Magnetization directions and geometries of helical microswimmers for linear velocity-frequency response

Recently, there has been much progress in creating microswimmers or microrobots capable of controlled propulsion in fluidic environments. These microswimmers have numerous possible applications in biomedicine, microfabrication, and sensing. One type of effective microrobot consists of rigid magnetic helical microswimmers that are propelled when rotated at a range of frequencies by an external rotating magnetic field. Here we focus on investigating which magnetic dipoles and helical geometries optimally lead to linear velocity-frequency response, which may be desirable for the precise control and positioning of microswimmers. We identify a class of optimal magnetic field moments. We connect our results to the wobbling behavior previously observed and studied in helical microswimmers. In contrast to previous studies, we find that when the full helical geometry is taken into account, wobble-free motion is not possible for magnetic fields rotating in a plane. Our results compare well quantitatively to previously reported experiments, validating the theoretical analysis method. Finally, in the context of our optimal moments, we identify helical geometries for minimization of wobbling and maximization of swimming velocities.

Figure 1

Rotation of the helix by angular velocity $\mathit{\Omega }$ can be characterized by the precession angle $\beta$.

Figure 2

(a) Helix with two turns, pitch $P$, and radius $R$. The helical axis is along $\widehat{\mathbf{x}}$, and the helix is symmetric about a ${180}^{\circ }$ rotation about $\widehat{\mathbf{y}}$. The principal axes of the rotational mobility matrix $\mathbf{M}$ are ${\widehat{\mathbf{e}}}_1,{\widehat{\mathbf{e}}}_2$, and ${\widehat{\mathbf{e}}}_3$. (b) Angles $\varphi$ and $\theta$ used for expressing the magnetic field direction $\widehat{\mathbf{H}}$ in spherical coordinates. As specified in text, the angles may be used relative to the symmetry $\left(x,y,z\right)$ axes or the principal (1,2,3) axes.

Figure 3

For a rotation axis along the helical axis $\left(\widehat{\mathbf{x}}\right)$, the torque $\mathbf{N}$ is not in the $x$ direction, and the moment $\mathbf{m}$ and field $\mathbf{H}$ lie in a plane perpendicular to the torque. As the rotation frequency changes, the angle $\mathbf{H}$ shifts to make a different angle $\gamma$ with $\mathbf{m}$.

Figure 4

The swimming velocity vs frequency for the stable solutions of a rotating helix with two turns, $P=4R,\chi =1$, and a moment along the $y$ direction. The lower branch corresponds to rotation around an axis with precession angle $\beta ={85}^{\circ }$, while the upper branch corresponds to rotation around an axis with precession angle $\beta =4.9^{\circ }$

Figure 5

The (a) swimming velocity and (b) precession angle vs frequency for the stable solutions of a rotating helix with two turns, $P=4R$, and a moment along the $z$ direction, which is perpendicular to the helical axis but not the first principal axis. The inset to (a) magnifies the low-frequency regime, illustrating the transition from tumbling with rotation about the $y$ axis at the smallest frequencies to wobbling rotation with varying precession angle at frequencies above the critical frequency ${\omega }_c$. The inset to (b) shows a log-log plot of the precession angle vs frequency. The straight line has slope $-1$, demonstrating the relation $\beta \sim {\omega }^{-1}$.

Figure 6

The swimming velocity (a) and precession angle (b) vs frequency for the stable solutions of rotating helices with moment along the $z$ direction. From left to right [for (a), as measured by location of sketched helices], and bottom to top [for (b)], curves correspond to helices with: two turns, $P=4R$ [same case as in Fig. 5]; two turns, $P=2R$; one turn, $P=5R$; one turn, $P=3R$; one turn, $P=2R$. Curves are only plotted for frequencies below the step-out frequency. As the aspect ratio (length/diameter) of the helix decreases, a greater portion of frequencies below step-out exhibit tumbling rotation.

Figure 7

The precession angle of a helix with moment along the $y$ direction, perpendicular to the first principal axis, as a function of ratio of helical pitch and radius $\left(P/R\right)$, for helices with different numbers of turns: one turn (red, thick solid line), two turns (orange, thick dashed line), four turns (green, thin solid line), and eight turns (blue, thin dashed line).

Figure 8

The velocity:frequency ratio of a helix with moment along the $y$ direction, perpendicular to the first principal axis, as a function of ratio of helical pitch and radius $\left(P/R\right)$, for helices with different numbers of turns: one turn (red, thick solid line), two turns (orange, thick dashed line), four turns (green, thin solid line), and eight turns (blue, thin dashed line).

Figure 9

(a) The maximum velocity of a helix with moment along the $y$ direction, perpendicular to the first principal axis, as a function of ratio of helical pitch and radius $\left(P/R\right)$, for helices with different numbers of turns: one turn (red circles), two turns (orange squares), four turns (green diamonds), and eight turns (blue triangles). (b) The step-out frequency of the same helices as a function of $P/R$.

Figure 10

(a) Geometry used to model helix of Ghosh et al. [39]. (b) Geometry used to model helix of Peters et al. [37].

Figure 11

Convergence study for modeling helix from [39]: velocity-frequency slope as a function of number of regularized Stokeslets used in discretization.

Figure 12

Comparison of velocity:frequency slopes calculated using resistive force theory (blue circles) and method of regularized Stokeslets (black squares) as a function of filament thickness. The helix geometry is from [39], but with the filament radius $a$ altered.