Interactions and pattern formation in a macroscopic magnetocapillary SALR system of mermaid cereal

Abstract

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 condensed matter 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.

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.

Fig. 2. Magnetocapillary interactions.

a The phase diagram for different regimes as a function of capillary Bond number Bo and magnetocapillary number $\mathcal{M}$. The left triangles (⊲) show the Cheerios regime, the right triangles (⊳) show the repulsive regime, and the circles (◯) show the mermaid regime, as observed in the experiments. The symbol sizes correspond to three disk sizes, a = 2.5 mm, a = 3 mm, and a = 3.5 mm, hence three different Bo in ascending order. The phase diagram is color-coded based on $l_{\mathrm{eq}}^{*}$ computed from the model, while the circles are color-coded based on $l_{\mathrm{eq}}^{*}$ value measured in the experiments. The symbols with solid outlines represent magnets with M = 9 A ⋅ cm² while the dashed-line outlined symbols represent disks with permanent magnets of M = 2.25 A ⋅ cm². While Bo is only varied by a in our experiments, $\mathcal{M}$ is varied with a, m, and M. The dashed lines indicate the model prediction for boundaries between different regimes. The dimensional parameters and equilibrium spacing for each experiment is provided in Supplementary Note 2. b The equilibrium configuration in the experiment for fully repulsive disks with $\mathcal{M}=2.1$, Bo = 1.52, and ϕ = 0.1425 (a = 3 mm, m = 0.073 g, M = 9 A ⋅ cm⁻², R = 6 cm, N = 57), also marked with the white border right triangle in the phase diagram. c The equilibrium configuration in the experiments for mermaid disks with $\mathcal{M}=1.22$, Bo = 1.52, and ϕ = 0.1425 (a = 3 mm, m = 0.092 g, M = 9 A ⋅ cm⁻², R = 6 cm, N = 57), also marked with the white border circle in the phase diagram.

Fig. 3. 2D pattern formation in the magnetocapillary mermaid regime.

a Patterns formed by magnetocapillary mermaid disks with $\mathcal{M}$ = 1.22 and Bo = 1.52 (a = 3 mm, M = 9 A ⋅ cm⁻², m = 0.092 g) for varying areal packing fraction, ϕ. The confinement radius R is 6 cm. The top row illustrates the experimental snapshots starting from the hexagonal lattice state to clusters, stripes, and labyrinths. The bottom row shows sample model results color-coded based on corresponding ϕ values in the experiments. The black scale bar shows 2 cm. b Simulation results of mermaid disks with the same parameters as (a) in a large square domain of 17 cm × 17 cm with a periodic boundary condition for varying packing fraction, ϕ. c The average radial distribution function RDF, $\overline{g}\left(r\right)$, from the periodic boundary condition simulations in a square domain of 17 cm × 17 cm as a function of $\overline{l}$ averaged over 25 independent simulations for varying ϕ. The error bars indicate one standard deviation of the simulation results.

Fig. 4. External control of equilibrium pattern.

a When a magnetic strip is placed below the bath, the disks are pulled downwards, increasing the capillary attraction force. If tuned correctly, fully repulsive disks can transition to mermaid disks under this additional force. b l_eq as a function of m from the magnetocapillary model for a = 2.5 mm, and M = 9 A ⋅ cm². The dashed line shows the physical value of m while the dotted dashed-line shows the effective m after applying the external magnetic force based on the l_eq value extracted from the experiments shown in the inset. c The hexagonal lattice state (on the left) transitions to the striped pattern (right) by adding the magnetic strip below the bath for disks with a = 2.5 mm, M = 9 A ⋅ cm², and m = 0.09 g (Bo = 1.05, $\mathcal{M}$ = 3.5) and ϕ = 0.18. The scale bar indicates 2 cm. d Sample magnetocapillary simulations for a = 2.5 mm, and M = 9 A ⋅ cm² showing phase transition from hexagonal lattice to stripes when m changes from 0.09 g to 0.115 g (Bo = 1.05 with $\mathcal{M}$ decreasing from 3.5 to 1.9) for ϕ = 0.18.