How dense can one pack spheres of arbitrary size distribution?

We present the first systematic algorithm to estimate the maximum packing density of spheres when the grain sizes are drawn from an arbitrary size distribution. With an Apollonian filling rule, we implement our technique for disks in 2d and spheres in 3d. As expected, the densest packing is achieved with power-law size distributions. We also test the method on homogeneous and on empirical real distributions, and we propose a scheme to obtain experimentally accessible distributions of grain sizes with low porosity. Our method should be helpful in the development of ultra-strong ceramics and high-performance concrete. Copyright c © EPLA, 2012 High-strength ceramics and ultra-high–performance concrete (UHPC) [1] require minimizing the porosity out of a compacted either reactive or sintered powder. Much research effort has been invested in the past to optimize this procedure and the most important factor turned out to lie in the adequate choice of the size distribution of the constituents. In fact, the broader this distribution, the smaller result the remaining voids. So, mixtures of very different grains with up to four orders of magnitude in grain size are typically used for UHPC. But what is the ideal partitioning? Which combined grain-size distribution would yield the most compact mixture? The key to answer this fundamental question posed by the practitioners is to be able to estimate the maximum density a given distribution can provide. This is precisely the aim of this letter. The maximal filling density has been studied for many specific types of packings [2] starting with the work of Fuller [3]. Besides the simple monodisperse case also exact results for some symmetric cases are known [4,5]. Various models have been proposed to deal with the superposition of two or three rather sharply peaked distributions, like the ones by Toufar et al. [6], Yu and Standish [7], or the various linear theories by De Larrard [8,9]. Also for continuous size distributions a hierarchical partitioning model was recently developed [10]. Most real grain-size distributions, used for dense packing, have a rather complicated shape (a)E-mail: [email protected] often being a superposition of various empirical functions. It is therefore of interest to develop a technique to obtain an upper bound for a packing having arbitrary distribution. Based on the insight that a completely space-filling packing of spheres, i.e., having unity volume density, can only be achieved with a generalized (random) Apollonian setup [11,12], we design a systematic technique to optimally fill the fines into the voids between larger grains and, by sweeping from the large end of any distribution, to finally obtain the highest density one could possibly attain with such distribution. Testing the technique on various real and artificial distributions we recover that power laws provide the highest densities. As illustrated in fig. 1, we discretize the distribution into bins by grain size. Grains are assumed to have spherical shape (disks in two dimensions) and are organized in each bin in the monodisperse closest-packing configuration. Then the gaps are filled with smaller ones, following an Apollonian packing. As rigorously proved by Tóth [13,14], in two dimensions, the most efficient monodisperse arrangement of disks is the hexagonal closest packing (hcp), with a density of ρhcp = 1/6π √ 3≈ 0.9069. With this arrangement, there are, per largest grain, two unitary voids to be filled with the traditional Apollonian packing (see fig. 1). Each bin b is characterized by four different parameters: the average radius, rb; the volume of grains, Vb; the effective volume, V b eff ; and the density of the arrangement of particles in the bin, ρb. Since we