Assembly, Disassembly, and Anomalous Propulsion of Microscopic

Motion can be created by applying oscillating or rotating magnetic fields to magnetic bodies. ... Figure 1. Microscopic magnetic helices. (a) Helical ...
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Letter pubs.acs.org/NanoLett

Assembly, Disassembly, and Anomalous Propulsion of Microscopic Helices Soichiro Tottori,† Li Zhang,‡ Kathrin E. Peyer,† and Bradley J. Nelson*,† †

Institute of Robotics and Intelligent Systems, ETH Zurich, Zurich, CH-8092, Switzerland Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Shatin NT, Hong Kong SAR, China

‡

S Supporting Information *

ABSTRACT: Controlling the motion of small objects in suspensions wirelessly is of fundamental interest and has potential applications in biomedicine for drug delivery and micromanipulation of small structures. Here we show that magnetic helical microstructures that propel themselves in the presence of rotating weak magnetic fields assemble into various configurations that exhibit locomotion and a change in swimming direction. The configuration is tuned dynamically, that is, assembly and disassembly occur, by the field input. We investigate a system that consists of two identical right-handed helices assembled at their center in order to model the motion of assembled swimmers. The swimming properties are dependent on both the component design and the assembly configuration. For particular designs and configurations, a reversal in swimming direction emerges, yet with other designs, a reversal in motion never appears. Understanding the locomotion of clustered chiral structures enables uni- and multidirectional navigation of this class of active suspensions. KEYWORDS: Magnetic, microfluidics, helical flagella, propulsion, low Reynolds number he field of robotics develops and investigates robotic mechanisms that can locomote in various environments across many orders of magnitudes in scale.1 One challenging area is propulsion in liquid at micro- and nanoscales, because the inertial forces employed by macroscopic swimming organisms do not play a role in this regime.2 Recently, researchers have proposed various types of propulsion methods that exploit both external and internal energy, such as chemical reactions,3−5 motile bacteria,6−9 light,10,11 acoustic waves,12,13 and magnetic14−19 and electric fields.20,21 The applications are mostly in biomedical fields, such as for drug delivery,22 biochemical detection,23 and micromanipulation.18,24,25 Magnetic fields are one of the promising ways for propulsion among these methods, because they do not require the use of toxic chemicals and operate through biological material without interference. Motion can be created by applying oscillating or rotating magnetic fields to magnetic bodies. However, the structural motion must be nonreciprocal in order to gain a net forward movement, a concept described as the “scalloptheorem” by Purcell.2 Symmetry-breaking motion is typically achieved by the following three ways: wavy motion of flexible filaments,14 rotation of chiral filaments that can be either rigid15,16,18 or flexible,17 and rotation that takes advantage of anisotropic boundary conditions, typically on solid surfaces.19,26,27 Among these, the first two swimming motions can move in three dimensions without constraints. The motion of a single swimmer has been intensely investigated, yet it is not well-known how multiple swimmers

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© XXXX American Chemical Society

move and interact with one another. For magnetically driven systems, the forces that counterbalance each other are primarily fluidic and magnetic forces. The interplay of these forces creates complex patterns and motions.28,29 Chemically driven colloidal systems can also exhibit complex motion behavior for assembled swimmers due to hydrodynamic effects.30,31 Therefore, understanding the locomotion behavior of assembled swimmers is important for designing and fabricating suspensions of active chiral particles. Here, we study multiple magnetic helical microstructures that swim in fluid in the presence of weak rotating magnetic fields (Figure 1a). Rotating helical microstructures propel at low Reynolds numbers by creating nonreciprocal motion,2 which is also found in swimming microorganisms, such Escherichia coli and their use of helical flagella.32 Magnetic forces can induce the assembly of multiple components, whereas fluidic forces disassemble them or limit the size of the assembly. Assembly occurs with various geometrical configurations, such as straight and bent chains and crosses, which significantly influence the swimming behavior of the assembly as a whole (Figure 1b). In this paper, we theoretically and experimentally investigate how swimming velocity changes depending on the design of swimmers and configurations of assembly. Received: June 4, 2013 Revised: August 15, 2013

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(rotation along the long axis) at higher frequencies and generates translational motion.16,18,33 Because of the ferromagnetic property of the surface, an attraction force is induced between the helical structures resulting in various assembly configurations. By controlling the direction, speed, and strength of the rotating field, multiple helical swimmers can be assembled and disassembled in a controlled manner. Figure 2 shows the assembly and disassembly of two swimmers with different designs: l and s (helix angle: 65°, 65°, diameter: 6.0 μm, 4.0 μm, number of turns: 4, 4, and total length: 35 μm, 24 μm, respectively; see also Supplementary Movie S1). First, the swimmer l swam toward one end of the swimmer s, resulting in magnetic assembly at the two ends (Figure 2a). The attractive force between the two swimmers was large enough to allow the two swimmers l and s to rotate and propel together. Subsequently, the field rotation was reversed, and the assembled swimmers moved in the opposite direction, with swimmer l apparently towing swimmer s (Figure 2b). At relatively low frequencies (56 Hz), the drag on swimmer s rose and eventually exceeded the magnetic interaction force between the two swimmers, leading to disassembly of swimmers (Figure 2c). Figure 2d shows the relative distance of two swimmers and input frequency as a function of time. The field strength was maintained at a constant value of 1.0 mT during the entire procedure. We estimate the magnetic force between the two straightchained swimmers by assuming no slip between the two swimmers. The condition of disassembly is described as, FT > Fmag, where FT is the force to tow the swimmer s, and Fmag is the

Figure 1. Microscopic magnetic helices. (a) Helical structures are propelled in a rotating weak magnetic field. (b) Clusters of multiple helices with various configurations are formed due to magnetic interaction. (c) A microscopic magnetic helix consists of a polymeric helical structure and ferromagnetic coating on the surface. (d) SEM image of an as-fabricated magnetic helical microstructure. The scale bar is 2 μm.

The helical swimmers consist of polymeric helical structures covered with ferromagnetic thin films, as schematically shown in Figure 1c. The fabrication process relies on the threedimensional lithography technique to make helical structures and electron beam evaporation of ferromagnetic thin films on the surfaces.18 A scanning electron microscope (SEM) image of a swimmer is shown in Figure 1d. In the presence of a rotating magnetic field, a magnetic torque is continuously applied to the helical bodies causing a rotational motion. The rotational motion starts from a tumbling motion (rotation along the short axis) at low frequencies and converges to a corkscrew motion

Figure 2. Assembly and disassembly of two swimmers (design: l and s) into and from a chain configuration. (a) Assembly into the chain configuration. The straight and round arrows indicate the translational and rotational directions, respectively. (b) A time-lapse image of the swimming chained assembly. (c) Disassembly of the two swimmers. After disassembly, the swimmer s became unsynchronized with the field rotation. The scale bars in a−c are 20 μm. (d) The input frequency of the rotating field and the relative distance between the ends of the two swimmers (indicated by the arrows in part c, t = 1.25). (e) Schematic illustration of the propulsion force Fl,s, towing force FT, and drag Dl,s on the two swimmers assembled into a chain configuration. B

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Figure 3. Assembled swimmers with various configurations. (a) Side configuration (assembly angle 45°). (c) Side configuration of three swimmers. The time-lapse images were taken every 1.0 s. The scale bar is 50 μm.

assemblies with nonstraight configurations showed varying swimming properties, highly influenced by their assembly angles. Because the rotational axes are not on the helical axes, the model using 6-DOF propulsion matrices are necessary to model the configurations where helices are assembled diagonally. For these diagonally assembled configurations, we conduct analytical calculation of swimming velocity of the twoassembled swimmers using 6-DOF propulsion matrices, as well as experimental measurement of swimming velocity of the assembled-swimmers directly fabricated using three-dimensional lithography. Our model consists of two identical helical swimmers (radius r, helix angles θ, number of turns N, filament thickness d, and handedness right) assembled symmetrically at the middle with an assembly angle of ϕ (|ϕ| ≤ 90°) and a connecting angle ε, as shown in Figure 4a. The two helices are connected at the middle with a distance from each helical axis of 2r. We assumed that the assembled swimmers always rotate along the long axis indicated as the long arrow between the two helices in Figure 4a.18,33 The swimming motion can be described using the 6-DOF propulsion matrix of the assembled swimmers by,

magnetic force between the two swimmers. Each swimmer creates propulsive force Fl or Fs and is counterbalanced by drag Dl or Ds, respectively (Figure 2e). The towing force is computed using the two degree of freedom (DOF) propulsion matrix:2,34,35 A B − As Bl Fmag = FT = l s 2πfdisassembly Al + A s (1) where parameters As,l and Bs,l are the components of the propulsion matrices of each swimmer, and fdisassembly is the disassembly frequency (see Supporting Information). The magnetic force Fmag can be estimated to be approximately 3.5 pN using theoretical values of the parameters As,l and Bs,l, and fdisassembly = 56 Hz. Helical swimmers can be assembled into a straight chain configuration as well as various other configurations, such as bent chains, crosses, and more. Figure 3a−c shows schematic illustrations and optical microscope images of the three types of assembled swimmers: (a) a cross with a small assembly angle (approximately 7°), (b) a cross with a large assembly angle (approximately 57°), and (c) an assembly of three swimmers (Supplementary Movie S2). The drifting that can be seen was due to the wall effect of the bottom surface.36,37 In Table 1, the

⎡ ⎤ ⎡ F ⎤ ⎢ A assembly Bassembly ⎥⎡ V ⎤ = ⎢⎣ ⎥⎦ ⎢ T ⎥⎢⎣ ⎥⎦ T ⎣ Bassembly Cassembly ⎦ Ω

Table 1. Normalized Swimming Velocity (Displacement per Rotation) of the Individual (l and s) and Assembled Swimmers with the Standard Deviation (n = 4) individual l individual s assembly (Figure assembly (Figure assembly (Figure assembly (Figure

The propulsion matrix of the assembled swimmers can be computed using that of the constituent swimmer, A and B, as follows.

normalized swimming velocity, v/f (μm)

configuration

2) 3a) 3b) 3c)

2.13 1.41 1.72 2.30 0.89 1.08

± ± ± ± ± ±

(2)

0.02 0.08 0.11 0.11 0.21 0.13

A assembly = R y(φ /2)R x( −ε)AR x(ε)R y( −φ /2)

(3)

Bassembly = R y(φ /2)(R x( −ε)AR x(ε)S{d} + R x( −ε)BR x(ε))R y( −φ /2)

(4)

where Rx, y are the rotation matrices around the x- and y-axes, respectively, and S{d} is the skew-symmetric matrix of the vector d = (d1d2d3), which is defined as:

normalized swimming velocities (displacement per rotation) of the assembled swimmers and individual swimmers l and s are summarized (with a constant field strength of 1.0 mT). To understand the variation in swimming velocity, we use 2and 6-DOF propulsion matrices for straight chain and nonstraight configurations, respectively.2,35 The velocity of the straight chain configuration vassembly (Figure 2, and Table 1) is modeled as: vassembly = −2πf(Bl + Bs)/(Al + As), where f is the rotational frequency (see Supporting Information). The

⎡ 0 − d3 d 2 ⎤ ⎥ ⎢ 0 −d1⎥ S{d} = ⎢ d3 ⎥ ⎢ 0 ⎦ ⎣−d 2 d1

(5)

and d = (0, −r, 0). C

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Figure 4. Dependence of assembly configuration on swimming velocity. (a) Schematic of the two helical swimmers assembled at the center with the assembly angle ϕ and the connecting angle ε. (b) Scaled theoretical and experimental swimming velocity of the assembled swimmers as a function of an assembly angle ϕ for different helix angles θ. The square (■) and circle (●) are experimental results of helix angle θ = 25° and 65°, respectively. The error bars show the standard deviations of three independent experiments. (c) Scaled simulated swimming velocity of the two swimmers (θ = 5°, 25°, 45°, 55°, 65°, and 85°) assembled at the center as a function of ϕ and ε.

The swimming velocity v is computed without the application of any external force (F = 0) by,

the helical axis to the symmetrical axis of assembled swimmers (between the two swimmers), and a variation of swimming velocity develops depending on the assembly angle as described by eq 2. At a particular range of assembly angles and helix angles (e.g., θ = 25°, − 90° ≤ ϕ ≤ −34° in Figure 4b), the swimming velocity of assembled two right-handed swimmers becomes negative, which implies that the assembled swimmers propel in the left-handed style, despite the fact that each component is right-handed. This reversal in swimming direction is attributed to the creation of an opposite propulsive force due to the combined effect of the shift and tilt of the rotational axis from each helical axis to the symmetrical axis of the assembled swimmers. We also modeled the swimmers assembled at other points by varying the connecting angle ε. The scaled swimming velocity is plotted for different helix angles θ = 5°, 25°, 45°, 55°, 65°, and 85°, separately, in Figure 4c. The reversal of swimming direction (negative velocity) is observed at most helix angles, yet at a certain range of helix angle (θ ≈ 55°), the reversal of swimming direction does not occur for any ϕ and ε (− 90° ≤ ϕ

v = −[b11 cos2 ϕ + (b22 sin 2 ε + b33 cos2 ε)sin 2 θ + (b13 cos ε + ra11 − ra 22)sin ϕ cos ϕ] /[a11 cos2 ϕ + a 22 sin 2 ϕ] × 2πf

(6)

where a11, a22, b11, and b22 are the elements of the 6-DOF propulsion matrices of a single helical swimmer (A and B) (see Supporting Information for detailed derivation). The interaction of flow between the two swimmers is neglected in this model. In Figure 4b, we plot the scaled velocity v/rω as a function of the assembly angle ϕ at the constant ε = 0, for various helix angles θ. Corresponding experiments were performed using the swimmers (r = 2.5 μm, θ = 25° and 65°, ϕ = −30°, −15°, 0°, 15°, and 30°) directly printed by three-dimensional lithography, instead of assembling individual swimmers, to precisely control the configuration (see Supplementary Figure S1). The center of rotation shifts from D

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≤ 90° and 0° ≤ ε ≤ 360°). To verify this result, we tested the suspension of swimmers designed in the range where the reversal of the swimming direction due to assembly does not occur (θ = 55°) in a rotating field (Figure 5 and Supplementary

In conclusion, we demonstrate that active magnetic helical structures suspended in liquid and in rotating fields propel and form a variety of assembly configuration. Assembly and disassembly of multiple components are dynamically controlled by tuning the input field. The assembled structures exhibit a change in swimming properties depending on their configuration. In the most prominent case, a reversal of swimming direction occurs because of the shift and tilt of the rotational axis. Our model suggests that anomalous propulsion can be controlled through the design of components, which implies that uni- and multidirection movement of active suspensions can be realized by tuning the design of particles that compose the suspension. The understanding of the dynamic behavior of these microscopic helices may open the route to novel selfassembly structures and can be utilized to design active suspensions for biomedical applications, such as drug delivery and imaging.42

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ASSOCIATED CONTENT

S Supporting Information *

Additional information is available on materials and methods, mathematical model, SEM images of assembled swimmers, and three videos. Movie S1 shows assembly and disassembly of two swimmers into and from a chain configuration. Movie S2 shows the swimming behavior of swimmers assembled into various configurations. Movie S3 shows swarming swimmers. All videos were recorded in real time. This material is available free of charge via the Internet at http://pubs.acs.org.

Figure 5. Swimming behavior of multiple swimmers. All swimmers including single swimmers and assembled swimmers (in the circle) swim in the right-handed fashion, except the one in the square. The scale bar is 200 μm.

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Movie S3). The assembly of swimmers and the variation of swimming velocity due to assembly were observed (circled in Figure 5); however, the reversal of swimming direction (squared in Figure 5) rarely occurred, as predicted by the model. Our model of the two swimmers connected at their center covers other symmetric configurations, such as two swimmers connected at another contact point, as long as the distance between the rotation axis and each helical axis is a constant r. The model does not cover asymmetric configurations. However, the asymmetric configurations can be considered as pairs of two conditions of connecting angles ε, which implies that the velocity of the asymmetric assembly falls between the two conditions of connecting angles ε. The more complicated configurations, such as three or more helices, are not easily modeled because of the unpredictable nature of the rotational axis of the assembly and interaction of magnetization. Although we have not observed such large aggregations in our experiments, at higher field strengths, where a larger assembly can emerge,26,38 the models for large assembly will be required. Assembly with broken ferromagnetic fractures also induces unpredictable motion of the suspension. We believe the reversal of swimming direction observed in Figure 5 is due to the assembly with those ferromagnetic fractures. Therefore, as the purity of the suspension is improved, the motion of the entire swarm is expected to be more controllable and reliable. In our system, most of the surface of the helical structures is coated with ferromagnetic thin films. Therefore, the swimmers are assembled at various points and configurations, which results in deviations in locomotion direction and speed. The assembly configurations can be controlled by introducing techniques used for colloidal self-assembly, such as nonuniform patterns39,40 or directional functionalization.41 The combination to these technologies will enable highly controlled motion of microscopic helices.

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank F. Qiu for experimental assistance and the FIRST lab of ETH Zurich for technical support. Funding for this research was partially provided by the Sino-Swiss Science and Technology Cooperation (SSSTC, Grant No. IZLCZ2_138898).

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