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Predictive Framework for Shape-Selective Separations in ThreeDimensional Zeolites and Metal−Organic Frameworks Eric L. First, Chrysanthos E. Gounaris, and Christodoulos A. Floudas* Department of Chemical and Biological Engineering, Princeton University, Princeton, New Jersey 08544, United States S Supporting Information *

ABSTRACT: With the growing number of zeolites and metal−organic frameworks (MOFs) available, computational methods are needed to screen databases of structures to identify those most suitable for applications of interest. We have developed novel methods based on mathematical optimization to predict the shape selectivity of zeolites and MOFs in three dimensions by considering the energy costs of transport through possible pathways. Our approach is applied to databases of over 1800 microporous materials including zeolites, MOFs, zeolitic imidazolate frameworks, and hypothetical MOFs. New materials are identified for applications in gas separations (CO2/N2, CO2/CH4, and CO2/H2), air separation (O2/N2), and chemicals (propane/propylene, ethane/ethylene, styrene/ethylbenzene, and xylenes).

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INTRODUCTION Zeolites and metal−organic frameworks (MOFs) are microporous materials commonly used for shape-selective separations in which molecules are separated on the basis of their shape and size. There are hundreds of zeolites,1 thousands of MOFs,2,3 and millions of hypothetical structures4,5 to consider; therefore, a quick screening method is needed to narrow down the field to the most promising candidates for detailed study. Recently, a number of screening methods have been developed on the basis of adsorption selectivity,5−12 diffusive selectivity,12,13 and pore geometry analysis.5,8,10,14 Watanabe and Sholl15 developed a hierarchical approach that considers all three screening criteria to identify candidate MOF membranes for gas separations. We previously developed a nonlinear optimization-based screening method16−18 in which the shape selectivity is predicted from the optimal rotation and translation of a guest molecule to percolate most comfortably through a zeolite’s portals. A guest molecule, depending on its size and shape, requires an activation energy to pass through a portal. Shape selectivity can then be evaluated by comparing the activation energies for two different species. Such an optimization-based approach enables a high-throughput screening for kinetic-based separations with greater sophistication than methods that consider only geometry (e.g., pore-limiting diameter) and less computational effort than detailed molecular simulations (e.g., molecular dynamics and Monte Carlo methods). Recently, we developed computational methods for the automated 3D characterization of zeolite19 and MOF2 porous networks via ZEOMICS and MOFomics, respectively. These methods are based on geometry, graph, and optimization algorithms. They automatically identify the portals, channels, © 2013 American Chemical Society

and cages of a microporous structure and describe the geometry and connectivity of the pores. Additionally, these methods calculate quantitative data including the pore size distribution, accessible volume and surface area, and largest-cavity and porelimiting diameters. In this Article, we extend our shape-selectivity screening method to MOFs and combine it with the 3D pore characterizations from ZEOMICS and MOFomics. To facilitate this, we have developed a mixed-integer linear optimization model to identify the minimum-energy pathway through a microprous material for each guest molecule. By considering all of the portals of a zeolite or MOF and the molecular pathways for the first time, we are able to improve our shape-selectivity predictions. We demonstrate our approach by screening databases of over 1800 zeolites and MOFs for applications of interest. As a result, we identify new potential materials for separating mixtures of gases (CO2/N2, CO2/CH4, and CO2/ H2), air (O2/N2), and chemicals (propane/propylene, ethane/ ethylene, styrene/ethylbenzene, and xylenes).

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METHODS

From pore-characterization methods ZEOMICS19 and MOFomics,2 we can readily identify all portals, channels, cages, and their connectivity in microporous materials. To determine the potential of a zeolite or MOF to be shape-selective toward a given guest molecule, it is necessary to calculate the activation energy required for it to pass through each portal of the microporous structure. We previously developed a method for calculating the portal activation energy for zeolites based on molecules passing through 2D portals.16−18 In those Received: February 9, 2013 Revised: March 26, 2013 Published: April 9, 2013 5599

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studies, 2D portal representations were taken from the Database of Zeolite Structures.1 Here, we use the portals identified by ZEOMICS19 and MOFomics,2 which may not be flat, simple rings. To account for this fact better, we take the loop of each portal at the end of each channel and project it onto a plane normal to the channel direction. We then apply our host−guest interaction model for zeolites (model 1) or MOFs (model 2) to calculate the activation energy for each portal. The model for zeolites accounts for both Lennard-Jones interactions as well as portal flexibility, which we have previously demonstrated18 to improve the accuracy of our predictions. The model for MOFs does not include flexibility because the variety of atom types creates a large number of possiblities for interatomic interactions. However, it has been shown that flexibility can play a large role in some structures,20 and we intend to incoroprate flexibility for MOFs as suitable parameters become available in the literature. The elemental parameters used in the models include van der Waals radii from the Cambridge Structural Database21 and Lennard-Jones coefficients from the Universal Force Field (UFF).22 The complete list of elemental data is provided in the Supporting Information, Table S1. UFF was selected for its completeness and popularity in the literature; however, the proposed models can accommodate other force field parameters if more suitable parameters are available for an application of interest. We have chosen not to incorporate electrostatic interactions in the proposed models to limit additional nonconvexities. For host oxygen atoms in zeolites, a radius of 1 Å is used because we found that it more accurately represents the oxygen radius in a tetrahedral environment.18 We use kTO = 2092 kJ/(mol Å2) and kOO = 430 kJ/(mol Å2) for the flexibility parameters of the zeolite frameworks.23 The notation used in models 1 and 2 is described at the end of the paper.

by constraints 4 and 5. The guest molecule is kept from translating outside of the portal by constraint 6. The extent of atom squeezing is computed by constraint 7 from the van der Waals radii, ri and rj, and the separation, dij, between guest and host atoms, which is calculated by constraint 8. When dij ≥ ri + rj, no atom squeezing is required and δi and δj take a value of 1. Otherwise, the radii of the atoms must be reduced from their van der Waals radii so that the guest molecule can fit through the portal. In this case, δ < 1 and a positive contribution is made to the ES energy term. A lower bound of δL = 0.1 is used to avoid numerical difficulties. The amount of portal distortion is calculated by constraints 9 and 10, which measure the T−O and O−(T)−O distances in the host. When these distances differ from their nominal values, flexing has occurred, and a positive contribution is made to the EZ energy term. Atoms of the portal may be only pushed outward, not pulled inward. This is imposed by constraint 11, where parameters f j, gj, and hj are calculated from the nominal host atom positions. Only the oxygen atoms of the portal are allowed to move. The metal atoms, which are behind the oxygen atoms and are strongly supported by the crystal matrix, are anchored in place by constraints 12 and 13.

The host−guest interaction model for MOFs (model 2) is based on the rigid portal formulation for zeolites developed in Gounaris et al.17 It has been modified so that all host atoms interact with the guest rather than only oxygen atoms. Objective function 14 minimizes the total strain energy as a result of atom squeezing. The rotation and translation of the guest molecule are computed by constraints 15 and 16 and constrained to the interior of the portal by constraint 17. The extent of atom squeezing is calculated from the separation between guest and host atoms by constraints 18 and 19. Once we have calculated the portal activation energies in a zeolite or MOF, we consider the pathways that a guest molecule may take through the 3D structure. The connectivity of a microporous structure is established by the pore characterization methods in terms of junctions, which are portals, cages, or channel intersections where a guest molecule can change its direction of travel.2,19 The junctions are connected by segments of channels, and molecules must pass through portals to get from one junction to another. We define a pathway to be a route through the junction connectivity graph from a junction to its periodic image in a neighboring unit cell. Such a path implies an infinite connected sequence in the periodic connectivity graph representing the crystal. We postulate that the energy of a pathway is equal to the maximum energy of any portal along the path through which the guest molecule must pass. This assumption is valid in the limit of fast transport through the structure so that the bottleneck in the path is the ratelimiting step. This is analogous to an extension of the pore-limitingdiameter metric used by others8,24 to characterize microporous materials. Because guest molecules tend to travel along the pathway with minimum energy, it is suitable to use the energy of this pathway in subsequent calculations of shape selectivity. An interpretation of these assumptions is that transport limitations are dominated by the most difficult portal through which guest molecules must pass in order to progress through the material. Consider the pore network in Figure 1. The three portals have energies such that EC > EA > EB. The channel passing through portals A and B is the only one that extends across the

The flexible portal formulation of the host−guest interaction model for zeolites (model 1) was developed in Gounariset et al.18 The objective function 1 of the nonlinear optimization model minimizes the total strain and distortion energies, ES and EZ, respectively, required for a guest molecule to pass through a 2D portal. The strain energy is calculated using a Lennard-Jones-type potential by constraint 2, and the distortion energy is calculated using a quadratic potential by constraint 3. The model allows rotation (defined by the variables ϕ, θ, and ψ) and translation (defined by the variables xt and yt) of the guest molecule within the interior of the portal so that the portal activation energy is minimized. The coordinates of the rotated and translated guest molecule projected onto the xy plane of the portal are calculated 5600

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for edge (i, j) in crystallographic direction k ∈ {a, b, c}. The variables tk are equated to the summation of the transit vectors for the selected edges through constraint 25, and they indicate the direction of the minimum-energy pathway. The binary variables bk encode the signs of tk, and constraints 26 and 27 require that for each dimension k, bk takes a value of 1 if tk = 1 and a value of 0 if tk = −1 (it is free otherwise). This informs constraints 28 and 29 such that variables yk contain the absolute values of tk. Constraint 30 requires that at least one of the absolute values is positive (i.e., that the transit vector sum is nonzero). As an illustration, consider the connectivity graph with six junctions in Figure 2. Each edge in the figure represents two directed edges, both

Figure 1. Pore network with three portals having energies of EC > EA > EB. The channel passing through portals A and B extends across the unit cell boundary. unit cell boundary, so it is necessary to traverse this channel to progress through the microporous structure. Because portal C can be avoided and portal B is more permissive than portal A, the ratelimiting portal is portal A. Therefore, the minimum-energy pathway passes through portals A and B and has an energy value of EA, the cost of the most difficult portal on the pathway.

Figure 2. Example of a periodic connectivity graph with six junctions, A−F. Each edge represents two directed edges in opposite directions, both of which have the same cost (shown in the edge labels). of which have the same cost. For example, edge (A, B) has cost cAB = 4 and transit vector mAB = (0, 0, 0). Edge (A, C) has cost cAC = 1 and transit vector mAC = (−1, 0, 0). The minimum-energy pathway consists of edges (A, B), (B, C), and (C, A), which has a cost of E = 7 and a transit summation of t = (1, 0, 0). By using well-known methods in mathematical optimization such as integer cuts or solution pool techniques, we can generate not only the minimum-energy pathway but also a rank-ordered list of alternate lowenergy pathways. This could be used in future work to take into account entropic contributions, which has been considered previously by Gounaris et al.18 Note that we previously developed a method to map out the accessible space to a guest molecule of a given diameter by pruning the junction connectivity graph to remove unreachable vertices and edges.2,19 By taking portal activation energies into account, we extend this approach to guest molecules of arbitrary shape. Given a portal activation energy E, we determine the accessibility by choosing a cutoff value of the Boltzmann factor, exp(−E/RT), to prune the connectivity graph for a guest molecule of any size or shape at any temperature T.

The mixed-integer linear optimization (MILP) model 3 is formulated to identify the minimum-energy pathway for a guest molecule in a microporous structure. In this model, the objective is to minimize the pathway energy subject to the pathway being a connected path through the periodic connectivity graph from a junction to its image in a neighboring unit cell. The connectivity graph is described using the static graph representation used by Höfting and Wanke,25 where each node represents a junction and each edge has an associated transit vector that encodes the number of unit cells in each of the three crystallographic directions that the edge spans. Then, a path from a junction to one of its images in the periodic graph appears as a simple cycle in the static graph with a nonzero transit vector sum. Each edge of the connectivity graph is associated with a binary variable xij, which takes on a value of 1 if the directed edge (i, j) is part of the path and 0 otherwise. Each edge (i, j) also has an associated cost cij, which is equal to the portal activation energy required to traverse that edge. Constraint 21 requires that the pathway energy, E, be at least as large as the largest cost of any selected edge. Because the pathway energy is being minimized, E will be exactly equal to the maximum portal activation energy along the selected pathway. Constraints 22 and 23 enforce that a simple path is selected, where each node may have at most one incoming and one outgoing edge selected. Constraint 24 requires that a node has an outgoing edge selected if and only if it has an incoming edge selected, which ensures that a connected cycle is found. The transit vector of each edge is encoded in the parameters mijk ∈ {−1, 0, 1}, each of which contains the component of the transit vector

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RESULTS AND DISCUSSION In Gounaris et al.,16 the shape selectivity between two molecules A and B was defined as the absolute difference between the Boltzmann factors of their portal activation energies. In later work,17 this was extended to consider multiple possible pathways for zeolites with more than one ring. In an analogous manner, we define the shape selectivity, γAB, in the presence of multiple possible pathways p, as in eq 34, where Emp is the energy of pathway p for guest molecule m. Values for γAB approach 0 for molecules with similar pathway energies and 1 for molecules with high selectivity. ⎛ min E A ⎞ ⎛ min E B ⎞ p p p p ⎟ − exp⎜ − ⎟ γAB = exp⎜⎜ − ⎟ ⎜ ⎟ RT RT ⎝ ⎠ ⎝ ⎠

(34)

The energies are calculated using the minimum-energy pathway approach discussed previously. The host−guest interaction models (models 1 and 2) have been implemented in GAMS 23.9,26 and the 3D minimum-energy pathway model (model 3) has been implemented in CPLEX 12.1.27 To solve 5601

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provided in the Supporting Information, Tables S8 and S9. Generally, the pore-limiting diameters of the top structures for each separation fall within a range of less than 1 Å, suggesting a correlation between the shape selectivity and pore-limiting diameter. However, not all structures with pore-limiting diameters in that range exhibit shape selectivity. Most of the materials that we have identified are novel materials that have not been previously proposed in the literature for these applications. The results refine previous work16,17 by using minimum-energy pathways instead of minimum-energy portals. Furthermore, the fact that this work utilizes 3D geometric information from ZEOMICS and MOFomics enables us to estimate guest−host interactions better as well as consider obstructions along channels. To that end, more accurate shapeselectivity predictions are obtained. The applications are discussed in more detail below. CO2/N2 Separation. The discovery of novel materials for carbon capture from flue gas at power plants is an important research area. Zeolites and MOFs have been evaluated on the basis of adsorption selectivity,12,36 working capacity,37 heat of adsorption,38 and parasitic energy.11 Wilmer et al.10 considered correlations among several structural and chemical characteristics, and Watanabe and Sholl15 performed a hierarchical screening based on structural analysis, pore analysis, and molecular simulation. Here, we consider the metric of shape selectivity to identify zeolites and MOFs with the potential to be highly selective at separating CO2 from N2. We have identified zeolites, including MAR, ANA, TOL, and NON, and MOFs, including hypothetical MOFs 31733, 20085, and 8640, as having the greatest potential to separate CO2/N2 mixtures at 300 K, with shape selectivities of 0.40−0.44. The topperforming zeolites and MOFs have similar shape selectivities and decline at higher temperatures (Figure 3), reflective of the size similarity of the circular footprints of CO2 and N2. The absence of shape selectivity in FAU, the framework type of the zeolite most often considered for adsorption-based carbon capture, zeolite 13X,39 indicates that it may not be the dominant factor for this sorbent. CO2/CH4 Separation. It is highly desirable to remove carbon dioxide from natural gas because it reduces the energy content and can be corrosive.40 Zeolites and MOFs have been proposed as adsorbents for natural gas sweetening and have been considered in the literature.10,36,41 We have identified zeolites and MOFs with high shape selectivity for this separation. Zeolites DOH, GIU, SOD, and FRA all have shape selectivities in the range of 0.74−0.83 at 300 K. The topperforming MOFs at 300 K include hypothetical MOF 31090, ZIF-12, hypothetical MOF 31091, and ZIF-11 with shape selectivities in the range of 0.67−0.77. Figure 4 shows that there are zeolites and MOFs with moderately high shape selectivity across a range of temperatures. CO2/H2 Separation. Steam reforming of natural gas is responsible for about 95% of the hydrogen produced in the United States, according to the U.S. Department of Energy.42 In the first step of the process, steam reforming produces carbon monoxide and hydrogen from methane and water: CH4 + H2O → CO + 3H2. Next, additional hydrogen is generated through the water−gas shift reaction: CO + H2O → CO2 + H2. An important final step is the separation of hydrogen from carbon dioxide through pressure-swing adsorption. Through our shape selectivity screening method, we have identified zeolites and MOFs that are highly suitable for this separation. The top structures all have shape selectivities of near unity at

nonlinear models 1 and 2, we initialize a local optimization solver (CONOPT 328) from multiple initial points corresponding to different rotations and translations of the guest molecule. For nonlinear guest molecules, we iterate over 125 orientations (5 values for each angle ϕ, θ, and ψ), and for all guest molecules, we choose initial translations corresponding to 9 points around the center of each channel. Once energies are computed, we obtain the shape selectivity as a function of temperature for each guest molecule in each microporous material by varying the temperature over a range of values. We have applied our proposed shape-selectivity screening method to 196 zeolites and 1690 MOFs with accessible pores from ZEOMICS29 and MOFomics,3 respectively. The zeolites are silica structures from the Database of Zeolite Structures.1 The MOFs include structures from the Crystallography Open Database (COD),30,31 the Cambridge Structural Database (CSD),32,33 and user submissions to MOFomics. In addition, we consider zeolitic imidazolate frameworks (ZIFs)34 and hypothetical MOFs.5 The complete list of structures studied is provided in the Supporting Information, Tables S2−S7. We have selected separation applications of interest from several areas including the separation of gases (CO2/N2, CO2/CH4, and CO2/H 2), air (O 2/N2 ), and chemicals (propane/ propylene, ethane/ethylene, styrene/ethylbenzene, and xylenes). These separations involve 14 different guest molecules, the structures of which were obtained from PubChem Compound,35 for a total of 26 376 host−guest pairs. The top 10 zeolites and top 10 MOFs identified for each separation at 300 K are tabulated in Tables 1 and 2, Table 1. Shape Selectivity for Zeolites at 300 Ka CO2/N2

a

CO2/CH4

MAR 0.44 ANA 0.44 TOL 0.41 NON 0.40 LOS 0.38 MVY 0.34 FAR 0.34 AFG 0.31 MSO 0.28 SGT 0.27 propane/ propylene

ethane/ethylene

JBW IHW SAT BIK AFN ESV CDO NAB ITW CGF

CAS BRE LOV EAB AHT BCT CGF VNI NSI MVY

0.92 0.91 0.84 0.79 0.67 0.67 0.67 0.64 0.62 0.55

DOH GIU SOD FRA FAR LOS MVY TOL ANA AHT

0.83 0.78 0.75 0.74 0.60 0.60 0.53 0.45 0.44 0.40

0.80 0.79 0.71 0.68 0.67 0.66 0.54 0.54 0.51 0.46

CO2/H2 RUT 1.00 LIT 0.99 MEP 0.99 LTN 0.99 LIO 0.99 MTN 0.98 AST 0.98 SGT 0.93 MSO 0.92 AFG 0.90 styrene/ ethylbenzene HEU WEI PAR CGS LAU MFI OBW SVR DAC PON

1.00 0.89 0.88 0.45 0.22 0.21 0.21 0.14 0.14 0.07

O2/N2 AFG SGT NON MSO AST MTN MAR ANA TOL MEP

0.14 0.13 0.13 0.13 0.10 0.09 0.08 0.08 0.08 0.08

xylenes OBW MEL MWW MTT OWE FER SZR IMF TON LAU

0.96 0.95 0.92 0.92 0.87 0.81 0.80 0.79 0.72 0.71

Top 10 structures for each separation.

respectively. The shape selectivity of selected top-performing zeolites and MOFs for each separation is plotted as a function of temperature in Figures 3−10. These plots show that there is generally not a single structure with the highest shape selectivity at all temperatures. Portal activation energies and pore-limiting diameters for each of the top structures are 5602

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Table 2. Shape Selectivity for MOFs at 300 Ka,b CO2/N2 H-31733 H-20085 H-8640 H-24755 H-22671 H-33915 H-22653 HUKRIO BIMDEF HIXDOH01 propane/propylene H-30749 H-30785 H-30794 H-30778 H-21207 H-30717 H-31407 H-5518 H-32907 H-32935 a

0.40 0.40 0.40 0.39 0.39 0.39 0.38 0.38 0.38 0.37

0.99 0.98 0.98 0.96 0.96 0.95 0.92 0.92 0.85 0.85

CO2/CH4

CO2/H2

H-31090 0.77 ZIF-12 0.69 H-31091 0.68 ZIF-11 0.67 H-31075 0.67 H-28183 0.61 H-31095 0.60 HUKRIO 0.57 H-20574 0.55 QOPHEI 0.55 ethane/ethylene

XECJAQ 1.00 H-23429 1.00 C-4308624 1.00 ZIF-7 1.00 H-7932 1.00 HODWUS 1.00 DOQRUW 1.00 H-33192 1.00 H-8089 1.00 ZIF-9 1.00 styrene/ethylbenzene

REGJIW H-32575 H-33150 H-31723 H-31064 H-31076 H-22075 H-31746 H-20518 ENCUCC10

0.93 0.93 0.93 0.93 0.93 0.93 0.88 0.84 0.83 0.79

H-22188 H-32211 H-32176 H-32147 H-21707 H-32149 H-31397 H-32156 H-32168 H-32194

O2/N2

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.99

H-20520 HIXDOH01 H-7562 BACMOH10 H-8568 H-22643 H-22659 H-22653 H-22699 H-31055 xylenes H-28604 H-28637 H-18081 H-18068 H-18058 H-18054 H-18047 H-18030 H-28633 H-18078

0.14 0.14 0.13 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.98

Top 10 structures for each separation. bCOD structures are labeled as C-(number), and hypothetical MOFs are labeled as H-(number).

Figure 3. Shape-selectivity plot of selected zeolites (solid curves) and MOFs (dashed curves) for CO2/N2 separation.

low to moderate temperatures (Figure 5), reflective of the large difference in molecular sizes. The most shape-selective zeolites at 300 K are RUT, LIT, MEP, LTN, and LIO, and the top MOFs are CSD structure XECJAQ, hypothetical MOF 23429, COD structure 4308624, and ZIF-7. Two of our top structures, ZIF-7 and ZIF-9, both with shape selectivities of near unity, were also identified by Battisti et al.36 as strong candidates for this separation. O2/N2 Separation. Air separation has great commercial significance for producing N2 and O2 with high purities. Separation using zeolites and MOFs based on pressure-swing adsorption can be a more economical alternative to cryogenic distillation.43,44 We have identified several zeolites and MOFs for air separation based on shape selectivity; however, the shape selectivities of the top structures tend to be low except at very low temperatures (Figure 6) because of the similarity in the

circular footprint sizes of the two molecules. The most selective of the zeolites (AFG, SGT, NON, and MSO) and MOFs (hypothetical MOF 20520, CSD structure HIXDOH01, and hypothetical MOF 7562) at 300 K have shape selectivies of only 0.13−0.14. Propane/Propylene Separation. Hydrocarbon separation is an important application for the chemical and petrochemical industries, and microporous materials can play a critical role. The separation of propane/propylene mixtures has been considered in zeolites,45 Fe-MOF-74,46 HKUST-1,47 and ZIFs.48 We have obtained very promising results, indicating that top-performing zeolites and MOFs are highly shapeselective for this separation over a range of temperatures (Figure 7). Top zeolites at 300 K include JBW, IHW, SAT, and BIK, with shape selectivities in the range of 0.79−0.92, and top 5603

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Figure 4. Shape-selectivity plot of selected zeolites (solid curves) and MOFs (dashed curves) for CO2/CH4 separation.

Figure 5. Shape-selectivity plot of selected zeolites (solid curves) and MOFs (dashed curves) for CO2/H2 separation.

Figure 6. Shape selectivity plot of selected zeolites (solid curves) and MOFs (dashed curves) for O2/N2 separation. 5604

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Figure 7. Shape selectivity plot of selected zeolites (solid curves) and MOFs (dashed curves) for propane/propylene separation.

Figure 8. Shape selectivity plot of selected zeolites (solid curves) and MOFs (dashed curves) for ethane/ethylene separation.

Figure 9. Shape selectivity plot of selected zeolites (solid curves) and MOFs (dashed curves) for styrene/ethylbenzene separation. 5605

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Figure 10. Shape-selectivity plot of selected zeolites (solid curves) and MOFs (dashed curves) for xylene separation.

separations involving three or more components. The top zeolites and MOFs are depicted in Figure 10 over a range of temperatures and show moderately high to high shape selectivity for this separation. The top zeolites at 300 K are OBW, MEL, MWW, and MTT with shape selectivities of 0.92− 0.96. The most selective MOFs at 300 K are hypothetical MOFs 28604, 28637, 18081, and 18068 with shape selectivities of around 0.98.

MOFs (hypothetical MOFs 30749, 30785, and 30794) have even higher shape selectivites, in the range of 0.98−0.99. Ethane/Ethylene Separation. Although cryogenic distillation is the traditional and primary method for the separation of ethane/ethylene mixtures, microporous materials have the potential to relieve the high energy costs of such processes. Ethane/ethylene separation has been considered in zeolites,7,49 Fe-MOF-74,46 and ETS-10.50 We have identified additional zeolites and MOFs with the potential to be selective in ethane/ ethylene separation, with top-performing structures having moderately high to high shape selectivity across a range of temperatures (Figure 8). At 300 K, the top zeolites are CAS and BRE with shape selectivities of 0.80 and 0.79, respectively, and the top MOFs are CSD structure REGJIW and hypothetical MOFs 32575, 33150, and 31723, all with shape selectivities of 0.93. EAB, the fourth most shape-selective zeolite at 300 K with a shape selectivity of 0.68, was also identified through a large-scale computational screening by Kim et al.7 as a top structure for ethane/ethylene separation. Styrene/Ethylbenzene Separation. Styrene is produced comercially from ethylbenzene, making the separation particularly important. The MOFs MIL-47 and MIL-53(Al) have been shown to be capable of separating these molecules.51 We have determined that several zeolites and MOFs, indicated in Figure 9, have high potential for this application across a range of temperatures. Top zeolites at 300 K are HEU, WEI, and PAR, with shape selectivities of in the range 0.88−1.00, and top MOFs at 300 K, with shape selectivities of 1.00, include hypothetical MOFs 22188, 32211, and 32176. Xylene Separation. The separation of xylene isomers by distillation is challenging because of their similar boiling points. As a result, microporous materials, particularly zeolites, play an important role in their separation industrially. The separation has been studied extensively in the literature in materials such as MFI-type zeolites,52 FAU-type zeolites,53,54 and MOFs including MIL-47, MIL-53(Al), and HKUST-1.55 Often, the desired product in xylene separation is para-xylene, and it is the isolation of para-xylene from its other two isomers, meta-xylene and ortho-xylene, that we focus on here. To evaluate structures, we calculate the shape selectivity between para-xylene and its other two isomers separately and then combine them using a geometric mean, which we found is a reasonable approach for

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CONCLUSIONS

We have developed a novel computational framework for the calculation of shape selectivity in 3D zeolites and MOFs. The method, based on mathematical optimization, considers the minimum-energy pathway that a guest molecule can follow to traverse a 3D microporous structure. This approach can be extended to consider the entropic effects of pathway multiplicity and also to incorporate electrostatic and binding interactions, though such enhancements will require an efficient construction to avoid increasing computational complexity. The method presented in this work is suitable for the highthroughput screening of large databases of materials and can be used to inform subsequent computational or experimental studies. We envision applying shape selectivity in conjunction with other metrics as part of a hierarchical computational screening procedure for identifying novel materials for molecular separations in which detailed molecular simulations are reserved for only the best candidates. We have applied our method to databases of 196 zeolites and 1690 MOFs (including ZIFs and hypothetical MOFs) to separations of gases, air, and chemicals. For each of the eight applications that we have studied, we have identified new zeolites and MOFs with the potential for shape selectivity as well as reconfirmed several structures suggested elsewhere in the literature for these separations. Additionally, the topperforming structures vary over different temperature ranges, indicating that there may not be a single best structure in all cases. These insights can be used to suggest that additional consideration should be given to the most promising structures to improve the performance of these separations. 5606

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ASSOCIATED CONTENT

S Supporting Information *

Complete lists of elemental data, zeolite and MOF structures studied, and portal activation energies for top materials. This material is available free of charge via the Internet at http:// pubs.acs.org/.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: fl[email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We gratefully acknowledge support from the National Science Foundation (NSF EFRI-0937706, NSF CBET-1263165). E.L.F. is also thankful for his National Defense Science and Engineering Graduate (NDSEG) fellowship.

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NOTATION USED IN MODELS 1 AND 2

Indices

i ∈ 1, 2, ..., I=atoms in the guest molecule j ∈ 1, 2, ..., J=atoms in the host structure (JO = set of host oxygen atoms) Decision Variables

ϕ, θ, ψ ∈(−π, π] = Euler angles for the rotation of a guest molecule xt ∈ (minj ∈ J xhj , max j ∈ J xhj)⎫ ⎪ O O ⎬ = translation of guest yt ∈ (minj ∈ J yhj , max j ∈ J yhj) ⎪ ⎭ O O molecule projection on xy plane δi,δj∈[δL,1] = required distortion of guest and host atoms xhvj ∈ [xhj − rj , xhj + rj]⎫ ⎪ ⎬ = position of host atoms in yhvj ∈ [yhj − rj , yhj + rj]⎪ ⎭ distorted portal Auxiliary Variables

ES = strain energy from squeezing atom spheres EZ = distortion energy from flexing the portal xi, yi = position of rotated and translated guest molecule projection dij = distance between guest and host atoms dTO = distance between two adjacent atoms of the portal j dOO = distance between two successive O atoms of the portal j Parameters

x°i , y°i , z°i = coordinates of guest atoms in an arbitrarily oriented molecule xhj, yhj = position of host atoms in a portal projected onto an xy plane ri, rj = effective atomic radii of guest and host atoms εi, εj = Lennard-Jones parameters kTO, kOO = interatomic potential parameters dTO,nom , dOO,nom = nominal T−O and O−(T)−O distances j j f j, gj, hj = parameters constraining host O-atom positions, j ∈ JO

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