Zeolites, curved polyhedra, 3-connected nets and the structure of glass

Transcription

1 1 Zeolites, curved polyhedra, 3-connected nets and the structure of glass Sten Andersson Sandforsk, Institute of Sandvik S Löttorp, Sweden Abstract Two zeolite type structures of sesquioxide compositions are derived with 3- connectors of curved saddle type polyhedra. A structure of glass is discussed, with the Thurston model of seven rings for the hyperbolic plane. Introduction It is convienent to use polyhedra in designing descriptions of structures of zeolites. Cubes, truncated octahedra, hexagonal prisms are some. That approach is not easy to use to derive 3-connected nets that are typical zeolite structures. We have derived curved polyhedra of saddle type for designing nets and structures. An oxide composition for such a net would be B 2 O 3. Examples of three connected nets exist with B 2 O 3,SrSi 2 and ThSi 2. 1 A new type of curved polyhedra Recently we described a group of new types of curved polyhedra that were derived using the Alhambra tilings(ref 1). All in order to describe a most exotic shape of one of the forms of pyrite. Saddle polyhedra that exactly describe periodic minimal surfaces were described some time ago(ref 2). In order to describe 3-connected nets that are of typical zeolite structure we realized we needed a new kind of polyhedra. Of the two kinds of conformations shown in fig 1.1 we use almost entirely the one to right. Both the conformations occur in the normal structure of B 2 O 3.

2 2 a b Fig 1.1 The two conformations in B 2 O 3 We make a saddle with six connectors with the conformation from fig 1.1b, and combine four such saddles to a saddle-like polyhedron as in fig 1.2a. This also has the topology of the adamantane molecule which is a piece chopped out of the diamond structure, and showed in 1.2b in form of the net. And as we showed recently(ref 3) we can by bringing the spheres closer together show the relations from the net in 1.2a to the surface in 1.2c. Which is the D surface, with the adamantane molecule in between in 1.2b. Regular adamantane coordinates were used in eq 1.2. But we derive unit cell size, positional parameters in 6f and 4e from the 3-connecter geometry using the group P43m in equations 1.1, with sphere diameter as unit: a= 6 +1 x 6 = 6 2a 1.1 x 4 = 6 4a

3 3-1 e 2 ((x - 1)2 + (y - 1) 2 + (z - 1) 2 ) e 2 ((x + 1)2 + (y - 3) 2 + (z - 3) 2 ) e 2 ((x - 3)2 + (y + 1) 2 + (z - 3) 2 ) e 2 ((x - 1)2 + (y - 5) 2 + (z - 5) 2 ) e 2 ((x - 5)2 + (y - 1) 2 + (z - 5) 2 ) e 2 ((x + 3)2 + (y - 1) 2 + (z - 5) 2 ) e 2 ((x - 1)2 + (y + 3) 2 + (z - 5) 2 ) e 2 ((x - 1)2 + (y - 1) 2 + (z - 9) 2 ) e 2 ((x + 1)2 + (y + 1) 2 + (z - 7) 2 ) e 2 ((x - 3)2 + (y - 3) 2 + (z - 7) 2 ) = 1 3 Fig 1.2a Saddle-like polyhedron with six rings. b Adamantane molecule, calculated with 1.1. Fig 1.2c Same as 2b, with smaller boundaries.

4 4 Each saddle in this figure is very similar with a saddle derived from the D minimal surface and as shown in fig 22 in ref 2. Eight such saddle-like polyhedra are now put together in fig 1.3 in a cubic structure with a well sized cavity built of 12 rings. Mark that if you rotate the saddle like polyhedra you join them with the conformation of fig 1.1a. And you have a new structure. We can also get the structure as a mathematical function as developed in ref, and this is shown as net-surface picture in fig 1.3.b. The equation is in 1.3. The coordinates are double sphere units, and after 6f and 4e using the group P43m, as in 1.1. Fig 1.3a Zeolitic structure of 3-connectors. Possible composition B 2 O 3.

5 5 SumGD,{n,1.22,11,6.9},{m,1.22,11,6.9},{p,1.22,11, 6.9} + SumGD,{n,1.22,11, 6.9},{m,-1.22,11, 6.9},{p,-1.22,11, 6.9} + SumGD,{n,-1.22,11, 6.9},{m,1.22,11, 6.9},{p,-1.22,11, 6.9} + SumGD,{n,-1.22,11, 6.9},{m,-1.22,11, 6.9},{p,1.22,11, 6.9} + SumGD,{n,"2.44,11,6.9},{m,0,11,6.9},{p,0,11,6.9} + SumGD,{n, 0,11,6.9},{m,"2.44,11,6.9},{p,0,11,6.9} + SumGD,{n, 0,11,6.9},{m, 0,11, 6.9},{p,"2.44,11,6.9} + SumGD,{n, 2.44,11, 6.9},{m,0,11,6.9},{p,0,11,6.9} + SumGD,{n, 0,11,6.9},{m, 2.44,11,6.9},{p,0,11,6.9} + SumGD,{n, 0,11, 6.9},{m, 0,11, 6.9},{p, 2.44,11, 6.9} = 1/2 1.3 GD = e "((x-n)2 +(y-m) 2 +(z-p) 2 ) Fig 1.3b So a hypothetical carbon form is there related in topology to zeolites, a challenge for an organic solid state chemist to prepare. On the other hand hydrophobic silica zeolites are prepared directly from solutions containig some organic template molecule. In this case we need of course boron oxide instead of silica. However it is well worth to remember what Kittel says we should not overemphasize the similarity of the bonding of carbon and silicon. Carbon gives biology, but silicon gives geology and semiconductor technology (ref 4, p 72).

6 6 We continue with an 8-ring saddle as shown in fig 1.4a and in b we show the corresponding saddle from the Neovius periodic minimal surface(ref 2 and 5). This saddle is a single surface element of that surface. Which, among other things, means that this element is repeated via two-fold axes(the straight lines in fig 1.4b) to build the entire Neovius surface. Fig 1.4a b We now continue to build the 3-connected net and in fig 1.5 we have joint six saddles from1.4a to build a polyhedron of saddle type. The equation is in 1.4 and the constant is.9. This is now a curved polyhedron built from 3-connecting spheres which we can continue to form a complete net using F symmetry. This is actually started in fig 1.5 where the farthest off spheres belong to the next polyhedron. The 3D part of the net with a cavity built of 9 rings is shown in fig 1.6. The space group is Fm3m, and spheres are in positions 32 f with x=.138 and 48 h with x=.194.

7 7 e -((x -1.38)2 + (y -1.38) 2 + (z -1.38) 2 ) + e -((x -1.38) 2 + (y -1.38) 2 + (z ) 2 ) + e -((x )2 + (y -1.38) 2 + (z -1.38) 2 ) + e -((x ) 2 + (y -1.38) 2 + (z ) 2 ) + e -((x )2 + (y ) 2 + (z -1.38) 2 ) + e -((x ) 2 + (y ) 2 + (z ) 2 ) + e -((x -1.38)2 + (y ) 2 + (z -1.38) 2 ) + e -((x -1.38) 2 + (y ) 2 + (z ) 2 ) + e -((x -1.94)2 + (y -1.94) 2 + (z) 2 ) + e -((x ) 2 + (y -1.94) 2 + (z) 2 ) + e -((x -1.94)2 + (y ) 2 + (z) 2 ) + e -((x ) 2 + (y ) 2 + (z) 2 ) + e -((x -1.94)2 + (y) 2 + (z -1.94) 2 ) + e -((x -1.94) 2 + (y) 2 + (z ) 2 ) + e -((x)2 + (y -1.94) 2 + (z -1.94) 2 ) + e -((x) 2 + (y -1.94) 2 + (z ) 2 ) e -((x )2 + (y) 2 + (z -1.94) 2 ) + e -((x ) 2 + (y) 2 + (z ) 2 ) + e -((x)2 + (y ) 2 + (z -1.94) 2 ) + e -((x) 2 + (y ) 2 + (z ) 2 ) + e -((x )2 + (y ) 2 + (z) 2 ) + e -((x ) 2 + (y ) 2 + (z) 2 ) + e -((x )2 + (y ) 2 + (z) 2 ) + e -((x ) 2 + (y ) 2 + (z) 2 ) + e -((x)2 + (y ) 2 + (z ) 2 ) + e -((x) 2 + (y ) 2 + (z ) 2 ) + e -((x)2 + (y ) 2 + (z ) 2 ) + e -((x) 2 + (y ) 2 + (z ) 2 ) + e -((x )2 + (y) 2 + (z ) 2 ) + e -((x ) 2 + (y) 2 + (z ) 2 ) + e -((x )2 + (y) 2 + (z ) 2 ) + e -((x ) 2 + (y) 2 + (z ) 2 ) = const I figures 1.5 and 1.6 the constant from equation 1.4 is.9.

8 8 Fig 1.5 Six saddles form a saddle type polyhedron with const=.9. Fig 1.6 The net of polyhedra from fig 5, build a cavity of 9-rings. From the net in fig 1.6 we bring out a surface in Fig 7.1 by changing the constant to.5.

9 9 Fig 1.7 Changing constant to.5 brings out a surface we analyze below in Figs 1.8a,b,c. The saddle like polyhedron in fig 1.8a has twelve catenoids that connects to other saddle polyhedra to build the complete surface in fig 1.7. This is somewhat similar with the geometry of the FRD surface(ref 3), but more complex. In the case of FRD the polyhedral background to the whole surface is the cube octahedron. Here the corresponding saddle polyhedron in fig 1.8a has an inside as shown in fig1.8b which is the P surface. And the inside is extended to an outside via trigonal monkey saddles as shown in fig1.8c.

10 10 Fig 1.8a Saddle type polyhedron from fig 5 shows K 0. b The inside of 8a. Fig 1.8c Between 8a and b shows part of P surface + trigonal monkey saddles.

11 11 2 On a glass structure Boron oxide is one of the best glass formers ever and we shall now describe a triangular glass instead of a tetrahedral. Glass is a perfect isotropic material. which means that if you are located inside the glass, it looks identical in every direction. We say this is typical for a space of constant negative Gaussian curvature. Which we now desribe. The hyperbolic plane is a surface characterized by having constant negative Gaussian curvature, or K= 1. We know that the case for K=1 is the sphere, and for K=0 there is the Euclidean plane. And for minimal surfaces K varies between 0 and 1. The hyperbolic plane has fascinated man ever since it was invented more than a century ago. In their classic article from 1984 Thurston and Weeks(ref 6) uses the hyperbolic plane to find suitable pathes to study the mathematics of higher space. They also show useful pictures of slices of the hyperbolic plane in that publication(page 103, bottom left). Thurston recommends an important way to build the plane which is of particular interest for us. Make seven triangles meet in a point and glue them together and keep on doing so and you get a surface, entirely built of saddles, and that keeps on intersecting until it fills space. We have done a bit of such a surface as shown in fig 2.1. Fig 2.1 The hyperbolic plane after Thurston There is another way doing a model of the hyperbolic plane, still using Thurston s idea. We make seven-rings with three connectors and join these seven-rings edge to edge. Due to this single edge sharing we cannot follow the algoritm we started with the connectors always being perpendicular(fig 1b above). We now get all sorts of angels and we start with building a saddle as seen in fig 2.2. And has the similarity that it intersects itself in this case at

12 12 every connector as it is a local three fold axis. The hyperbolic plane as a surface fills space intersecting itself everywhere. Fig 2.2 a. Model of the hyperbolic plane that consists of saddles and 3- connectors. References 1 S. Andersson, Z. f. Anorganische und Allgemeine Chemie. 631,499(2005) 2 S.T. Hyde and S. Andersson, Z. Kristallogr. 168, 221(1984). 3 S. Andersson Mathematics of some simple cubic structures, C. Kittel, Introduction to Solid State Physics, 7 th edition, page 72, Wiley, E.R. Neovius, Bestimmung zweier Speciellen Periodische Minimal flächen, J.C. Frenckel & Sohn, Helsinki, W.P. Thurston and J.R. Weeks, Scientific American, 251, 94(1984)