Geometrical properties of disordered packings of hard disks

We present experimental and theoretical results for geometrical properties of 2D packings of disks. We were mainly interested in the study of mixtures with disk size distribution which are of more practical interest than equal disks. Average geometrical properties, such as packing fraction or coordination number do not depend on the composition of the mixture, contrary to what would be expected from 3D experiments. We show the existence of a local order in the relative positions of grains with different sizes ; this local order may modify the physical properties of the packing. An approximate theoretical expression for the packing fraction c of 2D close packings is given. It implies the knowledge of the average area of quadrilaterals of the network drawn from the real contacts only. For equal disk disordered packings, it yields the limit c = 03C02/12~ 0.822. J. Physique 47 (1986) 1697-1707 OCTOBRE 1986, Classification Physics Abstracts 61.40 81.20E 81.30H 81.90 Except by numerical simulations, it is difficult to study the geometrical properties of hard sphere packings, particularly if we consider the real contacts between grains, which are important for transport or mechanical properties of granular materials. Therefore, besides its interest for structural modelling in two dimensions, the study of hard disk packings on a plane can be a good tool for easier approach and understanding of the geometry of 3D packings. Such a 2D study is especially interesting when the disk size distribution forbids a regular pavement of the plane : experimental equal disk packings, when they are dense, exhibit ordered domains, then their structure depends on the contruction mode. In this paper we study the effect of disk size distribution on geometrical and structural properties of 2D disordered packings. 1. Order-disorder in 2D hard disk packings. It is difficult to give a precise and formal definition of a random packing as it would not easily take into (+) Permanent address : C.E.N. Saclay, Service de Physique Th6orique, 91191 Gif-sur-Yvette Cedex, France. account the steric exclusions, which, for compact packings, may lead to long range geometrical correlations [1]. On the other hand, generating points according to a Poissonian distribution law is not difficult. Starting from this remark, Stillinger et al. [2] have proposed an algorithm to construct a random packing of 2D equal disks : a set of points are randomly distributed on a plane ; then, considering these points as centres, equal disks are grown. During the expansion process, any couple of overlapping disks is shifted symmetrically along its axis so to realize a tangential contact. Triplets and more complicated overlappings are rearranged following a similar procedure which minimizes the sum of the squares of the displacements. According to these authors, the final result should be a dense random packing and not the dense ordered packing (triangular lattice). On this basis, in a beautiful experiment, Quickenden and Tan [3] have built equal disk packings in the following way : small disks are put at random on a plane isotropically stretched rubber sheet, in very loose packing. Then, one lets the rubber shrink and a photograph is taken. The sheet is stretched again and the process is repeated n times, the disks at the Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198600470100169700