Arrangements of spheres in transverse confinement
Much work has been done on the study of structures formed by both hard and soft spheres held in transverse confining potentials [1, 2, 3]. These structures self-assemble under the action of their confining potentials, forming interesting columnar structures. Here we will present an overview of these structures and their formation.
Firstly, we consider a horizontal line of contacting spheres held in a transverse cylindrically symmetric confining potential. This system buckles under compression, or when tilted away from the horizontal once a critical tilt angle is exceeded. Initially these buckled structures contain only nearest-neighbour contacts between spheres, however there is a structure at a critical point of compression (referred to as the doublet structure) at which next-nearest neighbours contacts form and more complex structures develop.
For the case of hard spheres at compressions below the doublet structure, it is possible to formulate a continuous description of the displacement profile formed by the buckled line of spheres. The profiles are well described by numerical solutions of a second-order differential equation. These solutions are discussed in detail, and shown how they may be approximated by the well-known Jacobi, Whittaker and Airy functions.
Beyond this doublet structure, in the case of both hard and soft spheres, the more complicated structures emerge. These structures are explored computationally, employing the Morse-Witten model to study the assembly of the soft sphere structures.
 Non-Equilibrium Self-Assembly of Monocomponent and Multicomponent Tubular Structures in Rotating Fluids, Taehoon Lee, Konrad Gizynski, Bartosz A. Grzybowski, https://doi.org/10.1002/adma.201704274
 Equilibrium configurations of hard spheres in a cylindrical harmonic potential, J. Winkelmann, A. Mughal, D. Weaire, S. Hutzler, https://doi.org/10.1103/PhysRevE.99.020602
 1D Colloidal chains: recent progress from formation to emergent properties and applications, Xinlong Gan, Andreas Walther, https://doi.org/10.1039/D2CS00112H