Packing Spheres Tightly: Influence of Mechanical Stability on Close-Packed Sphere Structures

Many experiments and simulations of packings of monodisperse hard spheres report a dominance of the face-centered cubic structure in the hexagonally close-packed limit, even though it has no significant energetic or entropic gain over other close-packed configurations. Combining simulations and experiments, we demonstrate that a simple mechanical instability which occurs during the packing process may play an important role in selecting the face-centered cubic structure over other close-packed alternatives. Our argument is supported by detailed quantitative analyses of key configurations in sphere packings and highlights the importance of the packing dynamics. The proposed mechanism is elementary and should therefore play a role in a wide range of sphere systems.

Figure 1

(a) Layer of triangularly arranged spheres. (b) $A$-$B$-$A$-$B$-type layering of the hcp, and (c) $A$-$B$-$C$-type layering of the fcc packing, expressed by color coding.

Figure 2

(a) Setup of the simulations conducted. (b) Averaged distribution of spheres over height at discrete times, showing the filling of the layers. Broken lines indicate the theoretical positions of the ordered layers. (c) Number of ordered spheres $n_{\mathrm{fcc}}$ and $n_{\mathrm{hcp}}$ in the third and fourth layers averaged over 100 runs. (d) Preference $P$ of fcc ordering in the third and fourth layers and for individual spheres.

Figure 3

(a) Geometrical situation and force transfer in fcc and hcp with coloring as in Fig. 1. (b) Stability map of the fcc and hcp pyramid under tip force and drainage. (c) Experimental observation of stable fcc and the rearranging hcp pyramids of bubbles.

Figure 4

(a) Frequency of occurrence of the angle between the $x$ direction and path of permanently ejected spheres (dashed arrow) in layer three. The three dominant angles correspond to the three valleys in layer two, identified by three dotted arrows. (b) Frequency of occurrence of the angle between the ejecting path (dashed arrow) and direction of force (dash-dotted arrow) exerted from the sphere from the fourth layer on the ejected sphere in the third layer.