Authors: Martin E. Glicksman

Abstract: Space-filling in kinetically active 3-d network structures, such as polycrystalline
solids at high temperatures, is treated using topological methods. The theory developed represents individual network elements—the polyhedral cells or grains—as a set of objects called average N-hedra, where N, the topological class, equals the number of contacting neighbors in the network. Average N-hedra satisfy network topological averages for the dihedral angles and quadrajunction vertex angles, and, most importantly, act as “proxies” for real irregular polyhedral grains with equivalent topology. The analysis provided in this paper describes the energetics and kinetics of grains represented as average N-hedra as a function of their topological class. The new approach provides a quantitative basis for constructing more accurate models of three-dimensional grain growth. As shown, the availability of rigorous mathematical relations for the curvatures, areas, volumes, free energies, and rates of volume change provides precise predictions to test simulations of the behavior 3-d networks, and to guide quantitative experiments on microstructure evolution in three-dimensional polycrystals.

1025

Authors: Tamás Réti, Enikő Bitay

Abstract: In recent years, several attempts have been made to characterize the geometric structure
of fullerenes by means of topological shape factors in order to predict their physical properties and
stability. In this paper, we present a simple method to estimate the stability of fullerenes on the
basis of quantitative topological criteria. This approach is based on the concept of the generalized
combinatorial curvatures defined on the set of simple graphs embedded on a closed surface without
boundary (sphere, torus, projective plane, Klein bottle). It is shown that starting with the computed
generalized combinatorial curvatures several novel topological indices can be generated. From
computations performed on a set of C40 and C60 fullerenes, we concluded that the four topological
shape factors tested (Λ(-1), (-1), Λ(1) and (1)) could be successfully used to preselect the most
stable fullerene isomers.

439

Authors: Tamás Réti, Ibolya Zsoldos

Abstract: In order to simulate the polyhedral grain nucleation in alloys, 3-D cell population growth
processes are studied in space-filling periodic cellular systems. We discussed two different methods
by which space-filling polyhedral cellular systems can be constructed by topological
transformations performed on “stable” 3-D cellular systems. It has been demonstrated that an
infinite sequence of stable periodic space-filling polyhedral systems can be generated by means of a
simple recursion procedure based on a vertex based tetrahedron insertion. On the basis of computed
results it is conjectured that in a 3-D periodic, topologically stable cellular system the minimum
value of the average face number 〈f〉 of polyhedral cells is larger than eight (i.e. 〈f〉 > 8). The
outlined algorithms (which are based on cell decomposition and/or cell nucleation) provide a new
perspective to simulate grain population growth processes in materials with polyhedral
microstructure.

579

Authors: Martin E. Glicksman, Paulo Rangel Rios, Daniel Lewis

Abstract: The multiplicity and variety of grain shapes in three-dimensional polycrystalline metals
makes their energetic and kinetic analyses difficult. To help simplify the analysis of isotropic
polycrystals, average N-hedra (ANHs) (N=3,4,5,…∞) were created as a set of regular polyhedra,
consisting of N identical faces, which act as topological “proxies” for analyzing the corresponding
class of irregular grains containing mixed faces of the same number. This paper outlines a further
generalization of the ANH concept that extends three-dimensional analysis to include the growth or
shrinkage of a small population of grains embedded in a textured matrix.

625

Authors: Tamás Réti, Enikő Bitay

Abstract: In several fields of materials science space-filling polyhedral systems are generally used
for modeling and characterizing the microstructure of polycrystalline and cellular materials. In this
paper a simple quantitative method designated to classify 3D triply periodic, space-filling, cellular
systems is outlined. The concept of the proposed method is based on the known analogy between
the combinatorial structure of 3D space-filling polyhedral systems and of 4D polytopes. For
classification purposes various topological shape indices are defined and tested. It is demonstrated
that using two appropriately selected shape factors (asymmetry and compactness coefficients) a
global combinatorial classification of cellular systems can be performed.

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