Geometry, combinatorics, computation with Zeolites

Designer zeolites
Igor Rivin
Temple University
Department of Mathematics

All work joint with Mike Treacy (ASU)

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...

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?

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.

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.

• 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.

• 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.

• 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.

• 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.

• 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.


• 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.

• 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.

• Aquarium keeping

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

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

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

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

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
http://internet-mobil.na.xom.com/mobil_research/mapped/cumene.html 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

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)
Mordenite (MOR)

Attests to the versatility of these materials and the
exceptional selectivity provided by the specific crystal structure

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

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.

The number of known zeolites is
growing

LTL Framework

Pt/ K-zeolite L converts n-hexane to benzene with high yield and efficiency

Construction of LTL Model

Construct
Cancrinite
cages

Stack Cross-connect
Cancrinite cages Cancrinite columns

But instead…

Construct
Cancrinite
cages

Stack Cross-connect
Cancrinite cages Cancrinite columns

Definitely Not LTL

191_2_13

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

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”

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.

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)

The LTL framework has P6/mmm symmetry

36 T-atoms and 72 oxygen atoms per unit cell

The LTL framework has P6/mmm symmetry

All of the symmetry operations can be generated by mirrors

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.

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

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

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

Generalizing to other space
groups

“Roadmap” viewgraph from 1991

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.

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.

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

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

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 6.62.6.62.6.62

• Related to O'Keeffe's
γ –Si, bcc(8)
• Post-diamond
high-pressure form
of carbon
206_1_170

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.

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.

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

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

Unpredictable Convergence Rates

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.

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.

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

Combinatorial Explosion
The number of graphs tends to increase exponentially with increasing n

N = Α × Βν

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

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

Spacegroup Pm3m is “productive”
12 out of 13 binodal graphs refine well

Pm3m, 2 T- atoms

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.

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 g.cm-3
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
4.4.4.4.4.24
Parent member of the progression is sodalite 4.4.4.4.4.24
4.6.4.6.4.24

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.

P6/mmm produces some very pretty frameworks
P 6 / mmm,3 T - atoms

191_3_123

[001]

[100] [1 1 0]

Enormous channels are possible


a = 41.1Å
c = 9.7 Å

FD = 6.75 T-atoms/10003

191_4_1955 [001]

[100]

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

Likely candidate
a = 18.35Å
c = 17.56 Å
FD = 16.4 T-atoms/10003

4.4.4.6.8.12
4.4.4.6.6.8
4.4.6.6.6.6
4.6.4.6.6.12
191_4_5828 [001]
Vertex symbols
UBGB = 0.005 eV/TO2 suggest simple
polyhedra
Cancrinite and D8R

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

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!)

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)
• http://www.hypotheticalzeolites.net

Spheres tell us a lot about
zeolites

229_5_8058871
Maximum included sphere
Largest freesphere Packing: He, Ne, Ar, Kr, Xe

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)

Geometry, combinatorics, computation with Zeolites

“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).

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)

"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°

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.

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.

Interthreaded Cristobalite

Two frameworks do not cross-connect

Interthreaded Δ1 cristobalite framework
Pn3m, No. 224

This framework exists! [Sn5S9O2] . [HN(CH3)3]2 — Parise and Ko (1995)

Interthreaded FAU framework
Pn3m, No. 224

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.

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.

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!)

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.

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

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!

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.

ZSM-10: plausible low-energy frameworks

Two frameworks with 5-rings have even lower energy than LTL
When refined as pure SiO2

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).

Delaunay Triangulation of a set of points

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.

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!

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?

Cross-link defect that connects the
interthreaded cristobalite frameworks

“Wormhole” defect that cross-connects two parallel frameworks

Cogwheels of double 3-ring prisms
Pm3m, 3 T- atoms

225_3_32

225_3_27

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.

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

Correlation between topological density
and framework density

Correlation is strongest for lowest energy refinements

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.

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.

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

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.

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 4.42.4.62.62.261024
1 4 9 16 24 35 52 67 78 101 138 4.4.4.82.4.82
1 4 9 18 30 39 46 60 86 121 160 4.42.8.8.8.8

Geometry, combinatorics, computation with Zeolites

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Geometry, combinatorics, computation with Zeolites