Characterization of maximally random jammed sphere packings. II. Correlation functions and density fluctuations

In the first paper of this series, we introduced Voronoi correlation functions to characterize the structure of maximally random jammed (MRJ) sphere packings across length scales. In the present paper, we determine a variety of different correlation functions that arise in rigorous expressions for the effective physical properties of MRJ sphere packings and compare them to the corresponding statistical descriptors for overlapping spheres and equilibrium hard-sphere systems. Such structural descriptors arise in rigorous bounds and formulas for effective transport properties, diffusion and reactions constants, elastic moduli, and electromagnetic characteristics. First, we calculate the two-point, surface-void, and surface-surface correlation functions, for which we derive explicit analytical formulas for finite hard-sphere packings. We show analytically how the contact Dirac delta function contribution to the pair correlation function $g_2\left(r\right)$ for MRJ packings translates into distinct functional behaviors of these two-point correlation functions that do not arise in the other two models examined here. Then we show how the spectral density distinguishes the MRJ packings from the other disordered systems in that the spectral density vanishes in the limit of infinite wavelengths; i.e., these packings are hyperuniform, which means that density fluctuations on large length scales are anomalously suppressed. Moreover, for all model systems, we study and compute exclusion probabilities and pore size distributions, as well as local density fluctuations. We conjecture that for general disordered hard-sphere packings, a central limit theorem holds for the number of points within an spherical observation window. Our analysis links problems of interest in material science, chemistry, physics, and mathematics. In the third paper of this series, we will evaluate bounds and estimates of a host of different physical properties of the MRJ sphere packings that are based on the structural characteristics analyzed in this paper.

Figure 1

Disordered sphere configurations: (a) overlapping spheres, (b) an equilibrium hard-sphere liquid, (c) an MRJ sphere packing.

Figure 2

The schematic depicts events that contribute to the structure characteristics from Table 1 when single points, balls, or points at a given distance (dashed lines) are placed randomly in the sample (a) for a packing of hard spheres forming a two-phase random medium or (b) for the point process of the sphere centers.

Figure 3

Two-point correlation functions $S_2\left(r\right)$ for overlapping spheres ($\varphi =0.636$), equilibrium hard-sphere liquids ($\varphi =0.478$ and $\varphi =0.50$, data from Ref. [39]), and MRJ sphere packings ($\varphi =0.636$). The dashed lines indicate the limits of the curves. The distance $r$ is rescaled by the diameter of a single sphere $D$. The slope at $r=0$ (indicated by a dashed line) is proportional to the specific surface $s$; see Secs. 2a1 and 3a.

Figure 4

Surface-void correlation functions $F_{sv}\left(r\right)$ (rescaled by the specific surface $s$) for overlapping spheres ($\varphi =0.636, s=2.21/D$), an equilibrium hard-sphere liquid ($\varphi =0.478, s=2.87/D$), and MRJ sphere packings ($\varphi =0.636, s=3.81/D$). For details, see Fig. 3.

Figure 5

Surface-surface correlation functions $F_{ss}\left(r\right)$ (rescaled by the square of the specific surface $s$) for overlapping spheres ($\varphi =0.636, s=2.21/D$), an equilibrium hard-sphere liquid ($\varphi =0.478, s=2.87/D$), and MRJ sphere packings ($\varphi =0.636, s=3.81/D$). For details; see Fig. 3. The insets magnify $F_{ss}\left(r\right)/s^2$ at $r/D=2$, where the derivative of $F_{ss}$ is discontinuous for the MRJ sphere packings in contrast to the equilibrium hard spheres.

Figure 6

Spectral densities of the overlapping spheres (bottom), equilibrium hard-sphere liquid (center), and MRJ sphere packings (top): For the hard spheres, they are calculated by a Fourier transformation of either the autocovariance (solid line) [see Eq. (11)] or directly of the sphere packings themselves (crosses) [see Eq. (14)]. For the overlapping and equilibrium hard spheres, the dashed horizontal lines indicate the value in the infinite wavelength limit. The MRJ state is hyperuniform; therefore, the structure factor vanishes for $k\to 0$. For both hard-sphere systems, each analyzed packing contains 10 000 spheres. The dashed vertical lines indicate the zeros of the spectral densities that are universal for all disordered hard-sphere packings.

Figure 7

Complementary cumulative pore-size distribution $F\left(\delta \right)$ in MRJ sphere packings compared to those in overlapping and equilibrium hard spheres as well as face-centered-cubic (fcc) and body-centered-cubic (bcc) lattices: the solid (black) and dashed (blue) lines show the analytical curves for overlapping and equilibrium hard spheres, respectively. They are in excellent agreement with simulation results (black and blue points). The MRJ packings produced by the Torquato-Jiao (TJ) sphere-packing algorithm are also compared to results from molecular dynamics (MD) simulations; see Ref. [29]. The (quantitative) difference can be explained by slightly different global packing fractions. The inset shows that there is a strong variation in $F\left(\delta \right)$ if the TJ results are restricted to packings with either slightly larger or smaller global packing fractions than the average global packing fraction.

Figure 8

The exclusion probability $E_V\left(r\right)$ of the sphere centers in MRJ packings is compared to those in overlapping spheres, equilibrium hard spheres, and crystalline sphere packings (fcc or bcc). The point patterns are compared at unit density (the unit of length is given by ${\rho }^{-1/3}$, where $\rho$ is the number density); for details, see Fig. 7.

Figure 9

The number probability function $f_R\left(N\right)$ is rescaled by the square root of the number variance ${\sigma }_R^2\left(N\right)$ and plotted as a function of the normalized number of points $N$ inside a spherical observation window of radius $R$ that is randomly placed in the sample. For (a) overlapping spheres, (b) equilibrium hard spheres, and (c) MRJ sphere packings, the rescaled number probability functions are compared to the probability density function of the normal distribution (solid black line). For overlapping spheres, $f_R\left(N\right)$ corresponds to a Poisson distribution (dashed colored lines). It converges for an increasing test sphere radius $R$ to a normal distribution, but slowly compared to the equilibrium and MRJ sphere packings. For the hard-sphere systems, $f_R\left(N\right)$ can well be approximated by Gaussian probability density functions already for $R>1.5D$ (represented by solid circles instead of open squares), where $\lambda /D\approx 1.03$ and $\lambda /D\approx 0.937$ for the equilibrium or MRJ sphere packings, respectively.

Figure 10

The number variance ${\sigma }_N^2\left(R\right)$ for MRJ sphere packings as a function of the radius $R$ of the test sphere is compared to those of overlapping and equilibrium hard spheres. The solid black line shows the analytical curve for the overlapping spheres, which is proportional to $R^3$. The data for the equilibrium hard spheres agree well with a polynomial $b_2{R}^2+b_3{R}^3$ (solid blue line) but can clearly not be described by the fit of only a parabola (dashed line). This is in contrast to the MRJ data, which agrees well with a parabola (solid red line). Moreover, the extrapolation of the parabola is in perfect agreement with the results from MD simulations of MRJ packings with up to ${10}^6$ spheres; see Ref. [29].

Figure 11

A two-dimensional section through the spheres $B_R\left({\mathit{c}}_{\mathit{j}}\right)$ (solid line) and $B_r\left(\mathit{x}\right)$ (dashed line): The dashed red line indicates points at a distance $r$ from $\mathit{x}$ that lie inside sphere $j$; if the radii $r$ and $R$ and the distance $\delta :=\parallel \mathit{x}-{\mathit{c}}_j\parallel$ are given, the cosine of the angle $\omega$ follows from the law of cosines; this, in turn, allows for the computation of the area $A\left[\partial B_r\left(\mathit{x}\right)\cap B_R\left({\mathit{c}}_{\mathit{j}}\right)\right]$.

Figure 12

Two-body contributions $F_{sv}^{*}\left(r\right)$ to the surface-void correlation functions (rescaled by the specific surface $s$) for overlapping spheres (for which it is constant and equal to $\varphi =0.636$; the specific surface area is $s=2.21/D$), an equilibrium hard-sphere liquid ($\varphi =0.478, s=2.87/D$), and MRJ sphere packings ($\varphi =0.636, s=3.81/D$). For the MRJ packings, the dashed (green) line indicates the slope at $r=0$, which is strictly positive for the jammed sphere packings in contrast to the other two systems. The slope can be related to the mean number of contacts $\overline{z}$; see Eq. (B3).

Figure 13

Two-body contributions $F_{ss}^{*}\left(r\right)$ to the surface-surface correlation functions (rescaled by the square of the specific surface $s$) for overlapping spheres (for which it is constant unity; $\varphi =0.636, s=2.21/D$), an equilibrium hard-sphere liquid ($\varphi =0.478, s=2.87/D$), and MRJ sphere packings ($\varphi =0.636, s=3.81/D$). For the MRJ packings, the dashed (green) line shows the functional value at $r=0$, which is strictly positive for the jammed sphere packings in contrast to the equilibrium hard spheres. The value can be related to the mean number of contacts $\overline{z}$; see Eq. (B4). Moreover, for the MRJ packings, the value $F_{ss}^{*}\left(D\right)/s^2=0.9924\left(1\right)$ is close to unity (in contrast to the equilibrium hard-sphere liquid). The insets magnify $F_{ss}^{*}\left(r\right)/s^2$ at $r=D$, where the derivative of $F_{ss}^{*}$ is discontinuous for the MRJ sphere packings in contrast to the equilibrium hard spheres; see also the insets of Fig. 5 for the same finding at $r=2D$. Note that $F_{ss}^{*}\left(r\right)=F_{ss}\left(r\right)$ for $r>D$.