Ab Initio Adsorption Isotherms for Molecules with Lateral Interactions

Article pubs.acs.org/JPCC

Ab Initio Adsorption Isotherms for Molecules with Lateral Interactions: CO2 in Metal−Organic Frameworks Kaido Sillar,*,†,‡ Arpan Kundu,† and Joachim Sauer† †

Institut für Chemie, Humboldt Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany Institute of Chemistry, University of Tartu, Ravila 14a, 50411, Tartu, Estonia

‡

S Supporting Information *

ABSTRACT: Adsorption of carbon dioxide in the metal− organic framework CPO-27-Mg (Mg-MOF-74) is examined. We use accurate quantum chemical ab initio methods (wave function-type electron correlation methods for cluster models combined with density functional theory for periodic systems) to calculate gas−surface site and gas−gas interactions. At 298 K, the “zero-coverage” enthalpy and Gibbs free energy of CO2 adsorption on Mg2+ sites are −46 and −9 kJ/mol, respectively; for linker sites these values are −30 and +5 kJ/mol, respectively. For full monolayer coverage lateral interactions from nearby molecules contribute −6 and −5 kJ/mol to the adsorption enthalpy for CO2 at Mg2+ and linker sites, respectively. The predicted heats of adsorption and free energies of adsorption agree within 2.6 and 0.8 kJ/mol, respectively, with experimental values well within chemical accuracy limits (4.2 kJ/mol). We use two different ways of calculating isotherms from equilibrium constants for individual sites and interaction energies: (i) a Langmuir model, augmented with the mean-field (MF) approximation for lateral interactions, and (ii) grand canonical Monte Carlo (GCMC) simulations on a lattice of sites, which agree very well with each other. We use GCMC data to examine how different isotherm models (Langmuir, dual-site Langmuir, Sips, Toth, and mean-field) fit them. We conclude that the MF model yields the best fit over a wide pressure range with physically meaningful parameters, i.e., adsorption constants for individual sites and lateral interaction energies. bar)3,13,15,18 which is mainly attributed to a high concentration of accessible strong adsorption sites (undercoordinated metal cations).10 The key characteristics for the adsorption properties are adsorption isotherms, θ = f(P, T), and their reliable prediction is prerequisite to a rational design of improved materials with properties optimized for a specific target. The heart of any isotherm model is the description of the gas−solid interaction. In the Langmuir model the one and only parameter is the equilibrium constant for the binding of a gas molecule to an isolated adsorption site. This model is valid for very low surface coverages and homogeneous surfaces. At higher loadings the adsorbed molecules have additional interaction with neighboring molecules, and the Langmuir predictions deviate from experiment. These lateral interactions add to the adsorption equilibrium constant and binding to the surface becomes coverage-dependent. That is, the surface coverage is a function of the surface coverage itself (θ = f(θ, P, T)), and analytical isotherm models can only approximate this behavior. The widely used empirical Freundlich, Sips (Langmuir−Freundlich), and Toth isotherm models19 can fit experimental data in

1. INTRODUCTION Carbon capture is seen as a potential near-term technology for reduction of the carbon dioxide emission in carbon-based energy production by retrofitting existing power plants with postcombustion CO2 separation units.1 A major challenge in achieving this is reducing the energy consumption/cost of separating large quantities of gas mixture components, mainly CO2 from N2. Among the gas mixture separation techniques, selective adsorption by porous solids has the advantage that tailoring the adsorbent properties combined with optimization of the separation process makes it possible to lower the energy needed.2,3 Metal−organic frameworks (MOFs) are a new class of porous solids that have high potential to outperform classical adsorbents (silica gels, zeolites, activated alumina, or carbons) because they can have much larger surface areas (exceeding even 6000 m2/g) and their chemical composition can be varied widely to find the optimal gas−solid interaction strength.4−9 Numerous MOFs have been screened for CO2 adsorption. 3,10−14 As representative system we examine CO 2 adsorption in the metal−organic framework CPO-27-Mg,15 which is also known as Mg-MOF-74 or Mg/DOBDC (DOBDC stands for 2,5-dioxido-1,4-benzenedicarboxylate).16,17 The magnesium member of the CPO-27-M isostructural series (M = metal ion) has been shown to selectively adsorb CO2 at low partial pressures (below 1 © XXXX American Chemical Society

Received: March 24, 2017 Revised: May 22, 2017

A

DOI: 10.1021/acs.jpcc.7b02806 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

Figure 1. (A) Conventional unit cell of CPO-27-Mg doubled in the c-direction (2xcuc) with one CO2 molecule on a Mg2+ site (ball and stick style) and the other on a linker site (stick style). (B) 6B model cut from the 2xcuc structure. (C) Distances (in picometers) between the neighboring CO2 molecules adsorbed on Mg2+ and linker sites in monolayer coverage. The CO2 molecule drawn in yellow is considered when calculating the adsorption energy for the linker site.

allows deriving adsorption constants for individual sites and lateral interactions from measured isotherms.

wider range than the Langmuir isotherm, but these models are not thermodynamically consistent,20 and thus, the parameters of these models cannot strictly be regarded as thermodynamically meaningful quantities of adsorption. We have shown21−23 that experimental isotherms can be reproduced when adsorption energies are calculated ab initio with chemical accuracy, and then used in a Langmuir model augmented with a Bragg−Williams model for lateral interactions. The latter is also known as mean-field (MF) approximation, and hereafter we refer to these isotherms as computed MF isotherms. In this study, we use first quantum chemical ab initio calculations and molecular statistics to calculate MF isotherms for CO2 in the metal−organic framework CPO-27-Mg. These isotherms closely agree with the experimental ones, if the experimentally determined site availability is taken into account. In case of CPO-27-Mg, all the Mg2+ metal ion sites are equivalent and, thus, form a homogeneous surface for adsorption. This makes it possible to study the effect of lateral interactions on adsorption without the influence of surface heterogeneity.19 In the second part of this study, we use the same ab initio energies for molecule−surface and molecule− molecule interactions as we have used for the mean-field calculations as input for grand canonical Monte Carlo (GCMC) simulations on a lattice of adsorption sites.24 This way we obtain isotherms without having to assume an isotherm model such as the Langmuir model.24 The very close agreement we reach between the mean-field and GCMC results supports the use of the former also for isotherm predictions for systems with significant lateral interactions. For the system studied we find that CO2···CO2 lateral interactions have a significant effect basically from the beginning of the surface filling when the adsorbed amount increases linearly with increasing pressure (Henry region). This linear behavior is generally associated with adsorption on isolated sites. In the third part of this study we use the GCMC ab initio isotherms to test different isotherm models (Langmuir, dualsite Langmuir, Sips, and Toth) that are often applied to fit experimental data and compare their performance with the MF isotherm. We reach the conclusion that the mean-field model yields the most stable results for the fitting parameters. This

2. MODELS AND METHODS 2.1. Models. The CPO-27-Mg framework consists of onedimensional pores. The cross sections of the pores have hexagonal shape with Mg2+ ions located at the corners and connected with DOBDC4− anions that form the walls of the channels (Figure 1). For CPO-27-Mg two different adsorption sites have been identified by neutron diffraction, the Mg2+ ion and the DOBDC4− linker sites,25−27 which will be referred to as Mg2+ and linker sites, respectively. In the primitive unit cell (puc) there are six sites of each type (Mg2+ and linker), and in the conventional cell (cuc) there are 3 times as many. At the full monolayer coverage all these sites are populated with adsorbed molecules. The finite-sized models for the empty MOF, the MOF with one CO2 adsorbed on Mg2+ site, and the MOF with one CO2 on a Mg2+ site and one CO2 on a linker site were cut out from structures of the conventional unit cell that was doubled in the c-direction, 2xcuc (Figure 1A). These model systems, named 6B (Figure 1B), consist of six consecutive Mg2+ cations and benzene rings. They are large enough to include a major share of the long-range effects.22,28 For adsorption on the Mg2+ site, the CO2 molecule marked in yellow is not present, and it is the “gray red” CO2 molecule in Figure 1B that is adsorbed. For adsorption on the linker site, the “gray red” CO2 molecule is always present, and it is the “yellow” CO2 molecule that is adsorbed. 2.2. Quantum Chemical Calculation of Structures and Energies. The hybrid MP2:PBE+D2 adsorption energy, ΔEaMP2:PBE+D2, was obtained according to the subtractive scheme.29,30 ΔEaMP2:PBE+D2 = ΔEapbcPBE+D2(S) + ΔEaHL−corr(C)

(1)

ΔEaHL−corr(C) = ΔEaMP2(C) − ΔEaPBE+D2(C)

(2)

ΔEpbcPBE+D2 (S) a

where is the adsorption energy calculated with PBE+D2 under periodic boundary conditions (pbc) for the full PBE+D2 periodic structure (S), and ΔEMP2 (C) are a (C) and ΔEa adsorption energies for finite-sized 6B model systems C calculated by MP2 and PBE+D2, respectively (Figure 1B). B


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Table 1. Carbon Dioxide Adsorption Energies (kJ/mol) Calculated with Different Methods on Periodic PBE+D2 Structures and the 6B Model Cut Out from These Periodic PBE+D2 Structuresa Mg2+ site PBE+D2 PBE//PBE+D2 D//PBE+D2 MP2/CBS//PBE+D2 ΔEHL‑corr ΔCCSD(T)/CBSc finald

linker site ΔE

b

6B

pbc

−37.7 −18.4 −19.3 −46.9 −9.2 1.9

−41.5 −20.3 −21.2

6B

pbcb

ΔELR

−24.4 −4.7 −19.7 −33.9 −9.5 2.9

−26.2 −5.5 −20.7

−1.8 −0.8 −1.0

LR

−3.8 −1.9 −1.9

−48.8

−32.8

MP2 and CCSD(T) energies are corrected for BSSE. The long-range and high-level correction energies, ΔE and ΔEHL‑corr, are calculated according to eqs 4 and 2, respectively. bPeriodic boundary conditions. cCalculated on smaller models, see Supporting Information section S2. d Hybrid MP2/CBS:(PBE+D2)+ΔCCSD(T)/CBS at the PBE+D2 structure. a

LR

2.4. Enthalpy and Gibbs Free Energy of Adsorption. For site i at a temperature T, ΔHa,i(T) and ΔGa,i(T), respectively

The MP2 adsorption energies are extrapolated to the complete basis set limit, denoted as CBS. Alternatively, the hybrid MP2:PBE+D2 energy can be written as ΔEaMP2:PBE+D2 = ΔEaMP2(C) + ΔEaLR (S , C)

ΔHa, i(T ) = D0, i + ΔEthermal, i(T ) − RT

(3)

with the long-range correction to the MP2 cluster energy defined as ΔEaLR (S , C) = ΔEapbcPBE+D2(S) − ΔEaPBE+D2(C)

ΔGa, i(T ) = D0, i − RT ln

Q gas

(T )

(4)

(7)

are given by the interaction energy between the CO2 molecule and the MOF surface at 0 K, D0,i, i.e., including the zero-point energy changes, ΔEZPV,i

The final estimate for the adsorption energy ΔEafinal = ΔEaMP2:PBE+D2 + ΔEaCCSD(T)

Q isurface

(6)

(5)

D0, i = ΔEa,final i + ΔE ZPV, i

also includes higher order correlation effects, ΔECCSD(T) , that a are estimated as ΔCCSD(T)/CBS = CCSD(T)/CBS − MP2/ CBS for smaller models (see the Supporting Information for details of the models) and added to the MP2/CBS:PBE+D2 adsorption energy. Following previous work on CH4 in CPO-27-Mg,22 lateral interactions energies, ELat, are calculated by CCSD(T)/CBS as dimer formation energies for isolated pairs of CO2 molecules. These are single-point calculations with the positions of the dimer atoms cut out from the periodic PBE0+D2 structures for the primitive unit cell loaded with 12 CO2 molecules. The 12 CO2 molecules were placed into the primitive unit cell according to diffraction data by Queen et al.26 and the whole structure optimized. The hybrid PBE0 functional is used because it improves the description of the CO 2 ···CO 2 interaction energies as test calculations for the gas-phase dimers31 have shown (see Supporting Information, Table S1). 2.3. Vibrational Frequencies. For the adsorbed CO2 molecules, they are obtained as follows. In the 6B model system we fix the hydrogen atoms of the benzene rings and the atoms on the border of the model system for which the neighboring atoms in the periodic crystal are missing. The inner part of the model system and the position of adsorbed CO2 molecules are then optimized with MP2 using a split-valence polarization (SVP) basis set. The vibrational frequencies are calculated only for the adsorbed molecules and their motions relative to the surface (partial Hessian). The lowest external vibrational mode that corresponds to the hindered rotation of CO2 is approximated by a one-dimensional free rotation (see Supporting Information section 3 for more details). This way we take the observed32−34 enhanced CO2 motion at the adsorption site into account. The effect of different approximations to the vibrations on thermodynamic functions see Supporting Information sections S4 and S5.

(8)

and the thermal energy change upon adsorption, ΔEthermal,i(T). The pressure−volume work is given by RT, and Qgas and Qsurface i are the partition functions of carbon dioxide in the gas phase and on the surface, respectively. The latter, Qsurface , is calculated i from the populated (discrete) vibrational energy levels that takes into account that the adsorbed molecules are characterized as bound systems, and thus, their energy states on the surface are quantized. 2.5. Adsorption Isotherm Models. The Gibbs free energy of adsorption defines the adsorption equilibrium constant, Ka,i K a, i = e−ΔGa,i / RT =

Q isurface Q gas

e−D0,i / RT

(9)

which, in turn, gives the Langmuir adsorption isotherm θ=

na = nMax

∑ i

K a, iP 1 + K a, iP

(10)

where θ is the surface coverage, n and nMax are the absolute amount of gas molecules on the surface at a given pressure and maximum amount of molecules possible to adsorb on all the sites considered, respectively. As a limiting case for θ → 0 this simplifies to the Henry isotherm a

θa, i = K a, iP

(11)

The absolute amount of pure gas adsorbed on sites i calculated with the mean-field adsorption isotherm, na,MF (eq 12), depends i also on the surface coverage θi = C

nia,MF K iMF(θ )P = nMax 1 + K iMF(θ )P

(12) DOI: 10.1021/acs.jpcc.7b02806 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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Table 2. Zero-Point Vibrational Energy Changes, ΔEZPV, Thermal Energy Changes, ΔEthermal, Adsorption Enthalpies, ΔHa, Adsorption Entropy Terms, TΔSa, Adsorption Gibbs Free Energies, ΔGa (All in kJ/mol), and Adsorption Equilibrium Constants, Ka (1/atm) Calculated for CO2 Adsorption on Different Adsorption Sites in CPO-27-Mg

The adsorption sites are no more assumed to be isolated from each other (as in the classic Langmuir model) because the mean-field adsorption equilibrium constant, KMF i , includes the coverage-dependent effective adsorbate−adsorbate “lateral” interaction energy term, Eeff Lat,i K iMF(θ ) = e−ΔGa,i(θi)/ RT = eff av E Lat, i(θi) = E Lat, iNθi

Q isurface Q

gas

eff

e−(D0,i + E Lat,i(θi))/ RT

298 K

(13)

Mg ΔEZPV ΔEthermal ΔHa TΔSa ΔGa Ka

(14)

Eav Lat,i

with and N being the average lateral interaction energy between two neighboring molecules and the number of interacting neighbors, respectively. Under the mean-field approximation, the effective lateral interactions of a molecule adsorbed at site i become a linear function of surface coverage, θi (see the Supporting Information for further derivation).

2+

2.9 2.3 −46.1 −36.9 −9.2 41.3066

343 K linker

Mg

1.6 3.3 −30.4 −35.5 5.1 0.12861

2+

3.0 −45.7 −42.1 −3.7 3.6349

linker 4.0 −30.0 −40.5 10.4 0.02596

smaller initial isosteric heats of adsorptions of −40.5 and −39.7 kJ/mol, respectively. Our calculated Gibbs free energy of adsorption of −9.2 kJ/mol obtained for CO2 binding to Mg2+ sites at 298 K deviates by less than 1 kJ/mol from the experimental value (−10.0 kJ/mol) which we obtain as an average from fitting two different sets of measured low-coverage experimental adsorption data9,15 (up to loading 0.5 CO2 per Mg2+ ion) with an MF-isotherm model (see section 3.3.2 below). At a loading of approximately one CO2 molecule per Mg2+ ion, the experimental heat of adsorption drops to 24−32 kJ/ mol, which is assigned to adsorption at linker sites.15,26,37,38 The calculated CO2 adsorption enthalpy for linker sites, −30.4 kJ/mol, is in very good agreement with the latest measurement26 (−32 kJ/mol) and with MP2-derived force field result (about −30 kJ/mol).35 With increasing surface coverage the interaction energies between CO2 molecules at neighboring sites add to the total adsorption energy. The contribution from each pair of molecules (see Figure 1C) to the total lateral interaction is presented in Table 3. For the full monolayer the maximum lateral interaction contributions from nearby molecules on the surface are −6.0 and −5.4 kJ/mol for CO2 at Mg2+ and linker

3. RESULTS AND DISCUSSION 3.1. Ab Initio Mean-Field Isotherms. 3.1.1. Adsorption Energies and Thermodynamic Functions. The different contributions to the final estimate for the adsorption energy are listed in Table 1. They are obtained as PBE+D2 −corr ΔEa,final (pbc) + ΔEa,HL (6B) + ΔEa,CCSD(T)/CBS i = ΔEa, i i i (15)

Adsorption on Mg2+ and linker sites is controlled by dispersion which contributes 50 and 80%, respectively, to the PBE+D2 adsorption energy. The periodic PBE+D2 adsorption energies for CO2 on isolated Mg2+ sites calculated here match exactly the values calculated with B3LYP+D* for loading of one CO2 per Mg2+ in our previous study, −41.5 kJ/mol,28 and are very close to density functional theory (DFT) calculations with a van der Waals functional, which yielded −42.6 kJ/mol.35 Previous studies found that MP2 predicts 2−5 kJ/mol stronger CO2 binding to Mg2+ sites than DFT+D2.28,35 Here, this difference (the high-level correction in our hybrid MP2:PBE +D2 scheme) is even larger, 9.2 kJ/mol, which is due to structure relaxation effects. Our MP2 calculations are performed at PBE+D2 structures that are fully optimized both for the empty surface and the surface with adsorbed CO2, whereas in ref 35 the MOF structure is kept frozen on adsorption of CO2. For the MP2 calculations in ref 28, one CO2 molecules was removed from the puc with six CO2 molecules keeping all other atoms fixed. The MP2 and CCSD(T) corrections have a different sign, and the difference between the final estimate and the PBE+D2 result is 7.3 and 6.6 kJ/mol for the Mg2+ and linker sites, respectively. Hence, use of the PBE+D2 adsorption energies would lead to severely underestimated adsorption isotherms. We have shown previously23 that already 4 kJ/mol difference in the (Gibbs free) energy results in a factor of 2 difference in the predicted adsorbed amount. Calculated thermodynamic functions for CO2 adsorption on Mg2+ and linker sites in CPO-27-Mg are presented in Table 2. The CO2 adsorption enthalpy for Mg2+ sites, −46.1 kJ/mol, calculated here for 298 K, agrees within chemical accuracy limits (±4.2 kJ/mol) with the low-coverage CO2 isosteric heat of adsorption of 43.5 kJ/mol measured recently by Queen et al.26 and with the average of different previous experiments (44.5 kJ/mol).28 Alternative ab initio approaches that use grand canonical Monte Carlo simulations based on force fields fitted on MP235 or B2PLYP-D2 DFT results36 predict somewhat

Table 3. Distances R (pm) between Neighboring CO2 Molecules Adsorbed on Mg2+ and Nearby Linker Sites in CPO-27-Mg, Lateral Interaction Energies for a Pair of CO2 Molecules, Epair Lat , and Average Lateral Interaction Energies Per One CO2, Eav Lat (kJ/mol) sites

R C···Oa

R C···Cb

c Epair Lat

d Eav Lat

no. of neighbors

Mg2+···Mg2+ linker···linker Mg2+···linker

364.3 372.7 309.3 301.0 372.1 400.8

466.3 427.9 396.7 363.5 438.7 491.1

−2.81 −2.24 −4.70e −4.12 −1.68 −0.55

−1.405 −1.119

2 2

−1.057

3

a OCO···C(O)2 distance between the closest C and O atoms in neighboring CO2 molecules. b(O)2C···C(O)2 distance between the centers of the two neighboring CO2 molecules. cCCSD(T)/CBS energy per pair of molecules; PBE0+D2 structure with loading of 12 CO2 molecules in the primitive unit cell (puc). dCalculated according to eq S3. eNote that this part of lateral interaction energy is already included in the adsorption energy for linker sites, see section 2.1 and Figure 1.

D


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The Journal of Physical Chemistry C sites, respectively, reaching as much as 16.5% of the adsorption energy for the linker sites (−32.8 kJ/mol). 3.1.2. Comparison with Measured CO2 Adsorption Isotherms. The most direct comparison between ab initio prediction and experiment is at the level of isotherms because calculated data points can be compared with experimental ones without any fitting. All measured adsorption isotherms are excess quantities, whereas our calculations yield the absolute adsorbed amount. This difference is negligible for relatively low pressures but cannot be ignored for conditions where the gasphase density becomes comparable to the adsorbed phase density, i.e., at high pressures. The excess adsorbed amounts, nσ, are calculated from the absolute surface coverage, na, by subtracting from it the amount of CO2 gas, ng, contained in the same volume, Vamna, as the adsorbate occupies on the surface

n σ = n a − ng

(16)

ρ g V mana M

(17)

ng =

where M is a molar mass of CO2, Vam is an excluded molar volume of adsorbed carbon dioxide (98 cm3/mol) calculated from the average distance between CO2 molecules at Mg2+ and linker sites (Table 3), and ρg is the experimental CO2 density at the temperature and pressure of adsorption.39,40 The computed MF isotherms (eq 12) with the adsorption equilibrium constants calculated ab initio for isolated Mg2+ and linker sites predict for an ideal material an adsorbed CO2 amount that is about 15% higher than that measured by Dietzel et al.,15 see Figure S5 in the Supporting Information. In the pressure range from 0.2 to 1 bar, the adsorbate−adsorbate interactions increase the calculated surface coverage further by 20% on average and the amount of CO2 adsorbed in fully activated perfect crystals at 298 K and 1 bar is 11.4 mmol/g. In the real material not all adsorption sites may be accessible for adsorption. This is often evidenced by a rapid decrease of the isosteric heat of adsorption at loadings lower than one adsorbed molecule per metal ion site,9,15,26,37,38 by an inflection point in the adsorption isotherm at loading of 76.5% metal ion sites,15 or by obtaining lower maximum adsorption capacities than expected for the ideal material when fitting measured adsorption data with analytic isotherm models. Figure 2 shows that isotherms measured in wide pressure range at 298 and 343 K are reproduced by computed MF isotherms for ideal structures after the latter are scaled down to take into account that only 76.5% of the sites in these crystals are available for adsorption, as suggested for this particular sample15 (see Figures S6 and S7 in Supporting Information for comparisons with additional measurements). The calculated CO2 MF isotherm at 313 K also agrees very well with the adsorbed amounts obtained from GCMC simulations using an ab initio force field (Figure S6 in Supporting Information). In the latter approach, quantum effects on nuclear motion are not taken into account and MOF structure is assumed to be fixed. The absolute and excess surface coverages in Figure 2 are virtually the same up to 5 bar. Hence, it is justified to use absolute adsorption models for gas adsorption on Mg2+ sites only (see below). With increasing pressure (starting from about 10 bar) the density of adsorbed CO2 phase increases rapidly until liquefaction, which results in progressively increasing differences between the absolute and excess isotherms. Calculated excess isotherms do not reach the measured adsorbed excess amounts because our current model includes

Figure 2. Computed mean-field (lines) and experimental (open symbols) isotherms for CO2 adsorption in CPO-27-Mg for two different temperatures, 298 K (green) and 343 K (red). The experimentally determined availability (76.5%) of adsorption sites is assumed (ref 15). Solid lines are for absolute and dashed lines are for excess adsorbed amounts.

only two types of adsorption sites that make up the monolayer (Mg2+ and linker). Molecules adsorbed in the pore center are not included. Evidence for the latter at high loadings has been provided by neutron diffraction.26 3.2. Ab Initio GCMC Simulations on a Lattice of Sites and Analysis of the Mean-Field Model. 3.2.1. Ab Initio GCMC Simulation on a Lattice of Sites. The GCMC simulation method for a lattice of adsorption sites24 offers the possibility to calculate isotherms point by point without assuming any isotherm model or making any approximation on the lateral interactions as we did with eq 12. Figure 2 in ref 24 shows that the GCMC simulations reproduce the experimental data points. The Gibbs free energy of adsorption, ΔGa,i(θi), for a given coverage can be obtained from the calculated adsorption isotherm (θi, P) data points: ⎛θ ⎞ ΔGa, i(θi) = −RT ln⎜ i ⎟ ⎝P⎠

(18)

Figure 3 shows the Gibbs free energies of adsorption as a function of coverage calculated from the GCMC simulated isotherms using eq 18 for two different temperatures, 298 and 343 K. The free energies appear to be constant up to filling almost half of the Mg2+ sites, but the change in derivatives of these free energies (calculated numerically by central difference) from negative to positive indicates that at certain coverage the adsorption free energy reaches a minimum. 3.2.2. Analysis of the Mean-Field Model. For further analysis we will derive an expression for the Gibbs free energy of adsorption under mean-field approximation. We rewrite the mean-field isotherm (eq 12) E


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energy derivative with respect to surface coverage to zero (for details see the Supporting Information): θmin, i = 1 +

(19)

Exchanging (1 − θi) and P, taking logarithm of both sides, and substituting the expression of KMF from eq 13 we obtain i eff ⎛ Q surface ⎞ D0, i E Lat, ⎛ θi ⎞ i i ⎜ ⎟ ln⎜ ⎟ = ln⎜ − + ln(1 − θi) gas ⎟ − ⎝P⎠ RT RT ⎝ Q ⎠

(20)

The first two terms on the right-hand side of the above equation are negative of the Gibbs free energy of adsorption of an isolated site i, −ΔGa,i, divided by RT. Multiplying both sides of eq 20 by −RT and using eqs 14 and 18 we obtain for the Gibbs free energy of adsorption under the MF approximation. av ΔGa, i(θi) = ΔGa, i + E Lat, iNθi − RT ln(1 − θi) eff = ΔGa, i + E Lat, i(θi) + TScnfg(θi)

(22)

Equation 22 shows that the free energy minimum shifts toward lower coverages with increasing temperature if Eav Lat,i is attractive (negative). Consistent with this prediction, with the temperature increase from 298 to 343 K the GCMC simulated position of the free energy minimum shifts from a total coverage of 16% to 10% CO2/Mg2+. From the coverages at these minima it is possible to calculate the lateral interaction energies utilizing eq 22. The latter are −2.9 and −3.2 kJ/mol at 298 and 343 K, respectively, which are within 4% and 12%, respectively, of the original ab initio values for Mg2+ sites (Table 3) used in our GCMC simulations. Hence, the meanfield model reproduces the lateral interactions used as input for the GCMC simulations. According to eq 22 a minimum appears only if the lateral interactions are attractive (Eav Lat,i < 0) and stronger than the thermal energy, i.e., |Eav Lat,i| ≥ RT. If the attractive lateral interaction are weaker than thermal energy, |Eav Lat,i| < RT, then, θmin,i < 0, which is not a physically meaningful solution. As the surface coverage cannot be more than unity (θmin,i > 1) there will be also no free energy minimum if the molecules repel each other on the surface, i.e., Eav Lat,i > 0, or if the molecules do not interact on the surface (Eav Lat,i = 0) as the Langmuir model assumes. The GCMC simulations show that at 298 K and at surface coverage of one CO2 per Mg2+ 95% of all the metal ion sites are populated, whereas only 5% of all the linker sites are filled with CO2. Almost complete saturation of Mg2+ sites and the start of the population of energetically much weaker linker sites brings about a large change in the adsorption free energies. This coverage is marked by the inflection point in the adsorption free energy increase with increasing surface coverage (Figure 3, top), and it is complemented by the pronounced maximum in the derivative of the free energy as a function of the loading (Figure 3, bottom). Consequently, the position of this maximum can be used for determining the number of adsorption sites available in experimental adsorption measurements. After saturation of the metal ion sites only the lateral interactions between gas molecules adsorbed on the linker sites add to the adsorption free energy as the interactions between molecules at linker and Mg2+ sites remain constant. The former interaction energy (−2.2 kJ/mol) is slightly weaker than the thermal energies (−2.5 and −2.9 kJ/mol at 298 and 343 K, respectively), and because of the absence of the minimum on adsorption free energies, it becomes impossible to determine the lateral interactions energies for the linker sites. 3.2.3. Ab Initio GCMC Isotherms for Adsorption on Mg2+ Sites Only. To study the performance of the different models without surface heterogeneity effects in the following we consider gas adsorption on the Mg2+ sites only. This is justified because the population of the linker sites starts only at 0.04 and 0.3 bar for 298 and 343 K, respectively (Figure S8 in Supporting Information), after 80% and 70% of Mg2+ sites are occupied. We use simulated GCMC adsorption data up to a maximum linker site population of 1%. Thus, the CO2 pressure is kept below 0.04 and 0.3 bar for 298 and 343 K, respectively. These conditions are the most relevant from the technological

Figure 3. Top figure shows ΔGa(θ) obtained from GCMC simulation as a function of total coverage (θ) at 298 K (green) and 343 K (red). The bottom figure shows the variation of the numerical derivative of ΔGa(θ) with θ at those temperatures. The black dotted line represents the value 0. The position of the minimum has been obtained graphically, i.e., where the green (298 K) or red (343 K) solid line meets the black dotted line. Green and red dashed lines are the position of the shallow minimum at 298 K (at 16% CO2/Mg2+) and 343 K (at 10% CO2/Mg2+), respectively. Upon increase of temperature, the position of the minimum is found to shift further to lower coverage. The pronounced maxima of the derivatives at 100% CO2/Mg2+ signifies the saturation of Mg2+ sites.

θi = K iMFP 1 − θi

RT av E Lat, iN

(21)

The first term of the right-hand side of the above equation, ΔGa,i, is the coverage-independent part and related to the adsorption equilibrium constant, Ka,i, by eq 9. The second term, Eeff Lat,i (θi), and third term, TScnfg(θi), are the coverage-dependent free energy contribution coming from the lateral interaction energy and the configurational entropy, respectively. The position and conditions for the minimum, eq 22, are derived from the mean-field analysis (eq 21) by setting the free F


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The Journal of Physical Chemistry C point of view because the major applications of MOFs like gas storage and separation are related to gas binding just at the open metal ion sites. Figure 4 shows the adsorption of CO2 at conditions where only the Mg2+ sites are populated. The GCMC simulated

Figure 4. Comparison of Henry (violet dashed line) and Langmuir (black broken line) isotherms with GCMC simulated isotherms (green triangles) for CO2 at 298 K. Figure 5. Top: ΔGa(θ) obtained from GCMC data points (green triangles) at 298 K. The violet dashed line is the coverage-independent free energy of adsorption, ΔGa,i. The black broken and red solid lines are the adsorption free energies obtained from the Langmuir model and mean-field models, respectively. Bottom: contributions of the different coverage-dependent terms to the free energy. The black broken line is the configurational entropy contribution, TScnfg(θ), and the red broken line is the mean-field lateral interaction contribution, Eeff Lat(θ). The red solid line is the sum of these two terms, and the black dotted line represents the value 0. The green squares are the lateral eff , calculated from the GCMC simulations interaction terms, ELat according to eq S2 in Supporting Information section S6.

adsorbed amounts are compared with Henry and Langmuir isotherms (eqs 11 and 10, respectively) that neglect all adsorbate−adsorbate interactions. These isotherms are calculated using the same gas−solid interaction energy (ΔGa) which results in virtually the same linear increase of the adsorbed amounts up to 2% population of the Mg2+ sites. From that coverage on the Langmuir isotherm starts to deviate from the GCMC reference which continues to increase linearly until 40% of the Mg2+ sites are filled. Figure 5 shows the corresponding adsorption free energies (calculated from GCMC isotherm data points, eq 18, and analytically, eq 21) together with the lateral interactions and configurational entropy terms. The latter (black broken line, bottom) starts to increase immediately from the zero coverage as filling of the surface hinders adsorption. At the same time also the contribution from the lateral interactions (red broken line, bottom) increase, but this provides additional stabilization for the adsorbed phase due to the attraction between the CO2 molecules on the surface. Because the adsorbate−adsorbate interactions cancel the increase of the configurational entropy the total adsorption free energy (red solid line, bottom) is virtually constant until a coverage of 40% of Mg2+ sites is reached. The analytical free energies of adsorption calculated with the mean-field approximation (eq 21, red line in Figure 5, top) corroborate very well with that obtained from GCMC simulated isotherm points (green triangles), but the minimum of the adsorption free energy curve is shallower and it appears at lower coverage (12% of Mg2+ sites occupied). At moderate Mg2+ site coverages there is a difference because the averaged mean-field predicts a smaller number of pairs of adsorbed molecules, and thus, there are less attractive lateral interactions (0.3 kJ/mol at most) than in GCMC simulations where the effective lateral interactions, Eeff Lat, are calculated exactly using the pair probability functions, φi,j. In all, the very good agreement between GCMC and mean-field models justifies the use of MF isotherms for adsorption studies. 3.2.4. Comparison with Gases with Weak Lateral Interactions. At 298 K, the CO2···CO2 interaction energy is stronger than the thermal energy, whereas for CO and N2, which we have studied before,23 the lateral interaction energies,

−0.34 and −0.35 kJ/mol, respectively, are less than 20% of RT. Figure 6 shows the coverage-dependent adsorption free

Figure 6. Adsorption free energies (kJ/mol) calculated from the experimental isotherms as a function of coverage for CO2 (green triangles), CO (blue squares), and N2 (red diamonds). The 76.5% scaling for the availability of sites is assumed (refs 15 and 41). The black broken line shows the configurational entropy contribution, TScnfg.

energies, ΔGa(θ), calculated from the experimental isotherms, for these gases as a function of the coverage relative to the equilibrium adsorption free energy of each gas, ΔGa. The latter is obtained by linear extrapolation to the zero coverage of the first four points of the adsorption free energies, calculated according to eq 18 from experimental isotherms.15,41 Thus, G


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Table 4. Parameters for the Analytical Isotherm Models (See Supporting Information Section S11 for Analytical Expressions) Obtained from Fitting GCMC Adsorbed Amounts up to Denoted Different Populations of the Mg2+ Sites at 298 and 343 K 298 K isotherm models b

ab initio Henry Langmuirc dual-site Langmuir Sips Toth mean-field

parameters Ka (1/atm) KH (1/atm) K (1/atm) K1 (1/atm) K2 (1/atm) S1 S2 T1 T2 K* (1/atm) L −L RT (kJ/mol)

0.2

a

0.8 41.307

42.870 50.464 23.217 23.217 109.543 0.881 c c 41.243 1.620 4.0

343 K a

0.2

a

b

0.7a 3.635

68.064 47.377 7.153 281.703 0.763 536.556 1.644 41.805 1.549 3.8

b

3.645 4.359 1.990 1.990 6.148 0.904 15.649 2.106 3.626 1.352 3.9

5.518 3.567 0.787 9.860 0.791 8.780 1.615 3.593 1.413 4.0

Highest CO2 surface coverage (population of Mg2+ sites) that was used in fitting. bAdsorption equilibrium constant directly obtained from quantum chemistry and molecular statistics, see eq 9 and Table 2. cNo reasonable fit at this coverage. a

To the first approximation, the coverage is obtained from a Langmuir isotherm or, as a further simplification, just by the pressure, i.e., assuming a Henry isotherm. This yields an expression K*MF as basis for fitting.

only the lateral interaction and configurational entropy terms are plotted. For CO and N2 the lateral interactions are so weak that there is virtually no influence on adsorption free energies; that is, for these gases the adsorption free energy starts to increase from zero coverage closely following the configurational entropy increase of the ideal Langmuir model. This is clearly different from CO2 adsorption where the adsorption free energy increase starts only at 20% loading of Mg2+ sites. After that coverage the lateral interactions do no longer counterbalance the more rapidly increasing configurational entropy. Nevertheless, such a plot makes it possible to qualitatively assess the presence of adsorbate−adsorbate interactions in the system. 3.3. Test of Different Isotherm Models for Fitting Experimental Data. Determining experimental values for enthalpies, entropies, and free energies of adsorption from measured isotherms requires fitting. Most often Henry, Langmuir, dual-site Langmuir, Sips, or Toth isotherm models are used (see Supporting Information section S11 for analytical expressions).13,19,38,41,42 Fitting measured adsorption data with a mean-field isotherm has the advantage that separate parameters can be found for the gas−surface and for the coverage-dependent lateral interactions. 3.3.1. Mean-Field Adsorption Isotherm for Fitting. For that, in the mean-field adsorption equilibrium constant, eq 13, is written as K MF(θ ) =

Q surface −D0 / RT Lθ e e Q gas

K *MF = K *e Lθ* = K *e LK

Q surface −D0 / RT e Q gas

(23)

(24)

which is essentially the Langmuir adsorption equilibrium constant (eq 9). The parameter L in the second exponential term is the effective average lateral interaction energy, eq 14, multiplied by the surface coverage. L=

av E Lat, iN

RT

≈ K ′e L ′ P

(26)

The surface coverage that was a function of itself and the pressure becomes now a function of the pressure only. This makes the MF isotherm directly usable for fitting with two parameters K* and L or K′ and L′. 3.3.2. Fitting Pure Gas Adsorption Data. Table 4 shows the parameters of different isotherm models (see Supporting Information section S11 for analytical expressions) obtained from fitting the same CO2 adsorbed amounts calculated with GCMC. The fitting has been done for data points up to different loadings. The parameters for the Langmuir, dual-site Langmuir, Sips, and Toth isotherms are very dependent on the fitted rangethe parameters for the adsorbent−adsorbate interaction may vary more than 50% for fits using low (20%) or high (up to 80%) populations of Mg2+ sites. Nonetheless, all these isotherms fit perfectly with the GCMC data. The parameters for the isotherms depend on the surface coverage that is used for fitting because different shares of lateral interactions are present, and hence, the extrapolated and interpolated adsorbed amounts predicted by these isotherms are different. The Henry constant is in better agreement with the original adsorption equilibrium constant (used as input for GCMC) than the fitted Langmuir parameter because the lateral interactions compensate the missing configurational entropy term in the Henry isotherm (see Figure 5 above). On the contrary, the parameters for the MF isotherm, eq 26, are much less sensitive to the surface coverage used for fitting, differing less than 1.5% only. Moreover, the parameter K* that characterizes adsorbate binding to the surface is very close to the ab initio equilibrium adsorption constant used as input for the GCMC simulations. The adsorption Gibbs free energies calculated from parameters K* (−9.25 and −3.65 kJ/mol for 298 and 343 K, respectively) reproduce within 0.04 kJ/mol the values that were used for simulations of the adsorbed amounts (Figure 7, Table 2). The adsorption enthalpies, −45.9 and −46.4 kJ/mol, determined from the temperature dependence of the K* for the fits up to coverage of 0.2 and 0.8 of the Mg2+ sites, respectively,

with separate parameters for adsorbent−adsorbate (D0) and adsorbate−adsorbate (L) interactions. For the former, K* is introduced

K* =

*P /(1 + K *P)

(25) H


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adsorbate interactions in the system directly based on the adsorption measurements. From the application point of view the CO 2 ···CO 2 interactions at the surface increase the surface coverage by 25% on average in the range of partial pressures from 0.1 to 0.5 bar (i.e., compositions of natural and biogas, respectively) at 343 K; that is, in addition to gas−solid interactions also the gas−gas interactions must be overcome for regenerating the adsorbent. MF-isotherm models can be used in both waysto predict isotherms based on calculated ab initio free energies and lateral interactions or to obtain experimental values for thermodynamic functions of adsorption by fitting measured adsorbed amounts. With respect to fitting, the MF-isotherm model compared to others has the advantage of having clearly separated physically meaningful parameters for the adsorbate− adsorbent and adsorbate−adsorbate interactions. The first is independent of the pressure and is a true characteristics of gas− solid interaction strength. The lateral interactions are taken into account as a function of pressure instead of being dependent on the surface coverage. This makes it possible to directly fit measured adsorption data with nonlinear regression methods and obtain more accurate thermodynamic functions for adsorption together with an estimate for adsorbate−adsorbate interactions.

Figure 7. Amounts of CO2 adsorbed only on Mg2+ sites in CPO-27Mg at 343 K calculated with GCMC including exact treatment of lateral interactions (triangles) (ref 24). The solid red line is the meanfield isotherm fit of the GCMC adsorbed amounts, the broken green line is the Langmuir isotherm calculated using K*, and the dotted red line is surface coverage calculated with the Langmuir model with Ka.

are also in good accordance with the original adsorption enthalpy at 298 K, −46.1 kJ/mol. The ΔGa and ΔHa values that are obtained from the fitted MF-isotherm parameters can reproduce up to 99.6% of the entropy loss upon adsorption that was originally calculated with ab initio. These comparisons show that from the fitted parameters of the MF isotherm it is possible to get reliable thermodynamic functions of adsorption. The most important advantage of fitting (experimental) adsorption data with MF isotherms is the possibility to get an estimate for the lateral interaction energies. The effective adsorbate−adsorbate interaction energies for CO2 (−3.8 to −4.0 kJ/mol) calculated from the fitting parameter L show little variation with the fitting range or the temperature. The fitted values are close to the directly calculated value to the interaction energy between two of these molecules adsorbed on Mg2+ sites (−2.8 kJ/mol, for CO2, Table 3).

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.7b02806. Quantum chemical calculation of structures and energies, calculations of higher order correlation effects on small model systems, vibrational frequencies, comparison of partition functions used for the calculation of adsorption isotherms, comparison of thermodynamic functions calculated with different methods, mean-field approximation for lateral interactions, coverage-dependent adsorption free energy and the appearance of its minimum, determination of the number of available adsorption sites for real samples, comparison of different calculated and measured CO2 adsorption isotherms, the occupation of Mg2+ and linker sites at different CO2 pressures, and analytical expressions for isotherm models (PDF)

4. CONCLUSIONS Previously, it has been shown that accurate quantum chemical calculations beyond density functional theory in combination with molecular statistics yield adsorption isotherms in close agreement with experimental ones.21−23 This was shown for CH4, CO, and N2 adsorption in CPO-27-Mg/Mg-MOF-74 and H2 in MOF-5 where lateral interactions are weak. Here, we have shown that this methodology is also applicable to systems with significant lateral interactions such as CO2 adsorption in CPO-27-Mg. The isotherms are obtained from free energies for individual adsorption sites and interaction energies between CO2 molecules at neighboring sites, either point by point from GCMC simulations on the lattice of sites or by adopting the mean-field Langmuir model. Both approaches yield results in close agreement with each other and with experiment. For CO2 adsorption in CPO-27-Mg, the lateral interactions are prominent basically from the beginning of the surface filling. Binding next to already adsorbed CO2 molecules is energetically more favorable. The attractive adsorbate−adsorbate interactions reduce the initial increase of the configurational entropy, and consequently, the adsorption free energy starts to increase after filling 40% of Mg2+ sites instead of increasing from zero coverage which is expected for completely isolated adsorption sites. Thus, by plotting ΔGa(θ) = −RT ln(θ/P) versus θ, it is possible to assess the presence of the adsorbate−

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Kaido Sillar: 0000-0002-3434-3867 Arpan Kundu: 0000-0001-5351-3254 Joachim Sauer: 0000-0001-6798-6212 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Dedicated to Professor Frerich J. Keil (Hamburg) who has made significant contributions to simulation of sorption and separation in nanoporous systems. This work has been supported by German Science Foundation (DFG) within the priority program 1570 “Porous media” and with a Reinhart I


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Koselleck Grant to J.S. A.K. is a member of the International Max Planck Research School “Functional Interfaces in Physics and Chemistry”. K.S. is supported by the Estonian Ministry of Education and Research (IUT20-15 and PUT1541). We thank Eric D. Bloch, Jarad A. Mason, and Jeffrey R. Long for providing the adsorption isotherm data for CO and N2 CPO27-Mg.

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K


Ab Initio Adsorption Isotherms for Molecules with Lateral Interactions

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