Figure 1

(a) Sketch of the experimental setup. A Petri dish filled with a liquid is placed at the center of horizontal Helmholtz coils. When a current $i$ is injected in the coils, a vertical and uniform magnetic field $B$ is produced through the air-liquid interface. There, floating objects were placed on the liquid surface before switching the field on. (b) Pictures of typical configurations obtained with an increasing number $N$ of floating magnetic objects for a strong magnetic field ($\mathrm{Bo}=2.05$; see text). Large structures exhibiting a 5-fold symmetry are often encountered. (c) Four pictures of $N=28$ floating bodies experiencing two “on-off” cycles. Starting from an ordered structure under a magnetic field (left picture), the system collapses when the field is switched off. Thereafter, the field is again increased and again switched of: The initial ordered structure is not recovered.Reuse & Permissions

Figure 2

Illustration of forces acting on the beads in our experiments. Top: Interaction between two floating bodies at the air-liquid interface. Due to wetting of the bodies, deformations of the air-liquid interface implies the existence of a horizontal attractive force $F_{\gamma }$. The magnetization of each sphere induces a repulsive magnetic force $F_m$. The bodies are found at some equilibrium interdistance $r^{☆}$ where $F_{\gamma }=-F_m$. Bottom: On a curved surface which is slightly vibrated (reduced acceleration $\alpha \approx 1$), it is possible to determine the magnetic Bond number ${\mathrm{Bo}}_m$ characterizing the magnetic repulsion between beads by measuring the competition between $F_m$ and the attractive force $F_a$ due to surface curvature (see text).Reuse & Permissions

Figure 3

Dimensionless equilibrium interdistance $r^{☆}/D$ between two floating magnetized spheres as a function of the control parameter $\sqrt{{\mathrm{Bo}}_m}=i/i_0$. Each dot represents an experiment. The gray region is forbidden since it corresponds to overlapping spheres. Thick curves correspond to a fit of the data with a model proposed in the text. Arrows indicate the way the system behaves around the hysteresis loop in the model [Eq. (7)]. Vertical lines corresponds to separation events of beads in repeated experiments, providing a broad distribution of bead detachment events. Numbered arrows are discussed in the text.Reuse & Permissions

Figure 4

The continuous curves represent the attractive interaction potential $U_{\gamma }$ [Eq. (3)] as a function of the dimensionless interdistance $x=r/D$ ranging between 1 (contact) and 4. The dots represent the approximated potential [Eq. (4)] as proposed in the main text. The agreement is excellent in the considered range of $x$ values.Reuse & Permissions

Figure 5

The dimensionless interaction potential $U/mgD$ as a function of the dimensionless interdistance $r/D$ for different strengths of the magnetic repulsion. The parameters of Eq. (6) are fixed by the fit of the data presented in Fig. 3. Depending on ${\mathrm{Bo}}_m$, a primary and a secondary minimum can be observed. From top to bottom curves, one has $i/i_0=$ 0.50, 0.45, 0.40, 0.35, and 0.30.Reuse & Permissions

Figure 6

Dimensionless equilibrium distances $r_{jk}^{☆}/D$ for $N=3$ floating spheres, as a function of the control parameter $i/i_0$ for a typical experiment. Each symbol represents a pair. The gray region corresponds to overlapping spheres. Thick curves represent the behavior obtained numerically when the model [Eq. (6)] is implemented for 3 particles. Numbered arrows indicate relevant steps discussed in the main text. Typical configurations are shown. Near arrow 1, repulsion dominates capillary attraction. One observes three beads located at the vertices of a regular triangle. Near arrow 3, three beads come into contact. As illustrated, a nonsymmetrical configuration is obtained in most cases, and this remains unchanged for a vanishing magnetic field. Near arrow 5, two different scenarios exist: either complete detachment restoring the system into a regular configuration or a partial detachment into a capillary dipole plus a single bead. The detachment takes place at high $i/i_0$ values.Reuse & Permissions

Figure 7

Probability distribution function (PDF) of interdistances $r_{jk}/D$ obtained from 500 numerical simulations. Each simulation starts from a regular configuration at high magnetic field values. The repulsive interaction is then slowly decreased to zero. The interdistances are measured for final configurations. A peak appears at $r_{jk}/D=1$ since beads are in contact. However, a broad distribution of $r_{jk}/D$ values around 1.12 is observed meaning that the system lost its symmetry.Reuse & Permissions