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Ab Initio Prediction of Adsorption Isotherms for Gas Mixtures by Grand Canonical Monte Carlo Simulations on a Lattice of Sites Arpan Kundu,† Kaido Sillar,†,‡ 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: Gibbs free energies of adsorption on individual sites and the lateral (adsorbate−adsorbate) interaction energies are obtained from quantum chemical ab initio methods and molecular statistics. They define a Grand Canonical Monte Carlo (GCMC) Hamiltonian for simulations of gas mixtures on a lattice of adsorption sites. Coadsorption of CO2 and CH4 at Mg2+ sites in the pores of the metal−organic framework CPO-27-Mg (Mg-MOF-74) is studied as an example. Simulations with different approximations as made in widely used coadsorption models such as the ideal adsorbed solution theory (IAST) show their limitations in describing adsorption selectivities for binary mixtures.

W

experiment as long as open metal ion sites were not involved. For the latter, improvements have been achieved with polarizable force fields9 and with force-field parameters fitted to the results of quantum chemical calculations10 (see refs 11−13 for recent examples). Adsorption isotherms in very good agreement with experiment have been obtained.12 An alternative approach14−17 relies on ab initio Gibbs free energies, ΔG, for adsorption complexes at individual surface sites and the corresponding equilibrium constants,

hile crystalline microporous materials, zeolites, have been employed for decades for the selective adsorption and separation of gases, e.g., for air separation,1 with metal− organic frameworks (MOFs)2,3 new materials have become available with a high potential for gas storage, e.g., of energy carrying molecules H2, CH4,4 and separations, e.g., CO2 capture from gas mixtures.5 Prerequisite to a rational design of improved materials with optimized properties for a specific target is the reliable ab initio prediction of adsorption isotherms and selectivity coefficients with no other input than the positions of the atoms. Adsorption isotherms, θ = na/nmax = f(P), describe the adsorbed amount na as a function of the gas pressure, P. The surface coverage θ is the ratio of the absolute adsorbed amount na and the maximum adsorbed amount, nmax. For binary mixtures, the selectivity coefficient is defined as S1,2 = (x1/y1)/(x2/y2), where xc and yc denote adsorbed and gas phase mole fractions of component c, respectively. The simulation of adsorption isotherms requires the calculation of Gibbs free energies of adsorption. Since simulation cells on the order of a thousand atoms are needed, this is a computationally challenging task. Grand-Canonical Monte Carlo (GCMC) simulations4,6 neglect quantum effects on nuclear motion (zero-point vibrational energy) and require millions of energy calculations to sample the configuration space. This is only feasible if parametrized potentials (force fields) are used. Most of them are “effective” potentials parametrized on experimental data that carry possible imperfections of the samples used, and the simulations are carried out for experimentally7,8 determined structures that are kept frozen. In spite of these limitations, force-field-based GCMC simulations have been able to describe single component adsorption in MOFs in close agreement with © XXXX American Chemical Society

K a = e−ΔG / RT

(1)

and makes use of model isotherms, e.g., the Langmuir isotherm, θ=

na K aP = nmax 1 + K aP

(2)

to include the configurational entropy resulting from the distribution of adsorbed molecules over the sites. The free energies for an individual site are calculated from vibrational partition functions QVib, which corresponds to an analytical sampling for a Taylor expansion of the full potential energy surface around a local minimum. ΔG = ΔEel + ΔEZPV − RT ln

surf Q Vib gas gas gas Q Rot QTransQ Vib

(3)

The electronic energy, ΔEel, the zero-point vibrational energy, ΔEZPV, and the vibrational energies for the partition functions Received: May 15, 2017 Accepted: June 2, 2017

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The Journal of Physical Chemistry Letters QVib are calculated at the energy-minimum structure directly by density functional theory. This way the neglect of quantum effects on nuclear motions is avoided. Moreover, single point energy calculations with reliable wave function methods are affordable for the few stationary points,18 and accuracies within 1 kJ/mol can be achieved.17,19 For improved electronic energies of stationary points, we use the hybrid MP2:(DFT+D) + ΔCCSD(T) method18 (MP2: Møller−Plesset second order perturbation theory; DFT+D: Dispersion-corrected Density Functional Theory; CCSD(T): Coupled Cluster theory with Single and Double substitutions and perturbatively treated Triple Substitutions). The final electronic energy is obtained from (i) the MP2 energy of a large cluster model C, ΔEMP2(C), (ii) a long-range correction, ΔELR(pbc, C), which is evaluated as the difference between the DFT energies (we use PBE+D220,21) of the periodic system (pbc − periodic boundary conditions) and the cluster model, and (iii) a CCSD(T) correction, ΔECCSD(T)(C′), which is evaluated as the difference between the CCSD(T) and MP2 energies for a small cluster model (see refs 17 and 19 for details):

all adsorbate−adsorbate interactions explicitly. This makes it a powerful approach specifically for mixtures. In particular, we will use this method to examine the applicability of the widely applied25 Ideal Adsorbed Solution Theory (IAST).26 Since measurements of coadsorption equilibria27,28 are scanty, mixture isotherms are still estimated from pure gas data, mostly applying IAST.25 The latter is based on the assumption that the adsorbed phase behaves like an ideal solution of the adsorbed components, implying that the intermolecular interactions between the molecules of different components are the average of those within the individual mixture components. The Lattice Gas Hamiltonian for molecules adsorbed at sites i and j with pair wise lateral interactions Eij is (see section S2 of the Supporting Information): M

H(n1, ..., nM) =

i=1 c=A M

+

ΔEel = ΔE MP2(C) + ΔELR (pbc, C) + ΔE CCSD(T)(C′)

Table 1 shows the results. The improvement of the electronic energy compared to PBE+D2 is well outside the chemical Table 1. Changes on Adsorption of the DFT Energy Obtained with Periodic Boundary Conditions, ΔEPBE+D2(pbc), the Final Electronic Energy, ΔEel, the ZeroPoint Vibrational Energy, ΔEZPV, the Thermal Energy, ΔET, and the Gibbs Free Energy, ΔGT Calculated for CO2 and CH4 on Different Sites in CPO-27-Mg (kJ/mol) for Different Temperatures (T = 298 and 343 K)a CO2b −41.5 −46.9 −3.8 +1.9 −48.8 +2.9 2.3 3.0 −9.2d −3.6

Mg ΔEPBE+D2(pbc) ΔEMP2 ΔELR ΔECCSD(T) ΔEel ΔEZPV ΔE298 ΔE343 ΔG298 ΔG343

Linker

Mg

Linker

−26.2 −33.9 −1.8 2.9 −32.8 +1.6 3.3 4.0 5.1 10.5

−27.1 −28.3 0.0 0.5 −27.8 2.0 1.9 2.4 3.9 8.5

−18.0 −14.1 −5.2 1.0 −18.3 1.4 2.2 2.8 10.9 15.1

K

∑ ∑ i ≠ j c , c ′= A

Eij(c , c′)ni(c)nj(c′) (5)

K

∑ ni(c) = ni c=A

(6)

Only one molecule can occupy a particular site and the maximum allowed value of ni is 1. It is 0 if the site is vacant. The vector ni has the components ni(c), c = A, ..., K. If the ith site is occupied by the gas species c, then ni(c) will be 1. Otherwise, it will be 0. For each temperature and pressure, 5 × 105 simulation steps are performed to reach equilibrium coverage starting from the empty surface. This is followed by a production run of 5 × 105 steps for determining one isotherm point. For very low pressures (below 0.01 and 0.1 bar at 298 and 343 K, respectively) 5 × 106 steps are used. The moves include exchange of adsorbed species with the gas reservoir (Glauber dynamics)29 and exchange of molecules between two different sites (Kawasaki dynamics)30 (see also section S3 of the Supporting Information). We consider coadsorption of a binary CO2/CH4 mixture in the pores of the metal−organic framework CPO-27-Mg,23 which is also known as Mg-MOF-74 or Mg/dobdc (dobdc -2,5dioxido-1,4-benzenedicarboxylate).31 The Mg2+ ions and the dobdc linkers have been identified as adsorption sites.32−34 Figure 1 shows the one-dimensional hexagonal pores of CPO27-Mg with two CO2 molecules in it. The positions of the Mg2+ and Linker sites can be generalized to form a rectangular lattice, which is used for GCMC simulations. In one direction, the sites run parallel along the pore; in the other direction the six sites of the hexagonal cross-section of the pore are aligned perpendicular to the pore direction. Hence, the CO2 molecules adsorbed on Mg2+ sites are surrounded by four CO2 molecules at Linker sites, and vice versa. Each CO2 molecule at an Mg2+ or Linker site has two neighbors of the same site type in the direction perpendicular to the pore. The configuration space was sampled on a 6 × 100 two-dimensional periodic lattice.

CH4c 2+

1 2

Here M is the total number of available adsorption sites in the simulation box. The gas components of the mixtures are represented by c, c′ = A, B, ..., K. The adsorption Gibbs free energy at site i, ΔGi, is given by eq 3. The occupancy number of component c at site i, ni(c), is related to the total occupancy ni,

(4)

2+

K

∑ ∑ ΔGi(c)ni(c)

a

Also shown are the contributions to the electronic energy (eq 4): the MP2 result for the cluster, ΔEMP2, the long-range correction, ΔELR, and the CCSD(T) correction, ΔECCSD(T). bReference 19. cReference 15. dObserved −9.95 kJ/mol.19,22,23

accuracy range for CO2 (7.3 and 6.6 kJ/mol more binding at the Mg2+ and Linker sites, respectively), but only marginal for CH4. The hybrid method we apply here is generally applicable to molecule−surface interactions including adsorption in MOFs and zeolites as long as CCSD(T) is applicable,10 otherwise so-called multireference methods have to be used. Here, we present GCMC simulations for a lattice of adsorption sites that still rely on ab initio free energy calculations for adsorption complexes at individual sites, but are independent of any analytical isotherm model and, differently from mean-field (Bragg-Williams)24 models, treat 2714

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The lateral interactions energies, Eij, are calculated by Coupled Cluster theory with complete basis set extrapolation for pairs of adsorbed molecules (CCSD(T)/CBS) at fixed structures taken from periodic DFT+dispersion calculations (see refs 15 and 19 for details). To demonstrate that our ab initio lattice-GCMC method yields isotherms in close agreement with experiment (see Figures 2 and S3), adsorption simulations have been performed

Figure 2. Experimental23 (open symbols) and calculated (GCMC, filled symbols; see section S4 of the Supporting Information for numerical data) isotherms for CO2 adsorption in CPO-27-Mg for 298 (blue triangles) and 343 K (red triangles). The calculated isotherms are scaled (76.5%) according to experimentally determined site availability.

for pure CO2 with the ΔG values of Table 1. Both Mg2+ ions and organic linkers have been considered as adsorption sites with CCSD(T) lateral interaction energies from ref.19 (−2.81, −2.24, and −0.55··· −4.70 kJ/mol for CO2/Mg2+−CO2/Mg2+, CO2/Linker−CO2/Linker, and CO2/Mg2+−CO2/Linker interactions, respectively). Linker sites can only be occupied if there is already a neighboring Mg2+ site occupied, which requires constraints on the moves (see section S3 of the Supporting Information). In Figure 2 the GCMC isotherms calculated for the ideal material have been scaled by 0.765 to take into account that, in real samples, only this fraction of sites is available for adsorption due to, e.g., structural imperfections.23 After having shown for CO2 that our ab initio lattice-GCMC method yields isotherms in close agreement with experiment, we apply it to CO2/CH4 coadsorption. For a 10:90 mixture of CO2 and CH4, the top and middle panels of Figure 3 show the adsorbed amounts of CO2 and CH4 (coadsorption isotherms) and the selectivity as a function of the pressure, respectively, whereas the bottom panel shows the dependence of the CO2/ CH4 selectivity (298 K, total pressure of 5 bar) on the gas phase composition. Only Mg2+ sites are taken into account because below 1 bar partial pressures the selective adsorption of CO223,36,37 has been attributed to these sites. For the full GCMC simulations (named “mixture” in Figure 3) the interaction energies (kJ/mol) are −2.81 (CO2···CO2), −0.55 (CH4···CH4), −1.07 (CO2···CH4 “short”) and −0.22 (CO2··· CH4 “long”); see Figure S1 of the Supporting Information. We performed additional GCMC lattice simulations to examine the effect of different approximations for adsorbate−

Figure 1. (A) Conventional unit cell of CPO-27-Mg doubled in cdirection with one CO2 molecule on an Mg2+ site (ball and stick) and the other on a Linker site (stick). (B) Positions of the adsorbed CO2 molecules in the monolayer. Bottom: Part of the lattice used for GCMC simulations. The box shows the part that is explicitly presented in panel B.

Table 1 shows the energy and Gibbs free energy data for CH4 and CO2 which are taken from ref 15 and ref 19, respectively. For CH4, harmonic vibrational energies have been obtained from a partial Hessian, assuming that CH4 molecules are freely rotating also in the adsorbed state.15 Harmonic vibrational energies for adsorbed CO2 are also obtained from a partial Hessian for the CO2 atoms only.19 To take into account the enhanced CO2 motion at the adsorption site35 the vibrational mode that corresponds to the hindered rotation of CO2 is approximated by a one-dimensional free rotation. The calculated ΔG value for CO2/Mg2+ at 298 K (−9.2 kJ/mol) deviates less that 1 kJ/mol from the experimental value (−10.0 kJ/mol) obtained from fitting19 two different sets of measured low coverage adsorption data.22,23 2715

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interactions. Simulations on the gas mixture are also performed without any of the interaction terms, i.e., the second term in the Hamiltonian (eq 5) is missing. The two components compete for the adsorption sites according to their Gibbs free energies of adsorption. This model is known as the competitive “Langmuir” model.38 Finally, the adsorbed amounts calculated from the single component isotherms are also included (“pure gas”). Separate simulations are performed for each gas component and, hence, there is no competition for adsorption sites. Molecules of one component can bind to all surface sites that are not occupied with this component without taking into account that the sites may be (partially) occupied by a molecule of the other component. Thus, the “pure gas” isotherms yield larger uptake than those obtained with the “mixture” model (Figure 3, top). In the latter case, the strongly binding component, CO2, occupies most of the sites and leaves fewer sites for CH4 adsorption. In turn, though to a smaller extent, the adsorbed CH4 curtails the number of free adsorption sites for CO2. Compared to the explicit treatment of all lateral interactions (“mixture”), the stronger CO2···CH4 interactions in the “IAST” simulations provide more relative stabilization to adsorbed CH4 molecules and, hence, their surface coverage is overestimated. The selectivity for CO2 adsorption, SCO2,CH4, (Figure 3, middle panel) increases with increasing pressure for the “mixture” model. The strongest lateral interactions are between CO2 molecules, and as more CO2 is adsorbed, the increasing share of CO2···CO2 interactions makes the CO2 adsorption progressively more favorable than CH4 adsorption. There are gradually fewer sites left for CH4 adsorption and, thus, the CO2 population on the surface increases more rapidly than that of CH4. With “IAST” simulations, overestimated CH4 surface concentrations result in an increasing underestimation of CO2 selectivity with increasing pressure. Without the competition for adsorption sites (“pure gas”), the adsorbed amount of the weakly binding component (CH4) can increase without the limitations from the adsorbed CO2 molecules. As a result, with increasing pressure the overestimation of the amounts of adsorbed CH4 increases, which is the reason for the progressive decrease of the CO2 selectivity. GCMC simulations without any lateral interactions (competitive “Langmuir”) do not show any pressure dependence of the selectivities. At zero coverage, when both gases bind to the empty surface (all the adsorbed molecules are isolated), the selectivity is determined by the ratio of the adsorption equilibrium constants Ka,c of the pure gases and, with the ΔGc values from Table 1, all models predict the same selectivity coefficient, 70.6 at 343 K and 199 at 298 K. The latter is in good agreement with the initial selectivities of 210−220 obtained with IAST by Dietzel et al.22 and by Bao et al.39 As Figure 3, bottom shows, without any interactions between adsorbed molecules (competitive “Langmuir”), the selectivity does not depend on the composition of the gas phase either. For GCMC simulations with all interactions explicitly included (“mixture”), the CO2 selectivities increase from 201 for minimal CO2 content up to 460 for almost pure CO2 with minor CH4 impurities. This can be attributed to the increase of CO 2 ···CO 2 interactions that make adsorption of CO 2 increasingly more favorable. Too strong CO2···CH4 interaction energies in IAST results in a serious underestimation of CO2 selectivities for gas mixtures containing more than 0.5% CO2, whereas they are overestimated for methane-rich mixtures.

Figure 3. GCMC results for a 10:90 mixture of CO2 and CH4. Filled symbols: all adsorbate interactions taken explicitly into account (“mixture”); black crosses: “IAST” predictions; open symbols: “pure gas” predictions. Top: Adsorbed amount of CO2 (red, triangles) and CH4 (green, squares) as a function of pressure at 343 K. Black crosses and squares: IAST. Middle: Pressure dependence of CO2/CH4 adsorption selectivity calculated from “pure gas” data (open triangles), “mixture” data (filled triangles), and according to IAST (black crosses). Circles: simulations without any lateral interaction (competitive “Langmuir”). Bottom: Dependence of the selectivity on the composition of the gas mixture (298 K, 5 bar). Symbols same as before; reverse triangles: simulations without CO2−CH4 interactions, “no CO2−CH4 int”.

adsorbate interactions for binary gas mixtures. For mimicking the “IAST” model the CO2···CH4 lateral interactions are set to −1.68 kJ/mol, the average of the CO2···CO2 and CH4···CH4 2716

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The selectivities obtained with the “no CO 2 ···CH 4 interactions” approximation are underestimated for low CO2 content and overestimated (by 135 units or about 29%) for the mixtures with high CO2 content. Since the CO 2···CH4 interaction energy, −1.1 kJ/mol, is twice as large as the CH4···CH4 interaction, −0.55 kJ/mol, the first adsorbed CO2 molecules would provide more stabilization for the CH4 molecules on the surface. Without this stabilization, the surface coverage of the weakly binding CH4 is underestimated, which results in overestimation of CO2 coverage, and, hence, overestimation of the CO2 selectivity. On the contrary, for mixtures with only minor quantities of the more strongly binding component (CO2), the surface is mainly covered with CH4. The few CO2 molecules cannot profit from CO2···CH4 interactions (which are neglected) with the many CH4 molecules around, whereas CH4 profits from CH4···CH4 interactions (which are included). For a given potential energy surface, the proposed GCMC method on a lattice of adsorption sites is computationally much faster than GCMC sampling over points of the full configuration space. Moreover, when quantum chemical methods such as DFT are used, the tedious fitting of force field parameters is avoided. Another advantage is that more accurate methods such as MP2 or CCSD(T) can be used for lateral interactions and Gibbs free energies of the individual sites, which yields adsorption isotherms in close agreement with experiment corresponding to an accuracy of 1 kJ/mol. However, the approach then becomes computationally expensive and would be the method of choice for benchmarking more approximate approaches such as DFT or force field based mixture simulations. Here, our ab initio GCMC results are used to test popular coadsorption models. The IAST model is shown to overestimate adsorption of the component with weaker lateral interactions. The IAST selectivity for CO2 impurities (less than 0.5%) in methane are overestimated, whereas it is severely underestimated in CO2-rich mixtures. We conclude that the gas separation capability of the real material can be much better than predicted by IAST. Based on the insight gained here, in future studies improved coadsorption models may be suggested. With the ab initio lattice GCMC simulations presented here, the necessary tool is available for testing them.

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ACKNOWLEDGMENTS This work has been supported by German Science Foundation (DFG) within the priority program 1570 “Porous media” and with a Reinhart 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).

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REFERENCES

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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.jpclett.7b01205. Figure with illustration of lateral interactions, definition of grand canonical ensemble for the lattice gas model, description of different types of Monte Carlo moves, and numerical isotherm data (PDF)

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Letter

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

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

The authors declare no competing financial interest. 2717

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DOI: 10.1021/acs.jpclett.7b01205 J. Phys. Chem. Lett. 2017, 8, 2713−2718