Clustering of planar ring-like charged molecules in constrained spaces

We obtain an exact expression for the potential energy of a system of identical uniformly charged circular ring-like charged molecules that are coplanar with each other. It is found that the potential energy develops a cusp at a particular distance that corresponds to the two uniformly charged circular rings touching each other at one point. We use the resulting potential to study systems of an arbitrary number of such molecules in constrained spaces such as quasi one-dimensional spaces. It is found that such ring-shaped molecules tend to aggregate/cluster together rather than settling for a state with uniform density in which the uniformly charged circular rings occupy (in average) more or less the same area.