CColloidal trains
Mahla Mirzaee-Kakhki , Adrian Ernst , Daniel de las Heras , Maciej Urbaniak ,Feliks Stobiecki , Andreea Tomita , Rico Huhnstock , Iris Koch , Jendrik G¨ordes ,Arno Ehresmann , Dennis Holzinger , Meike Reginka , and Thomas M. Fischer ∗ Experimentalphysik X, Physikalisches Institut, Universit¨at Bayreuth, D-95440 Bayreuth, Germany. Theoretische Physik II, Physikalisches Institut, Universit¨at Bayreuth, D-95440 Bayreuth, Germany. Institute of Molecular Physics, Polish Academy of Sciences, 60-179 Pozna´n, Poland. Institute of Physics and Center for Interdisciplinary Nanostructure Scienceand Technology (CINSaT), University of Kassel, D-34132 Kassel, Germany (Dated: November 18, 2019)Single and double paramagnetic colloidal particles are placed above a magnetic square patternand are driven with an external magnetic field precessing around a high symmetry direction of thepattern. The external magnetic field and that of the pattern confine the colloids into lanes parallelto a lattice vector of the pattern. The precession of the external field causes traveling minima of themagnetic potential along the direction of the lanes. At sufficiently high frequencies of modulationonly the doublets respond to the external field and move in direction of the traveling minima alongthe lanes, while the single colloids cannot follow and remain static. We show how the doublets caninduce a coordinated motion of the single colloids building colloidal trains made of a chain of severalsingle colloids transported by doublets.
Biomimetics is used to implement biological functionsto artificial devices, fulfilling tasks in a non biologicalenvironment. Well known examples are artificial swim-mers [1–3] and active systems [4] that can be used to e.g.transport a load. Microscopic dynamics can also be in-spired by large scale transport systems such as trains. Arailroad train is powered either by a separate locomotiveor by multiple units of self propelled equally powered car-riages. In nature the motility of family members of ani-mals trailing behind each other is neither concentrated inthe animal heading the trail nor is it distributed equallyamongst family members. Young goslings trailing behindone of their parents need less but not zero power to followtheir more powerful mother goose [5]. When elephantstravel they walk in a line placing their youngest in be-tween the grownups with a grownup at the head and atthe tail. In the spirit of other bioinspired magnetic col-loidal dynamics [1–3, 6, 7] we generate a biomimetic trainof a collective ensemble of paramagnetic colloids. Singlecolloids are too weak to move on their own along the lineand must be assisted to move by pushing of the train witha larger paramagnetic colloidal doublet. The train is con-fined to an effectively one dimensional lane created via acolloidal potential which is generated by the combinationof a magnetic square pattern and an external magneticfield. The power of each unit in the train is generatedby modulating the external field on a control loop. Col-loidal trains can only move above a square lattice if asufficiently flat potential valley is created by orientingthe external field roughly in direction of a primitive unitvector of the square magnetic pattern.We illustrate the doublet assisted motion of a trainof single colloids using a square magnetic lattice [8, 9],Fig. 1(a). In the experiments, single paramagnetic col-loidal particles or doublets of particles move on a plane above a thin Co/Au layered system with perpendicularmagnetic anisotropy lithographically patterned via ionbombardment [9–11]. The pattern is a square lattice ofmagnetized domains with a mesoscopic pattern latticeconstant a ≈ µ m, see a sketch in Fig. 1(a). The pat-tern is magnetized in the ± z -direction, normal to thefilm. The pattern is spin coated with a 1 . µm poly-mer film that serves as a spacer between pattern andthe colloids. The paramagnetic colloidal particles (diam-eter 2 . µ m) are immersed in water. A uniform time-dependent external magnetic field H ext of constant mag-nitude ( H ext = 4 kAm − ) is superimposed to the non-uniform and time-independent magnetic field generatedby the pattern H p ( H p ≈ − ). The external field isstrong enough such that some of the paramagnetic par-ticles self-assembly into doublets due to induced dipolarinteractions between the single colloidal particles. Thedoublets then align with the direction of the externalfield. Our control space C is the surface of a sphere thatrepresents all possible orientations of the external field.We vary the external field H ext ( t ) with time t performingperiodic closed modulation loops in C .Both H p and H ext create a potential U ∝ − H ext · H p .The potential is a periodic function of the position of thecolloids in the magnetic lattice and it depends paramet-rically on the orientation of the external field in controlspace C . At every time during the modulation loop, thecolloids are attracted toward the minima of the poten-tial. Full details about the computation of U and themotion of single colloids are given in Refs. [8, 9]. Here, webriefly explain the points relevant for the present study.For a square lattice, there exist four special points (fencepoints) in C . The four points represent four directionsof H ext which are parallel and antiparallel to the latticevectors of the square pattern. If the modulation loop a r X i v : . [ c ond - m a t . s o f t ] N ov FIG. 1: Schematic of the setup. (a) Single and double spherical colloidal particles are placed on top of a lithographic magneticsquare pattern (lattice constant a ≈ µ m) of up (white) and down (black) magnetized domains along the z − axis. An isolatedsingle colloid (gray) does not move at the modulation frequency applied in the experiments. Doublets are able to respond to theexternal field and can also induce motion of single colloids. The color (blue, green, and orange) of the doublet and of the singletrepresents the position at three different times ( t , t , and t ) during the modulation loop. (b) Magnetic potential U createdby the pattern and the external potential for four different times ( t i , i = 1 , , ,
4) during one modulation loop (red,blue,green,and orange) of the external field. The ratio between the large and the small barriers of the potential is ∆
U/δU ≈
10. (c) Eachpoint in control space C (gray sphere) corresponds to a different orientation of the external field. The experimental modulationloop is highlighted in purple. The loop winds around a fence point (cyan) of control space. The colors of the four potentials in(b) and the coloring of the moving colloidal positions (a) correspond to the four colored points on the loop in control space.FIG. 2: Scaled magnitude of the velocity of singlets (orangesquares) and doublets (blue circles) as a function of the scaleddriving frequency. The black solid line is a fit of the singletvelocity using the generalized Adler equation (1). The verti-cal lines indicate the scaled experimental frequency ω ext andthe topological frequency ω t below which the motion is topo-logically protected. of H ext in C winds around one of the fence points, thenthe minima of the potential move one unit cell abovethe square pattern [8, 9]. The motion is topologicallyprotected, with the set of winding numbers around thefence points defining the topological invariants.If H ext points in the direction of a fence point, themagnetic potential U is effectively one-dimensional with valleys along the direction perpendicular to H ext . Forexample, let the lattice vectors of the magnetic patternpoint along x and y , Fig. 1(a). The magnetic potential U exhibits deep valleys along x , Fig. 1(b), when H ext points along − y , Fig. 1(c). If H ext slightly deviates fromthe direction of a fence point, see modulation loop inFig. 1(c), then secondary minima of U appear along thedirection of the valleys, Fig. 1(b). The variation of thepotential along the valley δU is much smaller than in thetransversal direction ∆ U . In our experimental setup, wefind ∆ U/δU ≈
10. Modulating the external field direc-tions along a loop that encloses a fence point, Fig. 1(c),causes the minima of U to travel by one unit vector ofthe lattice upon completion of the loop. The frequency ofthe loop ω can be chosen such that single colloids cannotfollow the potential minimum on their own but doubletsmove in direction of the traveling minima. If a doubletis on a collision course with a single colloid, then thedoublet can render the single colloid mobile and drive itthrough the potential valley, Fig. 1(a).In Fig. 2 we plot the speed of an isolated single colloidand that of an isolated doublet versus the driving angularfrequency ω of the modulation loop in C . At low frequen-cies, lower than the topological critical frequency ω t , themotion is adiabatic and topologically protected. Singletsand doublets follow the potential minimum at all times.Hence, the displacement caused by a loop is topologi-cally locked to the primitive lattice vector and particlesmove at the topological speed given by the lattice con-stant a times the frequency, i.e., v t = ωa/ π . However, FIG. 3: Time sequence of microscope images of the pattern illustrating the motion of colloidal trains. Each train has beenartificially colored differently. The images correspond to the times (a) t = 0 s, (b) t = 8 s , (c) t = 13 s (d) t = 15 s (e) t = 17 s(f) t = 23 s (g) t = 25 s and (h) t = 30 s. The period of one modulation loop is 2 π/ω exp ≈ .
12 s. Scale bar is 20 µ m. A sketchof the pattern has been superimposed in (e). A video clip recorded at twenty frames per second showing the motion of trainsand the non motion of singlets is provided in the supplemental material (adfig3a.mp4). A second video clip (adfig3b.mp4)recorded at sixty frames per second shows a time-resolved slow motion of the doublet during the course of a few modulationloops. at higher frequencies ω > ω t the speed drops below v t ,mostly due to viscous and adhesive forces impeding themotion. For singlets, the speed decreases with increasingfrequency until the critical frequency ω c is reached. At ω c the isolated single colloids stop moving. The speedof the singlets is well described by a generalized Adlerequation [12] v/v t = ω ≤ ω t − (cid:113) ( ω − ω t )(( ω c − ω ) ω t + ωω c )( ω c − ω t ) ω if ω t < ω < ω c ω ≥ ω c (1)The force due to the potential acting on a doublet isroughly twice the force acting on a single colloidal parti-cle. The viscous friction on the doublet, however, is lessthan twice the friction of a single colloid because of hy-drodynamic interactions [13]. Hence, the doublet can stillmove at frequencies higher than the critical frequency ofthe singlets, and we have a regime where the doubletmoves while the singlet is at rest. The experiments areperformed at an angular frequency of ω ≈ s − > ω c ,such that singlets do not move, and doublets move witha speed of roughly one eighth of the topological speed( v d /v t ≈ . x -direction with the doublets pushing the sin-glets. Nothing particular happens to the cyan train with FIG. 4: Speed of the train (scaled with the speed of a doublet v d ) versus the load, i.e. the number of singlets in the train n s .Data for different numbers of doublets in front of the train: n d = 0 (orange), n d = 1 (blue), n d = 2 and 3 (yellow). one doublet and one singlet moving through the field ofview at the doublet speed v d . The green train with onedoublet and three singlets moves with half the doubletspeed, Fig. 3(a-b), collects two further singlets, Fig. 3(c),and stalls, Fig. 3(d), until the two singlets close to thepushing doublets form a second doublet, Fig. 3(e). Thisincreases the power of the train such that it resumes tomove, Fig. 3(e-f) at the doublet speed. In Fig. 3(g) athird doublet is formed leaving only one singlet in thegreen train before it exits the field of view. Interestingly,when the green train passes the red singlets (sitting on FIG. 5: Time sequence of microscope images showing a dou-blet (green) moving a singlet (green) out of its way. The othersingle colloids (red) remain at rest. Scale bar is 10 µ m. Theimages correspond to the times (a) t = 0 s, (b) t = 2 s, and(c) t = 4 s. A video clip of the event recorded at twentyframes per second is provided in the supplemental material(adfig5.mp4). the next track to the right) the front immobile red sin-gle colloid is mobilized and performs a single translationby one unit vector in the positive x -direction, comparethe position of the red colloids next to the green train inFigs. 3(e) and 3(f). The yellow train originally consistsof one doublet and two singlets, Fig. 3(a). It collectstwo further singlets, Figs. 3(b,c), and then moves as onedoublet and four singlets train at a relatively slow speed v ≈ . v d through the image. No further doublets areformed from the singlets of the yellow train as it moves.In Fig. 4 we plot the speed of a train as a function ofboth the number of singlets in front of the train and thenumber of doublets at its tail. A train with no doubletis immobile and a train with more than one doublet canpush up to five single colloids at the unloaded doubletspeed. For one doublet we see a gradual transition frommotion at the doublet speed v d for trains with up to twosinglets toward no motion for trains with five singlets.Remarkably, none of the trains ever derails. This isdue to the special properties of the confining colloidalpotential which are inherited from the square pattern.In Fig. 5 we show the colloidal motion on a glass slidewith no magnetic confining pattern. Doublets also movewhen performing a modulation loop without the mag-netic pattern (albeit not by a unit vector) and singletsare generically at rest. However due to the absence ofthe confining potential, when one doublet moves onto asinglet, the singlet does not stay on track but is pushedto the side to let the doublet pass.So far we have shown the coordinated motion of col-loids in one direction. However, by changing the globalorientation of the driving loop (winding around otherfence points in C ) we can force the doublet to move alongany of the four symmetry directions of the magnetic pat-tern. Hence, the doublet-induced motion of single col-loids can potentially be used to arbitrarily set the po-sition of the singlets across the pattern. A step in thisdirection is shown in Fig. 6 where a complex modulationloop is programmed to clean the surface of singlets. FIG. 6: Surface cleaning. (a) Schematic of the trajectory of adoublet. Sequence of microscope images showing the cleaningof an area from single colloids (orange) by meandering dou-blets (blue) taken at times t = 0 (b), t = 2 min (c), and t = 6min (d). The scale bar is 20 µ m. A video clip of the event isprovided in the supplemental material (adfig6.mp4). Our colloidal trains are immersed into a low Reynoldsnumber liquid where the propulsion of shape changingobjects is governed by the area enclosed by the loopin shape space of the object [14, 15]. Swimmers areable to move by changing their shape. In contrast ourbiomimetic colloidal trains are driven by the shape ofthe potential created by the pattern and the externalfield which creates the topological nature of this classi-cal non-adiabatic phenomenon. We have shown in ref-erences [8, 9, 16–20] that, like other classical topologi-cal transport phenomena [21–27], there exist similaritieswith quantum mechanical topological transport [28, 29].We have demonstrated how long range many-body in-teractions between the well separated colloidal particlescan help sustain the topological nature of the transportup to higher frequencies of driving. 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