HYPOTHETICAL ZEOLITE FRAMEWORK STRUCTURES AND WEAVINGS

Zeolites are intriguing minerals. Their periodic microporous structures offer high internal surface areas and consequently they have attracted much attention in the chemical industry for applications as molecular sieves and catalysts. Their tectosilicate structure comprises corner-sharing silicate SiO₄ tetrahedra, with open channels and pores that allow water (at a minimum) to diffuse reversibly through them. Quartz, although a tectosilicate also, does not allow water to diffuse, and so is not classified as a zeolite. Zeolite structures, usually determined by x-ray and/or electron crystallography, are quite beautiful to behold and the mathematical subtleties underpinning their framework regularity can be quite mesmerizing. There are now about 250 known zeolitic materials, but there are, in principle, an infinite number of possible framework structures if infinitesimal densities (i.e., arbitrarily large internal pores) are permitted.

Hypothetical zeolites can be enumerated mathematically, by treating zeolite frameworks as graphs, where atoms are “vertices,” and bonds are “edges.” Omitting oxygen atoms, graphs can be described as four-valent mappings of Si-Si connections. Conversely, omitting Si atoms, graphs become six-valent (octahedral) mappings of O-O connections. Given the space group and the number of crystallographically unique vertices, zeolitic graphs can be generated by exploring all combinations of edges (bonds) keeping those graphs that are 4- or 6-connected, and then embedding the graphs in 3D by lowering the framework energy under a simple regular-tetrahedral (or, octahedral) cost function. Such mathematical procedures can generate some beautiful and surprising structures, including interlocking and woven structures.

In this talk we present some of our results for zeolitic structures and present a recent extension of our methods to enumerate 3D periodic weaves of threads and networks.