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Flexibility of ideal zeolite frameworks
V. Kapko, C. Dawson, M. M. J. Treacy* and M. F. Thorpe
Received 9th March 2010, Accepted 19th June 2010
First published as an Advance Article on the web 29th June 2010
DOI: 10.1039/c003977b
We explore the flexibility windows of the 194 presently-known zeolite frameworks. The flexibility
window represents a range of densities within which an ideal zeolite framework is stress-free. Here,
we consider the ideal zeolite to be an assembly of rigid corner-sharing perfect tetrahedra. The
corner linkages between tetrahedra are hard-sphere oxygen atoms, which are presumed to act as
freely-rotating, force-free, spherical joints. All other inter-tetrahedral forces, such as coulomb
interactions, are ignored. Thus, the flexibility window represents the null-space of the kinematic
matrix that governs the allowable internal motions of the ideal zeolite framework. We show that
almost all of the known aluminosilicate or aluminophosphate zeolites exhibit a flexibility window.
Consequently, the presence of flexibility in a hypothetical framework topology promises to be a
valuable indicator of synthetic feasibility. We describe computational methods for exploring the
flexibility window, and discuss some of the exceptions to this flexibility rule.
I. Introduction
Zeolites are remarkable oxide framework nanomaterials that
have important applications in the petrochemical and fine
chemical industries. Zeolites are structurally related to quartz
in the sense that they are periodic frameworks comprising
corner-sharing SiO4 tetrahedra. Each tetrahedron is connected
to four others via the four oxygen atoms that are located at the
tetrahedral apices. Zeolite frameworks are distinguished from
quartz, and other dense silicates, by their microporosity. Their
frameworks contain periodic arrays of channels and pores that
allow small molecules, such as water and light hydrocarbons,
to diffuse through them. Consequently, zeolites have large
internal surface areas accessible by such molecules.
Zeolites are not constrained to a pure SiO2 composition.
Typically, they are aluminosilicates, with trivalent aluminium
substituting isomorphously for a tetravalent silicon, with the
charge being balanced by extra-framework cations, such as
Na+ and H+, which are not physically part of the framework.
Other elements such as beryllium, boron, iron, phosphorous,
germanium, cobalt and zinc can also be substituted into the
framework tetrahedral position. These cations influence the
structure and chemistry of the framework. For example, for a
T–O distance d (where T represents the tetrahedral atom, Si,
Al etc.), the tetrahedron edge length L is the nearest-neighbour
O–O distance given by L ¼ffiffiffiffiffiffiffiffi8=3
pd. Thus, for a charge-neutral
SiO4 tetrahedron, we have d E 0.161 nm and L E 0.263 nm,
whereas for an anionic AlO�4 tetrahedron we have dE 0.173 nm
which corresponds to a larger tetrahedron with edge length
L E 0.282 nm. This mixture of tetrahedron sizes, and their
distribution, modifies the precise details of the framework
structure. Additionally, the extra-framework cations will affect
the chemistry, and the associated coulomb forces will, to a
small extent, also influence the structure. In particular, the
presence of protons (H+) renders a zeolite framework into a
solid Brønsted acid, a most valuable property for the zeolite-
based catalytic cracking of hydrocarbons.
Because of their usefulness, there are considerable ongoing
efforts to make new zeolites, and to better understand their
properties. Furthermore, the recent development of large
databases containing millions of hypothetical zeolite frameworks
demands improved computational tools that will rapidly and
reliably sift through these vast databases to identify the
potentially useful frameworks.1–3 Perhaps of even more
importance, but seemingly more difficult to achieve, is the
development of computational tools that generate recipes for
the selective synthesis of targeted hypothetical frameworks.
The ZEBEDDE program has been an important development
in this area.4
In this paper we examine a remarkable mechanical property
of ideal zeolite frameworks—their flexibility. We define an
ideal zeolite framework as one that is built using rigid, perfect,
corner-connected tetrahedra with freely rotating spherical
joints at the vertices. Here we show that almost all of the
known zeolite frameworks (those that have been synthesized
or occur as minerals) lack mechanical rigidity as ideal zeolite
frameworks over a range of densities. Within this range of
densities, which is referred to as the flexibility window,5 there is
no stress within the idealized framework. However, many
hypothetical zeolite frameworks lack this flexibility, at least
when constructed with only one type of tetrahedron, such as in
a pure SiO2 composition. Thus, many hypothetical frameworks
contain stressed tetrahedra, even when the nominal energy of
formation, as calculated by GULP6 for example, is low. This
raises the interesting possibility that the existence of a flexibility
window in an idealized hypothetical zeolite framework is a
strong indicator that the framework is realisable. This has
important implications in ascertaining which new zeolites are
most likely to be synthesized in the future.
It has been long known that some idealized zeolite frame-
works lack rigidity. Pauling, in his 1930 paper on the structureArizona State University, Department of Physics, P.O. Box 871504,Tempe, AZ 85287-1504, USA. E-mail: [email protected]
This journal is �c the Owner Societies 2010 Phys. Chem. Chem. Phys., 2010, 12, 8531–8541 | 8531
PAPER www.rsc.org/pccp | Physical Chemistry Chemical Physics
of sodalite, remarked that the SOD framework lacked rigidity
over a wide range of densities.7 In fact, the SOD framework
has one of the largest flexibility windows obtained for an ideal
zeolite framework. The experimental flexibility of zeolite rho
(RHO framework) is also well known,8 and its range is
comparable to that for SOD. However, many zeolite frame-
works are found to be rigid when constrained to their nominal
space group symmetry. For example, in the cubic space group
Pm�3m, the flexibility window of the SOD framework has zero
width, with a unique solution for the atom coordinates and cell
dimension. More recently, Sartbaeva et al.5 showed that when
the symmetry is reduced to P1, and as pure SiO2 composition,
many of the known zeolite frameworks exhibit such a flexibility
window. In addition, they noted that real zeolite materials
tend to adopt a structure that is close to the low density end of
the window. This preference for low density configurations
was ascribed to coulomb repulsions, mainly between frame-
work oxygen atoms.9 Simple models that provide insight and
unifying principles are particularly important when trying to
make sense of large amounts of data; in this case structural
data associated with real and hypothetical zeolites. It is in this
spirit that this paper is written.
The flexibility of zeolites arises from the fact that their
frameworks are locally isostatic; that is, the six degrees of
freedom per tetrahedron (three translational, three rotational)
are exactly matched by the six constraints imposed by the
sharing of the oxygen atoms (three constraints for each of four
oxygen atoms, divided by two since every oxygen is shared
between two tetrahedra). It would seem at first that the
equality of degrees of freedom and constraints would mean
that zeolite frameworks are fully determined geometrically.
However, the periodicity introduces six additional degrees of
freedom globally to the framework.10 Three of these can be
discarded as they correspond to simple translations of the
structure along the three orthogonal axes of the underlying
lattice. The remaining three must correspond to internal
mechanisms of the periodic structure. (They cannot be
three trivial rotations, since such motions are generally
incommensurate with the underlying periodic lattice.) The
presence of additional crystallographic symmetries (mirror
and glide planes, rotation and screw axes) also adjusts the
numbers of degrees of freedom as well as the constraints,
potentially increasing the surfeit of degrees of freedom over
constraints. (We are always guaranteed at least three more
degrees of freedom than constraints in three dimensions.10)
This excess of degrees of freedom over constraints is then
responsible for the generally under-determined nature of
zeolite structures, creating a null-space of solutions to the
kinematic matrix that are stress-free.
The computational algorithms needed to identify and
systematically explore the flexibility window cannot use the
ubiquitous gradient methods that minimize energy or cost
functions. Since we ignore coulomb terms in our ideal zeolite
framework, the potential energy is zero and flat within the
window. Within the flexibility window the framework is
stress-free and there are no restoring forces to guide any
search of this space. Rigid unit modes (RUM model) have
been studied extensively in tetrahedral frameworks by Dove
and co-workers.11,12 The rigid-unit modes are closely related
to the flexible folding modes studied here. Generally speaking,
rigid-unit modes exist as states at the surface of the flexibility
window where the spring forces between tetrahedra vanish. In
principle, such modes can be used to explore the denser states
of the flexibility window. The GASP program by Wells13 uses
geometric algebra to identify stress-free frameworks, and
explores the flexibility window by incrementally modifying
the structure while constraining those configurations to lie
within the window. Depth-first-search methods using GASP
within the flexibility window allow some systematic exploration.
However, neighbouring points in the flexibility window, based
on those structures having closely similar densities, do not
guarantee that those structures are topological neighbours. In
addition, multiple distinct structures can correspond to the
same density, yet necessitating large cooperative motions in
order to transform one structure into the other. Essentially,
most neighbours can be reached only by ‘unfolding’ the
framework, and then initiating a new, topologically distinct,
folding mechanism. Consequently, such searches of the
flexibility window tend to be hit-or-miss and incomplete.
A promising approach to the study of framework flexibility
has been developed for mechanical engineering applications by
Pellegrino14 and Guest and Hutchinson10 to study rigidity in
deployable structures and periodic trusses. The method uses
singular value decomposition (SVD) to systematically explore
the null-space of flexible systems. This method is readily
adapted to the study of ideal zeolite frameworks.
In this paper, we examine the flexibility characteristics of all
of the 194 zeolite frameworks that have been approved to date
by the Structure Commission of the International Zeolite
Association. We present results using both the GASP program
of Wells13 and a newly-implemented program ZeNuSpEx
(Zeolite Null Space Explorer) that is based on the SVD
methods. We discuss the flexibility characteristics of the
SOD framework, as well as examples of frameworks that
appear to be intractably rigid. This work establishes the
validity of this approach and sets the stage for future work
which will apply the principles learnt from real zeolites as a
sieve to sort through hypothetical zeolites for promising new
zeolite structures.
II. Computational methods
Zeolite frameworks were initially modeled using the GASP
computational tool (Geometric Analysis of Structural
Polyhedra), which was developed by Wells et al.13 This tool
had been used successfully in a previous study of zeolite
flexibility.5 Frameworks are treated as periodic networks of
rigid, corner-connected tetrahedral units that are formed by a
T-atom (such as silicon, aluminium, etc.) that has four oxygen
atoms at the regular tetrahedral vertices to form a TO4 unit.
To find relaxed atom configurations satisfying constraints that
are consistent with rigid tetrahedra, GASP attempts to match
oxygen atom positions to a template (represented by the
appropriate number of regular tetrahedra with an implied
T-atom at the center) within the unit cell. Periodic boundary
conditions are applied, but no further symmetry constraints
are included in order to allow maximal exploration of the
configurational space. Each oxygen atom is tethered to two
8532 | Phys. Chem. Chem. Phys., 2010, 12, 8531–8541 This journal is �c the Owner Societies 2010
corresponding template tetrahedra by a fictitious harmonic
spring with a natural length of zero. Hard-sphere repulsion
between oxygen atoms that are on different tetrahedra is
imposed to prevent steric clashes. We used an oxygen radius
of 0.135 nm, which is the value used by the Atlas of Zeolite
Framework Types.15 A cost function is then generated that
penalizes both the bending and stretching of the springs, as
well as any intertetrahedral oxygen atom overlaps. GASP uses
the limited memory Broyden–Fletcher–Goldfarb–Shanno
(BFGS) algorithm, which is a quasi-Newton method,16 to
minimize this function. A configuration is considered relaxed
when all T–O bond lengths have reached 0.161 � 0.0001 nm (a
typical silicon-oxygen bond distance in zeolites) and all
O–T–O bond angles have reached 109.471 � 0.0011 with no
oxygen overlaps.
GASP attempts to explore the flexibility window by
incrementally varying unit cell parameters, thereby adjusting
cell volume and density, and using the fractional coordinates
of the previously-obtained relaxed configuration as an initial
guess. The primary input is usually the cell dimensions and
coordinates obtained from the International Zeolite Association
database.17 The oxygen atoms are then moved a small distance
(typically, 0.01 nm or less) from their initial positions and each
tetrahedral template is placed at the geometric center of the
four corresponding oxygen atoms. The positions of the atoms
and the template are then refined until conditions for relaxation
are met or the minimization algorithm fails. This process is
repeated numerous times at each given density to determine
whether relaxed configurations can be found for a specific set
of unit cell parameters.
An alternative approach to exploring the flexibility window
using GASP is based on a breadth-first-search algorithm. This
algorithm explores the flexibility window by exploring a
finely-spaced grid of unit cell parameters, starting from a
known relaxed configuration and checking whether or not its
neighbouring configurations can also be relaxed. Once all of
the neighbours of the starting configuration have been
checked, then those neighbours that remain flexible are also
checked for their flexible neighbours. The algorithm stops
when it has exhausted all relaxed configurations accessible
from the starting point. This method has been used to build
two-dimensional projections of the flexibility window for
the frameworks MTN and SOD, which are presented and
discussed in the following sections.
A third method of probing the flexibility window involves
modification of the GASP approach. In this approach,
neighbouring oxygen atoms and T-atoms are attached to
one another by springs of the correct natural length for
forming unstressed perfect tetrahedral units. For each
tetrahedron, there a four springs associated with the T–O
bond distances, and six springs associated with the O–O edge
lengths. No angular constraints are necessary, since we are
allowing the tetrahedra to rotate about the oxygen force-free
‘spherical joints’. As before, periodic boundary conditions are
applied without any additional internal symmetry constraints.
A cost function is produced that includes each spring term as
well as hard-sphere repulsion terms between oxygen atoms,
and minimization of this function is undertaken by refining
input coordinates, using the same quasi-Newton method used
by GASP, until tetrahedral distance constraints are satisfied to
within 0.0001 nm. This stringent tolerance is needed in order
to find the near-perfect geometry. The results of these simulations
were used to check the GASP simulations, and also to generate
new relaxed configurations from which to restart the GASP
program.
While these methods have proven effective in finding
numerous relaxed conformations for almost all known zeolite
structures, the configuration space to be explored within the
flexibility window is enormous and it is difficult to ascertain
the extent to which our simulations efficiently explore this
space. Almost all structures exhibit multiple folding paths
from a low-density, maximum symmetry configuration. As a
result, we notice variations at the high-density end of the
window depending on the folding path followed. It is clear
that multiple relaxed configurations with significantly different
folded topologies can exist at any given density and even for a
given set of unit cell parameters. There is also the possibility
that ranges of relaxed configurations are topologically isolated
from one another when constraints are placed on the unit cell
parameters. Despite the power of the above tools, it is
apparent that a more systematic exploration of the flexibility
window is necessary to make final, definitive statements in this
regard.
A promising approach, which has been developed in the
field of mechanical engineering for the analysis of deployable
structures, such as spacecraft antennae, involves analysis
of the compatibility (or kinematic) matrix C.14 This is a
mathematical construct that relates the vector d, formed from
the generalized atomic displacements (displacements of the
‘joints’), to the vector e, which is the vector formed from the
extensions of the bonds (‘bars’),18
Cd = e. (1)
The compatibility matrix is found from the set of constraint
conditions (rj � ri)2 = L2
ij, where Lij is the bond length. It
has b rows, one for each bond in the unit cell (allowing for
periodicity), and, for zeolites, b+ 6 columns. Although zeolite
frameworks are locally isostatic, that is, b = 3N for N oxygen
vertices, there are six additional degrees of freedom associated
with the unit cell dimensions. It is important to include the
effects of small changes in the unit cell parameters in the
displacements (see ref. 10 for details). Within the flexibility
window, the extension vector is zero, e = 0. Displacements
that lie within this strain-free flexible regime are the null
eigenvectors of the eigenvalue equation Cd = ld. The null
eigenvectors are those solutions with eigenvalue l = 0, and
correspond to the strain-free folding mechanisms of the
framework. The kinematic matrix C for periodic locally
isostatic frameworks, such as zeolites, has more degrees of
freedom than constraints. This is because the generalized
displacement vector not only represents the displacements of
all the atomic coordinates, but also includes the six components
corresponding to changes in the unit cell parameters.
Consequently, C is not a square matrix, having six more
columns than it has rows. Frequently, topological symmetry
within the framework renders some of the constraints
degenerate, and consequently there can be more than six null
This journal is �c the Owner Societies 2010 Phys. Chem. Chem. Phys., 2010, 12, 8531–8541 | 8533
eigenvalues. This tends to happen at the maximum symmetry
condition of the unit cell, which almost always corresponds to
the maximum unit cell volume. Three of the null eigenvectors
correspond to trivial translations along the lattice axes. Thus,
there remain at least three null eigenvectors for every zeolite
framework that correspond to strain-free displacements.10 Of
interest to us are the finite-amplitude null eigenvectors that
define the flexibility window.
Our newly-developed program ZeNuSpEx (Zeolite Null
Space Explorer) uses singular value decomposition (SVD)
methods to find non-trivial null vectors of the kinematic
matrix.14 These null vectors can be added, with small amplitude,
to the original coordinate vector space to produce a new,
nearly-relaxed configuration. This configuration is then
relaxed again by energy minimization (to eliminate second
and third order deformations), resulting in a unique set of
relaxed coordinates that is a topological neighbour of the
original set. By following the null eigenvectors in this way,
we can, at least in principle, explore systematically the
flexibility window, eliminating the need for random moves.
In ZeNuSpEx, we have adopted the form of the kinematic
matrix introduced by Guest and Hutchinson10 for studies
of periodic trusses in mechanical engineering applications.
Using a hybrid of energy minimization and SVD techniques,
ZeNuSpEx allows systematic exploration of the flexibility
window. The issues still confronting our implementation of
this new approach will be discussed later.
III. Results and discussion
A Two-dimensional example; the kagome lattice
As a two-dimensional example of framework flexibility, we
first examine the kagome lattice (Fig. 1). This lattice has been
studied in more detail elsewhere,10,19,20 and comprises a plane
tiling of corner-connected equilateral triangles. The lattice is
equivalent to that found in a graphite sheet, where each
carbon atom has been replaced by an equilateral triangle. In
Fig. 1a, the primitive unit cell contains two triangles, and the
unit cell volume is the maximum allowed that is consistent
with perfect, unstrained, equilateral triangles. If we assume
that the triangle corners are interconnected by force-free pin
joints, then the triangles can rotate cooperatively to produce
higher-density configurations, such as those shown in Fig. 1b.
These configurations are still stress-free, and represent states
that lie within the flexibility window. In this particular folding
mechanism, the symmetry constraints dictate that alternate
triangles rotate in opposite directions. The internal degrees of
freedom of the structure can be augmented by increasing the
size of the unit cell so that there are more triangles per unit
cell. In Fig. 1c, the cell is doubled in area by transforming the
cell in Fig. 1a to a non-primitive rectangular unit cell to
enclose four triangles per cell. A second alternative folding
mode emerges as a result of this cell-doubling. The earlier
mode in Fig. 1b is still available, but can only be accessed by
expanding back to the maximum volume (Fig. 1a) and then
initiating the alternative folding mechanism corresponding to
Fig. 1b. Fig. 1d shows the new mode that occurs when the cell
is quadrupled to enclose 8 triangles per unit cell. This increase
in modes occurs because the number of symmetry constraints
per triangle is being reduced as the unit cell area is increased.
Zeolite frameworks are three-dimensional versions of such
four-valent frameworks. The four corners of each tetrahedron
are interconnected by force-free ‘spherical joints’, enabling
a more complex class of folding mechanisms to evolve.
Alternatively, zeolites can be viewed as six-valent networks
of oxygen atoms, where each oxygen atom is connected to six
others in an approximate octahedral arrangement. This latter
approach, while unconventional, is useful for exploring the
flexibility of these frameworks. We examine several zeolite
examples below.
B Examination of the known zeolites
Table 1 presents the lowest and the highest relative framework
densities, rmin and rmax (in units of T-atoms per nm3),
bounding the flexibility window for all of the 194 known-to-
date zeolite frameworks. The results are for frameworks
represented as pure SiO2, and we do not include the results
when different T-atoms (such as boron, aluminium or germanium)
are substituted at various sites. Different T-atoms have different
T–O and O–O tetrahedral distances, and this is found to alter
the flexibility window, as we shall discuss later. The flexibility
parameter,19 FO = Vmax /Vmin R rmax / rmin, is a simple
measure of the relative width of the flexibility window. For
the results presented in this table, at each density the unit cell
volume is held constant while the six unit cell parameters (cell
edge lengths a, b and c, and cell interaxial angles a, b and g) areallowed to vary. This introduces five additional degrees of
freedom which were included as adjustable parameters
along with the list of coordinate variables in the kinematic
matrix. This Table represents results from both GASP and
Fig. 1 Four configurations of the two-dimensional kagome lattice.
(a) Minimum density state (i.e.maximum unit cell volume), with p6mm
plane group symmetry. (b) A snapshot of the collapse mode when
there are two triangles per unit cell (p31m plane group symmetry). (c)
Snapshot of an alternative collapse mode when there are four triangles
per rectangular unit cell (p2gg plane group symmetry). (d) Snapshot of
an additional collapse mode when there are eight topologically distinct
hexagons per oblique unit cell (p2 plane group symmetry). If we allow
overlap of the equilateral triangles, the density can become infinite.
Adopted from ref. 19.
8534 | Phys. Chem. Chem. Phys., 2010, 12, 8531–8541 This journal is �c the Owner Societies 2010
ZeNuSpEx, and we report the extremes of density found
by either program. (As mentioned earlier, with improved
eigenvalue-following procedures, the ZeNuSpEx program will
be the more reliable tool.) The Table reveals that 182 of the
194 known zeolite frameworks are flexible when formed with a
pure SiO2 composition. The 12 inflexible frameworks tend to
have features such as double 4-rings, and more than one 5-ring
meeting at a vertex. The residual deformations are always
small, B0.01 nm or less.
C Shapes of the flexibility windows
The flexibility window exists in a six-dimensional space that is
defined by the six unit cell parameters, and is hard to represent
graphically. Nevertheless, it is instructive to examine the
boundaries of two-dimensional projections of the flexibility
windows (Fig. 2). The results for three well known cubic
zeolites, FAU, SOD and MTN are presented in Fig. 2a–d.
The yellow areas represent the flexibility windows. Fig. 2a–b
show the variation in relative density r/r0 of the FAU and
SOD frameworks as a function of the axial ratio c/a. r0 is thenominal density of the pure silica material as determined by
distance-least-squares fitting by the DLS76 program.21 The c/a
ratio represents a tetragonal distortion of the topologically
cubic cells. We have held b= a and a= b= g= 901 for these
2D plots. The numbers decorating the boundaries of the
windows are the number of extra-tetrahedral (non-codimeric)
oxygen–oxygen contacts that define the boundary. Such
contacts almost always delimit the high-density boundaries
Table 1 Minimum (rmin) and maximum (rmax) framework densities (T-atoms per nm3) for idealized zeolites with SiO2 composition. The flexibilityparameter FO is the ratio rmax/rmin. Entries with a dash ‘‘—’’ indicate that there is no flexibility window for this framework as a pure SiO2
composition. In many cases, flexibility is restored with a mixed composition, e.g. as an alumino-silicate or germano-silicate etc
CODE rmin rmax FO CODE rmin rmax FO CODE rmin rmax FO CODE rmin rmax FO
ABW 17.49 30.06 1.72 DDR 17.54 18.91 1.08 LTN 16.49 19.79 1.19 SAT 16.33 19.14 1.17ACO 16.40 25.89 1.58 DFO 14.78 16.53 1.12 MAR 16.70 27.51 1.65 SAV 15.20 17.24 1.13AEI 14.97 18.41 1.23 DFT 17.57 22.71 1.29 MAZ 16.63 18.65 1.12 SBE 13.61 16.05 1.18AEL 18.03 24.45 1.36 DOH 17.80 20.76 1.17 MEI 14.56 15.42 1.06 SBN 16.60 19.02 1.15AEN 20.88 22.36 1.07 DON 16.18 23.18 1.43 MEL 16.94 20.66 1.22 SBS 13.66 15.64 1.14AET 17.49 22.79 1.30 EAB 15.92 18.82 1.18 MEP 18.41 21.09 1.15 SBT 13.67 15.63 1.14AFG 16.73 26.17 1.56 EDI 16.26 19.24 1.18 MER 16.08 25.29 1.57 SFE 16.96 19.72 1.16AFI 16.67 21.49 1.29 EMT 13.13 16.83 1.28 MFI 16.94 21.21 1.25 SFF 18.03 19.99 1.11AFN 17.80 20.00 1.12 EON 16.78 18.74 1.12 MFS 17.66 18.78 1.06 SFG 17.44 19.38 1.11AFO 18.19 24.44 1.34 EPI 18.21 21.31 1.17 MON 17.60 23.33 1.33 SFH 16.33 18.14 1.11AFR 14.99 18.31 1.22 ERI 15.94 18.80 1.18 MOR 16.94 19.58 1.16 SFN 16.41 17.87 1.09AFS 14.49 16.99 1.17 ESV 16.37 21.99 1.34 MOZ 17.00 18.66 1.10 SFO 15.15 17.25 1.14AFT 14.76 17.58 1.19 ETR 15.91 18.20 1.14 *MRE 18.27 26.41 1.45 SFS 16.36 18.62 1.14AFX 15.03 17.46 1.16 EUO — — — MSE 16.17 17.91 1.11 SGT 17.58 19.38 1.10AFY 14.45 16.06 1.11 EZT 16.90 20.28 1.20 MSO 17.74 22.64 1.28 SIV 16.09 24.49 1.52AHT 19.38 28.26 1.46 FAR 16.82 26.12 1.55 MTF 20.95 22.61 1.08 SOD 16.58 27.98 1.69ANA 19.20 27.01 1.41 FAU 13.13 16.85 1.28 MTN 18.18 20.87 1.15 SOF 16.99 17.41 1.02APC 17.49 22.47 1.29 FER 17.51 22.64 1.29 MTT 18.17 20.94 1.15 SOS 16.58 21.39 1.29APD 17.93 27.69 1.54 FRA 16.63 25.66 1.54 MTW 18.29 20.24 1.10 SSF 16.36 21.23 1.30AST 15.78 27.38 1.74 GIS 16.08 26.94 1.67 MVY — — — SSY 16.85 19.91 1.18ASV 19.80 21.58 1.09 GIU 16.84 27.79 1.65 MWW 16.30 17.33 1.06 STF 17.00 19.88 1.17ATN 17.38 29.94 1.72 GME 15.02 17.46 1.16 NAB 16.25 25.15 1.55 STI 16.63 20.54 1.24ATO 18.46 22.16 1.20 GON 17.86 23.55 1.32 NAT 16.89 19.00 1.12 *STO 17.95 21.86 1.22ATS 15.91 26.85 1.69 GOO — — — NES 18.08 18.08 1.00 STT 17.89 17.89 1.00ATT 16.86 25.43 1.51 HEU 17.18 20.67 1.20 NON 18.87 19.93 1.06 STW — — —ATV 18.81 25.06 1.33 IFR 16.84 19.43 1.15 NPO 16.02 19.00 1.19 –SVR 16.48 18.43 1.12AWO 18.15 22.25 1.23 IHW 18.59 20.31 1.09 NSI 18.98 22.18 1.17 SZR 17.45 24.08 1.38AWW 16.52 25.60 1.55 IMF 17.42 19.76 1.13 OBW 12.97 17.19 1.32 TER 17.02 20.77 1.22BCT 18.89 36.82 1.95 ISV — — — OFF 15.96 18.35 1.15 THO 16.37 18.10 1.11*BEA 15.27 16.36 1.07 ITE 15.78 21.08 1.34 OSI 17.86 25.44 1.42 TOL 16.73 24.36 1.46BEC 16.03 17.13 1.07 ITH 17.78 19.18 1.08 OSO — — — TON 18.08 24.29 1.34BIK 18.61 34.97 1.88 ITR — — — OWE 17.03 19.43 1.14 TSC 13.15 16.17 1.23BOF 17.69 26.06 1.47 ITW 17.49 19.59 1.12 �PAR — — — TUN 17.38 19.69 1.13BOG 15.46 18.24 1.18 IWR 15.52 18.35 1.18 PAU 15.81 17.64 1.12 UEI 18.18 21.13 1.16BPH 14.49 16.82 1.16 IWS 10.06 11.27 1.12 PHI 16.09 24.47 1.52 UFI 15.03 18.00 1.20BRE 17.97 20.65 1.15 IWV — — — PON 17.95 20.91 1.16 UOS 17.23 19.59 1.14BSV 18.68 19.86 1.06 IWW 16.49 18.39 1.11 PUN — — — UOZ 19.74 21.44 1.09CAN 16.72 24.67 1.47 JBW 18.52 25.66 1.39 RHO 14.30 24.26 1.70 USI 16.05 17.98 1.12CAS 18.70 26.42 1.41 JRY 18.70 22.94 1.23 �RON 19.22 20.56 1.07 UTL 16.54 16.69 1.01CDO 20.40 20.40 1.00 KFI 14.93 17.99 1.20 RRO — — — VET 22.00 22.00 1.00CFI 17.44 19.10 1.10 LAU 17.75 20.95 1.18 RSN 16.87 20.40 1.21 VFI 14.80 17.83 1.20CGF 18.81 20.85 1.11 LEV 15.90 18.32 1.15 RTE 17.05 19.56 1.15 VNI — — —CGS 16.49 20.31 1.23 LIO 16.75 22.63 1.35 RTH 16.05 18.73 1.17 VSV 16.96 20.90 1.23CHA 15.01 17.19 1.15 �LIT 16.71 26.20 1.56 RUT 17.70 21.94 1.24 WEI 16.07 21.73 1.35�CHI 19.90 24.42 1.23 LOS 16.84 27.15 1.61 RWR 19.34 23.57 1.22 �WEN 16.84 20.52 1.22�CLO 10.89 13.69 1.26 LOV 16.84 20.60 1.22 RWY 8.31 14.03 1.69 YUG 18.13 20.82 1.15CON 15.85 17.90 1.13 LTA 13.95 17.33 1.24 SAF 17.96 18.75 1.04 ZON 17.53 21.84 1.25CZP 21.95 26.42 1.20 LTF 16.86 18.24 1.08 SAO 14.20 16.18 1.14DAC 17.43 22.69 1.30 LTL 16.77 19.23 1.15 SAS 15.23 18.69 1.23
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of the window. The low-density boundaries tend to be
defined by the onset of T–O stretching. As noted earlier
by Sartbaeva et al.,5 real materials adopt densities close
to the minimum allowed by the window. This inflation was
explained in terms of coulomb repulsions between non-
codimeric oxygen atoms, that is, those oxygens beyond the
six immediate oxygen neighbours. This inflationary mechanism
had been postulated earlier by O’Keeffe for all oxide frame-
work materials.9
From these two-dimensional plots, the flexibility parameters
for both FAU and SOD are greatest when the unit cell is cubic.
The low density cubic structure for SOD is portrayed in
Fig. 3a. For SOD a second high density state can occur when
the cell has a tetragonal shape with c/a= 1.27 (Fig. 3b). When
all six unit cell dimensions are allowed to vary, higher density
states are accessible. A remarkable example is shown in
Fig. 3c, which depicts a cell in lowest symmetry, P1, with a
density of 29.67 T-atoms per nm3, a value that exceeds that of
quartz (27.71 T-atoms per nm3).
An alternative two-dimensional representation of the
flexibility of SOD is to examine the flexibility as a function
of one of the cell angles (Fig. 2c). As the relative density
increases towards 1.27, the cell angle is increasingly
constrained to lie close to 901. For r/r0 = 1, the range of cell
angles is widest, with a = 90 � 8.51.
Fig. 2 Two dimensional representation of flexibility windows for three topologically cubic zeolite framework types. The flexibility regions are
depicted by the yellow areas. The numbers along the boundary are the number of non-codimeric oxygen contacts that define the boundary. (a) r/r0versus c/a for FAU. (b) r/r0 versus c/a for SOD. (c) r/r0 versus cell angle a for SOD. Windows for different fixed c/a ratios are shown. Note the
discontinuities within the flexibility window for c/a = 1.05. The window region is continuous with narrow channels connecting the three main
areas. (d) c versus a for MTN. Constant density contours are shown. The real material lies at the intersection of the lines c = a and r = r0(black dot).
Fig. 3 Representations of the SOD framework as a pure SiO2
composition at different densities. (a) Minimum density in cubic space
group Pm�3m, 16.61 T-atoms per nm3. (b) Maximum density in cubic
space group I�43m (a subgroup of Pm�3m), 29.67 T-atoms per nm3. (c)
Maximum density in triclinic space group P1, 30.39 T-atoms per nm3.
A hard-sphere radius of 0.135 nm for oxygen has been assumed.
8536 | Phys. Chem. Chem. Phys., 2010, 12, 8531–8541 This journal is �c the Owner Societies 2010
Data for the MTN framework is presented in a slightly
different format (Fig. 2d). Here, we plot the c-axis length
against the a-axis length. Contours of constant density are
overlaid. In addition, a diagonal line corresponding to a= c is
indicated. This line represents those structures adopting a
cubic cell shape, but does not guarantee that the underlying
space group symmetry of the cell contents are cubic. As a rule,
the structural symmetry in this plot is everywhere triclinic, P1,
even though the topological symmetry is cubic Fd�3m. The
MTN framework is classified as a clathrate, comprising a high
density of 5-rings, with a small fraction of 6-rings (see Fig. 4).
The structure of ZSM-39, which is a pure silica material with
the MTN framework, is reported to have the full Fd�3m space
group symmetry. That structure is indicated by the dark spot
near the center of the plot.MTN is unusual among the zeolites
studied in that it adopts a structure that lies outside the
flexibility window. One possibility is that the pure-silica
ZSM-39 material is actually a mosaic of cubic cells, whose
internal arrangements are not strictly periodic. The topology
remains periodic of course, it is just the precise atom locations
that do not repeat periodically. In this model, a grain of
ZSM-39 is not strictly crystalline, but is instead one large
multiple unit cell. The loss of strict periodicity, reduces the
constraints on the system, potentially extending the flexibility
window. The lack of strict periodicity would be expected to
show as an increase in diffuse scattering and a lengthening of
the atomic displacement factors, but no such effects have been
reported.
Even in two-dimensional plots, the flexibility windows
reveal complex shapes. The boundary for MTN (Fig. 2d)
exhibits pronounced involuted forbidden regions. Not evident
in such plots is the fact that within the flexibility window,
neighbouring points may be topologically remote from each
other. To transform between neighbouring points, while
remaining within the flexibility window at all times, the frame-
work needs to unfold to a high symmetry state (almost always
the maximum cell volume) and then to initiate a new folding
mechanism back to the desired point.
Some zeolite networks are extraordinarily flexible.
Frameworks ABW and AST (see the Table) can be collapsed
to 60% of their maximum volume before non-codimeric
oxygen atoms start to overlap. The most flexible zeolites
we have found, BCT and BIK, have flexibility parameters
close to 2, which almost reaches the theoretical upper limit
for face-centered cubic packing for oxygen atoms, 35.93
T-atoms per nm3. In real materials, interatomic forces will
prevent such high densities being reached at atmospheric
pressure.
There is a weak correlation between flexibility parameter
and homogeneity of ring size. A ring is a closed path through
connected tetrahedra within the framework, and has the
property that there is no shorter path between any two
tetrahedra on the ring than the path on the ring itself. The
ring size is the number of tetrahedral atoms visited by the ring.
The average ring size in tetrahedral zeolitic frameworks is
B5.5.22 The most flexible frameworks tend to have rings that
are all near this value (5, 6 and 8; 7-rings are rare in zeolites).
This observation is perhaps counterintuitive, since isolated
large rings are generally more flexible than small rings. For
example, a 3-ring is isostatically rigid, whereas an 18-ring
‘necklace’ is flexible in isolation. It appears that if a zeolite
contains large size rings (such as 10, 12 and larger) then it also
tends to include rings of small sizes (3, 4 and 5 rings).23 As a
rule, it appears that the combination of large and small rings
renders frameworks less flexible. One notable exception to this
generalization is the RWY framework. It has 3-, 8- and
12-rings, yet also has a remarkably large flexibility parameter
of 1.69, equal to that for SOD. This equality may be no
coincidence. RWY can be generated from the SOD framework
by the isomorphic substitution of each T-atom site by a
(locally rigid) ‘super-tetrahedron’ built from a tetrahedral
arrangement of four T-atoms. Such a substitution expands
some of the ring sizes, while introducing 3-rings that are
entirely within the super-tetrahedra, yet maintains the under-
lying framework flexibility.
Twelve of the 194 zeolites failed to relax as pure silica
structures. Those frameworks are indicated in the Table by
the dashed entries, since the window boundaries are undefined.
Most of these frameworks become flexible when some of the Si
T-atoms are substituted by other elements, such as boron
(dTO E 0.148 nm), aluminium (dTO E 0.173 nm), germanium
(dTO E 0.176 nm) or phosphorous (dTO E 0.154 nm). We
briefly discuss examples of this below. Substituting tetrahedra
of different size into the framework introduces additional
degrees of freedom, whilst the number of length constraints
remains unchanged, allowing those frameworks a chance to
Fig. 4 Three variations of the dodecahedral cage that occurs in the MTN framework for a cubic shaped cell, c = a. The unit cell is outlined.
(a) Structure in standard cubic space group Fd�3m. (b) Minimum density flexible structure, with c= a. (c) Maximum density flexible structure, with c= a.
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become flexible. However, some structures, such as goosecreekite,
GOO, resolutely remain inflexible (see discussion below).
It is worth remarking at this juncture that in an earlier
study,2 during the course of optimising the structures of
hypothetical frameworks, it was found that a simple harmonic
potential acting on T–O distances, O–T–O and T–O–T angles
alone, would allow non-codimeric oxygen atoms to overlap.
This tended to happen in the more complex topologies, those
with more than three unique T-atoms per fundamental region.
It was necessary to introduce repulsion terms between the
non-codimeric oxygen atoms in order to remove the flexibility,
and to inflate the unit cells towards their maximum volume.2
A similar coulomb repulsion force was also invoked in a study
on zeolite flexibility to explain the preference for maximum
unit cell volume.5
In Fig. 5 we present maps of the T–O–T angle and its mean
square root deviation from the preferred value (here, taken to
be 1451) for the FAU and DFT frameworks. Contours of
constant angle are shown. In the case of FAU, the configuration
with the maximum volume reproduces the experimental
preferred angles. However, it is a different state of affairs
for the DFT framework. It adopts a state close to the
maximum volume, despite the fact that a denser state with
mean Si–O–Si angle of 1451 is available within the flexibility
window.24 As a rule, pure silica zeolite frameworks tend
to adopt a structure that maximally inflates the unit cell,
whilst retaining perfect tetrahedra. This is consistent with
the argument that non-codimeric oxygen atoms in these
frameworks tend to repel each other, thereby expanding the
framework.
D Mixed tetrahedral types
Most zeolite materials have compositions with more than one
type of tetrahedral atom. For example, natural zeolites tend to
be aluminosilicates. A significant family of aluminophosphates
have been synthesized, and some unusual frameworks
have been synthesized with cobalt, gallium and arsenic atoms
in the tetrahedral positions. In addition, we have found
that some frameworks will only exhibit flexibility when tetra-
hedra of different sizes are present at specific sites in the
framework.
The recently-reported IWS framework has seven crystallo-
graphically distinct T-atoms.25 Experimental data reports that
sites 2, 4 and 6 are occupied by Ge with probability 32%
(Fig. 6). (Here, we are following the site-ordering reported in
ref. 25.) The Ge–O bond is longer (B0.176 nm) than the Si–O
bond (B0.161 nm). We find that IWS cannot be relaxed as a
pure silica (or germania) composition. By exploring all
27 = 128 possible combinations of Ge and Si tetrahedra
on the seven sites we find that the framework can become
flexible only if a Ge atom resides on site 4. The flexibility
window is largest if there is Ge on both sites 2 and 4. This
is consistent with the experimental evidence, and implies that
the composition of the IWS type-material, ITQ-26, is optimized
for flexibility.
Goosecreekite presents an interesting exception to the
flexibility rule. It has an aluminosilicate composition, and is
found as a mineral.26 The framework, GOO, can not be
relaxed as a pure silicate, or as an aluminosilicate. Examination
of the reported structure26 reveals that all of the SiO4 and
AlO�4 tetrahedra are slightly distorted, with the Si3 site being
distorted the most. Si3 is in close proximity to a divalent Ca2+
cation, which balances the framework charges associated with
the neighbouring Al1 and Al2 framework atoms (Fig. 7). Most
aluminosilicate zeolites are charge balanced by monovalent
cations such as Na+ and K+. The GOO example indicates
that strong cationic charges may stabilize distortions in the
framework tetrahedra. Thus, divalent cations may be helpful
for stabilizing frameworks that are not too far from being
flexible.
The recently-approved framework MVY offers a similar
example. The charge-balancing K+ extra-framework cations
in the type material, ZSM-70, cannot be ion-exchanged.27
They adhere tenaciously to the framework. Possibly, this
strong adherence is a result of their important role in stabilizing
the framework.
Fig. 5 Maps of the mean T–O–T angle and its mean square root
deviation d from the preferred value, which here is assumed to be
1451). (a) FAU framework. The outer perimeter is the boundary of the
flexibility window when c/a is the only unit cell degree of freedom. The
contours within the window are the lines of constant mean T–O–T
angle. (b) Contours of constant standard deviation in T–O–T angle
relative to 1451. (c) DFT framework. Both the mean and standard
deviation contours are shown on the same plot. The experimental
structure, DAF-2, is indicated by the black dot.
8538 | Phys. Chem. Chem. Phys., 2010, 12, 8531–8541 This journal is �c the Owner Societies 2010
Zeolite beta has long been a zeolitic material of great
interest because of its three-dimensional intersecting 12-ring
channels. Further, one of the polymorphs, *BEA is chiral, with
left- and right-handed helical 12-ring channels.28–30 Despite
concerted efforts, neither of the end-member polymorphs has
yet been synthesized in topologically pure form. Zeolite beta
can be thought of as an intergrowth of two end-member
framework types, polymorph A (*BEA) and polymorph B.
Polymorph B does not yet have an official framework code
assigned, and so here we will refer to it by an unofficial
lowercase code, beb. Crystallographic studies of zeolite beta
materials, which examine the stacking sequences carefully,
suggest that there is a slight preference for the beb
polymorph.29 Our own studies of flexibility find that the
flexibility parameter for pure silica form of beb is FO = 1.13,
which is bigger than that for *BEA, FO = 1.07. This flexibility
only emerges when we lower the space group symmetry from
(chiral) tetragonal P4122, to triclinic P1 (the chiral topology,
however, is still present in the lower-symmetry representation).
There are nine crystallographically unique T-sites in each
polymorph. We find that adding a single aluminium atom to
the *BEA framework makes the flexibility window disappear,
whereas for beb, five of the nine sites allow, or extend, the
flexibility. When substituting two aluminium atoms, relaxed
configurations were found in 2 out of 32 cases for *BEA and
12 out of 32 cases for beb. It appears that the beb framework
type is slightly more flexible than the *BEA framework type.
The detailed role of Al T-atoms on the flexibility of the zeolite
beta end-member frameworks is still not fully surveyed. It is
clear that for materials that are about 10% Al composition,
the flexibility argument suggests that the beb framework is the
preferred configuration, whereas *BEA will be more flexible as
a highly siliceous material. One question yet to be resolved
concerns the flexibility of the disordered intergrowth structure.
So far, we have examined only the pure end-member
polymorphs. This will entail examination of large structures
with extended cells along the c-axis to accommodate random
intergrowths.
Interrupted frameworks. The interrupted frameworks, those
with a ‘‘–’’ dash in front of the three-letter framework type
code, are frameworks containing tetrahedra with at least one
vertex that is terminated by a hydrogen atom and not
connected to another tetrahedron. The oxygen atom at the
terminated vertex is unconstrained by any neighbouring
tetrahedron. Fully-tetrahedral (tectosilicate) zeolites are
everywhere locally isostatic, that is the number of coordinate
degrees of freedom is equal to the number of constraints.
Interrupted frameworks (inosilicates) are locally hypostatic at
the interrupted tetrahedra, that is the number of coordinate
degrees of freedom is greater than the number of constraints.
Intuitively, one would expect interrupted frameworks to
exhibit increased flexibility. Surprisingly, the Table shows that
the seven interrupted frameworks have a spread of flexibility
parameters similar to those for the regular frameworks. At the
high end is -LIT with a flexibility parameter of 1.56, whereas
-PAR does not show flexibility in the pure silica form.
Thermodynamics of flexible frameworks. Since the presence
of a flexibility window indicates that the framework can be
formed by perfect regular tetrahedra, then the framework
energy of formation will also be low. This follows because
most zeolite structures are known to comprise near-perfect
tetrahedra. Tetrahedral distortions will cost energy. We would
expect a siliceous framework that possesses a flexibility
window to also have a low framework energy. As we have
found, a low framework energy does not necessarily guarantee
the presence of a flexibility window. Goosecreekite (GOO) is
such an example.
The large databases of hypothetical zeolites that are now
emerging tend to be sorted according to framework energy.1–3
However, a low framework energy does not guarantee that the
framework is synthetically viable. The presence of a flexibility
Fig. 6 Unit cell of the germanosilicate framework of IWS. The pure
SiO2 composition cannot be formed without straining T sites T2 and
T4. The material ITQ-26 has germanium atoms at these sites. The
larger GeO4 tetrahedra at these particular sites restore flexibility to the
IWS framework.
Fig. 7 Fragment of the Goosecreekite structure. This aluminosilicate
zeolite framework is unusual in that it exhibits no flexibility window.
The framework can not be formed from perfect silicon and aluminium
tetrahedra, without forcing the oxygen atoms to overlap (the oxygen
atoms have a nominal radius of 0.135 nm, and are not shown to scale
here). It is likely that the divalent calcium cations and trivalent Al
cations in the vicinity of each Si3 tetrahedron serve to stabilize the
tetrahedral distortions inevitably present in the GOO framework.
This journal is �c the Owner Societies 2010 Phys. Chem. Chem. Phys., 2010, 12, 8531–8541 | 8539
window for a given zeolite topology indicates that there is a
potentially large configuration space that corresponds to
the same zeolite topology. It is known that vibrational
entropy, which explores local configurations, is an important
consideration in zeolite formation.31
However, in our model of an ideal zeolite framework that is
in a configuration inside its flexibility window, the framework
energy is zero. Apart from the bond length constraints, there
are no forces acting on an ideal zeolite framework that is in a
flexible state. There are no forces guiding it towards any
preferred configuration. The energy landscape within the
window is a flat null space. The absence of harmonic forces
means that if the framework is given kinetic energy, the null
eigenvectors do not oscillate about a mean position, but can
grow in amplitude as the motion progresses. Vibrational
modes will still be associated with the stiff springs within the
tetrahedra, but we are assuming that the springs are too rigid
to activate such modes. The growing modes are related to the
flexible folding modes. At a non-zero temperature, there will
be a kinetic energy per unit cell associated with the framework.
The resulting motion allows the system to explore the available
states within the window. This is analogous to an ideal
gas, except that a zeolite framework has significantly more
constraints on the atomic motions.
At maximum volume, the unit cell is almost always in its
maximum symmetry state. There is only one known exception
to this rule in crystallography,32 and no exceptions have
yet been found for zeolite frameworks. At this maximum
symmetry state, many of the distance constraints become
degenerate and there is an increase in the number of null
eigenvectors in this maximum symmetry state. Once one of the
null eigenvectors is excited at large amplitude, symmetry is
lost, and most of the null states vanish. For example, the FAU
framework has 39 null eigenvectors at maximum volume
where it has Fd�3m symmetry. The number of null modes
drops significantly once any one of these states is excited with
a large amplitude.
IV. Conclusions
We have examined the flexibility of ideal zeolite frameworks,
by treating them as idealized assemblies of perfect, rigid,
tetrahedra that are interconnected at their corners via
force-free spherical joints. The flexibility of 194 known zeolite
frameworks has been analyzed, and we find that 182 of them
are flexible when modeled as a pure SiO2 composition. Most of
the 12 exceptions become flexible once a mixed-tetrahedron
composition is allowed. This establishes an important new
characteristic of zeolite frameworks that further unifies zeolites
as an important and unusual group of similar framework
structures.
Despite the fact that the most of the configurations inside
the flexibility window are not accessible in real zeolites due to
their high energy, zeolites with larger flexibility windows have
a much higher possibility of appearing during synthesis, since
most currently known zeolites (SOD, FAU etc) do indeed have
large flexibility windows.
Singular value decomposition of the kinematic matrices is
emerging as a powerful tool for exploring the flexibility
window. Reliable eigenvalue-following algorithms are needed
in order to navigate efficiently through the bifurcation points
encountered while following folding mechanisms within the
flexibility window.
We emphasize that the ideal zeolite frameworks studied here
are idealized geometric representations of the real structures.
Even in quartz, the SiO4 tetrahedra deviate slightly from
perfection. The quartz framework is isohedral, possessing
one topologically unique intertetrahedral angle (T–O–T) that
adopts an angle around 1441 at room temperature. The
maximum volume state for quartz would adopt an angle of
154.61, which elevates the framework energy. Most zeolite
frameworks are not isohedral, possessing several topologically
nonequivalent T–O–T angles, not all of which can settle at the
preferred angle of B1441. Such geometrically conflicted
structures can express a wide range of T–O–T angles, along
with small distortions of the tetrahedra. These deviations are
caused primarily by electronic interactions. In addition, alumino-
silicate compositions, in addition to introducing tetrahedra of
different sizes, also introduce framework charge that is
balanced by extra-framework cations, such as H+, Na+ and
K+. These additional electrostatic terms introduce opportunities
for additional framework distortions. Nevertheless, the structural
differences between ideal zeolite frameworks and the real
zeolite materials are remarkably small.
The near-universal flexibility of ideal zeolite frameworks is
emerging as an important topological property of their
structures. Flexibility provides us with a useful selection
criterion for evaluating whether or not a hypothetical zeolite
framework is realisable. A desirable extension of this work is
the development of a method to infer flexibility from the graph
of the framework.
This is a relatively unexplored area that is highly relevant to
all oxide framework materials, and promises to be an exciting
area of research.
Acknowledgements
The authors are grateful to Asel Sartbaeva (University of
Oxford), Stephen Wells (University of Warwick) and Simon
Guest (University of Cambridge) for stimulating discussions.
We also acknowledge the financial support of the National
Science Foundation grants Nos. DMR-0703973 and DMS-
0714953, and the Donors of the American Chemical Society
Petroleum Research Fund for partial support of this research.
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