Densest columnar structures of hard spheres from sequential deposition

The rich variety of densest columnar structures of identical hard spheres inside a cylinder can surprisingly be constructed from a simple and computationally fast sequential deposition of cylinder-touching spheres, if the cylinder-to-sphere diameter ratio is $D\in [1,2.7013]$. This provides a direction for theoretically deriving all these densest structures and for constructing such densest packings with nano-, micro-, colloidal, or charged particles, which all self-assemble like hard spheres.

Figure 1

The densest structures and their phyllotactic diagrams for various cases of $D\in (1,2]$: (a) the achiral planar zigzag at $D=1.800$, (b) the chiral single helix at $D=1.940$, (c) the chiral double helix at $D=1.992$, and (d) the achiral doublets, which have two spheres located on one $z$ position, at $D=2.000$. For each case, the required periodicity $\left|\mathbf{V}\right|=(D-1)\pi$ along $s$ is indicated by a solid arrow.

Figure 2

Plot of the vertical separation $\Delta z$ against the arc length $\Delta s$ for spheres in contact at various values of the diameter ratio $D$. The regimes $D\in [1,2)$ and $D⩾2$ correspond to the absence and presence of a solution for $\Delta z=0$, respectively. As $D\to \infty$, the relation between $\Delta z$ and $\Delta s$ approaches ${\left(\Delta z\right)}^2+{\left(\Delta s\right)}^2=1$ for the 2D packing of circular disks.

Figure 3

Plot of the maximum volume fraction $V_{F,\max }$ against the diameter ratio $D$, showing a complete overlap of data points between this deposition algorithm and simulated annealing (data supplied by Mughal [11]). The inset shows that the differences in $V_{F,\max }$ between the two algorithms are of the order of ${10}^{-3}$, i.e., less than $1\%$ of percentage differences, and thus comparable to the numerical uncertainties in the simulations. The occasional peaks in $|\Delta V_{F,\max }|$ are due to the relatively larger numerical uncertainties in the proximity of a structural transition.

Figure 4

(a) Sequence of structures as $D$ increases from 2.040 to 2.155: 321 symmetric at $D=2.040$ $\to$ type 2 line slip, e.g., at $D=2.010$ $\to$ type 3 line slip, e.g., at $D=2.142$ $\to$ 330 symmetric at $D=2.155$. (b) Two examples of periodicity breaking at $D=2.350$ and $D=2.620$; in each case, the region that does not belong to the periodic structure is highlighted with a rectangle.

Figure 5

Simultaneous deposition of $u=1,2,3$, and 4 spheres at regular spacings of $\phi$ in each deposition step. All the metastable curves for lower density packings correspond to cases of $u>1$. The symmetric structures are indicated along with their phyllotactic notations.