Jammed hard-sphere hcp crystals permeated with trivacancy tunnels

Supported by simple table-top experiments involving stackings of ball bearings and theoretical analysis, we have discovered crystal packings of identical hard spheres that are permeated by a high concentration of large tunnels and yet are jammed (mechanically stable). We show that starting with a strictly jammed hexagonal close-packed (hcp) crystal of identical hard spheres, removal of certain subsets of those spheres can produce mechanically stable vacancy arrangements involving compact (equilateral triangle) trivacancies such that they produce linear trivacancy tunnels. These tunnels can extend over the entire macroscopic length of the hcp medium, and their width is sufficient to allow contained "test" hard spheres with diameters less than 5 - 1 = 1.23606 ... to migrate over that entire length without contacting the static tunnel-wall spheres. A search for the stable (strictly jammed) periodic framework that hosts the highest density of parallel trivacancy tunnels has identified a structure exhibiting a packing fraction φ = π / 32 = 0.55536 ..., which is equal to 3/4 of the maximum monovalent sphere packing fraction φ max = π / 18 = 0.74048 .... In that periodic arrangement, filling the interior of the contained tunnels with movable unit-diameter spheres may approach the greatest possible "rattler" density within jammed monovalent sphere systems subject to periodic boundary conditions. It will be of interest to study the physical and chemical properties of these anisotropic porous crystal structures. Our findings may have practical implications for engineered separation and catalytic processes.