  When particles are deposited at a fluid interface they tend to aggregate by
capillary attraction to minimize the overall potential energy of the system. In
this work, we embed floating millimetric disks with permanent magnets to
introduce a competing repulsion effect and study their pattern formation in
equilibrium. The pairwise energy landscape of two disks is described by a
short-range attraction and long-range repulsion (SALR) interaction potential,
previously documented in a number of microscopic systems. Such competing
interactions enable a variety of pairwise equilibrium states, including the
possibility of a local minimum energy corresponding to a finite disk spacing.
Two-dimensional (2D) experiments and simulations in confined geometries
demonstrate that as the areal packing fraction is increased, the dilute
repulsion-dominated lattice state becomes unstable to the spontaneous formation
of localized clusters, which eventually merge into a system-spanning striped
pattern. Finally, we demonstrate that the equilibrium pattern can be externally
manipulated by the application of a supplemental vertical magnetic force that
remotely enhances the effective capillary attraction.
