Analytical interaction potentials for disks in two dimensions

Compact analytical forms are derived for the interactions involving thin disks in two dimensions using an integration approach. These include interactions between a disk and a material point, between two disks, and between a disk and a wall. Each object is treated as a continuous medium of materials points interacting by the Lennard-Jones 12-6 potential. By integrating this potential in a pairwise manner, expressions for the potentials and resultant forces between extended objects are obtained. All the results are validated with numerical integrations. The analytical potentials are implemented in LAMMPS and used to simulate two-dimensional suspension of disks with an explicit solvent modeled as a Lennard-Jones liquid. In monodisperse disk suspensions, a disorder-to-order transition of disk packing is observed as the area fraction of disks is increased or as the solvent evaporates. In bidisperse disk suspensions being rapidly dried, stratification is found with the smaller disks enriched at the evaporation front. Such "small-on-top" stratification echoes the similar phenomenon occurring in three-dimensional polydisperse colloidal suspensions that undergo fast drying. These potentials can be applied to a wide range of two-dimensional systems involving disk-like objects.