Fig. 1: Experimental setup and regimes of the magnetocapillary interaction potential.

a Schematics of two magnetocapillary disks of radius a and mass m each embedded with a small permanent magnet of magnetic dipole M. b In certain parameters regimes, two magnetocapillary disks find an equilibrium state defined by a finite spacing. The scale bar indicates 2 mm. c Sample magnetocapillary pairwise interaction potentials U_p versus interdisk spacing l. The three characteristic regimes are shown here (corresponding to the sequel snapshots in the inset): the locally attractive or Cheerios (red) regime, the mermaid (violet) regime, and the fully repulsive (blue) regime. For all three cases a = 3 mm and M = 9 A ⋅ cm² with decreasing masses of m₁ = 0.11 g, m₂ = 0.092 g, and m₃ = 0.080 g, respectively. Note that if the disks are started sufficiently far apart, they will always repel, as the magnetic repulsion decays at a slower rate than the capillary attraction. The circle (◯) and diamond (◇) markers indicate the locations of energy extrema for stable and unstable equilibria when l > 0, respectively. d Interaction potential for the sample mermaid regime. Inset: The experiment shows that when the mean spacing in an annular 1D array drops below l_cr, disks begin to spontaneously pair.