W. Brostow (1,2), V. M. Castaño (1,2)

(1) Department of Materials Science, University of North Texas, Denton, TX 76203-5310, USA; brostow@unt.edu
(2) Instituto de Fisica, Universidad Nacional Autonoma de México A.P. 1010, Queretaro , Qro. 76000, Mexico; castano@marta.phys.unt.edu meneses@servidor.unam.mx

ABSTRACT

Methods of dealing with crystal structures are described in detail in every textbook of Materials Science and Engineering. Structures of non-crystalline materials (amorphous solids, liquids, dense gases) are no less important, but usually treated in textbooks briefly – if at all. Voronoi polyhedra have been employed for describing non-crystalline structures in a way at least complementary with other descriptions, and often more meaningful. The structure is divided into a set of polyhedra in a unique way, with a polyhedron assigned to each center (atom, ion, polymer chain segment). The Voronoi diagram (the set of polyhedra for a given set of centers) constitutes the basis of an informational theory of amorphous solids and liquids; the theory includes also the structure representation in terms of the radial distribution function. The Voronoi diagram and its mathematical dual, the Delaunay diagram, also make possible the distinction between a solid and a liquid, thus capturing the essence of the difference between these two states of matter. Pattern recognition can be performed by analyzing shapes of the Voronoi ployhedra. A procedure for contraction of short edges and small faces of the polyhedra is available; it provides slightly simplified structures, but with a large gain in the perspicuity. The pattern recognition method is also applicable to crystals, thus making possible comparisons of crystalline and non-crystalline structures in a simple way. The relation of the Voronoi polyhedra to the radial distribution function, a well-known tool used in diffractive imaging of materials to elucidate their spatial structure, is analyzed as well. Finally, alternative methods for characterizing the structures of materials are discussed.