contains a detailed derivation of the underlaying theory which is confirmed by numerical simulations. Simulations are also used to demonstrate the robustness of the transport against thermal fluctuations. Experiments on the four fold symmetric pattern were made up later and are presented in [P1].
Colloidal topological insulator Finally the connection between our colloidal system and the various topological insulators that I discussed in section2.1can be established.
In [P4] I apply the theory which was developed in the previous publications to design a colloidal topological insulator. It is based on a hexagonal magnetic pattern that has edges towards a stripe pattern. The modulation can be chosen such that only the motion above the hexagonal pattern becomes nontrivial while the stripe pattern remains trivial.
The colloids on the bulk of the hexagonal pattern are transported in closed circles. That is after completing one circular orbit they return to their initial position. Thus there is no net transport and the bulk is insulating. Close to the edges towards the trivial stripe pattern, however, the particles cannot complete their full orbits and have to perform skipping orbits. Just like in the semiclassical picture of the quantum Hall effect (see figure 2.2), this results in robust helical edge states that transport colloidal particles along the edge between the two patterns. Above that we also implemented an analogue of the quantum spin Hall effect where particles of opposite susceptibility (paramagnets and diamagnets) are transported into opposing direction along the same edge.
On one hand this is an interesting new approach for stable transport of colloidal particles.
The particles can be transported along arbitrary edges into any desired direction and even around corners of these edges. In addition it is possible to transport multiple particles along multiple edges into different directions, all with one and the same external modulation. On the other hand our system is also an interesting model system. It is among the first experimental visualizations of the semiclassical picture of the quantum Hall effect. Often colloids are used as model systems for atoms [37] or molecules [38].
Here I go one step further and use my colloidal system as a model for the motion of electrons.
2.3 Summary and discussion
In this chapter I presented a novel approach for the driven dissipative transport of magnetic colloidal particles. I could show that the transport of colloids in our system is topologically protected. Modulation loops LC of the external driving field can be divided into topological classes that cause different transport of colloids in action space.
The transport is therefore topologically protected by winding numbers ω(LC) of the modulation loops in control space.
Similarly to the protected robust edge channels in topological insulators, topological pro-tection guarantees robust colloidal transport in our system. The topologically protected transport is a promising tool to overcome the drawbacks of former colloid transport sys-tems, based e.g. on thermal ratchets. Our system allows to transport big collections of particles in a precise and dispersion free manner. Furthermore in a three fold symmetric
pattern we have independent control over two different types of particles, paramagnetic and diamagnetic colloids.
To describe the transport in our system and to show that it is topologically protected we developed a new theoretical framework. This theoretical approach is quite different to the ones used in other topologically protected systems. Instead of Chern number associated to the bulk band structures our explanation is based on the topology of the stationary surface, which is described by its genus.
Despite these different approaches we can establish a connection to the other topologi-cally protected systems. The colloidal topological insulator shows the same phenomenol-ogy as all the others: Particles are only transported along the edge between a trivial and a nontrivial pattern.
Discussion and outlook This last analogy might give rise to two interesting open ques-tion. First, is it possible to describe the colloidal system within the theoretical framework of the quantum Hall effect. One possibility might be the fact that we can assign a sta-tionary field8 Hsext(xA) to every pointxA on the torus of action space A. This defines a vector bundle [48] which is similar to the mapping between the complex Bloch wave function|um(k)i andkon the torus of the Brillouin-zone and could provide an analyses of the colloidal transport similar to other topological insulators.
The second question is of rather speculative nature. In our system we mainly deal with topologically protected bulk transport. Nonetheless we can find a design that facilitates transport in edge states. In reverse conclusion, one could ask if it’s possible to construct a topological insulator such that it exhibits topological bulk modes.
Another intriguing aspect of our system is its dissipative character. So far topological phases are mainly discussed in Hamiltonian systems [71]. It is possible to engineer open quantum systems with dissipation. Diehl and co-workers showed that they can extract topologically protected edge states from such a system [71]. But the final edge modes are nonetheless dissipation free. In contrast to this our colloidal system and especially the deterministic ratchet motion is intrinsically dissipative. It cannot be described by an effective Hamiltonian.
The developed theory is not limited to this specific colloidal transport system. In princi-pal it can be extended to any, not necessarily magnetic, system with the same symmetry.
An example that we realized in our working group in Bayreuth is a macroscopic steel pump [72]. There the magnetic pattern is generated by millimeter sized permanent magnets. The external field is emulated by two opposing strong magnets with a tunable orientation. In this macroscopic system we can transport millimeter sized steel spheres (behave like paramagnets) and diamagnetic superconducting particles in the same way we transport colloids in the mesoscopic system. This setup has the advantage that we can simultaneously visualize the particle motion and the external field direction. Fur-thermore it is possible to observe a special type of ratchets in theS6 symmetric pattern, which was not possible in the colloidal system due to experimental limitations which are discussed in [P1].
8The stationary fieldHsext(xA) is the external magnetic field that renders the pointxAstationary.
2.3 Summary and discussion
In addition to the above discussed aspects there is probably a lot of room left for further investigations based on the colloidal transport system and the new theoretical approach.
So far we could experimentally only confirm our theory by using paramagnetic and diamagnetic colloids. These are suitable to examine the behavior of maxima and minima of the colloidal potential U. However, an experimental verification for the predicted behavior of saddle points U remains elusive. A composite particle built of four dipolar particles (two paramagnets and two diamagnets) in a octupolar configuration might be a promising approach for a saddle point seeker. Within certain approximation, these particles should go to the saddle points of the potential U and can be used to study their behavior.
Another interesting question for future investigations might be the impact of particle-particle interactions and manybody effects. So far the colloidal suspension was highly diluted. Hence different particles moved at sufficient distance and did not influence each other. In reference [40] Pietro Tierno and Thomas Fischer showed that manybody effects can be an important factor in transporting colloidal particles above periodic potentials. The open question is how these multi particle effects can be combined with the topological framework of this thesis. In combination with the possibility of two independent transport directions this might offer interesting routes to construct novel modes of colloidal motion.
Chapter 3
Magnetic guidance of magnetotactic bacteria
The previous chapter dealt with the topologically protected transport of passive micro particles. In this chapter I am following a different route based on the motion of active particles. Hence the challenge is no longer to induce motion but rather to rectify the otherwise non-directional motion of the active particles.
The motion of passive or Brownian particles is governed by Brownian dynamics. The equilibrium thermal fluctuations of the particles are driven by random collisions with molecules of the surrounding solvent [73]. In contrast to passive particles active or self-propelled particles are consuming energy from their environment, which is converted into directed motion of the particles [10]. Their motion is therefore governed by both, Brownian dynamics and the capability of active swimming. The constant consumption of energy drives them into a far from equilibrium state [74].
Active mesoscopic particles were originally studied as a model for the swarming behavior of macroscopic animals, such as birds and fishes [75]. In fact the non-equilibrium nature of these particles gives rise to novel collective phenomena which cannot be observed with passive particles [76]. Examples are swarming, clustering to living crystals [77] or active turbulence [78]. Beyond that active particles are ideal model systems to study aspects of far from equilibrium physics [10,79].
The ability of active self propulsion is widespread in biological microorganisms such as bacteria. In their natural habitat they use swimming for the efficient search for nutrition and to avoid toxic substances [80]. One of the best studied examples is the bacterium Escherichia Coli [34]. It is propelled by a bundle of helical flagella which are rotated by a molecular motor. The motion follows a run and tumble pattern. Periods of straight active swimming (run) are interrupted by actively induced but random reorientations (tumble). Using chemotactic sensing E.Coli can navigate in chemical gradients. Run phases are longer when the bacterium swims towards favorable conditions and shorter when it swims in the wrong direction [81].
Like in countless other examples, nature is the inspiration for man-made objects that mimic the behavior of biological microorganisms. The engineering of such artificial mi-croswimmers with precisely designed properties is particularly interesting for studying their motion and understanding the fundamental mechanisms behind it. Meanwhile there are various different approaches for artificial self-propelled particles [10]. The most prominent among them are active Brownian particles. They are driven by a force of constant magnitude whose orientation undergoes rotational diffusion dynamics. Most of such systems are so-called Janus particles, which are colloidal particles with a bro-ken symmetry due to local coating with catalytic materials. This asymmetry induces
local gradients in the solvent around the particle, which results in a selfdiffusiophoretic propulsion of the particle [82]. An example is the partial platinum coating which causes a gradient of H2O and O2 in a hydrogen peroxide solvent. Other realizations of artificial active particles are based on external fields, for example chiral particles that are rotated by external magnetic fields in a way that they act like propellers [83]. Other externally driven microparticles are rotating colloids [84] or self assembled colloidal wheels [85]
in the proximity of a confining wall. Despite the huge effort that is recently invested in the investigation of active particles, their applicability in realistic environments, a fundamental prerequisite for real applications, is still in its infancy [10].
Here and in publication [P5], I present a novel approach that might solve some aspects of this problem. I achieved the precise guidance of single bacteria along arbitrary lines of mechanical instability. I am using magnetotactic bacteria, a special type of actively swimming bacteria that additionally has a built-in permanent magnetic moment. This remarkable property is used to manipulate the bacteria with the magnetic fields of a garnet film. Like this it is possible to study the interplay of active motion and the presence of external constraints. These can be chosen such that a stable guidance of the bacteria is achieved.
This chapter starts with the introduction of magnetotactic bacteria. Following this I explain how the interplay of active swimming and the built-in magnetic moment enables the stable guidance of bacteria in the heterogeneous magnetic field of garnet films. Apart from the need for swimming, the guiding process itself is passive. It does not require any active feedback of the magnetic fields on the swimming behavior of the bacteria. The guidance only relies on a careful balance between propulsion and magnetism.
3.1 Magnetotactic bacteria
Figure 3.1: a) Dark-field scanning-transmission electron microscope (STEM) image of an uncultured magnetotactic bacterium. The white magnetosome chain along the elongated cell body is clearly visible. b)High magnification image of the ferrite crystals inside the magnetosomes. Picture adapted from [86].
Magnetotactic bacteria are aquatic unicellular prokaryotes [87]. They were first
discov-3.1 Magnetotactic bacteria
ered in 1970 by Richard Blakemore [88]. In this thesis I used the bacterium Magneto-tacticum gryphiswaldense[89]. These bacteria have an elongated shape with few microns in length (∼3µm) and approximately one micron in diameter. An example is shown in figure3.1. They possess two helical flagella located at the opposite poles of the cells [90].
These enable the bacteria to actively swim back and forth. The change of direction is achieved by reversing the sense of rotation of the flagella.
The unique characteristic, that distinguishes magnetotactic bacteria from other conven-tional bacteria, is the so-called magnetosome chain (see figure3.1). Magnetosomes are vesicles of magnetite crystals (Fe3O4) that are surrounded by lipid bilayers. They have a typical size of 35-120 nm (figure3.1b). The magnetosomes are arranged in a chain that is oriented along the long axis of the bacterium [86]. This arrangement maximizes the re-sulting permanent magnetic moment which is of the order ofm≈1.5·10−15Am2[91,92].
The natural habitat of magnetotactic bacteria is the sediment at the bottom of lakes. In the oxic-anoxic transition zone they find the required oxygen concentration for perfect growth conditions. Magnetotactic bacteria use their unique constructional property to tweak the navigation towards the optimal oxygen concentration. The predominant point of view is that the magnetosome chain and therefore the bacterium is passively aligned with the geomagnetic field and swims actively along the field lines [93].
This so-called magnetotaxis is inherently linked to aerotaxis. The bacteria swim in arun and reverse pattern back and forth along the field lines to find the optimal oxygen con-centration. This combined process is also known asmagnet-aerotaxis [94] and provides an efficient way of navigating in oxygen gradients. The process of scanning the oxygen concentration is reduced from the full three dimensional space to the one dimensional field lines.
There exist two different types of magneto-aerotaxis, axial and polar magneto-aerotaxis [93]. In the first case the magnetic field is only used to align the bacteria. These bacteria navigate along the field lines by sensing the oxygen gradient1. In the second case, the polar magneto-aerotaxis, the bacteria are not only aligned by the field, but the field direction also determines the direction of motion. Therefore these bacteria only need to sense the local oxygen concentration. If it is too high (low) they e.g. swim parallel (anti parallel) to the field direction. The polar magneto-aerotaxis is the predominant mechanism under natural conditions, while the axial type is only present in cultivated bacteria.
The geomagnetic field lines are pointing upwards in the Southern hemisphere and down-wards on the Northern hemisphere. For this reason there exist two opposite polarizations of polar magnetotactic bacteria [86]. On the Northern hemisphere they preferentially swim parallel to the field lines and are calledNorth-seeking. On the southern hemisphere the situation is inverted, theSouth-Seekingbacteria living there are preferentially swim-ming anti parallel to the field direction2.
1In reality bacteria cannot measure the oxygen gradient but only its concentration. The gradient is determined by comparing the concentrations at two different locations.
2Strong pulses of a magnetic field (H ≈50 kA/m) can invert this property. North seeking bacteria can be converted into South seeking bacteria and vice versa [95].
Magnetotactic bacteria attract a lot of multidisciplinary scientific interest in various ar-eas, even beyond biology. For example the precisely controlled synthesis of biomineral crystals in the magnetosomes makes them interesting for material scientist. Under-standing how the mineralization works in the bacteria might help to improve synthesis processes in the lab [96].
They are also of interest for physicists. The possibility to manipulate them with external magnetic fields facilitates the study of the interplay of biological control mechanisms and external constraints. In the next section I show that it is possible to impose external constraints such that the bacteria are forced to swim along predefined arbitrary paths.