Sphere packing in a cylinder is a three-dimensional packing problem with the objective of packing a given number of identical spheres inside a cylinder of specified diameter and length. 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.
Appearance in science
Columnar structures appear in various research fields on a broad range of length scales from metres down to the nanoscale. On the largest scale, such structures can be found in botany where seeds of a plant assemble around the stem. On a smaller scale bubbles of equal size crystallise to columnar foam structures when confined in a glass tube. In nanoscience such structures can be found in man-made objects which are on length scales from a micron to the nanoscale.
Botany
Columnar structures were first studied in botany due to their diverse appearances in plants. 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, viruses, microtubules, and the notochord of the zebra fish.
One of the largest flowers where the berries arrange in a regular cylindrical form is the titan arum. This flower can be up to 3m in height and is natively solely found in western Sumatra and western Java.
On smaller length scales, the berries of the Arum maculatum form a columnar structure in autumn. Its berries are similar to that of the corpse flower, since the titan arum is its larger relative. However, the cuckoo-pint is much smaller in height (height ≈ 20 cm). The berry arrangement varies with the stem to berry size.
Another plant that can be found in many gardens of residential areas is the Australian bottlebrush. It assembles its seed capsules around a branch of the plant. The structure depends on the seed capsule size to branch size.
Foams
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. By putting the resulting foam column under forced drainage (feeding it with surfactant solution from the top), the foam can be adjusted to either a dry (bubbles shaped as polyhedrons) or wet (spherical bubbles) structure.
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.
In the zigzag structure the bubbles are stacked on top of each other in a continuous w-shape. For this particular structure a moving interface with increasing liquid fraction was reported by Hutzler et al. in 1997. This included an unexpected 180° twist interface, whose explanation is still lacking.
The first experimental observation of a line-slip structure was discovered by Winkelmann et al. in a system of bubbles.
Further discovered structures include complex structures with internal spheres/foam cells. Some dry foam structures with interior cells were found to consist of a chain of pentagonal dodecahedra or Kelvin cells in the centre of the tube. For many more arrangements of this type, it was observed that the outside bubble layer is ordered, with each internal layer resembling a different, simpler columnar structure by using X-ray tomography.
Nanoscience
Columnar structures have also been studied intensively in the context of nanotubes. Their physical or chemical properties can be altered by trapping identical particles inside them. These are usually done by self-assembling fullerenes such as C60, C70, or C78 into carbon nanotubes, but also boron nitride nanotubes
Such structures also assemble when particles are coated on the surface of a spherocylinder as in the context of pharmaceutical research. Lazáro et al. examined the morphologies of virus capsid proteins self-assembled around metal nanorods. Drug particles were coated as densely as possible on a spherocylinder to provide the best medical treatment.