Sphere packing in a cylinder
For cylinders with diameters on the same order of magnitude as the spheres, such packings result in what are called columnar structures.
These problems are studied extensively in the context of biology, nanoscience, materials science, and so forth due to the analogous assembly of small particles (like cells and atoms) into cylindrical crystalline structures.
The book "Columnar Structures of Spheres: Fundamentals and Applications"[1] serves as a notable contributions to this field of study.
Authored by Winkelmann and Chan, the book reviews theoretical foundations and practical applications of densely packed spheres within cylindrical confinements.
Columnar structures appear in various research fields on a broad range of length scales from metres down to the nanoscale.
On a smaller scale bubbles of equal size crystallise to columnar foam structures when confined in a glass tube.
[2] D'Arcy Thompson analysed such arrangement of plant parts around the stem in his book "On Growth and Form" (1917).
But they are also of interest in other biological areas, including bacteria,[3] viruses,[4] microtubules,[5] and the notochord of the zebra fish.
[6] One of the largest flowers where the berries arrange in a regular cylindrical form is the titan arum.
On smaller length scales, the berries of the Arum maculatum form a columnar structure in autumn.
A further occurrence of ordered columnar arrangement on the macroscale are foam structures confined inside a glass tube.
They can be realised experimentally with equal-sized soap bubbles inside a glass tube, produced by blowing air of constant gas flow through a needle dipped in a surfactant solution.
[8] Due to this simple experimental set-up, many columnar structures have been discovered and investigated in the context of foams with experiments as well as simulation.
Many simulations have been carried out using the Surface Evolver to investigate dry structure or the hard sphere model for the wet limit where the bubbles are spherical.
Lazáro et al. examined the morphologies of virus capsid proteins self-assembled around metal nanorods.
These microrods are created by densely packing silica colloidal particles inside cylindrical pores.
By solidifying the assembled structures the microrods were imaged and examined using scanning electron microscopy (SEM).
It is used to describe arrangements of leaves of a plant, pine cones, or pineapples, but also planar patterns of florets in a sunflower head.
[22][1] The differences between uniform and line-slip structures are marginal and difficult to spot from images of the sphere packings.
Thus, each line slip is related to two adjacent uniform structures, one at a higher and one at a lower diameter ratio
[22][1][24] Winkelmann et al. were the first to experimentally realise such a structure using soap bubbles in a system of deformable spheres.
[10] Columnar structures arise naturally in the context of dense hard sphere packings inside a cylinder.
In-between these discrete diameter ratios are the line slips at a lower packing density.
The rich variety of such ordered structures can also be obtained by sequential depositioning the spheres into the cylinder.
Mughal et al. also discovered that such structures can be related to disk packings on a surface of a cylinder.
using linear programming and discovered 17 new dense structures with internal spheres that are not in contact with the cylinder wall.
[26] A similar variety of dense crystalline structures have also been discovered for columnar packings of spheroids through Monte Carlo simulations.
A further dynamic method to assemble such structures was introduced by Lee et al.[28] Here, polymeric beads are placed together with a fluid of higher density inside a rotating lathe.
With increasing rotational speed, the centripetal force then pushes the fluid outwards and the beads toward the central axis.
A comprehensive theory to this experiment was developed by Winkelmann et al.[29] It is based on analytic energy calculations using a generic sphere model and predicts peritectoid structure transitions.
