Minimal geometric requirements for micropropulsion via magnetic rotation

Controllable propulsion of microscale and nanoscale devices enhanced with additional functionality would enable the realization of miniaturized robotic swimmers applicable to transport and assembly, actuators, and drug delivery systems. Following biological examples, existing magnetically actuated microswimmers have been designed to use flexibility or chirality, presenting fabrication challenges. Here we show that, contrary to biomimetic expectations, magnetically actuated geometries with neither flexibility nor chirality can produce propulsion, through both experimental demonstration and a theoretical analysis, which elucidates the fundamental constraints on micropropulsion via magnetetic rotation. Our results advance existing paradigms of low-Reynolds-number propulsion, possibly enabling simpler fabrication and design of microswimmers and nanoswimmers.

Figure 1

Controlled achiral rigid microswimmers. (a) The microswimmers' three-bead structure contains a plane of symmetry making it achiral. (b), (c) Schematic of flexibility tests to quantify microswimmers rigidity under (b) impulsive reorientation and (c) steady rotation. The microswimmer's structure is measured by the three-dimensional distances between each pair of beads $(d_{12},d_{23},d_{13})$. (d) Percentage change in $(d_{12},d_{23},d_{13})$ during impulsive reorientation. (e) Percentage change in $(d_{12},d_{23},d_{13})$ during swimming with data points captured once per cycle when the swimmer is nearly perpendicular to the line of sight. In (d) and (e) distances change less than 5% indicating rigid geometry. (f) Trajectories of achiral microswimmers showing swimmers controlled to swim in different patterns and make sharp turns.

Figure 2

Schematic for analyzing swimmer dynamics in body-fixed frame. ${\mathbf{e}}_1,{\mathbf{e}}_2$, and ${\mathbf{e}}_3$ are the principal axes of $\mathbf{M}$; ${\mathbf{e}}_2$ and ${\mathbf{e}}_3$ are perpendicular to the symmetry planes of the swimmer. For steady rotation, the swimmer angular velocity $\mathit{\Omega }$ coincides with the magnetic field rotation, and the magnetic field $\mathbf{H}$ and its angular velocity are constant in the body-fixed frame. The magnetic field is perpendicular to its angular velocity.

Figure 3

Response of swimmer to rotation frequency. (a) The rotational axis (demarcated by red dashed line) changes relative to swimmer's orientation as rotation frequency changes with constant field strength. The swimmer's rotation transitions from a symmetrical axis (low frequencies) to a non-symmetrical axis (high frequencies). (b) Swimming speed of seven different achiral microswimmers as frequency and field strength are varied with their ratio held constant. Equation (2) predicts a linear relation; linear fit data and ratio of frequency to field strength are reported in Table 1. Due to the random fabrication process, each swimmer has a different geometry hence different swimming speed. (c) Swimming speed of the six different microswimmers as frequency is varied for constant field strength, including the microswimmer in (a) shown as swimmer 6. Error bars in (b) and (c) denote standard errors estimated from position uncertainty in image analysis and observation time used to measure velocity.

Figure 4

Modeled magnetic field. Density plot of magnetic field in $xy$ plane created by the coil system modeled using comsol multiphysics. The small box at the center shows the area where the magnetic field is near uniform $(<2\%$ variation).

Figure 5

(a) Schematic of discretized surface of the swimmer. Each of the 717 points represents a regularized Stokeslet on the surface of the swimmer. (b) Angular reorientation of the swimmer during the reorientation test (solid blue line), along with simulated angular reorientation for different magnitudes of the magnetic dipole (green dotted, red dashed, turquoise dash-dotted lines). The best fit corresponds to a moment of $4.05\times {10}^{-15}$ J/T.