Densest helical structures of hard spheres in narrow confinement: An analytic derivation

The emergence of helicity from the densest possible packings of equal-sized hard spheres in narrow cylindrical confinement can be understood in terms of a density maximization of repeating microconfigurations. At any cylinder-to-sphere diameter ratio D∈(1+3/2,2), a sphere can only be in contact with its nearest and second nearest neighbors along the vertical z-axis, and the densest possible helical structures are results of a minimized vertical separation between the first sphere and the third sphere for every consecutive triplet of spheres. By considering a density maximization of all microscopic triplets of mutually touching spheres, we show, by both analytical and numerical means, that the single helix at D∈(1+3/2,1+43/7) corresponds to a repetition of the same triplet configuration and that the double helix at D∈(1+43/7,2) corresponds to an alternation between two triplet configurations. The resulting analytic expressions for the positions of spheres in these helical structures could serve as a theoretical basis for developing novel chiral materials.