Dynamics of groups of magnetically driven artificial microswimmers

Magnetically driven artificial microswimmers have the potential to revolutionize many biomedical technologies, such as minimally invasive microsurgery, microparticle manipulation, and localized drug delivery. However, many of these applications will require the controlled dynamics of teams of these microrobots with minimal feedback. In this work, we study the motion and fluid dynamics produced by groups of artificial microswimmers driven by a torque induced through a uniform, rotating magnetic field. Through Stokesian dynamics simulations, we show that the swimmer motion produces a rotational velocity field in the plane orthogonal to the direction of the magnetic field's rotation, which causes two interacting swimmers to move in circular trajectories in this plane around a common center. The resulting overall motion is on a helical trajectory for the swimmers. We compare the highly rotational velocity field of the fluid to the velocity field generated by a rotlet, the point-torque singularity of Stokes flows, showing that this is a reasonable approximation on the time average. Finally, we study the motion of larger groups of swimmers, and we show that these groups tend to move coherently, especially when swimmer magnetizations are uniform. This coherence is achieved because the group center remains almost constant in the plane orthogonal to the net motion of the swimmers. The results in the paper will prove useful for controlling the ensemble dynamics of small collections of magnetic swimmers.

Figure 1

Three views of the three-sphere magnetic microswimmer, shown in the body-fixed frame of reference. $\mathbf{m}$ is the magnetic moment. $\alpha$ and $\mathrm{\Phi }$ are the azimuthal and polar angles used to parametrize the magnetic moment orientation.

Figure 2

Propulsion speed of the artificial swimmer as a function of the magnetic moment direction. (a) Direction parametrized by the spherical coordinates $\alpha$ and $\mathrm{\Phi }$. Parts (b) and (c) show the same dependence plotted on the unit sphere. The black $\times$ in (a) and the red line in (b) and (c) indicate the moment direction used in simulations herein.

Figure 3

Streamlines and magnitude of the disturbance fluid velocity field produced by the three-sphere swimmer at steady state, time-averaged over one period of the magnetic-field rotation.

Figure 4

Trajectory of two interacting microswimmers resulting from a rotating magnetic field with an axis of rotation perpendicular to the swimmer plane. Initial swimmer separation is 4 spherical diameters. The black line indicates the path of the mean position of the two swimmers.

Figure 5

Comparison of the propulsion velocity of two interacting swimmers to a single swimmer.

Figure 6

Streamlines and magnitude of fluid velocity field produced by two interacting swimmers, time averaged over a period of the magnetic-field rotation. The velocity field shows the plane orthogonal to the magnetic-field axis of rotation.

Figure 7

Comparison of the velocity of swimmer 1 due to hydrodynamic interactions with swimmer 2 as predicted using the Stokesian dynamics procedure and the rotlet velocity field approximation. (a) Direct comparison of the $y$-component of the swimmer velocity between the two models. (b) Normalized error of the velocity from the rotlet approximation.

Figure 8

Trajectory of two microswimmers resulting from a rotating magnetic field with an axis of rotation aligned with the line initially separating the swimmers. Initial swimmer separation is 6 spherical diameters.

Figure 9

Simulation results for 16 identical interacting swimmers initially placed in a plane orthogonal to the magnetic-field rotation direction. Parts (a) and (b) give trajectories of the bodies in the $xy$ and $xz$ planes. The black line indicates the path of the mean position of the swimmers. Filled circles indicate the starting position, while the swimmer bodies indicate the final position and orientation. (c) Standard deviation of the position coordinates from the mean position coordinate. (d) Distance of each swimmer from the group center of mobility as a ratio of the initial distance from this center.

Figure 10

Simulation results for 16 interacting swimmers of differing magnetic moments initially placed in a plane orthogonal to the magnetic-field rotation direction. (a) Direction of magnetic moment vectors with respect to the direction-parametrized propulsion speed previously shown in Fig. 2. Parts (b) and (c) give trajectories of the bodies in the $xy$ and $xz$ planes, respectively. The black line indicates the path of the mean position of the swimmers. (d) Standard deviation of the position coordinates from the mean position coordinate. (e) Distance of each swimmer from the mean position as a ratio of the initial distance from the mean.

Figure 11

Simulation results for 16 identical interacting swimmers initially placed in a plane containing the magnetic-field rotation direction. Parts (a) and (b) give trajectories of the bodies in the $xy$ and $xz$ planes. The black line indicates the path of the mean position of the swimmers. Filled circles indicate the starting position, while the swimmer bodies indicate the final position and orientation. (c) Standard deviation of the position coordinates from the mean position coordinate. (d) Distance of each swimmer from the mean position as a ratio of the initial distance from the mean.

Figure 12

Simulation results for 16 interacting swimmers initially placed in a plane containing the magnetic-field rotation direction. (a) Direction of magnetic moment vectors with respect to the direction-parametrized propulsion speed previously shown in Fig. 2. Parts (b) and (c) give trajectories of the bodies in the $xy$ and $xz$ planes. The black line indicates the path of the mean position of the swimmers. Filled circles indicate the starting position, while the swimmer bodies indicate the final position and orientation. (d) Standard deviation of the position coordinates from the mean position coordinate. (e) Distance of each swimmer from the mean position as a ratio of the initial distance from the mean.