Optimizing the Design of a Simple Three-Sphere Magnetic Microswimmer

TL;DR

This work addresses nonreciprocal propulsion at low Reynolds number by proposing a minimal yet functional microrobot: a three-sphere swimmer composed of magnetizable beads connected by two elastic springs actuated by an oscillating magnetic field. The key mechanism is hysteretic collapse and detachment of bead pairs, which, together with higher-order hydrodynamic interactions, produces a net displacement per cycle despite overdamped dynamics. The authors develop an analytical description of the two-sphere hysteresis, extend it to the full three-sphere system, and quantify propulsion through a configuration-space loop governed by the mobility matrix; they further optimize swimmer geometry and the driving field using CMA-ES to reach speeds around $20\,\mu$m/s under realistic constraints. The results demonstrate that simple magnetic actuation with a geometrically simple swimmer can achieve significant propulsion and potentially enable independent control of multiple microrobots, offering a practical pathway toward microinvasive biomedical applications and flexible microrobotic platforms.

Abstract

When swimming at low Reynolds numbers, inertial effects are negligible and reciprocal movements cannot induce net motion. Instead, symmetry breaking is necessary to achieve net propulsion. Directed swimming can be supported by magnetic fields, which simultaneously provide a versatile means of remote actuation. Thus, we analyze the motion of a straight microswimmer composed of three magnetizable beads connected by two elastic links. The swimming mechanism is based on oriented external magnetic fields that oscillate in magnitude. Through induced reversible hysteretic collapse of the two segments of the swimmer, the two pairs of beads jump into contact and separate nonreciprocally. Due to higher-order hydrodynamic interactions, net displacement results after each cycle. Different microswimmers can be tuned to different driving amplitudes and frequencies, allowing for simultaneous independent control by just one external magnetic field. The swimmer geometry and magnetic field shape are optimized for maximum swimming speed using an evolutionary optimization strategy. Thanks to the simple working principle, an experimental realization of such a microrobot seems feasible and may open new approaches for microinvasive medical interventions such as targeted drug delivery.

Figures (7)

• Figure 1: Two magnetic spheres (red) linked by a finitely extensible spring (silver-blue). The top snapshot shows the nonmagnetized configuration with the spring in its undeformed state. In the bottom constellation, a magnetic field is applied along the axis connecting the centers of the two spheres. It induces the magnetic moments $\boldsymbol{m}$ (green arrows) in the spheres, leading to an attractive magnetic force, which contracts the spring. Due to the finite extensibility of the spring, the spheres cannot be brought arbitrarily close together.
• Figure 2: (a) Two magnetizable spheres are subject to the dimensionless potential $U_2$, see \ref{['eq:2sphere_potential']}, when linked by a FENE spring and exposed to an external magnetic field of rescaled strength $\beta$. $r$ denotes the rescaled center-to-center distance between the spheres. The extensibility of the spring is set to $\epsilon=0.75$. When the magnetic field strength $\beta$ is varied, different local minima (circles) appear. For $\beta=0.15$ (blue), there is only one stable configuration close to the rest length of the spring, $r\approx1$. When increasing $\beta$ to $0.25$ (green), the minimum moves slightly to the left and a second minimum appears in a compressed state of the spring, $r\approx0.25$, due to the induced magnetic attraction. For $\beta = 0.3$ (orange), the magnetic attraction dominates, leading to the disappearance of the minimum on the right so that the compressed state is the only stable minimum. (b) Bifurcation diagram for this configuration in the plane spanned by the rescaled length of the spring $r$ and the rescaled magnitude of the external magnetic field $\beta$, see \ref{['eq:beta_bif']}. 1. When the magnetic field is increased, the stable equilibrium position (solid purple line) moves to smaller lengths $r$ of the spring, until the solution vanishes in a saddle-node bifurcation (red triangle). 2. From there, the spheres collapse to a compressed state of the spring. 3. When the field is decreased again, the spheres remain in the collapsed state for magnetic field amplitudes $\beta$ lower than those of the previous event of collapse, implying hysteresis. 4. Finally, another saddle-node bifurcation appears, and the spheres abruptly detach and reseparate. At intermediate field strengths $\beta$, between both saddle-node bifurcations, two stable equilibria coexist, separated by an unstable solution (dashed purple line). The three different field strengths $\beta$ in (a) are marked by horizontal, dotted, colored lines. Corresponding minima in the potential in (a) are indicated by circled, colored dots.
• Figure 3: (a) Location of the bifurcations depending on the extensibility $\epsilon =\ell_{\text{max}}/\ell$. With increasing extensibility, the two bifurcation values of the distance $r$ between the centers of the spheres corresponding to collapse (blue dash-dotted line) and detaching (green dashed line) further separate. The detaching position has the asymptotic behavior of $r = x/\ell = 4(1-\epsilon)/3$ (gray dotted line) when the extensibility approaches unity. At $\epsilon = 1/\sqrt{3}\approx0.6$, both curves corresponding to saddle-node bifurcations merge into a cusp bifurcation. Below this critical value of the extensibility, no bifurcations occur. In (b) the magnetic field strengths $\beta = h/\sqrt{k\ell^5}$ at which the bifurcations occur are shown for different extensibilities. The individual bifurcation diagrams for the marked points are shown in (c), together with the curve traced out by all points of bifurcation (green dashed line).
• Figure 4: (a) Color map of the net displacement $\Delta x_2$ after one full cycle of the magnetic field for swimmers with different ratios of spring stiffness $c_1/c_2$ and rest lengths $\ell_1/\ell_2$ (phase space). For all swimmers $c_2 = 10$, $\ell_2 = 25$, and in the fully compressed state the center-to-center distance is still larger than $\ell_{i,\text{max}} - \ell_i = 4$. The magnetic field is ramped up from $0$ to $1$ with the very slow rate $\text{d} h/\text{d} t = 10^{-3}$ and then back down again to $0$. Except for the points of bifurcation, the swimmer is almost always in equilibrium. The three markers each label one swimmer that represents one of the three dynamical states in this phase space. Their trajectories in configuration space are shown in (b). (b) Color map of the natural logarithm of the infinitesimal displacement $f$ per area in configuration space $(x_{12}, x_{23})$ calculated according to \ref{['eq:curvature']} from the sixth-order expansion of the mobility matrix \ref{['eq:mobility_matrix']}. Three loops in configuration space traced out during one cycle of the magnetic field by the three-sphere swimmers with the different parameters as marked in (a).
• Figure 5: (a) One cycle of the trajectories of a three-sphere microswimmer with optimized parameters $c_1=6.63$, $c_2=8.22$, $\ell_1=13.7$, $\ell_2=17.0$ and $\ell_{i,\text{max}} = \ell_i - 4$. The net displacement accrued over one cycle of the magnetic field is $\Delta x_2 = 0.791$ in dimensionless units. This gives an average speed of $v = 0.408$ over the full cycle or $v = 18.6µm\per s$ in physical units when using the scales described in Sec. \ref{['sec:evolution']}. In each of the four stages of the cycle, a snapshot of the swimmer consisting of three spheres (red) and two springs (silver-blue) is shown, where the spheres are depicted to scale. (b) The driving magnetic field $h(t)$ oscillates between $h_{\text{min}} = 0.52$ and $h_{\text{max}} = 0.884$ with angular frequency $\omega = 3.24$.