# Propulsion of Flexible Polymer Structures in a Rotating Magnetic Field

*Written by Kevin Tian, AP 225, Fall 2011*

--Ktian 17:05, 9 November 2011 (UTC)

**Title:** Propulsion of Flexible Polymer Structures in a Rotating Magnetic Field

**Authors:** Piotr Garstecki1, Pietro Tierno, Douglas B Weibel, Francesc Sagués and George M Whitesides

**Journal:** Journal of Physics: Condensed Matter 21 204110

**Keywords:** propulsion, electromagnetism, deformation, soft-lithograpy, fluid dynamics, PDMS

## Paper Summary

The essential concept behind this paper is the proposal of a new method of propulsion for abiological structures in fluids of low Reynolds numbers. The basic idea is based off of elastic propellers that use rotational mechanisms in order to propel micro-organisms through fluids. Taking inspiration from these biological mechanisms, the authors designed flexible, planar polymer structures with a permanent magnetic moment. When placed in the presence of an external, uniform and rotating magnetic field, the structures deform into structures with helical symmetry (essentially meaning they are chiral), cause linear translations through fluids of Reynolds numbers between <math>10^{-1}</math> and 10.

An example of the concept can be seen in Figure 1, where observe that with polydimethylosiloxane (PDMS) elastic swimmers doped with ferromagnetic powders and cured such that there is a permanent magnetic moment, when placed in a rotating magnetic field (accomplished by a magnetic stir plate) will begin to move on their own (so to speak).

Though a cute design and result, there are a number of potential applications for such methods of propulsion in designing for motions of nano-, micro, and meso-scale devices/objects, as well as the study of other self-propelled systems.

## Experimental Details

**The 'Swimmers' and the System**

It is mentioned by the authors that a general difficulty in exploiting chiral structures for generating motion has been the fabrication of such objects, especially at smaller length scales (micron and below). However the unique aspect of this new design is that fabrication involves making a planar object, which is significantly easier to do. It is claimed that the technique and the underlying physics are both scalable.

[Note: However this claim is never quite substantiated. I believe there are a multitude of considerations and problems that may arise with scaling, so it's definitely a non-trivial if not misleading claim. It would be interesting to see if there are problems that may arise in using these designs at smaller/larger length scales]

For simplicity, the structures fabrications have been referred to as 'swimmer' and are approximately 2mm long. The environment these swimmers are placed under can be observed in Figure 2a and Figure 2b. The swimmers are placed in a petri dish of fluid high enough to accomodate their thickness but not length (so they're approximately parallel to the dish bottom). However although the swimmers *do* work in the situation illustrated in Figure 2a and can act as a good qualitative verification of results, for the purposes of characterization the simple magnetic stirrer was less than ideal. This is due to the non-uniformity of the magnetic field generated that resulted in additional forces (in addition to hydrodynamic forces) on the swimmer.

Thus in order to have more control over the magnetic field, characterization (for the entire quantitative portion of the experiment), a setup closer to Figure 2b is used, where 3 electromagnetic coils were used to create the rotating magnetic field of uniform distribution (over the area the properties/performance of the swimmers was tested).

**Fabrication**

The authors (generously) provided a fairly detailed list of materials that were used in the experiment. This is in section 2.1 and will not be reproduced for brevity.

The general fabrication process is illustrated in Figure 3. The fabrication of the swimmers is boils down to a single-step soft lithography process with ferromagnetic particles inside in order to magnetize the final structure. The fabrication is essentially as follows:

Master and Molds

- A transparent mask is made with the structure design etched into the mask
- The structure design is then transferred onto SU8 photoresist (spun onto a silicon wafer) via photolithograpy
- This involves developing the wafer in propyleneglycol methylethylacrylate (PGMEA) to have a "master" with the design being embossed in the profile of the SU8

- The master is silanized by vapor phase deposition of (tridecafluoro-1,1,2,2-tetrahydrooctyl)-1-trichlorosilane for 3h @ <math>25^\circ C</math>.
- PDMS pre-polymer mix (10:1 ratio of base to curing agent) onto the wafer and set to cure for 4h @ <math>65^\circ C</math>
- This creates molds that can be cut out with a scalpel, peeled away from the master surface and trimmed (Figure 3a)
- The molds are then treated with oxygen plasma and silanized in the same fashion as the master

Swimmers

- The molds are now used to fabricate the swimmers. The mold indentations are filled with PDMS.
- This PDMS mix is admixed with ferrite powder (25% w/w)

- Excess prepolymer mix is scrapped off (Figure 3b)
- The PDMS is left to cure for 3h @ <math>60^\circ C</math> with an external magnet positioned such that the field was perpendicular to the mold surface (Figure 3c)
- After curing the structure is carefully released, yielding a single swimmer!

General Properties

- Magnetic dipole oriented perpendicular to the plane of the body
- General schematic of swimmer presented in Figure 3d and 3e
- Typical dimensions varied between:
- 2-10mm length
- 1-3mm width
- 100-200 <math>\mu m</math> thickness

**Petri Dish Experiment Details**

As in figure 2b, an external magnetic field was generated with three coils. Each coil had inner diameter 4cm, outer diameter 7cm, ~1100 turns of 4mm thick Cu wire. Magnetic field rotation was in the x-z plane (whilst x-y plane was the plane of swimmer motion). To produce uniform magnetic field in the x-y plane, two of the coils were arranged in a Helmholtz configuration (both aligned on common axis with separation equal to their radius).

A waveform generator (TTi TGA1244) created the rotating magnetic field. The waveform generator was connected to a current amplifier (IMG STA-800). Magnitude of current through the perpendicular coild systems was adjusted such that the magnetic field rotated uniformly in the plane with constant field amplitude of <math>H=10 \times 10^3 ~or ~11 \times 10^3 A~m^{-1}</math> (for the range of angular frequencies tested, <math>\Omega<200 s^{-1}</math>. A teslameter was used to measured the magnetic field intensity and uniformity (51662DE-Leybold, Germany).

The Petri Dish was 3.7cm in diameter filled with fluid mixtures (ethylene glycol, EG (1,2-ethanediol) or a mixture of EG and glycerol) which had a range of viscosities from <math>16 \times 10^{-3}~-~0.3~Pa~s</math>. The dish was positioned directly above the z-coil. Tweezers were used to position swimmers directly in the region of uniform external magnetic field.

## Results

**Condition for Propulsion**

We know that for low Reynolds number, as per the expression for Stoke's drag, we know that the force on the fluid is has a linear relation to the resulting flow field. Thus it follows that in order for rotating objects to yield translational motion, the object must form a chiral structure about its axis of rotation. This can be achieved by either rotating a static, non-planar shape that already satisfies this chiral criterion, or by rotating an elastic shape that spontaneously deforms into a chiral shape (Figure 4a and 4b respectively).

The exact mechanism of propulsion of helical structures are described well by G.I.Taylor's Educational Movies in Fluid Dynamics. In essence the helix can be divided into segments that are approximately cylindrical. These segments, when considered flowing through a fluid independently do not experience isotropic viscous drag. After some reasoning (that can be easily seen with a force body diagram) one notes that the net motion of the cylinder is not parallel to the applied force. If we then consider all these segments as part of the helical body we see that, although the body segments have force components in directions perpendicular to the rotational axis, the speed contains parallel components. Since the structure has a lack of inversion symmetry, due to its chirality, these parallel component's do not sum to zero.

If one observes the motion of these swimmers in a viscous fluid then we observe that there is a net linear displacement (Figure 4c).

**Synchronous rotation vs 'Tumbling'**

It has been noted that in general a swimmer can be rotating synchronously with the magnetic field, and the arms deform due to viscous forces, and translates normally through the fluid. However by making several mechanical considerations of the system, one notes that there is a critical rotational frequency (for a given fluid viscosity, magnetic field strength, and swimmer geometry) for which above this limit the swimmer cannot synchronously follow the rotation of the field and the motion devolves into a back-and-forth motion. Thus above this critical frequency there is a decrease in the swimmer efficiency. An example of when this "rocking motion" occurs can be seen in Figure 4d, which is at a significantly higher frequency than in 4c, and we note the lack of as large a net translation as for lower rotational frequency.

This mode of motion is referred to as "tumbling" since it does not contribute to efficient translation.

Further characterizations of speed with respect to frequency was performed for various viscosities. There were two distinct dynamic regimes that there observed.

- i) Velocity of the swimmers were linearly related to rotational frequency (low freq)
- ii) Velocity decreased with increasing rotational frequency (high freq)

It was noted that the critical rotational frequency increased with magnetic field intensity and decreased with increasing viscosity. It follows well with the prediction <math>\Omega_C \propto {H \over \eta} </math>, the background of which is not covered in this paper.

**Speed**

According to the linearity of the Stokes equations, a rotating collection of angular oblique bodies will achieve net speed dependent only on geometry and rotational frequency, but not fluid viscosity. In a similar fashion swimmer speed should only be proportional to the linear speed of the rotating arms. However this argument assumes that all points of the swimmer are contributing a net speed, and that there is a viscous torque influenced by swimmer geometry. Both introduce viscosity dependency.

The authors avoid the second complication (by saying it is difficult to analyze) and focus entirely on the first by assuming deformation is described by Hooke's law and is approximately proportional to viscous torque at low speeds/low viscous torques. [Note: This doesn't seem very plausible to me.]

If this is true then it is expected that for a constant rotational frequency, that increasing viscosity (and thus structural deformation) will increase the translational speed of the swimmer. The results of this experiment can be observed in Figure 5. As can be noted there is a distinct decrease of the speed (at the same angular frequency) for increasing viscosity, which the authors describe as "slight". I disagree with this evaluation and say it is rather significant compared to what speeds they are traveling at.

Regardless this opens up the question of what shape is most efficient for swimming at low Reynolds numbers. The design was modified in arm length, and the speed at various rotational frequencies is tested and shown in Figure 6. Figure 6a has longer arms, 6b is the original design and 6c has shorter arms. As one can see, at the higher frequencies, the longer arms design was clearly faster than the original design. Though there are some curious effects that are not adequately explained (such as why at low rotational frequencies the longer arms design is slower than the original) the message is clear. Geometry is essential to the design of the ideal swimmer, though what exact specifications are needed for ideal performance is up in the air.

## Discussions & Conclusion

The authors bring up an interesting discussion about the slight change in the way one must engineer solutions to problems on the low Reynolds number realm. Particular how it is not simply "what is the ideal propeller" but rather "what structure will deform into an ideal propeller?". The way the authors have designed their swimmers is an experimental platform for more such deformation-based self-propulsion. Since their platform involves planar objects, further redesigns can be scaled down into even smaller realms, since they did not approach the limit of photolithography and soft lithography.

It was only mentioned in the last stages of the paper that there *were* considerations that weren't mentioned that complicate their claim of scalability. Material choice (for the mechanical properties) and flow fluctuations, and how they affect the propulsion of these structure was not touched upon but acknowledged as a problem on smaller length scales.

Though interesting in demonstration, the use of magnetic fields came with an inconvenience in the form of constraints. It was required that the magnetic field be uniform, otherwise the motion would be complicated by additional forces. The details of these additional forces was not touched upon but are non-trivial considerations. The authors simply note that one potential solution is simply to increase the magnet size relative to the swimmers (doable for scaling down swimmers).

Nonetheless the general technique has been proposed and shown to be effective as an approach to developing means for self-propulsion. The novelty of the technique lies in it's capability to exploit what was previously inaccessible (or rather just extremely difficult) to fabrication. Namely the full 3D structure that nature seems to have endless access to.