Theory of cylindrical dense packings of disks

We have previously explored cylindrical packings of disks and their relation to sphere packings. Here we extend the analytical treatment of disk packings, analyzing the rules for phyllotactic indices of related structures and the variation of the density for line-slip structures, close to the symmetric ones. We show that rhombic structures, which are of a lower density, are always unstable, i.e., can be increased in density by small perturbations.

Figure 1

Left: A triangular lattice with a periodicity vector vector $\mathbf{V}$ between two lattice points, shown by a black arrow. Right: The excised section wrapped onto a cylinder of diameter $\left|\mathbf{V}\right|/\pi$.

Figure 2

A symmetric triangular lattice on to which is inscribed the periodicity vector $\mathbf{V}$ with phyllotactic indices [$l=5$, $m=3$, $n=2$]. By convention the periodicity vector is confined to the shaded region (see text). The diagram shows the Cartesian unit vectors $\widehat{\mathbf{x}}$ and $\widehat{\mathbf{y}}$. The unit primitive lattice vectors of the triangular lattice (${\mathbf{a}}_{\mathbf{i}}$) and the unit basis vectors (${\mathbf{b}}_{\mathbf{i}}$) used in the text are shown by small red and black arrows, respectively. Also shown is an equilateral triangle: the phyllotactic indices are related to the length of the triangle sides and segments thereof, as indicated.

Figure 3

The three disk packings discussed here are as follows: (a) symmetric packings (where the disks are centered on a triangular lattice, see Fig. 3, (b) affine structures, and (c) line-slip packings. The black arrow is the periodicity vector $\mathbf{V}$. The region between the dashed lines can be excised and wrapped on to a cylinder of the appropriate diameter.

Figure 4

A rhombic lattice onto which is inscribed the periodicity vector $\mathbf{V}$ with phyllotactic indices $[p=3,q=2]$. The primitive lattice vectors ${\widehat{\mathbf{u}}}_{\mathbf{0}}=(1,0)$ and ${\widehat{\mathbf{v}}}_{\mathbf{0}}=[cos\left(\theta \right),sin\left(\theta \right)]$ are shown by the red arrows.

Figure 5

(a) A triangular lattice onto which is inscribed a periodicity vector $\mathbf{V}$ with phyllotactic indices $[l,m,n]$. A rhombic lattice can be formed by disregarding the lattice lines in the (b) $\widehat{{\mathbf{a}}_{\mathbf{1}}}$ direction, (c) $\widehat{{\mathbf{a}}_{\mathbf{2}}}$ direction, or (d) $\widehat{{\mathbf{a}}_{\mathbf{3}}}$ direction. The rules that connect the two sets of indices are given in the text. (e) If contact BD is broken and BD is lengthened, a new contact AC is eventually established.

Figure 6

(a (a) An unperturbed rhombic lattice (with sides of length 1) onto which is inscribed a periodicity vector ${\mathbf{V}}_{\mathbf{0}}$ with phyllotactic indices $[p,q]$. The angle between the rhombic lattice vectors ${\widehat{\mathbf{u}}}_0$ and ${\widehat{\mathbf{w}}}_0$ is given by ${\theta }_0$; the angle between the $\mathbf{V}$ and ${\widehat{\mathbf{u}}}_0$ is denoted by $\gamma$. (b) A perturbation of the rhombic lattice. The lattice is divided into $p$ strips and within each strip the rhombic angle deviates from ${\theta }_0$. In fact, such a perturbation would involve a rotation of the resulting structure which, for convenience, is not shown in the image.

Figure 7

A plot of $F\left({\theta }_0\right)$ over the range $\pi /3\le \theta \le 2\pi /3$, which determines stability according to Eq. ((15)).

Figure 8

(a) Symmetric packing $[5,4,1]$, (b) line-slip in the $-\widehat{{\mathbf{a}}_{\mathbf{2}}}$ direction, and (c) symmetric packing $[6,4,2]$. Note that for the line-slip the blue disks remain fixed while the yellow disks are displaced. The region between the dashed lines, in each case, can be excised and wrapped onto a cylinder of the appropriate diameter.

Figure 9

Diagram used to analyze the effect of a line-slip. A change in the vertex angle by $\delta \theta$ leads to a variation in the length of the periodicity vector $\mathbf{V}$ and the density.

Figure 10

A plot of line-slip densities (black dotted curves) against the first order approximations (red lines) close to the symmetric packing $[5,4,1]$.

Figure 11

Expansion up to second order for the density of line-slip packings close to the symmetric structure $[5,4,1]$. The solid lines show the densities of the six line-slip structures while the corresponding dashed line of the same color shows the second order expansion given by Eq. ((17))