Ferromagnetic Microswimmers

We propose a model for a novel artificial low Reynolds number swimmer, based on the magnetic interactions of a pair of ferromagnetic particles: one with hard and the other with soft magnetic properties, connected by a linear spring. Using a computational model, we analyze the behavior of the system and demonstrate that for realistic values of the parameters involved, the swimmer is capable of self-propelling with average speeds of the order of hundreds of micrometers per second.

Figure 1

Diagrammatic representation of the system consisting of a hard and a soft magnetic particle connected by an elastic element. The magnetic forces experienced by the particles are shown for two orientations of the external magnetic field $\mathbf{H}$.

Figure 2

Trajectories of the hard bead [(a), red], the soft bead [(b), blue], and their center of mass [(c), black], after application of an elliptically rotating external field. As well as rapid oscillatory motion, the dipole pair follows a directional linear displacement (“swimming”) along the arrow towards the lower-right corner [24]. Inset (i) shows the mean speed of the center of mass as a function of time. Inset (ii) shows the trajectory of the swimmer in its configurational space where $\theta$ is the angle between the elastic element and the horizontal axis and $l$ is its instantaneous length. The parameters of the simulation are $H_y^0=1\mathrm{kOe}$, $H_x^0=5\mathrm{kOe}$, $f=230\mathrm{Hz}$, $2K_1/M_1=10\mathrm{kG}$, $2K_2/M_2=0\mathrm{kG}$, $M_1=M_2=1.4\times {10}^6{\mathrm{A}\mathrm{m}}^{-1}$, $2R_1=3.2\mu \mathrm{m}$, $2R_2=1.6\mu \mathrm{m}$, $r_0(\mathrm{\text{relaxed spring length}})=20\mu \mathrm{m}$, $k_s=3.0\times {10}^{-3}{\mathrm{N}\mathrm{m}}^{-1}$, $\eta ={10}^{-3}\mathrm{Pa}\mathrm{s}$, $\mathrm{Re}\approx {10}^{-3}$.

Figure 3

Particle trajectories (solid lines) at different phases of the field cycle. The diagram shows the orientations of the magnetization for each particle (top single arrows), the magnetic gradient forces (double-line arrows), and the orientation of the dipole pair at the same points of the cycle. The anisotropy easy axis of the hard particle coincides with the line connecting the particles.

Figure 4

Dimensionless speed of translational displacement of the swimmer, $v/(f{r}_0)$, as a function of the spring constant $k_s$ (circles) and magnetization $M_1=M_2=M$ (triangles), both scaled against the viscous drag. The values of the other parameters are as in Fig. 2. The right-hand axis shows the absolute speeds attained by the swimmer.