Modeling rigid magnetically rotated microswimmers: Rotation axes, bistability, and controllability

Magnetically actuated microswimmers have recently attracted attention due to many possible biomedical applications. In this study we investigate the dynamics of rigid magnetically rotated microswimmers with permanent magnetic dipoles. Our approach uses a boundary element method to calculate a mobility matrix, accurate for arbitrary geometries, which is then used to identify the steady periodically rotating orbits in a co-rotating body-fixed frame. We evaluate the stability of each of these orbits. We map the magnetoviscous behavior as a function of dimensionless Mason number and as a function of the angle that the magnetic field makes with its rotation axis. We describe the wobbling motion of these swimmers by investigating how the rotation axis changes as a function of experimental parameters. We show that for a given magnetic field strength and rotation frequency, swimmers can have more than one stable periodic orbit with different rotation axes. Finally, we demonstrate that one can improve the controllability of these types of microswimmers by adjusting the relative angle between the magnetic field and its axis of rotation.

Figure 1

Three views of the body-fixed frame for analyzing swimmer dynamics. (a) three-dimensional view; (b) ${\mathbf{e}}_1$ pointing out of page; (c) ${\mathbf{e}}_3$ pointing out of page. We work in the principal axes of the mobility submatrix $\mathbf{M}$, ${\mathbf{e}}_1$, ${\mathbf{e}}_2$, and ${\mathbf{e}}_3$; ${\mathbf{e}}_2$ and ${\mathbf{e}}_3$ are perpendicular to the symmetry planes of the swimmer. The magnetic moment $\mathbf{m}$, angular velocity $\mathit{\Omega }$, and magnetic field $\mathbf{H}$ are also shown.

Figure 2

Rotation of magnetic field ($\mathbf{H}$) at configuration angle $\alpha$ with rotation vector $\mathit{\omega }$. ${\mathbf{H}}_{\parallel }$ is the component of $\mathbf{H}$ in the direction of $\mathit{\omega }$. ${\mathbf{H}}_{\perp }$ is the component of $\mathbf{H}$ perpendicular to $\mathit{\omega }$.

Figure 3

Angles $\varphi$ and $\theta$ used for expressing the magnetic field direction $\widehat{\mathbf{H}}$ in spherical coordinates.

Figure 4

Helical trajectory taken by swimmer. The two curves (solid blue and dashed red) are the trajectories of the centroids of the middle and upper beads, which represent two possible choices of origins for the body-fixed frame. In the body frame the rotation around the origins ($\mathit{\Omega }$ and ${\mathit{\Omega }}^{\prime }$ for the middle and upper beads, respectively) and instantaneous velocity of the origins ($\mathbf{v}$ and ${\mathbf{v}}^{\prime }$ for the middle and upper beads, respectively) are constant. The instantaneous velocity depends on the choice of origin, but the rotation does not. The swimmer is depicted at two times separated by the rotation period $2\pi /\Omega$, during which it translates by one helical pitch (shown by dimension arrows at left) and rotates back to the same orientation. The average swimming velocity is the pitch divided by the rotation period and is the same for either choice of origin.

Figure 5

Schematic of the geometry of the swimmer corresponding to the stable solution (transparent geometry with dashed boundaries) and perturbed solutions (opaque geometry with solid boundaries). Note that the magnetic field and its angular velocity stay unchanged in the laboratory frame after perturbation.

Figure 6

Discretized surface of a three-bead swimmer. Each of the 717 spheres on the surface represents one regularized stokeslet.

Figure 7

1-D locus of allowable directions of the (a) magnetic field and (b) rotation axis in the body-fixed frame for $\alpha ={80}^{\circ }$. The area marked with purple dots on the surface of the sphere in panel (a) shows direction of $\mathbf{H}$ which are stable according to Eq. (35). The colors or symbols in (a) represent the azimuthal angle of the direction of the magnetic field in the 1-2 plane. The colors and symbols of (b) match those of the corresponding direction of magnetic field in (a).

Figure 8

1-D locus of allowable directions of the (a) magnetic field and (b) rotation axis in the body-fixed frame for $\alpha ={90}^{\circ }$. The area marked with purple dots on the surface of the sphere in panel (a) shows direction of $\mathbf{H}$ which are stable according to Eq. (35). The colors or symbols in (a) represent the azimuthal angle of the direction of the magnetic field in the 1-2 plane. The colors and symbols of (b) match those of the corresponding direction of magnetic field in (a).

Figure 9

1-D locus of allowable directions of the (a) magnetic field and (b) rotation axis in the body-fixed frame for $\alpha ={100}^{\circ }$. The area marked with purple dots on the surface of the sphere in panel (b) shows direction of $\mathbf{H}$ which are stable according to Eq. (35). The colors or symbols in (a) represent the azimuthal angle of the direction of the magnetic field in the 1-2 plane. The colors and symbols of (b) match those of the corresponding direction of magnetic field in (a).

Figure 10

Mason number as a function of the azimuthal angle $\varphi$ for $\alpha ={90}^{\circ }$. Color spectrum represents the values of the azimuthal angle $\varphi$ and is identical to the color coding used in Fig. 8. Intersections with the horizontal lines, at different Mason numbers, specify solutions for different experimental realizations. Here, the horizontal lines at Mason numbers of 0.08, 0.12, and 0.17 correspond to frequencies of 10, 15, and 20 Hz for $m=\sqrt{3}\times {10}^{-14}\text{J/T}$, and $H=10.11$ mT.

Figure 11

Dimensionless swimming velocity vs Mason number for different values of $\alpha$.

Figure 12

Dimensionless swimming velocity versus magnetic field's configuration angle ($\alpha$) at $\mathrm{Ma}=0.058$. In region I, no swimming motion is expected; regions II and III have one positive and negative solutions, respectively, corresponding to swimming in the same and opposite direction of the rotation vector.

Figure 13

Dimensionless swimming velocity versus magnetic field's configuration angle ($\alpha$) at $\mathrm{Ma}=0.125$. Regions I to III are defined as in Fig. 12. In region IV there are two possible swimming velocities with opposite directions. In region V, there are two possible swimming velocities, but in the same direction.

Figure 14

Dimensionless swimming velocity versus magnetic field's configuration angle ($\alpha$) at $\mathrm{Ma}=0.167$. Regions are defined in the same fashion as in Figs. 12 and 13.

Figure 15

Dimensionless swimming $V_s^{\prime }$ (blue squares) and precession angle $\beta$ (red circles) vs Mason number at $\alpha ={90}^{\circ }$.