Analytical Ab-Initio Based Modeling of the Adsorption Isotherm - The

Article Cite This: J. Phys. Chem. C XXXX, XXX, XXX−XXX

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Analytical ab initio-Based Modeling of the Adsorption Isotherm Ankita Jadon,†,‡,§ Nathalie Girault,† Bruno Piar,† Denis Petitprez,‡,§ and Sidi M. O. Souvi*,†,§ †

PSN-RES, Institut de Radioprotection et de Sûreté Nucléaire (IRSN), Cadarache, St Paul Lez Durance 13115, France Univ. Lille, CNRS, UMR 8522, PC2APhysicochimie des Processus de Combustion et de l’Atmosphère, Lille F-59000, France § Laboratoire de Recherche Commun IRSN-CNRS-Lille1 “Cinétique Chimique, Combustion, Réactivité” (C3R), Cadarache, Saint Paul Lez Durance 13115, France ‡

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S Supporting Information *

ABSTRACT: An analytical ab initio-based model for adsorption isotherm is presented. As input data, this model takes the adsorption Gibbs free energies of different possible coverages within a reasonably large unit cell. In this way, the lateral interactions between adsorbates and their dependence on temperature are included. The self-dependence of surface coverage is taken into account via the introduction of heterogeneity of adsorption and the use of an appropriate configurational partition function. The model reproduces the isotherm and selectivity (for binary gas mixtures) of ab initio-based grand-canonical Monte Carlo simulations. The presented isotherm has a moderated dependence on the width of the chosen ab initio unit cell, particularly for high surface loading. This isotherm seems to be a very good approximation for the grandcanonical solution, depending on the accuracy and completeness of the ab initio data. initio level of calculations.3,7 However, this method is twice computationally demanding due to its dependence on both the level of ab initio calculations and the Monte Carlo sampling. We present here an analytical expression of the adsorption isotherms based on ab initio calculations, without any a priori assumption of the nature of the isotherm. This model does not presume the homogeneity of adsorption while increasing the coverage. We confront this expression to the most conceptually exact method, which is the GCMC approach (also based on ab initio data) using the same ab initio data provided in refs 3 and 7. There is no direct comparison to the experiment at this stage.

1. INTRODUCTION The characterization and understanding of the physicochemical processes governing the gas-surface interactions (adsorption) are one of the ultimate aims in the field of surface chemistry. The key indicator of the consistency of our understanding and knowledge of these processes is our ability to predict the isotherm of adsorption. For decades, several analytical “experiment based” models of the adsorption isotherms have been proposed: Langmuir, Freundlich, Sips, and Toth,...1 However, these models suffer from their relatively narrow range of applicability, especially the Langmuir model, and the others, the lack of thermodynamic consistency of their parameters.2,3 At present, the use of ab initio methods in the area of surface chemistry has become routinely common due to the enhancement in their reliability and the exponential increase of the computational means. These methods allow investigating the reactivity at the atomic scale within large possibilities of discretization of the problem under study. They offer detailed insight (at the atomic scale), which cannot be neglected or bypassed when aiming to investigate macroscopic phenomena and quantities such as the adsorption and its isotherm. Recently, Sillar et al.4−6 have pointed out the ability of the mean-field approximation to reproduce experimental isotherms, when the lateral interactions are calculated with high-level ab initio calculations. At present, the most conceptually accurate approach is the grand-canonical Monte Carlo (GCMC) method when combined with the high ab © XXXX American Chemical Society

2. MODELS AND ANALYTICAL DERIVATIONS The total surface is divided into unit surfaces (Su). There are no constraints preventing different Su from having different coverages from each other. One can picture the total surface as a combination of differently covered surfaces that are named as S0, S1, S2,..., Smax (Sn stands for Su with n molecules adsorbed on it; see Figure 1). The unit cell should be chosen large enough (large Su) to avoid interactions between adsorbates at the low coverage limit, particularly when interested in the isotherm near the Langmuir region. Far from this region, a reasonably reduced Received: July 18, 2018 Revised: September 12, 2018 Published: September 24, 2018 A

DOI: 10.1021/acs.jpcc.8b06856 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

being the total of Su under study and Nu the number of active sites per Su), is θTOT (ads/site) =

N gad

max

=

NuNTOT

∑ dc = 1

n(dc) θdc Nu

(3)

Nad g

stands for the number of adsorbed molecules. This equation expresses the total adsorption isotherm (which is heterogeneous, in terms of surface coverage) as an average of individually homogeneous coverages. This expression evidences the self-dependence of the isotherm. This approach has been extended to the case where the gaseous phase is constituted of a mixture of species; see the Section S2 of the Supporting Information. Similar to the first part, one can consider the total surface as a distribution of Su with different coverages of species gA, gB,... In terms of degenerated configurations (dc), every dc is now distinguished by nA(dc), nB(dc),..., Δdc r G(T), and Ω(dc) for the number of adsorbed molecules of species gA, the number of adsorbed molecules of species gB,... in any configuration belonging to dc, the Gibbs free energy of the reaction of formation (dcr: S0 + nA(dc)gA + nB(dc)gB··· → Sc; ∀c ∈ dc) of any configuration belonging to dc, and the degeneracy of dc (the number of c belonging to dc), respectively. In such a mixture (see the Section S2 of the Supporting Information), the equilibrium constant of the reaction dcr, the coverage ratio for a given dc, and the total adsorption isotherm θXTOT of a given species gX are

Figure 1. Illustration of the coexistence of different coverages at the equilibrium.

unit cell could be acceptable, as will be shown later. One considers the formation reaction of Sn, nr: S0 + ng → Sn, g stands for a molecule in gaseous phase. Every coverage (Sn) can be characterized independently by ab initio calculations. This way, we include the lateral interactions at every level of surface loading and allow an evaluation of the energy gain between successive adsorptions. Furthermore, the dependence of these lateral interactions on temperature can be taken into account; since one considers the molecular partition functions of Sn at every stage of the surface loading, see the Section S1 in the Supporting Information. In the Supporting Information (Section S1), we discuss the fact that a given level of coverage Sn might correspond to several configurations (c) (local minima of the internal energy at 0 K) and among these c some might be energetically degenerated. Therefore, we refer to these degenerated Sn as an ensemble of degenerated configurations (dc). To every dc correspond n(dc), Δdc r G(T), and Ω(dc) for the number of adsorbed molecules in any configuration belonging to dc, the Gibbs free energy at temperature T and standard pressure (P°) of the reaction of formation of any configuration belonging to dc (dcr: S0 + n(dc)g → Sc; ∀c ∈ dc) and the degeneracy of the dc (the number of configurations belonging to the dc, with n(dc) and Δdc r G(T)), respectively. At global equilibrium, all of these reactions (dcr) will be in a local equilibrium and different spots of the total surface might have different numbers of molecules adsorbed per Su in different configurations. The characterization of such a global equilibrium requires solving a grand-canonical problem. For this reason, we introduced a definition of the chemical potential for every possible configuration within the Su. The thermodynamic formulas presented in the Supporting Information are based on the available literature.8 In the Supporting Information (Section S1), we show that at the equilibrium, the amount of different dc (θdc) is given by the equations Kedc =

θdc =

Kdc e

Ω(dc) P°n(dc)

dc ji −Δr G zyz zz expjjjj z k RT {

θdc =

(4)

Kedc ∏ g PgnI(dc) I

I

max

1 + ∑dc ′= 1 Kedc′ ∏ g PgnJ(dc ′)

(5)

J

and X θTOT =

N gad X

NuNTOT

max

=

∑ dc = 1

nX (dc) θdc Nu

(6)

respectively. And the selectivity vis-à-vis the given species gX and gY is selectivity (X /Y ) =

X θTOT Yg

Y

Y θTOT Yg X

(7)

where YgX stands for the mole fraction of gX species. Note that in the expression of θdc (eqs 2 and 5), the ideality of the gaseous phase has been assumed. Whenever this assumption does not hold (at very high pressure, for example), one should replace the partial pressure in these expressions (eqs 2 and 5) by the fugacity of different species. One can notice that these expressions depend on only few parameters: beside partial pressures and temperature, they depend on Nu, related to the unit cell Δdc r G(T) obtainable form ab initio calculations, and Ω(dc). The degeneracy is a “key parameter”, and either one is able to predict it a priori (on the basis of symmetrical consideration, for example) and save time avoiding redundant calculations or one calculates systematically all possible configurations and considers them one by one with no degeneracy.

Pgn(dc)Kedc max

P°(nA (dc) + nB(dc) +···)

ij −Δdc y r G (T ) z zz expjjjj zz RT k {

J

(1)

1 + ∑dc = 1 Pgn(dc)Kedc

Ω(dc)

Kedc =

(2) dc

stands for the equilibrium constant of r at temperature (T) and Pg for the partial pressure. We show also (in Section S1) that the isotherm (θTOT), in terms of the total number of available sites (NTOT × Nu: NTOT B


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whereas for linker sites these Gibbs free energies ΔrGad L are 5.1 and 10.5 kJ/mol at 298 and 343 K, respectively. The lateral interaction energies are in average −1.405, −1.119, and −1.057 kJ/mol for Mg−Mg, linker−linker, and Mg−linker, respectively. One should note that a part of Mg−linker interaction energy is already included in the ΔrGad L , since the adsorption on linker takes place after the adsorption on Mg sites; see refs 3 and 7. In this example (in both the current work as well as the Sauer et al. one), the lateral interactions are considered independent of the temperature. Technically, such an approximation is not needed when calculating the partition function at every stage of coverage. Using these energies we calculate Δdc r G step by step and evaluate Kdc e , θdc and the adsorption isotherm according to eqs 1−3, respectively. The interactions considered in refs 3 and 7 can be taken into account within a relatively reduced Su containing four Mg sites and four linker sites (Nu = 4); see Figure 2.

It is also worth mentioning the connection between this model at a very low coverage limit and Langmuir formalism. Actually, at this limit, only the first adsorption takes place (isolated adsorption) and one has max = 1. The general isotherm expression (eq 3) becomes θTOT (ads/site) =

N gad NuNTOT

=

n(1) θ1 Nu

(8)

One can also choose the size of the unit cell in such a way that n(1) = 1 and Nu = 1. One obtains θTOT (ads/site) = θ1 =

PgKe1 1 + PgKe1

(9)

which is the classical Langmuir formalism. One should highlight the fact that there are no technical limitations on the number of gaseous species included in the mixture or on the number of adsorbed molecules that make those general expressions applicable even in the case of multilayer adsorption. However, treating multilayer cases might require very large ab initio data and thus numerous ab initio calculations to be performed. By contrast, the extrapolation of such data via this model is almost cost free (the isotherm and selectivity are obtained straightforward with no iterations process and no time of calculation). Furthermore, the size of the ab initio data can be reduced considerably when targeting a specific region of the isotherm, as discussed in the next section.

Figure 2. Representation of the unit cell used in the isotherm adsorption for CO2 in Mg-MOF-74.

3. RESULTS AND DISCUSSION In this section, we compare the isotherm and selectivity calculated via these analytical expressions to those obtained with GCMC simulations3,7 concerning the adsorption of carbon dioxide and methane in the material organic framework CPO-27-Mg (Mg-MOF-74). The choice of this system is due to the availability of both ab initio data and GCMC results. In this work, we did not perform additional ab initio calculations. We use the energies reported by Sauer et al.3,7 However, these ab initio data are given in terms of Gibbs free energies of isolated adsorptions and lateral interaction energies, the adequate input for mean-field approximation and/or GCMC simulations. Hence, a “reconstruction effort” is needed to express these data in terms of input parameters of this analytical expression: identification of different configurations dc and Ω(dc) as well as the evaluation of the corresponding 7 Δdc r G. Furthermore, in their GCMC simulations, Sauer et al. have corrected the gas nonideality but the used parameters are not provided in the cited article. In the current work, the nonideality is taken into account using the fugacity coefficients of the mixture (CO2 and CH4) proposed by Spycher et al.9 3.1. Isotherm: Modeling of CO2 in Mg-MOF-74. The Mg-MOF-74 pores consist of successive hexagonal rings. Every ring contains six Mg sites and six linker sites. The Mg sites constitute local minima of adsorption energy for CO2, whereas the linker sites become local minima only once the neighboring Mg site is occupied. The Gibbs free energies of isolated adsorption of a CO2 molecule on Mg sites ΔrGad Mg and on linker sites ΔrGad L (with the neighboring Mg site being occupied), as well as the energies of interactions between neighboring adsorbed molecules have been calculated at very high ab initio level.3,7 Precisely, the ΔrGad Mg at a Mg site has been calculated to be −9.2 and −3.6 kJ/mol at 298 and 343 K, respectively,

The smallest “acceptable” cell should allow considering the first adsorption as an isolated one and introducing progressively the different identified interactions while increasing the surface loading. In Figure 2, the sites labeled Mg1 and Mg2 (or L1 and L2) are on the same ring and the adsorbed molecules on these sites have a mutual Mg−Mg (or linker−linker) interaction. The case is the same for the sites labeled Mg1′ and Mg2′ (or L1′ and L2′). The only possible Mg−Mg or linker−linker interactions are Mg1−Mg2, Mg1′−Mg2′, L1−L2, and L1′− L2′. However, the Mg−linker is possible for all combinations: Mg1−L1, Mg2−L1, Mg1′−L1, Mg2′−L1, and so on. Statistically speaking, there are 256 c (there are eight sites; each site can be either occupied or unoccupied: 28). However, ad due to the difference between ΔrGMg and ΔrGLad, the adsorptions on the linker sites are negligible until the Mg sites are completely saturated. So the adsorption of the first four molecules can be considered on the Mg sites, leading to 15 c (excluding the bare Su) distributed in 5 dc and later on the linker sites (leading similarly to 15 c in 5 dc); see Table 1. The only interaction affecting the first reaction Gibbs free energy is the isolated adsorption on an Mg site (Δ1r G = ΔrGad Mg) and the product (S 1 ) is of quadruple degeneracy, Ω(1) = 14 = 4 , due to the existence of four equivalent Mg sites in the Su. For the second reaction (adsorption of two molecules), there are six possibilities: (i) adsorption on different rings (M1−M1′, M1−M2′, M2−M1′, and M2− M2′); these four configurations are degenerated and do not include any lateral interaction, whereas (ii) the adsorption on the same ring (M1−M2 and M1′−M2′, which are degen-

()

C


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the two methods. Our calculations show that for this system (monocompound gaseous phase, CO2), the effect of gas nonideality correction on the isotherm remains negligible for the studied pressure range; see Tables S1 and S2 in the Supporting Information. Figure 3B shows the fashion in which the surface coverage evolves with respect to pressure. As said earlier, the adsorption isotherm is the average of different homogeneous coverages. It can be seen from this figure that when the ratio nCO2/nMg equals 1 (conventionally known as the point where the monolayer is completed), less than 65% of the surface is actually homogeneously monolayer (θ5 < 0, 64 at 298 K). The rest of the surface has an actual ratio nCO2/nMg different from 1 and mainly covered with θ4 and θ6, which corresponds to the ratio of 0.75 and 1.25, respectively, the contributions of the other coverages being negligible. At 343 K, only half of the surface is actually homogeneously monolayer (θ5 ≈ 0, 53) when the ratio nCO2/nMg equals 1; see Table S1 in Section S3. One can also notice that every region of the isotherm depends only on a reduced number of homogeneous coverages to be acceptably described. Such a characteristic allows, in some cases, reducing the ab initio effort when targeting a specific region of the isotherm. 3.2. Selectivity: CO2/CH4 in Mg-MOF-74. We applied the above analytical expressions to investigate the selectivity of the Mg-MOF-74 vis-à-vis CO2 and CH4. The results are then compared to the GCMC simulations performed by Kundu et al.7 In their study, Kundu et al. considered only the metallic centers; the adsorption on the linker, taking place at very high pressure, has been considered irrelevant for the expected applications. The interaction energies taken into account in their study are: (i) −9.2 and −3.6 kJ/mol for the isolated adsorption of CO2 (ΔrGad CO2) at 298 and 343 K, respectively; (ii) 3.9 and 8.5 kJ/mol for the isolated adsorption of CH4 (ΔrGad CH4) at 298 and 343 K, respectively; (iii) −1.405 and −0.275 kJ/mol for the average of “homogeneous” lateral CH4 2 interactions ECO CO2 and ECH4, respectively; and (iv) −0.535 and −0.11 kJ/mol for the average of “heterogeneous” lateral long interactions Eshort CO2−CH4 and ECO2−CH4, respectively. The smallest unit cell in which these interactions can be discretized consists of three Mg centers (Nu = 3); see Figure 4. Using such a reduced cell implies that when considering an occupation of one site, its equivalent in the other half of the ring is imposed symmetrically.

Table 1. Summary of Interactions Involved in Different dc (at Every Stage of Surface Loading) and the Corresponding Degeneracya dc

1

2

3

4

5

6

7

8

9

10

n(dc)

1

2

2

3

4

5

6

6

7

8

ΔrGad Mg ΔrGad L

1 0 0 0 0 4

2 0 2 0 0 2

2 0 0 0 0 4

3 0 2 0 0 4

4 0 4 0 0 1

4 1 4 0 3 4

4 2 4 2 6 2

4 2 4 0 6 4

4 3 4 2 9 4

4 4 4 4 12 1

Mg−Mg L−L Mg−L Ω(dc) a

Note that from the 6th to 10th dc the fourth lateral Mg−L interaction is already included in ΔrGad L.

erated) is stabilized by twice that in the Mg−Mg interaction (M2 having two neighboring occupied M1). Hence, the second reaction can be treated in two dc: the Gibbs free energy for the first is Δ2r G = 2ΔrGad Mg + 2EMg−Mg with n(2) = 2 and Ω(2) = 2, whereas for the second dc, the Gibbs free energy is Δ3r G = 2ΔrGad Mg with n(3) = 2 and Ω(3) = 4. Table 1 lists the different interactions involved in every dc (at different levels of coverage) and the corresponding degeneracy. The calculated isotherms at 298 and 343 K using this analytical expression (eq 3 within gas ideality correction ref 9) and the one obtained with GCMC method7 are presented in Figure 3A. One can notice the excellent agreement between

Figure 3. (A) CO2 isotherm adsorption in CPO-27-Mg at 298 and 343 K according to the presented analytical expression (full and dashed lines) and calculated with GCMC7 (circles and squares), respectively. “Ana.Fuga.C” for analytic with corrected ideality via the fugacity coefficients, ref 9. (B) Illustration of the evolution of the surface coverage: decomposition of the analytical isotherm into homogeneous coverages.

Figure 4. Representation of the reduced unit cell constituted of three Mg sites (half of a ring) used in the selectivity study of CO2/CH4 in Mg-MOF-74. D


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according to the eq 7. The results (with and without gas nonideality corrections) are compared to other selectivity modeling7 (in which the gas nonideality has been taken into account); Figure 5.

Given the nature of the interactions, there are 26 c (excluding the bare Su) distributed in 10 possible dc in this cell; see Table 2. Among them six are homogeneous and four are heterogeneous; see Section S4. Among the homogeneous ones, for CO2 one has the following Table 2. Summary of Interactions Involved in Different dc (at Every Stage of Surface Loading of Coadoption of CO2/ CH4 in Mg-MOF-74) and the Corresponding Degeneracy dc

1

2

3

4

5

6

7

8

9

10

nCO2(dc)

1

2

3

0

0

0

1

1

2

1

nCH4(dc)

0

0

0

1

2

3

1

1

1

2

ΔrGad CO2

1

2

3

0

0

0

1

1

2

1

ΔrGad CH4

0

0

0

1

2

3

1

1

1

2

2 ECO CO2

0

1

3

0

0

0

0

0

1

0

4 ECH CH4

0

0

0

0

1

3

0

0

0

1

Eshort CO2−CH4

0

0

0

0

0

0

1

0

1

1

Elong CO2−CH4

0

0

0

0

0

0

0

1

1

1

Ω(dc)

3

3

1

3

3

1

3

3

3

3

l Δ1G = Δ G ad o o r r CO2 dc = 1: m o o nCO2(1) = 1; nCH4(1) = 0 and Ω(1) = 3 n l ad CO2 o Δr2 G = 2Δr GCO + ECO o 2 2 dc = 2: o m o nCO2(2) = 2; nCH4(2) = 0 and Ω(2) = 3 n l 3 ad CO o o Δr G = 3Δr GCO2 + 3ECO22 dc = 3: m o o nCO2(3) = 3; nCH4(3) = 0 and Ω(3) = 1 n These three reactions should be similarly considered for the methane adsorption: 4r, 5r, and 6r. The four heterogeneous reactions are

l Δ7 G = Δ G ad + Δ G ad + E short o o r r CO2 r CH4 CO2 −CH4 :o m o nCO2(7) = 1 ; nCH4(7) = 1 and Ω(7) = 3 n dc = 8 l long ad ad o Δ8r G = Δr GCO + Δr GCH + ECO o 2 −CH4 2 4 :m o o nCO2(8) − 1 ; nCH4(8) = 1 and Ω(8) = 3 n dc = 9 l ad ad CO2 o Δr9G = 2Δr GCO + Δr GCH + ECO o o 2 4 2 o o long short :o m + ECO2−CH4 + ECO2−CH4 o o o o o = 2 ; nCH4(9) = 1 and Ω(9) = 3 n (9) o CO2 n l CH4 ad ad o o Δ10 r G = Δr GCO2 + 2Δr GCH4 + ECH4 o o o o o long short o + ECO + ECO o o 2 −CH4 2 −CH4 o o dc = 10: m = n n (10) 1 ; (10) o CO2 CH4 o o o o o o = 2 and Ω(10) o o o o o =3 n These dc, the involved interactions, and the corresponding degeneracy are listed in Table 2. These Gibbs free energies of reaction, as well as the degeneracies, have been used to calculate the selectivity, dc = 7

Figure 5. (A) Pressure dependence of the CO2/CH4 adsorption selectivity in Mg-MOF-74 at 343 K and a gas ratio of 1:9, calculated according to this analytical expression (dotted line “Ana.Ideal.G” within the gas ideality assumption and solid line Ana.Fuga.C for corrected ideality via fugacity coefficients, refErreur! Signet non défini.) and reported from ref 7, triangles for GCMC calculations, crosses for ideal adsorbed solution theory, and circles for a competitive Langmuir. (B) Dependence of the selectivity on the composition of the gas mixture (298 K, 5 bar), same symbols as (A).

Our calculations show that for this system, the effect of gas ideality correction on the selectivity is not negligible, particularly at a high mole fraction of CO2. The comparison of the selectivity evolution with respect to both the total pressure (at 343 K and the mole fraction ratio CO2/CH4 = 1:9, Figure 5A) and the CO2 mole fraction (at 298 K and 5 bar, Figure 5B) evidences the good agreement between the GCMC simulations and the analytical results, particularly when gaseous phase nonideality is taken into account. The maximum deviation of our results, for both isotherm and selectivity, with respect to GCMC ones is 1.25%. This deviation can be due to the probable different treatments of the nonideality of the gaseous phase in these studies. Finally, the sensitivity of the results with respect to the size of the unit cell has been investigated. We considered a larger Su, a complete ring with six Mg sites. In this large Su, there are E


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The Journal of Physical Chemistry C 729 configurations (36) distributed in 95 different dc instead of 27 c in 11 dc for the reduced cell (only three Mg sites). Figure 6 shows that the reduced cell reproduces very well the results obtained with the large cell. This means that the

sampling is performed. The presented analytical expression depends on the former; however, it allows saving the effort of sampling, which could be computationally expensive, particularly at low temperature and gas mole fractions. Given the conception of this model (consideration of chemical potentials for different possible coverages and the use of an appropriate configurational partition function) and its agreement with the GCMC results, it could be seen as an analytical expression of the grand-canonical solution within given ab initio data. Hence, this analytical expression allows a direct comparison of ab initio calculations to experimental observations (isotherm) and fills somehow the gap between atomic and macroscopic scales, since it makes the effort be focused on the accuracy of the ab initio calculations. Moreover, it has the advantage to give a detailed picture of the state of the surface in terms of coverages at any conditions (gas pressure, temperature), which is mandatory when investigating the surface’s reactivity. However, the most important limitation of this analytical expression remains the exhaustiveness and accuracy of the ab initio data.

■

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.8b06856.

■

Derivation of the isotherm expression for monocompound gaseous phase, (ii) derivation of isotherm and selectivity expressions for multicompound gaseous phase, and (iii) numerical values of different coverages, the total isotherms, and selectivity (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +33442199309.

Figure 6. Effect of the unit cell size on the sensitivity, solid line Ana.Fuga.C for the reduced cell and dotted line “Ana.L.C.Fuga.C” for the large cell. (A) Pressure dependence of the CO2/CH4 adsorption selectivity in Mg-MOF-74 at 343 K and a gas ratio of 1:9, (B) dependence of the selectivity on the composition of the gas mixture (298 K, 5 bar).

ORCID

Sidi M. O. Souvi: 0000-0002-6349-762X Notes

The authors declare no competing financial interest.

■ ■

ACKNOWLEDGMENTS We would like to thank A. Kundu, K. Sillar, and J. Sauer for providing the GCMC results presented in this paper.

reduced cell contains the most important configurations. These results highlight a moderate dependence of the proposed expressions on the size of the unit cell.

4. CONCLUSIONS The ab initio methods have become routinely common in the area of surface chemistry. Beyond the accuracy of these methods or their appropriateness to different systems for which they are usually used, the need of an easy extrapolation of their results to the equilibrium observable is real. We propose an analytical expression of the gas-surface equilibrium solution within a given ab initio data. At this stage, we confront this expression to the most conceptually exact method, which is the GCMC approach (also based on ab initio data) and point out the excellent agreement between these approaches. Technically, the GCMC converges toward the thermodynamic solution of the equilibrium within two conditions: (i) all interactions, as well as their dependence on the temperature, are taken into account, and (ii) enough

REFERENCES

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Analytical Ab-Initio Based Modeling of the Adsorption Isotherm - The

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