Paper detail

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 correlation functions that can be rigorously related to 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 contacts between spheres in the MRJ packings translate 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. These packings are hyperuniform, which means that density fluctuations on large length scales are anomalously suppressed. Moreover, 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, we will evaluate bounds and estimates of a host of different physical properties of the MRJ sphere packings based on the structural characteristics analyzed in this paper.