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Prediction of CO2 Adsorption Properties in Zeolites Using Force Fields Derived from Periodic Dispersion-Corrected DFT Calculations Hanjun Fang,† Preeti Kamakoti,‡ Ji Zang,† Stephen Cundy,‡ Charanjit Paur,‡ Peter I. Ravikovitch,‡ and David S. Sholl*,† †

School of Chemical & Biomolecular Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0100, United States ExxonMobil Research and Engineering, Corporate Strategic Research, 1545 Route 22 East, Annandale, New Jersey 08801, United States

‡

S Supporting Information *

ABSTRACT: We demonstrate a new approach to develop transferable force ﬁelds describing molecular adsorption in zeolites by combining dispersion-corrected density functional theory (DFT) calculations and classical atomistic simulations. This approach is illustrated with the adsorption of CO2 in zeolites. Multiple dispersion-corrected DFT methods were tested for describing CO2 adsorption in sodium-exchanged ferrierite. The DFT-D2 approach was found to give the best agreement with high level quantum chemistry results and experimental data. A classical force ﬁeld for CO2 adsorption in siliceous zeolites was then developed on the basis of hundreds of DFT-D2 calculations that probed the full range of accessible volume in purely siliceous chabazite (Si-CHA) via random sampling. We independently performed experiments with Si-CHA measuring CO2 isotherms and heats of adsorption by microcalorimetry. Excellent agreement was obtained between adsorption isotherms predicted with our ﬁrst-principles-derived force ﬁeld and our experiments. The transferability of this force ﬁeld was examined using available adsorption isotherms for CO2 in siliceous MFI and DDR zeolites, again with reasonably good agreement between calculated and experimental results. The methods demonstrated by these calculations will be broadly applicable in using molecular simulations to predict properties of adsorbed molecules in zeolites and other nanoporous materials. force ﬁelds can provide an accurate description of adsorbate/ adsorbent interactions. In this paper, we examine an approach to developing ﬁrstprinciples-based force ﬁelds for molecular adsorption in zeolites that addresses two of the limitations of earlier ﬁrst-principlesbased methods. In earlier work, cluster models of zeolites and other porous materials were used for quantum chemistry calculations.15−32 Throughout our calculations, we use a fully periodic framework of the zeolites of interest. Perhaps more importantly, earlier work relied on quantum chemistry calculations performed at a small number of speciﬁc adsorption conﬁgurations to ﬁt classical potentials.15,16,18−21,24−32 This approach is appropriate if the conﬁgurations chosen for quantum chemistry calculations represent all of the important degrees of freedom on the overall PES, but ensuring that this requirement is satisﬁed is challenging. We have taken an alternative approach of performing quantum chemistry calculations for a large number of conﬁgurations scattered throughout the zeolite pore space and used these results to develop a classical force ﬁeld. We illustrate our approach for adsorption of CO2 in siliceous zeolites.

1. INTRODUCTION Zeolites exhibit high adsorption capacities and thermal stability and have served as an important type of adsorbents for a wide variety of adsorbates.1,2 As an important complement to experiments, molecular simulations such as grand canonical Monte Carlo (GCMC) and molecular dynamics (MD) have been applied for predicting adsorption isotherms, heats of adsorption, and other thermodynamic properties of guest molecules in various nanoporous materials.3−9 Most GCMC and MD simulations of zeolites use force ﬁelds obtained by ﬁtting adsorption isotherms to existing experimental data.10−13 This can limit the extension of simulation methods to new materials and can also lead to diﬃculties when adsorption does not sample all relevant regions of the adsorbate’s potential energy surface (PES).14 Recently, ﬁrst-principles-based force ﬁelds have been proposed to predict adsorption properties of gas molecules including H2, CH4, CO2, and NH3 in zeolites15−17 and other nanoporous adsorbents including MOFs,18−23 COFs,24−29 ZIFs,30,31 and silicon nanotubes.32 In this approach, force ﬁeld parameters are obtained by ﬁtting the ﬁrst-principles calculated interaction energies and interatomic distances (usually at MP2 or DFT levels) to classical potential functions. If they can be developed in a reliable way, ﬁrst-principles-based © 2012 American Chemical Society

Received: March 13, 2012 Revised: April 19, 2012 Published: April 19, 2012 10692

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Figure 1. Na+ cation sites in Na-FER viewed along the main (left) and perpendicular (right) channels. For more information about the nomenclature of cation sites, see refs 40, 41.

The accuracy of the force ﬁeld determined as outlined above will of course be controlled by the accuracy of the ﬁrstprinciples method upon which it is based. It is not computationally feasible to examine hundreds of adsorption conﬁgurations in periodic structures using high level methods such as MP2 or coupled cluster calculations. It is, however, possible to achieve this task using density functional theory (DFT). Unfortunately, traditional local or semilocal density functionals such as PBE, B3LYP, etc., cannot properly describe dispersion (or van der Waals) interactions that play a dominant role in the adsorption of molecules such as CO2 in zeolites. In the past decade, several DFT approaches that can treat dispersion interactions more accurately have been developed. Below, we examine the suitability of the empirical dispersioncorrected DFT (DFT-D) method proposed by Grimme33−35 and the ab initio van der Waals density functional (vdW-DF) approaches developed by Langreth and Lundqvist and by Michaelides et al.36−39 for predicting CO2 adsorption in zeolites. In section 2, we assess the accuracy of various dispersion-corrected DFT methods by computing adsorption energies of CO2 on sodium-exchanged ferrierites (Na-FER) where high-level DFT/CC (density functional theory/coupled cluster) results and experimental data are already available and can be used for comparison.40−42 On the basis of these benchmark calculations, we choose an optimum DFT method for describing CO2 adsorption in zeolites. This approach is then extended in section 3 to the task of developing a reliable classical potential for CO2 adsorption in siliceous zeolites. We develop a force ﬁeld on the basis of hundreds of DFT calculations for CO2 in siliceous chabazite (Si-CHA). In section 4, we demonstrate that this force ﬁeld is transferable to other systems by applying it to simulate adsorption in two other siliceous zeolites (MFI and DDR).

FER samples with lower Si/Al ratios (i.e., with higher Na+ concentrations).40,42 Previous calculations at the PBE level indicate that the larger heats of adsorption are due to the formation of bridged adsorption complexes for CO2 on socalled “dual cation sites”, where the CO2 molecule can eﬀectively interact with two alkali cations.40 Conﬁgurations in which CO2 interacts with only one cation are referred to as single cation sites. A combination of experiment and theory indicates that single cation sites exist preferentially in Na-FER samples with high Si/Al ratios, while dual cation sites dominate in Na-FER samples with low Si/Al ratios.40,42 Cejka et al. observed better agreement between calculated and experimentally determined heats of adsorption using a DFT/CC method for dispersion corrections (based on PBE geometries) than using DFT alone.41 Here, we assess two types of dispersion-corrected DFT approaches by comparing the adsorption energies of CO2 on Na-FER with the DFT/CC and experimental data. The ﬁrst dispersion correction we considered is Grimme’s DFT-D2 method,34 in which a semiempirical dispersion potential (∼C6R−6) is added to the conventional Kohn−Sham DFT energy with a suitable damping function ( fdamp) at small atomic distances. The dispersion correction has the form Nat − 1

Edisp = −s6

Nat

∑ ∑ i=1

j=i+1

C6ij R ij 6

fdamp (R ij)

(1)

C6ij

where Nat is the number of atoms in the system, denotes the dispersion coeﬃcient for the atomic pair ij, Rij is the atomic distance, and s6 is a global scaling factor for each DFT functional (e.g., 0.75 for PBE and 1.05 for B3LYP).34 This method adds a negligible computational cost to the corresponding DFT calculation without the dispersion correction. We also examined the van der Waals density functional (vdW-DF) method, which allows for an ab initio description of dispersion interactions.36−39 In this method the exchange−correlation energy is given by

2. FIRST-PRINCIPLES CALCULATIONS FOR CO2 ADSORPTION ON NA-FER 2.1. Background on CO2/Na-FER and First-Principles Methods. To assess the applicability of various dispersioncorrected DFT methods for CO2 adsorption in zeolites, we focus ﬁrst on CO2 in Na-containing ferrierite (Na-FER). A range of experimental and quantum chemistry results are available for this system.40−42 Experiments show that larger heats of adsorption were obtained when CO2 adsorbs on Na-

Exc[n] = Ex GGA [n] + Ec LDA [n] + Ec nl[n]

ExGGA

(2) LDA

where n is the electron density, and and Ec are the GGA-exchange and LDA-correlation energies, respectively. The third term represents a non-local correlation functional that accounts for the dispersion interaction. In the original vdW-DF scheme, the revPBE functional is chosen for the GGA portion, 10693

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Table 1. Adsorption Energies (in kJ/mol) of One CO2 Molecule on Na-FER from Diﬀerent DFT Methodsa Na+ siting

PBEb

DFT/ CCb

PBE

PBED2

vdW-DF

P6/T1 I2/T2 M7/T3 I2/T4 P8/T4 M5/T4

−23 −29 −29 −28 −26 −28

−43 −47 −49 −46 −45 −48

−21 −27 −27 −29 −25 −29

−46 −50 −54 −52 −50 −53

−65 −68 −72 −70 −73 −72

(−65) (−66) (−70) (−69) (−74) (−71)

P6/T1···P6/T1 I2/T2···I2/T2 M7/T3···P8/T2 I2/T4···M5/T4 M5/T4···I2T2

−25 −40 −32 −35 −37

−47 −57 −60 −58 −59

−26 −37 −36 −36 −33

−58 −67 −72 −67 −65

−68 −82 −82 −77 −78

(−70) (−79) (−80) (−77) (−74)

vdW-DF (optB88)

vdW-DF2 Single Cation Sites −56 (−56) −57 (−58) −61 (−62) −60 (−57) −58 (−59) −59 (−61) Dual Cation Sites −60 (−60) −74 (−73) −74 (−72) −72 (−68) −73 (−72)

vdW-DF (optPBE)

vdW-DF (optB86b)

Exp.

−60 −63 −66 −65 −66 −66

(−64) (−65) (−70) (−67) (−67) (−68)

−70 −73 −78 −76 −75 −76

−64 −65 −68 −68 −68 −69

44,c 50d

−69 −78 −84 −79 −77

(−68) (−78) (−83) (−80) (−75)

−76 −88 −89 −84 −86

−70 −80 −82 −79 −78

52,c 59d

a

Values in parentheses are calculated using SIESTA. bPrevious results in refs 40, 41. cHeats of adsorption derived from isotherms through extrapolation (ref 40). dHeats of adsorption measured directly by calorimetry (ref 42).

performed using the unit cells doubled along the c direction, since the dimension toward this direction is small and the adsorbed CO2 molecule will interact with Na+ and its periodic image when a single unit cell is used. We ﬁrst employed DFT with the PBE functional to reproduce earlier calculations40,41 of CO2 adsorption on single and dual Na+ cation sites. The adsorption energies were deﬁned using

since it predicts negligible binding in the vdW complexes due to exchange alone.36 Later, the same group proposed a second version of this approach, called vdW-DF2, which employs a more accurate semilocal exchange functional (rPW86) and a large-N asymptote gradient correction in determining the vdW kernel.37 Klimes and co-workers also improved the original vdW-DF by optimizing the exchange functional and found that employing the optB88, optPBE, and optB86b GGA functionals could yield good performance on the s22 data set test and other general tests for a broad range of solid materials.38,39 An eﬃcient implementation of these vdW-DF methods has been proposed by Soler and co-workers.43 These two types of approaches have been extensively applied to a variety of systems where dispersion interactions are important.44,45 They have been applied in adsorption and catalysis of various guest molecules in zeolites, including alcohols in HZSM-5,46−48 water in zeolite clusters,49 and alkanes in chabazite.50,51 Pulido et al.52 applied DFT-D2 to account for dispersion interactions of CO2 in proton exchanged ferrierite (H-FER) and showed good agreement with experimental data. However, to the best of our knowledge, a systematic assessment of DFT-D2 and vdW-DF methods for CO2 in zeolites has not been reported previously. 2.2. Models and Computational Details. Siliceous FER zeolite has an orthorhombic unit cell (Immm space group) with composition Si36O72 and four distinguishable framework T-sites (T1 to T4). In Na-FER, framework Al atoms may reside at all of these four T-sites. Consistent with earlier treatments, the Na+ cation sites investigated here include P6/T1, I2/T2, M7/ T3, I2/T4, P8/T4, and M5/T4 (Figure 1).40,41 Each pair of these single cation sites with a suitable Na+−Na+ distance could form dual cation sites, including P6/T1···P6/T1, I2/T2···I2/ T2, M7/T3···P8/T2, I2/T4···M5/T4, and M5/T4···I2/T2.40,41 Two types of Na-FER models were employed in the calculations, NaAlSi35O72 (Si/Al = 35) and Na4Al4Si32O72 (Si/Al = 8), to represent single and dual cation sites, respectively. We used the same unit cell parameters as in earlier calculations (a = 19.1468 Å, b = 14.3040 Å, and c = 7.5763 Å), which were previously obtained from ﬁtting equilibrium volume of the siliceous FER unit cell with the Murnaghan equation of state using the PBE functional.53 The unit cell parameters are kept ﬁxed in all calculations. Calculations for Na+ at P6/T1 and P6/T1···P6/T1 sites were

Eads = ECO2 − zeolite − (ECO2 + Ezeolite)

(3)

where ECO2−zeolite, ECO2, and Ezeolite are the total energies for the adsorption complex, isolated CO2 molecule, and isolated zeolite, respectively. We then performed similar calculations using the PBE-D234 and various vdW-DF methods including vdW-DF(revPBE),36 vdW-DF(rPW86),37 vdW-DF(optB88),38 vdW-DF(optPBE),38 and vdW-DF(optB86b).39 The vdWDF(revPBE) is the original vdW-DF, and the later developed vdW-DF(rPW86) is also called vdW-DF2. Geometry optimizations and energy calculations were calculated using VASP.54,55 Additionally, vdW-DF, vdW-DF2, and vdW-DF(optB88) methods have recently been implemented in SIESTA,56,57 which is also based on periodic boundary conditions, but using norm-conserving pseudopotentials and localized basis sets. We performed calculations using SIESTA as a comparison with our VASP results. In VASP, electron−ion interactions were described with the projector augmented wave (PAW) formalism.58,59 For valence electrons a plane wave basis set was applied with an energy cutoﬀ of 400 eV. In SIESTA, norm-conserving Troullier− Martin pseudopotentials60 were used to describe the atomic cores. Double-ζ plus polarization (DZP) basis sets were used; these basis sets have been optimized with a soft conﬁnement potential.61 Basis set superposition error (BSSE) through the counterpoise correction62 was included when calculating CO2− zeolite interaction energies. The equivalent plane-wave cutoﬀ of the fast Fourier transform grid (i.e., mesh cutoﬀ) was set to 300 Ry. In both VASP and SIESTA calculations, a single k-point centered at the Γ-point of the unit cell was used, and geometry optimizations were converged until forces were smaller than 0.03 eV/Å. 2.3. DFT Calculation Results. The results of our DFT calculations for CO2 adsorption in Na-FER are summarized in Tables 1 and 2. We ﬁrst compare our PBE data with previous 10694

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Table 2. Geometry Parameters (in Å) of CO2 Adsorption Complexes on Na-FER from Diﬀerent DFT Methodsab Na+ siting

PBEc

PBE

PBE-D2

P6/T1 I2/T2 M7/T3 I2/T4 P8/T4 M5/T4

2.33 2.34 2.36 2.34 2.31 2.37

2.41 2.38 2.39 2.37 2.34 2.38

2.46 2.43 2.47 2.42 2.42 2.43

P6/T1···P6/ T1

2.33/3.06

2.38/2.97

2.46/2.54

7.6 2.47/2.50

7.6 2.50/2.52

7.3 2.53/2.53

7.3 2.58/3.01

7.3 2.66/2.52

7.3 2.63/2.59

7.4 2.34/2.48

7.1 2.45/2.40

7.1 2.45/2.45

6.2 2.37/2.50

6.4 2.39/2.47

6.5 2.45/2.48

6.8

6.6

6.6

I2/T2··· I2/T2

M7/T3···P8/ T2 I2/T4···M5/ T4 M5/T4···I2T2

vdW-DF

vdW-DF2

Single Cation Sites 2.38 (2.37) 2.38 (2.41) 2.40 (2.39) 2.36 (2.35) 2.37 (2.38) 2.37 (2.36) Dual Cation Sites 2.37/3.17 2.34/2.95 (2.33/3.21) (2.30/3.01) 7.8 (7.8) 7.6 (7.6) 2.53/2.57 2.50/2.50 (2.55/2.56) (2.51/2.53) 7.3 (7.4) 7.3 (7.3) 2.66/2.55 2.60/2.52 (2.66/2.55) (2.61/2.53) 7.1 (7.1) 7.1 (7.1) 2.51/2.41 2.41/2.37 (2.48/2.40) (2.42/2.36) 6.4 (6.5) 6.4 (6.5) 2.42/2.52 2.38/2.45 (2.39/2.52) (2.37/2.46) 6.6 (6.7) 6.6 (6.7)

2.45 2.46 2.44 2.44 2.39 2.41

(2.44) (2.47) (2.42) (2.42) (2.39) (2.39)

vdW-DF (optB88)

vdW-DF (optPBE)

vdW-DF (optB86b)

2.39 2.38 2.42 2.40 2.36 2.38

2.41 2.41 2.43 2.39 2.39 2.39

2.40 2.39 2.43 2.39 2.37 2.39

2.38/2.86

2.40/2.88

7.5 2.52/2.53

7.5 2.51/2.52

7.3 2.65/2.54

7.3 2.64/2.52

7.1 2.43/2.41

7.1 2.43/2.41

6.4 2.41/2.46

6.5 2.40/2.46

6.6

6.6

(2.37) (2.38) (2.41) (2.36) (2.36) (2.38)

2.39/2.90 (2.37/2.89) 7.5 (7.5) 2.50/2.51 (2.52/2.54) 7.3 (7.3) 2.58/2.52 (2.63/2.53) 7.1 (7.1) 2.43/2.41 (2.39/2.41) 6.4 (6.5) 2.38/2.48 (2.37/2.44) 6.6 (6.7)

For single cation sites, distances between Na+ cation and the oxygen atom of CO2 are given. For dual cation sites, the ﬁrst line refers to the distances between Na+ cation and the two oxygen atoms of CO2 and the second line refers to the Na+−Na+ distance. bValues in parentheses are calculated using SIESTA. cPrevious results in the reference.40,41 a

results40 using the same functional to validate our calculations. The deviation between our results and earlier calculations for the distance between the Na+ cation and the oxygen atom of CO2 (Na+−O) is within 0.1 Å for CO2 on single cation sites. Similar geometries were also obtained for most dual cation sites, with slight deviations (0.12 and 0.2 Å) for Na+−O and Na+−Na+ distances, respectively. A large deviation is found for CO2 on the M7/T3···P8/T2 site. In our case, the CO2 molecule is located almost midway (2.66 and 2.52 Å) between the two Na+ cations, whereas Pulido et al. reported CO2 is closer to the Na+ on M7/T3 than on P8/T2 (2.58 and 3.01 Å). The deviations in geometries on dual cation sites are probably due to diﬀerent distributions of Al atoms in the zeolite models, where four Al atoms exist per unit cell. Only two Al positions were explicitly determined in the original work. Nevertheless, the PBE adsorption energies from our calculations are very close to previous calculated results, with maximum deviations of 2 and 4 kJ/mol for CO2 on single and dual cation sites, respectively. Geometric characteristics from the dispersion-corrected methods are similar to those with PBE, with slightly longer Na+···CO2 distances. The change in Na+···CO2 distance is more obvious for PBE-D2, from 0.05 to 0.08 Å, depending on the cation sites. The PBE-D2 results show that on the P6/T1···P6/ T1 site CO2 is favorably located midway (2.46 and 2.54 Å) between the two Na+ cations, which lead to a smaller Na+−Na+ distance (7.3 Å) with respect to other methods (7.5−7.6 Å). The CO2 adsorption energies vary signiﬁcantly among these methods. On single cation sites, the adsorption energy for the same cation site generally increase in the order of PBE < PBED2 < vdW-DF2 < vdW-DF(optB88) ≈ vdW-DF(optB86b) < vdW-DF < vdW-DF(optPBE). On dual cation sites, the same tendency was also found, except that the vdW-DF results were close to those from the vdW-DF (optB88) and vdW-DF

(optB86b). The SIESTA results with BSSE corrections are very close to the corresponding VASP results. As we discuss further below, caution must be used in comparing the calculated adsorption energies of CO2 on NaFER with experimentally measured heats of adsorption. The DFT calculations only consider the local minimum conﬁgurations of CO2. Experimental heats of adsorption, however, are an average of all possible CO2 conﬁgurations in the pore, even at low loading, because of thermal eﬀects.51 As a result, the heat of adsorption includes both enthalpic and entropic contributions. A more reliable way to evaluate the accuracy of the DFT methods is to compare our DFT results with those from the DFT/CC method, which has been demonstrated to give results comparable to the “gold standard” CCSD(T) method.63 Among the dispersion-corrected DFT methods, PBE-D2 gives the closest adsorption energies with respect to DFT/CC, with deviations of 3−6 kJ/mol for single cation sites and 6−12 kJ/mol for dual cation sites. The other dispersioncorrected calculations give adsorption energies that substantially overestimate the DFT/CC values. Among the vdW-DF methods, vdW-DF2 performs better than the others, although the overestimation of the adsorption energy is still large (11− 17 kJ/mol). In contrast to the DFT/CC method, which needs a preliminarily established set of correction functions for each cross-species interaction,52,64 the DFT-D2 method is more easily implemented.65 On the basis of this observation and the reasonable agreement between DFT-D2 results and the more challenging DFT/CC calculations, we chose to use PBE-D2 as the basis for the force ﬁeld ﬁtting described in the next section. 10695

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distribution for both periodic and nonperiodic systems.67−69 These charges were computed using the program available at ddec.sourceforge.net as described by Manz and Sholl. To complete the speciﬁcation of the CO2−zeolite force ﬁeld, the scaling factors s6 and s12 must be deﬁned. As shown below, these factors are ﬁtted to allow the closest correspondence between the DFT results and the classical force ﬁeld in a leastsquares sense. This is similar to the concept in Grimme’s work, where s6 only depends on the density functional used and was determined by least-squares optimization of interaction energy deviations for standard benchmark sets.34 3.2. Force Field Fitting Procedures. We performed a large number of DFT calculations for CO2 adsorption in SiCHA in order to derive a classical force ﬁeld. Si-CHA was chosen because its small unit cell (Si12O24 per unit cell) reduces the computational demands associated with periodic DFT calculations. The rhombohedral unit cell (R3̅ m space group) of Si-CHA was ﬁrst fully optimized at the DFT level. PBE-D2 gives lattice constants of a = b = c = 9.333 Å and α = β = γ = 94.34°, close to the experimentally determined data, a = b = c = 9.229 Å and α = β = γ = 94.3°.70 DFT calculations were performed with a loading of one CO2 per unit cell. In all DFT calculations, a CO2 structure consistent with the EPM2 potential (linear structure and 1.149 Å for CO bond length) was used. A total of 400 adsorption conﬁgurations of CO2 within the zeolite framework were randomly generated with a restriction that there should be no interatomic overlap between the atoms of CO2 and zeolite framework. More speciﬁcally, only conﬁgurations where the minimum interatomic distance satisﬁed Rmin > 2.3 Å were used. The adsorption energies at the PBE-D2 level were calculated for all 400 CO2 conﬁgurations within the Si-CHA framework using eq 3. For each adsorption conﬁguration, the adsorption energy of CO2 was also evaluated on the basis of eq 4. In these calculations Coulombic interactions were calculated by Ewald summation.71,72 Ewald summation was performed with the realspace cutoﬀ of 12 Å and a relative error of 10−6. For these calculations, the 1 × 1 × 1 unit cell was extended to a 3 × 3 × 3 model so that the minimum length in each of the coordinate direction is larger than 24 Å. The repulsive and attractive vdW terms without scaling factors, C12ij/Rij12 and −C6ij/Rij6, were also determined separately for each CO2 adsorption conﬁguration. Table 3 gives the C12ij and C6ij coeﬃcients for each

3. DFT-DERIVED FORCE FIELDS FOR CO2 IN SILICEOUS ZEOLITES 3.1. Deriving Force Fields from Dispersion-Corrected DFT Results. Having selected an appropriate dispersioncorrected DFT method (DFT-D2), we now turn to the challenge of using DFT calculations to derive a robust potential for adsorbed molecules. For purposes of illustration, we restrict our attention to adsorption of CO2 in siliceous zeolites, although the methods described above could readily be adapted to other adsorbents. In this section, we show how a large collection of DFT calculations for CO2 adsorption in siliceous chabazite (Si-CHA) can be used to derive a transferrable potential for CO2 adsorption in siliceous zeolites. It is important to use a CO2−CO2 potential that correctly captures the phase behavior of pure CO2, so we used the EPM2 potential,66 which was developed for this purpose. As in most classical simulations of adsorption in zeolites, we assume that the zeolites are rigid. Therefore, in order to fully specify the energy of adsorbed molecules, we only need to deﬁne the CO2−zeolite interactions. To this end, we assume the interactions between each atom in CO2 and a zeolite atom are represented by pairwise van der Waals and Coulombic terms qiqj C ij C ij E FF(R ij) = EvdW + ECoul = s12 1212 − s6 6 6 + R ij R ij R ij (4) Here, Rij is the distance between atoms i and j, C6ij and C12ij are the attractive and repulsive coeﬃcients, qi and qj represent the charges for atoms i and j, respectively, and s6 and s12 are global scaling factors that depend on the ﬁrst-principles method used in the force ﬁeld ﬁtting. The attractive portion of the vdW interactions, −s6C6ij/Rij6, is based on Grimme’s empirical dispersion expression in the DFT-D2 method.34 The damping function in the original dispersion expression, which was used to avoid nearsingularities for small interatomic distances, is not considered here because the repulsive term, s12C12ij/Rij12, is included. The C12ij parameters were deﬁned by (R 0i + R 0 j)6 C12 ij ij = 2 C6 i

(5)

R0j

where R0 and are the van der Waals radii of atoms i and j. This relation is based on the fact that the vdW terms in our force ﬁeld are similar to the Lennard-Jones (LJ) 12−6 potential with the form ⎡⎛ ⎞12 ⎛ ⎞6 ⎤ σij σij Aij Bij E LJ(R ij) = 4εij⎢⎢⎜⎜ ⎟⎟ − ⎜⎜ ⎟⎟ ⎥⎥ = 12 − R R ij R ij 6 ⎝ R ij ⎠ ⎦ ⎣⎝ ij ⎠

Table 3. Attractive and Repulsive Coeﬃcients (C6ij and C12ij) and the Sum of van der Waals Radii (R0i + R0j) for Each Cross Species between CO2 and Si-CHA cross species Si−C Si−O Oz−C Oz−O

(6)

where εij is the depth of the potential well and σij is the interatomic distance at which the potential is zero. The repulsion coeﬃcient Aij is determined so that the potential has a minimum at the distance equal to the sum of the van der Waals radii of the atomic pair. A set of parameters for C6 and R0 from elements H to Xe are available in Grimme’s work,34 and these values were adopted in our force ﬁeld ﬁtting. The Coulombic interactions between atoms in CO2 and the zeolite were treated by a point charge model employing density-derived electrostatic and chemical (DDEC) charges for the atoms of zeolite framework.67−69 DDEC charges reproduce the electrostatic potential of the material outside of its electron

C12ij (J nm12 mol−1) 2.03 1.04 2.63 1.31

× × × ×

−3

10 10−3 10−4 10−4

C6ij (J nm6 mol−1)

R0i + R0j (Å)

4.02 2.54 1.11 0.70

3.168 3.058 2.794 2.684

cross species interaction between CO2 and Si-CHA, which were derived from Grimme’s work and eq 5. The distances (1/Rij12 and 1/Rij6) for each cross species were measured and summed on the basis of a ﬁnite 5 × 5 × 5 model, where the CO2 molecule is located near the center of the model and its position relative to the zeolite framework is the same as in the original 1 × 1 × 1 model. This approximation is reasonable because the vdW contributions from the framework atoms that reside beyond the 5 × 5 × 5 model are negligible. Least-squares 10696

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ﬁtting was then used to determine values for the scaling factors s12 and s6 that minimized the deviation between the adsorption energies deﬁned by the force ﬁeld and the PBE-D2 values. This gave scaling factors of s12 and s6 of 4.43 and 0.98, respectively. Since the vdW terms of the force ﬁeld developed here have the same form as the 12−6 LJ potential, the s12C12ij and s6C6ij term in eq 4 can also be expressed with corresponding εij and σij values in eq 6 for each cross species interaction. These parameters are listed in Table 4. For simplicity, we will refer to

shown in Figure 2. The 400 PBE-D2 calculations span a broad range of adsorption energies, from ca. −26 to +69 kJ/mol. As expected from their derivation, the D2FF interaction energies are generally consistent with the PBE-D2 results. Some deviations between D2FF and PBE-D2 energies exist, especially when repulsive interactions are the dominating factor in determining the overall adsorption energy. This discrepancy is explicitly shown in Figure 2b, where the diﬀerence between the energies with the two methods (ED2FF − EPBE‑D2) is plotted as a function of the nearest interatomic distance, Rmin, between CO2 and Si-CHA. When comparing to all 400 DFT energies, the D2FF interaction energies for 88% (97%) of the CO2 conﬁgurations have deviations within ±4 (±8 kJ/mol). If only the conﬁgurations for which PBE-D2 yields a favorable (i.e., negative) adsorption energy are included, this fraction of states increases to 93% (>99%). As shown in Figures 2d,e, CLAYFF performs well for the most favorable adsorption conﬁgurations of CO2 but shows a systematic deviation from the PBE-D2 values for less favorable sites. The CLAYFF interaction energies for 78% (84%) of all 400 CO2 conﬁgurations have deviations within ±4 (±8 kJ/mol) of the PBE-D2 energies. If attention is restricted to the PBE-D2 conﬁgurations with negative adsorption energies, this fraction of states increases to 91% (97%) for CLAYFF. The mean absolute deviation between the force ﬁelds and our PBE-D2 results is 2.2 and 5.4 kJ/mol for D2FF and CLAYFF, respectively.

Table 4. D2FF and CLAYFF Parameters for CO2 in Siliceous Zeolites D2FF

CLAYFF

cross species

ε/kB (K)

σ (Å)

charge (e)

Si−C Si−O

51.38 40.17

3.63 3.51

Oz−C Oz−O

30.07 24.20

3.20 3.08

Si (2.21) Oz (−1.105) C (0.6512) O (−0.3256)

ε/kB (K)

σ (Å)

charge (e)

0.16 0.27

3.03 3.17

Si (2.10) Oz (−1.05)

46.90 79.35

2.96 3.10

C (0.6512) O (−0.3256)

the force ﬁeld derived in this way as D2FF. Also shown in Table 4 are force ﬁeld parameters for the Clay force ﬁeld (CLAYFF),73 which was developed for clays and related layered materials. Here, Lorentz−Berthelot mixing rules were used to calculate the ε and σ values for cross-species interactions. As in our calculations with D2FF, the EPM2 potential was used for CO2−CO2 interactions. A detailed comparison of the adsorption energies predicted with D2FF and CLAYFF with our PBE-D2 calculations is

4. CLASSICAL SIMULATIONS USING DFT-DERIVED FORCE FIELDS 4.1. CO2 in Si-CHA. To examine the applicability of the force ﬁeld developed above, we compared adsorption isotherms

Figure 2. (a) Comparison of the interaction energies of CO2 in Si-CHA for the D2FF and PBE-D2, (b) the diﬀerence in interaction energies (ED2FF − EPBE‑D2) as a function of the nearest interatomic distance between CO2 and Si-CHA, and (c) distribution of ED2FF − EPBE‑D2 for all 400 CO2− zeolite conﬁgurations. Similar to a−c, parts d−f show the results for CLAYFF. 10697

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predicted using this force ﬁeld with adsorption data from several siliceous zeolites. Adsorption isotherms, isosteric heats of adsorption, and Henry’s constants were predicted computationally using standard GCMC methods (see Supporting Information). We ﬁrst discuss results for Si-CHA using experimental data from our own experiments. A sample of siliceous Si-CHA74 was used for experimental measurements of isotherms and heats of adsorption. CO2 adsorption isotherms were measured in an Autosorb-1C volumetric adsorption apparatus from Quantachrome equipped with 1, 10, and 1000 Torr pressure transducers. The isotherms were measured at temperatures of 301, 313, and 323 K. The isotherms were also measured separately in Cahn gravimetric microbalance system. Prior to the adsorption measurements, the Si-CHA sample was degassed carefully at 673 K for 6 h under a vacuum of 1.33 × 10−7 kPa. To measure the diﬀerential enthalpies of adsorption at low loadings, an Autosorb 1-C system was coupled with a SENSYS Tian-Calvettype diﬀerential microcalorimeter from Setaram. The measured heat ﬂow signal after each successive dose of gas was integrated to obtain the diﬀerential enthalpy of adsorption simultaneously with the amount of gas adsorbed by the volumetric measurement.75 The microcalorimetric method oﬀers substantially higher accuracy compared to calculations of the heats of adsorption than using the Clausius−Clapeyron equation, especially at low loadings. The simulated and experimental isotherms for CO2 in SiCHA at three temperatures are shown in Figure 3a. The simulation results based on D2FF are in excellent agreement with the experimental data. CLAYFF also makes predictions that are in reasonable agreement with the experimental data but slightly underestimates the CO2 loadings. Figure 3b shows the heats of adsorption (Qst) as a function of CO2 loading at 301 K. Both D2FF and CLAYFF can reproduce the experimental data within the deviation of 2 kJ/mol for the loading range up to 2.0 kg/mol. A slight increase in Qst with loading can be observed, which is attributed to an increase of the adsorbate/adsorbate interactions. Adsorption properties in the Henry’s regime are summarized in Table 5. The agreement between simulated (D2FF) and experimental Henry’s constants (KH) is excellent for these temperatures. The zero-coverage heats of adsorption (Qst0) at 301 K from D2FF and CLAYFF are very similar, 23.6 and 23.8 kJ/mol, respectively. These values are almost constant for temperatures from 301 to 323 K (see Supporting Information). The deviation in the heat of adsorption between theory and experiment is ∼1 kJ/mol, which is acceptable accuracy. We extended our PBE-D2 calculations for CO2 in Si-CHA to identify minimum energy structures for adsorbed CO2, and these calculations identiﬁed multiple local minima. The conﬁguration where CO2 is perpendicular to the eightmembered ring (8MR) with a C atom located near the 8MR center gives the most favorable adsorption energy. This conﬁguration is shown in Figure 4. At the PBE-D2 level, the adsorption energy for this site was −31.3 kJ/mol. For the same structures, D2FF and CLAYFF give −30.4 and −31.9 kJ/mol, respectively. These results are similar to that for CO2 in the 8MR of siliceous FER zeolite, which has been recently reported to be −30.0 kJ/mol by the DFT/CC method.76 Previous MD simulations showed that the most energetically preferred sites for CO2 in DDR, another siliceous zeolite with 8MR windows, lie inside the 8MR.14

Figure 3. Comparison of simulated (D2FF and CLAYFF) and experimental (a) adsorption isotherms at 301, 313, and 323 K and (b) heats of adsorption as a functional of loading at 301 K for CO2 in SiCHA.

An important observation from these results is that the isosteric heat of adsorption is not identical to the adsorption energy of the minimum energy conﬁguration. As mentioned above, the heat of adsorption includes both enthalpic and entropic contributions. Our GCMC simulations show, not surprisingly, that CO2 is not always located in the 8MR, even at low loadings. Because the isosteric heat averages over all occupied states in the material, the isosteric heat is smaller than the magnitude of the most favorable adsorption energy. Speciﬁcally, the isosteric heat for CO2 in Si-CHA near room temperature is ∼7 kJ/mol smaller than the magnitude of the global minimum energy state. Recent studies of alkane adsorption on acidic zeolites using ab initio MD51 and conﬁgurational-bias Monte Carlo (CBMC)77 simulations show that the probability to ﬁnd an alkane molecule adsorbed on the acid site and form a stable reactant decreases with increasing temperature because of thermal eﬀects. For propane adsorbed on proton-exchanged chabazite (H−CHA), the probability was only 68% at 300 K, and the internal energy of adsorption was predicted to be ∼15 kJ/mol smaller than the minimum energy state. This reinforces our statement in section 2.3 that caution must be used in comparing DFT adsorption energies with isosteric heats of adsorption. 10698

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Table 5. Henry’s Constants (KH) and Isosteric Heats of Adsorption in the Limit of Zero Coverage (Qst0) of CO2 in Siliceous Chabazite (Si-CHA) Qst0 (kJ/mol)

KH (mol/kg/Pa)

a

temp (K)

D2FF

CLAYFF

exp

D2FF

CLAYFF

exp

301 313 323

2.27 × 10−5 1.59 × 10−5 1.21 × 10−5

2.06 × 10−5 1.43 × 10−5 1.09 × 10−5

2.41 × 10−5 1.59 × 10−5 1.16 × 10−5

23.6 23.0 23.6

23.8 23.4 23.4

22.5a

Diﬀerential enthalpy of adsorption measured in this work in adsorption microcalorimetry experiments.

Figure 4. The most energetically favorable conﬁguration for CO2 in the Si-CHA framework, optimized at the PBE-D2 level. The adsorbed CO2 sits in approximately the center of the 8MR. Distances (in Å) between the C atom of CO2 and O atoms of zeolite framework are shown.

4.2. CO2 in Other Siliceous Zeolites. To examine the transferability of our force ﬁeld, we also performed GCMC simulations of CO2 in two other siliceous zeolites, MFI and DDR. These materials were chosen because the CO 2 adsorption properties in these two zeolites have been widely investigated, and extensive experimental data are available.13,78−83 The simulated isotherms from D2FF and the experimental data are shown in Figure 5, and the low-coverage properties are summarized in Table 6. The simulated isotherms from CLAYFF are similar to the D2FF results and are included in the Supporting Information. GCMC was performed for these materials as described previously.84 The agreement between simulations and experiments is good for CO2 isotherms in MFI, with some slight deviations at low temperature (273 K). At this temperature, the predicted Henry’s constants (KH) are roughly a factor of 2 larger than the experimental value. At higher temperatures the agreement is better. The isosteric heats of adsorption in the zero coverage limit (Qst0) are predicted by D2FF to be 25−28 kJ/mol from 273 to 373 K, in good agreement with experimental data.78−80 For CO2 in DDR, D2FF quantitatively reproduces the adsorption isotherms at low and high pressures, and slightly overestimates the adsorbed amount at intermediate pressures for temperatures from 273 to 373 K. At 273 K, the calculated KH from D2FF and CLAYFF are about half of the experimental value, and the agreement becomes better at higher temperatures. Qst0 calculated with the D2FF method is close to the experimental data from Bergh et al.83 We note that estimating Qst0 from experimental adsorption isotherm data requires use of data from low pressures, which are more susceptible to

Figure 5. Comparison of calculated (D2FF) and experimental adsorption isotherms for CO2 in (a) MFI and (b) DDR. The experimental data are from Garcia-Sanchez et al.,13 Dunne et al.,79 Zhu et al.,80 Himeno et al.,81 and Bergh et al.82,83.

variations between experiments due to sample preparation and experimental methods than data from higher pressures. Considering the variations between experimental results from diﬀerent groups, the predictions based on our method are good.

5. CONCLUSIONS The development of accurate force ﬁelds is central to the use of molecular simulations to quantitatively predict the properties of molecules adsorbed in zeolites and similar nanoporous materials. To make the broadest possible use of these simulation methods, it is important that reliable force ﬁelds can be developed without simply ﬁtting existing experimental data. If this task cannot be accomplished, using simulations to 10699

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Table 6. Henry’s Constants (KH) and Isosteric Heats of Adsorption in the Limit of Zero Coverage (Qst0) of CO2 in Siliceous MFI and DDR Qst0 (kJ/mol)

KH (mol/kg/Pa) MFI

DDR

a

temp (K)

D2FF

CLAYFF

exp

D2FF

CLAYFF

273 303 338 373 273 298 323 348 373

11.66 × 10−5 3.72 × 10−5 1.26 × 10−5 0.52 × 10−5 7.43 × 10−5 2.93 × 10−5 1.32 × 10−5 0.68 × 10−5 0.38 × 10−5

12.05 × 10−5 3.73 × 10−5 1.24 × 10−5 0.51 × 10−5 3.47 × 10−5 2.92 × 10−5 1.29 × 10−5 0.69 × 10−5

6.78 × 10−5a 3.29 × 10−5b 1.19 × 10−5b 0.52 × 10−5b 11.08 × 10−5d 4.21 × 10−5d 1.83 × 10−5d 0.64 × 10−5d

27.0 26.6 26.2 25.7 25.7 25.2 24.9 24.7 24.4

27.0 26.6 26.2 25.7 27.0 26.4 26.0 25.7

exp 24.6a 25.0b 27.2c

32d 23.7e

Rees et al.78 bZhu et al.80 cDunne et al.79 dHimeno et al.81 eBergh et al.83

Notes

consider genuinely new materials in advance of experimental studies is of limited value. In this paper we have demonstrated methods that fulﬁll the goal of deriving force ﬁelds without using experimental data. These methods were illustrated with a relatively simple example, CO2 adsorption in siliceous zeolites, but they are applicable to a wide range of adsorbates and adsorbents. Our approach diﬀers from earlier eﬀorts to derive force ﬁelds for adsorption from quantum chemistry calculations by not making a priori assumptions about the adsorption conﬁgurations that will dominate. An extensive comparison between periodic dispersion-corrected DFT results and prior high level quantum chemistry calculations for CO2 in Na-FER showed that Grimme’s DFT-D2 method gives the best results among currently available methods. On this basis, DFT-D2 calculations were used to derive a force ﬁeld for CO2 adsorption in siliceous zeolites. To this end, the DFT-D2 adsorption energies of hundreds of adsorption conﬁgurations probing the entire accessible volume in Si-CHA were determined. These results were then used to obtain a classical force ﬁeld (denoted D2FF) that gave a mean absolute deviation of 2.2 kJ/mol relative to the full set of DFT-D2 energies. Adsorption isotherms for CO2 in Si-CHA, Si-MFI, and SiDDR were predicted with standard GCMC calculations using D2FF. The agreement between the predicted and experimental data was good in all cases. One useful observation from these calculations is that the isosteric heat of adsorption is systematically smaller than the magnitude of the adsorption energy in the molecule’s global energy minimum. This is not surprising, given that the isosteric heat is a thermodynamic average over all states available to the adsorbed molecule. For CO2 in Si-CHA, the diﬀerence between these two quantities is ∼7 kJ/mol. This observation highlights the fact that averaging over a wide range of states is necessary if careful comparisons between computational predictions and experimentally measured heats of adsorption are to be made.

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The authors declare no competing ﬁnancial interest.

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ACKNOWLEDGMENTS This work has been supported by ExxonMobil Research and Engineering. The authors thank Dr. Thomas A. Manz, Dr. Taku Watanabe, Dr. Iyad A. Hijazi, and Dr. Yu Wang for helpful discussions, and Karl Strohmaier and Elizabeth Walker for experimental samples of Si-CHA.

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ASSOCIATED CONTENT

S Supporting Information *

Additional details on GCMC simulations. This information is available free of charge via the Internet at http://pubs.acs.org.

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. 10700

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