The flexibility window in zeolites

LETTERS

The flexibility window in zeolites ASEL SARTBAEVA, STEPHEN A. WELLS, M. M. J. TREACY AND M. F. THORPE* Department of Physics and Astronomy, Arizona State University, Tempe, Arizona 85287-1504, USA * e-mail: [email protected]

Published online: 19 November 2006; doi:10.1038/nmat1784

oday synthetic zeolites are the most important catalysts in petrochemical refineries because of their high internal surface areas and molecular-sieving properties1,2 . There have been considerable efforts to synthesize new zeolites with specific pore geometries3,4 , to add to the 167 available at present. Millions of hypothetical structures have been generated on the basis of energy minimization5 , and there is an ongoing search for criteria capable of predicting new zeolite structures. Here we show, by geometric simulation6–8 , that all realizable zeolite framework structures show a flexibility window over a range of densities. We conjecture that this flexibility window is a necessary structural feature that enables zeolite synthesis, and therefore provides a valuable selection criterion when evaluating hypothetical zeolite framework structures as potential synthetic targets. We show that it is a general feature that experimental densities of silica zeolites lie at the low-density edge of this window—as the pores are driven to their maximum volume by Coulomb inflation. This is in contrast to most solids, which have the highest density consistent with the local chemical and geometrical constraints. Zeolites are crystalline microporous frameworks consisting of corner-sharing nearly perfect tetrahedra, connected by soft hinges at the oxygen atoms9 . The strongest forces in silica networks are those holding the rigid structural units (SiO4 tetrahedra) together and also those forces bridging these tetrahedra through the twofold coordinated oxygen atoms to form a framework, as shown in Fig. 1 for the framework of faujasite (FAU) zeolite. Today, more than 40% of oil conversion is achieved using catalysts based on faujasite (Zeolite Y)3 . Zeolite frameworks contain pores with the larger oxygen anions lining the insides of the pores. Structurally, a perfectly coordinated silica network of this type would be considered ‘well balanced’ (if the tetrahedra had ideal geometry), as the number of degrees of freedom exactly matches the number of constraints10,11 . Each of the tetrahedra in silica networks can be considered to be a rigid body and so has six degrees of freedom. The shared corners of tetrahedra each require three constraints that are shared between the two joining tetrahedra. As there are four corners, this leads to a total of six constraints per tetrahedron, exactly balancing the degrees of freedom, and leading to a rigid but stress-free network that is referred to as isostatic12 . This is known to be a particularly favourable situation, leading to a free-energy minimum where the strain (enthalpy) is minimized as the network remains intact10 . What was not clear until the present study was that such globally isostatic networks could exist over a range of densities leading to the flexibility window, which is the subject here.

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The flexibility window, for a framework of interconnected structural units, is a range of densities in which the structural units (for example, tetrahedra) can retain their ideal shape. As we shall see, the limits of the window are defined by steric clashes under compression and stretching of bonds on expansion. All of the real silicate frameworks we studied and discussed in this work show this flexibility window, which suggests that this is a pervasive property of tetrahedral-framework minerals, and of zeolites in particular. As the limits of the window depend on the specific geometry and topology of a particular framework, no general theory on the existence of the flexibility window has yet emerged. We have investigated the flexibility window in zeolite frameworks using geometric modelling (see the Methods section). We present the results for the faujasite framework in most detail. The simulated bond lengths and tetrahedral angles are shown in Fig. 2a,b. We observe a surprisingly wide range of densities in which the bond lengths and tetrahedral angles retain their ideal values, which defines the flexibility window. Outside this range the tetrahedra are deformed. Not only do the mean bonds and angles deviate from perfect values, but also the variation around the average values increases. Within the flexibility window, the bridging Si–O–Si angles (Fig. 2c) change monotonically. Three angles co-rotate, decreasing as density increases, and one antirotates in the sense that it rotates in the opposite direction13 . On expansion, none of the Si–O–Si angles reaches 180◦ , and the minimum density in the flexibility window is achieved when the framework reaches a jammed (locked) state, where further expansion is impossible without distorting the tetrahedral units. This jamming is a cooperative global effect14 in which attempts are unsuccessful in increasing the volume beyond this minimum density at the low-density edge of the flexibility window. In rare cases, the minimum density occurs when the distance between some adjacent tetrahedral centres is a maximum (linear Si–O–Si bond angle). Incidentally, minimum density does not necessarily mean that the pore apertures widen. Hinging effects can readily decrease their width. The maximum density in the flexibility window corresponds to contacts between codimeric oxygen atoms, that is second-neighbour oxygen atoms. The experimental framework density of pure-silica faujasite lies right at the low-density edge of the flexibility window, as shown in Fig. 2, indicating that the real structure is maximally expanded, consistent with minimal distortion of the tetrahedral units (Fig. 2b). This is in contrast with the commonly held view that, as Bernal15 put it, ‘In all coherent structures there is a tendency to arrive at minimal volumes’, as is most apparent in close-packed structures nature materials VOL 5 DECEMBER 2006 www.nature.com/naturematerials

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Figure 1 Structural fragment of faujasite (FAU) zeolite with a pure silica composition at different densities, viewed down the cubic [110] direction. The upper half of each fragment shows the rigid SiO4 tetrahedral units, with small spheres representing the corner-sharing oxygen anions. The lower halves show the oxygen anions drawn with their nominal van der Waals radius of 1.35 A˚ (ref. 3). a, At high density (16.3 T/1,000 A˚ 3 ), with codimeric oxygen atoms in contact. b, In the middle of the flexibility window (density 14.8 T/1,000 A˚ 3 ). c, At low density, with codimeric oxygen atoms mutually repelling (density 13 T/1,000 A˚ 3 ). The structure can expand no further without distorting the tetrahedral units. Panels a and c are at the extrema of the flexibility window.

such as metals. Here, however, we have a class of structures that tend towards maximal volume consistent with the framework constraints. A similar situation has been described for some ionic solids16 . These results for faujasite lead to two questions. First, is the flexibility window a universal property for zeolite frameworks? Second, are all silica frameworks maximally extended in nature? In Fig. 3a, we present results for zeolites that exist as pure silica. This includes the industrially important catalyst for dewaxing and isomerization—MFI (ref. 2). We show that all these structures do indeed possess a flexibility window. In addition, the experimentally observed density is very near to the low-density edge of the flexibility window. In Fig. 3b, we present results for all 14 cubic zeolite frameworks, modelled as pure silica and indexed by effective framework density9,17 . We use effective framework density (see the Supplementary Information) because not all the cubic zeolites exist as pure silica, and often contain some Al or other cations, which substitute for the Si cations. It therefore seems that framework structures are maximally extended consistent with maintaining undistorted tetrahedral connected units, and this could expedite the search for new zeolites for industrial purposes. Three mechanisms that prevent ‘collapse’ of zeolite frameworks—limitation by antirotating angles, steric limitation by channel contents and repulsion between oxygen atoms—have been suggested previously by Baur18 . In Fig. 2c, the bridging angles of faujasite have an antirotating behaviour. The set of structures we studied includes frameworks with both antirotating and co-rotating angles (see the Supplementary Information), and both classes show a flexibility window. Steric limitation by channel contents (for example, water molecules) may well be a factor, but some frameworks can be dehydrated but remain stable19,20 . The experimental densities given in Fig. 3a include structures refined with channel content, for example, MTN (ref. 21), and without, for example, FAU (ref. 22). The most general mechanism to explain the extension of the framework is Coulomb repulsion between the oxygen anions, producing an overall inflation of the structure. Considering the codimeric oxygen atoms (second-neighbour oxygen atoms whose steric contacts limit the folding mechanism at high density), the effect of increasing density is to bring these oxygen atoms closer

together; clearly, this would incur a considerable cost in Coulomb energy. Thus, Coulomb inflation provides the internal driving force that decreases the density to the edge of the flexibility window. There is experimental evidence for structural flexibility in aluminosilicate zeolite frameworks, for example analcime (ANA), that is shown on dehydration23,24 . It is hard to attach a significance to the high-density end of the flexibility window, as it corresponds to the hard-sphere collisions of codimeric oxygens. However, hydrostatic compression to such densities should lead to a change in the structure. Either the cell will change to a lower symmetry that allows further compression (for example, cubic to tetragonal) or, if no such change is possible, then rebonding may occur (as in the pressureinduced amorphization of quartz and cristobalite25,26 ) with the associated onset of irreversible behaviour. For example, the AST framework shows flexibility windows in both its cubic (Fig. 3b) and tetragonal (Fig. 3a) forms. The high-density limit of the cubic form (17.79 T/1,000 A˚ 3 ) corresponds to the low-density limit of the tetragonal form (17.80 T/1,000 A˚ 3 ), where the experimental tetragonal structure is found27 . Similarly, the ANA framework shows a very narrow window in the cubic form (Fig. 3b) and is experimentally observed to transform to a triclinic form with a small increase in density at about 1 GPa (ref. 24). The actual variations in bond lengths and angles observed in crystallographic refinements of silica zeolites exceed slightly those allowed by our definition of ‘perfect’ tetrahedra. Some of this variation may be due to the difference between crystallographic average positions and instantaneous atomic positions, whereas some is due to thermal motion. The key point, however, is that distortions of the tetrahedra are not required by the topology of the structure. The existence of the window is related to the mathematics of framework flexibility28 ; however, flexibility analysis generally assumes a starting structure in which all constraints are satisfied, whereas we are studying whether such a structure exists at a given density. It might be concluded from this survey of framework structures that any tetrahedral network structure can be relaxed to form a network of perfect tetrahedra, given the right density and boundary conditions. However, this is not the case. Many lowenergy hypothetical zeolite structures5 cannot be made perfect

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using geometric simulation at any density and so there is no flexibility window (see example in Supplementary Information). All of the real cubic zeolite frameworks, and other non-cubic silica zeolite frameworks (with six hexagonal and tetragonal unit cells) that we have studied, can be made perfectly tetrahedral over a range of densities that defines the flexibility window. Known pure-silica zeolite structures exist at the low-density end of this window, indicating that the frameworks are maximally extended. This maximal extension is nicely accounted for by Coulomb 964

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Figure 3 Flexibility window for zeolites depending on densities. a, Flexibility window for silica zeolites with cubic, hexagonal and tetragonal unit cells depending on experimental densities. Here, the density is the zeolite framework density, which is defined as the number of tetrahedral units per 1,000 A˚ 3 . In all cases the experimentally observed density lies at the lower edge of the flexibility window. b, Flexibility window for cubic frameworks simulated as pure silica composition depending on the effective density computed by distance-least-squares, (DLS; see the Supplementary Information). Cristobalite, quartz and amorphous silica also show the existence of flexibility windows of various sizes.

inflation, that is repulsion between the codimeric oxygen atoms. The ability of zeolites to support large open channels and pores, even when dehydrated, seems less surprising from this point of view. Whereas energy minimization methods5 are generating hypothetical structures at a tremendous rate, this has not been matched by a corresponding increase in the actual synthesis of new zeolite structures. This indicates a bottleneck related to the selection of candidate structures for synthesis. The degree of tetrahedral distortion29 does seem to be a significant factor, and we are at present preparing a systematic survey of millions of hypothetical zeolites5 in search of flexibility-window-possessing candidates for synthesis as potential novel catalytic materials. The presence of a flexibility window in silica zeolites, including industrially important ones such as FAU and MFI, suggests that this nature materials VOL 5 DECEMBER 2006 www.nature.com/naturematerials


LETTERS is an important property for any hypothetical zeolite framework if it is to be a viable synthetic target.

METHODS We used the geometric analysis of structural polyhedra (GASP) method, previously applied successfully to model quartz structures7,8 and zeolite compression mechanisms6,30 , to idealize the SiO4 units. In this method, we tether the atoms in a SiO4 tetrahedron by springs to the vertices of a template having ideal tetrahedral geometry, and adjust the positions of the atoms and templates during relaxation to minimize the distortion of these springs (see Supplementary Information for details). We have investigated all 14 realizable cubic zeolite frameworks, modelled as pure silica. As not all of these frameworks exist as pure silica, we have also modelled several orthorhombic, tetragonal and hexagonal frameworks that do exist as pure silica, to give additional perspective. We have also simulated several computer-generated hypothetical frameworks5 so as to compare real and hypothetical structures. We relaxed each structure while retaining the same number of atoms in the unit cell, and the atomic positions were not constrained to crystallographic sites. In addition, we note that cubic unit cells have a single variable cell parameter, whereas tetragonal and hexagonal cells have two. The oxygen atoms were given a hard-sphere radius of 1.350 A˚ (ref. 9), and the tetrahedra were given a bond length of 1.610 A˚ as in silica. We have defined tetrahedra to be perfect, with the very strict criteria that they have bond lengths within the range 1.610 ± 0.001 A˚ and bond angles within the range 109.471 ± 0.001◦ .

Received 13 July 2006; accepted 29 September 2006; published 19 November 2006. References 1. Bellussi, G. in Proc. 14th Int. Zeolite Conf., S. Africa (eds van Steen, E. W. J., Callanan, L. H., Claeys, M. & O’Connor, C. T.) (Rondebosch, South Africa, 2004). 2. Marcilly, C. Present status and future trends in catalysis for refining and petrochemicals. J. Catal. 216, 47–62 (2003). 3. Corma, A., Diaz-Cabanas, M., Martinez-Triquero, J., Rey, F. & Rius, J. A large-cavity zeolite with wide pore windows and potential as an oil refining catalyst. Nature 418, 514–517 (2002). 4. Degnan, T. The implications of the fundamentals of shape selectivity for the development of catalysts for the petroleum and petrochemical industries. J. Catal. 216, 32–46 (2003). 5. Treacy, M., Rivin, I., Balkovsky, E., Randall, K. & Foster, M. Enumeration of periodic tetrahedral frameworks. ii. Polynodal graphs. Micropor. Mesopor. Mater. 74, 121–132 (2004). 6. Gatta, G. & Wells, S. Rigid unit modes at high pressure: An explorative study of a fibrous zeolite-like framework with edi topology. Phys. Chem. Mineral 31, 1–10 (2004). 7. Wells, S., Dove, M. & Tucker, M. Finding best-fit polyhedral rotations with geometric algebra. J. Phys. Condens. Matter 14, 4567–4584 (2002).

8. Sartbaeva, A., Wells, S., Redfern, S., Hinton, R. & Reed, S. Ionic diffusion in quartz studied by transport measurements, SIMS and atomistic simulations. J. Phys. Condens. Matter 17, 1099–1112 (2005). 9. Baerlocher, C., Meier, W. & Olson, D. Atlas of Zeolite Framework Types (Elsevier, Amsterdam, 2001). 10. Phillips, J. Topology of covalent non-crystalline solids ii: Medium-range order in chalcogenide alloys and A—Si(Ge). J. Non-Cryst. Solids 43, 37 (1981). 11. Thorpe, M. Continuous deformations in random networks. J. Non-Cryst. Solids 57, 355–370 (1983). 12. Chubynsky, M. & Thorpe, M. Self-organization and rigidity in network glasses. Curr. Opin. Solid State Mater. Sci. 5, 525–532 (2001). 13. Baur, W. Self-limiting distortion by antirotating hinges is the principle of flexible but noncollapsible frameworks. J. Solid State Chem. 97, 243–247 (1992). 14. Donev, A., Torquato, S., Stillinger, F. & Connelly, R. Jamming in hard sphere and disk packings. J. Appl. Phys. 95, 989–999 (2004). 15. Bernal, J. & Carlisle, C. The range of generalized crystallography. Sov. Phys. Crystallogr. 13, 811–831 (1969). 16. O’Keeffe, M. On the arrangement of ions in crystals. Acta Crystallogr. A 33, 924–927 (1977). 17. Baerlocher, C., Hepp, A. & Meier, W. in DLS-76, a FORTRAN Program for the Simulation of Crystal Structures by Geometric Refinement (Institut fur Kristallographie and Petrographie, ETH, Zurich, 1978). 18. Baur, W. in Proc. 2nd Polish–German Zeolite Colloquium (ed. Rozwadowski, M.) 171–185 (1995). 19. Sani, A., Cruciani, G. & Gualtieri, A. Dehydration dynamics of ba-phillipsite: An in situ synchrotron powder diffraction study. Phys. Chem. Mineral. 29, 351–361 (2002). 20. Vezzalini, G., Alberti, A., Sani, A. & Triscari, M. The dehydration process in amicite. Micropor. Mesopor. Mater. 31, 253–262 (1999). 21. Gies, H. Studies on clathrasils. 6. crystal-structure of dodecasil-3c, another synthetic clathrate compound of silica. Z. Kristallogr. 167, 73–82 (1984). 22. Hrilijac, J., Eddy, M., Cheetham, A., Donohue, J. & Ray, G. Powder neutron-diffraction and Si-29 MAS NMR-studies of siliceous zeolite-Y. J. Solid State Chem. 106, 66–72 (1993). 23. Cruciani, G. & Gualtieri, A. Dehydration dynamics of analcime by in citu synchrotron powder diffraction. Am. Mineral. 84, 112–119 (1999). 24. Gatta, G., Nestola, F. & Ballaran, T. B. Elastic behavior, phase transition, and pressure induced structural evolution of analcime. Am. Mineral. 91, 568–578 (2006). 25. Binggeli, N. & Chelikowsky, J. Structural transformation of quartz at high pressures. Nature 353, 344–346 (1991). 26. Tsuchida, Y. & Yagi, T. New pressure-induced transformations of silica at room temperature. Nature 347, 267–269 (1990). 27. Caullet, P., Guth, J., Hazm, J., Lamblin, J. & Gies, H. Synthesis, characterization and crystal-structure of the new clathrasil phase octadecasil. Eur. J. Solid State Inorg. Chem. 28, 345–361 (1991). 28. Whiteley, W. Counting out to the flexibility of molecules. Phys. Biol. 2, S116–S126 (2005). 29. Zwijnenburg, M., Simperler, A., Wells, S. & Bell, R. Tetrahedral distortion and energetic packing penalty in zeolite frameworks: Linked phenomena? J. Phys. Chem. B 109, 14783–14785 (2005). 30. Gatta, G. & Wells, S. Structural evolution of zeolite levyne under hydrostatic and non-hydrostatic pressure: geometric modelling. Phys. Chem. Mineral. 33, 243–255 (2006).

Acknowledgements This work was supported by NSF grants NIRT-0304391 and DMR-0425970. We acknowledge M. Foster for discussions. Correspondence and requests for materials should be addressed to M.F.T. Supplementary Information accompanies this paper on www.nature.com/naturematerials.

Competing financial interests The authors declare that they have no competing financial interests. Reprints and permission information is available online at http://npg.nature.com/reprintsandpermissions/

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1 Supplementary methods 1.1 GASP All simulations were carried out using the program GASP [1, 2, 3, 4], which models the rigid-unit motion of tetrahedral frameworks. The GASP algorithm models the structure simultaneously with a collection of atoms (Si, O) and a set of geometric templates with ideal tetrahedral angles and a center – vertex (Si–O) bond length appropriate to the system ˚ for silica). The offset between an atom and a template vertex is decomposed into (1.61 A components representing stretching of the bond and bending of the tetrahedral angles, as illustrated in Supplementary figure 1, and harmonic springs are applied to constrain each of these components. In an iterative process of relaxation, the atoms and templates move together so as to minimise the distortion in the structure. The algorithm includes hardsphere interactions between atoms, so as to forbid two atoms from approaching each other more closely than the sum of their radii. However no other long-range interaction, attractive or repulsive, was included, as the objective was to determine whether or not the atoms can be made to match the templates exactly; the bridging (Si–O–Si) angle was unconstrained. Simulations at different densities were carried out by taking an initial set of fractional coordinates for the atoms (e.g. from a crystal structure), setting the cell parameter to produce a given density, then geometrically relaxing the system. We found that there was a sharp distinction between cases where the atoms could be made to match the templates exactly, and those where some distortion remained. We defined a perfect match using the ˚ very strict criteria that bond lengths should differ from the ideal by no more than 0.001 A, ◦ and that the internal angles of the tetrahedron should vary by no more than 0.001 , which is close to the numerical limit of accuracy of the algorithm. A small random perturbation ˚ was applied to the atomic positions before relaxation. The relaxed atomic of about 0.01 A coordinates therefore correspond to an instantaneous snapshot of the system rather than to the crystallographic average positions.

1.2 Free boundary conditions simulations The only real cubic zeolite structure that did not display a window when simulated with periodic boundary conditions was the clathrasil MTN. This dodecasil consists of cages (i.e. it is a foam in that every point in space can be assigned to a single polyhedra or cage) and has multiple interlocking 5-rings. Simulations with periodic boundary conditions always displayed a small degree of tetrahedral distortion. Simulations of supercells of increasing size of 136, 1088, 3672 and 8704 polyhedra with periodic boundary conditions still displayed the distortion. We then simulated free boundary conditions by severing those bonds that crossed the boundaries of the simulation cell and placing the resulting truncated structure in a larger simulation box. The density of the truncated structure was controlled by fixing the coordinates of selected polyhedra at the corners. The density was calculated by systematically sampling points, using a Voronoi type construction, in the simulation cell, so as to obtain a density for the central bulk-like region while ignoring the surface region. This was tested against results for FAU where it was not necessary, and shown to be a reliable procedure as we obtained very similar results with either periodic or free boundary conditions. These supercells were all perfect with free boundary conditions; simulations at different densities showed a window as found for the other real cubic zeolites. On inspection of the periodic and non-periodic relaxed structures, it was clear that in the non-periodic structures, each 5-ring relaxed with a different pattern of tetrahedral tilts, whereas the regularity introduced by periodic boundary conditions required each 5-ring to have the same pattern of tilts. This implies that periodic simulations of this clathrasil using any method — not only our geometric approach — would tend to introduce an extra degree of strain in the structure.

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This is in line with the recent observation by O’Keefe (private communication) that the crystallographically symmetric positions for the atoms in MTN are not consistent with perfectly tetrahedral SiO4 geometry. Similar to this cubic framework, we found that several non-cubic structures were nearly perfectible with periodic boundary condition, but fully perfectible with free boundary conditions. Those were MEI, *BEA, IST, MWW and DOH. We have also investigated models of silica glass generated by the WWW bond-swapping algorithm [5] and described as ‘large well-relaxed [silica structures]’, as a means to study very large unit cells. We used structures containing 1000 and 4062 tetrahedra, both of which gave similar results; those for the 1000-tetrahedron model are discussed here. The structure as provided contained a great deal of tetrahedral distortion, with bond lengths of ˚ and O–Si–O angles of 109.46 ± 5.79◦. Given the method of generating 1.61 ± 0.04 A these structures, by relaxation with a potential, this could represent intrinsic strain of the tetrahedra (i.e. the bonding in the structure is not compatible with geometrically perfect tetrahedra); or it could represent distortion of an otherwise ideal tetrahedral network by the long-range interaction terms in the potential. Geometric simulation with periodic boundary conditions reduced the degree of distortion but still left the tetrahedra imperfect (bond ˚ bond angles of 109.47 ± 1.92◦). Geometric simulation with lengths of 1.61 ± 0.01 A, free boundary conditions, however, as for the case of MTN, did indeed produce a perfect structure, albeit over an extremely narrow window. The narrowness of the window is understandable since the distribution of bridging angles covers a full range from near 180◦ at some parts of the solid (so that further extension creates tension in a bond) to around 120◦, so that further compression leads to a steric collision in some parts of the solid, while at other parts the framework is expected to be jammed. This occurs because all possible kinds of allowed local conformations are present at some location in the glass when the system is large enough, using a Lifshitz type argument [6].

2 Supplementary notes 2.1 Effective framework density The effective framework density is defined as the number of tetrahedrally coordinated ˚ 3 . For non-zeolitic framework structures, values of at least atoms (T-atoms) per 1000 A 3 ˚ are generally obtained, while for zeolites with fully crosslinked frame19 to 21 T/1000 A works the observed values range from about 10 for structures with the largest pore volume to around 20.6. The framework density is obviously related to the pore volume but does not reflect the size of the pore openings. For some more flexible zeolite structure types, the framework density values can vary appreciably. A framework type is independent of chemical composition. Therefore, idealized framework data (cell parameters, coordinates of atoms at the center of the tetrahedra etc. ) were obtained from a distance-least-squares refinement [7] in the given (highest possible) symmetry for the framework type and we call it an effective density. The refinement was carried out assuming a (sometimes hypothetical) SiO2 composition and with the prescribed inter˚ In each case, the coordinates were first optimized within atomic distance for Si–O = 1.61 A. an approximate unit cell, and then the unit cell was refined. The space group, the cell dimensions and the atomic coordinates of a real material will depend upon its chemical composition, but they will be related to the crystallographic data listed for the framework type. If the symmetry is different, it will be a subgroup of this space group, and the unit cell parameters will be related by relatively simple geometric considerations.

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2.2 Classification by Baur Baur in 1992 [8] classified zeolites as collapsible if all the bridging angles co-rotate, or noncollapsible if some angles anti-rotate. By this criterion we would classify FAU, LTA, TSC, KFI, PAU, LTN and -CLO as non-collapsible. We note that the frameworks classified as ‘non-collapsible’ due to their antirotating angles do nonetheless collapse on compression, as shown by the example of FAU.

2.3 Hypothetical zeolites We have looked at several hypothetical zeolite structures. Supplementary figure 2 shows one of the hypothetical zeolites 229 4 60312. This structure was generated by Treacy et al [9] as a hypothetical zeolite. To the eye, the structure looks acceptable though there are slight distortions of some of the tetrahedra. However, geometric simulation does not show any window in which the tetrahedra can be made perfect, with either periodic or free boundary conditions. Since the structure has no flexibility window and cannot support geometrically perfect tetrahedra, we would predict that it cannot be synthesised and would reject it as a candidate pure-silica zeolite structure.

3 Supplementary figures

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Figure Supplementary figure 1: GASP overlays a bonded group of atoms with a template representing the ideal local tetrahedral geometry. The template is fitted to the atoms by a least-squares procedure. The mismatch between an atom and its template vertex is decomposed into components of bond stretching and tetrahedral angle bending, which are constrained by harmonic springs. Over multiple iterations of fitting templates and atoms, the structure is brought to perfect tetrahedral geometry; if not, some residual distortion remains. Red atoms represent oxygens, blue – silicon. Templates are represented by gray lines. Gay circles represent positions of template vertices. Tetrahedra have ideal shape when actual atom positions match the atom positions of templates, i.e. red atoms overlay gray circles.

References [1] Wells, S., Dove, M. & Tucker, M. Finding best-fit polyhedral rotations with geometric algebra. J. Phys.: Cond. Matter 14, 4567–4584 (2002). [2] Sartbaeva, A., Wells, S., Redfern, S., Hinton, R. & Reed, S. Ionic diffusion in quartz studied by transport measurements, SIMS and atomistic simulations. J. Phys.: Cond.Matter 17, 1099–1112 (2005). 3


Figure Supplementary figure 2: Structure of 229 4 60312 hypothetical zeolite in tetrahedral view. This structure is from the hypothetical zeolite database at http://www.hypotheticalzeolites.net [9]

[3] Gatta, G. & Wells, S. Rigid unit modes at high pressure: an explorative study of a fibrous zeolite-like framework with edi topology. Phys. Chem. Min. 31, 1–10 (2004). [4] Gatta, G. & Wells, S. Structural evolution of zeolite levyne under hydrostatic and nonhydrostatic pressure: geometric modelling. Phys. Chem. Min. 33, 243–255 (2006). [5] Vink, R. & Barkema, G. Large well-relaxed models of vitreous silica,coordination numbers, and entropy. Phys. Rev. B 67, 245201 (2003). [6] Lifshitz, I. The energy spectrum of disordered systems. Adv. Phys. 13, 483–536 (1964). [7] Baerlocher, C., Hepp, A. & Meier, W. DLS-76, a FORTRAN program for the simulation of crystal structures by geometric refinement. Institut für Kristallographie and Petrographie, ETH, Zürich (1978). [8] Baur, W. Self-limiting distortion by antirotating hinges is the principle of flexible but noncollapsible frameworks. J. Solid State Chem. 97, 243–247 (1992). [9] Treacy, M., Rivin, I., Balkovsky, E., Randall, K. & Foster, M. Enumeration of periodic tetrahedral frameworks. ii. polynodal graphs. Microporous and Mesoporous Mat. 74, 121–132 (2004).

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A lesson from virtual zeolites Although computational methods are generating a bewildering number of hypothetical zeolite structures, the selection of candidates for synthesis remains problematic. The presence of a flexibility window in the structure may turn out to be a useful criterion.

IGOR RIVIN

Figure 1 Faujasite tile. A representation of how the SiO4 tetrahedra are arranged with respect to one another in faujasite. The larger spheres at the centre of the tetrahedra are silicon atoms, the red spheres are the corner-sharing oxygen atoms. The orange SiO4 tetrahedron is the basic tetrahedron and lies within the fundamental tile, whereas the green ones are the neighbouring tetrahedra that lie outside the tile. The red lines are the edges of the tile.

is in the Temple University Mathematics Department, Wachman Hall, 1805 North Broad Street, Philadelphia 19122, USA. (Currently visiting the Stanford University Mathematics Department). e-mail: [email protected]

he name zeolite means boiling stone in ancient Greek. It stems from the fact that when this mineral is heated, water boils out of the pores. Thus porosity is the defining property of zeolites, giving them a dizzying variety of uses (petrochemical catalysis, molecular sieving, water purification, soil treatment and many more). There are 48 naturally occurring zeolites and 167 synthetic ones, not a small number in absolute terms. But considering the large host of applications, we should like to be able to generate more, with prescribed pore size and shape, so that they can ensnare specific molecules. Unfortunately, zeolite synthesis is a difficult process and we don’t know a priori which ones will actually be achievable. Asel Sartbaeva and colleagues, writing in this issue, evaluate real zeolite frameworks and find a criterion that may be useful to predict which hypothetical structures are realizable1. In addition their geometric computations based on empirical data offer insights into the very fundamental question of why zeolites have pores. Compared with the practically infinite number of synthesizable organic molecules, the number of zeolites available is miniscule, although the number of hypothetical zeolites obtained through energy minimization methods is growing at a tremendous rate. Mike Treacy, one of the authors of the paper in question, 15 years ago spearheaded a project to compute a census of all possible zeolites. The project is one of enormous complexity, as efficient generation and analysis of hypothetical zeolites turns out to use much of modern mathematics, computer science and of course chemistry. It is still gathering speed (the interested reader can marvel at the results at http://www.hypotheticalzeolites.net) and the article in this issue is one of the fruits borne by this project. The crystalline structure of zeolites consists of tetrahedral units, the so-called T-sites, which are 4-valent atoms, typically silicon or aluminium, connected to O-sites, which are oxygen atoms. So, for each of the 230 crystallographic groups there are four

T

regular frameworks, which admit a physical realization with low energy. Every crystallographic group Γ has a fundamental region — a tile whose image, repeated under the action of Γ, fills all the space. In each particular structure the number of T-sites in the tile is always the same. These are the ‘unique T-sites’; the hypothetical zeolite database consists of structures with a small (but growing) number of unique T-sites. But why do all materials made of these building blocks form porous frameworks? Sartbaeva and colleagues observe that whereas most materials appear to want to be as dense as possible, zeolites are just the opposite. All of the viable frameworks examined have low energy (that is, they can exist) over a wide range of densities but they seem to prefer a lower-density state. This is quite different from what happens in most classes of materials, from amorphous networks to metallic close-packed

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NEWS & VIEWS lattices. However, a situation similar to zeolites has been observed for some ionic liquids2. In fact, Sartbeva et al. suggest that the underlying reason for this behaviour might be electrostatic repulsion. In the geometrical models of zeolites computed by Sartbaeva and colleagues, the frameworks of interconnected tetrahedra can budge, and the extent of movement that does not cause distortions in the geometry of the tetrahedra constitutes a window of flexibility. Within this window the material’s density changes because the twisting or turning of tetrahedra makes the pores wider, but from the energy standpoint the material remains viable. The authors modelled all 14 real framework structures and focused in detail on simulating the annealing process of faujasite (the fundamental tile for faujasite is shown in Fig. 1), the most used catalyst in the petroleum refining industry. In each case, they found this flexibility window and discovered that the real structures always lie at the low-density end of this window. The authors’ explanation for this is based on considerations of electrostatic repulsion between oxygen atoms on adjacent tetrahedral, which has the effect of making the pores as large as possible. They also conjecture that the flexibility window may actually be a necessary condition for zeolite synthesis to be possible. Although this is plausible, the existence of the flexibility window is rather enigmatic, but it is likely that the explanation for it is purely geometric. A remarkable observation in the paper is that some structures with a narrow (or non-existent)

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flexibility window might undergo a phase transition to a less symmetric form (as illustrated in Fig. 2). Such a transformation may result in a small increase in density but it can give the framework more degrees of freedom to significantly expand (or create) the flexibility window. It is surprising to see how badly the structure wants to decrease its density, even at the cost of losing symmetry. This geometric phase transition strikes me as a very interesting and beautiful event, which is closely connected philosophically to the way mathematicians now study three-dimensional manifolds (following ideas on distortion, which also inspired the recent resolution of the Poincaré conjecture by Perelman). The fact that this occurs in the physical world is extremely satisfying. Currently, computational experiments are the only way for us to get insight into the basic questions of the physics and chemistry of zeolites. The article by Sartbaeva and colleagues seems to uncover some basic ingredients necessary for realizing the synthesis of yet undiscovered zeolites, which may one day lead us to designer zeolites. This is particularly exciting as we might be witnessing the birth of a new chemical technology era where we can imagine producing a zeolite with the precise pore shape needed to accommodate a specific molecule, and so facilitate completely new types of chemical processes. REFERENCES 1. Sartbaeva, A., Wells, A. S., Treacy, M. M. J. & Thorpe, M. F. Nature Mater. 5, 962–965 (2006). 2. O’Keeffe, M. Acta Cryst. A33, 924–927 (1977).

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