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QuantilVing the interstitial structure 01
non-crvstalline solids
Lilian P. Davila, Subhash H. Risbud and James F. Shackelford
Department of Chemical Engineering and Materials Science, University of California, Davis,
California 95616, USA
Abstract
The nature of interstitial geometry plays an important role in understanding the
medium-range structure ofnon-crystalline solids and related properties, e.g., gas trans­
port in silicate glasses. Our recent focus on quantifying this geometry (by analyzing
interstices in crystalline analogs) is reviewed. For example, the atomic structures of
crystalline silicas (analogs for vitreous silica) have been described as a packing offilled
oxygen polyhedra (Si04 tetrahedra) and empty ones (interstices ofvarious shapes) using
a graphics workstation. Computer simulation software permits the interstices to be
described quantitatively. For both the simplest crystalline polymorph (high cristobalite)
and the most common one (low quartz), the sum ofthe silica tetrahedra volumes plus the
CorrespondencefReprint request: James F. Shackelford, Department of Chemical Engineering and Materials Science, University
ofCalifomia, Davis, California 95616, USA. Fax: (530) 752-8058, E-mail: jft;hckel [email protected]
74
Lilian P. Davila et al.
volume ofassociated interstices are equal to the unit cell volume. The interstice in high
cristobalite (truncated tetrahedron) is relatively open, and the relatively tight interstices
in low quartz (tetrahedra, square pyramids, and triangular prisms) are ofthe same type,
with different distortions, found in calcium silicate (wollastonite, a crystalline analog for
CaSi03 glass).
This approach provides a useful perspective on the mimy forms (both crystalline and
non-crystalline) ofsilica, including the relationship ofthe interstices to the double helix
in low quartz. Furthermore, such characterization is discussed in terms ofthe relevance
to many technologically important applications ofglasses as well as to the next phase of
the research in which interstices in various computer-simulated glasses will be
systematically cataloged.
1. Introduction
The nature of interstitial structure is an important component of the description of
materials. Our approach derives from the utility of Bernal's canonical hole model of
liquids [1] in which interstitial voids are defined by connecting the centers of adjacent
atoms and leads to useful models of amorphous metals [2] and metallic grain boundaries
[3, 4]. (An advantage of characterizing structure as a stacking of polyhedra is that the
technique does not "break down" as one goes from crystalline to defect to completely
non-crystalline structures.)
Similarly, the interstitial structure of cristobalite helped to model the nature of gas
transport in vitreous silica [5]. Such studies have had, more recently, significant bearing
on understanding the behavior of molecular gases in molten silicates, a fundamental
aspect of volcanic eruptions [6] and the formation of the earth's atmosphere [7].
Oxygen polyhedra played a central role in the demonstration of medium-range
ordering in glasses by Gaskell, et al. [8]. Their neutron diffraction experiments showed
the presence of edge-shared Ca06 octahedra, comparable to the structure of the mineral
wollastonite. The interstitial structure of wollastonite was identified as part of a general
summary of the canonical hole set for non-metallic solids [9]. The interstices in
wollastonite were found to be distorted tetrahedra, square pyramids, and triangular
prisms.
Liebau [10] has pointed out that, in the first half of the 20th century, the structure of
silicate minerals were well characterized as a linkage of silica tetrahedra. During the
1970s and 1980s, a wide range of silicates were shown to have the silica tetrahedra
intricately linked to non-silicon metal-oxide polyhedra, often assembled as planar sheets
of edge-shared octahedra (e.g., wollastonite). In the course of studying crystalline
analogs of silicate glasses, we complete this trend by identifying the shapes of the
unfilled polyhedra (interstices) as well as the filled ones (e.g., Si04 and Ca06)'
Based on the set of constructable, convex polyhedra identified by Zalgaller [11], the
canonical hole set for non-metallic solids [9], such as silicates, consists of 44 simple
polyhedra (or 126 polyhedra if one includes the possibility of "compound holes"). This
is a much larger number than the eight simple, Bernal holes for metals [3] and is a
manifestation of the different· bonding (covalent versus metallic). The current study
builds on the previous identification of the canonical hole set [9] by using commercial
simulation software which permits the specific geometry of individual interstices to be
quantified.
Quantifying interstitial structure
2. Methods
Images of silica polymorphs and
Insight II. a graphic molecular mOt
studying biomolecular structures and
inorganic solids, focusing on stmctu
systems. Its capabilities have expll
crystalline microporous materials, mel
clusters supported on crystalline and IT
runs in the UNIX operating enviroDl
Graphics workstation. Insight Il.sofu
(MSI), formerly known as BIOSYM lot
Insight II 3.0.0 allows the user to a
crystalline silicates) from its library 81
Used in conjunction with Catalysis. am
be constructed. Catalysis 3.0.0 is a set:
and properties of catalysts, sorbents, am
Insight II and Catalysis allow
interatomic bonding potentials. Distino
visualize models of solid structures..
modules are mainly used in this interstii
silica.
The Solids_Builder module alloW!:
crystal structures. The module uses stu
libraries, or direct input to construct Cl)ll
surfaces). One must specify an asymnu
the space group symmetry operators in CI
The editing and display of tools j
Solids_Adjustment module. Among tin
groups of atoms or fragments, displa)l
central atom or about vertice atoms, andl
Catalysis 3.0.0 permits the cons;
equivalent sites in a structure. A semi-IT
was required, however, to ensure the
coordinates of the comers (oxygen ion II
and analyzed in terms of edge lengths ant
Interstitial polyhedral volume calc:
analysis. Because of the non-regularity
relatively straightforward determinatiOl
calculating an interstitial polyhedron von
By converting adjacent atomic coordinatl
edges, the volume of each tetrahedral voU
VIet =
1/61a • (b x c)1
To confirm the accuracy ofthe described!
relatively simple interstitial polyhedrom
were compared with published data and 1!l
Quantifying interstitial structure
75
2. Methods
Images of silica polymorphs and associated interstitial structure were created using
Insight II. a graphic molecular modeling software package initially developed for
studying biomolecular structures and later for the examination of atomic structures of
inorganic solids, focusing on structure-sensitive properties of catalyst and sorption
systems. Its capabilities have expanded into investigations of structures such as
crystalline microporous materials, metal and metal oxide surfaces, and metal atoms or
clusters supported on crystalline and non-crystalline inorganic matrices [12]. Insight II
runs in the UNIX operating environment and, in our case, on an Indigo 2 Silicon
Graphics workstation. Insight IIsoftware is a product of Molecular Simulations Inc.
(MSI), formerly known as BIOSYM Inc.
Insight II 3. O. 0 allows the user to call up various structures (e.g. crystalline and non­
crystalline silicates) from its library and manipulate structures using display graphics.
Used in conjunction with Catalysis. an MSI application program, molecular models can
be constructed. Catalysis 3. o. 0 is a set of programs developed for studying the structures
and properties of catalysts, sorbents, and inorganic solids [13].
Insight II and Catalysis allow precise modeling due to the use of realistic
interatomic bonding potentials. Distinct modules are used to construct, manipulate, and
visualize models of solid structures. The Solids_Builder and Solids_Adjustment
modules are mainly used in this interstitial study of crystalline silica analogs for vitreous
silica.
The Solids_Builder module allows one to import, manipulate, edit, and visualize
crystal structures. The module uses structural data from system libraries, user-supplied
libraries, or direct input to construct crystal, metal, or glass structures (solids, sheets, and
surfaces). One must specify an asymmetric unit of parent atoms, a unit cell lattice, and
the space group symmetry operators in order to construct a periodic structure.
The editing and display of tools for crystal structure models is provided via the
Solids_Adjustment module. Among the tools available are moving or deleting atoms,
groups of atoms or fragments, displaying Miller planes, creating polyhedra about a
central atom or about vertice atoms, and altering symmetry or connectivity.
Catalysis 3.0.0 permits the construction of a given n-ordered polyhedron at
equivalent sites in a structure. A semi-manual building process for interstitial polyhedra
was required, however, to ensure the generation of convex polyhedra. The atomic
coordinates of the comers (oxygen ion positions) of void polyhedra were saved to a file
and analyzed in terms of edge lengths and angles.
Interstitial polyhedral volume calculations were performed using simple vector
analysis. Because of the non-regularity of the polyhedra, vector calculations provided a
The process involved
relatively straightforward determination of their volume.
calculating an interstitial polyhedron volume as the sum of tetrahedral volume elements.
By converting adjacent atomic coordinates into vectors (a, b, e) representing polyhedral
edges, the volume of each tetrahedral volume element was calculated as
VIet
=
1/61a • (b x e)1
[1]
To confirm the accuracy of the described method, the volumes ofSi04 tetrahedra and the
relatively simple interstitial polyhedron (a truncated tetrahedron) of high-cristobalite
were compared with published data and geometrical equations.
Lilian P. Davila et al.
76
3. Results
Figure I shows an interstitial polyhedron (a truncated tetrahedron) in high
cristobalite, the simplest of the silica polymorphs, along with four of the adjacent Si04
tetrahedra. In this structure that is a common analog for vitreous silica, there are eight
interstitial polyhedra along with eight Si04 tetrahedra in the cristobalite unit cell [5].
Table I shows the correspondence between the sum of volumes of the eight filled (Si0 4 )
plus eight unfil1ed (interstice) oxygen polyhedra and the unit cell volume.
Quantifying interstitial S tl1lcture
Although cristobahte is at
ous silica [5J, the range of ring
smaller mterstices will also e~
density polymorph low quartz c
poly- hedra. The linkage of Sil
than in high cristobalite. This
c-axis [14 J, although less celf
volume. as viewed down the
c~annel defined by a siX-ring ..
FIgure 2). The six-ring loop
tetrahedra, and each three-rin o
helices. Table 2 shows the corr~
plus the 18 various shaped ir,
complexity of the polyhedral sh
Table 2. The interstitial space i
distorted triangular prism. The i
three smaller, distorted triangula
Computer simulated image of the interstitial oxygen polyhedron (a truncated
tetrahedron shown in yellow) in the cenler of the high cristobalite unit cell cube (white lines).
along with four adjacent silica tetrahedra (shown in orange). The red and yellow spheres
represent oxygen and silicon ions, respectively.
Figure 1.
Table 1. Cristobalite as a packing of oxygen polyhedra
Polyhedron
SiO. tetrahedron
Number
x
Volume/polyhedron [nm)] = Volume [nm 3 ]
8
0.001912
0.01530
8
0.04397
0.3518
Truncated
tetrahedron
0.3671
Note: Unit cell volume
=
(0.716 nm)3
=
0.3671 nm 3
F"Igure 2. The unit cell of low quIz vic
tetrahedra (in orange) and some of the ,
In Table 2 (shown in blue along the c
wllhm the channel defined by a 6-ring I
------
Quanlifying interstitial structure
77
Although cristobalite is arguably the most appropriate crystalline analog for vitre­
ous silica [5], the range of ring sizes in the non-crystalline material requires that some
smaller interstices will also exist.
Those interstices in the common and higher­
density polymorph low quartz can provide some indication of the nature of these smaller
poly- hedra. The linkage of Si0 4 tetrahedra in low quartz is substantially more complex
than in high cristobalite. This linkage is, in fact, a double helix when viewed along the
c-axis [14], although less celebrated than the double helix of DNA. The unit cell
volume, as viewed down the c-axis, is equivalent to three Si04 tetrahedra plus the
channel defined by a six-ring "loop" and two channels defined by three-ring loops (see
Figure 2). The six-ring loop is the c-axis projection of the double helix of Si0 4
tetrahedra, and each three-ring loop represents the overlap of two adjacent double
helices. Table 2 shows the correspondence between the sum of the three silica tetrahedra
plus the 18 various shaped interstices and the unit cell volume. Because of the
complexity of the polyhedral shapes, some scatter is seen in the volume calculations of
Table 2. The interstitial space in each three-ring loop of Figure 2 is represented by a
distorted triangular prism. The interstitial space in the six-ring loop is decomposed into
three smaller, distorted triangular prisms and one larger, central one. The double helix
Figure 2. The unit cell of low qutz viewed down the c.axis is equivalent to a cluster of 3 silica
tetrahedra (in orange) and some of the associated interstices (distorted triangular prisms) described
in Table 2 (shown in blue along the channel defined by a 3-ring loop and in yellow and purple
within the channel defined by a 6-ring loop)
-Lilian P. Davila e( al.
78
Quantifying interstitial structure
Table 2. Quartz as a packing of oxygen polyhedra
Si-filled
An alternate illustration II
5. A~ offset view of the dOll
~lJed mterstitial structure of III
In the channel defined by the sii
Interstice (Location)
Tetrahedron I
0.002134
Tetrahedron 2
0.002125
Tetrahedron 3
0.002134
Tetrahedron (Beneath Si-tetrahedron 1)
0.001692
Tetrahedron (Beneath Si-tetrahedron 1)
0.002084
Sq. pyramid (Beneath Si-tetrahedron 1)
0.004160
Sq. pyramid (Beneath Si-tetrahedron 1)
0.007669
Tetrahedron (Beneath Si-tetrahedron 2)
0.001691
Tetrahedron (Beneath Si-tetrahedron 2)
0.002108
Sq. pyramid (Beneath Si-tetrahedron 2)
0.004215
Sq. pyramid (Beneath Si-tetrahedron 2)
0.007655
Tetrahedron (Beneath Si-tetrahedron 3)
0.001694
Tetrahedron (Beneath Si-tetrahedron 3)
0.002078
Sq. pyramid (Beneath Si-tetrahedron 3)
0.004167
Sq. pyramid (Beneath Si-tetrahedron 3)
0.007666
Triangular prism (3-ring spiral)
0.008783
Triangular prism (3-ring spiral)
0.008778
Triangular prism (6-ring spiral)
0.006566
Triangular prism (6-ring spiral)
0.006583
Triangular prism (6-ring spiral)
0.006691
Triangular prism (6-ring spiral)
0.02235
0.006393
0.01561
0.01567
Figure 3. Doubled helix fonned bJ
connected by oxygen ions shown al
tetrahedra connected by oxygen ions sl.
0.01561
0.01756
0.04219
0.1130
2
Note: Unit cell volume = -./3/2 a c = 0.866 (0.4913 run/ (0.54052 nm) = 0.1130 nm l
fonned by Si04 tetrahedra is clearly seen with the c-axis in the plane of the page (see
Figure 3). The interstitial space between adjacent Si04 tetrahedra along the c-axis is
composed of a packing of two distorted tetrahedra and two distorted square pyramids
(see Figure 4).
F.igure 4. With the c-axis in the plane (
dIstorted tetrahedra and two distorted s u
slhca tetrahedra along the quartz double ~e
Quantifying interstitial structure
79
An alternate illustration of the interstitial geometry of quartz is provided in Figures
5. An offset view of the double helix looking down the c-axis shows the completely
filled interstitial structure of low quartz, with the addition of distorted triangular prisms
in the channel defined by the six-ring loop.
Figure 3. Doubled helix fonned by SiO. tetrahedra (one helix shown as purple tetrahedra
connected by oxygen ions shown as large red spheres with the other helix shown as orange
tetrahedra connected by oxygen ions shown as small red spheres).
Figure 4. With the c-axis in the plane of the page, one can seen how a set of interstices (two
distorted tetrahedra and two distorted square pyramids) pack into the space between two adjecellt
silica tetrahedra along the quartz double helix.
Lilian P. Davila et al.
80
Quantifying interstitial structure
Table 3. Polyhedra se
Metals·
8 polyhedra
b
Nonmetals 44 "simple"
28 sin
8 priSJ
8 antiJ
. or
126 total (inc
5 Plato
13 Arc.
8 prisrr.
8 antiPJ
92 "lal
"Ref. [3], 'Ref. [9], "Ref. [1
Figure 5. Offset view of the double helix in low quartz )'ooking down the c-axis with the interstitial
sttucture completely filled in. The interstices sandwiched between the helices (see Fig. 4) are no
longer visible in this view.
4. Discussion
(a)
It is interesting to note that the interstices found in low quartz are of the same type
found in wollastonite [9], viz. distorted tetrahedra, square pyramids, and triangular
prisms. Of course, the exact nature of the distortion is somewhat different in the two
cases. Quartz, like wollastonite, is a relatively tight structure with interstitial space
represented by a limited number of relatively small oxygen polyhedra. These relatively
small interstices are part of the complete set of polyhedra for non-metallic solids given
in Table 3. The 44 "simple" polyhedra are represented by the prisms and antiprisms
of Figure 6 and the "Zalgaller solids" of Figure 7 [15 J. The triangular prism is
comparable to Figure 6(a) but with the basal faces being triangles. The tetrahedron and
square pyramid are the first two polyhedra in Figure 7 (M 1 and M 2 in Zalgaller's
notation). Given that cristobalite and vitreous silica have similar densities, the truncated
tetrahedron shown in Figure I can be considered an average-size interstice in vitreous
silica and is polyhedron MlQin Figure 7.
It is useful to compare the sizes of the interstices found in the crystalline analogs of
vitreous silica with the distribution of interstitial solubility site sizes as determined by the
analysis of gas transport in vitreous silica and shown in Figur,l: 8 [16J. Assuming an
oxygen radius of 0.067 - 0.089 nm corresponding to a 50 - 75% covalent nature of the Si­
o bond f 17], the inscribed sphere diameters for regular polyhedra and the "doorways"
into those polyhedra are given in Table 4. As Figure 8 is based on gas transport
Figure 6.. Representative (a) prisn
pnsm, With basal faces compose
bcred nngs with 3 S /I...s 10 exp
crystallllle solids and /I =7 being il
[9].
are widely used as catalysts and
zeolrte stnlctures is the sodalj
poJ~hedron M I6 in Figure 7. S
v~nolls gases. [15]. Computer-go
~It,reous .calclum silicate [19].
systematically cataloging all int
Flleston and Garofalini [18].
stllt~y was done by a semi-manu'
of Interstitial sizes and shape~
a~t?mation of the process. Figu
diffUSIOnal paths in these relativl
t~lree adjacent interstices in higH
rings (hexagons) serve as doorwa
QuantifYing interstitial structure
81
Table 3. Polyhedra sets for interstices in metallic and nonmetallic glasses [15]
8 polyhedra (with up to 20 triangular faces)
Metals·
b
Nonmetals 44 "simple" polyhedra.
28 simple, convex regular polyhedra
8 prisms
8 antiprisms
. or
126 total (including "compound") polyhedra
5 Platonic solids
13 Archimedcan solids
8 prisms
8 antiprisms
92 "Zalgallerc solids"
"Ref. [3], "Ref. [9], ~ef. [11]'
experiments, the doorway sizes in
Table 4 are the more appropriate
comparison as the "sizes" of interstices
determined by gas probe atoms are
limited by the access of those atoms.
One sees that the values of the doorway
sizes in Table 4 are in good agreement
with
the range of interstitial sizes
(a)
(b)
given in Figure 8.
The most obvious applications
Figure 6. Representative (a) prism and (b) anti­
of this technique of quantifying
prism, with basal faces composed of n-mem­
interstitial structure would be for
bered rings with 3.::; n"::; I0 expected in non­
modeling
diffusional processes III
crystalline solids and n =7 being illustrated here
relatively open structures such as
[9].
zeolites and silicate glasses. Zeolites
are widely used as catalysts and molecular sieves. Among the best-known images of the
zeolite structures is the sodalite cage, a truncated octahedron [10] and shown as
polyhedron M l6 in Figure 7. Silicate glasses are well known for their permeability to
various gases [15]. Computer-generated models are available for vitreous silica [18] and
vitreous calcium silicate [19]. The next phase of the current research will focus on
systematically cataloging all interstices in a vitreous silica model equivalent to that of
Fueston and Garofalini [18]. The characterization of interstitial space in the current
study was done by a semi-manual technique, but the cataloguing of the expected range
of interstitial sizes and shapes in non-crystalline materials will be aided by the
automation of the process. Figure 9 illustrates the utility of this approach for monitoring
diffusional paths in these relatively open network silicates. The spatial relationship of
three adjacent interstices in high cristobalite (see Figure 1) is shown. Six-membered
rings (hexagons) serve as doorways between adjacent truncated tetrahedra (see Table 4).
82
Lilian P. Davila et al.
./1\.
~
~ ~
QuantifYing interstitial structure
~~.
<:L3Y~
Distributiol
Density
{Arbitrary Un
(Ms>
!M6>
Figure 8. Distribution of inl
probability distribution function
(Mil)
Table 4. Size of interstices an.
Interstice
Ii
Tetrahedron
o
Square Pyramid
0,
Triangular prism
o.
Truncated tetrahedron
o.
'Using an oxygen radius of 0.1
Si-O bond [17J.
(M17)
(Mu)
-$
,
,
\
)to
I
-..0(
I
I
(M2S)
Figure 7. The 28 simple, convex regular polyhedra, after Zalgaller [11].
Figure 9. A diffusional path •
The truncated tetrahedron in the cer
83
Quantifying interstitial slnzclure
mode = 0181 nm
Distribution
Density
(Arbitrary Units)
d He = 0.256 nm
nm
o
0.1
0.2
0.3
0.4
0.5
Interstitial Diameter (nm)
Figure 8. Distribution of interstitial solubility site sizes in vitreous silica as a log-nonnal
probability distribution function. detennined by the analysis of gas transport data [16)
Table 4. Size of interstices and their doorways (as inscribed spheres') for quartz and cristobalite
Interstice
Interstice dia. [run]
Doorway
Doorway dia. [run]
Tetrahedron
0.147 - 0.190
Triangle
0.128 - 0.172
Square pyramid
0.197 - 0.241
Square
0.197 - 0.241
Triangular prism
0.227 ­ 0.271
Square
0.197 - 0.241
Truncated tetrahedron
0.442 ­ 0.486
Hexagon
0.275 - 0.319
'Using an oxygen radius of 0.067 - 0.089 nm corresponding to 50 - 75% covalent nature of the
Si-O bond [ I7].
Figure 9. A diffusional path in high crystobalit illustrated by three, adjacent interstices.
The truncated tetrahedron in the center of the unit cell is equivalent to that in figure I.
84
Lilian P. Davila et at.
Acknowledgments
We thank T.E. Allis of the University of California, Davis and J.M. Newsam, N.
Khosrovani, and D. Khumayyis of Molecular Simulations, Inc. for experimental help.
Professor Stephen Garofalini of Rutgers University has provided numerous useful
discussions.
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i
i
i
J