THE MEAN NORMALIZED EULER CHARACTERISTIC OF A SIMULTANEOUSLY STARTINGAND GROWING 2D VORONOI TESSELLATION WITH POISSON-DISTRIBUTED NUCLEI

The Euler characteristic chi describes the relation between the number of comers, edges and areas of a polyhedron. For two-dimensional (2D) structures an Euler characteristic can be given as well. The 2D case c an be realized by the model of a simultaneously starting and circularl y growing 2D Voronoi tessellation with a Poisson distribution of nucle i. The triple points are equivalent to the comers; the grain boundarie s, circles and arcs to the edges; and the grains to the areas. These q uantitaties are deduced as a function of the fraction transformed, F, using probability theory. The results are the 'mean normalized numbers ' of comers, edges and areas, always referred to one nucleus. From thi s the 'mean normalized Euler characteristic' as a function of the frac tion transformed, X(F), can be calculated exactly. A growing 2D Vorono i tessellation consists of two different phases which are islands (tra nsformed regions) and seas (untransformed regions). Here a second rela tion is valid. The difference of the mean normalized numbers of island s and seas at a given F is also the above deduced 'mean normalized Eul er characteristic', X(F). The numbers of islands and of seas are simul ated by computer because they cannot be calculated analytically in an acceptable time. These numbers are divided by the number of nuclei for normalization. The sum of the normalized numbers of islands and of se as has a local minimum at about F=0.68. This seems to be the threshold of percolation.