Abstract
Fission and fusion processes of particle clusters occur in many areas of physics and chemistry from subnuclear to astronomic length scales. Here we study fission and fusion of magnetic microswimmer clusters as governed by their hydrodynamic and dipolar interactions. Rich scenarios are found that depend crucially on whether the swimmer is a pusher or a puller. In particular a linear magnetic chain of pullers is stable while a pusher chain shows a cascade of fission (or disassembly) processes as the selfpropulsion velocity is increased. Contrarily, magnetic ring clusters show fission for any type of swimmer. Moreover, we find a plethora of possible fusion (or assembly) scenarios if a single swimmer collides with a ringlike cluster and two rings spontaneously collide. Our predictions are obtained by computer simulations and verifiable in experiments on active colloidal Janus particles and magnetotactic bacteria.
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Introduction
Many phenomena in physics, chemistry and biology are governed by fission and fusion of particle clusters. These processes occur on widely different length and energy scales ranging from subnuclear to astronomic. Examples include the large hadron collider^{1}, nuclear fission and fusion^{2}, splitting and merging of atomic and molecular clusters^{3,4,5}, micronsize colloidal particles^{6,7}, cells^{8} and macroscopic granulates^{9} up to clustering in planetary debris disks^{10}. Fission and fusion can either happen spontaneously or be induced by a collision with another particle or entity. The fission and fusion processes provide valuable insight into the interactions and dynamics of the individual particles and are a promising pathway to fabricate new types of composite matter on the various scales.
In the past decade microswimmers have been studied extensively. These active soft matter systems consume energy and are, therefore, selfpropelled outofequilibrium autonomous systems. The interactions between many of those swimmers are either direct body forces or hydrodynamic ones as mediated by the solvent flow field created by the swimming motion. The latter depend on the details of the swimming mechanism and can be classified as either neutral, pushers or pullers^{11,12}. An important example for the former are magnetic dipole moments that enable to actuate and control the swimming path by an external magnetic field^{13,14,15,16,17,18,19,20}. In the absence of swimming, the structure of twodimensional^{21,22} and threedimensional^{23,24} dipole clusters has been explored recently, but the impact of activity on a swimmer cluster has been rarely studied^{25}. Recent simulation studies^{26,27} have revealed that the mean cluster size crucially depends on the strength of the selfpropulsion speed. Moreover, it has been shown that activity itself^{28,29,30,31} and chemotaxis^{32,33} can induce clustering^{28,29,30,31}. Hence, though the structure of microswimmer clusters have been widely studied, the process of induced and spontaneous fusion of such clusters is unexplored except for spontaneous fission of swimmers clusters where all hydrodynamics was neglected. Furthermore, the impact of hydrodynamic interactions (HIs) has not yet been studied for magnetic dipole swimmers.
In this communication, we explore both spontaneous and induced fission and fusion processes for magnetic microswimmer clusters. These processes may also be referred to as disassembly and assembly. We do this for the two basic swimmer classes pushers and pullers. The details of the HIs governing whether the swimmer are pushers, neutral or pullers^{12,34,35,36}. Pushers and pullers can be described as dipole swimmers whose flow fields are similar, but with opposite flow patterns. While pushers push fluid away from the body along their swimming axis and draw fluid in to the sides, pullers pull fluid in along the swimming direction and repel fluid from the sides. This has important consequences for the interactions between swimmers and give rise to qualitatively different fission and fusion scenarios, which we classify for small particle clusters. In detail, we study the spontaneous fission processes for two chosen types of clusters, linear chains and rings, representing the ground state of magnetic dipoles^{21,24}. While linear chains remain stable for pullers and neutral swimmers they are split in the case of pushers. However, ringlike clusters show fission events in any case independent on the details of the HI. A single swimmer hitting a ring cluster can end up in spontaneous and induced fission or in fusion. Finally, two fusing ring clusters reveal a wealth of complex joint dynamical modes. We use computer simulation techniques to obtain our predictions, which are in principle verifiable in experiments on active colloidal Janus particles and magnetoctactic bacteria.
Results
Spontaneous fission of a chain
Let us start using the simplest structure of a cluster—a one dimensional chain of N dipolar particles with a fixed dipole moment m, where all magnetic moments possess the same orientational direction aligned along their separation vector. So the total magnetic moment is M=Nm for all chains, which coincides with the ground state for small numbers of particles^{24}. By applying a selfpropulsion velocity v, directed along the dipole moments, to each single dipolar particle, a joint motion of the linear chain will emerge. Using the inverse of the endtoend distance
where r_{i}=[x_{i}, y_{i}, z_{i}] is the coordinate of the ith particle, we study the spatial extension of the chain for varied selfpropulsion velocity v, see Fig. 1a for the case of N=2, whereby, R_{E}(0) depicts the value for a passive chain and R_{E} is measured after a sufficiently long simulation time t to establish steadystate structures. While in the absence of HIs no changes in the chain conformation are observed^{37}, we identify three different behaviours for magnetic microswimmers. (i) Pullers, generating a contractile flow field, shrink the chain, R_{E}(0)/R_{E}>1, (ii) neutral swimmers hardly affect the configuration of the chain, R_{E}(0)/R_{E}≥1, and (iii) for pushers, generating an extensile flow field, the chain swells, R_{E}(0)/R_{E}<1. This even leads to fission of the chain R_{E}(0)/R_{E}→0, for reduced selfpropulsion velocities v/v_{0}≥0.32, where v_{0}=2m^{2}/πησ^{5} is the corresponding drag velocity caused by the magnetic dipole–dipole interaction for two aligned and touching dipoles, with diameter σ, in a headtotail configuration. This observed behaviour is the result of the superpositioned flow field of the microswimmers, which is contractile for pullers and extensible for pushers, see Fig. 1b.
The state diagram for chains of pushers spanned by the applied selfpropulsion velocity v and the number of microswimmers N within a chain is shown in Fig. 1c. This diagram reveals two distinct regimes. For low velocities the chain remains stable, while large selfpropulsion velocities lead to a fission. The fission process for chains of pushers happens stepwise as can be seen in the time series sketched in Fig. 1d and in the Supplementary Movies 1 and 2. The extensile flow field generated by pushers leads to a repulsion of the particles within the chain. Beyond a certain critical velocity, the chain of N particles will split into three different units—a remaining chain of N−2 pushers and repelled head and tail particle. To understand the symmetric stepwise mechanism we will address below the behaviour of the critical velocity leading to fission, as well as an analysis of all acting forces within a chain.
Let us now focus on the behaviour of the critical velocity leading to fission as a function of number of swimmers before we discuss the fission process itself. We have already shown that the extensile flow of the pushers can lead to fission although the particles are attracted due to their dipole–dipole interaction. By considering the total acting magnetic force on the head particle of a chain F_{1,} we can observe that the magnitude of the magnetic attraction grows with increasing chain length N. Here we compare the cases of soft magnetic dipolar particles with hard sphere dipoles. In Fig. 2, the force on the head particle F_{1} is normalized by F_{0}=6m^{2}/σ^{4}, where σ is the respective length scale provided by the hard core potential or the used WeeksChandlerAnderson (WCA) potential^{38}. The force F_{0} is the force due to the dipole–dipole interaction if a particle pair is in a headtotail configuration with a central distance σ. Hence for a larger number of pushers in the chain higher selfpropulsion velocities are necessary to split the chain. As a remark, we add these data to Fig. 1c as a dashed line, leading again to differences that are caused by the softness of the pair potential.
To determine why the chain splits in the middle, we focus on all acting forces on the microswimmers within a chain. Due to symmetry, torques are excluded for all chains and for the middle microswimmer in a chain with an uneven number of swimmers, the only unbalanced force is the force due to selfpropulsion, see Fig. 3a. The two ends are attracted by the dipole–dipole interaction and repelled by the Stokeslet. While for pullers there is a further attraction of the particles in the end of the chain, the extensile flow field generated by pushers lead to a further repulsion—which is proportional to the selfpropulsion velocity v. Hence, beyond a critical velocity v_{c} the chain of N≥3 particles will split into three distinct units, two individual swimmers and a remaining chain of N−2 swimmers, see again Fig. 1d. This remaining chain will show further fission events, since the critical velocity increases with an increasing number of particles, see again Fig. 1c.
Spontaneous fission of a ring
Now we consider the dynamical behaviour of an isolated ringlike cluster for different numbers of microswimmers N=3, 4 and 5 with vanishing total magnetic moment M=0 (refs 9,25,40). The activity of each particle leads to the formation of a cluster, which rotates with a constant angular velocity ω around the nonmoving center of mass. The angular velocity depends on the magnitude of the selfpropulsion velocity^{37}.
To study the fission of such a ringlike cluster, we analyse the selfpropulsion velocity dependence of the inverse radius of gyration R_{g} for the final configuration of the cluster, with respect to the initial radius of gyration R_{g}(0) for a passive cluster, see Fig. 4. The radius of gyration R_{g} is defined by
where is the center of the cluster. If each particle is attributed a fictive unit mass, this center can be interpreted as the center of mass. However, in the case of low Reynolds numbers, it is more appropriate to interpret it as the center of velocity. While R_{g}(0)/R_{g}>1 indicates shrinking, R_{g}(0)/R_{g}<1 implies swelling of the ringlike cluster. In the extreme case of fission, the inverse radius of gyration vanishes, R_{g}(0)/R_{g}→0.
Figure 4 shows that for ringlike clusters with N=3, 4, 5 magnetic microswimmers all swimmer types show a diverging radius of gyration, so fission, for large selfpropulsion velocity. Previous studies without HIs have shown that ringlike clusters are stable for all selfpropulsion velocities^{37}. However, the critical velocity to achieve fission as well as the general behaviour of the inverse radius of gyration strongly depends on the swimmer type and on the respective number of swimmers in the cluster. The fission of rings is a combined result of a swelling of the ringlike structure and a change in the orientation of the dipoles within the rings—with increasing velocity they point outwards, an effect which is enhanced by HIs.
Let us start with a ringlike cluster of N=3 microswimmers. Figure 5a shows the flow field of a single microswimmer and indicates the position of the two other particles within the ring by grey dots. While these additional particles are attracted in case of pushers, the neighbouring particles are repelled in case of neutral swimmers and pullers. Hence the rings show the expected shrinking and swelling behaviours, see again Fig. 4a. The velocity dependence of the inverse radius of gyration for pushers is nonmononotic. This is a result of an induced torque on each particle, Fig. 6a, which leads to a ring configuration where the dipoles point outwards. This state is nonfavorable for magnetic dipoles and, therefore, even pushers show a swelling behaviour for velocities v/v_{0}>2.7.
Ringlike clusters of N=4, 5 pushers and pullers show the opposite shrinking/swelling behaviour as rings of N=3 particles, see Fig. 5b,c. For pushers, the position of the nearest neighbour are now in the repulsive regime of the flow field, while the nearest neighbours of a pusher are attracted. As before the ring contracting for small selfpropulsion velocities, shows a nonmonotonic inverse radius of gyration as a function of velocity, see Fig. 4b,c, caused by the weaker magnetic attraction of the misaligned dipoles, see Fig. 6b,c.
The fission velocity for rings of pullers and neutral swimmers is monotonically increasing as a function of number of swimmers in the ring, while it is nonmonotonic for pushers, see Supplementary Fig. 1 for the fission state diagram for rings.
Induced fission and fusion
Let us now study the fusion scenarios of a ringlike cluster if it collides with a single swimmer. Initially, this swimmer approaches the center of velocity of the ring, here we choose a ring of N=5 swimmers, whereby all swimmers have the same selfpropulsion velocity, see Fig. 7a. As long as the ringlike cluster does not show any spontaneous fission the initial setup leads to an induced collision. In Fig. 7b, we summarize the resulting states for pushers, neutral swimmers and pullers and varied the selfpropulsion velocity for these collision events.
As seen before large selfpropulsion velocities will lead to a spontaneous fission of the ringlike clusters for all swimmers. If the initial ringlike cluster is stable and the single swimmer collides with the rotating ring, two different scenarios can be observed. Either the single swimmer fuses with the ring, creating a new stable ring of N+1 particles (see Supplementary Movie 3), or the impact leads to an induced fission into various fragments of different size and shape. While the induced fusion emerges for all considered types of microswimmers, the induced fission does not occur for pushers due to the instability of the initial ring beyond a selfpropulsion of v/v_{0}>0.3.
Spontaneous fusion of two ringlike clusters
Spontaneous pattern formation and pair interaction of rotating particles and vortex arrays, has been aroused a lot of interest in the past few years, in experiments^{39,40,41} and in theory^{42,43,44,45} revealing the importance of the HI in this type of systems. It has been proved that magnetic rotors at finite Reynolds number exhibit pattern formation and hydrodynamic repulsion, while a pair of interacting rotors in the Stokes limit exhibit hydrodynamic attraction.
In this section, we will consider a pair of rings A, B with N=5 particles with an initial distance d_{AB}. Due to the selfpropulsion of the individual swimmers the rings can be treated as active rotors^{42}. While the center of velocity for an isolated ring is nonmotile, a pair of rotors/rings interact via the longrange HI leading to a cooperative motion of their centers of velocity. The actual characteristics of this motion depends on the vorticity of the ringlike cluster: For opposite vorticities the HI leads to a linear translation of the pairs, while for the same vorticity the rotors propagate on a spiral^{42,43,44}.
Let us start by considering rotors with the same vorticity. Ringlike clusters composed of neutral swimmers approach each other on bended lines, see dashed lines in Fig. 8a. For large selfpropelled velocities v/v_{0}>0.5 the spiral trajectories, in the case of pullers and pushers, correspond to predictions for active rotors with higherorders in the mutipole expansion^{42}. The centers of velocity of the individual rings propagate on shrinking (puller) or expanding (pusher) spirals, see Fig. 8a. In this case, we have the contribution of the two mobility terms in the motion equation (see equation (5)), that leads to a radial (Stokeslet) and an azimuthal (rotlet) contribution in the cycledaveraged flow field for large distances d_{AB}≫R_{g}. Here it is important to notice that in our case we have a nonzero torque on each particle due to the dipole–dipole interactions, this makes our problem different from the past studied cases^{42,43}.
As a result, ringlike clusters of pullers or neutral swimmers will approach each other and collide, while rings of pullers would repel each other for selfpropulsion velocities v/v_{0}>0.5. However, this velocity is beyond the fission velocity of a single ring, therefore, we never observed this behaviour for rings of N=5 pusher. Note that ringlike clusters composed of N=3 and N=4 pushers have shown the expected propagation^{42,43} for selfpropulsion velocities v/v_{0}>0.5. For the velocities that allows stable rings, v/v_{0}<0.3, they behave similar to neutral swimmers. This threshold, v/v_{0}∼0.5, is determined by the velocity dependence of the contractile flow profile.
Figure 9 shows the emerging states after the encounter between the two rings for all microswimmer types. As expected from the center of velocity trajectories, we observe fusion scenarios for small selfpropulsion velocities for all swimmers. For pushers in the velocity range 0≤v/v_{0}≤0.27, the two rings will collide and stick together but not change their shape—this conformation will be referred to as a dimer (Supplementary Movie 4). Due to the vorticity of the individual rings the dimer will still rotate. The same structure can be found for neutral swimmers (0≤v/v_{0}≤0.67) and pullers (0≤v/v_{0}≤0.5). As shown before large selfpropulsion velocities destabilize the rings and spontaneous fission is observed. However, neutral swimmers and pullers exhibit even more distinct emerging states. While the neutral swimmers just show a collision induced fission for selfpropulsion velocities 0.67<v/v_{0}≤2.5, pullers exhibit two different fusion states. For selfpropulsion velocities 0.5<v/v_{0}≤0.8, the two rings fuse into a single ring with N=10 particles (Supplementary Movie 5). As a last state, we find a state with two counterrotating microswimmers close to a ring of N=8 pullers for 0.8<v/v_{0}≤1. These two counter rotating microswimmers are trapped by the superposition of hydrodynamic and dipole–dipole interactions (Supplementary Movie 6).
Now, we focus on rotors with opposite vorticity. For opposite vorticity pullers and neutral swimmers show converging lines in the full range of velocities. While we would find diverging lines for pushers for v/v_{0}>0.5, we could not observe this due to the spontaneous fission above v/v_{0}≥0.27 for rings of N=5 pushers. Ringlike clusters composed of N=3 and N=4 pushers have shown the expected propagation^{42,43} for selfpropulsion velocities v/v_{0}>0.5. For velocities that guarantee stable rings of pushers behave similar to rings of neutral swimmers and approach each other on converging lines, see Fig. 8b. For the corresponding velocity regime the contribution due to the mobility terms are larger, in the farfield, compared with the activity terms in the equation of motion (see equation (10)). Therefore, clusters of all swimmers types will collide even if they have opposite vorticities.
For pushers in the regime 0≤v/v_{0}≤0.2, we observed an emerging coupled dimer, see Fig. 10 (Supplementary Movie 7). Here the two rings are coupled and translate together on a linear path in the absence of any global rotation. For larger velocities, 0.2<v/v_{0}<0.3 the rings can exchange particles and moves with a complex combination of toddling and rotation (Supplementary Movie 8). Neutrals swimmers exhibit the same two states, coupled dimer for v/v_{0} ≤ 0.33 and the toddling and rotation state for 0.33<v/v_{0}≤0.5. In addition to that the fusion into one ring containing all particles is possible for 0.5<v/v_{0}≤0.67. Below the critical velocity for the spontaneous fission of an individual ring, the neutral swimmers exhibit some collision induced fission into various fragments of different size and shape. In contrast to pushers and neutral swimmers, the collision of two ringlike clusters of pullers show the fusion into a single ring and the induced fission as well show a new conformation. The new state, 0<v/v_{0}≤0.5, shows a ring of N=8 pushers and a chain of two swimmers orbiting around the ring against its direction of rotation (Supplementary Movie 9).
Discussion
In conclusion, we have introduced the concepts of spontaneous and induced fission and fusion for clusters of magnetic microswimmers and exemplified these processes by studying linear chains and rings upon increasing the selfpropulsion as well as ring–ring collisions. A wealth of dynamical processes governed by the excluded volume, magnetic dipole interactions and by the swimmer type, that is, whether the particles are pushers or pullers, were obtained. The type of swimming even dictates the qualitative scenario. As an example, linear magnetic chains do not show fission for pullers but do so for pushers. A ringlike swimmer cluster which is stable for selfpropelled particles without HIs^{37} shows fractionation both for pushers and pullers. This demonstrates how a complex active swimmer can be composed of individual entities generalizing recent work where the active spinners were formstable^{46,47,48,49,50,51,52,53}. Two fusing magnetic ring clusters exhibit complex joint dynamical modes including a coupled dimer motion and a combination of toddling and rotation. This complex selforganization can be used to obtain final translational speed out of an initial pure rotational motion.
Very recent experiments have analysed the motion of selfassembled clusters for active Janus particles. For magnetic dipoles it has been shown that the actuation by an oscillating external magnetic field enables the propulsion of structurally nondeforming chains and rings up to N=5 particles. While this study focused on single clusters, another recent study of nonmagnetic Janus spheres showed that assembled pinwheels can perform a robust propulsion and that two pinwheels with the same chirality synchronize their motion^{25}. However, up to now the main focus in experiments using artificial swimmers, was to enable the propulsion of individual clusters without changing their structure^{54,55}. Our study provides ideas about how this cluster structure can be tuned systematically. Moreover, in case of magnetotactic bacteria the main recent research goal is the steering of these swimmers by an external magnetic field^{56,57,58}. The recent experimental developments provide evidence that our predictions about fission and fusion, can be verified in granular and colloidal magnetic swimmers^{14,25} as well as in magnetotactic bacteria^{56,57,58}.
For the future, it would be interesting to put the magnetic microswimmer clusters into further external fields such as an external shear flow or an aligning external magnetic field^{14,59}. We expect that the fission and fusion processes will be tunable by external fields, which would facilitate an exploitation of our modes for various applications. These include drug delivery by magnetic clusters, assisted by selfpropulsion and guided by external magnetic fields, where the dynamical modes can help to surmount specific barriers and overcome geometric constrictions. Our results can also help to develop novel magnetorheological fluids whose viscoelastic behaviour can be steered by activity and by magnetic fields^{60}. In fact, the cluster size of the aggregates largely determines the shear viscosity and the magnetization of the fluid such that combined smart magnetoviscous material properties can result by changing the clustering kinetics. Interestingly, it was recently shown that active colloids may exhibit an effective negative viscosity^{61}, which is another marked nonequilibrium feature as induced by selfpropulsion. Combining this finding with the rich dynamical cluster scenarios found here will open new doors to construct active composite materials with ‘intelligent’ and unconventional properties.
Methods
Model
We consider N spherical magnetic microswimmers in three spatial dimensions. The position of the ith swimmer is given r_{i}=[x_{i}, y_{i}, z_{i}] and its selfpropulsion velocity v_{i}=v is directed along the unit vector =[sin θ_{i} cos φ_{i}, sin θ_{i} sin φ_{i}, cos θ_{i}]. A magnetic dipole moment m_{i}=m is directed along the same axis. We model each swimmer as a soft core dipole using a WCA U^{WCA} and a point dipole interaction potential U^{D}.
where r_{ij}=r_{j}−r_{i} is the position of swimmer j relative to particle i and r_{ij} their respective distance. The particles are moving in a viscous fluid restricted to the low Reynolds number regime. Each particle creates a longrange flow that disturbs the flow field around the other particles. These interactions are the socalled HIs, which couples the translational and rotational dynamics between microswimmers.
We describe the flow field around each miroswimmer as a linear combination of fundamental solutions of the Stokes equations^{62}. We split these HI in two contributions: The first one is related to the mobility which is controlled by the dipole–dipole interactions, these interactions create a Stokeslet and a rotlet in the fluid where the particle is placed. Then, the kcomponent of the velocity field created for the ith microswimmer due to the presence of a microswimmer j is,
where is the Oseen tensor, is the force and is the torque on jth particle. The simplest swimmer considered here is the neutral microswimmer which is basically an active particle^{37} that only interact hydrodynamically via the mobility tensor.
The second contribution due to HI is related to the activity and takes account the fluid flow that the microswimmer produces while moving in the fluid. To keep the system as simple as possible, we add a force dipole describing the farfield distortions and a source quadrupole to describe the nearerfield distortions to the equations of motion. The kcomponent of the velocity field created for the ith microswimmer is given by
where
are the derivatives of the Oseen tensor^{62,63}. In equation (6), from left to right, the first term corresponds to a point force dipole, a second order tensor which takes into account the selfpropulsion with and σ_{0}=3πησ^{2}v is the hydrodynamic dipole strength where η is the viscosity and v the selfpropulsion velocity. Since an isolated microswimmer is force free, we assume that particles are moving due to an effective internal selfpropulsion force f_{0} oppositely acting to the force generated by the Stokes drag on the surface of the sphere, therefore, f_{0}=±3πησ v. If this force is antiparallel to the particle’s magnetization σ_{0}<0 the swimmer is extensile while if f_{0} is pointing in the same direction of the magnetization σ_{0}>0 the microswimmer is contractile. The last term corresponds to a source quadrupole, a second order tensor given by (ref. 62).
Summarizing the HI we have,
Microswimmers move in the low Reynolds number regime, therefore the corresponding equations of motion for the positions r_{i} and orientations are given by
with U=U^{D}+U^{WCA}.
Numerical methods and parameters
We solve the equations of motion, equations (10 and 11), using the thirdorder AdamsBashforthMoulton predictorcorrector method with a time step Δt=10^{−4}. The time is measured in units of =3πησ^{3}/ with fluid viscosity η, and particle diameter σ, which is our length scale and the energy scale from the WCA potential. The simulation time is t_{f}∼5 × 10^{3} to allow the clusters to achieve new steadystate structures. The initial configurations for the cluster conformations, either chains or rings, are two dimensional and they are generated by a minimization of the potential energy for given N passive spherical dipoles. We vary number of particles for a chain configuration as N=2, 3, …, 10 and use N=3, 4, 5 for ring configurations. For all studies, the magnetic dipole strength m^{2}/(σ^{2})=2 is fixed and the selfpropulsion velocity v is applied instantaneously on each particle. The velocity is normalized by v_{0}=2m^{2}/πησ^{5}, the velocity corresponding to the drag velocity caused by the dipole–dipole interactions for a chain of two dipoles in a headtotail configuration.
Data availability
Data available on request from the authors.
Additional information
How to cite this article: GuzmánLastra, F. et al. Fission and fusion scenarios for magnetic microswimmer clusters. Nat. Commun. 7, 13519 doi: 10.1038/ncomms13519 (2016).
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Change history
17 January 2017
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Acknowledgements
This work was supported by the science priority program SPP 1726 of the Deutsche Forschungsgemeinschaft (DFG). A.K. gratefully acknowledges financial support through a Postdoctoral Research Fellowship (KA 4255/11) from the DFG. F.G.L. acknowledges the financial support of Conicyt Postdoctorado2015 74150045 and helpful discussions with Rodrigo Soto, Adam Wysocki, Arnold Mathijssen and Marco Leoni.
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F.G.L, A.K., and H.L. designed the research, analysed the data and wrote the paper; F.G.L. carried out the simulations.
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Supplementary information
Supplementary Information
Supplementary Figure 1 (PDF 23 kb)
Supplementary Movie 1
Spontaneous fission of a chain of pushers with N = 7 swimmers and v/v_{0} = 1.10. (AVI 165 kb)
Supplementary Movie 2
Spontaneous fission of a chain of pushers with N = 8 swimmers and v/v_{0} = 1.20. (AVI 191 kb)
Supplementary Movie 3
Induced fusion for pullers and v/v_{0} = 1.66. (AVI 544 kb)
Supplementary Movie 4
Spontaneous fusion of two ringlike clusters with same vorticity into a rotating dimer in the case of pullers and v/v_{0} = 0.33. (AVI 1161 kb)
Supplementary Movie 5
Spontaneous fusion of two ringlike clusters with same vorticity into a single ring in the case of pullers and v/v_{0} = 0.50. (AVI 873 kb)
Supplementary Movie 6
Spontaneous fusion of two ringlike clusters with same vorticity into a ring of N = 8 and two individual counter rotating microswimmers for pullers and v/v_{0} = 0.83. (AVI 1240 kb)
Supplementary Movie 7
Spontaneous fusion of two ringlike clusters with opposite vorticity into a nonrotating but translating coupled dimer for neutral swimmers and v/v_{0} = 0.50. (AVI 915 kb)
Supplementary Movie 8
Spontaneous fusion of two ringlike clusters with opposite vorticity into toddling and rotating clusters for pushers and v/v_{0} = 0.27. (AVI 1554 kb)
Supplementary Movie 9
Spontaneous fusion of two ringlike clusters with opposite vorticity into a ring of N = 8 and a counter rotating chain of N = 2 pullers and v/v_{0} = 0.83. (AVI 1194 kb)
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GuzmánLastra, F., Kaiser, A. & Löwen, H. Fission and fusion scenarios for magnetic microswimmer clusters. Nat Commun 7, 13519 (2016). https://doi.org/10.1038/ncomms13519
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DOI: https://doi.org/10.1038/ncomms13519
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