Heterometallic Metal-Organic Frameworks of MOF-5 and UiO-66 Families: Insight from Computational Chemistry
Fabien Trousselet, Aurélien Archereau, Anne Boutin, François-Xavier Coudert
PPublished as:
J. Phys. Chem. C , 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594, 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594
J. Phys. Chem. C , 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594, 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594
Heterometallic Metal–Organic Frameworks of MOF-5 and UiO-66Families: Insight from Computational Chemistry
Fabien Trousselet, † Aurélien Archereau, † , ‡ Anne Boutin, ¶ and François-Xavier Coudert ∗ , † † Chimie ParisTech, PSL Research University, CNRS, Institut de Recherche de Chimie Paris, 75005 Paris,France ‡ École Normale Supérieure, PSL Research University, Département de Chimie, 24 rue Lhomond, 75005Paris, France ¶ École Normale Supérieure, PSL Research University, Département de Chimie, Sorbonne Universités –UPMC Univ Paris 06, CNRS UMR 8640 PASTEUR, 24 rue Lhomond, 75005 Paris, France
E-mail: [email protected]
Abstract
We study the energetic stability and structural fea-tures of bimetallic metal–organic frameworks. Such het-erometallic MOFs, which can result from partial sub-stitutions between two types of cations, can have spe-cific physical or chemical properties used for examplein catalysis or gas adsorption. We work here to providethrough computational chemistry a microscopic under-standing of bimetallic MOFs and the distribution ofcations within their structure. We develop a method-ology based on a systematic study of possible cationdistributions at all cation ratios by means of quantumchemistry calculations at the density functional theorylevel. We analyze the energies of the resulting bimetal-lic frameworks and correlate them with various disor-der descriptors (functions of the bimetallic frameworktopology, regardless of exact atomic positions). We ap-ply our methodology to two families of MOFs knownfor heterometallicity: MOF-5 (with divalent metal ions)and UiO-66 (with tetravalent metal ions). We observethat bimetallicity is overall more favorable for pairs ofcations with sizes very close to each other, owing toa charge transfer mechanism inside secondary buildingunits. For cations pairs with significant mutual size dif-ference, metal mixing is globally less favorable; and theenergy signifantly correlates with the coordination en-vironment of linkers, determining their ability to adaptthe mixing-induced strains. This effect is particularlystrong in the UiO-66 family, because of high cluster co-ordination number.
Metal–organic frameworks (MOFs) are a class ofnanoporous materials constructed in a modular ap-proach by the combination of inorganic nodes and or-ganic linkers. They have shown great promise for gasstorage and separation applications, catalysis, and drugdelivery. Their design, structure, and properties can be varied by modification of the organic linkers, whichcan have different lengths, topologies, and geometriesand can incorporate functional groups, for example toenhance preferential binding of guest substrates viaoptimized pore shapes/diameters for molecular separa-tion. Their topology, chemical stability, and catalyticproperties can also be tuned by modifying the natureof the coordination bonds involved.One of the recent directions in MOF research hasbeen the drive toward creating multifunctional MOFs(sometimes also called “smart” MOFs), by incorporat-ing several functions in a single material: either mul-tivariate MOFs or heterogeneous MOFs encompassingseveral functions by the cumulation of different chem-ical groups or active sites with different activities; or stimuli-responsive MOFs that respond to external stim-ulus by a change in their chemical or physical properties,developing new activity under stimulation. The mostnatural avenue for multivariate MOFs is to incorporate alarge number of different functionalities on their organiclinkers, by mixing functionalized linkers based on thesame backbone and bearing different chemical groups.Multivariate MOFs based on MOF-5 have been demon-strated that can contain up to eight distinct function-alities in one phase, with ordered framework backbonebut disordered functional groups. The combination ofa variety of functionalized linkers can also in some casesgive rise to novel and complex topologies, as was shownin the case of multivariate MOF-177. Another possibility to obtain multifunctional MOFs isto design heterometallic MOFs (or mixed-metal MOFs),with different metal cations in the inorganic clustersof the MOF. Relatively simple bimetallic MOFs havebeen reported early in the advancement of MOF re-search, usually combining a pre-formed coordinationcomplex (acting as a secondary building unit, or SBU)with a metal salt to build up a three-dimensional frame-work.
However, more complex heterometallic MOFscontaining larger numbers of cations, or two types ofSBUs, have only be recently reported. One striking ex-ample is that of Yaghi’s family mixed-metal MOF-74, a r X i v : . [ c ond - m a t . m t r l - s c i ] A p r ublished as: J. Phys. Chem. C , 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594, 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594
However, more complex heterometallic MOFscontaining larger numbers of cations, or two types ofSBUs, have only be recently reported. One striking ex-ample is that of Yaghi’s family mixed-metal MOF-74, a r X i v : . [ c ond - m a t . m t r l - s c i ] A p r ublished as: J. Phys. Chem. C , 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594, 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594 which are microcrystalline MOF-74 frameworks with upto 10 different kinds of divalent metals (Mg, Ca, Sr, Ba,Mn, Fe, Co, Ni, Zn, and Cd) incorporated into the struc-ture. These heterometallic MOFs can present an im-pact on the performances of the material, due to the ad-dition of the functions of their metal centers or throughsynergistic effects of the heterometals. This was demon-strated, for example, for CO capture in the CPM-200family of materials. Yet despite this interest in heterometallic MOFs fortheir performance in applications, a detailed descriptionand characterization of these heterometallic systems isstill superficial — as with disordered complex solids.Experimentally, access to the exact composition of thematerial can be obtained, but information on the distri-bution of metals is not available. It is to be noted thateven in the case of multifunctional MOFs, the mappingof the different functional groups (random, well-mixed,or clustered) is very difficult to determine. Yet, it isof particular importance: in solid state science, the exis-tence of correlated disorder drastically affects materialsphysical and chemical properties, and is key to a widerange of useful functionalities. In the MOF area, thereis a growing realization that such might be the casetoo. It was recently demonstrated that correlated dis-order — the presence of complex states arising from thedistribution of species within a crystalline material —is present in UiO-66(Hf) with linker vacancies. Whilethis correlated disorder can be quite hard to evidence,it strongly affects the physicochemical properties of amaterial. Theoretical and computational chemistry tools havebeen widely used in the understanding of disorder insolid state physics in general and inorganic materials inparticular. However, there have been few computationalstudies of heterometallic MOFs, and all of the publishedworks, to our knowledge, assume perfectly disorderedmetal cations and focus on the impact of the mixedmetals for specific properties such as adsorption. Suchan example is the study by Lau et al. of the impact ofpost-synthetic exchange of Zr by Ti in UiO-66(Zr) oncarbon dioxide adsorption. Here, we focus on describ-ing a computational methodology, based on quantumchemical calculations at the Density Functional The-ory (DFT) level, for the study of heterometallic MOFs.The methodology allows us to predict whether, for agiven combination of metal centers, one can expect ran-dom distribution of the cations or clustering; as well asto understand which physical/chemical features have adominant impact on the energy and why some specificsubstitution patterns are preferred. We showcase it ontwo archetypical families of MOFs, namely MOF-5 andUiO-66, and show how some simple chemical reasoningcan explain the trends observed.
In this work we study two families of bimetallic MOFsystems, where each cation site is occupied by one of twometal atoms. To model them, we use quantum chemistrycalculations at the Density Functional Theory (DFT)level, with the CRYSTAL14 software package. It de-scribes fully periodic structures, uses localized atom-centered basis sets and takes advantage of symmetry ofthe crystal structures. As such, it is well suited to theporous MOFs subject of this study. The basis sets wechose can be found in the software’s basis set online li-brary, and below we give the corresponding acronymsin this library:C:
C_6-31d1G_gatti_1994 H: H_3-1p1G_1994 O (in MOF-5):
O_6-31d1_gatti_1994 O (in UiO-66), Zr: basis sets used in Ref. Ti :
Ti_86-411(d31)G_darco_unpub Hf :
Hf_ECP_Stevens_411d31G_munoz_2007 (pseu-dopotential) Ce :
Ce_ECP_Meyer_2009 (pseudopotential) Be :
Be_6-211d1G_2012 Mg :
Mg_8-511d1G_valenzano_2006 Ca :
Ca_86-511d21G_valenzano_2006 Zn :
Zn_86-411d31G_jaffe_1993 Cd :
Cd_dou_1998 Sr :
Sr_HAYWSC-311(1d)G_piskunov_2004 Ba :
Ba_HAYWSC-311(1d)G_piskunov_2004 For each of the two frameworks, the exchange-correlation functional was chosen among several can-didates (at the Generalized Gradient Approximationlevel, hybrid or not) to ensure a good agreementwith experimental data (e.g. cell parameters, metal-oxygen coordination distances) on reference structuresMOF-5(Zn) and UiO-66(Zr). The chosen function-als were B3LYP for MOF-5 structures (which hasbeen well validated in the published literature ), andPBESOL0 for UiO-66 structures (which gives goodagreement with known experimental data, see Ta-ble S2). The use of Grimme-type dispersion correc-tions was originally tested, but as the bimetallic struc-tures studied here are all of similar density and inter-molecular distances, the effect of the corrections wasfound to be insignificant and results reported in thismanuscript are thus obtained without dispersion cor-rections.Reciprocal space sampling were carried out witha k -point mesh generated using the Monkhorst-Packmethod . Given the large sizes of unit cells, a × × mesh (sampling limited to the Γ point) was used inall cases, except for bimetallic UiO-66 samples fromsubstitutions in a conventional cell, where structure-dependent meshs were used (e.g. × × or × × ,depending on the cell shape) to ensure high accuracyresults. ublished as: J. Phys. Chem. C , 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594, 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594
Ba_HAYWSC-311(1d)G_piskunov_2004 For each of the two frameworks, the exchange-correlation functional was chosen among several can-didates (at the Generalized Gradient Approximationlevel, hybrid or not) to ensure a good agreementwith experimental data (e.g. cell parameters, metal-oxygen coordination distances) on reference structuresMOF-5(Zn) and UiO-66(Zr). The chosen function-als were B3LYP for MOF-5 structures (which hasbeen well validated in the published literature ), andPBESOL0 for UiO-66 structures (which gives goodagreement with known experimental data, see Ta-ble S2). The use of Grimme-type dispersion correc-tions was originally tested, but as the bimetallic struc-tures studied here are all of similar density and inter-molecular distances, the effect of the corrections wasfound to be insignificant and results reported in thismanuscript are thus obtained without dispersion cor-rections.Reciprocal space sampling were carried out witha k -point mesh generated using the Monkhorst-Packmethod . Given the large sizes of unit cells, a × × mesh (sampling limited to the Γ point) was used inall cases, except for bimetallic UiO-66 samples fromsubstitutions in a conventional cell, where structure-dependent meshs were used (e.g. × × or × × ,depending on the cell shape) to ensure high accuracyresults. ublished as: J. Phys. Chem. C , 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594, 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594
Geometry optimizations were performed with thestandard updating scheme in CRYSTAL14; stan-dard convergence criteria (maximally 0.0012 a.u. onatomic displacements during one optimization step,and 0.0003 a.u. on forces) in the MOF-5 case, whilefor UiO-66 higher convergence criteria were used(0.0005 a.u. and 0.0001 a.u. on displacements andforces respectively). Input files and DFT-optimizedstructures are available in the online repository at https://github.com/fxcoudert/citable-data . The procedure we use to design bimetallic structuresrelies on CRYSTAL14’s tools for the description of dis-ordered solids and solid solutions, which has beenapplied in previous work to describe binary inorganicsolids, such as binary carbonates with calcite structureor the binary spinel solid solution Mg(Al,Fe) O . Wedescribe it briefly below, and illustrate it in the case ofMOF-5:a. select a reference cell of the MOF studied (e.g.the primitive cell of MOF-5, containing 8 cationsites);b. list all possible cation substitutions within thiscell, starting from a homometallic one (2 pos-sibilities in the example);c. among these structures, identify symmetry-related equivalent structures and retain only one,ending up with p distinct bimetallic structures inaddition to the 2 homometallic structures (in theMOF-5 case, p = 20 );d. for each structure, determine the remaining sym-metry — the space group in each case is a sub-group of the original homometallic framework( F m ¯3 m for MOF-5);e. for each of the p structures, perform a full energyminimization (optimizing both unit cell parame-ters and atomic positions) within its own spacegroup.The number of structures thus generated (and thus thecomputational effort) grows exponentially with the sizeof the reference cell chosen, which can be the primitivecell, conventional cell or even a supercell — which willbe useful in the case of UiO-66 (Section 4).We then analyze the relative stability of bimetallicstructures with respect to homometallic ones, via their mixing energies . For a substitution pattern labeled j ( ≤ j ≤ p ) with a substitution rate x j ∈ ]0; 1[ of ele-ment A by element B, if the energy after relaxation isE ( j ) , we define the mixing energy as: E ( j ) m = E ( j ) − x j E B − (1 − x j ) E A (1)where E A and E B are the energies of the homometallicframeworks. A mixing energy E m < indicates that a crystal with this pattern is energetically stable withrespect to demixing into A- and B-based MOFs (at T =0 and P = 0 ).In addition to the mixing energies, we also extractfrom the DFT calculations other properties of bimetal-lic structures: topology descriptors depending on thesubstitution pattern’s topology, rather than on the ex-act atomic positions after relaxation; coordination dis-tances between cations and carboxylate oxygens (aver-aged spatially on the relaxed structure); and the distri-bution of electronic charge, measured by the Mullikenpartial atomic charges on cations in the relaxed struc-ture’s ground state.Finally, we also consider isolated clusters centered onmetal nodes, formed by replacing bridging ligands bynon-bridging formate groups. Mixing energies E m , de-fined as above and obtained from relaxing such 0-D sys-tems, reflect more directly the local effects governingthe mixing, independently from lattice effects. We re-laxed various clusters of type A n − B O, where n = 4 (MOF-5) or n = 6 (UiO-66) and A, B are two metalelements. MOF-5, also known as IRMOF-1, is a prototypicalmetal–organic framework, one of the first synthesized and widely studied ever since. Its secondary buildingunits consist of M O tetrahedra, with a central oxygensurrounded by 4 divalent cation (M ) sites forming atetrahedron, and 1,4-benzenedicarboxylate linkers (ab-breviated as “bdc”). Each edge of the tetrahedron facesa carboxylate group from a linker, with each oxygen co-ordinating one of the two edge’s cations, see Fig. 1(a).Linkers, connecting neighboring tetrahedra, are orientedalong either of 3 axes orthogonal to each other, so thatthe MOF-5 structure has cubic symmetry (space group F m ¯3 m ; see Fig. 1(b)).The MOF-5 framework can a priori sustain varioustypes of divalent cations: Zn is the cation present inthe “original” MOF-5, but variants with other metalssuch as Be , Mg , Ca , Cd , . . . have been con-sidered theoretically and synthesized experimen-tally. Conditions for polymetallicity in MOF-5, i.e.the coexistence of several cation types in the crystalstructure, have been investigated in several recent ex-perimental studies, both for fundamental aspects (re-garding chemical and mechanical stability of MOFs)and motivated by possible applications e.g. in cataly-sis and adsorption. In this section we deal with MOF-5 structures wheretwo types of divalent cations (Zn , Cd , Mg , Ca ,Sr , Ca , and Be ) occupy the cationic sites. Forvarious such pairs, we consider all bimetallic structuresobtained from substitutions within the primitive cell ofMOF-5 (see Section 2). Our goal is to find out whetherand which bimetallic patterns are energetically more fa- ublished as: J. Phys. Chem. C , 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594, 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594
Geometry optimizations were performed with thestandard updating scheme in CRYSTAL14; stan-dard convergence criteria (maximally 0.0012 a.u. onatomic displacements during one optimization step,and 0.0003 a.u. on forces) in the MOF-5 case, whilefor UiO-66 higher convergence criteria were used(0.0005 a.u. and 0.0001 a.u. on displacements andforces respectively). Input files and DFT-optimizedstructures are available in the online repository at https://github.com/fxcoudert/citable-data . The procedure we use to design bimetallic structuresrelies on CRYSTAL14’s tools for the description of dis-ordered solids and solid solutions, which has beenapplied in previous work to describe binary inorganicsolids, such as binary carbonates with calcite structureor the binary spinel solid solution Mg(Al,Fe) O . Wedescribe it briefly below, and illustrate it in the case ofMOF-5:a. select a reference cell of the MOF studied (e.g.the primitive cell of MOF-5, containing 8 cationsites);b. list all possible cation substitutions within thiscell, starting from a homometallic one (2 pos-sibilities in the example);c. among these structures, identify symmetry-related equivalent structures and retain only one,ending up with p distinct bimetallic structures inaddition to the 2 homometallic structures (in theMOF-5 case, p = 20 );d. for each structure, determine the remaining sym-metry — the space group in each case is a sub-group of the original homometallic framework( F m ¯3 m for MOF-5);e. for each of the p structures, perform a full energyminimization (optimizing both unit cell parame-ters and atomic positions) within its own spacegroup.The number of structures thus generated (and thus thecomputational effort) grows exponentially with the sizeof the reference cell chosen, which can be the primitivecell, conventional cell or even a supercell — which willbe useful in the case of UiO-66 (Section 4).We then analyze the relative stability of bimetallicstructures with respect to homometallic ones, via their mixing energies . For a substitution pattern labeled j ( ≤ j ≤ p ) with a substitution rate x j ∈ ]0; 1[ of ele-ment A by element B, if the energy after relaxation isE ( j ) , we define the mixing energy as: E ( j ) m = E ( j ) − x j E B − (1 − x j ) E A (1)where E A and E B are the energies of the homometallicframeworks. A mixing energy E m < indicates that a crystal with this pattern is energetically stable withrespect to demixing into A- and B-based MOFs (at T =0 and P = 0 ).In addition to the mixing energies, we also extractfrom the DFT calculations other properties of bimetal-lic structures: topology descriptors depending on thesubstitution pattern’s topology, rather than on the ex-act atomic positions after relaxation; coordination dis-tances between cations and carboxylate oxygens (aver-aged spatially on the relaxed structure); and the distri-bution of electronic charge, measured by the Mullikenpartial atomic charges on cations in the relaxed struc-ture’s ground state.Finally, we also consider isolated clusters centered onmetal nodes, formed by replacing bridging ligands bynon-bridging formate groups. Mixing energies E m , de-fined as above and obtained from relaxing such 0-D sys-tems, reflect more directly the local effects governingthe mixing, independently from lattice effects. We re-laxed various clusters of type A n − B O, where n = 4 (MOF-5) or n = 6 (UiO-66) and A, B are two metalelements. MOF-5, also known as IRMOF-1, is a prototypicalmetal–organic framework, one of the first synthesized and widely studied ever since. Its secondary buildingunits consist of M O tetrahedra, with a central oxygensurrounded by 4 divalent cation (M ) sites forming atetrahedron, and 1,4-benzenedicarboxylate linkers (ab-breviated as “bdc”). Each edge of the tetrahedron facesa carboxylate group from a linker, with each oxygen co-ordinating one of the two edge’s cations, see Fig. 1(a).Linkers, connecting neighboring tetrahedra, are orientedalong either of 3 axes orthogonal to each other, so thatthe MOF-5 structure has cubic symmetry (space group F m ¯3 m ; see Fig. 1(b)).The MOF-5 framework can a priori sustain varioustypes of divalent cations: Zn is the cation present inthe “original” MOF-5, but variants with other metalssuch as Be , Mg , Ca , Cd , . . . have been con-sidered theoretically and synthesized experimen-tally. Conditions for polymetallicity in MOF-5, i.e.the coexistence of several cation types in the crystalstructure, have been investigated in several recent ex-perimental studies, both for fundamental aspects (re-garding chemical and mechanical stability of MOFs)and motivated by possible applications e.g. in cataly-sis and adsorption. In this section we deal with MOF-5 structures wheretwo types of divalent cations (Zn , Cd , Mg , Ca ,Sr , Ca , and Be ) occupy the cationic sites. Forvarious such pairs, we consider all bimetallic structuresobtained from substitutions within the primitive cell ofMOF-5 (see Section 2). Our goal is to find out whetherand which bimetallic patterns are energetically more fa- ublished as: J. Phys. Chem. C , 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594, 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594 (a) (b)(d)(c)
Figure 1: (a,b) MOF-5(Zn) and (c,d) UiO-66(Zr) struc-tures, considered in this study. (a,c) show the corre-sponding structure’s individual SBU, plus 3 of the car-boxylate groups coordinating it; (b,d) depict the cor-responding framework, showing all SBUs from a con-ventional unit cell plus (some) linkers connecting them.In (b) and (d), O atoms from SBUs and H atoms arenot shown for clarity. Red, Black, Green and Blue standfor O, C, Zr, and Zn respectively. Covalent and metal-oxygen bonds are shown by continuous and dashed linesrespectively; fictitious metal-metal bonds are also shownto highlight the cluster shape.vorable than the homometallic ones. We show how thisdepends not only on substitution patterns but also onintrinsic properties of the cations considered, such asion size and electronic structure.
The first situation we address is the simplest, namelythat of mixing between two cations with similar size.We use here the example of Zn and Mg: their ionicradii are close (0.60 Å and 0.57 Å respectively ) andso are the lattice parameters and interatomic distancesin the respective MOF-5 derivatives (e.g., 1.94 Å and1.96 Å for the M–O distances; see Table S1). For thispair of elements, the mixing energies are reported asa function of composition in Fig. 2(a). They are al-ways negative, with values comprised between − and − kJ/mol (per primitive cell of 2 clusters), except inthe case where both clusters are homometallic (there, | E m | < kJ.mol − ). The overall symmetry of the plotindicates that the mixing energies are nearly invariantupon interchanging the role of the elements. This is alsoseen upon metal exchange in isolated, formate-cappedclusters (see Table 1).We show in Fig. 2(e) a plot of the mixing energy E m versus the number n (AB) (here A = Zn and B = Mg)of mixed tetrahedra edges , i.e. the number of edges,in a tetrahedral metal cluster, that feature differentcations at both ends. The clear linear correlation showsthat the mixing energy is thus mainly determined byintra-cluster effects, namely of sum of pairwise interac-tions between neighboring cations. Furthermore, look-ing at metal-carboxylate coordination distances d Mg–O and d Zn–O in Fig. 3(a,b), one observes a surprising fea-ture: upon mixing, the smaller ion (Mg ) gets closer toits surrounding carboxylate oxygens, while the oppositeoccurs for Zn — though the magnitude of the effectis small, with changes of 1 pm at most. Further insightcomes from correlating these distances with the chargeson the respective cations: for both, the charge q M de-creases with increasing d M–O . This trend can be under-stood qualitatively within a classical picture of pointcharges. This also means that the Zn/Mg charge dif-ference, as that between the respective coordination dis-tances, is increased upon mixing — the correspondingvalue for pure structures being quite large already, 0.45 e from Table S1. This can be related to the ionizationenergies E e of Zn and Mg , i.e. for an element X theenergy cost of: X → X + 2 e − (2)Indeed, the double ionization energy of Zn exceeds thatof Mg by almost 3 eV. Thus, when both elements coexistin the same Zn n Mg − n O SBU, one may expect a smallelectron transfer from Mg atoms (from which valenceelectrons can be more easily removed) to Zn atoms.Possible mechanisms for this charge transfer (direct ex-change, superexchange involving the central O, or else)are not the purpose of this study; yet it seems to havean intrinsically chemical origin, and in any case it allowsthe system to gain energy from mixing, proportionallyto the number of mixed tetrahedra edges.
A radically different situation is found in (Cd, Zn) mixedframeworks, where both cations are of similar chemicalnature, featuring d electronic configurations and sim-ilar atomic charges in the homometallic MOF-5 frame-work (see Table S1). Yet they clearly differ in their size,and from this we may expect an important effect of theframework deformation to appear in the mixing energet-ics. As in the case of the (Zn, Mg) system, the plot ofmixing energies represented in Fig. 2(b) is almost sym-metric. Yet in this case the mixing energies are generallypositive, showing that heterometallic (Cd, Zn) MOF-5are energetically unstable compared to the separate ho-mometallic phases, i.e. mixing these two cations costsenergy. Although net charges on oxygens may show variations uponmixing of the same order as those on cations, we checked (seeFig. S4) that these variations depend mostly on the global com-position. They are either much smaller than charge variations oncations, or seem uncorrelated with the energy. ublished as:
J. Phys. Chem. C , 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594, 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594
J. Phys. Chem. C , 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594, 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594 n (Zn-Mg)-30-150 E m ( kJ / m o l ) x (Zn)-30-150 E m ( kJ / m o l ) n (Zn-Cd)020 0 0.5 1 x (Cd)01020(a)(e) (f)(b) n (Zn-Ca)-200 0 0.5 1 x (Zn)-30-20-100 0 2 4 n (Cd-Mg)-200 0 0.5 1 x (Cd)-200(c)(g) (h)(d) Figure 2: Bimetallic MOF-5 structures: mixing energy E m (in kJ.mol − ) versus (a–d): the fraction x (Zn) or x (Cd),of Zn or Cd respectively, at cationic sites; and (e–h) the number n (AB) of mixed tetrahedra edges (per cluster).Pairs of metal elements considered: (a,e): Zn and Mg; (b,f): Cd and Zn; (c,g): Zn and Ca; (d,h): Cd and Mg. Table 1: Mixing energies E m (in kJ.mol − ) for A B clusters of one MOF-5 tetrahedral metal centercapped with formate linkers. Symbol X (for the Sr Be and Ba Be cases) signals DFT calculationsthat yielded physically unrealistic structures where the cluster integrity is not retained.cation A
Be Mg Ca Sr Ba Zn Cd c a t i o n B Be — − . − . − . − . − . − . Mg − . — − . − . − . − . − . Ca − . − . − . − . − . Sr X − . − . — − . − . − . Ba X 1.0 − . − . — − . − . Zn − . − . − . − . − . — − . Cd − . − . − . − . − . − . —If we plot the mixing energy E m as a function of thenumber n (AB) of mixed tetrahedra edges, it doesn’thave the linear behavior previously observed in the caseof (Zn, Mg). This reflects the importance of latticestrains, induced by the difference in cation sizes at apositive energy cost; in each structure, strains dependon positions of all cations, and not only on the num-ber of neighboring cations of different type. This is alsoreflected in the mixing energies for the isolated metalclusters (Table 1): while the mixing energy for formate-capped Zn MgO and ZnMg O clusters were both clearlynegative ( − . and − . kJ/mol respectively), thoseof the Zn CdO and ZnCd O clusters are much smaller( − . and − . kJ/mol) due to larger deformation oftetrahedra.We thus wished to identify some key descriptors ofthe topologies of heterometallic structures (i.e. quan-tities depending on the Cd or Zn occupation of eachcationic sites, but independent of the mixing-inducedrelaxations) that may explain better the observed E m values. For that, let us imagine linkers as rigid bodies— an approximation actually reasonable when consid-ering intra-linker bond distances and angles in relaxedstructures. A linker’s position and orientation will relaxmore or less efficiently depending on its coordinating en-vironment, i.e. on which cation is coordinated by each ofthe 4 oxygens. For instance, if the linker is coordinatedby 2 cations of each type, one at each COO − groupin trans position, it may adapt to its environment bya small rotation around the benzene’s C axis. In con-trary, if the linker coordinates 2 cations of each type in cis configuration, such a rotation doesn’t help, so co-ordination and other bonds are expected to be morestrongly distorted, at a higher energy cost.Indeed, when plotting E m versus several types oflinker descriptors, in Fig. 4(a-d), one sees a quite clear Mean standard deviations of linker bond lengths, measured onall relaxed structures, are (cid:46) . pm; C–C=C bond angles deviatefrom 120 degrees by at most 2 degrees while C–O–C and intra-phenyl C–C–C bond angles show even less variation. ublished as: J. Phys. Chem. C , 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594, 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594
Be Mg Ca Sr Ba Zn Cd c a t i o n B Be — − . − . − . − . − . − . Mg − . — − . − . − . − . − . Ca − . − . − . − . − . Sr X − . − . — − . − . − . Ba X 1.0 − . − . — − . − . Zn − . − . − . − . − . — − . Cd − . − . − . − . − . − . —If we plot the mixing energy E m as a function of thenumber n (AB) of mixed tetrahedra edges, it doesn’thave the linear behavior previously observed in the caseof (Zn, Mg). This reflects the importance of latticestrains, induced by the difference in cation sizes at apositive energy cost; in each structure, strains dependon positions of all cations, and not only on the num-ber of neighboring cations of different type. This is alsoreflected in the mixing energies for the isolated metalclusters (Table 1): while the mixing energy for formate-capped Zn MgO and ZnMg O clusters were both clearlynegative ( − . and − . kJ/mol respectively), thoseof the Zn CdO and ZnCd O clusters are much smaller( − . and − . kJ/mol) due to larger deformation oftetrahedra.We thus wished to identify some key descriptors ofthe topologies of heterometallic structures (i.e. quan-tities depending on the Cd or Zn occupation of eachcationic sites, but independent of the mixing-inducedrelaxations) that may explain better the observed E m values. For that, let us imagine linkers as rigid bodies— an approximation actually reasonable when consid-ering intra-linker bond distances and angles in relaxedstructures. A linker’s position and orientation will relaxmore or less efficiently depending on its coordinating en-vironment, i.e. on which cation is coordinated by each ofthe 4 oxygens. For instance, if the linker is coordinatedby 2 cations of each type, one at each COO − groupin trans position, it may adapt to its environment bya small rotation around the benzene’s C axis. In con-trary, if the linker coordinates 2 cations of each type in cis configuration, such a rotation doesn’t help, so co-ordination and other bonds are expected to be morestrongly distorted, at a higher energy cost.Indeed, when plotting E m versus several types oflinker descriptors, in Fig. 4(a-d), one sees a quite clear Mean standard deviations of linker bond lengths, measured onall relaxed structures, are (cid:46) . pm; C–C=C bond angles deviatefrom 120 degrees by at most 2 degrees while C–O–C and intra-phenyl C–C–C bond angles show even less variation. ublished as: J. Phys. Chem. C , 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594, 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594 d (Cd-O)1.141.16 q ( C d ) d (Mg-O)1.51.51 q ( M g ) d (Zn-O)1.021.04 q ( Z n ) d (Zn-O)11.04 q ( Z n ) (a)(b) (d)(c) d (Mg-O)1.51.51 q ( M g ) d (Zn-O)0.9611.04 q ( Z n ) d (Cd-O)1.111.14 q ( C d ) d (Ca-O)1.51.53 q ( C a ) (e) (g)(h)(f) Figure 3: (A,B) bimetallic MOF-5 systems: Evolution of net charges q M (Mulliken atomic charges, in units of e )carried by the cations M = A, B versus distances d O–M to the coordinating carboxylate oxygens (in Å). Bothquantities are spatially-averaged. Pairs of elements (A,B) coexisting in the strutcures: (a,b): Mg and Zn; (c,d): Cdand Zn; (e,f): Zn and Ca; (g,h): Mg and Cd. Points with red circles correspond to homometallic structures. Thegeneral trend upon substitution is highlighted by a green arrow.correlation with the numbers n ( cis ) and n ( trans ) oflinkers in cis and trans configuration respectively –whereas the energy doesn’t seem to depend on otherlinker descriptors, such as the number n (M4) of link-ers coordinating 4 cations of the same type. To go fur-ther, we performed a multi-variable analysis, assuming alaw of the form E m ( j ) ,pred = a n ( AB ) ( j ) + a n ( cis ) ( j ) + a n ( trans ) ( j ) (see Supporting Information for details).This provides optimal coefficients a i defining a predicted y = E m ( j ) ,pred , such that a linear regression of E ( j ) m ver-sus E m ( j ) ,pred gives a very good correlation coefficient( r > . ). The predicted mixing energy is, as expected,increased by heterometallic edges ( a = 3 . ), and by cis linkers ( a = 2 . ) but (to a smaller extent) decreasedby trans linkers ( a = − . ).In contrast to the (Mg, Zn) case, coordination dis-tances d Zn–O of the smaller cation (Zn) are largerin mixed structures than the value d Zn–O in pureMOF-5(Zn), while d Cd–O is decreased upon mixing [seeFig. 3(c,d)]. This indicates a repartition of mixing-induced strains between these two types of bonds, inorder to minimize the strain-induced energy cost. Notethat other bond/angle degrees of freedom can also relaxin the mixing process, such as for example cation–cationdistances in a cluster (consistently with the rather large,positive a value).Charges on both cations show a similar behavior asin the (Mg, Zn) case, namely a rough decrease withincreasing coordination distance, as can be seen onFig. 3(c,d). Again, the (Cd, Zn) charge disparity is in-creased upon mixing. Yet here, considering the smalldifference ( < eV) between E e values of both ions,the observed charge transfer might have another ori-gin. It actually seems to result, at least partly, from thevariations in coordination distances (see previous para-graph), to which the charge degrees of freedom adapt. Assuming that the system gains energy from chargetransfer, in (Cd, Zn) bimetallic systems its amplitudeis too small for the resulting energy gain to balance thecosts of mixing-induced strains, hence explaining thatthese heterometallic structures are energetically unfa-vorable ( E m > ). Having identified on the cases of (Mg, Zn) and (Cd,Zn) pairs two main mechanisms impacting on the mix-ing of divalent cations in MOF-5, namely lattice strainsinduced by the difference in cation sizes and chargetransfer with intrinsically chemical origin, we then ad-dress more generic situations, when both these effectscome into play. For this we now turn to the two casesof (Ca, Zn) and (Cd, Mg) heterometallic MOF-5 struc-tures. In both cases, the first ion (A = Cd or Ca) isclearly larger than the second (B = Zn or Mg). Thus,in analogy to the (Cd, Zn) case, mixing-induced strainsare expected to lead to a reduction of the d A–O coor-dination distances, and to increase the charges q A . Yetthese two situations differ when considering the doubleionization energies of elements involved. For (Ca, Zn)systems, since E e (Zn) (cid:29) E e (Ca), one also expects anincrease in q Ca and decrease in q Zn ; while for (Cd, Mg)systems, E e (Cd)>E e (Mg) so in absence of mixing-induced strains one would expect a decrease in q Cd andincrease in q Mg .In the case of (Ca, Zn), mixing energies seen onFig. 2(c) are found always negative (for the 16 out of20 mixed structures that relaxed successfully) and notfully invariant upon Zn ↔ Ca interchange. E m again de-pends mainly on the number of mixed tetrahedra edges,but with somehow an influence of the role of linker de- ublished as: J. Phys. Chem. C , 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594, 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594
Be Mg Ca Sr Ba Zn Cd c a t i o n B Be — − . − . − . − . − . − . Mg − . — − . − . − . − . − . Ca − . − . − . − . − . Sr X − . − . — − . − . − . Ba X 1.0 − . − . — − . − . Zn − . − . − . − . − . — − . Cd − . − . − . − . − . − . —If we plot the mixing energy E m as a function of thenumber n (AB) of mixed tetrahedra edges, it doesn’thave the linear behavior previously observed in the caseof (Zn, Mg). This reflects the importance of latticestrains, induced by the difference in cation sizes at apositive energy cost; in each structure, strains dependon positions of all cations, and not only on the num-ber of neighboring cations of different type. This is alsoreflected in the mixing energies for the isolated metalclusters (Table 1): while the mixing energy for formate-capped Zn MgO and ZnMg O clusters were both clearlynegative ( − . and − . kJ/mol respectively), thoseof the Zn CdO and ZnCd O clusters are much smaller( − . and − . kJ/mol) due to larger deformation oftetrahedra.We thus wished to identify some key descriptors ofthe topologies of heterometallic structures (i.e. quan-tities depending on the Cd or Zn occupation of eachcationic sites, but independent of the mixing-inducedrelaxations) that may explain better the observed E m values. For that, let us imagine linkers as rigid bodies— an approximation actually reasonable when consid-ering intra-linker bond distances and angles in relaxedstructures. A linker’s position and orientation will relaxmore or less efficiently depending on its coordinating en-vironment, i.e. on which cation is coordinated by each ofthe 4 oxygens. For instance, if the linker is coordinatedby 2 cations of each type, one at each COO − groupin trans position, it may adapt to its environment bya small rotation around the benzene’s C axis. In con-trary, if the linker coordinates 2 cations of each type in cis configuration, such a rotation doesn’t help, so co-ordination and other bonds are expected to be morestrongly distorted, at a higher energy cost.Indeed, when plotting E m versus several types oflinker descriptors, in Fig. 4(a-d), one sees a quite clear Mean standard deviations of linker bond lengths, measured onall relaxed structures, are (cid:46) . pm; C–C=C bond angles deviatefrom 120 degrees by at most 2 degrees while C–O–C and intra-phenyl C–C–C bond angles show even less variation. ublished as: J. Phys. Chem. C , 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594, 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594 d (Cd-O)1.141.16 q ( C d ) d (Mg-O)1.51.51 q ( M g ) d (Zn-O)1.021.04 q ( Z n ) d (Zn-O)11.04 q ( Z n ) (a)(b) (d)(c) d (Mg-O)1.51.51 q ( M g ) d (Zn-O)0.9611.04 q ( Z n ) d (Cd-O)1.111.14 q ( C d ) d (Ca-O)1.51.53 q ( C a ) (e) (g)(h)(f) Figure 3: (A,B) bimetallic MOF-5 systems: Evolution of net charges q M (Mulliken atomic charges, in units of e )carried by the cations M = A, B versus distances d O–M to the coordinating carboxylate oxygens (in Å). Bothquantities are spatially-averaged. Pairs of elements (A,B) coexisting in the strutcures: (a,b): Mg and Zn; (c,d): Cdand Zn; (e,f): Zn and Ca; (g,h): Mg and Cd. Points with red circles correspond to homometallic structures. Thegeneral trend upon substitution is highlighted by a green arrow.correlation with the numbers n ( cis ) and n ( trans ) oflinkers in cis and trans configuration respectively –whereas the energy doesn’t seem to depend on otherlinker descriptors, such as the number n (M4) of link-ers coordinating 4 cations of the same type. To go fur-ther, we performed a multi-variable analysis, assuming alaw of the form E m ( j ) ,pred = a n ( AB ) ( j ) + a n ( cis ) ( j ) + a n ( trans ) ( j ) (see Supporting Information for details).This provides optimal coefficients a i defining a predicted y = E m ( j ) ,pred , such that a linear regression of E ( j ) m ver-sus E m ( j ) ,pred gives a very good correlation coefficient( r > . ). The predicted mixing energy is, as expected,increased by heterometallic edges ( a = 3 . ), and by cis linkers ( a = 2 . ) but (to a smaller extent) decreasedby trans linkers ( a = − . ).In contrast to the (Mg, Zn) case, coordination dis-tances d Zn–O of the smaller cation (Zn) are largerin mixed structures than the value d Zn–O in pureMOF-5(Zn), while d Cd–O is decreased upon mixing [seeFig. 3(c,d)]. This indicates a repartition of mixing-induced strains between these two types of bonds, inorder to minimize the strain-induced energy cost. Notethat other bond/angle degrees of freedom can also relaxin the mixing process, such as for example cation–cationdistances in a cluster (consistently with the rather large,positive a value).Charges on both cations show a similar behavior asin the (Mg, Zn) case, namely a rough decrease withincreasing coordination distance, as can be seen onFig. 3(c,d). Again, the (Cd, Zn) charge disparity is in-creased upon mixing. Yet here, considering the smalldifference ( < eV) between E e values of both ions,the observed charge transfer might have another ori-gin. It actually seems to result, at least partly, from thevariations in coordination distances (see previous para-graph), to which the charge degrees of freedom adapt. Assuming that the system gains energy from chargetransfer, in (Cd, Zn) bimetallic systems its amplitudeis too small for the resulting energy gain to balance thecosts of mixing-induced strains, hence explaining thatthese heterometallic structures are energetically unfa-vorable ( E m > ). Having identified on the cases of (Mg, Zn) and (Cd,Zn) pairs two main mechanisms impacting on the mix-ing of divalent cations in MOF-5, namely lattice strainsinduced by the difference in cation sizes and chargetransfer with intrinsically chemical origin, we then ad-dress more generic situations, when both these effectscome into play. For this we now turn to the two casesof (Ca, Zn) and (Cd, Mg) heterometallic MOF-5 struc-tures. In both cases, the first ion (A = Cd or Ca) isclearly larger than the second (B = Zn or Mg). Thus,in analogy to the (Cd, Zn) case, mixing-induced strainsare expected to lead to a reduction of the d A–O coor-dination distances, and to increase the charges q A . Yetthese two situations differ when considering the doubleionization energies of elements involved. For (Ca, Zn)systems, since E e (Zn) (cid:29) E e (Ca), one also expects anincrease in q Ca and decrease in q Zn ; while for (Cd, Mg)systems, E e (Cd)>E e (Mg) so in absence of mixing-induced strains one would expect a decrease in q Cd andincrease in q Mg .In the case of (Ca, Zn), mixing energies seen onFig. 2(c) are found always negative (for the 16 out of20 mixed structures that relaxed successfully) and notfully invariant upon Zn ↔ Ca interchange. E m again de-pends mainly on the number of mixed tetrahedra edges,but with somehow an influence of the role of linker de- ublished as: J. Phys. Chem. C , 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594, 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594 n (A2B2) n (M4)015 0 2 4 n ( cis )01020 E m n ( trans )01020 E m (a)(c) (d)(b) IRMOF-1(Cd,Zn)linker linkerlinkerlinker
CdZnr = 0.936 n ( trans )-20-10010 0 2 4 n ( trans ) -30-150 n ( cis )-20-10010 E m n ( cis ) -30-150 E m (e)(g) (h)(f) Ca-Zn mixingCd-Mg mixing r = -0.895 r = 0.719
Ca-Zn mixingCd-Mg mixing
Figure 4: Bimetallic MOF-5: mixing energy E m (inkJ.mol − ) versus number of linkers in various configu-rations. (a-d): for (Cd, Zn) mixed structures; (e,f): (Ca,Zn); and (g-h): (Cd, Mg) mixed structures. Abscisses n (M4), n (A2-B2), n ( trans ) and n ( cis ) refer to the con-figurations depicted in insets of (a-d).scriptors like n ( trans ) [see Fig. 4(e)]. A multi-variableanalysis, similar as that done above, confirms this with acoefficient a = − . [associated to n (AB)] while other | a i | values are at least 3 times smaller. Coordinationdistances and net charges, shown in Fig. 3(e,f), behavesimilarly as in the (Cd, Zn) case, with e.g. a chargedecrease for the (smaller and harder to ionize) Zn ion,roughly proportional to an increase in d Zn–O . It thus ap-pears that, when both size mismatch and charge trans-fer “push” in the same direction, they have a cooper-ative effect on the relaxation of atomic positions andcharges. When, as in the present case, the charge trans-fer effect is strong enough, the resulting energy gainovercomes strain-induced energy costs, and allows forgenerally negative mixing energies and a solid-solutionbehavior.In contrast, in the (Cd, Mg) system these two effectscompete. In consequence, a more contrasted situationis observed in Fig. 2(d,h), with E m taking both posi-tive and negative values depending on the compositionand configuration. While its dependence on the num-ber of mixed edges is unclear (Fig. 2(h)), the mixingenergy clearly tends to increase with n ( cis ) and to de- crease with n ( trans ), as was also seen e.g. for (Cd, Zn).Again, a multi-variable analysis confirms this trend: E m behaves almost linearly in function of an optimal vari-able y built on n ( cis ) and n ( trans ) only, with a corre-lation coefficient r = 0 . . The net charges on bothCd and Mg follows their ionization potentials (decreasein q Cd and increase in q Mg ), but with amplitudes typi-cally smaller than in the (Mg, Zn) or (Ca, Zn) systems.Thus the energy gain from charge transfer is smaller,but may be sufficient to have E m < if enough link-ers have environments allowing for efficient relaxations(e.g. many linkers in trans configuration and few in cis configuration). Aside from the structure-dependent signof E m , a further indication of the competition betweenboth effects, frustration of charge and strain degrees offreedom, is that cation charges q (M) do not decreasewith increasing coordination distance (see Fig. 3(g,h))in contrary to the previous cases. We presented in the previous sections some examplesof bimetallic MOF-5 systems with representative be-havior. We performed further simulations on additionalbimetallic systems in order to confirm the conclusionsreached. In particular, results of calculations on peri-odic systems for four other choices of cationic elementpairs are shown in the Supporting Information. They allexhibit similarity with the archetypical cases exposedabove. (Cd, Ca) bimetallic systems, where both cationsare of nearly same size, show a solid-solution behaviorsimilar to the (Zn, Mg) case. (Zn, Sr) systems show sim-ilarities with the (Zn, Ca) case, with even lower mixingenergies and larger displacements due the even largermutual differences in both charges and sizes of cations.In (Ca, Mg) and (Be, Mg) systems, which are charac-terized by large differences between cation sizes, mix-ing energies are very sensitive to linker environments.In particular, in the (Ca, Mg) case where the mutualcharge difference is negligible, they correlate well withchanges in coordination distances.In addition to these periodic calculations, we per-formed systematic calculations on isolated bimetal-lic A BO clusters, comprising 3 cations of one el-ement and one of another element, and with 1,4-benzenedicarboxylate linkers replaced by formates tocap the cluster. We performed these calculations for allcouples where A and B are either: Be, Mg, Ca, Sr, Ba,Zn, or Cd — hence a total of 42 different clusters. 40 ofthese lead to a stable structure, retaining the integrityof the inorganic cluster. We report in Table 1 their mix-ing energies. We can see that cation mixing is gener-ally favorable, with a negative mixing energy E m . It ishowever less favorable when both cations bear similarpartial charges, such as the cases of (Ca, Mg) and (Zn,Be), confirming that the charge imbalance, taken alone,drives an electronic reconstruction thanks to which theenergy is lowered upon mixing. For most pairs of ele-ments (A, B), mixing energies E m (A B) and E m (B A) ublished as: J. Phys. Chem. C , 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594, 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594
Figure 4: Bimetallic MOF-5: mixing energy E m (inkJ.mol − ) versus number of linkers in various configu-rations. (a-d): for (Cd, Zn) mixed structures; (e,f): (Ca,Zn); and (g-h): (Cd, Mg) mixed structures. Abscisses n (M4), n (A2-B2), n ( trans ) and n ( cis ) refer to the con-figurations depicted in insets of (a-d).scriptors like n ( trans ) [see Fig. 4(e)]. A multi-variableanalysis, similar as that done above, confirms this with acoefficient a = − . [associated to n (AB)] while other | a i | values are at least 3 times smaller. Coordinationdistances and net charges, shown in Fig. 3(e,f), behavesimilarly as in the (Cd, Zn) case, with e.g. a chargedecrease for the (smaller and harder to ionize) Zn ion,roughly proportional to an increase in d Zn–O . It thus ap-pears that, when both size mismatch and charge trans-fer “push” in the same direction, they have a cooper-ative effect on the relaxation of atomic positions andcharges. When, as in the present case, the charge trans-fer effect is strong enough, the resulting energy gainovercomes strain-induced energy costs, and allows forgenerally negative mixing energies and a solid-solutionbehavior.In contrast, in the (Cd, Mg) system these two effectscompete. In consequence, a more contrasted situationis observed in Fig. 2(d,h), with E m taking both posi-tive and negative values depending on the compositionand configuration. While its dependence on the num-ber of mixed edges is unclear (Fig. 2(h)), the mixingenergy clearly tends to increase with n ( cis ) and to de- crease with n ( trans ), as was also seen e.g. for (Cd, Zn).Again, a multi-variable analysis confirms this trend: E m behaves almost linearly in function of an optimal vari-able y built on n ( cis ) and n ( trans ) only, with a corre-lation coefficient r = 0 . . The net charges on bothCd and Mg follows their ionization potentials (decreasein q Cd and increase in q Mg ), but with amplitudes typi-cally smaller than in the (Mg, Zn) or (Ca, Zn) systems.Thus the energy gain from charge transfer is smaller,but may be sufficient to have E m < if enough link-ers have environments allowing for efficient relaxations(e.g. many linkers in trans configuration and few in cis configuration). Aside from the structure-dependent signof E m , a further indication of the competition betweenboth effects, frustration of charge and strain degrees offreedom, is that cation charges q (M) do not decreasewith increasing coordination distance (see Fig. 3(g,h))in contrary to the previous cases. We presented in the previous sections some examplesof bimetallic MOF-5 systems with representative be-havior. We performed further simulations on additionalbimetallic systems in order to confirm the conclusionsreached. In particular, results of calculations on peri-odic systems for four other choices of cationic elementpairs are shown in the Supporting Information. They allexhibit similarity with the archetypical cases exposedabove. (Cd, Ca) bimetallic systems, where both cationsare of nearly same size, show a solid-solution behaviorsimilar to the (Zn, Mg) case. (Zn, Sr) systems show sim-ilarities with the (Zn, Ca) case, with even lower mixingenergies and larger displacements due the even largermutual differences in both charges and sizes of cations.In (Ca, Mg) and (Be, Mg) systems, which are charac-terized by large differences between cation sizes, mix-ing energies are very sensitive to linker environments.In particular, in the (Ca, Mg) case where the mutualcharge difference is negligible, they correlate well withchanges in coordination distances.In addition to these periodic calculations, we per-formed systematic calculations on isolated bimetal-lic A BO clusters, comprising 3 cations of one el-ement and one of another element, and with 1,4-benzenedicarboxylate linkers replaced by formates tocap the cluster. We performed these calculations for allcouples where A and B are either: Be, Mg, Ca, Sr, Ba,Zn, or Cd — hence a total of 42 different clusters. 40 ofthese lead to a stable structure, retaining the integrityof the inorganic cluster. We report in Table 1 their mix-ing energies. We can see that cation mixing is gener-ally favorable, with a negative mixing energy E m . It ishowever less favorable when both cations bear similarpartial charges, such as the cases of (Ca, Mg) and (Zn,Be), confirming that the charge imbalance, taken alone,drives an electronic reconstruction thanks to which theenergy is lowered upon mixing. For most pairs of ele-ments (A, B), mixing energies E m (A B) and E m (B A) ublished as: J. Phys. Chem. C , 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594, 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594 are in the same range, but with small relative differences— apart from systems with cations of very different sizeor nature, such as (Ba, Sr) or (Ba, Zn). We can thus seethat while a wide variety of behavior can be expectedin heterometallic MOFs, in the absence of strain themixing of metal cations is in a vast majority of casesenergetically favorable.
The UiO-66 structure (see Figure 1) is formed withM O (OH) octahedra (later abbreviated M ), whereM is a tetravalent cation (M = Ti, Zr, Hf or Ce),connected to each other here again by bdc linkers. Ma-terials of the UiO-66 family are, like IRMOFs, an in-teresting playground for studying the consequences ofcation substitutions and possibility of bimetallicity. Par-tial Zr → Ti or Zr → Hf post-synthetic exchange (PSE)have been reported in the literature , with more ef-ficiency in the first case. In both cases the bimetallicsamples retained the UiO-66 structure, however no in-formation could be obtained on the spatial distributionof cations. It is not known whether intra-cluster mix-ing occurs, whether specific substitution patterns arefavored, in a word whether some order exists withinthe heterometallic samples. Post-synthetic replacementof Zr by Ce has also been achieved in UiO-66, butthis study revealed a change in oxidation state duringthe process, as well as a possible influence of ligand va-cancies. Regarding possible applications, partial post-synthetic exchange in UiO-66 was shown to improve itsefficiency in catalysis, as well as in CO adsorption. The standard UiO-66 structure has cubic symmetry anda F ¯43 m space group, with a primitive unit cell con-taining a single Zr octahedron while the conventionalcell contains four. If considering, as earlier for MOF-5,cation substitutions within a primitive cell, one obtains = 64 possible substituted structures; but taking intoaccount point group symmetries reduces the number ofsymmetry-inequivalent bimetallic structures to only 9(excluding homometallic cases).First we discuss bimetallic structures built from sub-stitutions within a primitive cell of UiO-66. For threepairs of metallic elements: (Zr, Ti), (Zr, Ce) and (Zr,Hf), all such structures have been relaxed, and theirmixing energies are depicted in Fig. 5. They are al-ways negative for (Zr, Hf) systems, which is consis-tent with our previous observations in MOF-5, giventhe very close sizes of Zr and Hf. The roughly lineardependence of E m on the number of bimetallic octa-hedra edges, shown for (Zr, Hf) systems in Fig. 5(b),indicates a small mixing-induced energy gain, possiblyfrom charge transfer. In contrast, mixing energies arepositive for the other element pairs, (Zr, Ti) and (Zr,Ce), where mixing is expected to occur at a substan-tial deformation cost (see Table S2). Given the quadru- x (Zr)0204060 E m (Ce, Zr)(Hf, Zr)(Ti, Zr)(a) -4 -2 0 2 4 6 V m E m n (AB)03060 E m (Zr, Ti)(Zr, Hf)(Zr, Ce) -0.3 0-8-40 (b) (c) Figure 5: Bimetallic UiO-66 structures (substitutionswithin a primitive cell), with coexistence of Zr and ei-ther Ti, Hf or Ce. Top: mixing energy E m versus Zr oc-cupancy on cationic sites. Bottom: E m versus (b) num-ber of bimetallic octahedra edges n (AB) (per octahe-dron) and (c) mixing-induced variation of the primitivecell volume (in Å ).ple ionization energies of the elements involved, chargetransfer effects are also expected to occur and even bemore important than for (Zr, Hf), but not sufficient tocounterbalance deformation costs. A further indicationfor the role of these energy costs comes from a compari-son with mixing energies computed on isolated clusters(see Table S4): while in the latter case mixing energiesare much lower for (Zr, Ce) clusters than for (Zr, Ti)clusters, the mutual difference is much reduced in peri-odic systems, presumably by lattice effects.The mixing-induced volume variation, V m , can be de-fined analogously to the mixing energy. Both quantitiesare plotted against each other in Fig. 5(c), for the sys-tems considered above. They correlate rather well in the(Zr, Hf) case, with moderate mixing-induced reductionin cell volume nearly proportional to the mixing-inducedenergy gain. This correlation is less obvious in the (Zr,Ti) and (Zr, Ce) cases, which also differ qualitativelyfrom each other: in the former the volume increasesupon mixing in most structures, while in the latter italways decreases. This difference may have the same ori-gin as the relatively lower mixing energies for (Zr, Ce)than for (Zr,Ti) systems, and indicate more efficient at-tractive interactions between neighboring cations in theformer case. ublished as: J. Phys. Chem. C , 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594, 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594
The UiO-66 structure (see Figure 1) is formed withM O (OH) octahedra (later abbreviated M ), whereM is a tetravalent cation (M = Ti, Zr, Hf or Ce),connected to each other here again by bdc linkers. Ma-terials of the UiO-66 family are, like IRMOFs, an in-teresting playground for studying the consequences ofcation substitutions and possibility of bimetallicity. Par-tial Zr → Ti or Zr → Hf post-synthetic exchange (PSE)have been reported in the literature , with more ef-ficiency in the first case. In both cases the bimetallicsamples retained the UiO-66 structure, however no in-formation could be obtained on the spatial distributionof cations. It is not known whether intra-cluster mix-ing occurs, whether specific substitution patterns arefavored, in a word whether some order exists withinthe heterometallic samples. Post-synthetic replacementof Zr by Ce has also been achieved in UiO-66, butthis study revealed a change in oxidation state duringthe process, as well as a possible influence of ligand va-cancies. Regarding possible applications, partial post-synthetic exchange in UiO-66 was shown to improve itsefficiency in catalysis, as well as in CO adsorption. The standard UiO-66 structure has cubic symmetry anda F ¯43 m space group, with a primitive unit cell con-taining a single Zr octahedron while the conventionalcell contains four. If considering, as earlier for MOF-5,cation substitutions within a primitive cell, one obtains = 64 possible substituted structures; but taking intoaccount point group symmetries reduces the number ofsymmetry-inequivalent bimetallic structures to only 9(excluding homometallic cases).First we discuss bimetallic structures built from sub-stitutions within a primitive cell of UiO-66. For threepairs of metallic elements: (Zr, Ti), (Zr, Ce) and (Zr,Hf), all such structures have been relaxed, and theirmixing energies are depicted in Fig. 5. They are al-ways negative for (Zr, Hf) systems, which is consis-tent with our previous observations in MOF-5, giventhe very close sizes of Zr and Hf. The roughly lineardependence of E m on the number of bimetallic octa-hedra edges, shown for (Zr, Hf) systems in Fig. 5(b),indicates a small mixing-induced energy gain, possiblyfrom charge transfer. In contrast, mixing energies arepositive for the other element pairs, (Zr, Ti) and (Zr,Ce), where mixing is expected to occur at a substan-tial deformation cost (see Table S2). Given the quadru- x (Zr)0204060 E m (Ce, Zr)(Hf, Zr)(Ti, Zr)(a) -4 -2 0 2 4 6 V m E m n (AB)03060 E m (Zr, Ti)(Zr, Hf)(Zr, Ce) -0.3 0-8-40 (b) (c) Figure 5: Bimetallic UiO-66 structures (substitutionswithin a primitive cell), with coexistence of Zr and ei-ther Ti, Hf or Ce. Top: mixing energy E m versus Zr oc-cupancy on cationic sites. Bottom: E m versus (b) num-ber of bimetallic octahedra edges n (AB) (per octahe-dron) and (c) mixing-induced variation of the primitivecell volume (in Å ).ple ionization energies of the elements involved, chargetransfer effects are also expected to occur and even bemore important than for (Zr, Hf), but not sufficient tocounterbalance deformation costs. A further indicationfor the role of these energy costs comes from a compari-son with mixing energies computed on isolated clusters(see Table S4): while in the latter case mixing energiesare much lower for (Zr, Ce) clusters than for (Zr, Ti)clusters, the mutual difference is much reduced in peri-odic systems, presumably by lattice effects.The mixing-induced volume variation, V m , can be de-fined analogously to the mixing energy. Both quantitiesare plotted against each other in Fig. 5(c), for the sys-tems considered above. They correlate rather well in the(Zr, Hf) case, with moderate mixing-induced reductionin cell volume nearly proportional to the mixing-inducedenergy gain. This correlation is less obvious in the (Zr,Ti) and (Zr, Ce) cases, which also differ qualitativelyfrom each other: in the former the volume increasesupon mixing in most structures, while in the latter italways decreases. This difference may have the same ori-gin as the relatively lower mixing energies for (Zr, Ce)than for (Zr,Ti) systems, and indicate more efficient at-tractive interactions between neighboring cations in theformer case. ublished as: J. Phys. Chem. C , 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594, 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594
Finally, linker configurations are expected to correlatewith the mixing energies in bimetallic UIO-66 struc-tures. Yet due to the small size of the cell considered(i.e. small number of mixed structures, with linker en-vironments much constrained by periodicity) we couldnot draw precise conclusions on the role of linker config-urations from these data alone – this led us to considera larger set of bimetallic structures, as described here-after.
In order to better identify the key factors impacting theenergy of bimetallic UiO-66 structures, and in partic-ular the role of linker configurations, we focus here onthe case of the bimetallic (Zr, Ce) UiO-66 and considerstructures where substitutions were carried out in theconventional cell of UiO-66, containing four M clus-ters. This allows for a much larger number of possi-ble configurations (about / (cid:39) , symmetry-independent structures). Among those, we focused ona subclass of 30 configurations, chosen to fulfill the fol-lowing two criteria: • all clusters, for fixed cluster composition, mini-mize the number of bimetallic octahedra edges;the motivation for this criterion is to focus onother factors influencing the energy, as well as onlow- rather than high-energy structures; • retaining by at least one (and in most cases sev-eral) point group symmetry, in order to minimizecomputation time.For all of the selected structures, we found positivemixing energies (Fig. 6) in the range of 10–40 kJ.mol − (per primitive cell), similar to those observed when sub-stitutions were considered in a primitive cell. Yet here,the larger data set allowed us to identify clear trendsconcerning the impact of individual variables (numberof bimetallic edges, linker configurations). Fig. 6(a-d)shows a clear correlation between the energy and thenumbers of 2 types of linkers: E m tends to be higherwith more linkers coordinating 4 atoms of the same type[descriptor n (M4), Fig. 6(a)], and lower with linkers co-ordinating 2 Ce on one carboxylate group and 2 Zr onthe other [descriptor n (A2B2), Fig. 6(c)]. It seems tocorrelate with the numbers of trans and cis linkers aswell, yet less clearly – same goes for the correlation be-tween E m and the number n (AB) of bimetallic octahe-dra edges (see Fig. S5). A multi-variate analysis con-firmed that the energy is more directly influenced bylinker descriptors n (M4) and n (A2B2), rather than byother linker- or cluster-descriptors.In other words, in bimetallic UiO-66 framework, ef-fects of size mismatch — when the two cations havesignificantly different size — make the linker configu-rations energetically inequivalent, as in MOF-5, but ina different way. This is due to the higher cluster coor-dination (12 ligands around a cluster, instead of 6 in n (A2B2) E m n (M4) E m n ( trans ) E m n ( cis ) E m linkerlinker (a)(c) (b)(d)
20 40 E m -4-3-2 V m ( Å ) d ( C e - O ) E m d ( Z r- O ) (e) (f)(g) Figure 6: UiO-66(Ce,Zr) structures (substitutionswithin a conventional cell). Right: Mixing energy E m (in kJ.mol − , per cluster) versus numbers of linkers ofspecific types (as indicated in Fig. 4) — all quantitiesare per primitive cell. Left: Plots of E m (abscisses) ver-sus: (e) mixing-induced volume variation V m (in A , perprimitive cell); and (f,g): average coordination distancesbetween either Ce (f) or Zr (g) and the carboxylate oxy-gens coordinating them (in Å).MOF-5). Indeed, linker rotations, which stabilize trans linkers in MOF-5, are more difficult in UiO-66: sucha rotation would bring a COO − group of the rotatedligand too close to a COO − group of a neighboring lig-and, and thus involve significant additional linker–linkerrepulsions. Instead, the optimal linker configuration isthe A B configuration, where each COO − group co-ordinates one type of cations: the corresponding relax-ation process, to adapt the cation size mismatch, is asmall-amplitude translation along the linker axis. Thisdoes not bring the translated linker too close to anotherlinker, and appears thus as the most efficient relaxationmechanism, compared to other linker configurations.We note that bimetallic (Zr, Hf) systems behave quitedifferently in this respect (see Fig. 5 and Fig. S6), withthe (negative) mixing energy nearly proportional tothe number of bimetallic octahedra edges. There, evenin a higly-coordinated framework, the quasi-absence ofmixing-induced strains leaves intra-cluster interactionsas the key mechanism dominating mixing energies.For further insight, we also analyzed some variablesquantifying the mixing-induced lattice distortions: rela- ublished as: J. Phys. Chem. C , 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594, 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594
20 40 E m -4-3-2 V m ( Å ) d ( C e - O ) E m d ( Z r- O ) (e) (f)(g) Figure 6: UiO-66(Ce,Zr) structures (substitutionswithin a conventional cell). Right: Mixing energy E m (in kJ.mol − , per cluster) versus numbers of linkers ofspecific types (as indicated in Fig. 4) — all quantitiesare per primitive cell. Left: Plots of E m (abscisses) ver-sus: (e) mixing-induced volume variation V m (in A , perprimitive cell); and (f,g): average coordination distancesbetween either Ce (f) or Zr (g) and the carboxylate oxy-gens coordinating them (in Å).MOF-5). Indeed, linker rotations, which stabilize trans linkers in MOF-5, are more difficult in UiO-66: sucha rotation would bring a COO − group of the rotatedligand too close to a COO − group of a neighboring lig-and, and thus involve significant additional linker–linkerrepulsions. Instead, the optimal linker configuration isthe A B configuration, where each COO − group co-ordinates one type of cations: the corresponding relax-ation process, to adapt the cation size mismatch, is asmall-amplitude translation along the linker axis. Thisdoes not bring the translated linker too close to anotherlinker, and appears thus as the most efficient relaxationmechanism, compared to other linker configurations.We note that bimetallic (Zr, Hf) systems behave quitedifferently in this respect (see Fig. 5 and Fig. S6), withthe (negative) mixing energy nearly proportional tothe number of bimetallic octahedra edges. There, evenin a higly-coordinated framework, the quasi-absence ofmixing-induced strains leaves intra-cluster interactionsas the key mechanism dominating mixing energies.For further insight, we also analyzed some variablesquantifying the mixing-induced lattice distortions: rela- ublished as: J. Phys. Chem. C , 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594, 120, 24885–24894,DOI: 10.1021/acs.jpcc.6b08594 tive volume variations, quantified by the mixing volume V m , and coordination distances d Ce–O and d Zr–O (hereagain to carboxylate oxygens) and correlate them withthe mixing energy. Fig. 6(e) indicates that the mixing-induced volume variation is always negative (as seenin subsection 4.1 for (Zr, Ce) systems), and the vol-ume reduction tends to be more important for lower-energy structures. As for coordination distances, shownin Fig. 6(f,g), they behave similarly as those of e.g. (Cd,Zn) MOF-5 structures: those of the smaller (Zr) ionincrease upon mixing while d Ce–O are decreased. Re-markably, we observe a clear linear correlation betweenthose distances and the mixing energy. These featuresconfirm the importance of lattice distortions in deter-mining the mixing energies and relative stabilities ofbimetallic structures, and the interplay between thesedistortions and linker environments.
In this study, we have proposed and used a DFT-basedmethodology for a systematic study of heterometallicMetal-Organic Frameworks, exemplified on the frame-works MOF-5 and UiO-66. Based on criteria of ener-getic stability, we could determine, among the cationpairs considered, which ones can lead to stable bimetal-lic MOF phases, and adressed the spatial distributionof metals in such phases.These results give a coherent picture of mixed-metalMOFs, with two essential aspects dominating their ener-getics. First, the coexistence of distinct types of cationsinside a SBU (cluster) leads to a charge transfer betweenthem, depending on their intrinsic chemical properties,and allows the system to gain energy upon mixing. Sec-ond, the difference between cation sizes induces localstrains; individual linkers can adapt to these strainsmore or less efficiently, depending on their coordinationenvironments, and the latter contribute significantly tothe structure’s total energy.When both effects have comparable importance, theycan either cooperate [case of MOF-5(Zn,Ca)], which am-plifies both intra-cluster charge transfers and the ener-getic stability of specific spatial distributions; or com-pete[case of MOF-5(Cd,Mg)], and result in systems withgenerally higher mixing energies and a more complexstructure-stability correlation. The cluster coordinationnumber is also an important factor: in UiO-66, the highcluster coordination makes linker relaxations less effi-cient than in MOF-5, but still of primary importance inconditioning the energies of mixed-metal structures.However the present study only unveiled these twomechanisms; further work, both experimentally and the-oretically, on a wider variety of bimetallic MOFs, wouldhelp to better understand the conditions for energeticstability. Future work will also be needed to addressthe question of thermodynamic stability, by accountingfor both configurational and vibrational entropy in het-erometallic structures. Cation (dis)ordering is also an aspect that could also be studied with a modeling atlarger scale, using (i) the DFT-based energy-structurerelationship to define potential energy terms and (ii)statistical energy sampling on larger systems to estimatevarious types of cation order parameters. More gener-ally, one could find inspiration from studies of configura-tional disorder in inorganic chemistry (alloys, ...). Thequestion of whether cations order in bimetallic MOFscould also be put in perspective with recent findings ofcorrelated disorder in ligand-defective UiO-66(Zr) ,with an underlying mechanism still only partially un-derstood.Finally, it is important to note that in real UiO-66samples, one ought to take into account common de-fects such as frequently-occurring linker vacancies ,with an average cluster coordination number that canbe closer to 11 than the nominally expected value of 12.In such a situation, not all cis linkers are equivalent,since those near a linker vacancy can rotate more eas-ily than in the case of MOF-5. The influence of linkerconfigurations on mixing energies could thus be less pro-nounced than in the fully-coordinated systems we con-sidered in our calculations. More importantly, we sus-pect that cation substitutions can occur more easily ata site close to one or several linker vacancies: as exem-plified in the extreme case of isolated clusters (see Ta-ble 1), the capping groups (e.g. HO − , H O, or formate)can move much more easily than the bdc linkers, in or-der to adapt a cation size mismatch. From this pointof view, ligand vacancies should not be avoided if oneaims at large cation substitution rates (e.g. to promoteor engineer catalysically-active sites). Another impor-tant aspect, not evoked here, is the mechanical stability,which is lower in systems with high rate of ligand va-cancies. Similarly, it may be worth investigating theconditions for mechanical stability, and more generallythe mechanical properties of bimetallic MOFs.
Acknowledgement
We acknowledge access tohigh-performance computing platforms provided bya GENCI grant (x2016087069).
Supporting Information Available:
Figures andtables of data on bimetallic MOF-5 and UiO-66 struc-tures. This material is available free of charge via theInternet at http://pubs.acs.org/ . References (1) Furukawa, H.; Cordova, K. E.; O’Keeffe, M.;Yaghi, O. M.
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