Statics and dynamics of magnetocapillary bonds

When ferromagnetic particles are suspended at an interface under magnetic fields, dipole-dipole interactions compete with capillary attraction. This combination of forces has recently given promising results towards controllable self-assemblies as well as low-Reynolds-number swimming systems. The elementary unit of these assemblies is a pair of particles. Although equilibrium properties of this interaction are well described, the dynamics remain unclear. In this paper, the properties of magnetocapillary bonds are determined by probing them with magnetic perturbations. Two deformation modes are evidenced and discussed. These modes exhibit resonances whose frequencies can be detuned to generate nonreciprocal motion. A model is proposed that can become the basis for elaborate collective behaviors.

Figure 1

Pair potential. (a) Notation. Two beads are placed at a water surface and submitted to a constant induction $\vec{B}=B_x\overrightarrow{e_x}+B_z\overrightarrow{e_z}$ as well as a small oscillating induction $\vec{\beta }$. Distance $d$ and angle $\theta$ characterize the system. (b) Magnetocapillary potential. Interaction potential $U=U_c+U_m$ in polar coordinates $(d,\theta )$, as given by Eq. (3), for typical parameters of the experiment ($D=500\mu \mathrm{m}$ and $B_z=3$ mT). Darker zones indicate potential wells. On the left half, vertical induction $\overrightarrow{B_z}$ generates an isotropic repulsion when $M_{\theta }=0$, which leads to a finite equilibrium distance $d_{\text{eq}}\approx 2.5$ mm. On the right half, cylindrical symmetry is broken when $M_{\theta }\ne 0$, due to the addition of a horizontal field $B_x=0.7$ mT. Here $M_{\theta }$ is the orthoradial magnetocapillary number introduced in Eq. (3). It is proportional to $B_x^2$.

Figure 2

Characterization of the system. (a) Beads magnetization. Magnetic field $H$ induces a magnetization $m$ in a 500-$\mu \mathrm{m}$ bead with negligible hysteretic behavior. Here $H$ is successively increased from 0 (1), decreased (2), and increased back to 0 (3). Values of $H$ are typical of the experiment. The inset shows a larger cycle up to saturation magnetization, with the same units. (b) Equilibrium distance. The dimensionless equilibrium distance ${\tilde{d}}_{\text{eq}}$ as a function of the radial magnetocapillary number $M_d$. The color map shows distance $d_{\text{eq}}$ over diameter $D$. The solid line corresponds to theoretical prediction without an adjustable parameter, obtained by numerically solving Eq. (4). Symbols represent different bead diameters.

Figure 3

Resonance spectrum. (a) Amplitude. Frequency response of distance $d$ and angle $\theta$ for typical values of the experimental parameters ($D=500\mu \mathrm{m}$ and $M_{\theta }/M_d\approx 0.8$). Radial oscillation of amplitude $\mathrm{\Delta }d$ happens for $\alpha =0^{\circ }$, while orthoradial oscillation of amplitude $\mathrm{\Delta }\theta$ happens for $\alpha ={90}^{\circ }$. Error bars are smaller than the symbols, with a typical error of $0.2\mu \mathrm{m}$ on the bead center. Data have been fitted with a resonance curve [see Eq. (10)], represented by solid lines. (b) Phase. Theoretical prediction for the phase in both cases, given by Eq. (11). The color map indicates the phase difference $\mathrm{\Delta }\psi$.

Figure 4

Resonance frequencies. Comparison between measured resonance frequency $f_{\text{expt}}$ and theoretical prediction $f_{\text{th}}$ for several values of induction fields $B_x$, $B_z$, and $\beta$ as well as several bead diameters. Two orientations $\alpha$ of $\vec{\beta }$ are explored that correspond to radial and angular oscillations, respectively, $\alpha =0^{\circ }$ on the left and $\alpha ={90}^{\circ }$ on the right. Theory and experiment agree well, with the exception of distances $d_{\text{eq}}/D\lesssim 2$ in the case $\alpha =0^{\circ }$, corresponding to light-colored points in the left graph.

Figure 5

Deformation cycles. Trajectories in plane $(\theta ,d/D)$ for ten periods of the oscillating induction with $\alpha ={45}^{\circ }$. Below both resonance frequencies, in-phase oscillations produce no loop. When the frequency is increased, loops appearing as oscillations get out of phase. Above both resonances, cycles disappear as the phase difference decreases to zero.