A van der Waals Density Functional Study - American Chemical Society

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Origins of CH4/CO2 Adsorption Selectivity in Zeolitic Imidazolate Frameworks: A van der Waals Density Functional Study Keith G. Ray,*,† David L. Olmsted,‡ Yao Houndonougbo,¶ Brian B. Laird,§ and Mark Asta‡ †

Department of Physics and ‡Department of Materials Science and Engineering, University of California, Berkeley, California, United States ¶ Department of Chemistry and Biochemistry, Eastern Washington University, Cheney, Washington, United States § Department of Chemistry, University of Kansas, Lawrence, Kansas, United States ABSTRACT: The energetics of methane binding are calculated in a set of five zeolitic imidazolate framework (ZIF) materials using the van der Waals Density Functional 2 (vdW-DF2) [Lee, K.; et al. Phys. Rev. B 2010, 82, 081101]. These results are compared to those from previous calculations for carbon dioxide in the same ZIFs [Phys. Rev. B 2012, 85, 085410] to examine the roles of electrostatic interactions, polarization, steric constraints, and hydrogen bonding in determining the CO2/CH4 adsorption selectivity. To isolate the effect of the chemical functionalization of the imidazolate linkers, the ZIFs considered share the same zeolite RHO topology and metal atom (Zn). Methane is found to be primarily bound by dispersion forces, and thus, site geometry and steric constraints have the greatest influence on its binding. These results are in contrast to those of carbon dioxide where electrostatic interactions play a sizable role. To quantify the relative importance of dispersion forces versus electrostatic contributions to the binding energies, we isolate these different contributions through an approach combining an analysis of the nonlocal contributions to the exchange−correlation energy and a decomposition of the charge density based on Bader analysis.

1. INTRODUCTION Zeolitic imidazolate frameworks (ZIFs) are crystalline nanoporous network materials formed by tetrahedrally coordinated transition-metal atoms bridged by organic imidazolate linkers. The bridging bonds in these linkers have a 145° angle, identical to the Si−O−Si bond found in zeolites; consequently, ZIFs are observed to form network topologies homeomorphic to those found in zeolites.1,2 A great variety of ZIFs have been already synthesized, involving different metal atoms, network topologies, and chemical functionalizations of the imidazolate linker.3 Thus, a large parameter space exists for the optimization of the properties of ZIFs for applications such as gas separation or storage. In particular, ZIFs have received considerable attention for application to separations in gas mixtures involving methane, for the upgrading of natural gas.4−6 For such applications, the adsorption selectivity of CH4 relative to that of other gas species, such as CO2, is a property of fundamental interest.7−9 Methane adsorption and selectivity in ZIFs have been studied both experimentally 10−14 and computationally5,9,11,15−17 for pure gases and gas mixtures relevant in the context of natural gas separations. Experimental studies in ZIFs have involved measurements of adsorption isotherms,10−12 breakthrough,12 and membrane selectivity.13 The understanding derived from these studies has been complemented by relevant investigations in other related metal−organic framework (MOF) systems,18−27 featuring data on the time-resolved uptake of methane and carbon dioxide24 as well as neutron © 2013 American Chemical Society

scattering to characterize CH4 and CO2 binding sites and occupations.28,29 The studies of ZIFs have shown that these frameworks can exhibit adsorption selectivities and uptakes that exceed the performance of commercial adsorbents used for natural gas separation18,21 and that the performance depends on several factors, including the choice of linker, functionalization, pore size, and metal ion.10,11,19−23 The understanding derived from experimental investigations has been supplemented with computational studies employing a variety of methods, from grand canonical Monte Carlo (GCMC) and molecular-dynamics (MD)5,9,11,15−17,30−37 simulations based on classical force fields to studies of binding energetics and geometries based on density functional theory (DFT), coupled cluster, and second-order Møller−Plesset perturbation theory (MP2).20,37−39 GCMC-simulated adsorption isotherms have reproduced experimentally measured trends in adsorption selectivity in several ZIFs. For separations involving CO2 and CH4 mixtures, in particular, gas selectivity in these materials is often attributed to stronger CO2 electrostatic interactions,9,11,16,20,30−34,40 specifically with an ionic framework,9 the linker,11,34 or polarizable functionalities,20,34 depending on the system. In the current study, we examine in detail the origin of the selective binding of CO2 relative to CH4 in five ZIFs, namely, Received: April 29, 2013 Revised: June 19, 2013 Published: June 21, 2013 14642

dx.doi.org/10.1021/jp404251m | J. Phys. Chem. C 2013, 117, 14642−14651

The Journal of Physical Chemistry C

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(ZIF-71), one −CHO and one −CH3 group (ZIF-93), one −CN and one −NH2 group (ZIF-96), and one −CH2OH and one −CH3 group (ZIF-97). The latter three ZIFs, -93, -96, and -97, each have two different functional groups on each imidazole linker. 2.2. van der Waals Density Functional Calculations. The most favorable binding sites obtained from the classical force field calculations reported in ref 42 are used to develop initial geometries for input into the first-principles calculations based on the vdW-DF2 method,44 as was done in our previous work on CO2.45 In the vdW-DF2 method, the exchange− correlation energy is divided into a semilocal exchange energy, a local correlation energy, and a nonlocal correlation energy.

ZIF-25, -71, -93, -96, and -97.41 These ZIFs share the same zeolite RHO topology but differ in the types of functional groups attached to the linker, allowing us to explore the effects of chemical functionalization on the relative binding energies of CH4 and CO2. This work builds on recent studies by Houndonougbo et al.42 involving experimental measurements and GCMC simulation calculations of adsorption isotherms for CH4 in this same set of ZIFs, as well as calculations of CO2/ CH4 adsorption selectivities using the ideal adsorbed solution theory (IAST).43 In ref 42, it was found that the CH4 adsorption per unit surface area was relatively constant with respect to ZIF functionalization, while previous investigations on CO241 in this set of ZIFs showed an increase of adsorption per unit surface area in those frameworks that featured two different functional groups on the same imidazolate ring. In these ZIFs with asymmetric functionalities, it was concluded that CO2 interacts more strongly with the linkers through electrostatic interactions, which enhances the selectivity relative to CH4. To understand in further detail the role of electrostatic and dispersion contributions to the selective binding of CO2 relative to CH4, we employ in the current work calculations based on the van der Waals Density Functional 2 (vdW-DF2) approach.44 We begin by presenting vdW-DF2 calculations of binding energies for CH4 in the ZIFs under consideration. The results are compared with those of our previous calculations of CO2 binding energies45 in the same systems with the same functional44 to highlight the energetic origins of adsorption selectivity. The binding energy results are complemented by an analysis of the relative importance of dispersion versus electrostatic and polarization contributions to the binding energies. For this purpose, we quantify the relative magnitudes of these different contributions through an approach combining an analysis of the nonlocal contributions to the exchange− correlation energy and a decomposition of the charge density based on Bader analysis.46−48 This analysis includes an estimate of the contributions to the binding energy resulting from the static polarizability of the gas molecule and functional groups. Such information can be useful for guiding the development of classical force fields and complements related studies undertaken with symmetry-adapted perturbation theory (SAPT).49−51

Exc = Ex + E lc + Enlc

(1)

where Ex is a semilocal exchange functional (taken from the PW86 parametrization52 in the vdW-DF2 method), Elc is a local correlation energy that is given by the local density approximation, and Enlc is the nonlocal contribution to the correlation energy, which can be written as Enlc =

1 2

∫ dr dr′ n(r)φ(r, r′)n(r′)

(2)

In eq 2, the kernel, φ, is a function of both r and r′ through a dependence on the electron densities (n(r)) at r and r′. Enlc is derived from the adiabatic connection fluctuation dissipation theorem through a series of approximations including an expression for the dielectric response based on the plasmon pole model.53 The vdW-DF2 differs from the original vdW-DF of Dion et al.53 in that it uses a different exchange functional and a changed parameter in Enlc representing a limit more appropriate for molecular interactions. It is worth noting that related functionals have been proposed involving different exchange functionals and have been shown to yield accurate results.54 We choose the vdW-DF2 because it yields more accurate binding energies and distances than the vdW-DF44 but still has no fitted parameters and therefore may be more transferable than other methods that introduce fitting to a particular reference data set. The vdW-DF class of methods has been previously applied to the calculation of binding energies, binding geometries, and vibrational frequencies of CO2 in MOFs and ZIFs40,45,55−58 as well as (H2)4CH4, C2H6, C3H8, and C4H10 in MOFs.59,60 To the best of the authors’ knowledge, the vdW-DF method has not been used previously for calculations of methane binding energies in MOFs or ZIFs. Because we are concerned with nonbonded interactions between the framework and guest CH4 molecule, which has an octupole as its lowest nonvanishing electric multipole, the dispersion interaction is expected to be of primary importance, such that use of the vdW-DF formalism is viewed to be essential for computing realistic binding energies within the formalism of DFT. vdW-DF2 calculations have been performed with the Vienna ab initio simulation package (VASP)61−65 using the projectoraugmented wave (PAW) method66,67 with Perdew−Burke− Ernzerhof (PBE)68 based potentials from the VASP library. The vdW-DF, as implemented in VASP, uses the efficient numerical algorithm for evaluating Enlc due Roman-Perez and Soler.69 The electronic wave functions are expanded in a plane-wave basis set with a cut-off energy of 400 eV, and only a single (Γ) kpoint is used in the Brillouin zone sampling due to the large size of the primitive unit cell. Forces on CO2 and CH4 molecules are relaxed to 0.01 eV/Å, and the atoms in the

2. MATERIALS AND METHODS Our aim is to elucidate the variations in CO2/CH4 binding selectivity resulting from changes in the functional group in a series of isorecticular ZIFs. We first use the results of the classical simulations reported in ref 42 to identify the important binding sites. Next, we relax the gas molecule position and orientation using the vdW-DF2 and separate the contribution to the binding energy due to dispersion from the nonlocal contribution to the correlation energy. Finally, we utilize a method of estimating the individual contributions to the electrostatic interactions due to static charge distributions and induced polarization to analyze trends in these contributions across functional groups. 2.1. ZIF Structures. We focus on a set of RHO topology ZIFs, each of which has a different set of functional groups on the 4 and 5 sites of the imidazole ring. With the exception of these functional group differences and a resultant 1.5% variation in lattice constants, the structures are identical.41 The chemical functionalities, depicted in the Abstract graphic, are as follows: two −CH3 groups (ZIF-25), two −Cl atoms 14643



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framework are held fixed at the experimentally determined positions41 to reduce the computational cost. We expect that framework flexibility would have the largest effect on CO2/CH4 binding sites, described below, that are sterically constrained. Among the sterically constrained cases, we find for CO2 in site C (site described below) of ZIF-97 and CH4 in site C of ZIF-93 and ZIF-96 that fixing the ZIF atom positions produces a difference of at most 2.0 kJ mol−1 in the binding energies compared with a flexible calculation. Binding energies are estimated to be converged to better than 0.3 kJ mol−1 with respect to the plane-wave cutoff energy, number of k-points, and the density of the real-space grid used to represent the charge density. In what follows, we will estimate the dispersion contribution to the binding energy by comparing Enlc in the combined gas molecule−ZIF system with the sum of the values for the isolated gas molecule and isolated ZIF. That is bind Enlc

=

gas (Enlc

+

ZIF Enlc )

−

gas−ZIF Enlc

binding sites. We thus consider the importance of this contribution in further detail in the current work for the ZIFs described above. To leading order in the induced charge densities, the contribution of electrostatic induction (i.e., induced static polarization) to the binding energy can be written as gas total ZIF E ind = E ind + E ind

(4)

where the two terms on the right-hand side are defined as gas E ind =−

∫ dr ρindgas (r)ΦZIF(r)

(5)

∫ dr ρindZIF (r)Φgas(r)

(6)

and ZIF E ind =−

ρgas ind

In eq 5, is the difference in the charge distribution of the gas molecule when bound to the ZIF relative to when it is an isolated molecule, and ΦZIF is the electrostatic potential associated with an isolated ZIF. Similarly, ρZIF ind is the induced charge density in the ZIF, and Φgas denotes the electrostatic potential of an isolated gas molecule. As above, the negative sign keeps with the convention of positive attractive binding energies. In the current work, we obtain ρgas ind by first determining a volume associated with the gas molecule using Bader analysis46−48 applied to the charge density calculated for the combined ZIF−gas system. This choice of volume is nonunique and, to some extent, arbitrary. However, because we are looking at only the induced difference in charge density and the electron density is small in the region midway between the gas molecule and the neighboring ZIF functional groups (i.e., the interactions are nonbonded in nature), our results are relatively insensitive to changes in the exact shape of the volume provided it is not enlarged to include charge density from neighboring molecules or functional groups. We extract the electron density in this volume and call it ρgas combined.. We also perform calculations on the isolated ZIF and isolated gas systems in the same binding geometry (but with the gas molecule at its isolated equilibrium geometry, unbent, and unstretched) and sum their charge densities. This is the charge density without any induction, from which we extract the same Bader volume, and we call the result ρgas summed isolated. We then define

(3)

Ebind nlc

where is the contribution to the binding energy from the ZIF nonlocal correlation energy and Egas−ZIF , Egas nlc nlc , and Enlc are the nonlocal correlation energies of the combined ZIF−gas molecule system, the isolated gas molecule, and the isolated ZIF, respectively. The definition in eq 3 keeps with the convention of positive binding energies defined by the heat released from reaction (i.e., attractive interactions). The value of Ebind nlc has been shown to depend largely on the low electron density regions surrounding the two bonded components.70 Although in the asymptotic limit Ebind nlc represents the dispersion energy and has the usual r−6 dependence,53 when there is a significant overlap of charge, there can be other contributions to Ebind nlc . For the nonbonded gas molecule−ZIF interactions considered in the current work, we expect small electron density overlap between the gas molecule and the framework atoms, and we thus equate Ebind nlc , defined in eq 3, with the contribution to the binding energy arising from dispersion interactions. 2.3. Electrostatic Induction Energy Calculations. Strong electrostatic framework−gas molecule interactions suggest that charge redistribution may be important to binding, and highly polarizable functional groups have been associated with high CO2/CH4 adsorption selectivity in previous work.20 Static charge redistribution, that is, electrostatic induction, can occur in the gas molecule due to the electrostatic potential of the ZIF and vice versa. To compute the contributions to the binding energy due to this polarization effect, we employ the following approach. To first order in the induced charge, the contribution to the gas−framework binding energy due to charge polarization can be obtained by considering the energy of the induced charge density in the ZIF in the electrostatic potential of the gas molecule (EZIF ind ) and the induced charge density on the gas molecule in the electrostatic potential of the ZIF (Egas ind). This induction term has been previously calculated using SAPT for CO2 in several ZIFs.49,50 The contribution from this energy was found to be less than 1 kJ mol−1 for the ZIFs considered.50 We note, however, that this result was obtained for ZIFs that did not display the type of asymmetric functionalization characteristic of three of the ZIFs considered here; the work in ref 41 suggests that such asymmetric functionalizations give rise to stronger electrostatic interactions. Further, the work in ref 50 presented induction energies averaged over several binding geometries, and the contribution could be larger for specific

gas gas gas ρind = ρcombined − ρsummed isolated

(7)

ρZIF ind

The induced charge density is defined similarly using the volume outside of the gas molecule Bader volume. It should be noted that the procedure described above includes the deformation of the gas molecule but not of the ZIF functional groups, which remain frozen at their experimentally determined positions in the calculations. In addition, we calculate the monopole and dipole moments of the induced charge density of the gas molecule as gas Δqmonopole =

∫ dr ρindgas (r)

(8)

and gas Δμdipole = ||

∫ dr [rρindgas (r)]||2

(9)

Δqgas monopole

where ∥x∥2 is the Euclidean norm of x. We find to be quite small, less than 0.016 e, although nonzero, which means 14644



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Figure 1. (a) CH4 binding energy landscape on a {100} slice of the cubic unit cell of ZIF-25 calculated using Lennard-Jones potentials. (b) RHO topology net of the ZIFs considered in this work with important binding sites A, B, and C labeled.

that Δμgas dipole would be origin-dependent. To remedy this, the Bader volumes could be slightly expanded or contracted to be made neutral. In practice, however, we rescale the negative charge, which is equivalent to changing the volume if the new choice of volume does not change the center of negative charge. With this modification, our definition of Δμgas dipole is independent of the origin. We also calculate the change in monopole and dipole moments of the ZIF functional group nearest to the gas molecule. As described above for the gas molecule, our definition of the change in dipole moment is made independent of the origin by rescaling the negative charge associated with the functional group. Included in the Bader volume associated with the functional group are the nearest carbon and nitrogen atoms on the imidazole ring so that we do not exaggerate the changes in the monopole and dipole with charge transfer between the ring and functional group. The relevant definitions for these quantities are FunctGrp Δqmonopole =

∫ dr ρindFunctGrp (r)

case of ZIF-25. Labeled in Figure 1a and b are the positions of three primary binding sites, A, B, and C, which are common to each of the five RHO ZIFs considered in this work. Sites A and B are each located in a different channel between Linde type A (LTA) cavities of the RHO topology ZIFs. Binding site A is within a ring of six Zn atoms and six imidazolate linkers in channels along ⟨111⟩ crystallographic directions, and B is within a ring of eight zinc atoms and eight imidazolate linkers in channels along ⟨100⟩ directions. Site C is located on the surface of the LTA pore, along ⟨110⟩ directions from the center of the pore. These framework locations correspond to the positions of the primary sites for CO2 binding in the same ZIFs, as identified by classical force fields and reported in ref 45. Binding energies taken from vdW-DF2 calculations for CH4 and CO2 in the three binding sites of ZIF-25, -71, -93, -96, and -97 are presented in Table 1. The results for CH4 were obtained in the present work, while those for CO2 are taken from ref 45. In every instance, the binding for CO2 is stronger than that for CH4, showing a universal selectivity in these materials for carbon dioxide over methane. The selectivity in binding energy ranges from a negligible 0.9 kJ mol−1 in site A of

(10)

and FunctGrp Δμdipole = ||

∫ dr [rρindFunctGrp (r)]||2

Table 1. CH4 and CO2 Binding Energies, Ebind total, Calculated with the vdW-DF Method, As Well As the Contribution of the Nonlocal Correlation to the Binding Energy, Ebind nlc , Tabulated in Units of kJ mol−1a

(11)

2.4. Binding Energy Decomposition. In the previous sections, we have described methods to evaluate the portion of the binding energy due to dispersion, Ebind nlc , and electrostatic induction, Etotal ind , giving us the following decomposition. bind Etotal

=

bind Enlc

+

total E ind

+

bind Eother

CH4 ZIF

Ebind total

Ebind nlc

bind Ebind total − Enlc

Ebind total

Ebind nlc

bind Ebind total − Enlc

A

−25 −71 −93 −96 −97 −25 −71 −93 −96 −97 −25 −71 −93 −96 −97

36.1 33.1 35.1 29.9 38.1 19.4 21.5 24.9 22.9 31.8 26.7 25.7 18.5 9.3 21.2

41.1 36.8 38.2 34.7 43.3 21.4 24.5 27.6 25.0 32.5 35.3 35.9 26.8 14.0 24.0

−5.0 −3.7 −3.1 −4.8 −5.2 −2.0 −3.0 −2.8 −2.1 −0.7 −8.5 −10.2 −8.3 −4.7 −2.8

40.5 39.9 36.0 36.3 44.3 22.6 31.2 39.4 37.1 44.8 30.8 29.9 41.7 41.8 24.1

42.2 45.2 44.9 42.9 44.8 26.6 35.4 27.2 37.9 25.9 41.4 37.9 45.3 43.3 21.1

−1.7 −5.4 −8.9 −6.6 −0.5 −3.9 −4.2 12.2 −0.8 18.9 −10.7 −8.1 −3.5 −1.5 3.0

(12)

In this decomposition, Ebinding other contains contributions from the electrostatic interaction between the unperturbed gas molecule and ZIF charge distributions, exchange repulsion, higher-order contributions from electrostatic induction, and the correlation energy associated with the local density approximation.

B

3. RESULTS AND DISCUSSION 3.1. Calculated Binding Energies. In ref 42, LennardJones potentials with parametrizations optimized for CH4 were used to calculate binding energy landscapes for CH4 in the five RHO ZIFs described in section 2.1. We summarize here the results presented in ref 42 to define the positions of the binding sites that will be discussed in the context of the vdW-DF2 calculations below. A binding energy landscape calculated with classical force fields is shown in Figure 1a for the representative

C

a

14645

CO2

site

CO2 values are taken from ref 45. dx.doi.org/10.1021/jp404251m | J. Phys. Chem. C 2013, 117, 14642−14651


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Figure 2. Binding geometries for CH4 and CO2 in site B for ZIF-25, -71, -93, -96, and -97. Induced polarizations due to gas molecule−framework interactions are given by yellow (blue) isosurfaces showing electron accumulation (depletion) at a density of 0.001 e Å−3. For each gas molecule and nearest functional group, boxed in blue, the magnitudes of the induced dipole moments of the associated Bader volumes are listed in units of D. A key to the atom colors can be found in the Abstract graphic (b) near the ZIF-93 linker.

ZIF-93 to 32.5 kJ mol−1 in site C in ZIF-96. The calculated results are qualitatively consistent with the trends in adsorption reported in ref 42, based on IAST calculations employing measured single-component adsorption isotherms. Specifically, the IAST results give the following CO2/CH4 adsorption selectivities at 1 bar and 298 K: ZIF-96, 10.2; ZIF-93, 8.2; ZIF97, 6.1; ZIF-71, 2.7; and ZIF-25, 2.5. If we order the ZIFs using the difference in the CO2 and CH4 calculated binding energies averaged across all of the binding sites, using the calculated results in Table 1, we obtain the same ranking (positive numbers indicate stronger CO2 average binding): ZIF-96, 17.8 kJ mol−1; ZIF-93, 12.8 kJ mol−1; ZIF-97, 7.3 kJ mol−1; ZIF-71, 6.8 kJ mol−1; ZIF-25, 3.9 kJ mol−1.

3.2. Dispersion Contributions. In Table 1, the nonlocal correlation contributions to the binding energy, Ebind nlc , are listed. These contributions are representative of the magnitude of the dispersion contributions to the binding energies, as described above. In site A, this energy varies with respect to ZIF functionalization by 8.6 kJ mol−1 for CH4 and 3.0 kJ mol−1 for CO2. In this site, the gas molecule is surrounded by a ring of six linkers and is in closest proximity with the atoms in the imidazole ring rather than those in the functional groups, as illustrated in the Abstract graphic (a). −1 In site B, Ebind for CH4 and 12.0 kJ nlc varies by 11.1 kJ mol −1 mol for CO2. As illustrated in Figure 2, the gas molecule in site B is located closer to the functional groups attached to the 14646



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Figure 3. CH4 in site C for the ZIFs considered. CH4 carbon to framework hydrogen distances are given in Å. The large variations in bond lengths highlight the importance of steric constraints. A key to the atom colors can be found in the Abstract graphic (b) near the ZIF-93 linker. bind As shown in Table 1, the trends in the difference Ebind total − Enlc for CH4 in ZIFs with distinct chemical functionalities differ from those of CO2. For CH4 in sites A and B, which do not show the pronounced steric constraints associated with site C; the variation in the magnitude of this energy difference is very small, only 2.1 kJ mol−1 for site A and 2.3 kJ mol−1 for site B. Thus, in these sites, nondispersive contributions to the binding energy are relatively insensitive to functional group identity. In −1 bind contrast, the variations in Ebind total − Enlc for CO2 are 8.4 kJ mol −1 in site A and 23.1 kJ mol in site B. Therefore, nondispersive interactions are important for CO2 binding and in governing the selectivity of CO2 over CH4 in these materials. In site B of ZIF-93 and -97, nondispersive contributions to the binding energy account for 15.0 and 19.6 kJ mol−1 of the difference in CH4 and CO2 binding energies, respectively. bind Because the magnitudes of the energy difference Ebind total − Enlc show large variations in site B across the ZIF series, we focus on this site to examine in more detail the factors contributing to this quantity. In what follows, we discuss the energy associated with induced charge polarization as defined in eqs 4−6, as well as the structural distortion of CH4 and CO2 and the changes in the gas molecule dipole moments. The energy of electrostatic induction in the gas molecule due to the ZIF electrostatic potential, Egas ind, is given in Table 2 for CO2 and CH4 in site B. For CH4, the values are, in general, smaller than those for CO2. The largest CH4 induced polarization energies are found in the asymmetrically functionalized ZIF-93 and -97 (3.7 and 3.0 kJ mol−1, respectively). Correspondingly, the induced gas dipole moments, Δμgas dipole, are also highest for CH4 in -93 and -97 (0.11 and 0.33 D, respectively). However, in every ZIF considered, the distortions in CH4 bond angles and lengths are relatively much smaller than the maximum found for CO2. For CO2, we see values of −1 Egas in ZIF-96 and -97. The ind as high as 5.2 and 10.6 kJ mol gas largest magnitudes of Δμdipole for CO2 are also obtained in these ZIFs, 0.72 D in ZIF-96 and 0.73 D in ZIF-97. The largest O−C−O bond angle and C−O bond length distortions are also found in ZIF-96 and -97, respectively. For CH4, the binding geometry in site B, as characterized by the center of mass and orientation of the molecule relative to the neighboring functional groups, is similar across the five different ZIFs. However, for CO2, the variations in electrostatic induction energy, molecule distortion, and induced dipole

imidazole ring than the rings themselves or the metal ions, consistent with the larger variation in the calculated binding energies across the ZIFs. In this geometry, the variation in Ebind nlc may be roughly attributed to the number of nearby functional groups and their effective frequency-dependent polarizability.71 In site B, the CO2 dispersion energies do not follow the same trend as CH4 because CO2 has different binding orientations and positions in this site, depending on the host ZIF, as shown in Figure 2. These differing CO2 binding geometries have varying numbers of nearby framework atoms; therefore, Ebind nlc , which depends strongly on the local coordination to members of the framework, will also vary.45 In contrast, the number of nearby functional groups to CH4 in site B is the same in each ZIF (although only the functional groups that have an induced charge density greater than 0.001 e Å−3 are shown in Figure 2). Site C is cup-shaped with the sides defined by four functional groups and the bottom defined by four hydrogen atoms, each bound to the carbon atom positioned directly between two nitrogens in an imidazole ring. As illustrated in Figure 3, larger functional groups force methane to bind further away from the center of the site. A larger binding distance results in a smaller Ebind nlc , and the variation in the distance is responsible for the large variation in Ebind nlc in this site across the ZIFs considered, 21.9 kJ mol−1 for CH4 and 24.2 kJ mol−1 for CO2. This cup geometry can enhance selectivity because the binding energy of CH4 is observed to be reduced by steric constraints in ZIF-93, -96, and -97, while there is only a hindrance in ZIF-97 for the linear CO2 molecule. For instance, CO2 is bound closest to the framework of ZIF-93 and has the highest binding energy in site C, with a 3.0 Å distance between an O atom in CO2 and a H atom in a HCO group. In site C, the binding energy for CH4 (CO2) in ZIF-93 and -96 is 18.5 (41.7) and 9.3 (41.8) kJ mol−1, respectively, showing a large influence on selectivity arising from steric constraints. 3.3. Electrostatic Contributions. We begin by examining bind the difference in Ebind total and Enlc in order to analyze the nondispersive contributions to binding energy that depend on the ZIF functionalization. These contributions include the electrostatic interactions between the charge distributions of the gas molecule and ZIF, as well as the exchange energy and local correlation energy associated with the induced polarization and (small) charge overlap. 14647



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Table 2. In Site B, the Maximum Change with Respect to Free Molecules in the Bond Angle between Neighboring CH Bonds in CH4 (Δ∠H−C−H) and between CO Bonds in CO2 (Δ∠O−C−O), the Respective Greatest Changes in Bond Length (ΔRC−H and ΔRC−O), the Induced Dipole Moments of the Bader Volumes Associated with These Molecules, and the Energies Associated with These Electronic and Ionic Distortions in the Field of the Fixed ZIF gas

ZIF

Δ∠H−C−H (°)

ΔRC−H (Å)

Δμgas dipole (D)

CH4

−25 −71 −93 −96 −97 ZIF

0.14 0.18 0.43 0.43 0.45

0.04 0.03 0.11 0.08 0.33

Δ∠O−C−O (°)

0.002 0.002 0.002 0.002 0.002 ΔRC−O (Å)

Δμgas dipole (D)

−25 −71 −93 −96 −97

0.11 0.33 0.06 1.70 0.05

0.001 0.001 0.002 0.000 0.009

0.01 0.02 0.24 0.72 0.73

gas CO2

Table 3. Data for the Functional Group in Each ZIF Closest to the Guest Gas Moleculea gas

ZIF

functional group

distance (Å)

ΔμFunctGrp dipole (D)

EZIF ind (kJ mol−1)

CH4

−25 −71 −93 −96 −97 −25 −71 −93 −96 −97

CH3 Cl CH3 NH2 CH2OH CH3 Cl CHO NH2 CH2OH

2.65 3.22 2.76 2.68 2.53 2.88 3.42 3.23 2.20 2.72

0.01 0.02 0.01 0.00 0.01 0.05 0.05 0.09 0.04 0.08

0.0 0.0 0.0 0.0 0.0 0.9 1.1 1.2 1.6 1.0

−1 Egas ind (kJ mol )

Egas ind

CO2

1.3 1.0 3.7 1.3 3.0 (kJ mol−1)

a

The chemical formula, proximity, and change in dipole moments of this functional group are given, as well as the energy due to the total change in the ZIF electron charge density in the electrostatic field of the gas molecule.

1.2 3.3 0.6 5.2 10.6

Table 4. CH4 and CO2 vdW-DF2 Binding Energies (kJ mol−1)a

moments in site B across the five ZIFs can be understood to be due to the differences in binding geometry illustrated in Figure 2 and discussed previously in ref 45. In ZIF-25 and -71, the CO2 is bound in a position that is symmetrically oriented with respect to the two functional groups, which are identical in these ZIF structures. By contrast, in ZIF-93, -97, and -96, which feature asymmetrically functionalized linkers, the CO2 molecule is asymmetrically oriented with respect to the two functional groups such that the positively charged C atom in CO2 is nearest the negatively charged O, N, and O atoms in the functional groups in -93, -96, and -97, respectively. For -96 and -97, the O atom in the CO2 is nearest hydrogen atoms in ZIF96 and -97 (H bond donors), at distances of 2.2 and 2.7 Å, respectively, which are indicative of hydrogen bonding.45 Hydrogen bonding is also consistent with the larger values of the induced CO2 dipole and induced polarization energy for these two cases.72 The interactions between the gas and functional group atoms give rise to the bond stretching reported in Table 2 for -97 and the bond bending reported for -96. In Table 3, the functional group nearest the gas molecule in the B site is listed for both CH4 and CO2 in each of the five ZIFs. Also listed are the nearest-neighbor distances between the atoms in the gas molecule and neighboring functional groups, as well as the induced dipole moment on the neighboring functional groups and the magnitude of the energy EZIF ind defined in eq 6. The dipole moments on the neighboring functional groups are an order of magnitude larger for the case of CO2 compared with those for CH4, in qualitative agreement with expectations, given that the lowest-order nonvanishing electrostatic moment for CO2 is that of a quadruple, while it is an octupole for CH4. Overall, however, the induced dipole moments on the functional groups in Table 3 are significantly smaller than those for the gas molecules in Table 2. Consistent with this finding, the values of EZIF ind in Table 3 are significantly smaller in magnitude than Egas ind. The largest value of the energy associated with induced polarizations in the ZIF is 1.6 kJ mol−1, obtained for CO2 in ZIF-96. For CH4 in all ZIFs considered, this energy is zero within the precision of our calculations. 3.4. Binding Energy Decomposition. In Table 4, we list for CH4 and CO2 in site B the calculated values of the binding

gas

ZIF

Ebind total

Ebind nlc

Etotal ind

Ebind other

CH4

−25 −71 −93 −96 −97 −25 −71 −93 −96 −97

19.4 21.5 24.9 22.9 31.8 22.7 31.2 39.4 37.1 44.8

21.4 24.5 27.6 25.0 32.5 26.5 35.4 27.2 37.9 25.9

1.3 1.0 3.7 1.3 3.0 2.1 4.4 1.8 6.8 11.6

−3.3 −4.0 −6.5 −3.3 −3.7 −5.9 −8.7 10.4 −7.6 7.4

CO2

a bind Etotal is the total binding energy, Ebind nlc is the dispersion contribution to the binding energy, Etotal ind is the contribution to the binding energy bind bind total from electrostatic induction, and Ebind other = Etotal − Enlc − Eind .

bind energy (Ebind total), the values of Enlc , which reflect the contribution of the dispersion interactions, and Etotal ind , defined by eqs 4−6, which gives a measure of the contributions due to induced polarization. In the last column of Table 4 is listed the quantity bind total total Ebind other = Etotal − Eind − Enlc , which represents exchange, local correlation, higher-order electrostatic interactions due to charge redistribution, and the electrostatic interaction between the unperturbed charge distributions of the ZIF and gas molecule. For CH4 in site B, we notice that Ebind total runs between 19.4 and 31.8 kJ mol−1, while the quantity Ebinding other varies by only 3.2 kJ −1 mol−1 across the five ZIFs, and Etotal ind varies by only 2.7 kJ mol . These results highlight the dominant contribution of dispersion interactions for CH4 binding in this site, as measured by Ebind nlc . −1 For CO2 in site B, Etotal ind varies between 1.8 and 11.6 kJ mol , showing that this energy can be a significant portion of the total binding energy and that this contribution is selective with respect to the adsorbed gas molecule (i.e., the contribution is larger in magnitude for CO2 than CH4). As discussed in the previous paragraphs, the variation in this quantity is due most likely to differences in the electrostatic interactions between the CO2 and ZIF, depending on chemical functionalities and their symmetry as well as hydrogen bonding in ZIF-96 and -97. For −1 CO2, Ebinding in other has the values of −5.9, −8.7, and −7.6 kJ mol ZIF-25, -71, and -96, respectively, which are opposite in sign to the values of 10.4 and 7.4 kJ mol−1 in ZIF-93 and -97, respectively. These differences are consistent with the expected

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groups can introduce steric constraints for CH4 that do not hinder CO2 binding.

variations in the electrostatic energy across these ZIFs. Specifically, in the symmetrically functionalized ZIF-25 and -71, CO2 relaxes to a position maximizing the dispersion energy, as we can see with the larger value of Enlc, and there are negligible electrostatic interactions between the unpolarized charge distributions. In ZIF-93 and -97, Enlc is smaller, and instead, electrostatic interactions with the framework are maximized by the position of the CO2, and an attractive Ebinding other is calculated. In ZIF-96, CO2 has a large attractive induced polarization energy as well as a large value of Ebind nlc . The results of the partitioning of the binding energies into dispersion, electrostatic induction, and a combination of other contributions including electrostatic and exchange are similar to what was reported by McDaniel et al.49,50 using SAPT. The nonlocal correlation energies for CH4 and CO2 in the five ZIFs considered here range from 58 to 117% of the total binding energies, and the electrostatic induction energies correspond to 4.5 to 26% of the total binding energy. These results are consistent with those found for CO2 in ZIF-8 and ZIF-71 in ref 50, which shows a roughly 5% contribution from induction and a roughly 160% contribution from dispersion (although this is not a direct comparison because the numbers in that study are averages over many CO2 positions and our results are for a particular binding site). Furthermore, in ref 49, SAPT-derived potentials show that the ratio of dispersion energy to electrostatic energy is much higher for N2 in several ZIFs than that for CO2 in the same ZIFs. This could be due to a smaller electrostatic energy associated with N2, which has roughly one-third of the electric quadrupole moment of CO2,73 as is suggested in ref 49. The electrostatic interaction for CH4 is found to be smaller than that for CO2 in our work, as expected because CH4 has vanishing electric quadrupole and lower moments.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We would like to thank Dr. William Morris and Dr. Colin Ophus for valuable discussions and for their assistance with Figure 1. The image in Figure 1b was supplied by Dr. William Morris. This work was supported as part of the Molecularly Engineered Energy Materials (MEEM), an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences under Award Number DE-SC0001342. This work made use of resources of the National Energy Research Scientific Computing Center, supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231.

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REFERENCES

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4. SUMMARY AND CONCLUSION In this work, the vdW-DF2 formalism was used to compute the binding energies of CH4 and CO2 molecules in three different binding sites across five differently functionalized RHO topology ZIFs. These calculations show that ZIF-93 and ZIF96 both bind CO2 more strongly than CH4 in sites B and C, by more than 14.2 kJ mol−1 and as high as 34.0 kJ mol−1. The selectivity in site B is determined to originate from enhanced electrostatic interactions between asymmetrically functionalized ZIFs and the electric quadrupole moment of the CO2, as opposed to the higher-order octupole moment of the CH4. Hydrogen bonding with the CO2 also occurs in ZIF-96 and -97. Site A is found to be comparatively nonselective to CO2 versus CH4 binding in all of the RHO ZIFs considered. The selectively in site C is determined to originate from steric constraints due to the geometry of the binding site and the different sizes of CO2 and CH4 molecules. For two of the ZIFs studied in this work (namely, ZIF-96 and -97), the contribution to the binding energy due to polarization is calculated to be on the order of 20−25% of the binding energy and is thus significant. From the specific results for the three binding sites in five RHO topology ZIFs, we conclude that chemical functionalization can have a large effect on CO2/CH4 adsorption selectivity. Having two different functional groups attached to each linker (asymmetric functionalization) as well as functional groups containing hydrogen bond donors are shown to increase the CO2 binding strength and selectivity over CH4. In addition, depending on the binding site geometry, larger functional 14649



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