Phyllotactic Description of Hard Sphere Packing in Cylindrical Channels

We develop a simple analytical theory that relates dense sphere packings in a cylinder to corresponding disk packings on its surface. It applies for ratios $R=D/d$ (where $d$ and $D$ are the diameters of the hard spheres and the bounding cylinder, respectively) up to $R=1+1/\sin (\pi /5)$. Within this range the densest packings are such that all spheres are in contact with the cylindrical boundary. The detailed results elucidate extensive numerical simulations by ourselves and others by identifying the nature of various competing phases.

Figure 1

(a) Symmetric packing $[5,4,1]$—the black and red arrows indicate the periodicity $\mathbf{V}$ and primitive lattice vectors, respectively; (b) square packing $[5,4]$; (c) symmetric packing $[9,5,4]$ which is connected to the structure $[5,4,1]$ by an affine shear; (d) a line-slip structure between the symmetric packings $[6,4,2]$ and $[5,4,1]$.

Figure 2

Area fraction of disk packings on the plane which are consistent with wrapping onto a cylinder of diameter $D^{\prime }=|V|/\pi$. Asymmetric packings are shown by (red) dashed curves; line-slip solutions are (black) continuous curves.

Figure 3

Comparison of simulated and analytic volume fractions for the densest packings—upper and lower curves, respectively. Dotted lines are a guide for the eye to the detailed correspondence. In the analytic results, the line-slip structures are identified by black dashed lines, and the red dashed-dotted lines denote asymmetric packing structures. The numbers 1, 2, and 3 (see text) denote the type of line slip observed (upper curve) or predicted (lower curve)—with $1\setminus 2$ denoting a degenerate case.

Figure 4

Simulated 3D structures for various values of $R$, with the corresponding rolled out phyllotactic diagram below (the red arrow denotes the periodicity vector).