Roundness of grains in cellular microstructures

Many physical systems are composed of polyhedral cells of varying sizes and shapes. These structures are simple in the sense that no more than three faces meet at an edge and no more than four edges meet at a vertex. This means that individual cells can usually be considered as simple, three-dimensional polyhedra. This paper is concerned with determining the distribution of combinatorial types of such polyhedral cells. We introduce the terms fundamental and vertex-truncated types and apply these concepts to the grain growth microstructure as a testing ground. For these microstructures, we demonstrate that most grains are of particular fundamental types, whereas the frequency of vertex-truncated types decreases exponentially with the number of truncations. This can be explained by the evolutionary process through which grain growth structures are formed and in which energetically unfavorable surfaces are quickly eliminated. Furthermore, we observe that these grain types are “round” in a combinatorial sense: there are no “short” separating cycles that partition the polyhedra into two parts of similar sizes. A particular microstructure derived from the Poisson–Voronoi initial condition is identified as containing an unusually large proportion of round grains. This microstructure has an average of 14.036 faces per grain and is conjectured to be more resistant to topological change than the steady-state grain growth microstructure.

Figure 1

A simulated microstructure resulting from a process of grain growth, assuming isotropic grain boundary energies and mobilities.

Figure 2

A typical set of grains from the grain-growth microstructure.

Figure 3

The relationship of a simple 3-polytope to the corresponding dual simplicial 3-polytope (double arrow) and of a polytope to the corresponding Schlegel diagram (single arrow).

Figure 4

The Schlegel diagram of a grain and the dual graph of the set of all faces; the dual graph vertex for the back face is not shown.

Figure 5

A 4-belt (left) and a 9-belt (right) of a grain shown on the Schlegel diagram along with their dual graphs. The back face belongs to the belt in both cases.

Figure 6

The Schlegel diagram of the smallest possible fundamental simple 3-polytope with a nontrivial 3-belt (indicated by the dotted red line), the dual triangulation with an empty triangle (bold red triangle), and an example from a three-dimensional evolved microstructure with this combinatorial type.

Figure 7

A simple 3-polytope with 13 faces that is seven truncations away from the cube. The collection of rooted truncation trees is on the right, with a truncation depth of three.

Figure 8

Evolution of the average number of faces per grain in time from a Poisson–Voronoi initial condition with constant grain boundary energy and mobility.

Figure 9

Evolution of the percentage of triangular faces in time from a Poisson–Voronoi initial condition with constant grain boundary energy and mobility.

Figure 10

Evolution of the fraction of grains that are ${\mathrm{flag}}^{*}$ in time from a Poisson–Voronoi initial condition with constant grain boundary energy and mobility.

Figure 11

Fraction of ${\mathrm{round}}_{\le k}$ grains in the Poisson–Voronoi, Evolved, and Round microstructures.

Figure 12

The frequencies of ${\mathrm{round}}_k$ grains for the Poisson–Voronoi, Round, and Evolved data structures.

Figure 13

Fraction of ${\mathrm{round}}_{\le k}$ grains in the Poisson–Voronoi, Evolved, and Round microstructures, restricted to $n=14$ faces, and compared to all possible grains with $n=14$ faces.

Figure 14

The average isoperimetricity of grains with a given number of faces for the Poisson–Voronoi, Round, and Evolved microstructures.

Figure 15

The average isoperimetricity of ${\mathrm{round}}_k$ grains for the Poisson–Voronoi, Round, and Evolved data structures. Error bars are the standard error of the mean, and classes with a single grain are indicated by an open marker.

Figure 16

Frequencies of various grain classes in the Poisson–Voronoi (V), Round (R), and Evolved (E) microstructures, where $n$ is the number of faces. Values for the Poisson–Voronoi and Round microstructures are vertically offset by 0.20 and 0.10, respectively.

Figure 17

Dodecahedron from the Evolved data set with several short edges.

Figure 18

Schlegel diagrams of the two polytopes with six faces. The flip of a bold edge gives the other polytope, the flip of a zigzag edge gives the same polytope, and the dashed edges are nonadmissible.

Figure 19

The frequencies of ${\mathrm{round}}_k$ grains in the Poisson–Voronoi and Evolved data sets, and for grains in the Evolved data set after flipping a fixed number of edges per grain. EEPG stands for edge errors per grain.

Figure 20

Percentage of tetrahedra, cubes, dodecahedra, and grains of type 0:$m$ over time.

Figure 21

${\mathrm{Flag}}^{*}$ grains with frequencies of at least 0.00001 in the Evolved microstructure as a function of penalty value. Color indicates the number of faces.

Figure 22

Three grains A, B, and C with the same $p$-vector $(0,4,4,2)$.

Figure 23

Fractions of the three grains with $p$-vector $(0,4,4,2)$ over time.

Figure 24

Three grains K, L, and M with the same $p$-vector $(0,6,0,8)$.

Figure 25

Examples of degenerate grains with 2-gon faces. (a) The trihedron and (b) the pillow.

Figure 26

Examples of degenerate grains with two faces sharing multiple edges.

Figure 27

Fraction of grains with a 2-gon face over time.

Figure 28

Extremal grain with 31 faces, type $1:4:26$, and $p$-vector $(0,4,13,7,5,2)$ from the Poisson–Voronoi microstructure.

Figure 29

Extremal grains with 38 faces, type $2:4:32$, and $p$-vectors $(0,9,8,12,5,3,1)$ and $(0,10,8,8,8,4)$ from the Evolved microstructure.

Figure 30

Extremal grain with 22 faces, type $9:4:9$, and $p$-vector $(0,7,7,3,3,0,2)$ from the Poisson–Voronoi microstructure.

Figure 31

Extremal grain with 21 faces, type $8:4:9$, and $p$-vector $(0,6,6,5,2,2)$ from the Evolved microstructure.

Figure 32

A 6-gon in a simple cellular microstructure (left) and its expansion into prisms over the 6-gon (right).

Figure 33

The star (in gray) and the link (in bold) of a vertex.