Development of Computational Methodologies for MetalâOrganic

Review pubs.acs.org/CR

Development of Computational Methodologies for Metal−Organic Frameworks and Their Application in Gas Separations Qingyuan Yang,† Dahuan Liu,† Chongli Zhong,*,† and Jian-Rong Li‡ †

Laboratory of Computational Chemistry and State Key Laboratory of Organic−Inorganic Composites, Beijing University of Chemical Technology, Beijing 100029, China ‡ College of Environmental and Energy Engineering, Beijing University of Technology, Beijing 100124, China 3.1.3. REPEAT Method 3.1.4. DDEC Method 3.1.5. Electrostatic Potential Energy Surface Method 3.2. Open-Metal-Sites Force Fields 3.2.1. Specific Interactions−Isolation Method 3.2.2. DFT/Coupled-Cluster Host−Guest PES Method 3.2.3. Nonempirical Model Potential Decomposition Method 3.3. Structural Characterization 3.3.1. Accessible Surface Area 3.3.2. BET Surface Area 3.3.3. Free Volume 3.3.4. Accessible Volume 3.3.5. Accessible Pore Size Distribution 3.3.6. Largest Cavity and Pore Limiting Diameters 3.3.7. Method to Identify Portal, Channel, Cage, and Their Connectivity 3.4. Breathing/Gate-Opening Description 3.4.1. Flexible Force Field Method 3.4.2. Phase Mixture Model Method 3.4.3. Osmotic Thermodynamic Model Method 3.4.4. Stress-Based Model Method 3.4.5. Osmotic Framework Adsorbed Solution Theory 3.4.6. Dispersion-Corrected DFT Approach 3.5. Regular Framework Flexibility Description 3.6. Concepts and Theories for Characterizing Separation Performance 3.6.1. Microscopic Selectivity 3.6.2. Adsorbility 3.6.3. Nonlocal On-Lattice Classical Fluid DFT 3.6.4. WDA-Based Three-Dimensional Classical Fluid DFT 3.6.5. Application of dcTST To Describe Diffusion of Molecules in MOFs 3.7. Methods for Constructing MOFs 3.7.1. Building Blocks-Based Approach 3.7.2. Zeolite Topology-Based Approach 3.8. Development of QSPR Models for MOFs 3.8.1. Adsorbility-Based QSPR Model for Gas Separations

CONTENTS 1. Introduction 2. Conventional Computational Methods 2.1. Atomic Partial Charge Calculation 2.1.1. Cluster-Based Quantum Mechanics Approach 2.1.2. Mulliken Population Analysis Approach 2.1.3. Molecular Charge Equilibration Method 2.1.4. Periodic Charge Equilibration Method 2.2. Force Fields 2.2.1. Force Fields for Adsorbate Molecules 2.2.2. Force Fields for MOFs 2.3. Molecular Simulation Techniques 2.3.1. Grand Canonical Monte Carlo Simulation 2.3.2. Molecular Dynamics Simulation 2.4. Ideal Adsorbed Solution Theory 2.5. Structural Characterization of Porous Solids 2.5.1. Pore Volume Calculation 2.5.2. Geometric Pore Size Distribution Analysis 2.6. Structural Optimization Method 2.7. Adsorption Enthalpy Calculation Method 2.7.1. Energy Difference 2.7.2. Ensemble Fluctuation 2.7.3. Revised Widom’s Test Particle 2.8. Separation Selectivity Calculation Method 2.8.1. Adsorption Selectivity 2.8.2. Permeation Selectivity 3. Development of Computational Methods and Concepts for MOFs 3.1. Atomic Partial Charge Estimation 3.1.1. Extended Charge Equilibration Method 3.1.2. Connectivity-Based Atom Contribution Method © XXXX American Chemical Society

B C C C D D E F F G G G H I I I I J J J K K K K L L L L

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Received: January 7, 2013

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dx.doi.org/10.1021/cr400005f | Chem. Rev. XXXX, XXX, XXX−XXX

Chemical Reviews 3.8.2. QSPR Model for Drug Encapsulation 3.8.3. QSPR Model for Isosteric Heat of Adsorption 3.8.4. QSPR Model for Hydrogen Uptake 3.9. Membrane-Based Selectivity Calculation 3.9.1. Pure MOF Membrane 3.9.2. Mixed-Matrix Membrane 4. Application of New Methods and Concepts in Gas Separations 4.1. Adsorption-Based Separation 4.1.1. Carbon Dioxide Capture 4.1.2. Other Gas Mixtures 4.2. Membrane-Based Separation 4.2.1. Carbon Dioxide Capture 4.2.2. Other Gas Mixtures 4.3. Computationally Proposed Strategies 4.3.1. Tailoring Pore Size and Shape 4.3.2. Chemical Modification 5. Concluding Remarks and Outlook Development of Approaches for Constructing Hypothetical MOFs Method Establishment for Reliable Atomic Partial Charge Estimation Modeling of MOFs with Highly Flexible Frameworks Modeling of MOFs with Open Metal Sites Modeling of Membrane-Based Separation Author Information Corresponding Author Author Contributions Notes Biographies Acknowledgments References

Review

and hypothetical materials, including MOFs.35,36,39−44 It allows subsequent experimental endeavors to be deployed on the promising candidates that demonstrate the highest level of performance. As a result, significant research effort has been made during the past decade in resorting to all kinds of computational approaches for investigating different properties of MOFs. To date, some excellent reviews have been published, focusing on summarizing the results from computational research on MOFs.22,27,45−58 All these articles related to various topics have contributed a lot to a better understanding of the performance as well as structure−property relationships for MOFs. Li et al.28 summarized some computational studies in their review, with focus on experimental studies of MOFs for separating various gas- and liquid-phase mixture systems. In a comprehensive overview of the experimental reports on MOFs for biomedical applications, Horcajada et al.22 also provided a brief description of the relevant molecular modeling studies. Meek et al.47 presented a review that concerns the use of MOFs for gas separation, catalysis, drug delivery, optical and electronic applications, and sensing, together with a short summary of the development in computational modeling of MOFs. In a review by Snurr and co-workers,49 particular attention was drawn to how the insights obtained from molecular simulations can be combined to develop design principles for specific adsorption applications of MOFs. Sholl and co-workers51 reviewed progress in the application of atomically detailed modeling methods for studying gas adsorption and transport in MOFs. Jiang et al.54 provided an extensive overview of the energy, environmental, and pharmaceutical applications of MOFs as well as other nanoporous materials. Krishna58 gave a tutorial review of the single-gas and mixture diffusion characteristics in a wide variety of crystalline meso- and microporous materials including MOFs. However, in all these reviews attention was mainly paid to introduction of the important results obtained in the computational studies. To the best of our knowledge, there is not a comprehensive review focusing on covering important advances in the computational methodologies developed so far with respect to MOFs. Considering that significant advances have been achieved to date in this area, which allow a deeper and more thorough understanding of the features of MOFs, it is highly necessary to present a comprehensive and critical review on them. Such a collection of the new advances would make the relevant researchers, particularly the newcomers, easier to understand the new methods and concepts developed for MOFs as a whole and thus easier to apply them in their work. Thereby, it would greatly stimulate the computational study on MOFs to contribute to the MOFs community. In this review, we devote our attention to highlighting both the successes and the limitations of various computational methodologies for MOFs. As an illustration, we provide a stateof-the-art review on modeling studies of gas separations in MOFs using these new methods and concepts. Since the newly developed computational methodologies are normally used in combination with the conventional ones, for the sake of continuity and clarity, some of the conventional methods that have been widely adopted in MOF studies are also briefly discussed here. As the modeling studies of MOFs emerged mainly in 200459 (with one exception in 200160), this review for the most part covers related work on MOFs from 200461 until the end of 2012. After this brief introduction as section 1, section 2 summarizes some conventional computational methods used

AL AL AM AN AN AO AP AP AQ AU AW AW AY AZ AZ BA BB BB BB BB BC BC BC BC BC BC BC BD BD

1. INTRODUCTION Under the impetus of some breakthrough studies,1−6 associated research on metal−organic frameworks (MOFs) has rapidly developed into one of the highly unique areas in chemistry, materials science, and multiple branches of engineering.7−15 This type of intriguing materials corresponds to hybrid crystalline solids with periodic nanoporous architecture in which inorganic metal-ions-based nodes are bridged by polytopic organic ligands through coordination bonds. The intrinsic nature of MOFs allows for intensive study of systematically modulating their pore dimensions, surface areas, framework topologies, and surface chemistry in an extremely broad range.16−18 This remarkable feature is largely absent in conventional porous materials such as zeolites.19 Thus, it allows MOFs to serve as a uniquely ideal platform for various specific applications including gas storage and separation.20−31 Experimentally, it would be very time consuming to explore the properties of thousands of MOFs reported in the literature. Detailed information that leads to the macroscopic properties observed in MOFs is also difficult to be addressed using only experimental methods. In contrast, molecular modeling provides a valuable complement to experimental studies of MOFs. It can isolate the key influencing factors at the molecular level as well as give deep insights into the underlying mechanisms involved.32−38 Furthermore, molecular modeling is a strong and attractive tool for large-scale screening of existing B


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to model and characterize MOFs. Section 3 discusses advances in computational methodologies for MOFs achieved to date, including new concepts, methods, and newly developed theories. Section 4 reviews research progress on studies of MOFs in the field of gas separation by applying these computational methods. Finally, a summary of current challenges of molecular modeling and some outlooks are provided.

∑ [V kQM − VkESP(q1, q2 , ···, qN )]2

y(q1 , q2 , ..., qN ) =

k=1 nc

+

∑ λj gj j=1

(1a)

where m is the total number of fitting points, N is the number of atoms in the cluster, VQM is the quantum mechanically k calculated ESP at point k, qi is the fitted partial charge of each atom (i = 1, 2, ...,N), nc is the total number of constraints, λj is the Lagrange multiplier (j = 1, 2, ..., nc) for constraint gj, and VESP is the fitted ESP at point k expressed by k

2. CONVENTIONAL COMPUTATIONAL METHODS 2.1. Atomic Partial Charge Calculation

Development of computational methods often requires evaluation of intermolecular interactions in which electrostatic forces may be the dominant long-range contributions. In most cases, to represent the electrostatic potentials of MOFs, assignment of partial charges for their framework atoms is a good compromise between computational effectiveness and accuracy. In addition, atomic partial charges can be regarded as a convenient representation of the asymmetric distribution of electrons in chemical bonds. Thus, they are also intuitively appealing ingredients for qualitatively understanding the physicochemical properties of the systems. However, it is worthwhile to note that they are not the formal charges with integer property but rather the fractions of an electron corresponding to the percentage of time that an electron is near each nucleus.62 At the same time, because these charges do not correspond to any unique physical property, they are also neither experimental nor formally quantum mechanical observables.63,64 As a consequence, there is no unambiguous rule that can be used to uniquely determine their values on a rigorous level. Before the advent of MOFs, multiple approximation methods with different partitioning schemes have been proposed to accomplish this target. Some of these conventional methods that have been commonly used in the MOF studies are outlined below. 2.1.1. Cluster-Based Quantum Mechanics Approach. Due to very good descriptions of intermolecular interactions using Coulomb potential, electrostatic potential (ESP) derived charges are perhaps the most widely adopted charge type in molecular modeling. They are usually obtained from the calculations performed on nonperiodic cluster models using a quantum mechanics (QM) approach on the basis of density functional theory (DFT). Specifically, a single-point energy DFT calculation is conducted on a representative cluster cleaved from the matrix of a certain MOF for which the dangling bonds are saturated by appropriate groups. In most cases, these groups are methyl ones which serve the purpose of reducing any artificial charge or spin effects introduced by truncating a chemical cluster that is representative of a full periodic system at the chosen point.65,66 For the cleaved clusters with the terminations connected by metal atoms, these locations are usually capped with lithium atoms to mimic the environmental effects of these metal atoms in the real systems.67,68 Then the ESP is evaluated at a large number of fine grid points that usually lie outside of the van der Waals (vdW) surface of the cluster. A curve-fitting procedure is further used to determine the set of ESP charges on the nuclei that would most closely reproduce the obtained ESP. In this procedure, the least-squares fit criterion is utilized to minimize the function defined by63

N

VkESP(q1 , q2 , ..., qN ) =

∑ i=1

qi rik

(1b)

where rik is the distance between atom i and point k. Usually a single constraint is used in the minimization by constraining the sum of the charges of all atoms to equal the total charge (qtotal, which is zero in most cases) on the cluster molecule N

g1(q1 , q2 , ..., qN ) =

∑ qi − qtotal = 0 i=1

(1c)

When the so-obtained ESP charges are transferred to the periodic system, the underlying assumption is that the electrostatic potential surrounding the isolated fragment will be similar to that of the periodic structure. Currently, four common algorithms have been employed to generate charges for various MOFs: Merz−Singh−Kollman (MK),69,70 restrained electrostatic potential (RESP),71 electrostatic potentials (CHELP),72 and electrostatic potentials using a grid-based method (CHELPG);73 these algorithms differ primarily in how the ESP sampling points are chosen. Among them, the CHELPG algorithm is the most popular electrostatic charge computational scheme for MOFs.74,75 The accuracy of the DFT-derived results also depends on the choice of functional and basis set. In contrast to the local density approximation (LDA), the exchange-correlation functional in the generalized gradient approximation (GGA) is expressed using both the local electron density and the gradient in the electron density.76 Thus, it is generally held that the GGA functionals can give more accurate results than the LDA ones. In the literature, Perdew−Wang (PW91)77 and Perdew− Burke−Ernzerhof functional (PBE) functionals78 are two of the most widely used GGA functionals. In addition, hybrid functionals are also often encountered, such as the popular Becke plus Lee−Yang−Parr (B3LYP) functional.79,80 The basis set is a set of functions used to describe the shape of the orbitals in an atom. Considering the accuracy, the smallest basis set used generally is 6-31G(d) (also written as 6-31G*) or the equivalent, while only a small increase in accuracy can be gained using very large basis sets.62 This is probably due to the fact that the functional is limiting accuracy more than the basis set limitations. Different combinations of the options raised above have been employed to obtain the ESP charges for various MOFs, several examples being B3LYP/6-31+G(d) with CHELPG,67,81 PBE/ 6-31G(d) with CHELPG,82,83 and B3LYP/6-31G(d) with RESP.84,85 For complexes involving heavy atoms (such as transition metal atoms), the approach of effective core potential (ECP) basis sets (such as LANL2DZ86−88) for the heavy atoms mixed with all-electron basis sets for the rest atoms is also C


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commonly adopted in the calculations for MOFs.89−91 LANL2DZ is a collection of double-ζ basis sets and contains the effective pseudopotentials to represent the core electrons as the potential for valence electrons. Thus, such an approach allows a significant reduction of computational cost. It has been shown that charge calculations based on all-electron basis sets and mixing ones can lead to similar results.92 On the other hand, since atomic vdW radii are used to construct the surface on which the electrostatic potential is evaluated, it is crucial to point out where these values are taken for the atoms, especially for the heavy atoms. Unfortunately, many charge calculations have not included this step thus far. Besides the default values set in various QM softwares, it is common to use the well-known radii reported by Bondi93 as inputs in DFT calculations94,95 (some studies96,97 with the radius of H taken from another source98). For the elements not treated by Bondi, as suggested by some researchers,40 their vdW radii can be assigned about 0.75 Å larger than their covalent values. Such a treatment is more consistent with Pauling’s approximation used in Bondi’s work, and the atomic covalent radii can be taken from the data reported by Alvarez and co-workers.99 In addition, a variation on the DFT procedure is the way that orbitals are constructed to reflect paired or unpaired electrons. For systems with a singlet spin, a restricted closed-shell method can be employed in which the same orbital spatial function is used for both α and β spin electrons in each pair. However, for systems with unpaired electrons, a spin-unrestricted open-shell method should be used to realize the spin-polarized states where two completely separate sets of orbitals are used for α and β electrons. This strategy has been used to compute the ESP charges for some special MOFs, such as those containing coordinatively unsaturated metal sites (CUS) (also called open metal sites).68,100−103 It should be noted that for singlet spin systems the charges obtained from the restricted closed- and unrestricted open-shell methods should be similar. Nevertheless, a disadvantage in the unrestricted open-shell method is that an error called spin contamination may be introduced into the calculations.62 To eliminate this problem, perhaps a restricted open-shell method can be utilized in which the paired electrons share the same spatial orbital. 2.1.2. Mulliken Population Analysis Approach. Another relatively less popular but important type of atomic partial charges is the Mulliken charges. On the basis of DFT calculations conducted on the periodic structures of MOFs, these charges are typically extracted from Mulliken population analysis,104 in which the fundamental assumption is that the overlap population is partitioned equally between each pair of orbitals. Each normalized molecular orbital (MO), ϕi, is expressed as a linear combination of a set of normalized basis functions ϕi =

∑ Ci ,mχm

orbitals are summed to give the net population of that basis function. Likewise, the overlaps for a given pair of basis functions are summed for all orbitals to determine the overlap population for that pair of basis functions.62,104 According to the equipartition assumption and defining Θi,m = Ci,m(Ci,m + ∑n≠m Ci,nSm,n), the partial charge on any atom A can be calculated as the difference between its atomic number (ZA) and the total gross population on this atom qA = ZA −

i

N

E(q1 , q2 , ...qN ) =

m

∑ Ci ,mCi ,nSm,n

m m>n

(2c)

⎡

∑ ⎢⎣EA0 + χA0 qA + A=1

+

1 2

N

1 0 0⎤ J q 2 AA A ⎥⎦

N

∑ ∑ A = 1 B = 1, B ≠ A

JAB qAqB

(3a)

where the first term represents a sum of the charging energy of each atom and the second term represents a sum of interatomic Coulomb interactions between pairs of atoms. EA0, χ0A, and J0AA are the ground state energy, atomic electronegativity, and idempotential (self-Coulomb integral) of atom A, respectively. The latter two physical quantities can be obtained directly from atomic data, χ0A = 1/2(IP + EA) and J0AA = (IP − EA), where IP and EA denote the ionization potential and electron affinity of atom A, respectively; qA is the charge of each atom to be determined; JAB is the Coulomb interaction between unit charges centered on atoms A and B, which depends on the distance between them. Considering that a shielding correction should be applied to a pair of atoms at close distance, JAB is estimated by the twocenter electron repulsion integral using the normalized s-type Slater functions. In the calculation, the valence orbital exponents ζ are related to the covalent radii of the atoms and their values are geometry (or charge) independent except

where Ci,m is the coefficient for each basis function χm. Normalization condition requires that the integral of a molecular orbital squared must satisfy Ni = Ni ∑ Ci2, m + 2Ni ∑

Θi , m

m∈A

Depending on the QM codes employed, different combinations of functionals and basis sets are used in the calculations for MOFs, such as GGA PW91 or PBE functionals combined with basis sets DNP105−108 or DND.109−111 The two basis sets are comparable to the Gaussian-type 6-31G(d, p) and 631G(d) basis sets, respectively. Despite successes in these studies, Mulliken population analysis has suffered much criticism, such as sensitive dependence on the type and size of basis set,97 tending to overestimate the covalent character of a bond (underestimating the bond ionicity and the charge separation between electropositive and electronegative atoms), etc.112 Thus, the charges obtained by this method are usually overestimated. In light of these fundamental deficiencies, Mulliken charges are generally not recommended for use in computing charges for interatomic potentials.113 However, due to its ease of understanding, Mulliken population analysis is often used to identify the electronic structure and charge transfer, consequently gaining detailed understanding of orbital interactions in the chemistry of MOFs.114,115 2.1.3. Molecular Charge Equilibration Method. The charge equilibration (QEq) method was first proposed by Rappé and Goddard III116 to estimate atomic partial charges for molecular systems. Its basic idea is that only geometry and experimental atomic properties are used to predict charges and their distribution is allowed to respond to changes in the environment. Then, the electrostatic energy (E) of a molecular system composed of N atoms can be expressed as

(2a)

m

∑ Ni ∑

(2b)

where Ni is the number of electrons in MO i and Sm,n is the overlap integral over three-dimensional space, ∫ χmχn dvm dvn. In Mulliken analysis, the contributions of a basis function in all D


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for H atoms (which will be shown later). Given a set of atomic positions, the first derivative of E with respect to qA leads to an atomic-scale chemical potential (electronegativity), χA, in the form of χA (q1 , q2 , ..., qN ) =

∂E 0 = χA0 + JAA qA + ∂qA

E(q1 , q2 , ..., qN ) = +

N

∑ JABqB

1 + 2

B≠A

(3b)

For an equilibrium system, it is required that the atomic chemical potentials of all N atoms are equal, which gives N − 1 equations. Combining the constraint for the total charge (eq 1c) and taking JAA = J0AA, the N unknown QEq charges can be determined by solving a system of N linear equations without iteration ⎡ 1 1 ⎢ ⎢ J21 − J11 J22 − J12 ⎢ ⋮ ⋮ ⎢ ⎢⎣ JN1 − J11 JN 2 − J12

⎤⎡ q1 ⎤ ⎡ qtotal ⎤ ··· 1 ⎥ ⎥⎢ ⎥ ⎢ ··· J2N − J1N ⎥⎢ q2 ⎥ ⎢ χ10 − χ20 ⎥ ⎥ ⎥⎢ ⎥ = ⎢ ⋮ ⋮ ⋮ ⋮ ⎢ ⎥ ⎥⎢ ⎥ ··· JNN − JNN ⎥⎦⎢⎣ qN ⎥⎦ ⎢ χ 0 − χ 0 ⎥ ⎣ 1 N⎦

N

∑ ∑

JAB qAqB

A = 1 B = 1, B ≠ A

N

∑ ∑ ∑ JAB,L qAqB

(4a)

L≠0 A=1 B=1

N 0 χA ,periodic (q1 , q2 , ..., qN ) = χA0 + χc , A + JAA qA +

∑ JABqB B≠A

(4b)

where χc,A is the lattice electronegativity defined as N

As described by Rappé and Goddard III, the QEq charges of all atoms are constrained within their chemically meaningful boundaries that correspond to emptying or filling the valence shell of electrons. If any atom is outside its range, its charge is fixed at the boundary and a reduced set of equations is further solved for the nonfixed atoms. On the basis of the ionization potential and electron affinity of H atoms, Rappé and Goddard III found that the calculated electronegativity is in poor agreement with those measured by other methods. To solve this problem, Rappé and Goddard III adopted a special treatment for this atom. By examining the charges on a number of hydrides, the effective charge parameter ξH and the idempotential JHH are represented as

χc , A =

N

∑ ∑ J′AB,L qB + ∑ ∑ L≠0 B=1

L≠0 B=1

qB RAB , L

(4c)

where J′AB,L calculated as J′AB,L = JAB,L − 1/RAB,L is the cloud penetration term that dies off rapidly with respect to the interatomic distance RAB,L. Thus, the lattice contribution in eq 4b is factored into a rapidly convergent penetration term summation and a more slowly convergent 1/RAB,L summation, which is also the case for eq 4a. The equalization conditions of chemical potentials given in eq 4b for all atoms in equilibrium generate N − 1 equations. The constraint for the total charge given in eq 1c is also required, and qtotal should be zero for periodic system. The computational procedure starts with an assumed initial charge distribution for all N atoms, and the Ewald summation technique is used to calculate the values for χc,A using eq 4c. Taking χA = χ0A + χc,A and JAA = J0AA, we have a system of N linear equations that can be written in a matrix form given by

0 ζH = 1.0698 + qH , JHH (qH) = (1 + qH /1.0698)JHH

(3d)

and the charging energy of one H atom in the first term of eq 3a is calculated by 1 0 2 J q (1 + qH /1.0698) 2 HH H

N

A=1 N

where the first and second terms are identical to those in eq 3a but the summation now is over the N atoms within the central unit cell; the last summation over L just adds the effect of the lattice. The index L refers to the periodic images of the unit cell with the central one denoted as L = 0. Similarly, taking the first derivative of E with respect to qA one obtains

(3c)

E H = E H0 + χH0 qH +

1 0 2⎤ 1 J q + 2 AA A ⎦⎥ 2

⎡

∑ ⎢⎣EA0 + χA0 qA

⎡1 ⎢ ⎢ J21 − J11 ⎢⋮ ⎢ ⎢⎣ JN1 − J11

(3e)

where χ0H = 4.5280 eV and J0HH = 13.8904 eV. As ξH and JHH are allowed to be charge dependent, an iterative scheme with initial guesses of qH for all H atoms is required in solving eq 3c using eq 3d for H-containing systems. Iteration is complete when this procedure becomes self-consistent. For this original QEq method, it should be pointed out that such a charge dependence is neglected when deriving eq 3b. At the moment, although the ESP charges are most widely used ones, there are still some studies using this method to obtain partial charges for the framework atoms of MOFs.117−119 2.1.4. Periodic Charge Equilibration Method. To include the effect of a lattice on charge distribution, Ramachandran et al.120 further extended the above QEq into a periodic charge equilibration (PQEq) method. For a periodic system containing N atoms per unit cell, its total electrostatic energy in the PQEq method is given by

⎤⎡ q1 ⎤ ⎡ 0 ⎤ 1 ··· 1 ⎥⎢ ⎥ J22 − J12 ··· J2N − J1N ⎥⎢ q2 ⎥ ⎢ χ1 − χ2 ⎥ ⎥ ⎢ ⎥⎢ ⎥ = ⎢ ⋮ ⎥ ⋮ ⋮ ⋮ ⋮ ⎥⎢ ⎥ ⎢ ⎥ JN 2 − J12 ··· JNN − J1N ⎥⎦⎣⎢ qN ⎦⎥ ⎣ χ1 − χN ⎦ (4d)

This set of linear equations is solved to yield a new charge distribution. As done in the original QEq method,116 if some charges exceed their bounds then a reduced set of equations is further solved for the nonfixed atoms. The overall solving procedure is performed iteratively until the charge distribution has converged. For the H-containing systems, the treatment is the same as that done in the QEq method. Applying the PQEq method to a periodic system just requires its lattice parameters and the positions of all the atoms in the unit cell. Because of the low computational cost but reasonable accuracy of PQEq, Sholl and co-workers41,43 recently adopted this method for the first time to assign charges for screening a large population of MOFs. E


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quantum effects,132 and it expresses the intermolecular (external) potential between a pair of quantum particles as

2.2. Force Fields

A set of potential models, also known as force fields, is the basic input required in molecular simulations to describe the energetic interactions. They play a crucial role in determining the reliability and accuracy of the simulation results. The force fields can be categorized into three components that are used to describe (i) adsorbate−adsorbate interactions, including the nonbonded and bonded parts, (ii) adsorbate−MOF interactions, and (iii) interactions among the framework atoms of MOFs. A general force field consists of bonded and nonbonded terms. The former includes the potentials that describe various intramolecular interactions, while the latter consists of the potentials that account for the vdW and electrostatic interactions. For the sake of computational efficiency and simplicity, most force fields treat the nonbonded term only by two-body interactions. Generally, the nonbonded interatomic interaction is often modeled as a combination of Lennard− Jones (LJ) and Coulomb potentials ⎡⎛ ⎞12 ⎛ ⎞6 ⎤ σij σij 1 qiqj U (rij) = 4εij⎢⎢⎜⎜ ⎟⎟ − ⎜⎜ ⎟⎟ ⎥⎥ + r 4πε0 rij ⎝ rij ⎠ ⎦ ⎣⎝ ij ⎠

1 P

U inter(rij) =

P

∑ ϕ(rijα) α=1

(6a)

where P is the number of beads on a polymer ring that is isomorphic to a quantum particle, ϕ(rαij ) is the adsorbate− adsorbate pair potential between bead α on molecule i and molecule j, and rαij is the distance between the two beads. In addition, the intramolecular potential for a quantum particle with mass m is given by U inter(ri) =

2πmP (βh)2

P

∑ |r iα − r iα + 1|2 α=1

(6b)

rαi

where is the position of bead α on ring i and when α = P, α + 1 = 1 as required for a ring polymer, h is Planck’s constant, and β = 1/(kBT), in which kB and T are the Boltzmann constant and temperature, respectively. The FH effective potential (UFH) can be obtained from a Taylor series expansion of the original Gaussian FH potential,133 and the one truncated at the quartic term (QFH) has the following form134,135

(5)

where U is the potential energy between a pair of atoms i and j at a separating distance rij, qi is the partial charge on atom i, εij and σij are the LJ potential well depth and diameter, respectively, and ε0 is the dielectric constant. For the cross LJ potential parameters between a pair of unlike atoms, the arithmetic mean (i.e., as done for diameter in the Lorentz− Berthelot) or geometric mean mixing rules are usually used to obtain their values. 2.2.1. Force Fields for Adsorbate Molecules. In most modeling studies of MOFs, the force fields for adsorbate molecules are taken from a library of well-established force fields reported in the literature, which were developed within their various thermodynamic bulk states. A systematic approach to parametrize these force fields is to match the simulated vapor−liquid equilibrium properties to the corresponding experimental data.121−123 For small molecules, such as N2 and CO2, their potential models are often developed using rigid geometrical structures; thus, only the nonbonded interactions with potential models like eq 5 are taken into account. For molecules with flexible conformation, more complex models are commonly used to represent them. In these models, potential functions for accounting for the intramolecular interactions are included. In addition, a socalled united-atom approach is also adopted to group the atoms in molecules into single interaction centers. A typical example is the transferable potentials for phase equilibria united-atom (TraPPE-UA) force fields developed for alkanes and other fluids.123 At sufficiently low temperatures, the quantum effects of some gases such as H2 can become very important due to their low masses. These effects are particularly significant when they are confined in the nanospace with dimensions comparable to their de Broglie thermal wavelengths.124 Consequently, such gases and their isotopes cannot be treated in a classical way under these conditions. Currently, according to the approaches employed, computational studies of the quantum effects of H2 in MOFs can be mainly divided into two groups: one is the use of the path integral (PI) formalism113,125 and the other is those using the Feynman−Hibbs (FH) effective potential.92,126−131 The former is essentially exact for capturing

UFH(rij) = U (rij) +

+

2U ′(rij) ⎤ β ℏ2 ⎡⎢ ⎥ U ″(rij) + rij ⎥⎦ 24μ ⎢⎣

⎡ ⎤ 4U ‴(rij) β 2ℏ4 ⎢ 15U ′(rij) ⎥ U ( r ) + + ⁗ ij ⎥⎦ rij 1152μ2 ⎢⎣ rij3 (6c)

where U(rij) is the classical potential, μ is the reduced mass between a pair of interacting adsorbate molecules i and j with μ = mimj/(mi + mj), rij is the separating distance, and ℏ = h/2π. The prime, double prime, etc., denote the first, second, and higher order derivatives with respect to rij, respectively. For adsorbate−MOF interaction, μ is usually set to the mass of the adsorbate molecule. It should be noted that neglecting the last term in eq 6c corresponds to the quadratic version. If U(rij) is a LJ potential (the first term in eq 5) one can obtain UFH

14 ⎡ ⎛ σij ⎞8⎤ β ℏ2εij ⎢ ⎛ σij ⎞ ⎜ ⎟ 22⎜ ⎟ − 5⎜⎜ ⎟⎟ ⎥ = ULJ(rij) + μij σij2 ⎢⎣ ⎝ rij ⎠ ⎝ rij ⎠ ⎥⎦ 16 ⎡ ⎛ σij ⎞10 ⎤ β 2ℏ4εij ⎢ 1987 ⎛ σij ⎞ 265 ⎜⎜ ⎟⎟ − ⎜ ⎟ ⎥ + 2 4 ⎢ 48 ⎜⎝ rij ⎟⎠ ⎥ μij σij ⎣ 24 ⎝ rij ⎠ ⎦

(6d)

By comparing the obtained results with those using the PI formalism as well as the classical potential, the quadratic FH effective potential has been shown to represent the quantum effects fairly well for H2 adsorption in MOFs at cryogenic temperatures while overestimating these effects at room temperature.127 Actually, molecular simulations have indicated that the quantum effects at room temperature are small enough to be neglected.127 In addition, the quartic form has been shown to be more accurate than the quadratic one in the case of confined fluids at low temperatures,134 and the situation becomes even worse for the latter with decreasing temperature. Generally, compared with the PI formalism, use of the FH effective potential method can lead to much less computational cost as well as a relatively easier implementation in the standard molecular simulation programs. F


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2.2.2. Force Fields for MOFs. Apart from the potential models discussed above for adsorbates, an atomistic representation is generally used to model the structures of MOFs in computational studies. To date, the majority of investigations have been performed using a rigid approximation for their structures. The advantage of such a treatment is that there is no need to consider the intraframework interactions. At the same time, due to the plentiful sources of metal ions and organic moieties, the structures of MOFs are theoretically unlimited. Thus, from a computational point of view, it is necessary to have general and accurate force fields that can be applied to account for the full range of interactions between all atoms of MOFs. Since the force fields for the adsorbate−adsorbate interactions have been well established, the main challenge lies in those for accurately describing the interactions of adsorbates with MOFs. In molecular simulations, the adsorbate−MOF interactions are generally calculated using the expression given in eq 5 (of course, there are also some specially developed force fields using Morse or other potentials for vdW interactions). The partial charges for the framework atoms of MOFs can be calculated according to the methods addressed in section 2.1 as well as some newly developed ones described in section 3.1. To date, the LJ parameters used for the framework atoms of MOFs have been taken from several generic force fields that are designed to cover the elements in the full periodic table or a subset thereof. These force fields include the DREIDING,136 universal force field (UFF),137 and optimized potential for liquid simulations all-atom (OPLS-AA) force fields,138 etc. Among them, the DREIDING and UFF are the most widely used force fields and have well (or reasonably well) reproduced the experimental data for different adsorbates in a great variety of MOFs.139−146 Despite their successful applications, it should not be assumed that the above force fields can be transferred to all MOF systems owing to the complex combination of atoms in these coordination compounds. Thus, many specific force fields have been developed with respect to the targeted gas/MOF systems using high-quality QM calculations,147−156 especially for materials with strongly specific interacting sites such as open metal sites or those with framework flexibility. Some of these representative force fields developed for open-metal-sitescontaining and breathing/gate-opening-type MOFs will be presented in detail in sections 3.2 and 3.4, respectively. In addition, some typical force fields developed to take into account the regular framework flexibility of MOFs will be described in section 3.5.

instead of using chemical potential, it is more convenient to use fugacity as input in practical GCMC simulations, which is equivalent to the pressure within a low-pressure domain. For pure-component simulations, molecules generally involve four types of trial moves,158 i.e., attempt to (i) translate an existing molecule that is randomly selected, (ii) rotate an existing molecule that is randomly selected (not necessary for spherical molecules), (iii) create a new molecule in a random position with a random orientation, and (iv) randomly delete an existing molecule. In a normal GCMC simulation method, these trial moves are handled by the normal Metropolis method,158 and insertion and deletion trial moves should be randomly selected with equal probability. For mixture systems, one of the components should be first selected at random (usually with equal probability), and the molecular number and fugacity corresponding to this component must be used in the acceptance rules. In addition, an attempt to exchange molecular identity is also often introduced as an additional type of trial move, so as to speed up equilibration and reduce statistical errors.159 In this trial move, one of the components is first selected at random and then one existing molecule that belongs to this component is randomly selected and converted to another component (if there are at least three components, it is also randomly selected). For small rigid molecules, the above normal GCMC simulation method is generally applicable. However, it is often inefficient when applying this method to larger, more complex rigid molecules or to adsorbents that molecules fit very tightly in their pores. A fraction of successful insertions that is too small (more significant at high loadings) causes a failure in sampling the grand canonical ensemble correctly. To overcome this problem, many improved simulation methods with bias techniques have been devised.160 In addition, the normal GCMC simulation method is also very time consuming for flexible or long-chain molecules with strong intramolecular interactions. For such systems, the most widely used method in MOF studies161,162 is the configurational-bias Monte Carlo (CBMC) scheme.163 Indeed, the CBMC method is applicable to simulate adsorption of both rigid and flexible chain-like molecules in MOFs.164 One of the basic items of thermodynamic information obtained from GCMC simulations is the adsorption isotherm. Before comparing the simulation results with the experimental data, one needs to distinguish two useful concepts: the absolute (nabs) and excess (nex) amounts adsorbed.50 The output from GCMC simulation is the absolute one, that is, the total number of molecules present inside the adsorbent. In contrast, the excess amount is the difference between the total amount adsorbed and the number of molecules that would be present in the same pore volume under the bulk conditions for adsorption. Experimental measurements usually report the latter. The relationship between the two quantities can be given by

2.3. Molecular Simulation Techniques

2.3.1. Grand Canonical Monte Carlo Simulation. With appropriate force fields available for describing various interactions, the grand canonical Monte Carlo (GCMC) simulation method is usually employed to investigate the adsorption properties of gases and their mixtures in MOFs. In this very versatile and powerful method, the temperature T, volume V, and chemical potential μ of each species are kept fixed while the number of gas molecules N in the adsorbed phase is allowed to fluctuate. Then, the amount of molecules adsorbed is calculated using a statistically averaged approach after the equilibrium stage. The chemical potential can be related to the bulk pressure or fugacity through an equation of state (EOS) such as the Peng−Robinson EOS157 or via a separate molecular simulation for the bulk phase. Actually,

nex = nabs − ρbulk Vpore

(7)

where ρbulk is the fluid density in the bulk phase at the same temperature and pressure for adsorption (which can be obtained from the library of thermodynamic data or calculated with the appropriate EOS) and Vpore is the pore volume of the adsorbent that can be measured by experiment or obtained using the computational method described in section 2.5.1. The origins of the difference between the absolute and the excess amounts can be attributed to two factors. One is the G


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except in the limit of vanishing concentrations of the adsorbates. To describe the diffusion behavior of mixtures in MOFs, there are several mathematically equivalent formalisms such as Onsager, Fickian, and Maxwell−Stefan (M−S).187 The Onsager theory of irreversible thermodynamics188 provides a fundamental basis for mixture diffusion in which a phenomenological description of diffusion is commonly employed in practice. From the viewpoint of determining M−S diffusivities, Krishna and co-workers187 suggested that it is much more convenient to define a modified Onsager matrix [Δ]. Each element Δij of this M−S matrix is calculated by

total bulk phase pressure. At low pressures, the bulk phase can be considered an ideal gas, and thus, the difference between the absolute and excess amounts is negligible but becomes larger with increasing bulk pressure due to greater deviation from the ideal state. The other one is the adsorption strength of the adsorbate itself. The poorer the adsorption strength, the larger is the difference between absolute and excess loading; this makes the excess to absolute (or vice versa) conversion particularly important when comparing experimental and computational data for poorly adsorbing gases such as H2 and N2 but much less important for more strongly adsorbing gases such as CO2.165 In addition, the absolute amount is more useful for statistical thermodynamic theories on the fluid adsorption in porous solids since they are generally formulated in the language of absolute thermodynamic variables.166 2.3.2. Molecular Dynamics Simulation. The diffusion properties of guest molecules in MOFs are also of great importance in practical applications. At the moment, experimental data for the molecular diffusivities in MOFs are very limited167−178 and most of them were reported for pure components with only a few for mixtures.179−183 As an accurate experimental characterization of molecular transports in MOFs is much more challenging, most of the information on this topic has come from molecular dynamics (MD) simulations. Actually, almost all experimental measurements conducted so far are combined with MD simulations to obtain a complete picture for diffusion of guest molecules. Although both the Einstein and the Green−Kubo relations can be used to calculate the diffusivity, the former is the one most widely used in studies of MOFs. In this case, the diffusivity is calculated by taking the slope of mean squared displacement (MSD) over a long time. For pure-component systems, there are three most common diffusivities:51 (i) self-diffusivity (Ds) that measures the motion of individually tagged molecules, (ii) transport diffusivity (also referred to Fickian diffusivity, Dt) that is the proportionality between the adsorbate flux J and its concentration gradient (∇c) in the Fickian law, J(c) = −Dt∇c, and (iii) corrected diffusivity (D0, which is equivalent to the Maxwell−Stefan diffusivity Đ) that is defined to correlate Dt with a term called thr thermodynamic correction factor (Γ), a logarithmic derivative of bulk phase fugacity f related to the adsorbate concentration (c) in the adsorbed phase184,185 ⎛ ∂ln f ⎞ ⎟ Dt (c) = D0(c)Γ, Γ = ⎜ ⎝ ∂ln c ⎠T

Δij =

1 d lim 2dN t →∞ dt

M

Dii =

1 d D0 = lim 2dN t →∞ dt

M

Dii =

cj ⎛ ∂ln f j ⎞ ⎜ ⎟(i = 1, 2, ..., M ) ci ⎜⎝ ∂ln ci ⎟⎠

(9b)

ck ⎛ ∂ln fk ⎞ ⎜ ⎟(i ≠ j , i , j = 1, 2, ..., M ) cj ⎜⎝ ∂ln cj ⎟⎠

(9c)

where ci is the concentration of species i in the adsorbed phase. Similarly, the thermodynamic correction factors present in the above two equations can also be determined from the mixture adsorption isotherms or calculated by mixture GCMC simulations using the fluctuation formula.189 Then, we can calculate the fluxes of all components through a MOF membrane by ⎡ J1 ⎤ ⎡ D11 D12 ⎢ ⎥ ⎢ ⎢ J2 ⎥ ⎢ D21 D22 ⎢ ⎥ = −⎢ ⋮ ⎢⋮⎥ ⎢ ⋮ ⎢J ⎥ ⎢ D D ⎣ M1 M 2 ⎣ M⎦

··· D1M ⎤⎡ ∇c1 ⎤ ⎥ ⎥⎢ ··· D2M ⎥⎢ ∇c 2 ⎥ ⎥ ⎥⎢ ⋮ ⋮ ⎥⎢ ⋮ ⎥ ··· DMM ⎥⎦⎢⎣∇cM ⎥⎦

(9d)

where Ji is the flux of component i and ∇ci is the concentration gradient of this component across the membrane. Defining that [B] is the inverse one of the M−S matrix [Δ], [B] = [Δ]−1, the M−S dif0fusivities (D̵ MS i ) and binary exchange 187 coefficients (D̵ MS i ) can be determined by ci ,sat θi D̵ ijMS = − (i ≠ j , i , j = 1, 2, ..., M ) cj ,sat Bij (9e)

(8b) 2

∑ [ri(t ) − ri(0)] i=1

∑ Δik k=i

ri(t ) − ri(0) 2

N

∑ Δij j=i

(8a)

i=1

(9a)

where Ni and Nj represent the number of molecules of components i and j, respectively, and ril(t) is the position of molecule l of component i at time t. The self-diffusivity of each component i can be determined according to eq 8b but with Ni replacing N. For an M-components system, the M−S matrix can be converted into the matrix of the Fickian diffusivities in which the elements are calculated by

N

∑

⎛ Ni ⎞ ⎜⎜∑ [r il(t ) − r il(0)]⎟⎟ ⎝ l=1 ⎠

⎛ Nj ⎞ × ⎜⎜∑ [r kj(t ) − r kj(0)]⎟⎟ ⎝k=1 ⎠

The values of Γ are fully determined once the adsorption isotherm is known or alternatively can be calculated directly by GCMC simulation using fluctuation formula.186 Ds and D0 can be determined using the formulas given by Ds =

1 d lim 2dNj t →∞ dt

(8c)

D̵ iMS =

where d is the spatial dimension of the adsorbents (for MOFs, usually d = 3), N is the number of molecules in the system, and ri(t) is the center-of-mass (COM) position of molecule i at time t. The above three diffusivities are in general not equal

1 Bii −

θj M ∑ j = 1, j ≠ i MS D̵ ij

(i = 1, 2, ..., M ) (9e)

where θi is the fractional occupancy of component i with a saturation loading ci,sat, θi ≡ ci/ci,sat. Self-diffusivity (Di,self) can be H


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pressure domain. Alternatively, if the single-component isotherms span a broad range of pressures (loadings), they can also be numerically integrated in eq 10b without the need of models for curve fitting.42 Then, the adsorption selectivity can be calculated using eq 15a described in section 2.8.1. Although many examples have shown good agreement between IAST predictions and mixture adsorption data obtained computationally and experimentally,31,94 there are still limitations on the applicability of this approach for MOFs. Sholl and co-workers42 illustrated that IAST fails to accurately predict selectivity when some volume within a material is completely inaccessible to one of the two adsorbing species due to pore-blocking effects. They demonstrated that this limitation can be overcome in principle by applying IAST separately to distinct volumes within the material. In addition, Cessford et al.191 recently concluded that for MOFs with no major heterogeneities, such as cavities of very different sizes, IAST can well predict the adsorption of mixtures in which adsorbates have similar sizes and adsorption strengths, while it is less accurate for mixtures involving adsorbates of disparate sizes or nonsphericities. The ability of IAST to predict mixture adsorption is particularly poor if there is a significant disparity in the strengths of interaction between molecules in a mixture of adsorbates of differing polarities.

calculated using eq 8b or using the following expression derived within the M−S formulation 1

Di ,self =

1 D̵ iMS

M

+ ∑j=1

θj D̵ ijMS

(9f) 164

D̵ MS i

As described by Krishna and co-workers, characterizes in the broad sense the interactions of component i with the pore wall of the adsorbent, and the self-exchange coefficient D̵ MS is a measure of the degree of correlations for pureii component diffusion. For mixture diffusion, D̵ MS ij describes the correlation effects between components i and j in the adsorbent. The lower value of D̵ ijMS implies a stronger correlation effect, and there is no correlation effect when the value becomes infinite. 2.4. Ideal Adsorbed Solution Theory

Ideal adsorbed solution theory (IAST) developed by Myers and Prausnitz190 is a widely used engineering thermodynamic method for predicting the adsorption equilibria of the components in mixtures. This thermodynamically consistent theory only uses pure-component adsorption isotherms as input and is exact in the limit of zero pressure. It is based on the elegant concept of creating an ideal multicomponent adsorbed phase by mixing pure adsorbed phases at a constant spreading pressure (grand potential density for three-dimensional systems) and temperature. The relationship between the bulk and the adsorbed phases is described by an expression analogous to Raoult’s law for vapor−liquid equilibria Pi = yP = Pio(π )xi (i = 1, 2, ..., M ) i

2.5. Structural Characterization of Porous Solids

2.5.1. Pore Volume Calculation. As noted previously, estimation of the pore volume of an adsorbent is a key step for conversion of absolute amount into the excess one (eq 7). For computational characterization of MOFs, the thermodynamic method of Myers and Monson166 is the most widely used technique. In this method, on the basis of the absolute adsorption second virial coefficient, the pore volume of the porous adsorbent is calculated by the following configuration integral

(10a)

where M is the number of components in the mixture, Pi is the partial pressure of component i in the bulk phase at total pressure P, yi and xi are the mole fractions of this component in the bulk and adsorbed phases, respectively, and Poi (π) is the hypothetical partial pressure of pure component i at the spreading pressure π. For each pure component, π and Poi are related via integration of the Gibbs adsorption isotherm πA = RT

∫0

Pio

nio(p)d ln p

Vpore =

(10b)

where is the pure-component adsorption isotherm, A is the surface area of the adsorbent, and R is the ideal gas constant. At equilibrium, all components have the same spreading pressure, π1 = π2 = ... = π. On the basis of the summation over the mole fractions of all components in the adsorbed phase being equivalent to one, the total amount ntotal of the mixture adsorbed is calculated by ntotal

M

=

xi o o n (Pi ) i=1 i

∑

(10c)

where noi (Poi ) is the amount of pure component i adsorbed at Poi (π). Then, the quantity of each species in the adsorbed phase can be obtained using ni = xintotal (i = 1, 2, ..., M )

∫ e−E(r)/k T dr B

(11)

where E is the fluid−adsorbent interaction of a single helium atom at position r and m is the mass of the adsorbent. Using a simple Monte Carlo (MC) technique, the integration is conducted over the entire adsorbent and the exponential is finite inside the pores but vanishes within the solid where E → ∞. It should be noted that Vpore calculated from eq 11 is the specific pore volume of the adsorbent. As described by Myers and Monson, T is set to 298 K, which usually is the reference temperature chosen for experimental determination of helium-based pore volume. This approach was for the first time adopted by Düren et al. for MOFs.192 Pore volume calculation can provide useful reference information for assessing to what extent the experimental MOF samples deviate from their perfect crystalline structures as well as whether they are well activated. 2.5.2. Geometric Pore Size Distribution Analysis. The pore size distribution (PSD) of a porous solid is one of the key parameters needed to understand the adsorption and transport phenomena of guest molecules in its networked nanoscale space. Consequently, there has been significant research effort described in the literature to find appropriate ways for this target. Among them, the efficient computational approach developed by Gelb and Gubbins193 for porous glasses was first adopted by Snurr and co-workers192 to characterize IRMOFs, and then it was widely used for other MOFs.12,61,82,194 The

noi (p)

1

1 m

(10d)

In order to perform the integrations required in eq 10b, appropriate models should be used for all pure components to fit their discrete sets of adsorption data, which are usually obtained over a finite range of pressures. There is no restriction on the choice of these models, but adsorption data should be fitted to an acceptable degree of accuracy, especially in the lowI


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satisfied. Final optimization quality depends on the force field adopted. At the moment, UFF force field is perhaps the most widely used one. In addition, this force field is commonly coupled with the QEq method to obtain charges for calculating electrostatic interactions.196 Since the electronic motions are not taken into account explicitly, the MM method allows us to optimize very large MOF systems such as MIL-101(Cr)197 that are operationally infeasible to process using the more expensive QM method. Currently, the QM method has evolved into a powerful tool to theoretically obtain the geometric equilibrium structures of MOFs. It has been considered to be capable of giving more accurate predictions than the MM method. In most QM-based optimizations, the procedures are mainly conducted within the scope of the DFT approach. The advantage of DFT lies in the easy and efficient implementation by using either localized basis functions (Gaussian-type orbitals) or delocalized ones (plane wave orbitals). In contrast to the MM method, the DFT procedure is primarily a theory of electronic ground-state configuration that determines, from the corresponding electron density, the energy of a collection of atoms at prescribed positions.62 Usually, the majority of standard DFT methods perform very well for geometric optimizations of MOFs. However, as we will see later, due to their drawbacks of inaccurate description of the long-range electron correlations, these standard methods sometimes can lead to erroneous results for MOFs where the contribution of dispersive (or vdW) forces plays a significant role.198 Due to the QM-based optimizations being very time consuming, a reasonable guess of the initial positions for the framework atoms of MOFs from the MM method can greatly accelerate the convergence speeds of these calculations. Moreover, when there are several possible positions for functionalizing MOFs with functional groups, an efficient operation is to optimize each modified form using the MM method. The locations for the functional groups are identified by selecting those in the optimized structure with the lowest energy.95 Then, a DFT-based optimization procedure can be further employed to refine the selected model.

basic idea is that the geometric PSD of a given adsorbent is calculated on the basis of a cumulative pore volume curve.193 Specifically, the pore range of interest is first divided into a number of small bins. Then, the calculations are accomplished by two MC procedures, both of which consist of a large number of cycles. In each cycle of the first procedure, a testing point P is randomly placed into the adsorbent, making sure that this point does not overlap any adsorbent atoms,195 as shown in Figure 1. Within this cycle, the second MC procedure is used to

Figure 1. Two-dimensional schematic depiction of the geometric pore structure of an adsorbent. Dotted red circle denotes the largest sphere that covers the point P, without overlapping with all framework atoms of the adsorbent.

probe all the pore space of this adsorbent, so as to find the largest sphere (red circle shown in Figure 1) that encloses point P without overlapping any adsorbent atoms. Once the largest sphere is found, the values of all the bins corresponding to this pore radius as well as smaller radii are incremented by one. At the end, a curve for the cumulative pore volume Vp(r) is obtained using the normalized bin distribution, which is a monotonically decreasing function of pore radius r. The PSD of the adsorbent can be acquired from the derivative −dVp(r)/dr using a mathematic differentiation method. 2.6. Structural Optimization Method

In the structures of many MOFs built from their experimental crystallographic data, there are often several superimposed equivalent positions, due to space-group symmetry of the structure, for the organic ligands or/and inorganic components. In particular, in contrast to neutron diffraction experiments, hydrogen atoms are commonly missing from experimental single-crystal X-ray diffraction data because the diffraction intensity in this approach is directly related to the atomic weight of the element. In addition, MOFs can be decorated with functional groups so as to examine the effects of functionalization on the structural properties of MOFs as well as on their performance for the application of interest. As a result, crystalline structures in these situations are usually obtained by a geometric optimization procedure. The initial atomic positions (sometimes the lattice parameters) in the structure are progressively adjusted to find the most favorable conformation, which corresponds to the one with a minimum on the potential energy surface. Such a structural refinement can be conducted using the molecular mechanics (MM) method, QM approach, or a combination of them. The MM method is a modeling technique that treats the system to be optimized at the atomic level. The overall potential energy of this system is defined for every set of positions of the atoms using the intermolecular/intramolecular potential functions in classical force fields.62 This method allows us to iteratively refine the geometry in a mechanical approach until some predefined convergence criteria are

2.7. Adsorption Enthalpy Calculation Method

Adsorption enthalpy ΔH (or isosteric heat of adsorption qst, ΔH = −qst) is one of the most important thermodynamic quantities for understanding the possible thermal effects related to adsorption. It can be used to compare the interaction strength of the adsorbates with various adsorbents. This property is also useful for assessing whether or not the pore surface of an adsorbent is homogeneous for adsorption. In experiments, the adsorption enthalpy is usually computed using the Clausius−Clapeyron equation or directly obtained by microcalorimetry measurements. On the basis of a survey of the literature, several alternative methods have been adopted to obtain the adsorption enthalpies in the computational studies of MOFs. 2.7.1. Energy Difference. On the basis of MC or MD simulation in the canonical (NVT) ensemble, the adsorption enthalpy of the adsorbate in the limit of zero coverage can be calculated from the potential energy difference199 ΔH = − qst = ⟨Ua⟩ − ⟨Us⟩ − ⟨Ug⟩IG − RT

(12)

where the condition of zero coverage is approximately represented by a single adsorbate molecule present in the simulation box, ⟨...⟩ denotes the ensemble average, Ua is the J


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lations are usually required to obtain statistically accurate values for the adsorption enthalpies, especially at very low loadings. 2.7.3. Revised Widom’s Test Particle. By relating Henry’s coefficient to the free energy of a guest molecule inside the adsorbent, Vlugt and co-workers201 suggested a revised Widom’s test particle method to calculate the adsorption enthalpy using

total potential energy that includes both the contributions of the adsorbate and adsorbent, Us is the potential energy of the empty adsorbent, and Ug is the potential energy of a single adsorbate molecule in the ideal-gas state which only depends on temperature. For rigid adsorbate molecules, Ug = 0. Similarly, the value of Us is also zero for adsorbents with rigid frameworks and without extraframework species. It should be noted that this equation neglects the nonideal behavior of the bulk phase. 2.7.2. Ensemble Fluctuation. The ensemble fluctuation method has been widely used to calculate the adsorption enthalpy, which is based on the energy and particle fluctuations occurring in GCMC simulations, and can be applied to the situation where the adsorbate loading in the adsorbent is nonzero. Considering a general case that a system consists of M components, the adsorption enthalpy of component i can be calculated using200 ⎛ ∂U ⎞ ΔHi = −qst, i = ⎜ ⎟ − ⟨Ug, i⟩IG − RT ⎝ ∂Ni ⎠ N j≠i

⎛ ∂U ⎞ ⎜ ⎟ = ⎝ ∂Ni ⎠ N j≠i

⎛ ∂U ⎞ ⎜⎜ ⎟⎟ ⎝ ∂βμk ⎠ μ

⎛ ∂U ⎞ ⎟⎟ ∂βμk ⎠ k=1 ⎝ M

∑ ⎜⎜

μj ≠ k

⎛ ∂βμk ⎞ ⎜ ⎟ ⎝ ∂Ni ⎠ N

j≠i

⟨(UN + U +)exp[−βU +]⟩N ⟨exp[−βU +]⟩

− ⟨UN ⟩N − ⟨Ug⟩IG − kBT

where U = UN+1 − UN is the potential energy change after inserting one test (ghost) particle into the system composed of N adsorbate molecules. The key feature of this method is that the values of ⟨UN+1⟩ and ⟨UN⟩ are simultaneously obtained in a single simulation in the NVT ensemble. For long-chain molecules or flexible molecules with strong intramolecular interactions, it is much more convenient to use the CBMC method to generate a configuration for the test particle. Then, the adsorption enthalpy can be calculated by201

(13a)

ΔH = (13b)

= f (U , Nk)

⟨(UN + U +) × W *exp[−βδ]⟩N ⟨W *exp[−βδ]⟩N

∂n2 ∂βμ2 ∂n2 ⋮ ∂βμM ∂n2

∂βμ1 ⎞ ⎟ ∂nM ⎟ ⎟ ∂βμ2 ⎟ ··· ∂nM ⎟ ⎟ ⋱ ⋮ ⎟ ⎟ ∂βμM ⎟ ··· ⎟ ∂nM ⎠ ···

⎛ f (N1 , N1) f (N1 , N2) ⎜ ⎜ f (N2 , N1) ⋱ =⎜ ⋮ ··· ⎜ ⎜ ⎝ f (NM , N1) f (NM , N2)

2.8. Separation Selectivity Calculation Method

··· ··· ⋱ ···

−1 f (N1 , NM ) ⎞ ⎟ f (N2 , NM ) ⎟ ⎟ ⋮ ⎟ ⎟ f (NM , NM )⎠

In separation processes, a good indication of the separation ability is the selectivity of a porous adsorbent, and there are two kinds of selectivities: adsorption selectivity and permeation selectivity. They are involved in adsorption-based and membrane-based separation processes, respectively. 2.8.1. Adsorption Selectivity. Adsorption-based separation is a physisorptive operation governed by a thermodynamic equilibrium process, which relies on the fact that guest molecules reversibly adsorb in nanopores at densities that far exceed the bulk density of the gas sources in equilibrium with the adsorbents.202 This thermodynamic separation process is mainly controlled by the competitive adsorption affinity of the pore surface of an adsorbent for various species in the mixtures. For separations in the pressure (PSA) or temperature (TSA) swing adsorption processes, the performance of a porous material is usually characterized by adsorption selectivity. This selectivity (Sads,A/B) for species A relative to B is calculated by

(13d)

where f(X,Y) = ⟨XY⟩ − ⟨X⟩⟨Y⟩ stands for the fluctuation of any X−Y pair, U is the total potential energy of the system, μi and Ni is the chemical potential and number of molecules adsorbed of component i, respectively, and qst,i is the isosteric heat of adsorption of that component. Again, the bulk phase is also assumed to be in the ideal-gas state. For pure-component adsorption, the above equations can be reduced to a more familiar formula ΔH = −qst =

(14b)

where δ is the energy difference between the total nonbonded part of the potentials and the nonbonded part that is used to select trial segments. As suggested by Vlugt et al., it is often convenient to use only part of the nonbonded energy to select trial segments, which leads to a modified Rosenbluth weight W*. For example, this is useful for charged systems, for which only the real-space part in the Ewald summation can be used to select trial segments. In addition, they also found that the adsorption enthalpy calculated by the energy difference method shown in eq 12 is extremely inaccurate for zeolites with extraframework cations.

(13c)

∂βμ1

(14a)

+

− ⟨UN ⟩N − ⟨Ug⟩ − kBT

j≠k

⎛ ∂βμ 1 ⎜ ∂ n ⎜ 1 ⎜ ⎜ ∂βμ2 ⎜ ∂n1 ⎜ ⎜ ⋮ ⎜ ⎜ ∂βμM ⎜ ⎝ ∂n1

ΔH = −qst =

⟨UN ⟩ − ⟨U ⟩⟨N ⟩ − ⟨Ug⟩IG − RT ⟨N 2⟩ − ⟨N ⟩2

⎛ x ⎞⎛ y ⎞ Sads, A / B = ⎜ A ⎟⎜⎜ B ⎟⎟ ⎝ xB ⎠⎝ yA ⎠

(13e)

As commented by Karavias and Myers,200 when the system’s density is close to that of a condensed phase, the fluctuation quantities may have convergence problems in GCMC simulations. This is due to the low acceptance ratios from creation−destruction attempts. In addition, long-time simu-

(15a)

where xA and xB are the mole fractions of A and B in the adsorbed phase, respectively, while yA and yB are the mole fractions of A and B in the bulk phase, respectively. Apart from K


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3. DEVELOPMENT OF COMPUTATIONAL METHODS AND CONCEPTS FOR MOFS

the adsorption selectivity, there are four other quantities frequently used as evaluation criteria:194 the uptake (Nads A ), working capacity (ΔNA), regenerability (R) of targeted component in the mixture, and sorbent selection parameter (S). The latter three quantities are defined as ΔNA =

N Aads

−

N Ades

3.1. Atomic Partial Charge Estimation

Atomic partial charges are usually required in the study of MOFs by molecular simulation; however, the various methods described in section 2.1 do not necessarily give identical results, and each of them has certain deficiencies. On the other hand, the number of MOFs is theoretically unlimited. Thus, it is very useful to have a general protocol that can be used to quickly estimate the charges with reasonable accuracy for a MOF without requiring the time-expensive QM calculations. Indeed, with the aid of an influencing factor isolation (IFI) method,205 a number of computational studies have shown that the electrostatic features of MOFs are of importance for their storage and separation properties toward the systems involving gases that are polar/polarizable.33,101,113,206−213 Therefore, significant research effort has been paid to improving existing methods or developing new methods for generating atomic partial charges or electrostatic potential energy surfaces (EPES) to account for the electrostatic potentials of MOFs. It is worth noting that because no atomic partial charges are used in the EPES method proposed for rigid framework, Watanabe et al.214 utilized it to unambiguously evaluate the reliability of various charge assignment methods (see section 3.1.5), including the CBAC, REPEAT, and DDEC methods, as well as the Hirshfeld approach.215 The last one is not involved in this review due to its seldom use for MOFs. 3.1.1. Extended Charge Equilibration Method. Although the QEq method (see section 2.1.3) can be used to rapidly assign atomic partial charges for MOFs, it was found that the disagreements with the CHELPG charges become large for heavy atoms (Z > 20) or atoms with partial charges typically greater than one.216 This is attributed to the fact that the neutral state ionization potentials and electron affinities of the atoms are used in this method. For the same reason, the aforementioned PQEq method, first adapted for studying MOFs by Sholl and co-workers, also tends to underestimate the charges, particularly for metal atoms and those connected to them.41 To overcome this problem, Wilmer and Snurr216 suggested a modification of the original QEq method by only changing the Taylor series expansions for the transition metal atoms to be centered about their oxidation charges. At the same time, the ionization potentials and electron affinities corresponding to their oxidation states are used in the calculations. It was found that such a preliminary treatment is crucial for obtaining good charges on the inorganic moieties of MOFs. Very recently, they presented a generalized version named the extended charge equilibration method (EQeq).217 In this method, the Taylor series expansion for the energy EA(Q) of an isolated (gas-phase) atom A is centered about some particular partial charge Q* (Q* is an integer). This expansion is truncated at the quadratic term which leads to

(15b)

R = (ΔNA /N Aads) × 100%

(15c)

ads 2 des S = (αAB ) /(αAB )(ΔNA /ΔNB)

(15d)

where A is assumed to be the targeted component, Nads A and NAdes are its uptakes under adsorption and desorption des conditions, respectively, and αads AB and αAB are the selectivities (calculated by eq 15a) under the adsorption and desorption conditions, respectively. In addition, ideal adsorption selectivity (Sideal,A/B) is also widely used to characterize the separation performance of MOFs, which is expressed as the ratio of the amounts (Npure) of pure components adsorbed at certain pressures (see eq 15e). It should be pointed out that usually there is a distinct difference between Sads,A/B and Sideal,A/B due to competitive adsorption of different components in the mixtures. Sideal, A / B =

Npure, A Npure, B

(15e)

2.8.2. Permeation Selectivity. Membrane-based separation is generally based on the size and shape of the adsorbates to be separated or on the interplay of their adsorption and diffusion properties in the membrane material.203 As a rule, the performance of a membrane is evaluated by two characteristic parameters: permeability and selectivity. The steady-state permeability (Pperm,i) of component i in the mixture reflects its transport rate through a given membrane. This quantity relates the net flux (Ji) to the membrane thickness (L) and the partial pressure (or fugacity) drop (ΔPi) across the membrane according to31 Pperm, i =

Ji × L ΔPi

(16a)

where ΔPi = Pfeed,i − Ppermeate,i with Pfeed,i and Ppermeate,i are the applied upstream and downstream partial pressures (or fugacities) of component i, respectively. The permeation selectivity (Sperm,A/B) is defined as the ratio of the steady-state permeabilities of two components in the mixture31 ⎛ Pperm, A ⎞ ⎛ J ⎞⎛ ΔP ⎞ ⎟⎟ = ⎜⎜ A ⎟⎟⎜ B ⎟ Sperm, A / B = ⎜⎜ P ⎝ perm, B ⎠ ⎝ JB ⎠⎝ ΔPA ⎠

(16b)

Similar to the ideal adsorption selectivity, a membrane’s ideal permeation selectivity (Sideal,A/B) is defined to be the ratio of pure-component fluxes under similar conditions204 Sideal, A / B =

EA(Q ) = EA(Q *) + χQ * (Q − Q *) +

Jpure, A Jpure, B

1 J (Q − Q *)2 2 Q* (17a)

(16c)

⎛ ∂ 2E ⎞ ⎛ ∂E ⎞ IQ *+1 + IQ * , JQ * = ⎜ 2 ⎟ χQ * = ⎜ ⎟ = 2 ⎝ ∂Q ⎠Q = Q * ⎝ ∂Q ⎠Q = Q *

In the cases where mixture effects are important, the ideal permeation selectivity often differs significantly from the real situation of a permeating mixture. This means that characterizing the selectivity of a membrane based on mixed-gas feeds is crucial.

= IQ *+1 − IQ * L

(17b) dx.doi.org/10.1021/cr400005f | Chem. Rev. XXXX, XXX, XXX−XXX

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where In is the nth ionization energy required to remove the nth electron after n − 1 have already been removed (I0 is the energy to go from a charge of −1 to 0, which is equivalent to the atom’s electron affinity), χn is the nth electronegativity, and Jn is the hardness (idempotential). Then, the total energy of the system composed of N atoms can be expressed as a sum of individual atomic energies and pairwise Coulomb interactions Esys(Q 1 , Q 2 , ..., Q N ) N

=

∑ [EAk(Q k*) + χQ * (Q k − Q k*) k

k=1

1 J *(Q − Q k*)2 + ECk + EOk ] 2 Qk k

+

N

ECk =

∑K

Q kQ m

m≠k

rkm

(17c)

, EOk Figure 2. Comparison of the atomic charges predicted by the EQeq method versus REPEAT charges for all atoms in 12 MOFs. Average charge difference between the two methods is ±0.16. Reprinted with permission from ref 217. Copyright 2012 American Chemical Society.

⎡ ⎛ Jkmrkm ⎞2⎛ ⎤ 2 Jkm rkm −⎜ ⎟ Jkm 1 ⎞⎟⎥ ⎢ K ⎠ ⎝ ⎜ = KQ k ∑ Q m e − ⎜K − ⎢ rkm ⎟⎠⎥⎦ K2 ⎝ m ⎣ N

(17d)

where K = 1/(4πεrε0) (εr is the dielectric strength and ε0 is the permittivity constant), ECk is the energy of the charge on the kth atom interacting with those of all other atoms, EOk is a pairwise damping term to prevent infinite charge separation when two atoms are brought arbitrarily close together, rkm is the distance between the kth and the mth atoms, and Jkm is the geometric mean of the chemical hardness of both atoms. For a periodic system, ECk can be handled using the standard Ewald summation technique or the direct summation only if the system composed of a certain number of unit cells of a MOF is large enough. For an equilibrium system, the following equations should be satisfied ∂Esys ∂Q 1

=

∂Esys ∂Q 2

= ··· =

purpose of large-scale screening of materials as well as for calculation of fluctuating charges during molecular simulations.217 3.1.2. Connectivity-Based Atom Contribution Method. Recently, our group proposed a strategy named the connectivity-based atom contribution (CBAC) method218,219 to quickly estimate the atomic partial charges of MOFs. The central idea is that although the number of MOFs is nearly infinite, the atomic types involved are quite limited. Thus, if the framework charges of a MOF can be estimated solely based on the information of its atomic types without requiring QM calculations it will make possible the computational screening of MOFs on a large scale. Starting from this point, it is assumed that the partial charge of a framework atom is determined by its bonding connectivity environment and the atoms with the same connectivity have identical charges in different MOFs. This approximate approach is similar to the well-known group contribution method that has been successfully used for calculating thermodynamic properties of fluids.220 In this CBAC approach, DFT calculations are first performed on the fragmented clusters cleaved from a certain number of representative MOFs with various pore sizes, topologies, and chemistries. Then, the ESP charges extracted by the CHELPG method are adopted as the training set to establish the general charges of the defined atomic types. The distribution of charge values for the same atomic type in different MOFs is found to fluctuate closely around its average value despite the existence of small differences. This indicates that the idea of the CBAC is a good approximation for framework charge estimation in MOFs. In addition, using the CBAC-based and DFT-derived charges, the reliability of this method has been further validated by good agreement between the simulated adsorption isotherms and the microscopic adsorption mechanisms of CO2 in MOFs.218,219,221 This validation was achieved both for those materials used and for those not used in developing this method. As commented by others,47 this CBAC method can greatly save both time and computational cost of QM calculations, further enabling a theoretically rapid screening of MOFs for targeted properties. Moreover, this simple method can be

∂Esys ∂Q N

(17e)

Combining eqs 17e and 1c, the charges Qi (i = 1, 2, ..., N) can be simultaneously determined by finding the minimum energy in eq 17c without iteration. For systems containing hydrogen atoms, two ad hoc parameters are employed: one is the global dielectric strength (εr) used to prevent infinite charge separation as well as to maintain a noniterative solution; the other is the zeroth ionization energy (I0) applied to H atoms which considers the physically unrealistic tendency for these atoms to be assigned negative charges when using the measured values. To validate the above method, Wilmer and Snurr compared the predicted EQeq charges against those derived by the REPEAT method (which will be introduced in section 3.1.3) for all atoms in 12 diverse MOFs. The results shown in Figure 2 indicate that the average difference between the charges obtained by the two methods is very small. It also showed that the EQeq method leads to correct ranking of these materials regarding their performance for CO2 adsorption at 298 K and 0.1 bar. This new general scheme can be applied to periodic systems without adding significant computational expense. It also does not require the iterative procedure that is needed in the PQEq method for minimizing the system’s energy in a self-consistent manner. Thus, the EQeq method is better suited for the M


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charges for buried atoms. The REPEAT error functional devised for a system containing N atoms is expressed as

readily extended by including new atomic types in the library, making it possible to estimate framework charges in a MOF solely based on its structure. Indeed, we recently achieved a considerable extension of our CBAC database.36 Figure 3

F({qj , δϕ}) =

∑ [ϕQM(rgrid) − (ϕq(rgrid) + δϕ)]2 grid

+λ[∑ qj − qtotal] j

+

⎡

∑ wj⎣⎢Ej0 + χj qj + j

δϕ =

1 Ngrid

1 00 2 ⎤ J q 2 j j ⎦⎥

(18a)

∑ [ϕQM(rgrid) − ϕq(rgrid)] grid

(18b)

where qj represents the charge on each atom (j = 1, 2, ..., N), ϕQM and ϕq are the electrostatic potentials at the position (rgrid) of each grid point obtained from the QM calculations and the fitted charges, respectively, ϕq is given by eq 1b and calculated using the Ewald summation method for the periodic system, and λ is the Lagrange multiplier used to ensure that the total charge of the system is equivalent to qtotal (zero in the case of periodic systems). In eq 18a, the third term is a sum of the physically motivated penalty multipliers introduced for systems with buried atoms, where {wj} are the weighting factors and χj and J00 j are the electronegativity and self-Coulomb interaction of each atom, respectively, which are set to the values used in the QEq method.116 Thus, this term can be omitted if there are no buried atoms. In addition, a single value of w can be used for all atoms or be individually adjusted to selectively turn on penalty corrections only for buried atoms. Minimizing the error functional, F, with respect to the independent variables {qj} and δϕ yields N + 1 equations. After some lengthy algebra, one arrives at a matrix problem from which the charges {qj} and λ can be calculated simultaneously. Campaña ́ et al. found that there are significant similarities by comparing the REPEAT charges for IRMOF-1 (also called MOF-5) with those ESP charges obtained by the cluster-based calculation methods. Furthermore, one great advantage of this REPEAT method lies in avoiding extracting chemically sensible fragments from a periodic structure, which can sometimes have nontrivial complications. However, although this new method perhaps is the most accurate technique for fitting ESP in the nonperiodic and periodic materials without buried atoms,222 it requires addition of fine-tuned constraints to accurately treat systems with buried atoms. Moreover, for the atoms far from the vdW surfaces of these systems, many different combinations of charges can give almost the same ESP. Thus, the resulting charges may not be chemically meaningful, as pointed out by Manz and Sholl.222 3.1.4. DDEC Method. Considering the limitations of the REPEAT method as well as other schemes, Manz and Sholl222 recently proposed a new approach to obtain ESP charges for molecular and periodic systems, which are named densityderived electrostatic and chemical (DDEC) charges. Their basic idea is that the net atomic charges (NACs) are optimized to reproduce both the chemical states of atoms and the local electrostatic potential outside a material’s electron distribution. Due to the use of reference charge densities derived from the distinct oxidation states of isolated atoms,43 this approach can give the results that are chemically meaningful.214 For a system with a set of atoms {A} at the positions {RA} in a reference unit cell, an optimization functional is proposed as

Figure 3. Comparison of the calculated adsorption selectivities for CO2 over N2 at 298 K and 0.1 MPa in 15 MOFs based on the CBAC and QM-based charges. Bulk composition of the mixture is CO2:N2 = 15:85. Reprinted with permission from ref 36. Copyright 2012 American Chemical Society.

shows a comparison of the adsorption selectivities for CO2 over N2 in 15 MOFs calculated from the CBAC charges and QM charges. Evidently, the two sets of charges lead to very similar selectivities. These observations demonstrate that this CBAC scheme is accurate enough to considerably speed up the computational screening of a large series of MOFs for mixture separation. This could not be feasible in a reasonable period of time if one would have a preliminary requirement of determining DFT-based charges. However, the CBAC method also faces a drawback that it cannot be applied to a MOF if the atom types are missing in the CBAC charge database. 3.1.3. REPEAT Method. As described in section 2.1.1, ESP charges are usually fit to reproduce the electrostatic potentials on the grid points that lie outside of the vdW surface of the cluster models. However, when a system contains some deeply buried atoms, where an atom is considered buried if the minimum distance of this atom to the system’s vdW surface is larger than its radius,222 the ESP charges for these atoms can fluctuate widely and even sometimes have nonchemically intuitive values. In addition, conventional ESP-fitting procedures developed for molecular systems generally will not work for periodic solids, because the ESP in the periodic system is ill defined up to a constant offset at each spatial position.223 Considering these problems, Campaña ́ et al.223 first proposed a simple and robust method to derive ESP charges, which can be applied to both the molecular and the periodic systems including MOFs. These charges are termed repeating electrostatic potential extracted atomic (REPEAT) charges. Its basic idea is to introduce a modified error functional that acts on the relative differences of the electrostatic potentials instead of the absolute values. In this method, a new parameter (δϕ) is introduced in the conventional ESP-fitting error functional, which can overcome the fundamental problem of the ESP having an ill-defined nature of the reference state. A new RESP-like penalty function is also introduced to avoid wide fluctuations in the fitted N


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G = χFchem + (1 − χ )FESP +

∫U λ(r)Θ(r)d3r

methods outlined above implicitly assume that the charges assigned for MOF atoms are sufficient to reproduce the electrostatic interactions relevant to the adsorbate molecules.43 On the other hand, the vdW parameters required in molecular simulations must be fixed using mixing rules or by fitting to experimental/QM data based on preset charges. Due to the integrated contribution of vdW and electrostatic interactions,214 it is not possible to determine which set of charges is the most (or least) appropriate one using the calculation results alone. To avoid these ambiguities, Watanabe et al.214 recently demonstrated that it is not necessary to assign atomic partial charges for nanoporous materials including MOFs, in which the main idea is to directly implement the electrostatic potential energy surface (EPES) computed from periodic DFT calculations into classical molecular simulations. Such a treatment can rigorously include information from the full electron density of the periodic adsorbent. For a given MOF, the EPES obtained from the plane-wave DFT calculation is first tabulated on a finely spaced grid in its unit cell. From this pretabulated grid, the electrostatic energy of a charged adsorbate atom at an arbitrary position can be obtained via interpolation. Then, with the dispersion contribution defined by the generic force fields such as the UFF, the total potential energy between an adsorbate molecule and this material (Uadsorbate−MOF) can be calculated as

(19a)

where Fchem is a measure of the distance between ρA(rA) and ρref A (rA,nA) subject to the constraint Θ(r) = 0 (the former is the density of each atom in the interacting system, while the latter is the one of that atom in a noninteracting reference state with the same number (nA) of electrons), FESP is the information distance between ρA(rA) and its spherical average ρavg A (rA), and χ is the weighting factor. As noted in this study, for optimum performance, Fchem should be weighted much less than FESP so that Fchem only becomes important when FESP defines a shallow landscape (i.e., for systems containing buried atoms). The Lagrange multiplier λ(r) enforces the constraint Θ(r) = 0 in which Θ(r) is given by Θ(r) = ρ(r) −

∑ ∑ ∑ ∑ ρA (rA) k1

k2

k3

(19b)

A

The reference unit cell has k1 = k2 = k3 = 0, and the summation over A means over all atoms in the unit cell. For periodic system, ki ranges over all integers with the associated lattice vector νi and rA = r − k1ν1 − k2ν2 − k3ν3 − RA. Setting the first derivative of G with respect to ρA(rA) to be zero, the following expressions can be obtained wA(rA) = [ρAavg (rA)]1 − χ [ρAref (rA , nA)] χ

(19c)

Nsite

Uadsorbate − MOF(r) =

ρA (rA) = wA(rA)ρ(r)/∑ ∑ ∑ ∑ wB(rB) k1

k2

k3

B

j=1

(19d)

(20)

Starting from the neutral atom densities ρavg A (rA,0) as initial estimates, the DDEC charges are solved iteratively. In each iteration, the current estimate of {wA(rA)} is used to calculate {ρA(rA)} by eq 19d. After calculating nA = zA − qA (zA is the nuclear charge, and qA is the NAC), the new reference density is computed by ρAref (rA , nA) = (1 − f )ρAτ (rA) + fρAτ+ 1(rA)

∑ qjVe(rj) + Udispersion,adsorbate − MOF

where Nsite is the number of interacting sites in the adsorbate molecule, qj is the partial charge on the jth interacting site, r is the position of this molecule in the MOF framework, and Ve(rj) is the ESP at the corresponding position rj of the jth interacting site. By taking the results from the DFT-derived EPES as the correct representation of the electrostatic potential, Watanabe et al. systematically assessed the accuracy of the CBAC, REPEAT, DDEC, and Hirshfeld215 charge methods. They also compared the performance of these methods for describing CO2 adsorption in four materials: IRMOF-1, ZIF-8, ZIF-90, and Zn(nicotinate)2. It was shown that the magnitudes of the Hirshfeld charges are quite small compared to those obtained using the rest of the charge methods. The DFT EPES-based limiting adsorption properties (Henry’s coefficient and isosteric heat of adsorption) and isotherms are most accurately reproduced by the simulations using the REPEAT charges. However, through a detailed analysis of these charges on the influence of the vdW multiplier, they showed that there is little physical significance in the individual REPEAT charges. The CBAC and DDEC charges can be used to reasonably reproduce the isotherms, although there are relatively large deviations for calculating the limiting adsorption properties. This strategy may be the most realistic representation of the EPES in the nanoporous materials available today. It provides a completely rigorous description of the electrostatic potential defined by a MOF if the framework atoms are assumed to be nonpolarizable toward the adsorbed molecules.214 As claimed by Watanabe et al., this EPES method provides a practical way to assess the reliability of various charge assignment methods as well as to test those models of dispersion interactions with greater rigor. However, it is not suitable for MOFs with flexible frameworks, since it is impractical to perform DFT calculations

(19e)

where ρAref(rA,nA) is a linear interpolation between the spherically averaged ground state densities of isolated atoms that have the closest lower (τ) and higher (τ + 1) integer number of electrons. It should be noted that these atoms belong to the same element, and f = nA − τ.224 Finally, avg ρref A (rA,nA) and ρA (rA) are combined using eq 19c to generate an improved estimate of {wA(rA)}. This procedure is complete when the NACs for consecutive iterations differ by less than a predefined tolerance. As described by Manz and Sholl,222 since the DDEC charges are optimized to resemble a chemical reference state, they exhibit better transferability than the charges that are simply optimized to fit ESP. Thus, the DDEC charges are useful in development of force fields for complex materials, particularly for those periodic solids containing buried atoms. In addition, for a subset of MOFs that has been screened out from largescale examinations using approximation methods such as the EQeq or CBAC methods, this DDEC method can be employed to make charge assignments with higher quality for further investigations. 3.1.5. Electrostatic Potential Energy Surface Method. For molecular modeling with classical force fields, there is a long-standing issue: the continuous electron density of a real material cannot be exactly represented by a finite collection of atomic partial charges.214 However, the charge calculation O


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Figure 4. (a) Comparison between experimental (points) and simulated isotherms (full lines) for C3H6 adsorption in Cu−BTC at four temperatures, where the simulation results are scaled by a factor of 0.84 to account for the imperfect experimental sample. (b) Typical snapshot of C3H6 molecules adsorbed in Cu−BTC at 323 K and 5 kPa. Notice the central molecule adsorbing onto the open Cu atom. Reprinted with permission from ref 241. Copyright 2012 American Chemical Society.

popular approach. A so-called multiscale simulation strategy can then be employed to examine the properties of such MOFs.238 It has been shown that this approach can successfully capture the experimental observations related to the open metal sites with a high level of accuracy. Below, we describe some typical methodologies proposed to develop the open-metal-sites force fields for MOFs. 3.2.1. Specific Interactions−Isolation Method. To match the DFT-derived potential energy profiles for some gases interacting with the open Cu sites in MOFs, Fischer et al.239,240 initially tuned the vdW potential parameters between them. By comparing the simulation results with those obtained using the generic force fields, they found that the so-obtained force fields give a much better description of the experimental adsorption isotherms of these gases. These force fields were also shown to have a reasonable degree of transferability among open Cu-containing MOFs. Later, they further applied this strategy to study the adsorption of olefins in Cu−BTC.241 It was shown that there is a significant discrepancy between the simulated potential curve and the DFT-derived one. After a detailed analysis, this observation was attributed to the “double counting” of the repulsive contribution to the adsorption energy by the fitted potential models. To overcome the drawbacks, Fisher et al.241 proposed a more accurate DFTbased method to develop force fields for describing the specific interactions between olefin molecules and the open metal sites in MOFs. To avoid many complex factors brought about by other olefin molecules, C 2 H 4 was chosen as the simplest representatives of olefins. On the basis of representative cluster models cleaved from Cu−BTC, DFT calculations at the PBE/ DNP level are used to obtain the interaction energies between the C2H4 molecule and the open Cu sites as a function of their distance r. Due to the inability of DFT for describing the dispersion interactions, the dispersion contribution is assumed to be precisely zero. At the same time, the electrostatic interactions between the adsorbates and MOFs are also neglected. Therefore, the energies calculated by DFT are divided into two contributions: (i) the repulsive interactions and (ii) the specific interactions between the Cu atom and the π orbitals of the CC bond. The repulsive interactions are computed using the classical LJ potential with the Weeks− Chandler−Andersen (WCA) approximation

to assess the EPES during every step of molecular simulations. Further, it would become very computationally expensive to tabulate fine EPES grids for MOFs with very large unit cell sizes. 3.2. Open-Metal-Sites Force Fields

The metal nodes in some as-synthesized samples of MOFs are partially coordinated by solvent molecules. After appropriate activations, removal of these molecules creates coordinatively unsaturated metal sites (also called open metal sites). It has been shown that these sites can promote the interactions of MOFs toward polar molecules,225 thus remarkably enhancing their gas storage capabilities and separation performance. Although the generic force fields such as UFF or DREIDING do surprisingly well for many prototypical MOFs,27,84,95,101,111,140 they fail to correctly account for the interactions of the fluids with MOFs featuring open metal sites.67,226−228 In addition, even for CH4, which is usually considered to be highly symmetric and nonpolar, both experiments and theoretical calculations have observed that these sites have nonnegligible influence on its adsorption in some MOFs at low temperatures.229,230 Such an influence is especially evident in the materials like M2(dhtp) (M = Mg, Mn, etc.) (alternatively labeled MOF-74(M),231 CPO-27(M),232 or M/DOBDC233) with high densities of open metal sites.234 In the literature, two distinct approaches have been used to develop the open-metal-sites force fields for MOFs. One approach is to combine the generic force fields with simplified models that rely on empirically refitting the related vdW potential parameters92,235 or scaling the partial charges of the framework atoms of MOFs.236 Although the pure-component experimental isotherms are reasonably reproduced, such reparameterized models are highly dependent on the quality of samples, which can vary widely under different conditions of synthesis and/or activation. Further, the selectivities have also been shown to be significantly underestimated when applying them to explore mixture adsorption behavior.237 Recent studies have also pointed out that the ability of this approach to correctly capture the realistic physics of the system and the transferability of the parameters is questionable.155 Due to the strong affinity of open metal sites toward polar molecules, the interaction distances between these sites and the atoms in the molecules are often much less than their combined hard-sphere (or vdW) diameters. Thus, using QM methods to develop force fields has become an increasingly P


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Figure 5. (a) CH4 adsorption isotherms in Cu−BTC at 77 K. (b and c) Contour plots of the potential energies between a CH4 molecule and the Cu−BTC framework on the (200) plane for DFT/CC-PES and UFF, respectively. Adsorption sites B and C (at the open Cu sites) are labeled. White space corresponds to the Cu−BTC framework. Reprinted with permission from ref 242. Copyright 2011 American Chemical Society.

⎧ r < rmin ⎪ULJ(r ) − ULJ(rmin) URep(r ) = ⎨ ⎪ r ≥ rmin ⎩0

where there are not any dispersion interactions, and generic force fields correctly describe the adsorbate−MOF dispersion and repulsion interactions. These mean that using more accurate QM methods to obtain the dispersion interactions would be a valuable addition to this method. Moreover, for many other adsorbate molecules (such as water), the electrostatic interactions between them and MOFs can play an important role in explaining the adsorption behavior involved. Therefore, explicitly taking into account the Coulomb potential in the developed open-metal-sites force fields is essential for the general applicability of this method. 3.2.2. DFT/Coupled-Cluster Host−Guest PES Method. Similar to the idea of the EPES method described in section 3.1.5, Chen et al.242 recently suggested directly implementing the potential energy surface (PES) of a MOF with open metal sites into classical molecular simulations. On the basis of the hybrid DFT/coupled-cluster (CC) calculations, the derived host−guest PES can be used to account for the adsorbate− MOF potential energy contributed from both the vdW and the electrostatic interactions. It can remove the ambiguity and inaccuracy that are caused by the classical force fields. To illustrate this DFT/CC method, adsorption of CH4 in Cu−BTC was selected by Chen et al. as an example. Since CH4 is a nearly spherical molecule, the orientation dependence of the CH4−MOF potential energy can be neglected, thus significantly reducing the computational costs. The CH4− framework PES is represented on a pretabulated threedimensional grid for the unit cell of Cu−BTC. After placing the carbon atom of a single CH4 molecule on a specific grid point with random configuration, the CH4−framework potential energy is calculated at the DFT/CC level from

(21a)

where rmin is the distance at the minimum point of the LJ potential. By subtracting the repulsive contribution from the DFT-derived interaction energies, an estimate of the actual (fully attractive) Cu−π interaction profile is obtained. Then, this profile is fitted by the following continuous function ⎧ ⎡ ⎛ ⎪ r ⎞⎤ UCu − π(r ) = D0⎨ exp⎢α⎜1 − ⎟⎥ ⎪ R 0 ⎠⎥⎦ ⎩ ⎢⎣ ⎝ B ⎫ ⎡α⎛ r ⎞⎤⎪ ⎛⎜ A ⎞⎟ − 2 exp⎢ ⎜1 − + ⎟⎥⎬ ⎪ ⎝r⎠ ⎢⎣ 2 ⎝ R 0 ⎠⎥⎦⎭

(21b)

where the first term on the right is the Morse potential used to describe the underlying attractive well while the second term is the power law used to capture the monotonically decreasing character of the resulting curve. In this equation, R0 is the distance corresponding to the position of the minimum of the Morse potential, D0 is the energy value at that minimum, α is a stiffness parameter that adds flexibility to the Morse function, and A and B are empirical parameters. It should be pointed out that the reference point for this specific interaction with the open Cu sites is the midpoint of the CC bond. By incorporating the obtained specific interaction potential into GCMC simulations, Fisher et al. predicted the adsorption of C3H6 in Cu−BTC. It was found that there is excellent agreement between the simulated adsorption isotherms and experimental data at all temperatures examined (Figure 4a). The microscopic adsorption mechanism is also well captured by simulations, as evidenced from a representative snapshot shown in Figure 4b. Olefin molecules are indeed adsorbed close to the open Cu sites in Cu−BTC with preferentially aligning their CC bonds perpendicular to the Cu−Cu axis, which corresponds to the most favorable orientation observed in the DFT calculations. Compared to the previous approach adopted by Fisher et al.,239,240 the major improvement of this new method lies in a more realistic and consistent consideration of the different contributions to the adsorption energy, which allows one to isolate the contribution from the specific metal−adsorbate interaction. It is expected that such a treatment can also be applicable to other MOFs with open metal sites. However, this method still bears several assumptions, such as DFT calculations correctly account for the specific Cu−π interactions

E int = E(Cu − BTC/CH4) − E(Cu − BTC) − E(CH4) + ΔE DFT/CC

(22a)

where E(CH4), E(Cu−BTC), and E(Cu−BTC/CH4) are the total energies of the isolated CH4 molecule, Cu−BTC framework, and CH4/Cu−BTC system, respectively, calculated at the DFT/PBE level with the periodic model and ΔEDFT/CC is the DFT/CC energy correction. The interaction potentials are evaluated on the equally spaced grid points, where the grid points in close contact with the framework are excluded. On the basis of these values, a refinement of the grid is performed by introducing a denser grid and taking into account the symmetry of the individual sites. Finally, to obtain the grid needed in GCMC simulations with desired grid spacing, a three-dimensional linear interpolation is used to determine the Q


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developed a novel computational approach to obtain accurate force fields for MOFs with open metal sites. The unique feature of this approach is based on a so-called nonempirical model potential (NEMO) methodology, which decomposes the total interaction energy obtained from Møller−Plesset second-order perturbation theory (MP2) calculations into the contributions from electrostatics, repulsion, dispersion, and polarization. Such an important treatment allows the potential parameters in the proposed force-field expressions, which closely match the terms from the NEMO decomposition, to be separately fitted to reproduce the MP2 calculation results. In this approach, the number of distinct atomic types is first identified for a MOF based on its symmetry (here 9 for MOF74(Mg)), from which a representative cluster is constructed for each of them (excluding the H atomic type). Each cluster should accurately reflect the local chemical environments of the targeted atomic type in the MOF. Then, one path composed of a set of configurations of a CO2 (or N2) molecule is generated for each cluster. For each configuration along each path, CO2 (or N2) approaches the specific atomic type in such a way that the total repulsive energy is dominated by the repulsive interactions between the guest molecule and the atoms of that atomic type. The separating distance (r) is defined as the one between the atomic type of interest and the nearest oxygen of CO2 (or nitrogen of N2). For each generated configuration on a given path, a MP2 calculation is performed. On the basis of the results, NEMO decomposition is utilized to get the individual components of the total interaction energy (Eint), i.e., the electrostatic (Eelec), polarization (Epol), and dispersive (Edisp) energies. The repulsive energy (Erep) is calculated as

potential energies between CH4 molecule and the Cu−BTC framework. The DFT/CC scheme is based on the pairwise representability of the DFT error (ΔE) which is defined as ΔE = ECCSD(T) − E DFT

(22b)

where ECCSD(T) and EDFT are the interaction energies calculated at the CCSD(T)/CBS and DFT/AVQZ levels, respectively. Within the DFT/CC method, ΔE is expressed as the sum of the atom−atom correction functions εij ΔE =

∑ εij(R ij) ij

(22c)

where Rij is the distance between atoms i and j and εij is obtained from a reproducible kernel Hilbert space interpolation. The correction functions are assumed to be transferable between the reference system and the real system of interest. Then, PBE/AVQZ and CCSD/CSB calculations are conducted on a series of reference sets. The values of εij for different atom pairs between CH4 and Cu−BTC are evaluated using the onedimensional potential energy curves obtained for a series of reference complexes. On the basis of their DFT/CC-based PES, Chen et al. performed GCMC simulations to study the adsorption of CH4 in Cu−BTC. They found that the shape of the isotherms, including the step and the maximum uptake, are captured accurately when compared to the available experiments, as shown in Figure 5a. Most importantly, the interaction of CH4 with the open metal sites is also correctly described, while it cannot be well captured by the generic force fields such as the UFF (Figure 5b). It was also found that neither pure DFT nor the empirical dispersion-corrected DFT-D approaches (DFTD2243 and DFT-D3244) are able to generate an EPES that can lead to correct prediction of the adsorption isotherm. Actually, similar observations have already been found in the previous study reported by some authors of this work.245 Later, an accurate description of CO2 and CO interactions with the different adsorption sites in Cu−BTC was also achieved on the basis of the DFT/CC calculations.246,247 Obviously, the above DFT/CC host−guest PES approach has the advantage over other methods based on classical force fields. Nevertheless, it remains unclear how it can be extended to study the adsorption of more complex adsorbates in MOFs. Similar to the limitation of the EPES method, this approach is also only applicable to MOFs with rigid frameworks. On the other hand, because of the structural diversities between different MOFs, it is hard to transfer the generated PES from one material to another. This means that expensive ab initio calculations are required to determine the PES for each new MOF. In contrast, force fields with mathematic representations for the potential are more convenient to use and are transferable to other systems. In light of this reason, Chen and co-workers248 recently parametrized a force field for this system using the above-derived PES in connection with a genetic algorithm. They also developed a force field suitable for describing the interaction of CO2 with open Mg sites. More encouragingly, their newly derived force-field parameters for CH4 in Cu−BTC have been shown to exhibit a degree of transferability to PCN-14, another MOF with similar open Cu sites. 3.2.3. Nonempirical Model Potential Decomposition Method. Aiming at examining the CO2 capture performance of MOF-74(Mg) from flue gas, Smit and co-workers249 recently

Erep = E int − Eelec − Epol − Edisp

(23a)

Once these interaction energies in all the paths are available, they are grouped for fitting the new force field consisting of repulsive Urep(r), attractive Uatt(r), and electrostatic Uelec(r) potentials U (r ) = U rep(r ) + U att(r ) + U elec(r )

(23b)

where a modified Buckingham potential is used to fit the repulsive energy Erep as given by ⎧∞ r < rmin U rep(r ) = ⎨ ⎩ A 0exp( −B0 r ) r ≥ rmin ⎪

⎪

(23c)

The attraction potential used to fit the total attractive energy, Eint − Erep − Tij(qiqj), is given by

U att(r ) =

C D + 6 5 r r

(23d)

ij

where T is the interaction energy tensor for the electrostatic term and qi and qj are the atomic partial charges. A0, B0, C, and D are the parameters to be fitted. The electrostatic part is calculated using the standard Coulomb potential with the charges estimated by a LoProp scheme. The charge of each atomic type in its representative cluster was directly transferred into the periodic system. The total charge of the MOF framework was balanced to be zero by assigning suitable charge to the H atomic type. Due to the small contribution to the total interaction energy, the H atomic type is not explicitly considered in the parametrization and the standard UFF parameters are adopted. Because the O atoms of CO2 dominate the interactions of CO2 with the MOF, UFF is also adopted for the C atom in CO2 molecules. R


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Figure 6. Comparison of simulated and experimental adsorption isotherms (a) and Henry’s coefficients (b) of CO2 in the MOF-74(Mg). Reprinted with permission from ref 249. Copyright 2012 Nature Publishing Group.

To confirm the reliability of the force field developed, Smit and co-workers conducted GCMC simulations to calculate the heat of adsorption, Henry’s coefficients, and adsorption isotherms of CO2 and N2 in MOF-74(Mg) with a perfect crystalline structure. They found that there is excellent agreement between the simulation results and the experimental data. Such agreement is best when they took into account the experimental evidence that not all Mg sites are accessible. In contrast, in the Henry regime, there is a significant deviation from UFF. Figure 6 shows an example of the comparison between simulated and experimental results for adsorption of CO2. Moreover, by refitting the potential parameters of the new additional atomic types, Smit and co-workers also demonstrated that their methodology is transferable to the isostructural materials of MOF-74(Mg) with different metals or organic ligands as well as other MOFs without open metal sites. This robust methodology for yielding accurate force fields provides a very useful foundation for predicting the adsorption properties of new MOFs with open metal sites, which in turn will help the design of suitable capture materials toward the application of interest. However, even with such elaborate procedures for fitting the potential parameters, it still remains challenging to create force fields for materials with low-lying unfilled electron shells because they can be highly correlated and/or induce chemical bonding.250 For such systems, the MP2 description is thus not adequate, such that even higher level QM calculations should be conducted, for which it would require more computational resources.

Figure 7. Definition of the accessible surface area. Reprinted with permission from ref 192. Copyright 2004 American Chemical Society.

particle, respectively. Then, the probe particle is placed on each point to test whether it overlaps with other atoms of the structure. For each pair of framework atom and probe particle, overlap is considered to have occurred if their distance is smaller than the diameter of a hard sphere related to them. Then, the accessible surface area (Sacc,i) associated with the targeted framework atom is calculated as Sacc, i =

Mi ,no − overlap Mi ,total

·π ·σi2

(24a)

where Mi,no‑overlap is the number of the random points without overlap and Mi,total is the total number of random points generated. Finally, the overall accessible area (Sacc) is given by

3.3. Structural Characterization

N

Sacc =

3.3.1. Accessible Surface Area. Surface area is one of the important structural properties for characterizing MOFs. Considering their well-defined crystalline structures, Düren et al.192 proposed a simple MC integration method to calculate the accessible surface areas of MOFs which is purely based on their geometrical topology. By rolling a spherical probe particle over the framework surface of a MOF, the accessible surface area is defined by the locus of the positions of the center of mass of this particle, as shown in Figure 7. In these authors’ method, the framework atoms of the MOF are examined in sequential order.195 The sizes of these atoms can be taken from the widely used generic force fields such as the UFF or DREIDING. Around each targeted framework atom i, many random points are generated on the surface of a hard sphere with a diameter of σi = σa + σp, where σa and σp are the collision diameters of this framework atom and the probe

∑ Sacc,i i=1

(24b)

where N is the total number of framework atoms. Using the structural properties of the material examined, such as its skeleton density and unit cell volume, this absolute value can be converted into a more meaningful one. It is worth noting that to fairly compare accessible surface area with experimental BET surface area it is necessary to choose a probe particle with a size in line with the gas used in the experimental measurement. Because experimental BET analysis is in most cases derived from N2 adsorption at 77 K, making a nitrogen-sized probe particle for the accessible surface area calculations consistent with the findings.251 The accessible surface area provides a useful structural property for judging the quality of a synthesized sample, giving a benchmark for the theoretical upper limit for the surface area S


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large differences between them for some materials with smaller pores. In a follow-up study, Düren et al.255 argued that the Langmuir surface area is not physically meaningful and should be avoided for materials with pores large enough to support multiple layers. Later, Bae et al.253 used the same approach to examine the applicability of BET surface area for MOFs and zeolites that contain ultramicropores ( 0). Then, the so-obtained value of Vm can be used to calculate the BET surface area in units of m2/g by SBET =

VmS0Na VSTP

(25b)

where Na is Avogadro’s number, s0 is the cross section area (16.2 Å2) of a N2 molecule at the liquid state, and VSTP is the molar volume of N2 at standard temperature and pressure (273 K, 1 atm), i.e., VSTP = 2.24 × 104 cm3(STP)/mol. Using the above two established consistency criteria to identify the appropriate pressure range, Walton and Snurr found that the estimated BET surface areas surprisingly agree very well with the accessible surface areas for a series of microporous IRMOFs. In contrast, when using the “standard” BET pressure range (0.05 < P/P0 < 0.3), there are relatively

Vacc = Msuccess × δV = f × Vbox T

(26)


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where Msuccess is the times of successful insertions, Vbox is the volume of the simulation box, δV is the differential volume, δV = Vbox/M (the space of the simulation box is sampled with equal probability), and f is the fraction defined by f = Msuccess/ M. By replacing the quantity Vpore in eq 7 with Vacc, the excess amount can become always positive. In addition, for a given porous material, the accessible volume is dependent on the properties of the adsorbate molecule used in the calculations, that is, this volume depends on the size of adsorbate molecule, the adsorbate−solid interaction potential, and the average orientation of the adsorbate molecule in the adsorbent at a given temperature. 3.3.5. Accessible Pore Size Distribution. Following the definition of accessible volume, Do et al.256 also proposed a new concept of accessible pore size for MOFs as well as for other porous materials. It is a local quantity that depends on the locations inside the accessible volume and thus can have a lower limit of zero. To obtain this quantity, a Tri-POD method was developed. Specifically, if the adsorbate molecule is randomly inserted into the simulation box at point A and the solid−fluid potential energy is nonpositive, this insertion step is considered to be successful. Then, it starts to search for the largest sphere that encloses this point and rests on three closest framework atoms. Any molecular probe residing inside this sphere will have a nonpositive potential energy. Supposing that the diameter of this sphere is Dj, the volume associated with this pore size is incremented by δV (see the definition in section 3.3.4). By repeating the above process M times, the pore size for each successful insertion is determined using the same procedure. Then, the accessible pore volume corresponding to the pore size Dj is simply calculated by Vacc, j = (Mj /M ) × Vbox

are mainly two pore sizes of 8.7 and 12.3 Å. By including the combined collision diameter (3.1 Å) between the argon and the hydrogen atom in the material, Do and co-workers found that the accessible pore sizes (11.8 and 15.4 Å, respectively) are in good agreement with TEM image analysis. It should be noted that they also presented the procedure for calculating the accessible geometrical surface area, which is defined as the area of the boundary of the accessible volume.257 Unlike the accessible volume that is a macroscopic quantity, the accessible pore size is a local variable that reflects the different sizes in different parts of the pores. Since it is determined within the accessible volume, the calculated APSD will also depend on the choice of probe particle. In addition, this method is consistent in calculation of accessible pore volume, pore size distribution, and accessible surface area, thus offering a single and robust theoretical framework for characterization of porous solids. 3.3.6. Largest Cavity and Pore Limiting Diameters. The pores in the periodic crystalline structures of MOFs show a great variety of shapes and connectivities. For a certain application, before performing extensive studies, it would be helpful if we can screen out a small number of MOFs with top performance from a large variety of candidates. In regard to kinetic separation, it would be very important to identify the critical factors that control diffusion of adsorbates in MOFs. Therefore, aiming at quantifying the pore features of a structure, Sholl and co-workers40 introduced an efficient computational method to characterize the pore-limiting (PLD) and largest cavity (LCD) diameters for materials with rigid frameworks, as shown in Figure 9. The PLD, also referred to as the maximum free diameter,258 is defined such that it is impossible for any hard sphere with a larger diameter to travel through this structure without overlapping one or more framework atoms. In contrast, the LCD, also called the maximum fixed diameter, is defined as the largest spherical particle that can be inserted into the pore of a structure without overlapping with any framework atoms. For some porous MOFs having multiple pore channels with limited interconnectivity, the pore containing the PLD is labeled the major pore, as depicted in Figure 9b. The largest cavity in the structure, denoted as the global cavity diameter (GCD), is not necessarily equivalent to the LCD along the major pore because it could be outside this pore in some situations. To obtain the above quantities, a hard sphere is used by Sholl and co-workers as the probe molecule and the unit cell of a MOF is divided into discrete grid points equally spaced along its three crystallographic axes. At each grid point, the distances from its center to all the framework atoms are calculated so as to search the maximum possible probe size on the basis of no overlaps with the framework atoms. The largest size observed for any grid point in this calculation defines the material’s LCD (or GCD). To determine the PLD, the connectivity of pores within the material is first established using the Hoshen− Kopelman (HK) algorithm.259 The calculation starts by recording a list of grid points at which probe particles with a given size could be located without overlapping framework atoms. Then the HK algorithm is used to identify the clusters formed by the grid points recorded in this list, where two grid points are assigned to the same cluster if they are adjacent under the periodic boundary conditions. Once finishing the cluster-labeling procedure, the resulting clusters are used to detect whether they form the so-called spanning clusters of connecting the grid points into a

(27)

where Mj is the number of successful insertions that have a pore size of Dj. The total accessible pore volume is a summation of all individual accessible volumes that correspond to all pore sizes. At the end, the accessible pore size distribution (APSD) is obtained from the histogram plot of the accessible pore volume Vacc,j against the corresponding accessible pore size Dj. In a subsequent article, this method was further improved and described in more detail by the same group257 and also used to analyze the APSD of IRMOF-1, IRMOF-10, and IRMOF-16. Figure 8 shows an example for the results obtained for IRMOF-1 with argon as the probe particle. Obviously, there

Figure 8. Plot of the accessible pore volume versus accessible pore size obtained for IRMOF-1 with argon as a probe particle. Reprinted with permission from ref 257. Copyright 2011 Springer. U


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Figure 9. (a) Two-dimensional schematic representation of a periodic nanoporous channel system. Pore-limiting (PLD) and largest cavity (LCD) diameters have been highlighted in one of the unit cells. (b) Schematic illustration of a single unit cell of a material where the global cavity diameter (GCD) is outside the major pore, which has been colored light gray. Reprinted with permission from ref 40. Copyright 2010 American Chemical Society.

features of MOFs. It is an extension and improvement of the method proposed previously by this group for full threedimensional characterization of zeolites.262 This novel approach can be used to automatically identify the portals, channels, and cages of MOFs as well as their connectivity. Their approach starts from identification of the chemical bonds among all framework atoms in a unit cell. Then, the framework of the targeted material is represented as a threedimensional periodic molecular graph, where each vertex is an atom and each edge is a chemical bond. Within this so-obtained molecular graph, paths of increasing length are constructed until they can be closed into cycles. Portals are defined to be “isometric” cycles for which there exists no short sequence of atoms cutting across the middle. This property is checked incrementally for each path to prune paths incapable of leading to a portal. After identification of portals, it begins to search the channels that link portals together. For a pair of portals, a nonlinear optimization model (Model 1 given in this work) is used to find the largest cylinder void of any framework atoms with end points on the planes of the portals. On the basis of the vdW radii of the framework atoms, the radius of the cylinder is maximized subject to none of these atoms overlapping the interior of the cylinder. Pairs of portals are discarded when they are not linked by a void cylinder with sufficiently large radius (greater than the vdW radius of hydrogen). Overlapping portals are merged, and the end points of all channels at each portal are aligned using a modified nonlinear optimization algorithm (Model 2). Cages in the structure are identified by making use of the three-dimensional Delaunay triangulation263 to find the large void spheres in a set of points space. This technique has also been used to analyze the structural properties of a large database of hypothetical MOFs.264 In the current approach, the radius of each sphere is reduced by the maximum vdW radius of its four constraining atoms as well as by avoiding overlap with any other framework atoms. Cages are defined by the union of several overlapping spheres to more accurately describe their shapes in a way that they have a radius at least 20% larger than that of any channel overlapped by them. The minimum radius for isolated cages is 20% larger than the vdW radius of hydrogen. To determine the connectivity between the channels and the cages, calculations are first conducted to find the intersections between the underlying geometric shapes that describe the channels and cages (that is, cylinders and spheres). Then, a connectivity graph for the MOF is constructed where each vertex is a junction (defined as a portal, cage, or channel

continuous diffusion path through the crystallographic unit cell in any of the dimensions. The PLD corresponds to the largest probe sphere diameter that is found to result in at least one spanning cluster. During the above procedure, the maximum cavity size within the accessible regions can also be obtained in the case that the largest cavity defined above does not fall within the major channel. Further, by visualizing the spanning cluster associated with the PLD, it offers a simple way to see the pores accessible to diffusing molecules. Such visualization can be used to locate and block the regions in a material that are inaccessible to guest molecules. This would be very useful when performing GCMC simulations, as done by this group in the study of rare gas separations in some ZIFs.42 Recently, Sarkisov and Harrison195 developed a similar method but used a rigid, nonspherical particle as the probe molecule. Its important feature is that the orientation of a probe molecule is explicitly taken into account for identifying the grid points at which this probe molecule could be located without overlapping framework atoms. To achieve this treatment, within each step in the outer cycles for examining all the grid points, another calculation with a certain number of cycles is carried out for each grid point. In each step of these inner cycles an orientation of the molecule is randomly selected. If a configuration is found featuring no overlaps with the framework atoms, the grid point currently being examined is recorded in the list of the grid points used for identifying the clusters; otherwise, it is excluded from all other considerations. The rest of the procedures are analogous to those described above. Similar to the method of Sholl and co-workers, this improved approach in essence is still based on purely geometric criteria. As suggested by Sarkisov and Harrison,195 a more accurate approach would be based on the free energy map of the porous space: considering systematic variation of the molecule orientation, interactions between the molecule and the structure, as well as the flexibility of the molecule. On the other hand, the LCD of a MOF reflects that an adsorbate molecule with size similar to this diameter will typically not be able to diffuse through the more constricted regions of the material’s pores.260 In contrast, the PLD characterizes the width of the bottlenecks that control what molecules can and cannot diffuse into the pores. Therefore, knowledge of these quantities can be used to estimate the potential of a material for a specific kinetic separation. 3.3.7. Method to Identify Portal, Channel, Cage, and Their Connectivity. Complementary to the methods introduced above, First and Floudas261 recently developed an automatic computational approach to characterize the pore V


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unique position. Therefore, a great deal of attention has been paid to developing various new theoretical tools/methods to explore the physicochemical and mechanical properties of such subsets of MOFs. The effort has greatly promoted the progress in evaluating and discovering the potential applications of MOFs for selective gas adsorption/separation. The following will introduce some advances in this field. 3.4.1. Flexible Force Field Method. The breathing/gateopening phenomena occurring in flexible MOFs usually take place at a very fast speed. Though experimental techniques can detect the signatures related to them, it is hard to elucidate the underlying microscopic mechanism in play. In this respect, molecular simulations can play an important role by enabling identification at the molecular level of the key factors that govern the observed structural transitions. For such simulations, one crucial prerequisite is the suitable flexible force fields that can be used to describe accurately the interactions within the whole system. Following the pioneering studies271,272 and overcoming the drawbacks of the force field proposed by Coombes et al.,273 Salles et al.274 first developed a robust force field for breathingtype MOFs in which MIL-53(Cr) is chosen as a representative. This material is built from chains of corner-sharing CrO4(OH)2 octahedra linked by terephthalate groups. It can display a reversible structural switching between the narrow (NP) and the large (LP) pore forms with a cell volume change up to 40%.265 In this new force field, the nonbonded interactions are calculated using eq 5 and Mulliken charges are used to calculate the electrostatic interactions. For the organic ligand, bond stretching and bending are described by a simple harmonic potential function, while both the bond torsion and the improper torsion are expressed by a periodic cosine potential function as given by

intersection) and each edge is a segment of channel linking the junctions together. To validate the above approach, First and Floudas performed a full characterization of the structures of the existing and hypothetical MOFs taken from several databases. Figure 10

Figure 10. Main pore system of ZIF-8. Channels (green) and cages (blue) are represented as unions of cylinders and spheres, respectively. Framework atoms are rendered as “ball-and-stick” where each atom is a small ball colored by type and each bond is a stick. Reprinted with permission from ref 261. Copyright 2013 Elsevier.

shows an example of the pore features obtained for ZIF-8. From the characterization results, they also calculated a variety of structural properties of the MOFs, including the PSD, accessible surface area, and volume as well as the PLD and LCD, giving deep insights into the gas storage and separation properties of the MOFs. This approach is the first computational characterization method that can be used to automatically identify a full threedimensional landscape for the pore architectures of MOFs. As can be seen from above, it proceeds with the crystallographic structure of a material as the only input instead of dividing it into arbitrary grid points. Identification of portals, cages, channels, and their connectivity is independent of the guest molecule. The channels are also allowed to be oriented in any direction due to the use of portal linkage. As claimed by First and Floudas, one application of this pore characterization method is that it can serve as a basis for more advanced investigations, such as to study the shape-selective separation and reaction selectivity within MOFs.

Uijbond =

1 kij(rij − r0)2 2

(28a)

Uijkbend =

1 kijk(θijk − θ0)2 2

(28b)

Torsion Uijkl = kijkl[1 + cos(nϕijkl − ϕ0)]

(28c)

where kij, kijk, and kijkl are the force constants for different interactions, r0 is the equilibrium bond length between atoms i and j, θ0 is the equilibrium angle involving atoms i, j, and k, n is the periodicity, ϕ is the dihedral angle, and ϕ0 is the factor phase. For the inorganic node, Cr−O bond stretching is described by eq 28a. Apart from a bond bending term, an additional torsion term is included for interactions between the inorganic and the organic parts. In addition, the nonbonded interactions between the atoms separated by exactly three bonds are described using the LJ potential. The organic moiety is treated with the consistent valence force field (CVFF)275 for both the intramolecular and the nonbonded LJ interactions. The other potential parameters are fitted empirically to reproduce the structural features of both empty NP and LP forms via energy minimization procedures. To validate the developed force field, Salles et al. conducted MD simulations for this material at 300 K. It was shown that this force field well reproduces the experimental structures of both the empty NP and the LP forms. The simulation results also successfully capture the experimentally observed two-step structural switching (from LP to NP and then NP to LP) induced by CO2 adsorption, as shown in Figure 11. From these

3.4. Breathing/Gate-Opening Description

In addition to a wide variety of chemical and structural features of MOFs, another fascinating property is the dynamic structural flexibility. The flexible frameworks of some MOFs can exhibit many unusual phenomena, such as “breathing”265 and “gateopening”266 behavior, upon adsorption of guest molecules267 or other stimuli, including pressure and temperature.268,269 The so-called breathing means a structural change with a drastic expansion/shrinkage in the cell volumes of MOFs, while the gate opening typically involves an abrupt structural transition of the materials from a nonporous to an open porous phase when passing a certain threshold (for instance, pressure). One of the common features associated with this peculiar behavior is the unusual adsorption isotherms with the existence of steps.270 Such intriguing properties open the way to an extreme richness of host−guest chemistry, putting this class of materials in a W


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proposed by Maurin and co-workers,107 that is, a flexible MOF is treated as a heterogeneous mixture of crystallites that has different structural phases during adsorption of gases. The phase mixture composition is considered as a function of pressure, and then this phase mixture model accounts for the overall adsorption process at each pressure by summing the contributions of all structural forms. A simple variation of this method was first adopted by Maurin and co-workers,284 initially proposed by Snurr et al.285 for studying adsorption of aromatic hydrocarbons in silicate, to investigate the structural transition of MIL-53(Al) upon CO2 adsorption. This material can exhibit similar breathing behavior as MIL-53(Cr). The authors’ method begins with GCMC simulations to obtain adsorption isotherms of CO2 at 300 K in both the NP and the LP forms of MIL-53(Al) with rigid structures. Then, the composite isotherm is constructed by following the curve in MIL-53(Al)-NP up to a transition pressure measured experimentally and then switching to the one in MIL-53(Al)-LP. Such obtained “composite” isotherm displays the experimental characteristic step with only slightly underestimating the loading before the transition of NP to LP as well as slightly overestimating the saturation amount. This simple approach has also been used to study the gate-opening behavior of ZIF-8, which is caused by a swing effect of the imidazolate ligands upon the adsorption of guest species.286−288 On the basis of the simulation results for the two rigid structures of ZIF-8 at ambient and high pressures (referred as ZIF-8AP and ZIF-8HP, respectively),289 the constructed composite isotherms were found to be in line with the experimental measurements for many gases.286,290,291 However, the step in the composite isotherm of CO2 in MIL53(Al) constructed above is too steep compared to the one observed experimentally. At the same time, it only considered one type (NP → LP) of structural transition. In contrast, the experimental results of Hamon et al.292 showed that both LP and NP forms can coexist in the two transition regions (LP to NP and then NP to LP) that appear at low- and intermediatepressure ranges, respectively. Therefore, in order to more accurately analyze the experimental adsorption isotherm, Maurin and co-workers107 put forth an analytical phase mixture model to build the composite isotherm using

Figure 11. Evolution of the unit cell volume of MIL-53(Cr) as a function of the CO2 loading calculated at 300 K. Filled circles represent simulated results and open circles are the experimental data. Reprinted with permission from ref 274. Copyright 2008 Wiley-VCH.

good agreements, they further explored the microscopic mechanisms associated with the breathing phenomena. It was found that there is a much larger free rotation of the phenyl rings during the breathing. The structural switching is predominantly governed by a change of the dihedral angle consisting of four atoms: metal Cr, the oxygen and carbon atoms in the carboxylic group, and the carbon atom in the phenyl ring connecting to the carboxylic carbon atom. Following this work, similar force fields with some variations have also been developed for the same material276−278 and several other highly flexible MOFs.279,280 The potential parameters in some force fields are obtained with more accurate QM calculations instead of empirical adjustments. Because the structural transition of NP to LP forms can be considered as a certain type of gate-opening process, similar strategies can also be applied to build flexible force fields for the in-depth study of the gate-opening phenomena that occur locally in some typical MOFs including ZIF-8.281−283 Currently, a large number of interesting MOFs with highly flexible frameworks have been synthesized whereas the related flexible force fields are still very limited. In addition, the flexible force field is usually MOF dependent, and a general one applicable to many MOFs is missing. Thus, it is quite necessary to put more effort into this aspect of MOF research, which would in turn provide more comprehensive molecular-level insights into the breathing/gate-opening mechanisms, stimulating rational design of new MOFs with improved performance. 3.4.2. Phase Mixture Model Method. In addition to the flexible force field method, another approach to describe the flexibility of MOFs is the so-called “composite” method

sim sim N (P) = XNP(P)NNP + [1 − XNP(P)]NLP

(29)

where P is the pressure, N represents the total amount adsorbed in the MOF, XNP and XLP are the fractions of the NP sim and LP forms (XLP = 1 − XNP), respectively, and Nsim NP and NLP

Figure 12. (a) Fraction of the NP (solid line) and LP (dashed line) forms of MIL-53(Cr) as a function of the pressure (semilogarithm plot). (b) Comparison of the simulated “composite” isotherm (open cycles) of CO2 in MIL-53(Cr) at 300 K based on the phase mixture model with the corresponding experimental data (filled circles). Reprinted with permission from ref 294. Copyright 2010 American Chemical Society. X


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factorizing the thermodynamic potential and configurational partition function into two contributions: one characterizing the framework structures themselves (i.e., the free energy), and the other describing fluid adsorption in each metastable phase involved in the transition. In this model, for a given phase k of the host structure at temperature T, the pressure-dependent profile of the osmotic thermodynamic potential (Ωos k ) upon pure-component adsorption is expressed by

correspond to the amounts adsorbed in the NP and LP forms, respectively. The values of XNP and XLP can be determined using experimental techniques, as done by Salles and coworkers.293 Alternatively, Ghoufi and Maurin294 proposed a computational approach to determine these quantities. It starts with performing m times of GCMC simulations for each form (LP and NP) with rigid structure in the pressure range of [P1, Pm] with a step of (Pm − P1)/(m − 1). It is thus possible to sim compute Nsim NP and NLP at all of the pressures examined. Then, a set of m NP fractions (as many GCMC simulations as XNP fractions) is further chosen in the range of [0, 1]. For each of them at the given pressure m possible values of N are calculated using eq 29. Considering the shape of the experimental isotherm in the regions of structural transition, the evolution of N is assumed to be linearly increasing with increasing pressure. Finally, by comparing the simulation results with the corresponding experimental data, the value of XNP at each adsorption pressure can be obtained using the interpolation method. On the basis of the proposed computational approach, Ghoufi and Maurin employed eq 29 to extract the fractions of the NP and LP forms of MIL-53(Cr) using the experimental adsorption isotherm of CO2 at 300 K, as shown in Figure 12a. Obviously, this figure distinctly points out that there are two structural transitions (LP → NP and NP → LP). It also unambiguously demonstrates that there is a phase mixture of NP and LP at both very low- and intermediate-pressure ranges. On the basis of these data, the constructed composite isotherm reproduces very well the experimental data, as evidenced from Figure 12b. These observations clearly emphasize that investigating such a complex host/guest system cannot be envisaged without taking into account the existence of the phase mixture. This phase mixture model provides an efficient approach to computationally interpret the macroscopic adsorption behavior of guest species in highly flexible MOFs as well as analyze their phase mixture composition within a wide range of pressure. It should be noted that this model does not have a particular physical meaning but only aims at taking into account the experimental evidence. Thus, it is difficult to apply this model without experimental assistance to extract the phase mixture composition. 3.4.3. Osmotic Thermodynamic Model Method. A substantial number of flexible MOFs exhibit S-shape or stepwise adsorption isotherms, sometimes accompanied by hysteresis loops. Therefore, besides experimental measurements, there also exists a need to develop/apply theoretical methods for understanding these phenomena. One direct approach is to use molecular simulations; however, the required flexible force fields are very scarce at the moment, and they are also not always appropriate, as evidenced from a recent study.294 QM approaches are powerful for examining the origins of this behavior;295−297 nevertheless, they are too computationally expensive to obtain related information in a large parameter space, leaving the thermodynamic picture beyond incomplete. Thus, Coudert et al.298−300 developed a generic osmotic thermodynamic model to understand adsorption in highly flexible MOFs that can exhibit clear structural transitions between a few metastable framework structures (as opposed to a phenomenon such as continuous swelling upon adsorption). The main idea is the use of a socalled osmotic subensemble to describe the full thermodynamic equilibrium between the host structures of MOFs. It relies on

Ωkos(T , P) = Fkhost(T ) + PVk −

∫0

P

Nkads(T , p)Vm(T , p)

dp

(30a)

where Fhost is the free energy of the empty host structure at zero k pressure, Vk is the unit cell volume, Nads is the absolute k adsorption isotherm in phase k with a fully rigid structure, and Vm is the bulk molar volume of the fluid, which can be obtained from thermodynamic data. The distinct parts (below and above the step) of a stepped adsorption isotherm are fitted over the whole range of pressure using appropriate equations. Such fitting procedures can generate a full “rigid-host” isotherm needed for each phase, as indicated by the dashed lines in Figure 13.

Figure 13. Schematic representation of the determination of free energy difference ΔFhost from an experimental adsorption isotherm (upper panel) by calculation of f i(P); the pressure-dependent part of the thermodynamic potential of the osmotic ensemble (lower panel). Reprinted with permission from ref 298. Copyright 2008 American Chemical Society.

For a MOF in equilibrium between two phases (labeled 1 and 2), it can plot the two functions f1(P) and f 2(P) from the fitted isotherms, which are defined by fk (P) = Ωkos(P) − Fkhost = PVk −

∫0

P

Nkads(p)Vm(p)dp (30b)

At the transition pressure Peq, the osmotic thermodynamic potentials of the two phases are equal. Thus, the free energy difference (ΔFhost) of phase 2 relative to 1 can be calculated by Y


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and chemical potential μ, the adsorption-induced stress σs is expressed as

ΔF host(T ) = F2host(Peq) − F1host(Peq) = f1os (Peq) − f 2os (Peq) (30c)

⎛ ∂Ω ⎞ σs(Vc) = −⎜ c ⎟ ⎝ ∂Vc ⎠

where Peq can be determined from the step in the experimental isotherm, as shown in Figure 13. If adsorption isotherms are available for different temperatures, both the transition energy (ΔUhost) and the entropy (ΔShost) differences between two empty host structures can be extracted from the free energies, ΔFhost(T) = ΔUhost − TΔShost. The osmotic thermodynamic OS potential difference (ΔΩOS = ΩOS 2 − Ω1 ) between the two phases upon gas adsorption can then be written as

∫0

−

∫0

P

P

N2ads(T , p)Vm(p)dp

N1ads(T , p)Vm(p)dp]

(31a)

where Ωc is the grand thermodynamic potential of the adsorbed phase per unit cell of the material and Vc is the unit cell volume. For anisotropic MOFs, such defined stress can serve as an overall scalar measure of the magnitude of the adsorption forces acting on their frameworks. The difference between this stress and the pressure represents the solvation pressure, Ps = σs − Pext, which determines the magnitude of the material’s elastic deformation in terms of the volume strain ε (ε = ΔVc/Vc, where ΔVc is the variation of the cell volume). For breathing materials, the stress−strain linearity described by Hooke’s law should hold only for their individual stable phases instead of in the vicinity of transitions. If the adsorption isotherm in each host phase can be fitted by a generic Langmuir equation, N(P) = NmaxKHP/(Nmax + KHP), the stress at a fixed temperature can be expressed as a function of pressure by

ΔΩos(T , P) = ΔF host(T ) + P ΔV −[

μ,T

(30d)

where the volume difference ΔV = V2 − V1. The relative stability of various possible structures at a given temperature and pressure can be determined by the sign of ΔΩos. The occurrence or absence of structural transitions is governed by the number of solutions of the equation ΔΩOS(P) = 0. This model has been successfully applied by Coudert and coworkers301 to understand the breathing behavior of MIL-53(Al) upon xenon adsorption. It was found that both adsorption isotherms in the LP and NP phases can be well fitted by the Langmuir equation. The occurrence of breathing is conditioned by two factors: one is the relative adsorption affinities (the ratio of Henry’s coefficients, KLP/KNP) of the gas in the LP and NP host phases; the other is ΔFhost for describing the intrinsic relative stability between the two phases. Similar observations have also been found for the same material and its aminomodified form upon adsorption of other gases.302,303 From these studies, it was concluded that the breathing effect in MIL53(Al) is a very general phenomenon and should be observed in a limited temperature range regardless of the properties of guest molecules. The above osmotic thermodynamic model can be directly extended to MOFs with multiple structural transitions by simply applying it to each structural transition. It allows one from a thermodynamic standpoint to reveal the relative stability of the metastable structures of flexible MOFs.302,304 Therefore, this method provides a useful tool for macroscopically predicting whether the structural transition occurs or not under various thermodynamic conditions. However, it does not give insight into the physical mechanisms of the phase transitions in flexible MOFs or explain the hysteresis loops observed experimentally.305 Moreover, it does not take into account the phase mixture effects in the MOFs undergoing structural transitions as described in section 3.4.2. 3.4.4. Stress-Based Model Method. Considering the limitations of the above osmotic thermodynamic approach, a stress-based model was proposed by Neimark et al.305 to address the issue of the physical mechanisms associated with the breathing transitions of MOFs. This instructive model considers the stress exerted on the material by the adsorbed molecules as a stimulus that triggers the structural transitions. It is basically a “thermomechanical” model302 in which the structural transition is supposed to take place when the stress approaches a certain critical threshold. In this new model, the stress is related to the adsorption isotherm of a gas in flexible MOFs. At a fixed temperature T

⎧ ⎡ ⎪ dN max ⎢ln(1 + KHP /Nmax ) σs(P) = RT ⎨ ⎪ ⎩ dVc ⎢⎣ ⎫ ⎛ KHP /Nmax ⎞⎤ dKH ⎛ ⎞⎪ P ⎬ −⎜ ⎟⎥ + ⎜ ⎟⎪ dVc ⎝ 1 + KHP /Nmax ⎠⎭ ⎝ 1 + KHP /Nmax ⎠⎥⎦ (31b)

where KH is the Henry’s coefficient that describes the adsorption affinity in that phase and Nmax is the corresponding saturation capacity per unit cell. The values of dNmax/dVc and dKH/dVc are fitted to reproduce the experimental transition pressure upon adsorption and desorption. Because the two quantities are apparently positive and negative, respectively, eq 31b indicates a nonmonotonic variation of σs (or Ps) during the adsorption process. To show how this stress model can inherently account for the adsorption−desorption hysteresis loop, adsorption of Xe in MIL-53(Al) at 220 K was selected by Neimark et al. as a showcase. This system exhibits two consecutive hysteretic breathing transitions between LP and NP phases, as reflected from the experimental isotherms shown in Figure 14 (top panel). This figure also shows the adsorption stress for both host phases (bottom panel), where σ*NP and σ*LP are the critical stresses associated with the NP → LP and LP → NP structural transitions upon adsorption and desorption, respectively. According to the adopted model, adsorption starts in the LP phase. At very low vapor pressures, the negative stress causes a structural contraction and eventually induces the first transition (LP → NP) when it reaches the critical stress σ*LP (line A1 in Figure 14). At higher vapor pressure, the second reverse transition (NP → LP) takes place when the stress approaches the critical stress σ*NP (line A2). A similar analysis can be performed for the desorption branch. In this figure, the vapor pressure at line D1 corresponds to the critical stress σ*LP exerted in the LP phase during desorption while the vapor pressure at line A2 corresponds to the critical stress σ*NP exerted in the NP phase during adsorption. Due to a lower value of the former compared to the latter, it results in the hysteresis loop having occurred in the high-pressure region. Z


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composition curves along both adsorption and desorption branches. Later, this model has also been successfully transferred to account for the breathing behavior of the MIL53 family upon adsorption of other gases as well as mechanical compression.302,303,306 It can be expected that a combination of this stress-based model and the osmotic thermodynamic model will give a more comprehensive understanding of the breathing behavior of other MOFs as well as the origins of the related hysteresis loops. However, the mechanism of the adsorption process in the MOFs with gate-opening behavior is not yet fully clarified. In addition, although this model can be used to examine phase transitions of flexible MOFs upon adsorption of pure gases, its applicability in the presence of mixtures still needs further exploration. 3.4.5. Osmotic Framework Adsorbed Solution Theory. By solely comparing pure-component adsorption isotherms, it seems that flexible MOFs exhibiting a gate-opening or breathing phenomenon are excellent candidates for gas separations. However, if the pores of a material are triggered to open by one component of a mixture, the other components might also be permitted to enter its structure, possibly negating the perceived selectivity. This illustrates the necessity to evaluate the separation performance of these MOFs using gas mixtures instead of pure gases. Toward this goal, Coudert and co-workers307−309 proposed an osmotic framework adsorbed solution theory (OFAST) that can predict the structural transitions and adsorption properties of mixtures in flexible MOFs with multiple metastable structures. This OFAST method couples the standard adsorbed solution theories with the thermodynamic equations of an osmotic ensemble in which the former is used to describe the adsorbed mixtures in each possible host structure while the latter is employed to model the equilibrium between guest-loaded host structures. Its central idea is that for each value of composition and pressure at fixed temperatures the most stable phase is simply the one with the lowest osmotic potential. Such a treatment allows one to determine the thermodynamic stability domain of each phase and subsequently predict the separation performance of flexible MOFs at given conditions of interest. Similar to eq 30a for pure components, the osmotic thermodynamic potential in a given phase k of a material upon mixture adsorption can be expressed by

Figure 14. (Top) Experimental adsorption and desorption isotherms of Xe in MIL-53 (Al) at 220 K (in blue), along with the Langmuir fits for the isotherms in the LP and NP host structures (black and red). (Bottom) Adsorption-induced stress for both host phases (black, LP; red, NP) and critical stress σ*NP and σ*LP determining the structural transitions upon adsorption and desorption (blue arrows; pressures noted as A1, A2, D1, and D2). Reprinted with permission from ref 305. Copyright 2010 American Chemical Society.

Again, a similar discussion can also be conducted to analyze the origin of another hysteresis loop which has occurred in the lowpressure region. On the basis of these results, it is clear that the structural transition pressure depends on the critical stress threshold of the host structure before the transition rather than on the condition of thermodynamic equilibrium between two phases. By assuming that the experimental sample is composed of a large number of crystallites with different sizes, the above stress model was also used by Neimark et al. to give a simplistic explanation of the phase mixture (coexistence of NP and LP phases) effect in MIL-53(Al) and further predict the phase

Figure 15. (a) Predicted pressure−composition adsorption phase diagram for CO2/CH4 mixture at 304 K in MIL-53(Al). (b) Predicted gateopening pressure in Cu(4,4′-bipy)(dhbc)2 at 298 K for the binary mixture of N2, O2, and CH4 as a function of mixture composition. Reprinted with permission from ref 307. Copyright 2009 American Chemical Society. AA


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gases in a large number of flexible MOFs as well as in a large space of thermodynamic parameters. Thus, the knowledge provided by this method is very helpful in guiding the search for optimal flexible materials and operating conditions related to gas separation. Nevertheless, it should be noted that this original OFAST can only be applied to situations where purecomponent experimental adsorption data are available for estimating the free energy differences between different phases of the material. In addition, it is unclear whether or not it could be applied to materials with a continuous nature of the physicochemical transformation. 3.4.6. Dispersion-Corrected DFT Approach. As described in previous sections, a significant deficiency in the commonly used DFT methods is the lack of a suitable treatment for dispersion interactions. Walker et al.310 applied such methods combined with a standard GGA functional like PBE to study the breathing behavior of the empty structure of MIL-53(Al). It was found that these calculations cannot stabilize its NP structure, and it opens up to give the LP form. In addition, with the cell parameters of both forms fixed, they also showed that by comparing their energies the optimized LP structure is more stable than the optimized NP one. This is in conflict with the experimental observation269 that the NP form is the low-temperature favored phase. From this deviation, Walker et al. speculated that the dispersion interactions may play an important role in stabilizing the NP phase at low temperatures. To verify this speculation, two approaches were used to introduce the missing long-range dispersion interactions into their DFT calculations. The first one (which can be called DFT-D2) is to supplement the forces from DFT with an empirical dispersion term proposed by Grimme.243 The second approach is to directly introduce the long-range dependency into DFT in a self-consistent manner, making the exchange-correlation functional explicitly nonlocal,311 which is often referred to as the van der Waals density functional, or vdW-DF. DFT calculations revealed that the low-temperature NP phase of MIL-53(Al) is indeed stabilized by the dispersion interactions, and the result is not sensitive to the approach used to take into account the dispersion contribution. On the basis of these results, Walker et al. further probed the nature of the stabilizing interactions. It was found that the attractive dispersion interaction exceeds the penalty incurred for increased short-range repulsion and less favorable electrostatic interactions in the inorganic chains, thus resulting in the stable NP form. This study also showed that the LP form is entropically stabilized at 300 K, and the LP form has more lowfrequency modes than the NP one which drives the transition at higher temperatures. Thus, they concluded that the exceptional bistable behavior of MIL-53(Al) is due to a competition between short- and long-range interactions and entropic factors. This computational study first highlights that dispersioncorrected DFT approaches are required to successfully understand the origin of the structural transitions of flexible MOFs. Such DFT calculations have also been found to play a crucial role in accounting for the breathing or gate-opening behavior of other flexible MOFs.297,312,313 Currently, inclusion of dispersion interactions into DFT calculations has become an active research area, and different approaches have been proposed to give deep insights into the structural properties of MOFs as well as their adsorptive properties toward various guest species.314−320

host ΩOS k (T , P , y) = Fk (T ) + PVk M

P

−

∫0 (∑ Niads,k (T , p , y)Vm,i(T , p , y))dp i=1

(32a)

where M is the number of components in the mixture, y = {y1, ads y2,..., yM} is the corresponding bulk composition, Ni,k is the adsorption amount of component i, and Vm,i is the partial molar volume of this component in the bulk phase. If the mixture and pure fluids in the bulk state can be considered as ideal then eq 32a can be simplified to yield host ΩOS k (T , P , y) = Fk (T ) + PVk

− RT

∫0

P

ads Ntotal, k(T , P , y)

p

dp

(32b)

Nads total,k

where is the total amount of all components adsorbed in the pores of phase k. Then, the osmotic thermodynamic potential difference between two phases can be written as ΔΩOS(T , P , y) = ΔF host(T ) + P ΔV − RT

∫0

P

ads Ntotal, k(T , P , y)

p

dp

(32c)

ΔNads total

where is the difference of the total adsorption amounts between the two phases and ΔFhost(T) is the host free energy difference which can be calculated from pure-component stepped isotherms according to the approach described in section 3.4.3. In addition, if IAST is used to determine the total quantity (Nads total,k) adsorbed in each possible phase k, a rapid prediction can be performed to study the coadsorption behavior in flexible MOFs with pure-component adsorption isotherms as the only input. To illustrate the capability of the OFAST method, Coudert et al.307 presented two examples on predicting the coadsorption in two flexible MOFs. Figure 15a shows the predicted pressure−composition adsorption phase diagrams for CO2/ CH4 mixture in MIL-53(Al) at 304 K. Obviously, the stable phase of the material is dependent on the interplay of the thermodynamic conditions (total pressure and mixture composition). Because pure CO2 can induce the material’s breathing behavior while pure CH4 does not at the temperature examined, there is a limiting molar composition for CO2 in the mixture (xlim = 0.12) below which no breathing occurs. The pressure for the first structural transition (LP → NP) evolves in a monotonic trend between ∼0.2 (for pure CO2) and ∼4 bar (for xlim), while the pressure for the second one (NP → LP) does not vary monotonically and reaches a maximum value at x(CO2) = 0.3. This means that for some CO2/CH4 mixtures the second transition can take place at a higher pressure than that for pure CO2. The predicted phase diagrams for the adsorption selectivity will be described in section 4.1. Figure 15b displays the evolution of the gate-opening pressures for three binary mixtures in Cu(4,4′-bipy)(dhbc)2. It can be found that there is a smooth and monotonical variation between the gate-opening pressures for the respective pure components, and the range of evolution of this pressure is large enough for O2/ CH4 and N2/CH4 mixtures. Later, this OFAST method was further applied to study this behavior in more detail by including the influence of other factors such as temperature.309 These computational studies indicate that OFAST can be used to rapidly explore the coadsorption properties of various AB


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fields are too repulsive at short distances. The LJ potential in the force field developed by Snurr and co-workers276 gives a good description at larger intermolecular distances, whereas the one in the force field of Greathouse and Allendorf325 gives too strong attractive interactions for benzene; this explains the too low self-diffusivity of benzene observed by Greathouse and Allendorf. In addition, Amirjalayer and Schmid also found that Coulomb interactions only have a very small contribution to the overall binding energies. For a channel-type material, Zn(tbip), Seehamart and coworkers323 developed a fully bonded flexible force field. The bond stretching, bond bending, bond torsion, and improper torsion are expressed using eqs 28a−28c. The equilibrium bond distances and angles are taken from the experimental crystal data, except for the C−H bond distances that are taken from the available standard optimized structures. The nonbonded interactions (expressed by eq 5) are calculated only between atoms separated by more than three bonds. Potential parameters are either adopted from generic force fields137,327 or other flexible force fields developed for MOFs.272,321,325 On the basis of this new force field, Seehamart et al.283 found that framework flexibility can yield significantly higher selfdiffusivities of C2H6 relative to the rigid framework and also lead to a maximum at intermediate loadings. After a detailed analysis, they attributed these differences to the reason that the forces exerted on the flexible lattice by C2H6 molecules become stronger with increasing loading, causing an enlargement of the narrow parts of the channels. This allows more molecules to diffuse into the central region of the channels, thus resulting in an increase of the self-diffusivity, while the steric hindrance effects at high loadings give rise to a decreasing trend. Later, Seehamart et al.328 extended this MD simulation study to examine diffusion of three equimolar mixtures (CO2/C2H6, CH4/C2H6, and CO2/CH3OH) in Zn(tbip). The results showed that the above trend found for the self-diffusivity of pure C2H6 also appears in the mixture. The framework flexibility enhances the self-diffusivity of CH3OH, while that of CO2 with the more slender molecule remains nearly unaffected. In addition, they also calculated the diffusion selectivity using the method described in section 3.9.1, and the results show different trends for these mixtures. Additionally, Yang and co-workers174 deveoped a fully bonded force field for another prototypical material, UiO66(Zr). In this force field developed for the dehydroxylated form, the potentials for describing the intramolecular interactions in the organic ligands and nonbonded interactions are the same as those used by Seehamart and co-workers (see above). All equilibrium bond distances and angles are taken from the DFT-optimized structure, and the force constants for intramolecular potentials are directly taken from the CVFF force field. The LJ potential parameters are taken from the DREIDING force field with atomic partial charges obtained from the Mulliken population analysis. The interactions between the inorganic and the organic parts are described by a torsion term and a bond bending contribution. The undefined potential parameters are manually determined to accurately reproduce the structural properties of the DFT-optimized lattice. On the basis of this force field, it was found that UiO66(Zr) behaves more like a “dynamic” framework when one probes the diffusion of the species trapped within its porosity. MD simulations accurately capture the magnitudes and profiles for both the self-diffusivity of CH4 and the transport diffusivity of CO2 observed from QENS measurements. The unusually

3.5. Regular Framework Flexibility Description

Parallel with development of the force fields described in sections 3.2 and 3.4, there is also increasing effort in building force fields for MOFs with regular framework flexibility. Depending on the approaches for treating the inorganic parts of MOFs, the force fields developed so far can be generally classified into nonbonded and fully bonded versions. In the former, only vdW and electrostatic interactions are used to describe the interactions between the atoms in the inorganic parts, ignoring directionality of bonding,271,321,322 while in the latter covalent bonds between these atoms are explicitly taken into account.174,323 Some of the representative force fields that have been applied in the diffusion studies are introduced below. IRMOF-1 is one of the MOFs that has received much attention in development of flexible force fields. By parametrizing the MM3 force field to account for the interactions associated with the Zn4O clusters, Amirjalayer et al.272 developed a fully bonded flexible force field. All parameters are derived from DFT calculations of model fragments and then tuned via MD simulations to reproduce both the geometry and the vibrational normal modes. Bond dipole parameters are generated from atomic charges fitted to the ESP using the MK scheme. In a subsequent MD simulation study,324 Amirjalayer et al. applied the so-built force field to investigate diffusion of benzene at a low loading in IRMOF-1 with and without framework flexibility. It was found that the self-diffusivity simulated using the flexible framework is in good agreement with the PFG NMR measurements.167 However, the diffusivity predicted in the rigid case is nearly an order of magnitude higher, which is also the case for the activation energies calculated. Greathouse and Allendorf271 developed a nonbonded force field to describe the framework flexibility of IRMOF-1. In this force field, eq 5 is used to model the nonbonded interactions between atoms. The LJ parameters for the atoms in BDC (benzene dicarboxylate) ligands are adopted from the CVFF force field275 with slight modifications, and a similar treatment is also used for describing the intramolecular interactions (bond stretch, angle bend, dihedral angle, improper angle) for BDC atoms. The parameters in this flexible force field are manually determined so as to obtain good agreement with the structural data for an empty IMOF-1. Similar to the study of Amirjalayer et al., Greathouse and Allendorf325 also applied their force field to examine diffusion of benzene in the MOF in which the CVFF force field is directly used to describe the adsorbate molecule. At the same low loading, it was found that the selfdiffusivity simulated in the flexible lattice is higher than that obtained in the rigid one and is also in much better agreement with the experimental value. These observations are in sharp contrast to the opposite findings of Amirjalayer et al. described above. The activation energy calculated with the flexible framework is in good agreement with the result of Amirjalayer et al.167 However, their force field underestimates the experimentally determined self-diffusivity by a factor of ∼8. Recently, Amirjalayer and Schmid326 calculated the potential energy curves for benzene interacting with different model systems that represent the inorganic units of IRMOF-1. On the basis of these DFT results obtained at the B2PLYP+D/ccpVTZ level, they benchmarked three different flexible force fields developed for this material. They found that the Buckingham potential in their MM3-type force field272 is able to reproduce the interaction strength of benzene with the vertex of IRMOF-1 to a high degree, while the LJ-based force AC


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Figure 16. Simulation results for separation of an equimolar mixture of CH4/H2 in IRMOF-9 at 298 K. (a) Macroscopic selectivity of CH4 over H2 as a function of pressure. (b) Contour plot of the microscopic selectivity of CH4 over H2 at 2 MPa in the plane through the catenated area in IRMOF-9. Reprinted with permission from ref 330. Copyright 2008 American Chemical Society.

material. However, due to the presence of nanostructures in MOFs, it can be envisaged that the selectivity should show a distribution, that is, the local selectivity should be different in the pores, which is local-region dependent. As a result, we can call such selectivity “local selectivity” or in a more general concept “microscopic selectivity”207,330 to distinguish it from the “macroscopic selectivity”. Apparently, the microscopic selectivity can show a clearer picture of the separation feature of a MOF than the traditional one, thus providing more detailed microscopic information for tailoring MOFs with improved separation performance. This concept was proposed by our group,207,330 and methods were also developed to obtain the values of the microscopic selectivity in MOFs using GCMC simulations. To obtain such selectivity, the unit cell of a MOF is first divided into a large number of grid points with appropriate spacing; then the occupying times of the COM of the molecules at each grid point are counted for each component in the mixture during simulation. For the simulation box consisting of multiple unit cells, if the molecules are adsorbed in the periodic image cells of the central one they are mapped into the central unit cell at the corresponding fractional positions. At the end of the simulation, the average number of molecules for each species at each grid point is statistically obtained from the ratio of the total occupying times of this component at that grid point to the total simulation steps used for sampling. On the basis of these results, the microscopic selectivity at each grid point is calculated according to eq 15a, in which the compositions for the two components in the bulk mixture are used. For the grid points occupied by framework atoms, the selectivities are set to zero. Then, a so-called “computer tomography for materials” (mCT)331 can be employed to visualize the distribution of the microscopic selectivity in the plane of each layer along any crystallographic axis of the unit cell. It should be noted that the distribution of such selectivity could be reliably estimated only when the number of adsorbate molecules in the system is large enough. To demonstrate the applicability of the above approach, we investigated the separation of an equimolar CH4/H2 gas mixture in IRMOF-9 at 298 K using GCMC simulations.330 This material has a 2-fold catenated (interpenetrated) structure where each framework features a primitive cubic topology with the octahedral Zn4O(CO2)6 clusters connected by 4,4′biphenyldicarboxylate (BPDC) ligands. Figure 16a and 16b shows the simulated macroscopic selectivity as a function of pressure and the distribution of the microscopic selectivity at 2 MPa, respectively. Obviously, the microscopic selectivity is

faster diffusivity of CH4 compared to CO2 is opposite to the trend previously observed in zeolites that possess narrow windows (LTA, DDR, and CHA).329 In a subsequent study, Yang et al.182 further investigated diffusion of the CO2/CH4 mixture in this MOF with the framework flexibility taken into account. This work revealed a rather unusual dynamic behavior, that is, the “slower” molecule (CO2) enhances the mobility of the “faster” one (CH4). Through careful analysis of the MD trajectories it was found that CH4 molecules in the mixture are more frequently pushed into the octahedral cages than in the single-component diffusion. Such a situation explains the faster diffusivity of CH4 as these molecules spend less time in the tetrahedral cages in which stronger interactions with the pore wall occur. The above studies demonstrate that even if the frameworks of MOFs exhibit regular flexibility it can have a nonnegligible impact on the diffusion properties of gases. Such an influence is especially significant for the structures with small pores or tight channels relative to guest size and/or the existence of strong guest−host interactions. Therefore, it is highly necessary to develop new force fields toward a wide variety of MOFs. It will allow us to have deeper insights into the diffusion properties of gases as well as to reveal the underlying mechanisms involved at the molecular level, which will in turn provide valuable information for rational design of novel MOFs relevant to gas storage and separation applications. 3.6. Concepts and Theories for Characterizing Separation Performance

In recent years, a variety of new concepts have been proposed for MOFs, providing better characterization and/or modeling of MOFs. On the other hand, in addition to computational tools, many theories have been improved/applied to understand the performance of MOFs. In this section, we will give a summary of the advances in this area. Since some new concepts, such as the CBAC, REPEAT, and DDEC charges, accessible volume and pore size, pore limiting diameter, etc., have been dealt with in sections 3.1−3.4, they will not be repeated in this section, which is also the case for theories. Therefore, here we only supplement those that have not been mentioned in previous sections. 3.6.1. Microscopic Selectivity. Although the adsorption selectivity of a mixture in a MOF can be measured experimentally or predicted by molecular simulations and other theoretical approaches such as IAST or OFAST, the resulting selectivity is usually a value averaged over the whole material. We can call such selectivity “macroscopic selectivity” since it represents the overall separation performance of a AD


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The physical meaning of the adsorbility can be considered as a measure of the adsorption energy density for a gas adsorbed in a given porous solid. It is expected that this new parameter can serve as a measure of the ‘‘compatibility’’ of two gases adsorbed, and thus, the difference (ΔAD) between the adsorbility of the two components in a mixture can be used to characterize the adsorption selectivity. As will be seen in sections 3.8.1 and 4.3.2, this quantity not only can be applied to establish good correlations with selectivity but also can be utilized to suggest some fundamental rules for designing new MOFs. 3.6.3. Nonlocal On-Lattice Classical Fluid DFT. Apart from molecular simulations, classical fluid DFT provides an alternative modeling tool for studying the thermodynamic and structural properties of confined fluids. Its key feature is that the thermodynamic quantities of an inhomogeneous fluid are built on the molecular density distribution (not the density of electrons used in the QM-based DFT) in the system, and the equilibrium density profile is extracted by minimizing a free energy functional.334 This approach has been widely used in studies of adsorbents with pores that can be approximately treated as simple geometries, such as smooth-walled cylinders, slits, and spheres.335 In regard to MOFs where the density profile requires a three-dimensional representation, there is also increasing interest in developing suitable versions for the related DFT studies. To the best of our knowledge, the first endeavor in this respect was conducted by Siderius and Gelb336 in which they introduced a new nonlocal classical DFT of finely discretized lattice fluids. This new method allows prediction of the adsorption of simple gases (constituent molecules can be described by a spherical model) in complex porous solids including MOFs. For a system of a lattice fluid interacting via a long-range potential, at a fixed temperature T and chemical potential μ, Siderius and Gelb proposed that the grand potential functional Ω can be written in a mean-field approximation (MFA) as

heterogeneously distributed with the highest one occurring in the small pores formed by the two metal clusters and BPDC ligands (see Figure 16b). In these regions, the selectivity ranges from 30 to 90, much higher than the average value of the macroscopic selectivity (∼15) shown in Figure 16a. The small pores formed by the ligands also show relatively high selectivity (∼20), with the lowest selectivity in the center of the large pores (∼4). Thus, this figure gives a clear picture of the selectivity distribution in the MOF. The above approach has also been successfully used to examine the separation of CO2/ H2 mixture in a series of IRMOFs and their catenated counterparts.207 Knowledge of this microscopic selectivity is very helpful for understanding the relationship between the structural properties of MOFs and their separation performance at the microscopic level, which is difficult to access by experiment if not impossible. At the same time, the microscopic information obtained for the selectivity can provide a theoretical foundation for guiding the future design of new MOFs with improved separation performance. This is also a powerful example to show how computer modeling can help to understand the characteristics of MOFs. 3.6.2. Adsorbility. The structure−property relationships are essential for rational design of new MOFs with improved properties. Since various influencing factors may have conflict contributions to the separation performance of a MOF, these fundamental relationships for MOFs are not well understood to date. As a result, design of MOFs for separation of a specific system is based, to large extent, on a trial-and-error procedure. Indeed, our previous studies35,36 as well as those from other groups194,332 showed that the selectivity of MOFs for a given mixture is affected by the interplay of various factors and cannot correlate strongly with the available single intrinsic properties of the materials. On the other hand, if a single parameter can be developed to characterize the main feature of a gas/MOF system of interest, quantitative models for the structure− property relationships may be established, which will contribute to the further understanding of these relationships for MOFs as well as to future design of new materials with tailored separation performance. To this target, we recently proposed a new concept to characterize nanoporous materials like MOFs, named “adsorbility” (AD).35 It is stimulated from the concept of the solubility parameter (also denoted by SP) in solution theory,333 which can be used as an estimate of the compatibility (mutual solubility) of two components in solution. This important parameter is defined as the square root of cohesive energy density (Ecoh = Hvap/VL), the energy of vaporization per unit liquid volume. Similarly, a definition was proposed by us for the “adsorbility” of an adsorbate in a nanoporous material as AD = Ead =

M

Ω[{ρi }] =

i=1

+

i=1

1 2

M

M

∑ ∑ ρi ρj ϕatt(|ri − rj|)(1 − δij) i=1 j=1

M

+

∑ [V ext(ri) − μ]ρi

(34a)

i=1

where M is the number of lattice sites, V is an external field applied to the system, and ρi is the occupancy fraction on lattice site i that locates at position ri. The four terms on the right side of eq 34a represent the ideal-gas Helmholtz free energy, excess hard-sphere Helmholtz free energy, mean-field attractive energy, and external field contributions, respectively. The weighted occupancy fraction (ρ̅i) is defined as ext

Q st0 ϕ

M

∑ fid [ρi ]ρi + ∑ f hsex [ρi̅ ]ρi

(33)

M

ρi̅ =

where Q0st is the isosteric heat of adsorption at infinite dilution in units of kJ/mol for an adsorbate and ϕ is the material’s free volume in units of cm3/cm3, which is also known as the porosity of a material. From the definition, Ead is the adsorption energy density based on specific free volume, thus having a similar physical meaning to the Ecoh. In our approach, instead of defining AD as the square root of Ead, we make them identical (see eq 33).

∑ ρj w(|ri − rj|; ρi̅ ) j=1

(34b)

where the weighting function (w) is obtained by applying Tarazona’s approach337 to a hard-sphere fluid on a cubic lattice, forcing it to have a direct correlation function similar to the Percus−Yevick result.338 In eq 34a, f ex hs is the excess Helmholtz free energy per lattice hard-sphere particle, which is derived from the on-lattice GCMC simulation results. For LJ fluids, the AE


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Figure 17. (a) Comparison of the adsorption isotherms of H2 in MOF-5 at 298 K obtained by the lattice-based classical DFT method and GCMC simulations. (b) Density distribution of H2 adsorbed in MOF-5 at 298 K and 179 bar. Gray and green isosurfaces correspond to a local density of 1.33 and 19.9 mol/L, respectively. Reprinted with permission from ref 336. Copyright 2009 American Chemical Society.

WCA perturbation method339 is used to separate the potential into the hard-sphere and attractive (ϕatt) components. By setting ∂Ω/∂ρi = 0 for all i, the resulting Euler−Lagrange equations are iteratively solved to obtain a minimum of Ω, from which the equilibrium density profile can be obtained. To examine the applicability of the above approach, Siderius and Gelb calculated the adsorption isotherms of H2 in MOF-5 at 298 K, where H2 is treated as a spherical LJ fluid. As shown in Figure 17a, the calculation results are in very good agreement with GCMC simulations over a wide range of pressure. On the basis of the obtained equilibrium density distributions, they observed that the low-density isosurface is close to the vdW surface of the material, as illustrated in Figure 17b for H2 adsorption at 179 bar. The energetically favorable adsorption sites are near the tetrahedrally coordinated zinc atoms, again agreeing well with those observed by experiments and other theoretical methods. This nonlocal on-lattice classical fluid DFT can dramatically reduce the computational demand required to model simple fluids confined in complex inhomogeneous materials due to use of discretization. It may be concluded that this new efficient method represents an important step forward for deeply understanding the adsorption behavior of fluids in MOFs. However, since the MFA is used for the attractive contribution to the excess free energy, which has been shown not very reliable at low temperatures,340 it is not clear whether this lattice-based DFT is applicable under such conditions. 3.6.4. WDA-Based Three-Dimensional Classical Fluid DFT. As described above, the lattice-based DFT proposed by Siderius and Gelb is constructed from a combination of the MFA for the attractive contribution and Tarazona’s weighted density approximation (WDA) for the repulsive contribution. Considering the drawbacks of the MFA at low temperatures, Liu et al.341 developed a three-dimensional classical fluid DFT to study the adsorption of pure LJ fluids in MOFs, where both the attractive and the repulsive contributions to the excess free energy are estimated using the WDA but with different weighting functions. By examining H2 adsorption in two materials (MOF-5 and ZIF-8), they found that the theoretical predictions agree well with the simulation results and experimental data at both high and low temperatures. Later, this DFT method was further extended by them to be applicable to the LJ fluid mixtures confined in threedimensional nanospace.342 We briefly introduce this method in the following paragraph.

Following the standard classical DFT, for an open system consisting of K components, at a fixed temperature T and chemical potential μi of each component i, the grand potential functional Ω[ρ(r)] is expressed as ex ex Ω[ρ(r)] = F id[ρ(r)] + Fhc [ρ(r)] + Fatt [ρ(r)] K

+

∑ ∫ [Viext(r) − μi ]ρi (r)dr i=1

(35a)

where ρ(r) denotes the local density vector, ρ(r) = [ρ1(r), ρ2(r), ..., ρK(r)], and Vext i (r) denotes the external potential for component i at position r. In this equation, the intrinsic Helmholtz free energy functional F[ρ(r)] is decomposed into three terms: ideal-gas (Fid), excess hard-core (Fex hs), and excess attractive (Fex att) contributions. The ideal-gas term is exactly known as K

F id[ρ(r)] = kBT ∑

∫ ρi (r){ln[ρi (r)Λ3] − 1}dr

i=1

(35b)

where Λ is the de Broglie wavelength and kB is the Boltzmann constant. For LJ fluid, the WCA perturbation scheme is used to split the potential into hard-core (ϕhc) and attractive (ϕatt) terms ⎧∞ r < rh ϕhc(r ) = ⎨ ,rh ⎩ 0 r < rh 1 + 0.2977T * = 1 + 0.33163T * + 1 + 0.0010471T *

(35c)

⎧− ε r < 21/6σ ⎪ ⎪ ϕatt(r ) = ⎨ 4ε[(σ /r )12 − (σ /r )6 ] 21/6σ < r < rc ⎪ ⎪0 r > rc ⎩

(35d)

where rh is the hard-core diameter, T* is the reduced temperature (T* = kBT/ε), and rc is the cutoff distance. The hard-core contribution is evaluated using the WDA as K ex Fhc =

∑ ∫ ρi (r)f hs(i) [ρhc̅ (1) (r), ···, ρhc̅ (K ) (r)]dr i=1

(35e)

where the weighted density ρ̅ h(i)s (r) for the hard-core contribution of component i is given by AF


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Figure 18. (a) Adsorption selectivity for CO2 over CH4 (50:50 in the bulk phase) in ZIF-8 at 298 K. Lines, DFT; symbols, simulation. (b) Density distribution of CO2 in the CO2/CH4 mixture adsorbed in ZIF-8 at 0.6477 bar, viewed along one crystallographic direction of the unit cell. Density units: mol/L. Reprinted with permission from ref 342. Copyright 2010 American Chemical Society. (i) ρhc ̅ (r) =

∫ ρi (r′)whc(i)(|r − r′|, ρhc̅ (i) )dr′

Figure 17b). These observations are confirmed by subsequent experimental and computational studies.344 To our knowledge, this work is the first and also the only report that classical fluid DFT is used to study the separation of mixtures in MOFs. The above results and those obtained in section 3.6.3 indicate that the classical fluid DFT can successfully predict both the adsorption and the separation of fluids in MOFs as well as give insight into the underlying mechanisms involved. In contrast to molecule simulations where the quantities of interest are obtained from an ensemble-average approach by following the configurational evolution of the fluid molecules during simulation, such DFT methods compute these quantities directly. However, they are only applicable to fluids with molecules that are spherical. For MOFs with strong electrostatic features, multiple-sites charged models are usually required for modeling the fluid molecules (such as CO2) so as to properly take into account their electrostatic interactions with MOFs. Thus, an extension of these methods for more complex fluids should lead to a more useful understanding of the relationships between the macroscopic phenomena and the fundamental physics. Since developing robust and accurate classical fluid DFT for MOFs is challenging and not straightforward, the related computational studies are still in an extremely early stage. 3.6.5. Application of dcTST To Describe Diffusion of Molecules in MOFs. Conventional MD simulation techniques have been widely used to understand the transport properties of pure fluids and their mixtures in various nanoporous adsorbents including MOFs. If the activation energy for diffusion is much higher than the thermal energy kBT, the diffusion process involved will be very slow, sometimes even becoming a collection of “rare events”. These phenomena are often observed when the sizes of guest molecules are close to the pore sizes of the adsorbents, where the molecular mobilities are severely restricted due to their strong confinement. In such situations, one of the difficulties encountered is that the diffusion processes occur outside the time scale accessible to conventional MD simulations, which are typically limited to diffusion rates on the order of ∼10−12 m2/s.345 As a result, enormous amounts of computational resources can be wasted without detecting one of these events. The approach of dynamically corrected transition state theory (dcTST) proposed by Beerdsen and co-workers345,346 is a powerful tool for efficiently solving this time scale problem. Apart from giving equivalent self-diffusivity where conventional MD simulations can apply, the dcTST is also applicable to the regime of very slow diffusion which MD fails to describe.

(35f)

(i) where the hard-core weighting functions whc (r,ρ) are 337 (i) determined by Tarazona’s approach. f hs in eq 35e is the partial molecular excess free energy of component i for a hardsphere fluid, which can be derived from an EOS for hard-sphere mixtures.343 Similar to eq 35e, the WDA is also used to evaluate the attractive contribution as given by

K ex Fatt =

∑ ∫ ρi (r)f att(i) [ρatt̅ (1) (r), ..., ρatt̅ (K ) (r)]dr i=1

(35g)

where the weighted density for component i is expressed as (i) ρatt ̅ (r) =

∫ ρi (r′)watt(i)(|r − r′|)dr′

(35h)

with the weighting function given by (i) (i) watt (r ) = ϕatt (r )/

∫ ϕatt(i)(r)dr

(35i)

(i) (i) (i) In eq 35g, f(i) att = f LJ − f hs , where f LJ is the partial molecular excess free energy of a LJ uniform fluid of component i. With these expressions for free energies, the equilibrium density profiles can be solved iteratively by minimizing the grand potential functional

∂Ω[ρ(r)] = 0 (i = 1, 2, ..., K ) ∂ρi (r)

(35j)

To validate the above WDA-based DFT, Liu et al. investigated the adsorption of CO2/CH4 and CO2/N2 mixtures in ZIF-8 and Zn2(BDC)2(ted) at room temperature, in which the adsorbate molecules are modeled as spherical LJ particles. In all calculations, good agreement is observed between the DFT predictions and the GCMC simulations both for the adsorption isotherms and the selectivities. Figure 18a shows a comparison of the selectivities for CO2 over CH4 in ZIF-8. In addition, from the obtained three-dimensional density distributions it was found that the density is low near the framework surface of both MOFs, which is similar to the result shown in Figure 17b for H2 adsorption in MOF-5. As an example, Figure 18b illustrates a two-dimensional density distribution of CO2 in the CO2/CH4 mixture adsorbed in ZIF8. Clearly, the regions near the mIM organic ligands are the preferential adsorption sites, followed by the corners around the metal clusters. This behavior is in sharp contrast to the inverse sequence found in many MOFs such as MOF-5 (see AG


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The value of κ is computed as the fraction of particles starting on top of the barrier with a velocity toward B that successfully reach this state. For classical systems, the value of κ is 0 < κ ≤ 1. For practical applications of MOFs in gas storage and separation, the diffusivity of gases in them is one of the important parameters to be considered. Thus, based on a combination of MD simulation and the dcTST, Liu et al.348 investigated the effect of interpenetration (catenation) on H2 diffusion in MOFs. MD simulation results on 10 IRMOFs with and without interpenetration show that catenation can reduce H2 self-diffusivity by a factor of 2−3 at room temperature. To further understand the relationship between H2 diffusivity and MOF structure, a dcTST study was performed for IRMOF-16 and its catenated counterpart IRMOF-15 at infinite dilution. Diffusion at such conditions generally can be fully accounted for by DTST s , i.e., the free energy contribution to self-diffusion, as the dynamical correction factor κ is the contribution to the diffusion of interparticle collisions. The obtained free energy profiles in IRMOF-15 and -16 are shown in Figure 19, where

Furthermore, evolution of the diffusion behavior via loading can be explained at the molecular level by analyzing the free energy profiles and lattice information provided by the dcTST, giving valuable insight into their mechanisms. Application of this method that was initially developed for examining the diffusion of fluids in zeolites345−347 was first carried out by Liu et al. to MOFs.348 In the dcTST formalism, the diffusion processes in confinement are considered as hopping events on a lattice, where the hopping from state A to state B (the lattice distance between them is set to λ) is limited by a free energy barrier between them. The dcTST defines a reaction coordinate q as the Cartesian coordinate along the axis parallel to the line connecting the centers of A and B. This quantity indicates the progress of the diffusion event from A to B and is chosen as the position of one atom of the diffusing molecules. Together with the dynamical correction factor κ, free energy profiles F(q) can be used to compute a hopping rate kA→B from state A to state B by kA → B = κ ×

=κ×

exp[−βF(q*)] kBT × q* 2πm ∫−∞ exp[−βF(q)]dq kBT × P(q*) 2πm

(36a)

where kB is the Boltzmann’s constant (β = 1/kBT), T is the temperature, m is the mass of the hopping particle, q* denotes the (assumed) location of the barrier between state A and state B, and P(q*) is the relative probability. This hopping rate kA→B in turn can be converted into a self-diffusivity Ds by Ds = kA → B × λ 2 = κ ×

kBT 2πm

× P(q*) × λ 2 = κ × DsTST

Figure 19. Free energy profiles F(q) of H2 in IRMOF-15 and IRMOF16 at room temperature and infinite dilution. Reaction coordinate is chosen along the z direction. For IRMOF-15 the pore centers are located at 5.36, 16.09, 26.82, and 42.92 Å, while for IRMOF-16 they are at 0, 21.49, and 42.98 Å. Reprinted with permission from ref 348. Copyright 2008 PCCP Owners Society.

(36b)

DTST s

where is the free energy contribution to the self-diffusivity, the part of the diffusion that is governed by the free energy barriers: influences of the confinement topology on the diffusion of the fluid molecules. In this theory, κ is regarded as a measure for the interaction between the fluid molecules themselves, and it corrects for recrossing events, i.e., for trajectories which cross the transition state from A but fail to end up in B.345 By calculating the diffusivity according to eq 36b, it can distinguish between the topology contributions (included in DTST s ) and the particle collision contributions (included in κ), thus leading to a better understanding of the diffusion behavior as a function of loading. To compute free energy profiles, NVT-ensemble MC simulations can be performed at the desired loading using the histogram sampling (HS) method.346 In this method, a histogram is made of the particle positions, mapped on the reaction coordinate q. This histogram can be converted into a free energy profile using βF(q) = −ln⟨P(q)⟩, where P(q) denotes the probability to find a molecule at a given position q according to the histogram. To calculate the value of κ, a large number of initial configurations at a specific loading are first generated using the MC method, in which one particle is constrained to the dividing surface and the other particles are allowed to move around freely. The generated configurations are then integrated using short unconstrained microcanonical (NVE) MD simulations, starting with velocities sampled from a Maxwell−Boltzmann distribution at the desired temperature.

the reaction coordinate q is simply the position of a tagged molecule along the Cartesian connection from one pore to another. The wells of the barriers correspond to the locations near the metal cluster and ligand region inside the pores, i.e., the energetically favorable adsorption sites, while the maxima correspond to the locations of the metal cluster and ligand region. Obviously, the number of barriers per unit cell in IRMOF-15 is larger than that in IRMOF-16, attributed to the catenation structure in the former. From the MD simulations it was found that H2 diffusivity in IRMOF-15 is about one-half that in IRMOF-16 (39.8/84.3) at infinite dilution. Liu et al. explained this observation by comparing the DTST obtained s from the dcTST for the two materials. As can be seen from this figure, the values of (kBT/2πm)1/2 × exp[−βF(q*)] are nearly identical for IRMOF-15 and IRMOF-16, since the barrier heights are nearly the same for the two materials. On the other hand, the integration in eq 36b for IRMOF-15 is approximately one-half of the one for IRMOF-16 and the value of λ in the former (the lattice distance between two adjacent pore centers) is also one-half of the one in the latter. Therefore, the dcTST gives consistent results with the MD simulations, while the structural effects can be revealed clearer by the former: catenation in IRMOF-15 leads to a decrease of λ, i.e., at a AH


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Figure 20. Visual summary of the hypothetical MOF-generation strategy. (a) Illustration of decomposing the structures of MOFs into building blocks and then recombining them to form new, hypothetical MOFs. (b) Illustration of the stepwise recombination of building blocks which are attached at their connection sites (purple Xs). These building blocks are also connected across periodic boundaries (ii, hashed circles indicate mirror images). H atoms are omitted for clarity. Reprinted with permission from ref 39. Copyright 2012 Nature Publishing Group.

atomic positions in the AIMD data. For gases (such as CH4 and Xe) with molecular sizes smaller than the crystallographic window size of ZIF-8, they showed that taking into account the framework flexibility with a full description of the window size distribution in the material leads to orders of magnitude improvement in prediction accuracy. In contrast, for gases with smaller molecular sizes such as H2, He, and Ar, framework flexibility has only a minor impact on their diffusions through ZIF-8. In the case of nonspherical adsorbate CO2, the approach results in a diffusivity several times larger than that using the rigid DFT-optimized structure. Moreover, the transition state (TS) for CO2 diffusion is found to be not simply located at the narrowest region of the pore. As a result, this work demonstrated that the ability to include framework flexibility in assessing the performance of MOFs is vital in applications in which diffusion plays an important role.

given length the number of barriers is increased, resulting in a smaller H2 diffusivity in IRMOF-15 than that in IRMOF-16. The above analysis demonstrates that the dcTST is a useful method to understand the relationship between the molecular diffusivity and the MOF structure, and this method may find wide applications in future studies of MOFs. In addition, transition state theory (TST) assumes that all particles that reach the barrier with a trajectory toward B do end up in B (κ = 1), and thus, the TST rate (DTST s ) is an upper bound of the one calculated by the dcTST. Actually, TST has also been used to analyze the effects of pore confinement on the diffusion behavior of many different gases as well as their separation mechanisms in various MOFs with narrow pore windows.40,42,349,350 For example, based on a combination of TST and MD simulations, Haldoupis et al.350 studied the effects of the framework flexibility of ZIF-8 on diffusion of five spherical-approximated gases (H2, He, Ar, CH4, and Xe) and one linear-represented gas (CO2). They first employed ab initio MD (AIMD) calculations to obtain information about the flexibility of the empty framework as well as to collect a large number of structural configurations of the material. Classical MD simulations were then performed for the gases under infinite dilution conditions in each structure with a rigid treatment. To calculate the net diffusivities using TST, the hopping rate of each species was connected to the activation energy of each molecule through the windows present in the structures. Haldoupis et al. also compared these results with those obtained in the rigid structures that are determined from neutron power diffraction measurement, DFT-based geometric optimization, and a time average of the

3.7. Methods for Constructing MOFs

Compared to conventional porous materials, one of the advantages of MOFs stems from use of modular building blocks that self-assemble into predictable crystal structures. As a result, hundreds of reports have successfully demonstrated the reticular design and synthesis of novel MOFs with such a modular building block approach. However, due to the great variety of the metal ions and organic ligands, it is difficult to synthesize all possible MOFs and characterize their properties for a specific application. In addition, although some novel MOFs have been designed computationally, they are usually performed on a very small scale, mainly based on the available topologies of one or several MOFs. Thus, there is great need to AI


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conducted a systematic screening on a very large scale to identify promising MOFs from 137 953 hypothetical MOF structures. These materials are generated using 102 building blocks that differ significantly in their geometries, number of connection sites, as well as chemical compositions. On the basis of the GCMC simulations, they screened out more than 300 MOFs that have storage performance surpassing the current world-record-holding material (PCN-14). Compared to the value reported experimentally for PCN-14 (∼230 vol(STP)/ vol),8 the best hypothetical MOF can exhibit a storage capacity up to 267 vol(STP)/vol. From a global view of the trends obtained for these possible MOFs, a clear linear relationship exists between the volumetric CH4 adsorption and the volumetric surface area. By examining the correlation between the CH4 storage capacity and the pore size of the materials, the best MOFs are identified as those with pore sizes of 4−8 Å. It can be expected that the above building blocks-based constructing approach is also applicable to identify new MOFs for other applications, thus allowing synthesis effort to be more efficiently focused on targeted structures. The large structural database generated by this approach also permits one to computationally seek deep insight into the structure−property relationships for MOFs. As described in this study, the approach can be further improved by adding information about the estimated thermodynamics of structure formation to further refine the best candidates. Moreover, if novel inorganic building blocks can be generated instead of taking the ones already known in the existing MOFs, the approach would become more powerful for accelerating the discovery of promising materials. 3.7.2. Zeolite Topology-Based Approach. Aiming at screening out promising carbon-capture materials, Lin et al.44 proposed another methodology to construct hypothetical MOFs, which may be called a zeolite topology-based approach. In this method, some ZIF-type materials are considered as parent compounds, which are a subclass of MOFs that have a pore topology similar to zeolites: transition metal atoms (M) replace the Si atoms, and imidazolates (IM) replace bridging oxides in the zeolite structures. Considering that hundreds of thousands of possible zeolites with different pore topologies exist in the zeolite database, Lin et al. applied this analogy of ZIFs to the zeolite database from which hypothetical materials are generated using the ZEOCC code.351 In the reported zinc/ IM-based ZIFs 352 that have the International Zeolite Association (IZA) zeolite topologies, the distance between zinc atoms and the center of IM rings is about 1.95 times larger than the Si−O distance in zeolites. On the basis of this observation, a hypothetical structure is generated by scaling the unit cell of the corresponding zeolite structure with the same factor, together with all the O and Si atoms being replaced by the IM groups and Zn atoms, respectively. Using this approach, a very large database of hypothetical materials is obtained with a large variety of structural topologies. To validate their approach, Lin et al. compared the resulting geometries with those of ZIF3 (the DFT topology) and ZIF-10 (the MER topology) reported experimentally. They showed that the observed differences in the geometries of the hypothetical and experimental structures do not translate into significant differences in the parasitic energy examined in their study. Although only the simplest organic ligand (imidazole) is used, it is expected that this zeolite topology-based approach can be extended to other organic ligands as well as to metal ions, leading to an even larger number of hypothetical zeolite-

have efficient approaches that can rapidly generate hypothetical MOFs on a large scale and then screen out promising materials before performing experimental synthesis and related property characterization. In the following we will introduce some progress achieved in this respect. 3.7.1. Building Blocks-Based Approach. Recently, Snurr and co-workers39 proposed a novel computational procedure to efficiently create hypothetical MOFs on a very large scale by recombining a large number of known modular building blocks. This approach does not require any extra MM-based or QMbased energy minimizations as well as the predefined symmetry and topology. The structure of a hypothetical MOF is built according to the simple geometric rules that govern how the building blocks are connected in the existing materials. The library of building blocks is derived from structures of the known MOFs that are extracted from their crystallographic data. A representative illustration for decomposing MOF structures into building blocks and then recombining them to form new hypothetical MOFs is shown in Figure 20a. According to their characteristics, the building blocks are divided into three types of structural units: inorganic, organic, and functional groups. In order to recombine the building blocks into crystals, each building block is manually assigned additional topological and geometrical information, where the former takes the form of numbered connection sites, while the latter takes the form of three “pseudoatoms” and a list of angles for every connection site in the building block. Any building block can be combined with any other building block provided that the geometry and chemical composition local to the point of connection is the same as those in the crystallographically determined structures. The algorithm in this approach allows one to consider all possible combinations of building blocks in all possible space arrangements. Figure 20b shows a representative example of stepwise recombination of building blocks. First, one of the inorganic building blocks is selected to be recombined with one organic building block by attaching their connection sites (Figure 20b, marked by a purple X). Then, when atomic overlap occurs at a particular step, another building block or a different connection site is chosen, until all possibilities are exhausted. At the same time, when two building blocks can form a periodic connection along one direction by comparing their orientations and coordinates, a periodic boundary is imposed instead of adding a building block (Figure 20b, steps ii and iv). The process repeats (iii to iv) until no more building blocks can be added, and then the crystal generation procedure is complete. This approach also can generate MOFs with interpenetrated (catenated) structure if enough space exists (v, black circles indicate atoms belonging to one of two interpenetrated frameworks). To validate their approach, Snurr and co-workers generated crystal structures that resemble some of the MOFs reported experimentally. Then, they compared these structures with their counterparts that are energetically relaxed by the UFF-based MM method and also with their experimentally measured structures. Their results demonstrate that both the relaxed and the unrelaxed structures are very similar to the experimental ones. It was also found that the adsorption isotherms of CH4 predicted from GCMC simulations agree well among the three structures of each examined MOF. Considering a representative application of MOFs for CH4 storage at room temperature and high pressure (35 bar), after the above validation step, Snurr and co-workers further AJ


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Figure 21. Limiting selectivity (a) and selectivity at 20 bar (b) vs 1/ΔAD from an equimolar CH4/H2 mixture. Red curves are obtained from the QSPR models, and the filled squares are the GCMC simulation data. Reprinted with permission from ref 35. Copyright 2012 Wiley-VCH.

Prior to developing the QSPR model, we performed GCMC simulations to examine the separation behavior of CH4/H2 mixture in 105 MOFs with diverse chemical composition, pore size, and topology. To find out the structure−property relationship for the investigated MOFs, a single-factor analysis is first performed, attempting to establish a correlation between their selectivities for CH4 over H2 at infinite dilution and some descriptors that are commonly reported to characterize the physicochemical properties of these materials as well as the strength of their interactions with the considered gases. The selectivity under such conditions is also called limiting selectivity, which reflects the intrinsic separation capability of a material. The descriptors tested involve the intrinsic energetic and structural features of MOFs, including the difference of the isosteric heats of adsorption (ΔQ0st) between CH4 and H2 at infinite dilution, the specific accessible surface area (Sacc), the free volume (Vfree), and the porosity (ϕ). The pore size is not a considered parameter since some MOFs contain more than one type of pore with different sizes. The results show that good correlation cannot be obtained between the selectivity of MOFs and the available single property, indicating the selectivity is affected by the interplay of various properties. However, on the basis of single-factor analysis, we found that ΔQ0st and ϕ are the descriptors most correlated to the limiting selectivity. Considering the definition of the adsorbility (AD = Q0st/ϕ) and its physical meaning, it can be expected that there should be an intimate relationship between the selectivity of MOFs and the difference between the adsorbilities (ΔAD = ΔQ0st/ϕ) of the two components. To give evidence for that, we conducted a correlation procedure based on the following expression where ΔAD is used as a single variable

like MOFs. As this approach mimics the zeolite chemistry, it cannot be used to generate materials with topologies previously unknown in the zeolite database. 3.8. Development of QSPR Models for MOFs

A quantitative structure−property relationship (QSPR) model is an efficient way to deal with a large amount of information. The method is widely used in the fields of chemical/biological sciences and engineering to reduce the costs of developing new chemicals with optimal performance. In QSPR modeling, a set of “predictor” variables (X) is related to the potency of the response variable (Y). The predictors consist of molecular descriptors which are selected to quantitatively represent the system’s structural properties, and their mathematical correlations are generally established by different approaches, including regression analysis, genetic algorithm, recursive partitioning, artificial neural network, etc. To date, only a few studies have been devoted to developing QSPR models for MOFs with respect to several applications. The lack of such studies might be mainly due to the difficulty in defining or identifying suitable descriptors for solid materials. Below, we present some typical QSPR models proposed for describing the properties of MOFs on several topics. 3.8.1. Adsorbility-Based QSPR Model for Gas Separations. Both experimental and theoretical investigations have already demonstrated that many MOFs show great potential for various gas separations. However, compared to the very rich database of MOFs, the number of materials examined is rather limited. From these studies it is difficult to accomplish a comprehensive and clear understanding of the structure− property relationships for MOFs, hampering development of new MOFs for a targeted application. Therefore, there is increasing demand to expand the scope of the current methodology for efficiently screening a much larger population of MOFs. On the other hand, if an efficient QSPR model is available to rationalize the separation properties of a large series of MOFs in a reasonable time, it would be very useful for guiding the tuning/design of advanced materials with outstanding performance. For this purpose, on the basis of the adsorbility introduced in section 3.6.2, our group developed a QSPR model to relate the separation performance of MOFs with their structures.35 In this work, an important practical system, a gas mixture of CH4 and H2, was selected as the targeted system, which is usually involved in the process of purification of synthetic gas obtained from steam reforming of natural gas.

SCH4 /H2 = a ×

⎛ 1 ⎞−b ⎜ ⎟ + 1.0 ⎝ ΔAD ⎠

(37)

where SCH4/H2 denotes the adsorption selectivity and a and b are the parameters to be fitted. In this model, a constraint is considered: when one component disappears, the selectivity must be equivalent to one. The results shown in Figure 21a indicate that the limiting selectivity correlates well with ΔAD. To further assess the interplay of ΔQ0st and ϕ, different combinations of them have been tested to build individual QSPR models. We found that the QSPR model using eq 37 gives the best performance. In addition, good correlation is also found at high pressure (see Figure 21b), although a little worse than that for the limiting AK


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selectivity. This demonstrates that ΔAD may also be useful in building such QSPR models at high pressures. Later, we found that eq 37 can be also successfully transferred to build the QSPR model for describing flue gas (CO2/N2) separation in a large database of MOFs under industrial operating pressure (1.0 bar).36 All these observations indicate that AD can grasp the main feature of a gas/MOF system, approximately representing the interplay of various influencing factors, and thus, ΔAD can be used to characterize the adsorption selectivity. Also, it is clear that in order to obtain large ΔAD, the difference in the interactions between the two adsorbates and the MOF should be as large as possible, and at the same time the MOF should have a low porosity. As we will see in section 4.3.2, a hypothetical MOF with promising performance for the separation of CO2 from N2 gas streams can be generated according to this strategy. These computational studies show that synthesizing MOFs with large ΔAD can serve as a general design criterion for development of MOFs as well as for preliminary large-scale screening of MOFs for targeted separations. 3.8.2. QSPR Model for Drug Encapsulation. Beyond the large amount interest in gas adsorption and separation, a new and an exciting development for MOFs lies in the biomedical area.22,353 In order for MOFs to be useful for biomedical applications, one of the prerequisites is that the material must have a biologically friendly composition. Thus, biocompatible and nontoxic nanoparticles of Fe(III)-based MOFs have been proposed as an alternative for the controlled release of challenging drugs. In the context of such potential applications, Maurin and co-workers354 recently established a QSPR model to systematically explore the influence of the functionalization of the MOF pore walls on their performance for drug encapsulation. In this study, the amphiphilic cosmetic caffeine is chosen as a probe drug; a series of flexible porous MIL88B(Fe) solids is selected as a representative of the Fe(III)based MOFs in which the terephthalate ligands bear different functional groups with various polarity, hydrophilicity, and acidity (−Br, −F, −2CF3, −CH3, −2CH3, −4CH3, −NH2, −NO2, −2OH, or simply H). In order to build a QSPR model, a wide range of molecular descriptors of different dimensionalities, including both 2D and 3D types, are first calculated to capture the topological, chemical, and electronic features of the organic ligands. A forward selection is then applied to identify the most correlated descriptors relevant to the caffeine uptakes that are evaluated by elaborate experimental techniques. By selecting two or three descriptors among the 14 identified, a total of 455 QSPR models are constructed using a multiple linear regression (MLR) method. For all these models, a combination of the leave-one-out (LOO) cross-validation and the y-scrambling procedure is further employed to internally validate their robustness. At the end, the so-obtained QSPR model, representing the predicted caffeine encapsulation capacity (y), can be expressed as a linear combination of three descriptors: hydrogen-bond donor properties (denoted as a_don), vdW surface (SMR_VSA3), and polar volume (vsurf_Wp1)

On the basis of the so-built model, Maurin and co-workers found that there is a rather good correlation between the experimental and the predicted caffeine uptakes, as shown in Figure 22. The resulting cross-validation correlation coefficient

Figure 22. Predicted vs observed caffeine encapsulation loading for the set of the functionalized MIL-88(Fe) materials (n = 10). Reprinted with permission from ref 354. Copyright 2012 Elsevier.

(Q2) of 0.84 ensures a reasonable quality of the model. Besides a prediction role for the drug uptake in the examined MOFs, this QSPR model can also be used to interpret the evolution of the encapsulation loading with the nature of the organic ligands from the descriptors involved. It provides evidence that the polarity and H-donor capacity of the organic ligands are the key factors which govern caffeine uptake into the investigated MOF series. This can be seen from a significant improvement of the uptakes observed in the −2OH- and −NH2-modified materials compared to others. For MOFs functionalized with groups that cannot form additional hydrogen bonds with the drug molecules, the encapsulation loading can be enhanced by increasing the vdW surface that interacts with these molecules (SMR_VSA3 descriptor). Such a conclusion can be reflected from the trend via increasing the number of methyl groups within the linkers. This work provides a QSPR approach to investigate the relationship between caffeine uptake and the structural properties of the parent material MIL-88(Fe) and its 10 functionalized forms. It can be expected that this QSPR model can be further improved when more experimental data in various MOFs with different topologies are considered and thus will be useful for examining more types of MOFs. In addition, the methodology is also applicable to build QSPR models for predicting and understanding the encapsulations of other drug molecules in MOFs. 3.8.3. QSPR Model for Isosteric Heat of Adsorption. The organic ligands of MOFs provide very ideal platforms to achieve high performance for a targeted gas separation by tuning their chemical functionality. For guiding development of promising materials, it is strongly desirable to have efficient methods to provide a direct connection between the structural chemistry of MOFs and the thermodynamic properties that determine the adsorption behavior of related gases. In this respect it is well recognized that the isosteric heat of adsorption (Q0st) of gases is one of the most essential thermodynamic properties related to their separation performance in MOFs. Therefore, using a genetic function approximation (GFA)

y = 6.48 × a_don + 3.16 × SMR_VSA3 + vsurf_Wp1 + 10.14

(38)

where all the descriptors have positive contributions. AL


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method, Amrouche and co-workers355 built a QSPR model based on four descriptors to predict the Q0st of gases at zero coverage in ZIF-type materials. A database of the Q0st for 11 gases (containing nonpolar, quadrupolar, and dipolar molecules) in 14 Zn-based ZIFs were considered, which are composed of three different topologies (SOD, RHO, GME) and 14 different functionalized imidazolate ligands. Considering the lack of experimental data and possible inconsistencies among the few available experimental sources, they performed GCMC simulations to calculate the values of the Q0st using eq 13e. Due to the observation that the metallic sites play a secondary role for gas adsorption, the molecular descriptors are only defined for the organic ligands in ZIFs. Owing to the nature of the isosteric heat of adsorption, two types of descriptors are required to reflect the information of the adsorbed gases and solids. On the basis of about 60 possible descriptors identified initially, the GFA approach is used to obtain a correlative equation composed of the most relevant descriptors. It starts by establishing an initial population of equations that is randomly chosen. Then, the population of equations evolves through iterative operations: selection, cross over, and mutation of equation’s terms. During this process, the constructed equations are scored using Friedman’s lack of fit (LOF) method. In the development of the QSPR model, the total database is divided into two subsets: one is the training set (90% of the total database) used to build the predictive models, and the other is the test set (the remaining 10% of the total database) used to select the best model. At the end, the GAF procedure leads to the following QSPR model

simulations, as shown in Figure 23. The mean absolute errors (MAE) between the QSPR and the GCMC results are about

Figure 23. Comparison of the simulated/experimental isosteric heat of adsorption of different gases in solids with the predicted ones obtained using the QSPR model. Reprinted with permission from ref 355. Copyright 2012 The Royal Society of Chemistry.

5.7 and 5.6 kJ/mol, respectively, while the mean absolute percentage errors (MAPE) are about 24.5% and 23.8%, respectively. Their analysis reveals that eq 39 is slightly more accurate for the family of solids with GME topology followed by the RHO and SOD ones. The QSPR model also indicates that the higher the gas polarity, the higher the increase of Q0st will be when |μOL| is increased. It should be pointed out that the above QSPR model is built on a hypothesis for the calculations of the isosteric heat of adsorption, that is, the gases can probe the overall pore structures of the ZIFs, although previous studies42 have shown a certain portion of the pores may not be accessible to some gas molecules. Despite that, the approach adopted in this study introduces a form of physical consistency check to select the equation that preserves some physical meaning. This approach and the resulting QSPR model can help to relate the Q0st, an important measure of adsorption ability, to the structural descriptors/properties of a MOF/gas system, which is useful for guiding selection of organic ligands in development of new materials for a specific application of interest. 3.8.4. QSPR Model for Hydrogen Uptake. Application of MOFs as H2 carriers is also one of the hottest topics that have attracted much attention since the appearance of MOFs. Understanding the relationship between the structural factors of MOFs and their H2 uptake capacities in a quantitative way would be helpful for development of MOFs as efficient storage materials. For that purpose, Kim et al.356 established a QSPR model to characterize the dependence of H2 uptake on the molecular structure of MOFs as well as to find out the key structural factors that are important for H2 adsorption. To build the QSPR model, Kim et al. considered many molecular descriptors, including the electrotopology and polar surface area. The electrotopological state indices for a particular atom result from the topological and electronic environments, while the polar surface area is defined as the surface area occupied by the polar atoms with the absolute value of partial charge greater than 0.2e. The isovalue surface area of ESP (electrostatic potential) is also used as a new descriptor to represent the molecular field of MOF pores. In their study, three types of hydrogen uptakes are defined: the final uptake

Q st0 = (λ1|Q OL| + λ 2[n fg ·H ])s + (λ3 ln Tb)g + (λ4H ·[|μOL |·|μg |])s/g

(39)

where the coefficients λi are the fitted parameters. In this model there are four descriptors that represent the solids: (i) dipolar moment of organic ligand (μOL), (ii) mean quadrupole moment of organic ligands (|QOL|), (iii) pore mean curvature H (H = 1/dp, where dp is the pore diameter) that describes how a surface deviates from a flat plane, and (iv) number of functional groups (nfg) that accounts for the number of nonaromatic functional groups in each organic ligand. μOL and | QOL| are calculated on the basis of the partial charges of the atoms in the ligands used in GCMC simulations. For the ZIFs with two different ligands, they proposed a unique value for μOL by a linear combination of the dipolar moments of the two ligands in which the pore and channel-accessible volume fractions are used as weighting factors. For gases, there are two descriptors: (i) dipolar moment of adsorbed gas (|μg|) and (ii) atmospheric boiling temperature (Tb) of the adsorbed gas. Thus, this QSPR model includes three types of contributions: the structures of the solids, gas properties, and gas−solid interactions. The polar nature of the solid is represented by | QOL| and |μOL|, and the vdW interactions are represented by H and nfg that take into account the solid topology and chemical nature of the organic ligands implicitly. In addition, the adsorbed gas contributions are represented by μg and Tb that are, respectively, a measure of its polar nature and cohesive energy. For both the test and the training sets of the database, Amrouche and co-workers found that the predictions based on the so-built QSPR model agree fairly well with the GCMC AM


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characterized as multiplication of the adsorption and diffusion contributions on the basis of mixed-gas feeds. For screening purposes, Krishna and van Baten previously suggested189 that if the permeate side is a vacuum (i.e., the downstream loading to be vanishingly small), the selectivity of zeolite membranes for a gas mixture could be approximated by

(FU), ultimate uptake (UU), and relative uptake (RU). These three quantities represent the H2 uptake at 77 K and 1 atm, the saturation value predicted by Langmuir−Freundlich fitting, and the relative ratio of UU over FU, respectively. From analysis of the calculation results with the isovalue surface area of ESP, 204 values for the descriptors are obtained, which are reduced further down to 110 values by applying the correlation matrix analysis. Among the reduced descriptors, the important molecular descriptors are chosen using the genetic algorithm. Finally, the QSPR models for the three uptakes obtained from the MLR are expressed by a linear combination of 2D and 3D descriptors as given by

Sperm, A / B =

(40a)

UU = 3.115897843X4 + 0.009243973X15 + 22.287571606X30 − 39.901306667

(40b)

RU = 2.065034894X4 + 0.005460121X15 + 12.766121949X30 − 22.387596901

DB ,self

×

qA qB

= Sdiff × Sads

(41a)

where Sperm,A/B denotes the permeation selectivity of species A over B, Sdiff is the diffusion selectivity, which is defined as the ratio of the self-diffusivities (DA,self and DB,self) of the two components, and Sads is the adsorption selectivity, which is obtained by the ratio of their molar loadings (yA and yB) adsorbed at the feed side. However, a disadvantage of this approach is that the mixture loadings used for calculating the diffusion and adsorption selectivities do not necessarily match each other. To overcome the above problem, Keskin and Sholl357 modified eq 41a to make it more suitable for describing the performance of pure MOF membranes under arbitrary conditions. In their proposed approach, two simple ways are adopted: one is that a single GCMC simulation is used to obtain the adsorption selectivity for a gas mixture at an arbitrary bulk composition and a specified feed pressure; the other is that a MD simulation is performed to evaluate the self-diffusivity of each component in the mixture at the loadings obtained from the GCMC simulation. Then, the resulting permeation selectivity of a pure MOF membrane can be predicted by

FU = 0.348488739X11 + 0.000767785X15 + 0.023889971X48 − 0.02134

DA ,self

(40c)

where X 11 and X 4 are the specific ESP(+0.009) and ESP(−0.005), respectively, X15 is molecular refractivity, X48 is the dipole moment, and X30 is the inverse of the molecular shadow area fraction. “Specific” means the value is normalized by the weight for conversion of an extensive value to an intensive one. With these QSPR models, Kim et al. found that the correlation between the observed and the calculated hydrogen uptakes is acceptable. On the basis of a recursive partitioning analysis, they demonstrated that the polarization and window size in the frameworks of MOFs could play key roles for H2 adsorption in MOFs. At low temperatures, the interaction between H2 molecules and MOFs is strongly related to the ESP given by the frameworks of MOFs, indicating that the ESP surfaces of the materials could provide information on their storage performance for H2. With increasing pressure, the window size becomes another important factor for determining the increment of hydrogen uptake. To our knowledge, this is the first QSPR model developed in connection with MOFs, although only 10 MOFs with different organic ligands and topologies were considered. This method combined with the proposed descriptors for predicting hydrogen uptake might be applicable to examine other MOFs.

Sperm, A / B =

DA ,self (qA , qB) DB ,self (qA , qB)

×

qA /qB yA /yB

= Sdiff × Sads

(41b)

where yA and yB are the mole fractions of components A and B in the bulk phase, respectively. Similarly, this approximate predictive model can only be used to give information at permeate pressures close to vacuum condition. To examine the accuracy of their approximate screening model, Keskin and Sholl compared the predicted results with those obtained from the detailed calculation approach. Within the range of the feed pressure examined it was found that this model can accurately predict the membrane selectivity for CH4/H2 mixture in MOF-5, while it is less accurate for the same mixture in Cu−BTC, as shown in Figure 24. For two other mixtures (CO2/H2 and CO2/CH4) this model shows poor agreement with the detailed calculations for CO2/H2 in Cu-BTC. After detailed analysis, Keskin and Sholl proposed a rule of thumb for screening MOF membranes in which IAST is used as a criterion: if IAST accurately predicts the mixture isotherm for the adsorbed mixture of interest, this approximate model can be expected to give results that are accurate enough for screening targets; otherwise, it is not able to judge its reliability, and detailed modeling approaches are required to predict permeation in which MD simulations are needed to determine mixture transport diffusivities over a wide range of loadings. In addition, under the conditions that the downstream loadings are negligibly small, Krishna and van Baten358 derived a simple approximate expression to determine the permeability of gases in a mixture through pure MOF membranes as given by

3.9. Membrane-Based Selectivity Calculation

3.9.1. Pure MOF Membrane. In the application of MOFs for membrane-based separations, there is a need to have a deep understanding of both the adsorption and the diffusion of adsorbates in the material. Detailed calculation approaches based on molecular simulations have been widely used for this target. These approaches consider a full treatment of mixture permeations, where diffusion is described by a matrix of macroscopic (Fickian) diffusivities. However, due to the extreme computational demand it is impractical to use them for screening a large number of pure MOF membranes at various operational conditions, although they can certainly be of value for detailed predictions on specific materials of interest. In order to rapidly assess the potential of MOFs as membranes, Keskin and Sholl357 recently proposed an efficient approximation method in which the selectivity of a membrane is AN


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have a trade-off between selectivity and permeability, as demonstrated by Robeson.359,360 Because of their unique properties, MOFs are considered excellent candidates for fabricating mixed-matrix membranes (MMMs) for gas separations,361,362 where MOF particles are incorporated into the polymer matrices. Considering the large variety of possible MOFs and polymers, if computational approaches can be utilized to reliably predict the performance of the polymer/ MOF composite membranes, it will be beneficial largely to guide experimental efforts. For this demand, Keskin and Sholl363 first introduced a modeling approach for predicting the MMM properties on the basis of macroscopic permeation models. In the existing theoretical studies of polymer/MOF MMMs, the models frequently employed are the Maxwell, Bruggeman, and modified Felske models.363−365 The Maxwell model describes the permeability (PMMM) of a gas species through MMMs as

Figure 24. Predicted membrane selectivities of IRMOF-1 and CuBTC for separation of an equimolar CH4/H2 mixture at room temperature. Reprinted with permission from ref 357. Copyright 2009 American Chemical Society.

c Pi = ϕDi ,self i fi

PMMM = Ppolymer ⎡ PMOF + 2Ppolymer − 2ϕ (Ppolymer − PMOF) ⎤ MOF ⎥ ×⎢ ⎢⎣ PMOF + 2Ppolymer + ϕMOF(Ppolymer − PMOF) ⎥⎦

(41c)

(42a)

where Pi is the permeability of component i in the mixture, ϕ is the fractional pore volume of the membrane material (which can be obtained using the method described in section 2.5.1), ci is the equilibrium pore concentration of component i at the upstream face of the membrane (which is defined in terms of the pore volume of a MOF), and f i is the bulk phase partial fugacity of that component. It should be pointed out that this equation is also applicable to calculate the permeability of single gases if the downstream face of the membrane is evacuated. For separations operated in the Henry regime, the permeability of a pure species A passing through a membrane can be approximately given by40 Pperm, A = KADA ,self

where PMOF and Ppolymer are the permeabilities in the dispersed (MOF crystals) and continuous phases (polymers), respectively, and ϕMOF is the volume fraction of the MOF particles in MMMs. Since nearby particles are assumed not to affect the streamlines around other particles,363 this model is applicable to low volume fractions of the filler (usually ϕMOF < 0.2). The Bruggeman model can be considered as an improved version of the Maxwell model as given by ⎡ (PMMM /Ppolymer) − (PMOF/Ppolymer) ⎤⎛ P ⎞−1/3 MMM ⎢ ⎥⎜⎜ ⎟ 1 − (PMOF/Ppolymer) ⎢⎣ ⎥⎦⎝ Ppolymer ⎟⎠ = 1 − ϕMOF

(41d)

This model is valid for a broad range of ϕMOF but has the same limitations as the Maxwell model. In addition, both of them belong to the models that treat ideal morphology, that is, there are no defects in the polymer−particle interface. Since both models are functions of volume fractions of the filler particles but not of the size and morphology of the particles, they are applicable to the situations where the filler particles are relatively isotropic and can be well dispersed in the polymeric matrix, as described by Keskin and Sholl.363 The modified Felske model is the one considering nonideal morphology that contains interface voids and polymer rigidification around particles. This model treats the dispersed particles as a core and the surrounding interfacial layer as a shell, in which the gas permeability is defined by

where KA is the Henry’s coefficient of that species and DA,self is its self-diffusivity in the limit of zero loading. Under such conditions, the ideal selectivity of the membrane can be approximately calculated as Sideal, A / B =

DA ,self DB ,self

×

KA = Sdiff × Sads KB

(42b)

(41e)

where Sads is equivalent to the ratio of the Henry’s coefficients of the two pure components and Sdiff is expressed as the ratio of their self-diffusivities. The above approximate models provide efficient approaches for a large-scale screening of the separation performance of pure MOF membranes for various mixture systems due to their simplicity and convenience. They avoid the need to calculate the transport diffusivities of mixtures as well as escape the complications that arise from fitting smooth functions to the adsorption and diffusion data. Using these models also does not require application of the SSK correlation to predict mixture diffusivities at arbitrary loadings, as usually done when applying M−S theory for evaluating the performance of membranes.357 3.9.2. Mixed-Matrix Membrane. Compared to the membranes based on porous solids, polymer membranes currently show a dominant role in the membrane market for mixture separations due to their ease of fabrication and low cost. Unfortunately, the existing polymer membranes usually

⎧ 1 + 2[(β − γ )/(β + 2γ )]ϕ ⎫ MOF ⎬ PMMM = Ppolymer ⎨ ⎩ 1 − [(β − γ )/(β + 2γ )]ϕMOFφ ⎭ ⎪

⎪

⎪

⎪

(42c) 2 φ = 1 + [(1 − ϕmax )/ϕmax ]ϕMOF

(42d)

β = (2 + δ 3)(PMOF/Ppolymer) − 2(1 − δ 3)(PI/Ppolymer) (42e)

γ = (1 + 2δ 3) − (1 − δ 3)(PMOF/PI) AO

(42f)


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Figure 25. Contour plots of the microscopic selectivity of CO2 over H2 in the planes through the catenated area in IRMOF-9 (upper two graphs) and through the Zn4O(CO2)6 cluster area in IRMOF-10 (lower two graphs) at 298 K and 1 bar. Case 1: Only switching off the electrostatic interactions between adsorbates-IRMOFs. Case 2: Including all the electrostatic interactions. Reprinted with permission from ref 207. Copyright 2009 Elsevier.

4. APPLICATION OF NEW METHODS AND CONCEPTS IN GAS SEPARATIONS

where PI is the permeability of the rigidified interphase layer, ϕmax is the maximum packing volume fraction of MOF particles, and δ is the ratio of outer radius of interfacial shell to core radius. This model can be converted into the Maxwell model if the values of ϕmax and δ are both equivalent to one. The method proposed by Shimekit et al.366 can be used to determine the values of ϕmax, δ, and PI. In the above approach introduced by Keskin and co-workers, the key feature is that the permeabilities of pure gases or gas mixtures in polymer phases are taken from the experimental data, while the intrinsic permeability properties of MOF phases are predicted using molecular simulations. Specifically, a combination of GCMC and MD simulations is employed to determine the steady-state fluxes of all components in a gas mixture through the MOF phase; these fluxes are then converted into their corresponding permeabilities using eq 16a. If the permeate pressure is close to vacuum condition, as done by Thornton et al,365 the gas permeabilities in the MOF phase can be alternatively determined according to eq 41c. Substituting the MOF-phase permeability of each gas into the permeation model, one can lead to its permeability through a specific MMM. For a gas mixture consisting of two species (A and B), the permeation selectivity of the MMM is calculated by SA/B = PA,MMM/PB,MMM. If these procedures are conducted for two pure gases, the ratio of their permeabilities is the ideal permeation selectivity (Sideal,A/B) of the MMM.

The above sections show that significant progress on development of computational methodologies has been achieved for MOFs during the past decade. These highlighted methods combined with conventional approaches have greatly promoted deep understanding of the structure−property relationships of MOFs. Among them, the separation performance of MOFs toward various practical applications is one of the fields that the most attention has been paid to. Here, we will summarize important progress on application of these computational methods for exploring the separation behavior of some industrially important gas mixtures in MOFs, based on both adsorption and membrane separations. 4.1. Adsorption-Based Separation

The adsorption-based separation is a reversible physisorption operation, which is mainly governed by the adsorption equilibrium effects due to the difference in the adsorbate− adsorbent interactions. To date, significant accomplishments have been made in understanding the structure-separation performance relationships for MOFs, including various industrially important systems. Among them, CO2 capture is a hot and timely topic. Therefore, we will summarize some progress on CO2 capture first, followed by summaries of the other gas mixtures. AP


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4.1.1. Carbon Dioxide Capture. As a subject of widespread public concern, the reduction of the anthropogenic CO2 emissions in the atmosphere has become one of the most urgent climate issues. On the basis of the large gas adsorption capacity and easy structural regulation of MOFs, significant computational effort has been made in recent years to examine the performance of MOFs for carbon capture and sequestration (CCS) from various gas mixtures. One issue that has received much attention is to assess MOFs for precombustion CO2 removal from its gas mixtures with CH4 or H2.165,367,368 The separations of the two systems are also of great economical and technological importance for upgrading low-quality natural gas (such as biogas and landfill gases) or purification of H2-rich syngas obtained from steam reforming of natural gas. Another important issue is to study the applicability of MOFs for postcombustion CO2 capture from flue gas emitted from power plants, which is mainly composed of CO2 and N2.164,369−374 A combination of the IFI method mentioned in section 3.1 with GCMC simulation and QM-level DFT calculation is a very effective way to explore the separation performance of MOFs for CO2-containing gas mixtures. This approach has been widely employed by our group in studies of a series of diverse MOFs.205−208,375 In 2006, we conducted the first computational study on the subject of CO2 capture, where CO2/CH4/ C2H6 mixtures were selected to examine the separation performance of Cu−BTC.205 On the basis of the IFI method introduced in this study for electrostatic feature characterization, we further investigated the effects of various electrostatic interactions on the separation of CO2 from H2 in three pairs of IRMOFs with and without catenation.207 Simulation results indicate that the enhanced selectivity at low pressures can be explained by the electrostatic interactions between CO2 and MOFs, and such an enhancement is much more pronounced in the catenated MOFs than that in the noncatenated counterparts. At higher pressures, the CO2− CO2 electrostatic interactions dominate the remarkably increasing trend of the selectivity, which can be further enhanced by including the CO2−MOF electrostatic contributions. In another study, we found that Cu−BTC exhibits a promising application for separating CO2 from a simulated flue gas (CO2/N2/O2).208 The selectivity for the CO2/N2 system shows an increasing trend with increasing pressure, which becomes the reverse trend when switching off all adsorbate− Cu−BTC electrostatic interactions. The electrostatic field inside the pores of Cu−BTC was also shown to be very important for CO2/O2 separation. Later, Calero and coworkers376 extended these studies with a more detailed analysis. They also demonstrated that Cu−BTC is a potentially good candidate for removal of CO2 from gas mixtures due to the strong electrostatic field existing within its structure. From these systematic studies it can be seen that the electrostatic property of MOFs plays an important role in separating CO2 from natural or flue gases. This leads to the conclusion that strengthening the electrostatic field gradient in MOFs should be an efficient approach to improve their separation performance for gases that are easily influenced by the electrostatic effects. As described in section 3.6.1, the microscopic selectivity has an inhomogeneous distribution in MOFs due to the complexity of their structures. In the study of CO2/H2 separation in MOFs207 we combined the microscopic selectivity calculation method with the IFI method to examine the effects of catenation and electrostatic interactions on the distribution of

such selectivity. Figure 25 shows an example for comparing the contour plots of the microscopic selectivity of CO2 over H2 in the catenated IRMOF-9 and the noncatenated IRMOF-10. In this figure, case 1 denotes the simulation results by switching off only the adsorbates−MOFs electrostatic interactions, while case 2 denotes the ones by considering all of the electrostatic interactions. Clearly, the influence of the electrostatic interactions between the adsorbates and the MOFs in the IRMOF-9 is much more significant than those in IRMOF-10. The largest effect occurs in the small pores formed by the two metal clusters and phenyl ligands in IRMOF-9, while for IRMOF-10 the most pronounced effect occurs around the Zn4O(CO2)6 inorganic clusters. In addition, the microscopic selectivities in the catenated area of IRMOF-9 are much higher than those around the Zn4O(CO2)6 clusters in IRMOF-10. In a subsequent study, we further investigated the microscopic selectivity distributions for separation of CO2 from CH4 in IRMOF-1 and its modified form IRMOF-(NH2)4.377 It was found that the microscopic selectivities around the organic ligands in IRMOF-(NH2)4 are much larger than those around the ligands in IRMOF-1. This observation is in good agreement with the DFT-calculated electrostatic field distributions in the two MOFs, that is, the ESP around the organic ligands in the modified material has a larger gradient and higher absolute values than that in IRMOF-1. Therefore, these studies demonstrate that the microscopic selectivity distribution is highly dependent on the pore size and topology as well as on the electrostatic feature of MOFs. A fine regulation of these properties can greatly enhance the performance of MOFs for gas separations. Generally, the adsorption isotherms in flexible MOFs exhibit stepped behavior, while such unusual phenomena can also be observed for CO2 and CH4 in rigid MOFs at low temperatures.378,379 Inspired by this striking feature that is thermodynamically favored, we conducted GCMC simulations to examine its effect on the gas mixture separation in a series of IRMOFs, where the CO2/N2 system was studied.380 Figure 26

Figure 26. Selectivity for CO2 from CO2/N2 mixture in IRMOF-10. Bulk phase composition: 15% CO2 and 85% N2. Reprinted with permission from ref 380. Copyright 2011 American Chemical Society.

shows an example for the simulated adsorption selectivity for CO2 over N2 in IRMOF-10 at seven temperatures. Obviously, the selectivity curves show remarkable steps below certain temperatures, and the steps are dramatically reduced with increasing temperature. The steeper the steps in the selectivity curve, the larger the selectivity for separating CO2 from the mixture. With the aid of the IFI method, we found that for these IRMOFs the CO2−CO2 electrostatic interactions have a AQ


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Figure 27. (a) CO2 (⧫) and CH4 (●) adsorption isotherms at 303 K for CO2/CH4 mixture (bulk composition CO2:CH4 = 50:50) adsorbed in MIL53(Cr) compared to the relative Raman intensity of CO2 (◊) and CH4 (○) and simulations (CO2, red full line; CH4, blue dash line). (b) Adsorbed amounts of CO2 (red) and CH4 (blue) and unit cell volumes of MIL-53(Cr) (green) as a function of the HMC steps, simulated for a 50:50 gas mixture at 303 K and 2 bar. Reprinted with permission from ref 292. Copyright 2009 American Chemical Society.

phase mixture model (eq 29) was then used to construct the mixture composite isotherms by combining the GCMC simulation data obtained in the LP and NP forms. They found that this composite approach leads to good agreement with the gravimetric measurements for both 50:50 and the CO2-rich 75:25 mixtures, as shown in Figure 27a for the mixture with the former bulk composition. Their calculations also demonstrated that the crystallites in the LP form contain both CH4 and CO2 molecules, while those in the NP structure only adsorb CO2 molecules. A HOMC simulation approach (a combination of standard GCMC and MD simulations with flexible force fields) was further performed to follow the evolution of the composition of the adsorbed mixture at 2 bar and 303 K, starting from the LP form with an equimolar CO2/ CH4 mixture. The results shown in Figure 27b give clear evidence that CH4 molecules are excluded from the NP form, which is filled exclusively by CO2 molecules. In addition, Hamon et al. revealed that coadsorption of CO2 and CH4 leads to a similar breathing behavior of MIL-53(Cr) as with pure CO2. Such a breathing effect is mainly controlled by the partial pressure of CO2, but increasing the CH4 content progressively decreases the transformation of LP to NP. Apart from the phase-mixture model, the thermodynamic approach OFAST proposed by Coudert and co-workers (see section 3.4.5) has also been applied to study gas separations in flexible MOFs.307−309 They first employed the IAST method to obtain coadsorption data from isotherms of pure CO2 and CH4 in the LP and NP structures of MIL-53(Al).307 Then, a phase diagram was constructed at 304 K using the OFAST method to illustrate the dependence of the CO2/CH4 selectivity on the total pressure and bulk mixture composition, as given in Figure 28. Obviously, at pressures below 0.2 or above 8 bar, the predicted CO2/CH4 selectivity is very low for all compositions because the structure is in the LP form. In contrast, the selectivity in the NP phase (central “island”) is much higher. Further, from the profile of the selectivity in this enclosed region, there is a certain range of pressure and mixture composition in which the material can have higher separation capability. Later, the study was extended by Ortiz et al. to explore the effect of temperature on mixture adsorption.309 From the predicted pressure−composition phase diagrams, they showed that the critical composition of CO2 in the CO2/ CH4 mixture that triggers structural transitions increases with increasing temperature. From the evolution of the temperature−pressure phase diagrams with mixture composition, they

significant effect on the separation of CO2 and N2, which play a dominant role for the steep steps that appear in the selectivity curves. It can be expected that such a strategy for gas mixture separation should also be applicable to other systems when the stepped phenomenon can occur for one of the components in the mixtures. As a unique subset of MOFs, ionic MOFs have received much attention in recent years due to their attractive properties for gas separations.381,382 In contrast to zeolites, the chargebalancing extraframework ions can be cationic or anionic type, depending on the chemical constituents of the MOFs. Jiang and co-workers conducted systematic computational studies to assess the performance of the ionic MOFs for carbon capture from various gas mixtures.211,383−385 They showed that the predicted adsorption selectivities in the ionic MOFs are much higher than those in the nonionic materials. On the basis of the IFI method, they further performed GCMC simulations to examine the separation behavior of three gas mixtures (CO2/ CH4, CO2/N2, and CO2/H2) in an anionic rho-ZMOF with Na+ cations, Na-rho-ZMOF.384 By switching off the charges of the Na+ ions and the framework, it was found that there is a significant drop of the selectivity relative to those in the charged framework, especially at the low-pressure region. These studies clearly demonstrate that the extraframework ions can substantially enhance the selectivity of CO2 over other gases, and thus, incorporation of extraframework ions into MOFs can be considered as a promising strategy to enhance the ability of MOFs for CO2 capture. Another interesting type of MOFs are those with highly flexible frameworks, such as MIL-53(M) (M = Al, Cr, Fe)10 and Cu(dhbc)2(4,4′-bpy).5 As noted previously, such MOFs can exhibit the unusual reversible feature of “breathing” or “gate opening” upon adsorption of guest species or the stimuli of temperature and pressure. In addition, their structural transformations are also dependent on the polar nature of guest molecules, which could pave the way to use them for separating gases with different polarities. Currently, most of the computational studies are performed for pure-component systems,386−390 and reports on gas mixtures are still very scarce. A typical study was conducted by Hamon et al.,292 where they examined the performance of MIL-53(Cr) for CO2 capture from CO2/CH4 gas mixtures. They used Raman spectroscopy to extract the fractions of LP/NP in MIL-53(Cr) upon adsorption of CO 2 /CH 4 mixtures with different bulk compositions. On the basis of the extracted information, the AR


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and Smit210 found that small amounts of water ( 190) for CO2/N2 separation over all conditions examined. In addition, Wu et al.36 recently examined the performance of 105 MOFs for CO2 capture from flue gas under industrial pressure conditions (1.0 bar). The flue gas was treated as a binary mixture of 15% CO2 and 85% N2, and the adsorption selectivities of all these materials were obtained from GCMC simulations in which the partial charges for the framework atoms were assigned by the CBAC method. This study discovered that simultaneously increasing Qst values and decreasing the free volume is a useful design rule for increasing the selectivity, as described in section 4.3.2. In addition to the above studies on the synthesized MOFs, Wilmer et al.419 performed a large-scale computational examination of over 130 000 hypothetical MOFs for the adsorptive CO2 separation applications via five adsorbent

Figure 29. Simulated and predicted mixture isotherms. Open symbol shows the simulated CO2/N2 (CO2:N2 = 14:86 in the bulk phase) mixture isotherms in Mg-MOF-74 at 313 K. Red line shows the mixture isotherm predicted by IAST. Blue arrow indicates the flue gas condition. Reprinted with permission from ref 249. Copyright 2012 Nature Publishing Group.

isotherms computed by GCMC simulations over a very wide range of fugacity. Therefore, apart from providing an accurate description of the interactions between adsobates and MOFs with open metal sites, such force fields can also be used in GCMC mixture simulations to validate the applicability of IAST for these systems. To our knowledge, this is the first study using the open-metal-sites force field to investigate separation of the CO2-containing gas mixture in MOFs with exposed cation sites. Rapid growth in the community of MOFs has led to the situation where the properties of most reported materials are scarcely explored beyond their initial discovery. To identify

Figure 30. Relationship between the heat of adsorption (Qst) of CO2 at 0.01 bar and the accessible surface area of MOFs for each case. These materials are the top 7.5% (∼10 000) MOFs in each case according to their sorbent selection parameters (S). Each plot is divided into 30 × 30 regions that are represented by a filled circle if more than 10 structures exist within the region. Color of each circle represents the average helium void fraction of all structures in that plot region. Reprinted with permission from ref 419. Copyright 2012 The Royal Society of Chemistry. AT


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4.1.2. Other Gas Mixtures. Separation of olefin and paraffin mixtures represents a class of most important and difficult chemical separations in the petrochemical industry.420 Energy-efficient adsorptive separation is widely considered as a promising alternative to the traditional cryogenic distillation technologies, but it requires a careful choice of suitable adsorbents. In contrast to the extensive studies for CO2 capture, to date, only a few computational investigations have been performed for separation of gaseous olefin/paraffin mixtures in MOFs.141,235,237,421−423 From examination of the existing studies, it seems that MOFs with open metal sites show better performance for separating such systems. Since the generic force fields like UFF and DREIDING are inadequate for describing the interactions between the π orbitals of olefin molecules and the open metal sites,67,164 the IAST method combined with pure-component experimental adsorption data is usually adopted to examine the separation properties of such MOFs.241,424,425 These studies demonstrate that some MOFs have great potential for separation of olefin/paraffin systems. Development of open-metal-sites force fields provides an opportunity to computationally predict the performance of MOFs with exposed cation sites for olefin/paraffin separations. For example, based on the force field developed using the specific interaction-isolation method (see section 3.2.1), Fischer et al.241 performed GCMC simulations to explore the separation of C3H6 from C3H8 in Cu−BTC, and the results were further compared with the IAST predictions using the experimental pure-component adsorption data. They observed that there is a reasonable qualitative agreement between the mixture isotherms obtained by the two methods, but some quantitative discrepancies are found in the propylene-rich part. In addition, the IAST method predicts constant selectivity in the entire composition range, while molecular simulations show increasing selectivity with increasing C3H8 mole fraction in the bulk phase. A significant difference is also discovered in the adsorption mechanism of the two species, where C3H8 molecules adsorbs mainly in the small pores of Cu−BTC, while C3H6 molecules also prefer to adsorb close to the metal sites. Another industrial issue of interest is the separation of rare gases because of their specific properties for commercial lighting, chemical analysis, medical applications, etc.426−428 Due to the full outer shell of valence electrons, the low chemical reactivities of the rare gases pose a unique challenge in the field of gas separation.429 To identify promising materials for such applications, van Heest et al.42 published an extensive computational survey of the capabilities of 3423 known MOFs for separating the two components in each of three binary mixtures (Kr/Ar, Xe/Kr, and Rn/Xe). For adsorptive separations, it also requires that gases can enter and leave the adsorbents at an appreciable rate. Thus, based on the single-gas Henry’s coefficients and self-diffusivities at infinite dilution plus the permeation selectivities estimated using eq 41e, a subset of 70 MOFs were identified to meet at least one of the six criteria proposed in this study. By correlating the PLDs (see the calculation method described in section 3.3.6) with the diffusion energy barriers, van Heest et al. performed a pore accessibility analysis for the full set of MOFs with regard to these gases. On the basis of the obtained inaccessible information, they further investigated the mixed-gas adsorption in the 70 materials identified, from which several promising materials were found for each separation. In particular, the material GUPJEG01 was predicted to be an excellent candidate

evaluation criteria (see section 2.8.1), aiming at giving a more complete picture of the structure−property relationships of MOFs than the previous studies.36,41,194 For such a huge database of crystalline structures generated using the building blocks-based approach (see section 3.7.1), the EQeq method described in section 3.1.1 was employed to assign the atomic partial charges for each MOF. Separation of four distinct systems was considered: natural gas (CO2:CH4 = 10:90) purification using PSA (case 1), landfill gas (CO2:CH4 = 50:50) separation using PSA (case 2) or VSA (case 3), and flue gas (CO2:N2 = 10:90) separation using VSA (case 4). They found many interesting trends for the five criteria from large-scale examination of their correlations with the heats of adsorption (Qst) of CO2, the gravimetric accessible surface areas (see the calculation method described in section 3.3.1), as well as the properties of the chemical functional groups. For example, the selectivity, CO2 working capacity, and sorbent selection parameter reach their maxima at optimal Qst values for all four cases. Each case has an optimal void fraction in which higher CO2 partial pressures in the gas phases favor greater void fractions while favoring lower optimal Qst. The correlations also indicate that higher CO2 partial pressures lead to optimal selectivities at larger pore diameters. By selecting the top 7.5% (∼10 000) MOFs in each case according to their sorbent selection parameters, Wilmer et al. further examined the relationships between the Qst, the surface area, and the void fraction for a given application. They found that for each case there is a narrow window of Qst values but a relatively broad range of surface area and void fraction combinations, as shown in Figure 30. It is expected that these structure−property relationships deriving from large-scale analysis can serve as a map for guiding future experimental effort on synthesizing MOFs with desirable structural characteristics for CO2 capture. The energy cost is one of the most important engineering parameters required to be considered for large-scale CO2 capture and storage in power plants. On the basis of a large number of hypothetical zeolite-like MOFs constructed using the zeolite topology-based approach (see section 3.7.2), Smit and co-workers44 computationally evaluated their feasibilities as adsorbents for separating CO2 from flue gas (CO2:N2 = 14:86) at 1 bar and 313 K. The atomic partial charges for each material were assigned using the CBAC method. They introduced a socalled parasitic energy as a performance metric to compare different materials. This quantity corresponds to the minimum electric load imposed on a power plant by a temperature− pressure swing capture process followed by compression. In addition, a state-of-the-art amine capture process with a parasitic energy of 1060 kJ/(kg CO2) was used as a reference to seek materials with significantly lower values. This study showed that although the overall parasitic energy of these hypothetical MOFs is higher than that for zeolites, there are still many materials that have optimized parasitic energies well below this reference system. The structures of these materials look very different from the optimal zeolite structures identified using the same computational strategy. Because the MOF structures are generated using a simple imidazole ligand, their adsorption selectivities toward CO2 are rather low, from which Smit and co-workers claimed that the ligands with higher selectivity will further reduce the parasitic energy. Thus, use of parasitic energy as a performance metric provides direct insight into the overall performance of a material, allowing one to rank adsorbents for a realistic carbon capture process. AU


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Figure 31. Top materials for Xe/Kr separation with selectivities calculated via IAST for equimolar bulk phase mixture. Stars indicate materials that exhibit reverse selectivity, that is, preferential adsorption of Kr relative to Xe. Reprinted with permission from ref 42. Copyright 2012 American Chemical Society.

Figure 32. Relationship between the Xe/Kr selectivities: (a) largest cavity diameters and (b) ratio of the large cavity (LCD) and pore-limiting (PLD) diameters for all hypothetical MOFs. In both figures, the selectivity has been cut off at 40 for clarity. Reprinted with permission from ref 264. Copyright 2012 The Royal Society of Chemistry.

atom, which decreases rapidly as cavity size increases (Figure 32a). At the same time, selectivity is highest when the LCD/ PLD ratio is between 1 and 2 (similar sizes for channels and cavities), as shown in Figure 32b. When relating a performance metric (the product of the selectivity and the adsorption capacity of Xe) to the LCD, Sirkora et al. observed that the materials best suited for Xe/Kr separation are still those with the LCD between 4 and 8 Å shown in Figure 32a. Besides the separation of the mixture systems discussed above, the computational methods developed with respect to MOFs have also been utilized to investigate adsorption of other gas mixtures in materials of this type. For example, Coudert308 applied the OFAST method to predict the evolution of the structural transitions of Cu(4,4′-bipy)(dhbc)2 upon adsorption of ternary CH4/O2/N2 mixture. It was found that the main factor determining the gate-opening pressure is the CH4 mole fraction in the mixture. Liu et al.330 performed a systematic molecular simulation study to examine the separation of CH4 from H2 in three pairs of IRMOFs with and without catenation. In this study, the method for calculating the microscopic selectivity was used to examine the selectivity distribution in these MOFs, as already described in section 3.6.1.

for separation of Xe from Kr, as shown in Figure 31. Interestingly, it was found that this material shows a reverse selectivity where adsorption favors the smaller component (Kr) of the two adsorbing species, which becomes even more significant with decreasing temperature. On the basis of a modified definition of the probe size for each adsorbate molecule, it was shown that this reverse effect can be well predicted by the accessible surface areas (see the calculation method described in section 3.3.1) of the material for Xe and Kr. The building blocks-based approach developed for constructing hypothetical MOFs (see the detail described in section 3.7.1) has also been employed for screening on a large scale the potential candidates for rare gas separations. On the basis of a database of over 137 000 MOFs generated, Sirkora et al.264 conducted a high-throughput computational study to identify promising candidates for separation of Xe and Kr (Xe:Kr = 80:20 in the bulk phase). They performed three-stage GCMC simulations with successively higher quality to examine the separation performance of these MOFs at 273 K. They found that the top 350 hypothetical MOFs have higher selectivities of Xe over Kr than the highest value reported for a Pd−MOF at that time.427 On the basis of the LCD and PLD calculated for each MOF, Xe/Kr selectivity is found to be sharply peaked when the LCD is slightly larger than a xenon AV


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4.2. Membrane-Based Separation

24000), indicating that an effective kinetic separation is applicable. On the basis of eq 41b, the membrane would have a permeation selectivity of 104−105. This unprecedented net selectivity is also higher than any known polymer or zeolite membrane reported at that time. Watanabe et al. also found the presence of slowly moving CH4 has a very weak impact on diffusion of CO2, and thus, the membrane exhibits excellent performance for CO2/CH4 separation. It can be expected that the research method adopted in this work is applicable to identify other MOF membranes with attractive properties for kinetic separations. Recently, the trend that both the adsorption and the diffusion selectivities favor CO2 in the same direction has also been observed for CO2/CH4 separation in another MOF.413 For design of membrane-based separation processes, it is desirable to have membranes with both high gas selectivity and permeability from an economic point of view.358 For that purpose, combination of eqs 41b and 41c provides an efficient approach for evaluating the separation performance of various MOF membranes.221,365,434 For example, Atci et al.435 assessed the performance of 15 different ZIF membranes for CO2 capture from binary CO2/CH4 and CO2/H2 mixtures. In their calculations, the atomic partial charges for all materials are assigned by the CBAC method except for ZIF-90 using the REPEAT method described in section 3.1.3, and the selfdiffusivities are predicted according to eq 9f with the Krishna and Paschek approach.187 It was found that only ZIF-90 membrane can exceed the Robeson’s upper bound for CO2/ CH4 separation, in which both the adsorption and the diffusion selectivities favor CO2 over CH4. However, it is a weakly H2selective membrane for CO2/H2 separation due to diffusion of H2 being strongly favored over CO2. Atci et al. also showed that ZIF-2 membrane can be a very promising candidate for CO2/ H2 separation, outperforming some typical zeolites (CHA, DDR, and MFI) and MOFs (MOF-177 and Cu-BTC) as well as other ZIF membranes, both from the permeability and from the permeation selectivity toward CO2. Therefore, these observations highlight the importance of choosing membrane material for a specific gas separation. Using a similar approach, Thornton et al.365 systematically investigated the properties of 7 ZIF membranes for gas separations involved in carbon-free energy generation. They demonstrated that ZIF-11 can meet the industrial feasibility targets for H2/CO2 separation, while ZIFs-8, -77, and -90 are attractive candidates for the CO2/CH4 system. However, for CO2 /N2 mixture, the predicted membrane properties of all ZIFs fall outside the target areas and do not exceed Robeson’s upper bound. Apart from deeply understanding the relationships between various diffusion coefficients that arise in the phenomenological M−S formulation,58,436−438 Krishna and co-workers also published a series of computational studies to evaluate the feasibility of various MOF membranes for CO 2 capture.164,183,358,439−441 On the basis of eqs 41b and 41c, they found that MOFs with large and open pore structures, such as MIL-53(Cr)-LP, IRMOF-1, and Cu-BTC, have high CO2 permeabilities but low permeation selectivities for CO2 over CH4.358 For ZIF-8 with narrow windows separating cages, due to the fact that H2 diffuses much more easily than CO2, the membrane exhibits a H2-selective separation behavior for their mixture. In another study, Krishna and van Baten164 computationally screened a large group of zeolites and MOFs for the adsorption-based and membrane-based separation of CO2 from binary CO2/CH4, CO2/N2, and CO2/H2 mixtures. They found

Compared to current technologies such as pressure swing adsorption, cryogenic distillation, and amine-solution absorption, membrane-based separation processes can be relatively more energy efficient and require lower investment cost in many industrial-scale processes.28,31 At the moment, identification and development of membranes that can exceed the Robeson’s upper bound for selectivity and permeability has become one of the research focuses in the field of MOFs. Due to the expense and challenge in preparation of membranes, computational methods can play an important role in characterizing membrane properties and thus greatly accelerate development of MOF-based membranes for practical applications. In contrast to the significant effort that has been devoted to adsorptive separations, the related investigation on MOF membranes using molecular modeling methods is still in an early stage. Similar to the sequence arranged in section 4.1, in the following we will first summarize some progress on CO2 capture and then deal with the other gas mixtures. 4.2.1. Carbon Dioxide Capture. Among the existing computational studies on MOF membranes for gas separation, removal of CO2 from various mixtures is perhaps the most popular topic, for which the first publication comes from Sholl and co-workers.430 This work and their other studies204,431 demonstrated that it is crucial to characterize the performance of MOF membranes using mixed-gas feeds rather than extrapolating their properties from the results for pure gases. To accelerate application of MOFs as membrane materials for CO2 capture, the new modeling methods introduced in section 3.9 have been employed to quantitatively evaluate their performance. Using the approximate model (eq 41b), which reflects the combined influence of adsorption and diffusion selectivities, Keskin and Sholl357 examined the membrane properties of COF-102 and five IRMOFs for CO2/CH4 and CO2/H2 separations. They found that none of these membranes has attractive performance for both gas mixtures. Similarly, Atci et al.432 studied removal of CO2 from binary CO2/CH4 and CO2/ H2 gas mixtures in bio-MOF-11 membrane. Since the higher adsorption selectivities can compensate for the lower diffusion selectivities, the membrane still exhibits relatively high permeation selectivities for the two mixtures. Thus, they suggested that bio-MOF-11 can be used as a promising membrane material for CO2 capture. The above compensation effect between the adsorption and the diffusion selectivities has also been observed in many other pure MOF membranes.221,433 Using a similar approach, Babarao and Jiang85 compared the separation performance of C168, MFI, and IRMOF-1 when they are used as membrane materials for CO2/CH4 mixtures. The permeation selectivity for CO2 over CH4 is found to be marginal in IRMOF-1, slightly enhanced in MFI, and greatest in C168 schwarzite. As noted above, membrane-based separations, which intrinsically rely on both adsorption and diffusion, would have a lower selectivity than adsorption-based separations. However, Watanabe et al.349 claimed that such a conflicting trend is not a universal property of all MOFs. To give evidence, they examined the separation of CO2/CH4 mixture in a microporous MOF membrane, Cu(hfipbb)(H2hfipbb)0.5, by a composite computational method that involves GCMC, MD, TST, and DFT calculations. They showed that apart from the favorable CO2 adsorption by a moderate selectivity (∼4), diffusion favors this gas by orders of magnitude larger than CH4 (∼2400− AW


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Figure 33. Permeation selectivities for CO2 plotted against the CO2 permeability for (a) CO2/N2, and (b) CO2/H2 mixture permeation across a variety of MOF and zeolite membranes at 300 K. Upstream fugacity is 1 MPa with a downstream condition of vacuum. Only some selected membrane materials are shown. Green line is the Robeson’s upper-bound curve for polymer membranes. Reprinted with permission from ref 164. Copyright 2011 PCCP Owners Society.

Owing to the large database of MOFs reported experimentally, the ability to efficiently isolate a small portion of them for further investigation is crucial when attempting to identify promising membrane materials for specific applications. Recently, Watanabe and Sholl43 applied a multistage computational approach to screen the membrane properties of 1163 known MOFs for CO2 capture from flue gas (CO2/N2). As shown above, the material with kinetic separation characteristics can exhibit high membrane-based separation performance. Therefore, at a first level of screening, the LCDs and PLDs of the MOFs are computed using the methods described in section 3.3.6. By defining a range (2.2−3.6 Å) for the PLD that is suitable for kinetically separating CO2 from N2, 359 interesting materials are identified. Then, the Henry’s coefficients and self-diffusivities of pure CO2 and N2 at infinite dilution in all of the selected MOFs are calculated using molecular simulations, in which the adsorbate−MOF electrostatic interactions are accounted for by the EPES method described in section 3.1.5. On the basis of eqs 41d and 41e, many MOF membranes are predicted to have very high permeabilities for CO2 and also exhibit high ideal selectivities for CO2/N2 separation compared to those of most polymer membranes. In addition to studies of pure MOF membranes, a growing research effort has been made over the past few years in incorporation of MOFs into polymeric matrices to enhance the permeability−selectivity properties of polymer membranes. Due to the large variety of MOFs and polymers, one key challenge in developing the MMMs is the requirement of efficiently screening their appropriate combinations toward specific separations of interest. In this respect, theoretical methods that can quantitatively predict the performance of different polymer/MOF combinations play an important role prior to performing experimental exploration. In 2010, Keskin and Sholl363 reported the first modeling study of MOF-based MMMs using an approach of atomistic simulations coupled with the Maxwell (eq 42a) and Bruggeman (eq 42b) models. Their predictions indicate that when a polymer membrane has a high selectivity for CO2 but low permeability, adding a MOF can enhance the membrane’s permeability slightly with little or no change in selectivity, and thus, the nature of the MOF appears to be not important. In contrast, if a polymer membrane is very permeable but has small selectivity it is

that the diffusion selectivity estimated using the ratio of selfdiffusivities is not good enough when the pore space of a material is close to be fully filled at the pressures of interest. This observation is particularly evident in the materials with strong interactions for CO2, such as MgMOF-74 (i.e., MOF74(Mg)) and ZnMOF-74. Thus, to give a more accurate estimate of the diffusion selectivities for such materials, Krishna and van Baten introduced an effective diffusivity to replace the self-diffusivity in eq 41b, which can be obtained from MD mixture simulations combined with the M−S formulation. Their calculation results reveal that zeolites such as NaX, NaY, CHA, and DDR are the preferred membrane materials when high CO2/CH4 permeation selectivities are sought, as shown in Figure 33a. For the cases of CO2/N2 and CO2/H2 separations, MgMOF-74 membrane offers both high permeability and permeation selectivity, as indicated in Figure 33b. Additionally, through careful analysis of the published experimental data, Krishna and van Baten 442 recently investigated the influence of diffusonal coupling on the permeation of a variety of binary systems including CO2/H2 and CO2/CH4 gas mixtures across different MOF and zeolite membranes. They found that the diffusional coupling effects occur when the less mobile species slows down its more mobile partner by not vacating an adsorption site quick enough for the latter to occupy that position. In the cage-type structures (such as ZIF-8, CHA, DDR, and LTA) in which the adjacent cages are separated by the narrow windows, the diffusional coupling effects are found to be negligible. The permeance of each component in the mixture can be reasonably estimated using eq 41c by replacing the mixture self-diffusivity with the single-gas M−S diffusivity. On the other hand, in structures that feature one-dimensional channels (such as MOF-74) or intersecting channels (such as MFI), the extent of the diffusional coupling effects is particularly strong. Therefore, this work demonstrated that an appropriate choice of materials with a high degree of diffusional coupling (or correlation) can result in the possibility of enhancing permeation selectivities. In addition, they showed that the diffusion coupling effects can also occur in mixtures in which molecular clustering exists due to hydrogen bonding. In such situation it is not possible to predict the M−S diffusivity of a species in the mixture and the mixture permeation behavior on the basis of pure-component permeation data alone. AX


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Figure 34. Performance of (a) ZIF-10- and (b) ZIF-11-based MMMs for CO2/CH4 separation. Straight line is Robeson’s upper-bound curve established for polymer membranes. Stars represent predictions of the Maxwell model for the MMMs with the volume fraction of ZIFs increasing from 0.1 to 0.4. Reprinted with permission from ref 444. Copyright 2012 American Chemical Society.

74(M) (M = Co or Ni) and IRMOFs-1, -8, -10, and -14 membranes for separating this mixture. It was shown that the two MOF-74 materials outperform the IRMOFs in terms of membrane selectivity, which is associated with the higher adsorption selectivities in the former. In the study of another MOF membrane,446 Zn2(BCD)2(ted), Keskin found that the mixture selectivity of CH4 over H2 is higher than the ideal selectivity (calculated using eq 16c) in the pressure range examined. She attributed the observation to two reasons: (i) CH4 reduces the concentration gradient of H2 across the membrane, and (ii) CH4 reduces the diffusion rate of H2. Similarly, Liu and Johnson447 studied the performance of a Zn(tbip) membrane and found that membrane selectivity strongly favors CH4 over H2 due to the extraordinarily large contribution from the adsorption selectivity. As noted previously, besides the requirement of high permeation selectivity, a desirable permeability is also indispensable for designing efficient membrane-based separation processes. Thus, combination of eqs 41b and 41c has been used to characterize, in a more comprehensive way, the performance of many pure MOF membranes. For example, Liu et al.448 compared the membrane properties of two mixedligands MOFs with (L5−L2) and without (L6−L2) catenation for CH4/H2 separation. They found that the former shows a higher permeation selectivity than its noncatenated counterpart, which also exhibits much better performance than ZIF-8 as well as zeolites (LTA and CHA) in terms of the selectivity and permeability toward CH4. Thus, Liu et al. suggested that catenation is a good strategy to improve the membrane properties of MOFs for gas separations. Using a similar computational approach, Krishna and van Baten164 screened the performance of a large collection of MOF and zeolite membranes for CH4/H2 separation. They showed that MOF74(Mg) and MOF-74(Zn) membranes exhibit similar CH4selective performance and offer the best combination of the permeation selectivity and permeability. In addition, based on eqs 41d and 41e, Sholl and coworkers40 computationally screened the membrane properties of 143 MOFs for CH4/H2 separation in the Henry regime. They predicted that a certain amount of MOF membranes have high H2 permeabilities relative to the known polymers as well as extraordinarily high ideal selectivities for H2 over CH4, such

crucial to select an appropriate MOF for fabricating the MMM. They also demonstrated that there is a wide range of polymers with both moderate selectivity and permeability, for which adding an appropriate MOF can yield large enhancement on their performance. Following the above study, several computational investigations have been carried out to evaluate the properties of various MOF-based MMMs for CO2 capture.364,365,434,443 As a typical representative, Yilmaz and Keskin444 assessed the performance of ZIF-based MMMs for CO2/CH4 and CO2/ H2 separations, in which eqs 16a and 41c are applied to calculate the pure gas and mixture permeabilities through ZIF phases, respectively, and both the Maxwell and the modified Felske (eq 42c) models are used to predict gas permeability through the MMMs. Compared to the modified Felske model, they found that the Maxwell model gives much better reproduction of the experimental permeabilities of the pure gases as well as the CO2/CH4 mixture through the ZIF-8- and ZIF-90-based MMMs. After validating the Maxwell model, Yilmaz and Keskin further predicted the separation performance of 360 new ZIF-based MMMs composed of 15 ZIFs and 24 polymers. They showed that adding weakly CO2-selective ZIFs into polymers can increase the permeabilities of all polymers toward CO2 without significantly changing their selectivities, as shown in Figure 34a for the ZIF-10-based MMMs. In contrast, using highly CO2-selective ZIFs as filler particles can significantly enhance both the CO2 permeability and the selectivity of some polymers, as shown in Figure 34b for the ZIF-11-based MMMs. For CO2/H2 mixture, because most ZIFs examined are highly H2 permeable, the main enhancement is observed in the H2 permeability of MMMs rather than the gas selectivity. 4.2.2. Other Gas Mixtures. Currently, a very limited number of molecular modeling studies have been reported to examine the performance of MOF-related membranes for separating other gas mixtures. Below we will briefly introduce some of the investigations conducted on several industrial mixture systems using the newly developed methods described in section 3.9. An important practical system that has received relatively intensive attention is the separation of CH4 from H2. On the basis of eq 41b, Keskin445 compared the performance of MOFAY


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Figure 35. Plot of H2/CH4 ideal selectivity against H2 permeability in the Henry regime for 143 MOFs (filled symbols) and 191 zeolites (empty symbols) at 298 K. Line is Robeson’s upper bound with an extrapolation (dash line part) for H2/CH4 separations with polymer membranes. Reprinted with permission from ref 40. Copyright 2010 American Chemical Society.

into 5 groups, Erucar and Keskin identified several MMMs with very high H2 selectivity and permeability relative to pure polymer membranes. They also suggested some useful strategies for selecting appropriate polymer/MOF combinations that can result in promising MMMs for CH4/H2 separation.

as the Cu(hfipbb)(H2hfipbb)0.5 membrane (labeled Cuhfb) shown in Figure 35. By plotting the estimated ideal selectivities against the PLDs of the MOFs, two distinct regimes are identified: one is limited by adsorption, and the other is controlled by diffusion. The materials with both high permeability and ideal selectivity fall into the diffusion-limited regime, and thus, their performance is mainly the result of an extremely selective kinetic separation. Considering the importance of rare gas separations in industry, Gurdal and Keskin449 investigated the performance of 10 MOFs as membrane materials for separation of binary Xe/ Kr and Xe/Ar mixtures. From the calculations based on eqs 41b and 41c they found that MOF-74(M) (M = Co or Ni) exhibit higher Xe permeability than all other MOFs examined. Using a similar strategy, Thornton et al.365 computationally explored the properties of 7 ZIF membranes for separating binary H2/N2 and O2/N2 mixtures, which are relevant to the process of hydrogen production from coal gasification and air separation for oxygen combustion of coal, respectively. It was found that ZIF-11 can meet industrial feasibility targets for H2/N2 separation, while ZIFs-8, -90, and -71 are attractive candidates for O2/N2 system. Apart from the above studies on pure MOF membranes, several computational studies have also been performed to examine the separation properties of MMMs for gas mixtures other than CO2-containing ones.365,446 For example, Yilmaz and Keskin444 evaluated the properties of 360 ZIF-based MMMs for H2/CH4 separation. On the basis of the calculations using the Maxell model (eq 42a), several ZIF membranes are identified to have the performance above Robeson’s upper bound. In another study,450 Erucar and Keskin found that the predictions using both the Maxwell and the modified Felske models are in good agreement with experimental data obtained for pure H2 and CH4 permeabilities in four different MMMs: Matrimid/IRMOF-1, PSF/Cu−BTC, PDMS/Cu−BTC, and Matrimd/Cu-BPY-HFS. After validating their approach, they further estimated the performance of 190 new MMMs for separation of H2/CH4 mixture, which are composed of 17 different MOFs and 7 different polymers. Polymers selected lie along Robeson’s upper bound.360 By categorizing the MOFs

4.3. Computationally Proposed Strategies

As described in the foregoing sections, a wide variety of the existing MOFs with different structural topologies, pore sizes, and chemical properties have been tested for their potential applications to separate diverse industrial gas mixtures. On the basis of the knowledge obtained from these molecular modeling studies, several strategies have been brought out for designing MOFs with enhanced separation performance. We here give a brief introduction to the main strategies proposed. 4.3.1. Tailoring Pore Size and Shape. The pore size and shape are normally the first consideration in selecting a porous adsorbent for a specific separation. A good example is the work of Düren and Snurr, who performed GCMC simulations for IRMOFs61 and found that the adsorption selectivity for nC4H10 over CH4 increases with decreasing cavity size. On the basis of IRMOF-1 with the smallest pore size (10.9/14.3 Å), they computationally designed a hypothetical material IRMOF993 with even smaller pore size (6.3/14.5 Å), which shows dramatically higher selectivity in comparison to that of the parent material. By comparing the results calculated with and without blocking the small side pockets in Cu−BTC, we also proved that the presence of the pockets can largely enhance the separation ability of the material.451 This strategy has also been applied to MOFs for experimentally preparing molecular sieve membranes toward kinetic separation of gases.452 Therefore, tailoring MOFs with appropriate pore size and shape is an effective approach to strengthen their performance for both adsorption- and membrane-based separation processes. It is often difficult to experimentally isolate pure phases of catenated and noncatenated versions of the same MOF, especially for those materials synthesized with expanded organic ligands.453 However, due to formation of additional small pores and adsorption sites, such catenation occurring in MOFs can become a beneficial factor for gas separations. From AZ


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by McDonald et al.,460 where the alkylamine functional groups are grafted on the open Cu2+ sites in the parent framework, H3[(Cu4Cl)3(BTTri)8 (CuBTTri). As described previously, the ionic MOFs can have significantly higher adsorption selectivity for gas mixture separation than the conventional MOFs. For such materials, exchange of counterions is an efficient strategy to tune their pore surface properties and thus enhance their selective adsorption performance. To understand how different extraframework ions would affect gas separations, Jiang et al. investigated the separation of CO2 and H2 in 7 cationexchanged rho-ZMOFs.400 At a given pressure, they found that the simulated selectivity largely follows the increasing order of the charge-to-diameter ratio of the cations. On the other hand, experimental studies also confirmed that ion exchange is a feasible approach to strengthen the separation performance of MOFs.461,462 Another strategy used widely for pore surface modification is to functionalize the organic ligands in MOFs. On the basis of our findings on the electrostatic characteristics of MOFs for gas separations, we did a series of computational studies to explore the possibility for enhancing the separation performance of MOFs using various functional groups.95,377 Through substituting the hydrogen in the BDC ligands in IRMOF-1 with four types of functional groups,377 we found that introducing stronger electron-donating groups leads to a higher adsorption selectivity for CO2/CH4 separation. In addition, by grafting different hydrophobic and hydrophilic groups onto the BDC ligands in UiO-66(Zr),95 we discovered that the materials modified using hydrophilic groups strongly outperform the parent solid as well as those functionalized with hydrophobic groups. This approach has also been successfully applied to other MOFs to enhance their separation performance.374,463 On the other hand, a typical experiment conducted by Yaghi et al.332 proved that the ZIFs with more polar functionalities in the organic ligands can achieve higher adsorption selectivity for CO2 over CH4 or N2. A second experimental example is that MIL-53(Al) functionalized with the amino group shows a drastically higher selectivity for CO2/CH4 separation than the nonmodified form.332,464 Besides the above approaches, we also proposed a route to achieve high separation performance of MOFs for CO2 capture by physically or chemically doping metal atoms onto organic ligands. It was found that introduction of Li atoms can induce charge transfer from them to framework atoms and also leads to higher adsorption selectivities for separating CO2 from various gas mixtures such as CO2/CH4465,466 and CO2/N2.380 On the basis of the IFI method, we found that such enhancement effect is due to the much stronger electrostatic interactions of CO2 with the modified structures. By examining the distribution of the microscopic selectivities, we also showed that for the organic ligands doped with metal atoms their chemical characteristics can change the preferential adsorption sites of CO2 in the parent material,467 resulting in a significant enhancement of the selectivities around the ligands. The ideas in these computational studies have recently been confirmed experimentally by Hupp and co-workers,468 where they first report an improvement in CO2/CH4 selectivity by doping Li into the MOFs synthesized by them. Additionally, a QSPR model was built in our previous study36 to predict the CO2/N2 adsorption selectivity in MOFs under industrial pressure condition (1 bar). The QSPR model is a function of the difference between the adsorbilities (ΔAD =

the distributions of the microscopic selectivities shown in Figures 16 and 25, it can be seen that the adsorption selectivities are indeed greatly enhanced in the catenated MOFs.207,330 Such enhancement on gas separation has also been found in other MOFs.209,211,448,454 Actually, catenation is also an alternative to tune the pore size of MOFs for kinetic separations, which can be easily incorporated into membranebased separations.202 As discussed in section 4.2.1, Keskin and Sholl363 computationally showed that the catenated MOF, Cu(hfipbb)(H2hfipbb)0.5, is an attractive membrane material for CO2/CH4 separation due to the kinetic effects, which was later confirmed experimentally by Bao and co-workers.455 For the same mechanism the material can also be utilized to separate linear and branched alkanes.456 Accordingly, both the experimental and the modeling studies have shown that forming rational self-catenated frameworks in MOFs is a feasible strategy for improving their separation efficiency. 4.3.2. Chemical Modification. Compared to conventional adsorbents, one of the most distinguishing characteristics of MOFs is the ability to precisely predesign their structures at the molecular level. On the other hand, for most MOFs, the pore surface characteristics play a crucial role in determining their separation properties for a particular target. From many in silico studies of MOFs involving chemical modifications, multiple approaches are available to achieve the improved separation performance of MOFs. The strong interactions of the open metal sites with many chemicals provide us an approach to realize property enhancement. For example, by replacing the intraframework Zn2+ ions with the Mg2+ ions, Krishna et al.437,439 computationally showed that both the adsorption and the permeation selectivities are higher in MOF-74(Mg) than those in the Zn version for separating CO2 from binary CO2/H2, CO2/CH4, and CO2/N2 mixtures. The reason can be attributed to the significantly higher binding energy of CO2 in the former, as indicated by the DFT calculations of Valenzano et al.319 Similarly, due to the stronger interaction for Fe2+ ions with the π-electron cloud of the “double bond” in olefin molecules, Long et al.425 found that MOF-74(Fe) shows much better separation performance than MOF-74(Mg) for C2H4 over C2H6 and C3H6 over C3H8 in both the adsorption- and the membrane-based separation processes. Open metal sites are usually created by removing axially coordinated solvent molecules in the as-synthesized MOFs through activations. Thus, anchoring functional organic molecules onto these active sites by postsynthesis is a useful approach to modify the chemical properties of the pore surfaces of MOFs. This strategy can be applied to achieve selective adsorption of guest molecules,457,458 for which a prototypical example is the work reported by Bae et al.459 Starting from the parent solid Zn2(BTTB)(DMF)2, they synthesized a novel material, Zn2(BTTB)(py-CF3)2, by replacing the DMF molecules coordinated to Zn2+ ions with a highly polar ligand py-CF3. Their IAST calculations indicate that the py-CF3modified MOF exhibits larger CO 2 /N 2 and CO 2 /CH 4 selectivities than the parent material as well as the material Zn2(BTTB) with open metal sites. Bae et al. explained this enhancement from a combination of two factors: one is the more attractive affinity of the −CF3 groups for CO2 than N2 or CH4, and the other is the more constricted pores of Zn2(BTTB)(py-CF3)2 that result in more strongly adsorbed CO2 due to the increased potential. A second example is the enhanced adsorption selectivity for CO2/N2 mixture reported BA


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ΔQ0st/ϕ) of the two components (see the description in section 3.8.1), where ΔQ0st is the difference of the isosteric heats of adsorption (ΔQ0st) between CO2 and N2 at infinite dilution in a MOF with porosity ϕ. Using the model, an interplay map was constructed, as shown in Figure 36, where the intervals between

mechanisms involved on the microscopic scale, which has been proved to be a powerful tool in complement to experiment. With the effort of researchers in the computational field, significant advances have been achieved in development of computational methodologies for MOFs, allowing a deep and thorough understanding of some phenomena observed. Such efforts will continue indefinitely and we believe will contribute even more in the future. Considering the development in the MOF field as well as the current urgent requirement of finding practical applications for them, we think the following are the main challenges that molecular modeling should be devoted to in the future. Development of Approaches for Constructing Hypothetical MOFs

Considering the richness in metal ions and organic ligands, the number of MOFs is nearly infinite. Therefore, a reliable approach is highly required that can construct hypothetical MOFs on a large scale quickly and further screen out the less promising candidates, leaving the outstanding structures for a target application. In this respect, the work of Snurr and coworkers39 has made great progress. Further effort is required to establish new approaches or improve the existing constructing methods as well as to give reliable predictions for various properties, from pure MOFs to their adsorption and separation performance.

Figure 36. Interplay map of ϕ and ΔQ0st for their impact on the CO2/ N2 adsorption selectivity at 0.1 MPa in MOFs, where the design strategy based on UiO-66(Zr) is also given. Reprinted with permission from ref 36. Copyright 2012 American Chemical Society.

Method Establishment for Reliable Atomic Partial Charge Estimation

the contour lines denote the different ranges of selectivity. From this figure some fundamental strategies can be generalized: (i) the influence of the porosity is evident only when ΔQ0st is large enough; in this case, decreasing the porosity is an efficient way to increase the selectivity; (ii) compared to the most usual approach that consists of pursuing a larger ΔQ0st, increasing this quantity and simultaneously decreasing ϕ seems to be a more appropriate route to enhance the selectivity. With these strategies, we conducted a computational design of new MOFs. Considering that in practical applications the MOF structures should be stable under humid environments, UiO66(Zr) was selected as the parent material, which has been shown to have an excellent structural stability469,470 as well as good performance for CO2/CH4 separation.95,182,470−472 We first considered its two functionalized forms,95 UiO-66(Zr)NH2 and UiO-66(Zr)-(CF3)2. The triangle in Figure 36 clearly emphasizes that there are different possible routes to enhance the selectivity: route 1 denotes that by increasing ΔQ0st with a similar porosity the selectivity can be largely improved; route 2 shows that with even similar ΔQ0st, a drastic decrease of porosity can also lead to an evident increase of the selectivity. However, if we take route 3 to simultaneously increase ΔQ0st and decrease ϕ, the largest enhancement on the selectivity can be reached. Starting from this conclusion, we designed a novel modified version with highly polar sulfonic groups, UiO-66(Zr)(SO3H)2. It was found that the so-built material shows both significantly increased ΔQ0st and decreased porosity, leading to an enhanced selectivity of 82 that is about 400% of the value for the parent material, as shown in Figure 36. This observation also indicates that such a predictive approach using the QSPR model is very useful to guide the synthesis effort toward identifying more appropriate materials.

The atomic charges in MOF frameworks are usually required as input in molecular modeling of MOF systems. For large-scale computational screening of MOFs, a rapid and reliable method to estimate them is essential. Unfortunately, all existing methods face some drawbacks, either requirement of extending charge database (CBAC218), addition of fine-tuned constraints for MOFs with buried atoms (REPEAT223), use of empirical ad hoc parameters (EQeq217), or limiting to rigid frameworks (EPES method214). Therefore, new methods that can overcome these problems are highly needed. Modeling of MOFs with Highly Flexible Frameworks

A lot of flexible MOFs have been synthesized and are shown to have very attractive advantages in both adsorption- and membrane-based separations. At the moment, the key problem that limits the computational study of flexible MOFs is the lack of a general force field; all developed flexible force fields are MOF dependent and thus cannot be fully transferred to other MOFs. It means that to study flexible MOFs we have to develop a special force field for each. Therefore, it would be wonderful if we could develop a general force field for flexible MOFs. In addition, the present molecular simulations mainly focus on adsorption and diffusion of pure gases in these MOFs;265,273,297 therefore, further systematic studies should be directed to understanding the mixture adsorption and diffusion in such MOFs, particularly the cooperative effects between structural flexibility and associated properties. Moreover, for practical applications, both the thermal and the chemical stabilities of MOFs are crucial; understanding the mechanical properties of MOFs is thus essential. Therefore, computational modeling of MOF stability and establishing the structure− stability relationships is of high priority. To date, besides the investigations performed using powerful quantum mechanical computational techniques,66,473−476 some classical reactive force fields (ReaxFF) that can model bond breaking and

5. CONCLUDING REMARKS AND OUTLOOK During the past decade, molecular modeling has contributed a lot to the understanding of the structure−property relationships for MOFs as well as to elucidation of the underlying BB


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through detailed molecular modeling studies, it not only can provide very useful strategies to synthesize MOFs for a specific applicatio but also can contribute to development of novel nanoporous materials in new application areas.

formation have also been developed to explore these issues.477,478 However, related studies in this respect are still at the early stage, and more effort is needed. Modeling of MOFs with Open Metal Sites

Due to the strong interactions with the open metal sites in MOFs, gas adsorption behavior usually cannot be accurately captured by simulations with generic force fields such as the UFF and DREIDING, especially in the low-pressure domain. Force fields derived from high-level first-principle calculations can offer a much more accurate description of gas−adsorbent interactions, giving much better agreement with experimental observations.240−242,248 This will open a door to gain an understanding of the competing nature of different gases. The mixture simulation results can also be used to check the applicability of IAST that is usually used to calculate the selectivity in MOFs with open metal sites.249 However, these highly desirable force fields are extremely scarce at the moment. In addition, Leclerc et al.479 recently showed that partial reduction of open metal sites Fe3+ to Fe2+ within MIL-100(Fe) occurs upon an appropriate thermal treatment. It has been confirmed that C3H6 molecules are strongly adsorbed on Fe2+ sites despite its weaker Lewis acidity compared to that of Fe3+, but the occurrence of Fe2+ sites does not modify the interaction with CO2 and C3H8. This is a very interesting experimental phenomenon, while there is no related modeling report, to the best of our knowledge, for gas separations in MOFs with open metal sites having mixed oxidation states. It can be expected that development of robust open-metal-sites force fields for MOFs is an area that is likely to see substantial activity in coming years.

AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]. Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Notes

The authors declare no competing financial interest. Biographies

Modeling of Membrane-Based Separation

Currently, it is still in the primary stages for computational evaluation of MOF-based membranes for gas separation, and thus, many fundamental problems exist. First, the skeletons of MOFs are usually treated as rigid in molecular simulations, while the existing modeling studies have revealed that even though the frameworks of some MOFs are not highly flexible, their framework flexibilities can greatly influence diffusion of guest molecules (see section 3.5). Thus, such a rigid treatment might deviate from the intrinsic nature of MOF membranes, leading to results that may deviate largely from the real ones. Second, some assumptions are adopted in MOF membrane simulations, such as perfect solids without defects and with the pores aligned in the same direction, no surface diffusion resistance, etc. Third, for evaluating MMMs, only macroscopic mathematic models can be used by a combination of the information on pure MOF and polymer membranes, neglecting many important influencing factors, such as the sizes and shapes of MOF particles and the real networks formed by the mutual impacts of MOFs and polymers. Finally, the thickness of the simulated membranes is limited to the nanometer scale, which is much thinner than the practical situation. Therefore, future efforts should be focused on these issues using more detailed molecular modeling. Without doubt, molecular modeling is playing an indispensable role in identifying the influencing factors and understanding the microscopic mechanisms that control the performance of MOFs at the molecular level. With the help of ever-growing computational sources, molecular modeling is becoming an extremely powerful and efficient tool for largescale in silico screening of MOFs to identify the bestperforming materials for target applications. Simultaneously,

Qingyuan Yang was born in China in 1976. He received his Ph.D. degree (2005) in Chemical Engineering at Beijing University of Chemical Technology; then he joined the group of Professor Chongli Zhong. He worked in the group of Professor Guillaume Maurin as a Postdoctoral Research Fellow at Université Montpellier 2 in 2010− 2011. He currently is Professor of Chemical Engineering at Beijing University of Chemical Technology. His main research interest focuses on employing computational techniques to model gas adsorption, diffusion, and separation in metal−organic frameworks.

Dahuan Liu was born in China in 1980. He received his Ph.D. degree in Chemical Engineering at Beijing University of Chemical Technology in 2006. Then he joined the group of Professor Chongli Zhong and was appointed Assistant Professor at Beijing University of Chemical Technology. His main research interest is gas adsorption and separation in metal−organic frameworks using computer modeling. BC


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Chongli Zhong was born in China in 1966. He studied chemical engineering at Beijing University of Chemical Technology and received his Ph.D. degree in 1993 under the supervision of Professor Wenchuan Wang. He worked as an Assistant Professor in Hiroshima University in 1995−1998 and joined the group of Professor J. de Swaan Arons at Delft University of Technology as a Postdoctoral Research Fellow in 1998. He joined Beijing University of Chemical Technology and was appointed Professor in 1999. His main research interests are focused on computational study of fluid adsorption and diffusion in nanoporous materials such as MOFs and COFs for gas storage, carbon capture, and separation of industrial gas mixtures.

Jian-Rong “Jeff” Li obtained his Ph.D. degree in 2005 from Nankai University under the supervision of Prof. Xian-He Bu. Until 2007 he was Assistant Professor at the same university. In 2004, he was also a research assistant in Professor Guo-Cheng Jia’s group at Hong Kong University of Science & Technology. In 2008, he worked as a postdoctoral research associate, first at Miami University and later at Texas A&M University; in 2010 he was an assistant research scientist at the same university. Since 2011, he has been Full Professor at Beijing University of Technology. His recent research interests focus on new porous materials for energy and environmental science.

ACKNOWLEDGMENTS This work was supported by the Natural Science Foundation of China (Nos. 21136001, 21121064, and 21276009), the Program for New Century Excellent Talents in University (No. NCET-12-0755), the Specialized Research Fund for the Doctoral Program of Higher Education of China (Contract 20110010130001), and the National Key Basic Research Program of China (“973”) (2013CB733503). We also sincerely appreciate Dr. Briant L. Davis (Professor Emeritus, South Dakota School of Mines and Technology) for his warmhearted help in improving the English usage and grammar of this paper. BD


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