Complete Topology of Cells, Grains, and Bubbles in Three-Dimensional Microstructures

We introduce a general, efficient method to completely describe the topology of individual grains, bubbles, and cells in three-dimensional polycrystals, foams, and other multicellular microstructures. This approach is applied to a pair of three-dimensional microstructures that are often regarded as close analogues in the literature: one resulting from normal grain growth (mean curvature flow) and another resulting from a random Poisson-Voronoi tessellation of space. Grain growth strongly favors particular grain topologies, compared with the Poisson-Voronoi model. Moreover, the frequencies of highly symmetric grains are orders of magnitude higher in the grain growth microstructure than they are in the Poisson-Voronoi one. Grain topology statistics provide a strong, robust differentiator of different cellular microstructures and provide hints to the processes that drive different classes of microstructure evolution.

Figure 1

Two topologically distinct six-faced grains.

Figure 2

Two topologically distinct grains that share $p$ vector (00222200…). In the first, two triangular faces are connected by an edge, whilst in the second they are not.

Figure 3

A typical grain and a corresponding Schlegel diagram (the embedding of the vertex-edge graph in the plane).

Figure 4

Vertices are labeled as they are initially encountered while traversing the graph following rules (1) and (2); the vector lists all vertices in the order in which they have been visited.

Figure 5

Schlegel diagrams of the eight most common grain topologies (Weinberg vectors) in the Poisson-Voronoi and grain growth microstructures. Listed for each topological type is a label, the frequency of occurrence $f$, the $p$ vector, the number of faces $F$, and the order $S$ of the associated symmetry group. The Weinberg vectors are tabulated in Tables SII and SIII of the Supplemental Material [18].

Figure 6

A log-log plot of the ratio of the frequencies of grains with a given symmetry group order $S$ ($\le 120$) from the grain growth microstructures $f_{\mathrm{GG}}^S$ and Poisson-Voronoi microstructures $f_{\mathrm{PV}}^S$. Note that the statistical errors for $S=3$, 32, 48, and 120 all exceed 30% because of their extreme rarity in the microstructures.