Statistical topology of cellular networks in two and three dimensions

Cellular networks may be found in a variety of natural contexts, from soap foams to biological tissues to grain boundaries in a polycrystal, and the characterization of these structures is therefore a subject of interest to a range of disciplines. An approach to describe the topology of a cellular network in two and three dimensions is presented. This allows for the quantification of a variety of features of the cellular network, including a quantification of topological disorder and a robust measure of the statistical similarity or difference of a set of structures. The results of this analysis are presented for numerous simulated systems including the Poisson-Voronoi and the steady-state grain growth structures in two and three dimensions.

Figure 1

Sections of the 2D cellular structure described by the swatches in (a) Table (a) and (b) Table (b), respectively. The swatch assigns strings to the vertices around a root, indicated by the black dots in the center of the figures. The vertex labeled A0 in (a) is the root of the region in (b); i.e., the roots of the two swatches are separated by two edges. Additionally, the labeling convention is inverted in (b) relative to (a), corresponding to a switch of the normal vector for the 2D network from out of the page to into the page.

Figure 2

Conventions for the labeling of vertices in 2D. The top and bottom rows correspond to the normal to the structure being oriented out of and into the page, respectively. Within a given row, the first figure indicates the labeling conventions around the root, while the second and third figures indicate the labeling conventions after a left or a right turn, respectively.

Figure 3

The 3D region described by the swatch in Table . The swatch assigns strings to the vertices around a root at the center of the four indicated cells.

Figure 4

Conventions for the labeling of vertices in 3D. The top and bottom rows correspond to positive and negative orientations of the space, respectively. Within a given row, the first figure indicates the labeling conventions around the root, while the remaining three figures indicate the labeling conventions after a turn of 0, 1, or 2, respectively.

Figure 5

From left to right, representative areas of the hexagonal, perturbed, evolved, Poisson, and flip 2D reference structures, as described in the text. The corresponding $k$ entropies increase from left to right.

Figure 6

$k$ entropies for the 2D evolved structure with a varying number of cells in the system. As the system size increases, the $k$ entropies approach a limiting curve that is characteristic of the evolved structure. The saturation on the right occurs from finite-size effects.

Figure 7

$k$ entropies of the five 2D reference structures with ${10}^5$ cells.

Figure 8

From left to right, representative volumes of the bcc, perturbed, evolved, and Poisson 3D reference structures; the flip structure is not illustrated for reasons described in the text.

Figure 9

$k$ entropies of the five 3D reference structures with roughly 9800 cells.

Figure 10

$k$ distances between distinct realizations of the 2D evolved structure. A curve shows the $k$ distances between two realizations with the same number of cells.

Figure 11

$k$ distances between a 2D evolved and 2D Poisson structure.

Figure 12

$k$ distances between a 2D evolved structure and the other four 2D reference structures. The evolved curve represents the distance between different realizations of the evolved structure and is shown for reference. All of the realizations contain ${10}^5$ cells.

Figure 13

$k$ distances between a 3D evolved structure and the other four 3D reference structures. The evolved curve represents the distance between different realizations of the evolved structure and is shown for reference. All of the realizations contain roughly 9800 cells.

Figure 14

(a) Our labeling scheme applied to the reference structure (three-coordinated Bethe lattice) for the 2D reconstruction, with a unique string for every vertex. (b) Reconstruction of the cellular structure in Fig. 1 from the the reference structure and the relations in Table (a).

Figure 15

(a) Our labeling scheme applied to the reference structure for the 3D reconstruction, with a unique string for every vertex. (b) Reconstruction of the cellular structure in Fig. 3 from the reference structure and the relations in Table . Dashed lines indicate the continuation of the adjoining surface.