1.=:====.1 37/661 (2), Fort P.O., Trivandrum , Kerala, India

Transcription

1 ~ _ Transworld Research Network 1.=:====.1 37/661 (2), Fort P.O., Trivandrum , Kerala, India lecnt1n...llell-cl1iiiilllne SIIIIIs, 1[20011: S11:...18IIHI28-8 QuantilVing the interstitial structure 01 non-crvstalline solids Lilian P. Davila, Subhash H. Risbud and James F. Shackelford Department ofchemical Engineering and Materials Science, University ofcalifornia, 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 transport 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 (Si0 4 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) , jft;hckel ford@ucdavis.edu

2 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 analogfor CaSi0 3 glass). This approach provides a useful perspective on the mimyforms (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 ofthe 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., Si0 4 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.

3 Quantifying interstitial structure 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 noncrystalline 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 setofprograms 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 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 ofedge 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 relatively straightforward determination of their volume. The process involved 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 ofthe described method, the volumes ofsi0 4 tetrahedra and the relatively simple interstitial polyhedron (a truncated tetrahedron) of high-cristobalite were compared with published data and geometrical equations. 75

4 76 Lilian P. Davila et al. 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 Si0 4 tetrahedra. In this structure that is a common analog for vitreous silica, there are eight interstitial polyhedra along with eight Si0 4 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. Figure 1. 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. Table 1. Cristobalite as a packing of oxygen polyhedra Polyhedron Number x Volume/polyhedron [nm)] = Volume [nm 3 ] SiO. tetrahedron Truncated tetrahedron Note: Unit cell volume = (0.716 nm)3 = nm 3

5 Quanlifying interstitial structure 77 Although cristobalite is arguably the most appropriate crystalline analog for vitreous 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 higherdensity 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 Si0 4 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)

6 78 Lilian P. Davila e( al. Table 2. Quartz as a packing of oxygen polyhedra Si-filled Interstice (Location) Tetrahedron I Tetrahedron 2 Tetrahedron 3 Tetrahedron (Beneath Si-tetrahedron 1) Tetrahedron (Beneath Si-tetrahedron 1) Sq. pyramid (Beneath Si-tetrahedron 1) Sq. pyramid (Beneath Si-tetrahedron 1) Tetrahedron (Beneath Si-tetrahedron 2) Tetrahedron (Beneath Si-tetrahedron 2) Sq. pyramid (Beneath Si-tetrahedron 2) Sq. pyramid (Beneath Si-tetrahedron 2) Tetrahedron (Beneath Si-tetrahedron 3) Tetrahedron (Beneath Si-tetrahedron 3) Sq. pyramid (Beneath Si-tetrahedron 3) Sq. pyramid (Beneath Si-tetrahedron 3) Triangular prism (3-ring spiral) Triangular prism (3-ring spiral) Triangular prism (6-ring spiral) Triangular prism (6-ring spiral) Triangular prism (6-ring spiral) Triangular prism (6-ring spiral) l Note: Unit cell volume = -./3/2 a c = ( run/ ( nm) = nm fonned by Si0 4 tetrahedra is clearly seen with the c-axis in the plane of the page (see Figure 3). The interstitial space between adjacent Si0 4 tetrahedra along the c-axis is composed of a packing of two distorted tetrahedra and two distorted square pyramids (see Figure 4).

7 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.

8 80 Lilian P. Davila et al. 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 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 nm corresponding to a 50-75% covalent nature of the Sio bond f17], 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

9 QuantifYing interstitial structure 81 Table 3. Polyhedra sets for interstices in metallic and nonmetallic glasses [15] Metals 8 polyhedra (with up to 20 triangular faces) Nonmetals b 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 "Zalgaller c 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 (a) (b) with the range of interstitial sizes given in Figure 8. The most obvious applications Figure 6. Representative (a) prism and (b) antiof this technique of quantifying prism, with basal faces composed of n-meminterstitial structure would be for bered rings with 3.::; n"::; I0 expected in noncrystalline solids and n =7 being illustrated here modeling diffusional processes III [9]. relatively open structures such as 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).

10 82 Lilian P. Davila et al../1\. ~ ~~. ~ ~ <:L3Y~ (Ms>!M6> (Mil) (M17) (Mu) --, \ I $ )to -..0(, I I (M2S) Figure 7. The 28 simple, convex regular polyhedra, after Zalgaller [11].

11 Quantifying interstitial slnzclure 83 mode = 0181 nm Distribution Density (Arbitrary Units) d He = nm nm o 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 Triangle Square pyramid Square Triangular prism Square Truncated tetrahedron Hexagon 'Using an oxygen radius of 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.

12 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. References I. Bernal, ld., 1964, Proc. R. Soc. (London), 280, Finney, J.L., and Wallace, J., 1981, J. Non-crystalline Solids, 43, Ashby, M.F., Spaepen, F., and Williams, D.; 1978, Acta Metallurgica, 26, Fitzsimmons, M.R., and Sass, S.L., 1989, Acta Metallurgica, 37, Shackelford, J.F., and Masaryk, J.S., 1978, J. Non-crystalline Solids, 30, Carroll, M.R., and Stolper, E.M., 1993, Geochimica et Cosmochimica Acta, 57, Chamorro-Perez, E., Gillet, P., Jambon, A., Badro, J., and McMillan, P., 1998, Nature, 393, Gaskell, P.H., Eckersley, M.e., Barnes, A.C., and Chieux, P., 1991, Nature, 350, Shackelford, J.F., 1996, J. Non-crystalline Solids, 204, Liebau, F., 1985, Structural Chemistry of Silicates, Springer, Berlin. I\. Zalgaller, V.A., 1969, Convex Polyhedra with Regular Faces, Consultants Bureau, New York. 12. Insight , 1995 and 1996, Users Guide (BiosymlMSI). 13. Catalysis 3.0.0,1995 and 1996, Users Guide (BiosymlMSI). 14. Palmer, D.e., 1994, Silica - Physical Behavior, Geochemistry and Materials Applications, Heaney, P.J., Prewitt, C.T., and Gibbs, G.V. (Eds.), Mineralogical Society of America, Washington, D.C., Shackelford, J.F., 1999, J. Non-crystalline Solids, 253, Nakayama, G.S., and Shackelford, J.F., 1990, J. Non-crystalline Solids, 126, ~ Shackelford, J.F., Revesz, A.G., and Hughes, H.L., 1985, Reactivity of Solids, Barret, P., and Dufour, L.-e. (Eds.), Elsevier, Amsterdam, Fueston, B.P., and Garofalini, S.H., 1988, J. Chern. Phys., 89, Abramo, M.e., Caccamo, e., and Pizzimenti, G., 1992, J. Chern. Phys., 96, 9083.