repelling magnetic forces must be balanced. This is done by the interplay of passive reorientation towards the line and active swimming against the magnetic forces. The swimming directions are thereby sorted on the opposite stripe domain of the magnetic pattern. Bacteria that swim parallel (anti parallel) to their magnetic moment can only be guided above downwards (upwards) magnetized stripes.
In the third and last case of the lowest magnetic field, the percentage of guided bacteria is again reduced. Instead a third type of motion, the ignorant bacteria, emerge. The magnetic field is no longer strong enough to guide these. Therefore they ignore the stripe pattern and follow their current swimming direction.
We complement our experimental observations with a theoretical model. By balancing magnetic and viscous forces and torques, the motion of the bacteria can be explained in terms of their relative fitness
f = Fpropulsion
Fmagnetic (3.1)
which is defined as the ratio of the propulsion force and the maximum magnetic force.
The best guidance is achieved for the ideal fitnessf = 2. For low fitnesses f <1, which corresponds to strong magnetic fields or non motile bacteria (f = 0), all bacteria will get trapped above the domain walls. For higher fitnesses f >2, the percentage of ignorant bacteria increases on the expense of the percentage of guided bacteria.
Therefore here it is not the strongest that triumphs over all the others. It is the fitter, but not the fittest bacteria that are guided best.
3.3 Discussion
I showed that the active motion of magnetotactic bacteria in combination with the magnetosome chain and the heterogeneous field of the garnet film can result in a novel pattern of motion. Bacteria are passively guided along (curved) lines of mechanical instability. The precise control over the motion of bacteria makes this system ready for potential applications.
This unusual combination of garnet films and the unique magnetotactic bacteria could serve to answer further interesting questions, both from biology and physics. One impli-cation is already clear from the given experimental results. Bacteria that are swimming parallel and anti parallel to their magnetic moment can only be guided above one of the stripe domains. Both types of bacteria are thereby moving under the same external conditions. They experience the same oxygen concentration and therefore also the same deviation from the optimal concentration. Under natural conditions the moments are aligned along the geomagnetic field and all moments point into the same direction. This implies that, despite the very same external conditions, these two types of bacteria would swim into different directions. In magnetotactic bacteria with polar magneto-aerotaxis this can only be the case for North-seeking and South-seeking bacteria. In consequence, our system can be used as a tool to sort North-seeking and South-seeking magnetotactic bacteria. Both types move on oppositely magnetized stripes.
Another subject of discussion is the validity of the assumption that the magnetic field does not induce an active feedback to the bacterial motion. The dominating point of view is that magnetotactic bacteria are only passively aligned along the geomagnetic field lines without any active reaction to the magnetic field [93]. However, it is still highly debated if the geomagnetic field is actually strong enough to completely align the bacteria. In Ref. [97] the authors show that magnetotactic bacteria can perform active magneto-aerotaxis if the magnetic field is too weak to align them. They sense the direction of the external field and use this information for navigation similar to the chemotaxis of E.Coli bacteria. In addition magnetotactic bacteria could even be able to sense magnetic forces due to external magnetic gradients [98]. In the natural environment this feature might be useful to avoid trapping at naturally occurring magnetic materials such as iron ore.
In the presented experimental setup we cannot observe any active feedback or at least a potentially present feedback was not of importance for the guiding process. The domain structures of the garnet films with their strong magnetic fields and gradients nonetheless seem to be perfect candidates to study potentially existing feedback mechanisms. Maybe different, more sophisticated patterns of magnetic domains could induce active feedback.
Once this turns out to be possible, one could dream of engineering magnetic patterns that allow to selectively manipulate bacteria. Possible manipulations could thereby include active inducing of swimming reversals or even the control over the swimming speed. Another interesting direction for future research could be the enhancement of the (passive or active) magnetic manipulation with additional control parameters. An additional oxygen gradient could for example allow to study the competition between these two influence factors, their interplay in the internal signal processing cascade of the bacteria and their impact on the swimming behavior.
Both approaches could be helpful to learn new aspects about magnetotactic bacteria.
Moreover they are also interesting for physical applications. Additional control mecha-nisms could significantly enhance the ability to guide and control the motion of magne-totactic bacteria.
Chapter 4
Defect dynamics in spin ice
So far I discussed the transport of active and passive colloidal particles. Beyond that the colloidal topological insulator was simultaneously a model system for the motion of electrons within the semiclassical picture of the quantum Hall effect. The work presented in this section is also based on colloidal particles but the interest no longer lies in the motion of the particles itself. Here, colloids are purely used as a model system for so-called spin ice. The collective behavior of the mesoscopic particles emulates the dynamics of the spin degrees of freedom in the spin ice crystal. This approach offers the possibility to access the otherwise hidden dynamics of the spins in real space and real time.
Spin ices are rare-earth pyrochlore oxides and form a class of exotic magnetic materi-als with remarkable properties. Its spins are geometrically frustrated, which prevents ordering of the spins and leads to a macroscopically degenerate ground state. This is observable as a residual entropy that persist even upon cooling of the material to ab-solute zero. Another striking feature is the existence of emergent magnetic monopoles in spin ice. Despite a huge effort that is invested in the search for magnetic monopoles, there is only one other experimental observation of magnetic monopoles in the synthetic magnetic field produced by a Bose-Einstein condensate [99,100].
However the spins and their dynamics in bulk spin ice can only be studied by indirect measurements, such as for example neutron scattering [101]. A direct experimental observation remains elusive. The design of two dimensional model systems, so-called artificial spin ices (ASI), avoids these limitations [43]. In publication [P6] I am using a special version of ASI, the artificial colloid ice. Colloids that are confined in bistable gravitational traps take the role of the spin degrees of freedom. The advantage is, that the colloids can be individually observed in real space and time.
In publication [P6] colloidal ice is used to study the dynamics of monopole defects. I experimentally observe the motion of monopole excitations in real time and examine the interaction between pairs of them. Furthermore I am using the system to design a universal logic gate based on the motion of monopole excitations. Transferring this to ordinary ASI or even real spin ice might foster novel circuitry based on magnetricity instead of electricity.
I will start this chapter by explaining frustration and how it results in the macroscopically degenerate ground states of water and spin ice. After that I give a brief introduction to the model systems of spin ice, artificial spin ice and colloidal ice. I will then summarize how colloidal ice is used to experimentally measure the interaction of charged excitations in publication [P6]. In the last part of the chapter I present a novel approach to restoring the residual entropy in colloidal ice.
4.1 Geometrical frustration
Figure 4.1: Antiferromagnetic spins on the corner of a triangle. The first two spins can be placed in opposite direction such that they minimize their energy. However it is not possible to place the third spin in a way that it simultaneously satisfies all pairwise interactions. This geometric frustration result in two possible lowest energy states.
Geometrical frustration is a widespread phenomenon in physical and biological systems.
It is a unifying concept with an important role not only in various fields of condensed matter, but also far beyond [102]. In physics the presence of frustration reaches from magnetic moments in disordered solids [103] to high temperature superconductors [104].
Moreover it is also important in biological systems, exemplified by the folding of pro-teins [105] or neural networks [106].
Frustration arises when geometric or topological constraints impede the simultaneous satisfaction of all local interactions [107]. A consequence of geometric frustration can be a multifold degenerate ground state. In water ice or spin ices this goes along with the observation of residual entropy [108,109].
A traditional example of frustration is the antiferromagnetic Ising triangle [107,110]. It consist of three Ising spins with antiferromagnetic coupling residing on the three corners of a triangle as shown in figure 4.1. The first two spins can be aligned in opposite directions. There is however no possibility to place the third spin anti parallel to the first two spins. The minimization of one of the nearest neighbor interactions is always violated. Like this the triangular geometry causes the system to be frustrated with two possible lowest energy states.