Local Origin of Global Contact Numbers in Frictional Ellipsoid Packings

In particulate soft matter systems the average number of contacts $Z$ of a particle is an important predictor of the mechanical properties of the system. Using x-ray tomography, we analyze packings of frictional, oblate ellipsoids of various aspect ratios $\alpha$, prepared at different global volume fractions ${\varphi }_g$. We find that $Z$ is a monotonically increasing function of ${\varphi }_g$ for all $\alpha$. We demonstrate that this functional dependence can be explained by a local analysis where each particle is described by its local volume fraction ${\varphi }_l$ computed from a Voronoi tessellation. $Z$ can be expressed as an integral over all values of ${\varphi }_l$: $Z({\varphi }_g,\alpha ,X)=\int Z_l({\varphi }_l,\alpha ,X)P({\varphi }_l|{\varphi }_g)d{\varphi }_l$. The local contact number function $Z_l({\varphi }_l,\alpha ,X)$ describes the relevant physics in term of locally defined variables only, including possible higher order terms $X$. The conditional probability $P({\varphi }_l|{\varphi }_g)$ to find a specific value of ${\varphi }_l$ given a global packing fraction ${\varphi }_g$ is found to be independent of $\alpha$ and $X$. Our results demonstrate that for frictional particles a local approach is not only a theoretical requirement but also feasible.

Figure 1

(a) Pharmaceutical placebo pills with $\alpha =0.59$. (b) Gypsum particles made with a 3D printer with $\alpha =0.40$. (c) Rendering of the particles detected in a x-ray tomogram. (d) The black wire frame indicates the Voronoi cells of the ellipsoids.

Figure 2

Contact number as a function of the global volume fraction. Lines correspond to Eq. (6), which is the numerical integration of the local theory presented here. The different symbols indicate preparation of the initial packing, which is then compactified for all but the loosest samples by tapping. The different aspect ratios and particle types are indicated by different colors; see Table 1.

Figure 3

Measuring the local contact number function $Z_l$ that describes how many contacts an average particle with a local area fraction ${\varphi }_l$ will form. In panel (a) each line corresponds to a single experiment, i.e., a single data point in Fig. 2. Each cross represents the average number of contacts a particle with this value of ${\varphi }_l$ (using a bin size of 0.02) will form. The colored lines in panel (b) are averages over all data sets (i.e., different values of ${\varphi }_g$) displayed in the upper panel. The black dashed lines are parabolic fits according to Eq. (2). The inset shows the theoretical result from Song et al. [7] for spheres compared with our sphere data.

Figure 4

Scaling properties of the local volume fraction distribution $P({\varphi }_l)$. (a) For a given global volume fraction (here ${\varphi }_g\approx 0.625$) the probability of finding a specific value of ${\varphi }_l$ does not depend on $\alpha$. (b) $P$ can be rescaled using ${\varphi }_g$ and the standard deviation of the local volume fractions $\sigma$. Shown here are all experiments with $\alpha =0.59$. The black lines in panels (a) and (b) are Gaussian fits using Eq. (4). (c) The standard deviations of the local volume fraction distribution depends only on ${\varphi }_g$. Panel includes all experiments shown in Fig. 2. The black line is a linear fit resulting in Eq. (5).