Sphere Packing - Unequal Sphere Packing

Unequal Sphere Packing

Many problems in the chemical and physical sciences can be related to packing problems where more than one size of sphere is available. Here there is a choice between separating the spheres into regions of close-packed equal spheres, or combining the multiple sizes of spheres into a compound or interstitial packing. When many sizes of spheres (or a distribution) are available, the problem quickly becomes untractable, but some studies of binary hard spheres (two sizes) are available.

When the second sphere is much smaller than the first, it is possible to arrange the large spheres in a close-packed arrangement, and then arrange the small spheres within the octahedral and tetrahedral gaps. The density of this interstitial packing depends sensitively on the radius ratio, but in the limit of extreme size ratios, the smaller spheres can fill the gaps with the same density as the larger spheres filled space.

When the smaller sphere has a radius greater than 0.4142 of the radius of the larger sphere, it is no longer possible to fit into even the octahedral holes of the close-packed structure. Thus, beyond this point, either the host structure must expand to accommodate the interstitials (which compromises the overall density), or rearrange into a more complex crystalline compound structure. Structures are known which exceed the close packing density for radius ratios up to 0.659786.