the CMFfit-fGM results deviate by less than 1% from the GCMC reference values.
We therefore recommend our computational protocol based on the GM mixing rule for general use when predicting co-adsorption isotherms from pure gas data. As IAST, our method which we name co-adsorption mean field theory (CAMT) needs fitted pure gas isotherms only, either from experiments or from simulation. The difference is that CAMT needs fitting with the mean field form that yields parameters for lateral interactions. CAMT involves the following steps:
(i) Mean field fitting of pure gas adsorption data, eqn (24), yielding Gibbs free energies for gas–surface interactions and gas–gas (A⋯A and B⋯B) interaction energies;
(ii) Competitive mean field calculation, eqn (6) and (8), of co-adsorption isotherms using pure gas data and A⋯B interaction energies approximated as the geometric mean of the single component A⋯A and B⋯B interaction energies, eqn (26).
Further improvement can be expected if a third, optional step is made which, however, requires input from computational methods:
(iii) Performing three structural optimizations using force fields or DFT for the adsorbent loaded both with the pure gases and the mixture, and calculation of the anisotropy factor in eqn (30) according to eqn (29). This does not necessarily require CCSD(T) calculations for the isolated pair interactions E^{av} (comput) at the periodic structures. As Table S4 in the ESI† shows, DFT-D calculations for the isolated pairs yield very similar values for f_{AB}.
With both the GCMC and CMF methods we have tested the arithmetic mean (AM) of A⋯A and B⋯B adsorbate–adsorbate interactions as approximation for A⋯B interactions and obtained the same results, which also agreed with the standard IAST protocol for predicting co-adsorption isotherms from pure component isotherms. For the CH_{4}/N_{2} and CO/N_{2} mixtures in which both gases have very weak lateral interactions, use of AM mixing or IAST shows very good agreement with the exact GCMC and CMF predictions, whereas for mixtures which contain the more strongly interacting CO_{2} (CO_{2}/N_{2} and CO_{2}/CH_{4}) substantial deviations from ideal behavior are found.
If GM mixing is applied instead of AM mixing, as the laws of intermolecular interactions such as the London formula for dispersion suggest, agreement with the exact CMF (and GCMC) results is perfect for CO_{2}/N_{2}. For CO_{2}/CH_{4} substantial improvement is reached, and the remaining deviation can be explained by the anisotropy of the lateral interactions in the mixed adsorbate layer. After introducing a factor (1/2 in this case) that accounts for this anisotropy, very good agreement is also reached for CO_{2}/CH_{4}.
For predicting co-adsorption isotherms and selectivities from pure gas data, we suggest a new computational protocol, co-adsorption mean field theory (CAMT) that like IAST starts from fitting isotherm expressions to pure gas adsorption data, but unlike IAST uses mean field theory for fitting and applies the GM mixing rule to approximate the interaction energies between different components in the CMF equations for the mixture.
Because of the generality of the GM for parameters describing intermolecular interactions, we expect an improvement compared to IAST in all cases in which adsorbate–adsorbate interactions are smaller than adsorbate–surface interactions. The anisotropy factor cannot be derived from pure gas data only; it requires atomistic structural optimizations for the pure gas and mixed adsorbate layers, using either force fields or DFT.
The present work has considered binary mixtures on homogeneous surfaces with identical surface sites. Future studies should aim at incorporating surface heterogeneity, for example linker sites for MOFs.
Footnotes |
† Electronic supplementary information (ESI) available: Mean field approximation for lateral interactions in a multicomponent mixture. Derivation of the geometric mean mixing rule. Additional figures for ideal mixtures. Lateral interactions and the anisotropy factor. Details of ideal adsorption solution theory. Expansion of the mean field equilibrium constant for pure gas in continued fraction representation. See DOI: 10.1039/c9sc03008e |
‡ Equal contributions. |
This journal is © The Royal Society of Chemistry 2020 |