Finding the grain size distribution that produces the densest arrangement in frictional sphere packings: Revisiting and rediscovering the century-old Fuller and Thompson distribution

By means of discrete-element methods, we investigate the joint effects of the grain size distribution (GSD) and contact friction on the structure of three-dimensional samples composed of spherical grains. Specifically, we compress these systems isotropically until jamming and then analyze their structure in terms of density, connectivity, coefficients of uniformity and curvature, and parameters of grading entropy. Our study focuses on power-law GSDs and particularly on the Fuller and Thompson distribution, proposed over a century ago. First, we show that, among the set of GSDs investigated, this particular distribution produces the densest and best-connected systems, falsifying a conjecture recently posed in the literature. Second, we find that the jamming packing fraction can be accurately predicted as a function of simple descriptors of the GSD, but among these descriptors the granular entropy concept proves to be the most useful. This allows for an alternative interpretation of both jamming and grading entropy concepts. Finally, we compare the Fuller and Thompson distribution with two well-known GSDs: that of the Apollonian sphere packing and that towards which granular systems evolve after intensive grain fracturing. Surprisingly, we find that these three GSDs are practically coincident in the limit of large size spans, despite having been introduced or discovered in different scientific contexts (i.e., engineering, mathematics, and earth sciences, respectively).