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Geometry, combinatorics, computation with Zeolites


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Geometry, combinatorics, computation with Zeolites

  1. 1. Designer zeolites Igor Rivin Temple University Department of Mathematics All work joint with Mike Treacy (ASU)
  2. 2. What is a zeolite? Zeolites (Greek, zein, "to boil"; lithos, "a stone") are hydrated aluminosilicate minerals and have a micro- porous structure. The term was originally coined in the 18th century by a Swedish mineralogist named Axel Fredrik Cronstedt who observed, upon rapidly heating a natural mineral, that the stones began to dance about as the water evaporated. Using the Greek words which mean "stone that boils," he called this material zeolite. We will describe what zeolites are to a mathematician shortly, but the important aspect of them for a material scientist is that they are very porous, and that’s what is responsible for most of the uses described below...
  3. 3. Summary • Zeolites are industrially important. There is a need for new zeolite structures. • Zeolite frameworks can be represented as directed, or colored, graphs, that contain information about site symmetry and bonded neighbors. • Given a space group, and number of unique T-atoms NT, all the possible graphs can be enumerated by a combinatorial analysis of all those site-site interconnections that are consistent with tetrahedral bonding. • Graphs can be “embedded” in real space by various methods, such as simulated annealing, to find the regular tetrahedral SiO2 frameworks. • A combinatorial explosion of graphs with increasing NT, limits the method as presently implemented, to NT ≤ 7 for high symmetry space groups. The percentage of viable frameworks drops off rapidly with increasing NT. • Describe the methods used and highlight the problems with imbedding. • Examples from high-symmetry space groups, Pm3m, P6/mmm, etc • Can we predict how to make hypothetical zeolites?
  4. 4. What are zeolites good for • Petrochemical industry Synthetic zeolites are widely used as catalysts in the petrochemical industry, for instance in fluid catalytic cracking and hydro-cracking. Zeolites confine molecules in small spaces, which causes changes in their structure and reactivity. The hydrogen form of zeolites (prepared by ion-exchange) are powerful solid-state acids, and can facilitate a host of acid-catalyzed reactions, such as isomerisation, alkylation, and cracking. The specific activation modality of most zeolitic catalysts used in petrochemical applications involves quantum- chemical Lewis acid site reactions. Catalytic cracking uses a furnace and reactor. First crude oil distillation fractions are heated in the furnace and passed to the reactor. In the reactor the crude meets with a catalyst such as zeolite. It goes through this step three times, each time getting cooler. Finally it reaches a step known as separator. The separator collects recycled hydrogen. Then it goes through a fractionator and becomes the final item.
  5. 5. What are zeolites good for? • Commercial and Domestic Zeolites are widely used as ion-exchange beds in domestic and commercial water purification, softening, and other applications. In chemistry, zeolites are used to separate molecules (only molecules of certain sizes and shapes can pass through), as traps for molecules so they can be analyzed. Zeolites have the potential of providing precise and specific separation of gases including the removal of H2O, CO2 and SO2 from low-grade natural gas streams. Other separations include: noble gases, N2, O2, freon and formaldehyde. However at present, the true potential to improve the handling of such gases in this manner remains unknown.
  6. 6. What are zeolites good for? • Nuclear Industry Zeolites have uses in advanced reprocessing methods, where their micro-porous ability to capture some ions while allowing others to pass freely allow many fission products to be efficiently removed from nuclear waste and permanently trapped. Equally important are the mineral properties of zeolites. Their alumino-silicate construction is extremely durable and resistant to radiation even in porous form. Additionally, once they are loaded with trapped fission products, the zeolite- waste combination can be hot pressed into an extremely durable ceramic form, closing the pores and trapping the waste in a solid stone block. This is a waste form factor that greatly reduces its hazard compared to conventional reprocessing systems.
  7. 7. What are zeolites good for? • Agriculture In agriculture, clinoptilolite (a naturally occurring zeolite) is used as a soil treatment. It provides a source of slowly released potassium. If previously loaded with ammonium, the zeolite can serve a similar function in the slow release of nitrogen. Zeolites can also act as water moderators, in which they will absorb up to 55% of their weight in water and slowly release it under plant demand. This property can prevent root rot and moderate drought cycles.
  8. 8. What are zeolites good for? • Animal Welfare In Concentrated Animal Growing facilities, the addition of as little as 1% of a very low sodium clinoptiloite was shown to improve feed conversion, reduce airborne ammonia up to 80%, act as a mycotoxin binder and improve bone density. See US Patents 4,917,045 and 6,284,232. Can be used in general odor elimination for all animal odors.
  9. 9. What are zeolites good for? • Medical Zeolite-based oxygen concentrator systems are widely used to produce medical grade oxygen. The zeolite is used as a molecular sieve to create purified oxygen from air using its ability to trap impurities, in a process involving the absorption of undesired gases and other atmospheric components, leaving highly purified oxygen and up to 5% argon. QuikClot® brand hemostatic agent, which continues to be used successfully to save lives by stopping severe bleeding, contains a calcium loaded form of zeolite.
  10. 10. What are zeolites good for? • Heating and refrigeration Zeolites can be used as solar thermal collectors and for adsorption refrigeration. In these applications, their high heat of adsorption and ability to hydrate and dehydrate while maintaining structural stability is exploited. This hygroscopic property coupled with an inherent exothermic (heat producing) reaction when transitioning from a dehydrated to a hydrated form, make natural zeolites useful in harvesting waste heat and solar heat energy.
  11. 11. What are zeolites good for? • Detergents The largest single use for zeolite is the global laundry detergent market. This amounted to 1.44 million metric tons per year of anhydrous zeolite A in 1992.
  12. 12. What are zeolites good for? • Construction Synthetic zeolite is also being used as an additive in the production process of warm mix asphalt concrete. The development of this application started in Europe (Germany) in the 1990s. It helps by decreasing the temperature level during manufacture and laying of asphalt concrete, resulting in lower consumption of fossil fuels, thus releasing less carbon dioxide,aerosols and vapours. Other than that the usage of synthetic zeolite in hot mixed asphalt leads to easier compaction and to a certain degree allows cold weather paving and longer hauls. When added to Portland Cement as a Pozzolan, it can reduce chloride permeability and improve workability. It reduces weight and helps moderate water content while allowing for slower drying which improves break strength.
  13. 13. • Aquarium keeping What are zeolites good for? Zeolites are marketed by pet stores for use as a filter additive in aquariums. In aquariums, zeolites can be used to absorb ammonia and other nitrogenous compounds. However, due to the high affinity of some zeolites for calcium, they may be less effective in hard water and may deplete calcium. Zeolite filtration is used in some marine aquaria to keep nutrient concentrations low for the benefit of corals adapted to nutrient-depleted waters. Where and how the zeolite was formed is an important consideration for aquariums. Northern hemisphere natural zeolites were formed when molten lava came in contact with sea water, thereby 'loading' the zeolite with Na (sodium) sacrificial ions. These sodium ions will speciate with other ions in solution, thus the takeup of nitrogen in ammonia, with the release of the sodium. In southern hemisphere zeolites, such as found in Australia, which were formed with fresh water, thus the calcium uptake on formation. Zeolite is an effective ammonia filter, but must be used with some care, especially with delicate tropical corals which are sensitive to water chemistry and temperature. Space hardware testing Zeolites can be used as a molecular sieve in cryosorption pumps for rough pumping of vacuum chambers which can be used to simulate space-like conditions in order to test hardware bound for space. Cat litter
  14. 14. Zeolites Are Important for Synthesis, Refining, and Environmental Processes Zeolite Catalyst Sales, $M, Constant $ 1995 2000 2005 Chemical (1) 180 280 350 Refining (2) 650 930 1,130 Environmental (3) 150 410 530 Notes 1) Aromatics and specialty organic synthesis 2) USY in FCC, hydrocracking, Other zeolites in FCC additives, saturation, isomerization and lubes 3) VOC, automotive Source: The Catalyst Group
  15. 15. Cumulative No. of Zeolite Patents and Publications (New Structures, Synthesis, Catalysis, Sorbents)/Year 1965 1970 1975 Patents 1980 Publications 1985 1990 1995 2000 Are there Opportunities for New Structures? 2005
  16. 16. Why zeolite catalysts? • Significantly better product selectivity • Greater activity – leading to higher throughputs and debottlenecking. • Environmental compatibility – Catalyst disposal – Reduction of byproducts • Growth in the variety of available zeolite structures and compositions. • Improved understanding of diffusivity and structure – property – function relationships. Advances in zeolite catalyst technology have changed the nature of the refining and petrochemical processes – requiring less separation, less energy, smaller reactors, and often simpler process configurations
  17. 17. Cumene Process sPA Cumene Purity 99.0% Cumene Yield 95% Bz/Propylene 8:1 Catalyst Life 12-18 mo. Zeolite Cumene Purity 99.97% Cumene Yield 99.7% Bz/Propylene 3:1 Catalyst Life 5 yrs + sPA-Based Cumene Process In 1986 (US only) • Generated 250M lbs of heavy aromatics as a cumene byproduct - put into gasoline • Generated 5M lbs of spent SPA catalyst as solid waste Special Handling Requirements • SPA catalyst disposal involves drying and directed explosive charges to dislodge it from reactors • Requires precise H2O addition
  18. 18. There are Five “Big” Zeolites Faujasite (FAU) MCM-22 (MWW) 12-MR, 3-dimensional 10- &12-MR, 2-dimensional Beta (BEA) 12-MR, 3-dimensional ZSM-5 (MFI) 12-MR, 1-dimensional Mordenite (MOR) 10-MR, 3-dimensional Attests to the versatility of these materials and the exceptional selectivity provided by the specific crystal structure
  19. 19. Factors Contributing to the Predominance of These Five Structures • Early discovery and development • Scaleability and low cost of manufacture • Early structure resolution – allows modelling • Hydrothermal stability - regenerability • Compositional and morphological versatility • Understanding the underlying catalytic chemistry and implications of molecular transport • Inability of other materials to match the broad selectivity advantages and activity of these structures
  20. 20. Are there Opportunities for New Structures? The Answer is Yes! Most Likely: – Supplementing existing catalysts to tailor selectivity – In new applications driven by changes in product demand or regulatory changes (e.g., benzene alkylation) – At the intersection of petroleum refining and petrochemical manufacturing – Where the materials provide new routes to existing processes (e.g., reaction and separation) – In specialty chemicals production where product margins can justify the cost of catalyst development and scale-up – Where there are needs for processing non-conventional feedstocks (e.g. biomass; heavy/dirty feedstocks; natural gas) – In advanced environmental catalysts (e.g. NOX and SOX reduction) However, almost all new zeolites are discovered by serendipity – by unpredictable synthetic methods, or as new minerals.
  21. 21. The number of known zeolites is growing
  22. 22. LTL Framework Pt/ K-zeolite L converts n-hexane to benzene with high yield and efficiency
  23. 23. Construction of LTL Model Construct Cancrinite cages Stack Cross-connect Cancrinite cages Cancrinite columns
  24. 24. LTL Framework
  25. 25. But instead… Construct Cancrinite cages Stack Cross-connect Cancrinite cages Cancrinite columns
  26. 26. Definitely Not LTL 191_2_13
  27. 27. Rotated Cancrinite Columns A simple connection error produced a new and interesting structure 191_2_14 191_2_13 LTL NOT-LTL Cross-linking creates 8-rings Cross-linking creates 4-rings apertures are sinusoidal apertures are smooth 12-ring 18-ring channels 18-ring channels a = 18.0 Å, c = 7.50 Å a = 21.4 Å, c = 7.66 Å
  28. 28. Defects in Zeolites Represent New Local Topologies Faujasite with a stacking fault M. Audier, J, M,. Thomas, J. Klinowski, D. A. Jefferson and L. Bursill, J. Phys. Chem. 1982, 86, 581. The elusive “Breck’s structure 6”
  29. 29. Challenge! How many ways can two Si atoms be interconnected in space group P6/mmm to produce regular tetrahedral zeolitic frameworks? • Billions+ ? – the combinations are almost infinite. • A handful ? – the combinations being limited by symmetry. It took 10 years to get an answer – there are exactly 48! 31 of which are close to regular tetrahedral.
  30. 30. Methods For Finding New Frameworks • Synthesis + Direct experimental methods (Single crystal, Rietveld, TEM). – MFI, beta (+ many others) • Modification of existing structures. – ALPO-8 • Model building – trial and error. – FAU framework • Permutation of connections between sheets, polyhedra. – Many examples by J. V. Smith • Applying symmetry operators to secondary building units. – Akporiaye & Price, Shannon • Distance least squares, simulated annealing. – DLS-76 (Hepp Baerlocher, Meier), ZEFSA (Deem & Newsam) • Permutation of symmetry operators. – Fischer et al. (1993) - search for low-density frameworks • Dense grid search. – O'Keeffe & Brese (1990) • Symmetry-Constrained Intersite Bonding Search (SCIBS). – Treacy et al. (1993, 1997, 2004), Klein (1996) • Polyhedral tiling. – Andries & Smith (1996), Delgado-Friedrichs (1999)
  31. 31. The LTL framework has P6/mmm symmetry 36 T-atoms and 72 oxygen atoms per unit cell
  32. 32. The LTL framework has P6/mmm symmetry All of the symmetry operations can be generated by mirrors
  33. 33. LTL Framework LTL fundamental region contains 2 T-atoms and 6 unique Oxygen atoms, and is bounded by 5 mirror planes. Each T-atom is connected to four oxygens The LTL framework is generated by the action of the mirror planes, much in the same way as a kaleidascope works.
  34. 34. The general site has 3 choices 2 1 • 2 must connect directly to 1, and to the top and side faces to ensure 3-D connectivity. • This leaves one free bond, and three connectible faces. Permutation over the three available bonds gives 3 new structures LTL
  35. 35. The basal mirror site has 3 choices 2 T-atom 2 is inside the fundamental region 1 • 1 must connect directly to 2. • Two of the oxygens attached to 1 must lie on the edges defined by intersecting (perpendicular) mirror planes to preserve regular tetrahedral symmetry. • The fourth bond to 1 is generated by reflection in the basal mirror plane. There are only 3 = 3.2.1 = 3 possibilities for atom 1 2 1.2.1
  36. 36. Nine topologies with T-atoms on the same sites as LTL There are exactly 48 binodal topologies in P6/mmm, 31 of which refine well
  37. 37. Generalizing to other space groups “Roadmap” viewgraph from 1991
  38. 38. Colored Graph Description – LTL Colored (or directed) graph. The “color” is the bond operator type Space group Number of unique T-atoms Atom site list Atom id operator Atom id o o’ ’ Mirror sites o and o’ are topologically distinct. Oxygen atoms are implied by the Si–Si bonds, and are therefore redundant.
  39. 39. Another Example – FER
  40. 40. Surprisingly, most of the effort is in refining the graphs The effort planned assumed that the refining would be quick and easy Imbedding the graphs in real space – refining them – has been arduous, and is an ongoing struggle. Speed and efficiency are low.
  41. 41. Regular tetrahedral force model This is an empirical model with lowest cost for the cubic diamond structure 2 Ångstrom U = K 1 (d TT - 3.05) 3 + K 2 (a TTT - 1.91063) radian This formula attempts to force T-atoms into a regular tetrahedral arrangement with TTT angles of 109.47° This generates reasonable approximations to zeolite frameworks. Predicts that cristobalite has lower energy than quartz, because real zeolites do not favor tetrahedral TTT angles because of the bridging oxygen atoms
  42. 42. Out of 6,471 uninodal graphs, only one was new! Space group #56. Pccn. a = 8.02 Å, b = 4.43 Å, c = 8.68 Å T-atom at: x = 0.071, y = 0.126, z = 0.151 T/1000 = 25.92 Out of 6,471 uninodal graphs, ~300 were plausible tetrahedral arrangements, and only one was truly new! mmt in O'Keeffe's database
  43. 43. γ High density phase – “ −silica” Ia3 FD = 26.76 T-atoms/1000Å3 TD10 = 1165.0 Coordination Sequence 1 4 12 27 49 77 109 148 194 244 301 Vertex Symbol • Related to O'Keeffe's γ –Si, bcc(8) • Post-diamond high-pressure form of carbon 206_1_170
  44. 44. Next steps We have an enormous number of graphs out to NT ≤ 7, but had succeeded only in imbedding the uninodal (NT = 1) graphs without the oxygen atoms. • Find a way to efficiently imbed graphs with NT ≥ 2 – increases the number of graphs exponentially. • Find a way to efficiently include the bridging oxygen atoms – potentially triples the degrees of freedom. Both of these goals dramatically increase the complexity of the problem to be solved.
  45. 45. Boisen-Gibbs-Bukowinski force model M. B. Boisen, G.V. Gibbs and M.S.T. Bukowinski, Phys. Chem. Minerals (1994) 21 269 – 284 2 UBGB = Aå (L - L0 ) O 2 + B å (OTO - OTO0 ) T 2 + C å (TO - TO0 ) T + Då å (L - L0 )(TO - TO0 ) T O + E å exp(- Fd OO +G) dOO >4• Empirical force field derived from ab-initio modelling of Si2O7, and fitting to quartz compressibility data. Most terms relate to the local T2O7 cluster. The non-codimer repulsion terms (dOO) are computationally expensive.
  46. 46. Stages of refinement Example, LTL, space group P6/mmm (191) two unique T-atoms Place atoms at the Refine T-atoms and Refine T-atoms, Refine T-atoms, geometric center of unit cell parameters O-atoms and O-atoms and their neighbors using “regular unit cell parameters unit cell using (barycentering). tetrahedral” forces Using Boisen-Gibbs- GULP (J. Gale) Refine unit cell on T-atoms Bukowinski force model using T-atoms
  47. 47. Refinement Method Parallel tempering with selective inheritance Parallel simulated annealing runs, with temperature swapping and with elements of a genetic algorithm Inherited “genes” • Temperatures are decreased, and swapped according to a Boltzmann factor • If logjammed, parameter lists are compared, and favorable “genetic” traits are selected from other annealing results
  48. 48. Unpredictable Convergence Rates
  49. 49. Preconditioning the frameworks for refinement SiGH – Silica General Handler (S. A. Wells) • SiGH is a symmetry-aware ‘offspring’ of GASP. It finds rapidly the conformations that preserve best the rigid tetrahedra.
  50. 50. SiGH as an efficient filter • SiGH finds rapidly the implausible frameworks – i.e. those for which there is no hope of ever identifying a tetrahedral flexibility window.
  51. 51. SiGH speeds up the rate of framework discovery by a factor of ~10 • It appears that there are very few “babies in the bathwater”, but it seems likely that some good frameworks will be discarded inadvertently
  52. 52. Combinatorial Explosion The number of graphs tends to increase exponentially with increasing n N = Α × Βν
  53. 53. Combinatorial Explosion The number of graphs tends to increase exponentially with increasing n Pm3m (225) The number of viable frameworks does not increase as rapidly with increasing n
  54. 54. Spacegroup Pm3m is “productive” Pm3m, 1T- atom 3 out of 3 uninodal graphs refine well 225_1_2 225_1_3 225_1_1 LTA SOD KFI UBGB = 0.007605 eV UBGB = 0.026605 eV UBGB = 0.021484 eV UGULP = -128.504213 eV UGULP = -128.562527 eV UGULP = -128.382971 eV
  55. 55. Spacegroup Pm3m is “productive” 12 out of 13 binodal graphs refine well Pm3m, 2 T- atoms
  56. 56. GULP evaluates stability from phonon eigenvalues Pm3m, 2 T- atoms 225_2_13 Tetrapod of double 3-ring prisms Some phonon eigenvalues are complex indicating that the framework is unstable in this space group and composition.
  57. 57. Some construction themes are obvious Pm3m, 3 T- atoms (with hindsight) Sodalite cages connected by chains of cubes. The chain length can be varied indefinitely UGULP = -128.1129 eV/TO2 UBGB = 0.08123 eV/TO2 FD = 5.83 T-atoms/1000 Å3 Density = 0.5817 Coordination sequences TD10 = 245.667 1 4 7 8 10 17 27 35 39 40 42 53 78 110 137 154 1 4 7 10 15 20 25 31 36 41 51 68 89 110 127 139 1 4 9 15 20 23 24 26 33 47 67 88 104 111 112 115 225_3_8 Vertex symbols Parent member of the progression is sodalite
  58. 58. Framework density tends to increase with increasing refinement energy P6 / mmm, 3 T-atoms 659 graphs out of 1150 refined with energy ≤ 1.0 eV/TO2 (BGB) The distribution of frameworks over energy is not uniform.
  59. 59. P6/mmm produces some very pretty frameworks P 6 / mmm,3 T - atoms 191_3_123 [001] [100] [1 1 0]
  60. 60. Enormous channels are possible P 6 / mmm,4 T - atoms a = 41.1Å c = 9.7 Å FD = 6.75 T-atoms/10003 191_4_1955 [001] [100]
  61. 61. Delicate low-density structures P 6 / mmm, 3 T - atoms a = 26.5 Å c = 7.26 Å This representation is cell-doubled 191_4_3295 [001] Assembly of decorated 12-rings or decorated 24-rings FD = 10.88 T-atoms/10003
  62. 62. Likely candidate P 6 / mmm,4 T - atoms a = 18.35Å c = 17.56 Å FD = 16.4 T-atoms/10003 191_4_5828 [001] Vertex symbols UBGB = 0.005 eV/TO2 suggest simple polyhedra Cancrinite and D8R
  63. 63. Higher energy structures are also interesting Pm3m, 3 T- atoms Many beautiful, but improbable frameworks emerge at higher energies 191_3_786 UBGB = 0.5 eV/TO2
  64. 64. Unembeddable frameworks are also interesting In ten, The smallest ring size is 10! ten Another structure, elv, has smallest ring size is 11. It cannot be drawn (yet!)
  65. 65. Our database is online and searchable Several characterization tools have been implemented, including • Interactive graphics • Powder pattern simulation • Bond lengths, angles, topology • Pore characteristics (by Delaunay triangulation) •
  66. 66. Spheres tell us a lot about zeolites 229_5_8058871 Maximum included sphere Largest freesphere Packing: He, Ne, Ar, Kr, Xe
  67. 67. Sphere packing
  68. 68. Sphere packing
  69. 69. What is next? • Extend method out to NT = 12 (ie MFI) and beyond. – Improved graph-filtering based on graph topology is needed. – Rapid graph-refinement strategies are still needed. – Computer cluster working on the problem. • Improve framework topology → microporous properties tools to help identify appropriate synthetic targets. • Implement search algorithms based on pore characteristics. • Can Delaunay triangulation work on a torus? • Solve the Apollonian problem. This will accommodate the different van der Waals radii of the framework atoms. • Implement search algorithms against powder patterns. • Do all ‘real’ zeolites have a flexibility window? (Thorpe, Kapko)
  70. 70. “Real” zeolites are flexible A. Sartbaeva, S. A. Wells, M. M. J. Treacy and M. F. Thorpe, The flexibility window in zeolites, Nature Materials 5 962–965 (2006).
  71. 71. A set of simple rules helps limit the number of combinatorial possibilities (1) No T-atom can lie on a 6-fold axis (2) No T-atom, or T-atom vertex, can lie on a vertex of the fundamental region (3) If a T-atom lies on a face of the fundamental region, then two (and only two) of the T-atom vertices lie on that same face. (otherwise it is planar)) • Connections to atoms outside the fundamental region must involve either a T-atom, or one of its vertices, that lies on a mirror (or on an edge defined by two perpendicular mirrors). (5) All T-atoms are connected to four other T-atoms. (6) Tetrahedra are denied edge- and face-sharing connectivities. • Each of the five faces of the fundamental region must have at least one bond connecting through it. (For 3-dimensional connectivity)
  72. 72. "γ-silica" comprises chiral space-filling units Ia3 206_1_170 There are equal numbers of left- and right-handed units One of the TOT bond angles is ~180°
  73. 73. Sphere packing
  74. 74. Pores are characterized automatically by Delaunay Triangulation Methods Empty circumspheres in SOD Delaunay triangulation identifies the empty circumspheres in an array of points. It is a natural and convenient method for identifying and characterizing the empty spaces (pores and channels) in zeolites. It also allows us to estimate pore opening diameters.
  75. 75. Zeolites as Colored Graphs • Frameworks are represented as graphs with four edges (bonds) from each vertex. • All 4-connected uninodal graphs look the same – the clover-leaf shape. • There are four distinct 4-connected binodal graphs. • Edges (bonds) are "colored" by the crystallographic operator (and its inverse) that defines the connection. • A combinatorial search is performed on all possible permutations of edge colorings.
  76. 76. Interthreaded Cristobalite Two frameworks do not cross-connect
  77. 77. Interthreaded Δ1 cristobalite framework Pn3m, No. 224 This framework exists! [Sn5S9O2] . [HN(CH3)3]2 — Parise and Ko (1995)
  78. 78. Interthreaded FAU framework Pn3m, No. 224
  79. 79. Coordination Sequence • The coordination sequence for a T-atom Sk is the number of T-atoms in the shell that is k bond lengths away. • Topological density TD10 can be defined simply as the sum of the first 10 entries of the coordination sequence Count T-atoms on expanding shell Coordination sequence is not necessarily Faujasite fragment unique to each framework.
  80. 80. Circuit Symbols and Vertex Symbols • Each T-atom has 6 interbond angles • Describe each of the six shortest loops connecting any pair of bonds • Example FAU – has one unique T-atom • Circuit/Vertex symbols are not necessarily unique to each framework.
  81. 81. Issues when atoms are not points Apollonian triangulation Eight solutions exist for circles, sixteen solutions for spheres. We believe that we have this problem solved (in principle!)
  82. 82. Lowest-energy 6 T-atom structure Pm3m, 6 T- atoms UGULP = -128.5184 eV/TO2 Clathrated assembly of sodalite cages (in a sodalitic arrangement), cancrinite cages double 6-ring prisms and cubes. 225_6_22665 Modified SOD + LTA + LTL.
  83. 83. Family of 3D defect structures Pm3m, 6 T- atoms a = 41.285 Å (doubled cell) 225_6_22585 225_6_22665 a = 41.633 Å UGULP= -128.4852 eV/TO2 UGULP = -128.5184 eV/TO2 (More stable!) • The third end-member of this particular series, ALL CAN/D6R units, has not yet been located in the data (confident it is there). • The SOD ⇐⇒ CAN/D6R transformation can occur in local pockets of 8 units at a time
  84. 84. Framework of ZSM-10 Known to be in P6/mmm with 6 unique T-atoms There were 18.4 million graphs with 6 unique T-atoms. This is the one!
  85. 85. Visual Comparison of Powder Patterns Favors Model A It is difficult to remove all extraframework K cations. A recent Rietveld refinement by D.L Dorset confirms A as the best fit.
  86. 86. ZSM-10: plausible low-energy frameworks Two frameworks with 5-rings have even lower energy than LTL When refined as pure SiO2
  87. 87. Correlation between BGB and GULP framework energies is linear at low energies P 6 / mmm, 3 T - atoms • Some of the scatter may be related to the vagaries of simulated annealing • The gradient is 1:6 at low energies, 1:1 at higher energies (EGULP > 0.7 eV).
  88. 88. Delaunay Triangulation of a set of points
  89. 89. The perpendicular bisectors define the Voronoi cells The edges of the Voronoi cell are equidistant from two points. Each Voronoi cell “belongs” to one point.
  90. 90. The empty circumcircles reveal the empty space Each circle touches three points, but does not enclose any points. These circles thus delineate the empty space – i.e. the pores!
  91. 91. Typo at a recent conference: Zeoltie? A combinatorial permutation of zeolite. ZeoTile Is more appropriate for a polyhedral tilings (O. Delgado Friedrichs & M. O’Keeffe?) OzElite J. C. H. Spence and D. J. Smith?
  92. 92. Cross-link defect that connects the interthreaded cristobalite frameworks “Wormhole” defect that cross-connects two parallel frameworks
  93. 93. Cogwheels of double 3-ring prisms Pm3m, 3 T- atoms 225_3_32 225_3_27
  94. 94. Establishing Connectedness in the General Case is Time-Consuming. For connectedness, there must exist a path of bonds connecting each atom A to its translated image A' in an adjacent unit cell. Further, there must exist a path connecting all dissimilar atoms. To prove that A and its image A' are not connected can involve (2n+1)3 unit cells, where n is the number of unique atoms in the unit cell. Significant speed-up is obtained by restricting the search to adjacent unit cells only. Some legitimate structures will be overlooked.
  95. 95. Comparison of GULP and BGB Refinements The GULP program and the Boisen-Gibbs-Bukowinski (BGB) Refinements produce subtle differences in frameworks BGB is a bonded-neighbour-only force-field
  96. 96. Correlation between topological density and framework density Correlation is strongest for lowest energy refinements
  97. 97. Comparison of two combinatorial methods O. Delgado Friedrichs et al (Nature 400 644 (1999)) demonstrated a combinatorial method based on tilings of polyhedra. Since many important zeolites can be thought of as being built from simple polyhedral units, the tiling method effectively pre-selects the connected sub-units (tiles) based on their likelihood of forming regular tetrahedral frameworks. In our method, the sub-unit is the isolated T-atom. ALL possible graphs are found for a given space group and number of unique T-atom by permuting all possible arrangements of T-atoms on special crystallographic sites. However, many of these graphs cannot be arranged as regular Tetrahedral frameworks. The likely topologies (ie based on the polyhedra implicit in the graph) are filtered out after each graph is created. The two methods must converge on the same frameworks, but from different starting points.
  98. 98. What is next? • The Structure Commission of the International Zeolite Association is planning to create a database of hypothetical zeolite frameworks that will be available to researchers on the web (perhaps by mid-2004). – Data of Smith, O’Keeffe/Delgado-Friedrichs, Bell/Foster, Deem, Treacy. • Extend method out to NT = 12 (ie MFI) and beyond. – Improved graph-filtering based on graph topology is needed. – Rapid graph-refinement strategies are needed. – Computer cluster working on the problem (plus Martin Foster). • Improved tools for cataloging frameworks (O’Keeffe leads the way) • Improve graphics tools for visualizing results! • Needed: Improved framework topology → microporous properties tools to help identify appropriate synthetic targets.
  99. 99. Outstanding issues for the database • Solve the Apollonian problem. This will accommodate the different van der Waals radii of the framework atoms. • How to handle elliptical apertures? • Implement search algorithms based on pore characteristics. • Implement search algorithms against powder patterns
  100. 100. Combinatorics of connections between crystallographic sites There are 14 connectable sites for tetrahedra in the P6/mmm fundamental region. The rules for interconnections depend on the site.
  101. 101. A Venn Diagram Framework! P 4 / mmm, 3 T- atoms Tetragonal trihedron I4/mmm (139) 3 unique T-atoms A 3-fold “paddle wheel” -128.238 eV/TO2 (from GULP) of bent 4-rings. 1 4 8 13 22 36 52 69 86 98 112 1 4 9 16 24 35 52 67 78 101 138 1 4 9 18 30 39 46 60 86 121 160