Size-Topology Relations in Packings of Grains, Emulsions, Foams, and Biological Cells

Particulate packings in 3D are used to study the effects of compression and polydispersity on the geometry of the tiling in these systems. We find that the dependence of the neighbor number on cell size is quasilinear in the monodisperse case and becomes nonlinear above a threshold polydispersity, independent of the method of creation of the tiling. These size-topology relations can be described by a simple analytical theory, which quantifies the effects of positional disorder in the monodisperse case and those of size disorder in the polydisperse case and is applicable in two and three dimensions. The theory thus gives a unifying framework for a wide range of amorphous systems, ranging from biological tissues, foams, and bidisperse disks to compressed emulsions and granular matter.

Figure 1

Tilings of space-filling systems and tessellations of particulate packings. In 2D, epithelial tissue of MDCK cells stained with GFP fused E-cadherin ($\mathrm{\text{scale bar}}=10\mu \mathrm{m}$) shown in (a) is compared with a random packing of bidisperse disks segmented by the navigation map shown in (b) (scale $\mathrm{\text{bar}}=2.2\mathrm{cm}$). The 2D confocal images in (c) and (d) represent slices of the 3D tessellations of spheres achieved by the compression of polydisperse droplets ($p=22\%$) and the tessellation of the undeformed droplets, respectively ($\mathrm{\text{scale bar}}=10\mu \mathrm{m}$).

Figure 2

For 2D systems, the relationship between the average area of cells with $n$ neighboring cells and $n$ is shown. Lewis’ cucumber data and law $A/⟨A⟩=1+0.246(n-6)$ [11]. Bidisperse discs with size ratio 1.22 chosen to avoid crystallization [30].

Figure 3

For 3D systems, (a) depicts the relationship between the average volume and $n$. This is plotted on a log-log scale in the inset, where the blue dotted line is the power-law fit (slope of 2.262) to the random 3D foam from Kraynik et. al [7]. (b) The relationship between widths of neighbor-number and cell-volume distributions. Each point represents one experiment; the legend indicates additional experiments not shown in (a). The dashed line is obtained from the granocentric model [25] by varying the particle size distribution from monodisperse ($p=0\%$) to 22% polydispersity.