Surface Wave Assisted Self-Assembly of Multidomain Magnetic Structures

An ensemble of magnetic microparticles at the liquid surface displays novel snakelike self-assembled structures induced by an alternating magnetic field. We demonstrate that these structures are directly related to surface waves in the liquid generated by the collective response of magnetic microparticles to the alternating magnetic field. The segments of the “snake” exhibit long-range antiferromagnetic ordering, while each segment is composed of ferromagnetically aligned chains of microparticles. The structures exhibit magnetic hysteretic behavior with respect to an external in-plane magnetic field and logarithmic relaxation of the remanent magnetic moment.

Figure 1

Schematic view of the experimental setup.

Figure 2

Snake structures generated by a vertical ac 110 Oe magnetic field. The size of the segments is determined by the magnetic field frequency $\nu$. Panel (a) corresponds to $\nu =17\mathrm{Hz}$ excitation; (b), (c), and (d) represent structures generated at 30, 50, and 70 Hz, respectively.

Figure 3

Top panel: Sketch of the instability mechanism. Arrows indicate the orientation of the magnetic moment in the segments. Bottom panel: Characteristic length of the snake $\lambda$ vs frequency $\nu$. Symbols show experimental data for 110 Oe field amplitude. The dashed line represents the dispersion relation for surface waves in deep water (no fitting parameters). The solid line represents the dispersion relation for surface waves in water, accounting for the magnetic contribution [see Eq. (1)] with $\alpha =0.28$. Top inset: Velocity field in the vicinity of the snake structure at $\nu =50\mathrm{Hz}$. Bottom inset: Pattern obtained by numerical modeling. The intensity of the dark color is proportional to the density of the magnetic particles. Parameters in Eq. (2) are $ϵ=1$, $b=2$, $\gamma =2.5$, $D=1$, $\beta =8.5$, and $\omega =3$, in the domain of $100\times 100$ dimensionless units.

Figure 4

Response of the snake to an in-plane magnetic field. The structure was generated by 50 Hz, 110 Oe driving field. White arrows designate directions of the magnetization vector at corresponding segments. Left panel: An in-plane dc magnetic field of 2 Oe was applied perpendicular to the structure. The resulting zigzag structure reflects the antiferromagnetic arrangement of the segments. Right panel: The in-plane dc field (10 Oe) was applied parallel to the snake structure.

Figure 5

Normalized $M$ vs in-plane magnetic field for the snake generated with a 40 Hz, 110 Oe external magnetic field. $M_{\max }$ corresponds to the effective magnetic moment at the 10 Oe in-plane dc field. At each point, the system was left to relax for 300 s after the in-plane magnetic field was changed. $M$ is normalized to its maximal value $M_{\max }$. The corresponding coercivity force ($H_c$) is about 3.1 Oe. Inset: Relaxation of $M$ from a single orientation state in a zero in-plane magnetic field (50 Hz, 110 Oe amplitude driving).