Characterization of maximally random jammed sphere packings: Voronoi correlation functions

We characterize the structure of maximally random jammed (MRJ) sphere packings by computing the Minkowski functionals (volume, surface area, and integrated mean curvature) of their associated Voronoi cells. The probability distribution functions of these functionals of Voronoi cells in MRJ sphere packings are qualitatively similar to those of an equilibrium hard-sphere liquid and partly even to the uncorrelated Poisson point process, implying that such local statistics are relatively structurally insensitive. This is not surprising because the Minkowski functionals of a single Voronoi cell incorporate only local information and are insensitive to global structural information. To improve upon this, we introduce descriptors that incorporate nonlocal information via the correlation functions of the Minkowski functionals of two cells at a given distance as well as certain cell-cell probability density functions. We evaluate these higher-order functions for our MRJ packings as well as equilibrium hard spheres and the Poisson point process. It is shown that these Minkowski correlation and density functions contain visibly more information than the corresponding standard pair-correlation functions. We find strong anticorrelations in the Voronoi volumes for the hyperuniform MRJ packings, consistent with previous findings for other pair correlations [A. Donev et al., Phys. Rev. Lett. 95, 090604 (2005)], indicating that large-scale volume fluctuations are suppressed by accompanying large Voronoi cells with small cells, and vice versa. In contrast to the aforementioned local Voronoi statistics, the correlation functions of the Voronoi cells qualitatively distinguish the structure of MRJ sphere packings (prototypical glasses) from that of not only the Poisson point process but also the correlated equilibrium hard-sphere liquids. Moreover, while we did not find any perfect icosahedra (the locally densest possible structure in which a central sphere contacts 12 neighbors) in the MRJ packings, a preliminary Voronoi topology analysis indicates the presence of strongly distorted icosahedra.

Figure 1

Maximally random jammed (MRJ) sphere packing and its Voronoi diagram. Among all jammed sphere packings (roughly speaking, the mechanically stable packings), the MRJ state is the most disordered one.

Figure 2

Distributions of the single Minkowski functionals $W_{\mu }$ of a three-dimensional Voronoi cell in a Poisson point process, an equilibrium hard-sphere system at a packing fraction $\varphi =0.48$, and a MRJ sphere packing: $\mu =0$ volume (left), $\mu =1$ surface area (center), and $\mu =2$ integrated mean curvature (right). The distributions are rescaled—like in Ref. [47]—by their mean $\langle W_{\mu }\rangle$ and their standard deviation ${\sigma }_{W_{\mu }}$ (see Table 1). The lines in the plot of the volume distributions are $\gamma$ distributions; generalized $\gamma$ distributions are fitted to the distributions of the surface area and the integrated mean curvature.

Figure 3

Joint distribution of volume $v$ and surface area $S$ of a single three-dimensional Voronoi cell in a logarithmic scale for a Poisson point process and a MRJ sphere packing; the joint distribution for the equilibrium hard-sphere liquid is represented by the blue contour plot. The unit of length is $\lambda =1/{\rho }^{1/3}$, where $\rho$ is the number density. Samples of a MRJ sphere packing and an overlapping sphere packing, where the sphere centers follow a Poisson point process, are depicted together with their Voronoi diagrams.

Figure 4

Correlation functions for a three-dimensional Poisson point process: pair-correlation function $g_2\left(r\right)$ (bottom); volume-volume correlation function $C_{00}\left(r\right)$ (center), which is the correlation function of the volumes of two Voronoi cells given that their centers are at a distance $r$; the mark correlation functions of the three different Minkowski functionals (top): $\mu =0$ volume, $\mu =1$ surface area, and $\mu =2$ integrated mean curvature. The radial distance $r$ is normalized by $\lambda =1/{\rho }^{1/3}$, where $\rho$ is the number density.

Figure 5

Correlation functions for equilibrium hard spheres with diameter $D$ at a packing fraction $\varphi =0.48$; for details, see Fig. 4.

Figure 6

Correlation functions for MRJ sphere packings of spheres with diameter $D$; the average packing fraction is $\varphi \approx 0.64$. For details, see Fig. 4. The pair-correlation function $g_2\left(r\right)$ is in agreement with previous results for MRJ sphere packings [39].

Figure 7

The cell-cell pair-correlation function $g_{vv}(r,v,v^{*})$ for a three-dimensional Poisson point process for either two large cells (bottom), a large and a small cell (center), or two small cells (top). As examples of large or small cells, the volumes were chosen such that their probability density is equal to $1/3$ of the maximum of the volume distribution; see Table 1 and Fig. 2. The curves are compared to the standard pair-correlation function $g_2\left(r\right)$ (dashed black line), which is trivially unity for the uncorrelated Poisson point process. The radial distance $r$ is normalized by $\lambda =1/{\rho }^{1/3}$, where $\rho$ is the number density.

Figure 8

The cell-cell pair-correlation function $g_{vv}(r,v,v^{*})$ for equilibrium hard spheres at a global packing fraction $\varphi =0.48$; for details, see Fig. 7.

Figure 9

The cell-cell pair-correlation function $g_{vv}(r,v,v^{*})$ of the MRJ sphere packings; the average packing fraction is $\varphi \approx 0.64$; for details, see Fig. 7.

Figure 10

Correlation functions for the one-dimensional Poisson point process: pair-correlation function $g_2\left(r\right)$ (bottom); volume-volume correlation function $C_{00}\left(r\right)$ (top). The distance $r$ is scaled (in this one-dimensional example) by the inverse of the number density $\lambda =1/\rho$. An example of a one-dimensional Voronoi diagram of a Poisson point process is depicted.