Using Shape Diversity on the Way to Structure-Function Designs for Magnetic Micropropellers

Felix Bachmann, Klaas Bente, Agnese Codutti, and Damien Faivre

Phys. Rev. Applied 11, 034039 – Published 15 March 2019

Abstract

Synthetic microswimmers mimicking biological movements at the microscale have been developed in recent years. Actuating helical magnetic materials with a homogeneous rotating magnetic field is one of the most widespread techniques for propulsion at the microscale, partly because the actuation strategy revolves around a simple linear relationship between the actuating field frequency and the propeller velocity. However, full control of the swimmers' motion has remained a challenge. Increasing the controllability of micropropellers is crucial to achieve complex actuation schemes that, in turn, are directly relevant for numerous applications. However, the simplicity of the linear relationship limits the possibilities and flexibilities of swarm control. Using a pool of randomly shaped magnetic microswimmers, we show that the complexity of shape can advantageously be translated into enhanced control. In particular, directional reversal of sorted micropropellers is controlled by the frequency of the actuating field. This directionality change is linked to the balance between magnetic and hydrodynamic forces. We further show an example of how this behavior can experimentally lead to simple and effective sorting of individual swimmers from a group. The ability of these propellers to reverse swimming direction solely by frequency increases the control possibilities and is an example for propeller designs, where the complexity needed for many applications is embedded directly in the propeller geometry rather than external factors such as actuation sequences.

Received 14 June 2018
Revised 21 December 2018

DOI:https://doi.org/10.1103/PhysRevApplied.11.034039

Physics Subject Headings (PhySH)

Actuating materials Biomechanics Drug delivery Magnetism Microstructure Swarming

Living matter & active matter Low Reynolds number swimmers

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Felix Bachmann¹, Klaas Bente^1,2, Agnese Codutti^1,3, and Damien Faivre^1,4,*

¹Department of Biomaterials, Max Planck Institute of Colloids and Interfaces, Science Park Golm, 14424 Potsdam, Germany
²Department of Nondestructive Testing, Bundesanstalt für Materialforschung und -prüfung, Unter den Eichen 87, 12205 Berlin, Germany
³Department of Theory & Bio-Systems, Max Planck Institute of Colloids and Interfaces, Science Park Golm, 14424 Potsdam, Germany
⁴University of Aix Marseille, CEA, CNRS, BIAM, 13108 Saint Paul lez Durance, France

^*damien.faivre@cea.fr

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Vol. 11, Iss. 3 — March 2019

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Images

Figure 1
Velocity-frequency relationship. The measured velocity (red dots) dependent on the applied frequency of the actuating magnetic field is shown in (a) and (d) for propellers 1 and 2, respectively. The theoretic fit on those data points is shown in blue (solid line) and green (dashed line), which also describes the observed branching (corresponding to different propeller configurations at the respective frequencies). The 3D reconstruction of the propellers is depicted in (b) and (e). To visualize the change in propeller configuration to the horizontal axis of rotation dependent on the frequency, microscope images are shown in the measured frequency regime for one respective branch ((a) and (d) on the bottom). The far most left image corresponds to the orientation for a constant field in between the measurements. Both propellers show a reversal of the velocity direction in the first branch (blue), when increasing the frequency: after a linear tumble regime (I, swimming in one direction) until f_tw, a nonlinear wobbling regime [II happens up to the step-out frequency f_so (III)]. During the wobbling, the long axis of the propeller starts to tilt more and more toward the horizontal axis of rotation and at a certain frequency, the propeller velocity reverses its sign – the propeller swims in the opposing direction. The respective velocity landscapes are shown in (c) and (f) as contour plots (red for positive velocities, blue for negative ones, and white for zero). They reflect the velocity possibilities (in m s⁻¹/1 Hz) of the respective propeller shapes. The green and blue lines are again the branches, since the propellers are restricted to certain rotation axes for increasing frequency. The rotational similarity of propeller 2 is reflected in its velocity landscape. As a result, the branches pass similar values for increasing frequency (in contrast to propeller 1).
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Figure 2
Frequency dependent axis of rotation (a). The orientation of a propeller toward its axis of rotation (red horizontal line) changes with increasing frequency for some geometries to balance the magnetic and hydrodynamic torques. As a result, the coupling, and therefore, the velocity changes with frequency, which can lead to an inversion of the propulsion direction. Branching schematic (b). Coming from the tumbling configuration at low frequencies (middle), two wobbling configurations are possible for higher frequencies between the transition frequency f_tw and the step-out frequency f_so. Here, the two possibilities of Propeller 1 are depicted (cf. Movies S7 and S8 in the Supplemental Material [23]): the first wobbling solution with an angle of $θ_{1}$ between the long axis of the propeller and the axis of rotation (red horizontal line) and the second solution/configuration (right side) with $θ_{2} = 180^{\circ} - θ_{1}$ .
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Figure 3
Scaling with the magnetic field. (a) Velocity-frequency dependence of propeller 1 at three different magnetic field strengths (0.5, 1, and 2 mT). Depending on the magnetic field, the propeller can swim in opposing directions for the same frequency, for example, at f = 20 Hz for 1 mT (blue diamonds) and 2 mT (red dots). This is possible since the frequency characteristics scale with the strength of the magnetic field. In the inset, the three curves are normalized on the respective field strength, and therefore, fall onto each other. (b) Theoretical velocity as function of the applied magnetic field strength and frequency. The characteristic frequencies (f_tw and f_so, dashed lines), and therefore, the propeller behavior scales linearly with the magnetic field strength. The three measurements at 0.5, 1, and 2 mT are depicted through the yellow, blue, and red lines, respectively. Additionally, microscope images of the propeller configuration are shown to illustrate the three different regions and the two arrows indicate the respective swimming direction: the linear regime (I), the wobble regime (II), and the regime after the step-out frequency (III, not included in theory, set to 0).
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Figure 4
Propeller isolation for randomly shaped propellers (a). Tracks of a group of randomly shaped propellers. They all swim to the left for the initially applied magnetic field frequency of f₁ = 20 Hz. The arrows mark the location when the frequency is increased to f₂ = 40 Hz. One propeller (green track) shows a reversal of swimming direction at this frequency and swims in the opposing direction, while the rest of the swarm propellers are still in their linear regime and continue to swim to the left. This allows a joint propulsion for one frequency and an isolation of a single propeller for another as can be seen in the Supplemental movie S11 [23]. The according v-f diagram is schematically shown on the left. Propeller selection schematic for identically shaped propellers (b). Isolation of a single propeller from a swarm of identical propellers that only differ in their magnetic moment modulus (increases from 1 to 4). At a frequency f₁, all propellers swim in the same direction. At frequency f₂, the propeller with the smallest magnetic moment (green, 1) has already changed its axis of rotation and has a negative velocity, while the rest of the swarm continues swimming in the original direction (positive velocity). At frequency f₃, all propellers except for the one with the highest magnetic moment (blue, 4) have reversed their swimming direction. At frequency f₄, all propellers swim in the opposing direction compared to f₁. This offers new possibilities for controlling swarms of propellers and single propellers at the same time.
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