Autocorrelation functions of granular media with free arrangement of spheres, spherical shells or ellipsoids

Abstract

The decomposition of natural media, such as snow and soil, into grains or pores of variable size and shape is an ambiguous task, depending on how the grains and their size are defined. Volume scattering of electromagnetic waves, adsorption of molecules on the grain surface, percolation of a fluid through the pores, or sedimentation of grains in a fluid do not depend on such definitions; they can be characterized by more general structural parameters, examples of which are the specific surface s or the correlation length pc, both being related to the spatial autocorrelation function (SAF). Therefore, physically meaningful structure information can be obtained even without specifying the grains, by adopting pc as an effective size. In order to get an understanding of how pc is related to the geometrical dimensions of simple particles, we compute the SAF and pc for single spheres, spherical shells, and for isotropically oriented ellipsoids; exact and approximate formulas are derived, plotted, and exact expressions of pc are found. A result is that pc is not related to the maximum particle extent, but in all cases studied, pc is close to the minimum, characteristic extent of the grain. The SAFs can be applied to the computation of volume scattering in the weak and strong fluctuation theory, respectively. Furthermore, it can be shown, based on an assumption of the free arrangement of impenetrable granules, that the SAF of an irregular medium is identical to the SAF of its particles. Thus the single-particle correlation is the dominant structure in media consisting of irregular granules. This result is in contrast to systems of spatially correlated particles, such as atoms in crystals or molecules in liquids. Although this work was driven by the need for a quantitative interpretation of remote sensing data of snow, the results may be applied to other disciplines as well.