Counter-rotation of magnetic beads in spinning fields
Jean Farago, Thierry Charitat, Alexandre Bigot, Romain Schotter, Igor Kulic
CCounter-rotation of magnetic beads in spinning fields
Jean Farago, Thierry Charitat, Alexandre Bigot, Romain Schotter, and Igor Kulić ∗ Université de Strasbourg, CNRS, Institut Charles Sadron UPR-22, Strasbourg, France (Dated: October 6, 2020)A magnetic stirrer, an omnipresent device in the laboratory, generates a spinning magnetic dipole-like field that drives in a contactless manner the rotation of a ferromagnetic bead on top of it.We investigate here the surprisingly complex dynamics displayed by the spinning magnetic beademerging from its dissipatively driven, coupled translation and rotation. A particularly stunningand counter-intuitive phenomenon is the sudden inversion of the bead’s rotational direction, from co-to counter-rotation, acting seemingly against the driving field, when the stirrer’s frequency surpassesa critical value. The bead counter-rotation effect, experimentally described in [
J.Magn.Magn.Matter , , 376-381, (2019)], is here comprehensively studied, with numerical simulations and a theoreticalapproach complementing experimental observations. I. INTRODUCTION
The broad availability of magnetic neodymiumbeads has led to an increased "table top" experimentalinterest in their self-assembly and individual dynamics[1–4]. Merely rolling such a bead on the top of the ta-ble leads to surprisingly complex behavior, with beadtrajectories that depend on the motion speed and in-clination to the local earth magnetic field [5]. In themicroscopic realm, the interaction of ferro- and super-paramagnetic beads with external fields and surfaceconfinement has been extensively investigated. Spa-tially inhomogeneous, dynamic magnetic fields thatare linearly propagating [6] or rotationally spinning[3, 7] along surfaces have been investigated. The con-finement of magnetic or magnetizable objects to solid[3, 6] and fluid interfaces [8, 9] under oscillating fieldsleads to an intricate phenomenology including self-assembly and self-propulsion [10].A permanently magnetized object placed in a non-uniform field, moves to minimize its magnetic free en-ergy via two mechanisms: a) By aligning its magneticmoment with the field direction and b) by moving to-wards the maximum of field intensity. However, whencoupled to a substrate the two otherwise independentdegrees of freedom become tightly coupled giving riseto subtle and often counterintuitive effects. Here weinvestigate one such easily reproducible, yet stunningeffect : A neodymium bead placed on top of a lab-oratory magnetic stirrer. When the stirrer’s magnetrotates at slow rates the bead naturally follows thefield. However, surprisingly, once the field rotates fastenough the bead inverts its direction and rolls, to thesurprise of the observer, in the opposite direction -against the driving field direction. This effect was re-cently described by Chau et al. [11] and in a relatedform by Gissinger [7]. ∗ Also at Leibniz Institute for Polymer Research (IPF), 01069Dresden, Germany.
In this paper, we revisit the experiments of Chauet al. [11] with a comprehensive approach which com-bines experiments, numerical simulations and theoret-ical analysis. The benefit of numerical simulations isto provide a complete description of the rotation ofthe beads, which are quite complicated to access ex-perimentally. We performed experiments, rather sim-ilar to the “opposite polarity case” described in [11],but simpler as we used a usual magnetic stirrer (com-monly found in chemical laboratories) to provide therotating magnetic field. We then wrote down the dy-namical equations of a magnetic bead moving on thehorizontal plane of the stirrer, assuming a viscous fric-tion between the bead and the stirrer. The featuresof the magnetic field of the stirrer have been carefullymodeled, in order to reproduce as faithfully as possiblethe experiments. We simulated the dynamical equa-tions obtained and by fitting two parameters of themodel, we were able to reproduce qualitatively andquantitatively our experimental results. A theoreticalanalysis of the motion allows us (i) to understand theradial stabilization of the bead in its counter-rotatingmotion, (ii) to confirm the asymptotic role of the freerolling at large driving angular velocities, and (iii)to show that counter-rotating motion is not possiblewhen purely paramagnetic beads (without remanentmagnetization) are used.The paper is organized as follows. In sections IIand III, the system is introduced and our experimen-tal results are presented. The dynamical equations ofthe system and the modeling of the rotating magneticfield are derived in section IV. Two typical motionsare then discussed in detail in section V. A theoret-ical discussion follows in section VI. For the sake ofcompleteness, equations for the rotational dynamicsof the bead in spherical coordinates are reported inAppendix IX A. a r X i v : . [ phy s i c s . c l a ss - ph ] O c t FIG. 1. Sketch of the system and the main notations usedthroughout the paper. θ and φ are used to orient the mag-netic axis n of the bead : n = cos( θ ) e r +sin( θ )(sin( φ ) e α − cos( φ ) e z ) . II. DESCRIPTION OF THE SYSTEM
The system, depicted in fig. 1, is a ferromagnetic(neodymium) sphere of mass m = 0 . g, radius R = 2 . mm and magnetic moment along a diam-eter µ ( t ) = µ n ( t ) (with n ( t ) a unit vector goingfrom the south pole of the magnet to the north pole),placed on the stirrer surface (substrate). The lat-ter is immobile in the laboratory frame ( O, e x , e y , e z ) and located at the height z = 0 . The position ofthe sphere is described by a two-dimensional vector r = x e x + y e y = r e r , its vertical coordinate stay-ing at z = R . In the following, the local polar vec-tor basis is denoted by ( e r , e α ) and the correspondingcylindrical coordinates by ( r , α , z ) .Below the substrate, at the coordinate z = − h m = − . mm, a permanent magnet, of approximatelyrectangular shape (with a length (cid:96) m = 56 mm, awidth w m = 40 mm, and thickness mm), rotatescounterclockwise around the axis Oz with a constantangular speed ω . Its time-dependent magnetic field B influences the spherical bead via the interaction po-tential V ( n , r , t ) = − µ n · B ( x , y , R, t ) . Notice thatthis interaction potential is an approximation whichassumes that the magnetic field can be considered con-stant over the volume of the sphere. III. EXPERIMENTAL RESULTS
We recorded the bead trajectories using a high-speed camera (Phantom Miro LC320S) at 200 fps,and extracted them using the tracking features of theopen-source software Blender, see fig. 2 [12]. We usedthe reflection of a laser beam on a mirror glued on the
FIG. 2. Example of a counter-rotating trajectory trackedby Blender (coordinates in pixels). The frequency of therotating magnet is ω / π = 10 . Hz. rotating magnet (cf. fig. 1) to measure precisely therotating frequency ω .At very low rotation frequencies, the bead istrapped above one of the two poles of the rotatingmagnet (where the accessible magnetic field is maxi-mum), and the bead’s motion is trivially a corotationat the same angular speed as the rotating magnet.The proper rotation of the sphere is largely dictatedby the requirement that the bead axis stays parallelto the local magnetic field. This proper motion yieldsa substantial friction between the bead and the hor-izontal slab of the stirrer. Beyond a certain angularvelocity ω c , the friction force is too high and the beadis no longer trapped in the magnetic potential. Thetypical value of v c is obtained by balancing the fric-tion force mγRω c with the magnetic force at stake µB/(cid:96) . We get ω c (cid:39) µB/ [ mγR(cid:96) ] . Fed with typicalnumerical values of our experiment (see values givenabove and in section IV A), we obtain ω c / (2 π ) (cid:39) Hz,which is in accordance with what we observed in ourexperiment. For frequencies slightly above this limit-ing value, the behaviour is complex, mainly chaotic.This window is however rather narrow, and as can beseen in fig. 2 for ω c / (2 π ) (cid:39) Hz, a regular regime(with some precession) sets in, where the global mo-tion of the bead is counter-rotating, with a pattern, whose amplitude is large at small frequencies andshrinks at higher frequencies (see fig. 9, which corre-sponds to ω / (2 π ) = 19 . Hz. We term these patterns“festoons” in the following because of their similaritywith the garland-like adornment of some architecturalfriezes [13].We measured as a function of the magnet rota-tion angular frequency ω (i) the mean radius r of the trajectories (ii) its standard deviation δr = (cid:112) (cid:104) r (cid:105) − (cid:104) r (cid:105) and (iii) the mean angular velocity ˙ α of the bead. FIG. 3. Blue circles : Mean radius (cid:104) r (cid:105) of the tra-jectory (left plot). Red squares : Standard deviation (cid:112) (cid:104) r (cid:105) − (cid:104) r (cid:105) (right plot). Both curves are plotted asa function of the magnet frequency ω / π . Only thecounter-rotative regime is shown. The dashed lines area guide for the eye.FIG. 4. Mean bead frequency (cid:104) ˙ α (cid:105) / π as a function ofthe magnet frequency ω / π . Notice that (cid:104) ˙ α (cid:105) is negativein the regime of counterrotation. The lines are a guidefor the eye and the inset is a zoom of the counterrotatingregion. The mean radius and standard deviations are shownin fig. 3. The large standard deviation at small fre-quencies come from the large festooning of the trajec-tory, as can be seen in the example of fig. 2. When the frequencies become too small, the festoons cannotgrow indefinitely and a chaotic behaviour is insteadobserved (for still lower frequencies, a co-rotativelocked motion is recovered). At larger frequencies, thefestoons are still present, but with smaller amplitudes.In fig. 4 we plot the revolution frequency of thebead (cid:104) ˙ α (cid:105) / π against the magnet frequency ω / π .The corotative low frequency regime is rather obvi-ous and described by ˙ α = ω , since it correspondsto the bead locked in one or another potential energyminimum, and constrained to follow the rotation ofthe magnet with quite a lot of frictional dissipation.At larger frequencies, after a chaotic transitional zone(where nothing relevant has to be reported, the beadbeing most of the time ejected from the stirrer table),the counterrotative region (characterized by negative ˙ α ), the typical values of | ˙ α | are one order of magni-tude smaller than ω , which is qualitatively explainedby the mechanism which allows the counter-rotation: On the one hand, a frequency locking occurs be-tween the rotation of the sphere around itself at anangular velocity, say (cid:104) ˙ φ (cid:105) , and ω : (cid:104) ˙ φ (cid:105) ∼ ω (Forthis qualitative argument, there would be no need todefine precisely ˙ φ , but a precise definition can be any-way given by looking at the definition of φ in fig. 1).On the other hand, at large values of ω , the effect ofthe magnetic field averages rapidly to zero, so the mo-tion must converge to a free rolling (albeit constrainedinto a circular motion) for which the friction on thetable, proportional to the coincidental point velocity ∼ r ˙ α + R ˙ φ , is approximately zero. As a result, wehave ˙ α ∼ − ω R/ (cid:104) r (cid:105) ∼ − ω ( R/(cid:96) ) , for which in ourcase we have moreover R/(cid:96) (cid:39) . . The other salientfeature of the right branch of fig. 4 is its bell shapewith a maximum around ω / π = 16 Hz. This min-imum signals a crossover between a complex regimewith few large festoons which act as shortcuts duringrevolutions (and therefore enhance the absolute valueof the revolution frequencies), and a second regime( ω / π > Hz) where the trajectories are close tocircles, with many small festoons. For this regime, theprevious arguments leading to ˙ α (cid:39) − ω ( R/(cid:96) ) applyand explain the enhancement of | ˙ α | . IV. MODELLING
To achieve a comprehensive description of the beadmotion in the counter-rotative regime, we develop adetailed theoretical model that we solve numerically.Let us term Ω the rotation vector of the bead inthe laboratory frame. We have for the time derivativein this frame d n /dt = Ω × n which implies that Ω = n × ( d n /dt ) + ˙ ψ n where ˙ ψ is the rotationvelocity of the bead around its magnetic axis n .The location of the center of the sphere is givenby the cylindrical coordinates ( r , α , z = R ) , suchthat the velocity of its center of mass is v G = ˙ r u r + r ˙ α u α . From the Koenig’s theorem, we get a uncon-strained Lagrangian L uncstr . ( n , r , α , ˙ n , ˙ r , ˙ α , ˙ ψ , t ) = m r + r ˙ α ] + 15 mR (cid:18) [ d n dt ] + ˙ ψ (cid:19) + µ n · B ( r , α , t ) (1)where we used the expression mR / for the iner-tia moment of the sphere with respect to one di-ameter. Notice that the constraint n = 1 affectsthe vector n for all times, so that the actual La-grangian which describes the frictionless dynamics is L = L uncstr − Λ( t ) n , the function Λ( t ) being theLagrange multiplier associated to this constraint.The dynamics of the sphere is affected by a possi-ble friction of the sphere on the table (it is not con-strained to roll only). This friction is modelled by aforce F fr proportional to the velocity V of the coinci-dental point I (the point of the sphere in contact withthe table at any instant) : F fr = − mγ V = − mγ ( v G − R Ω × e z ) (2)This friction can be incorporated in a Lagrangian de-scription by means of the so-called Rayleigh function F = m γ V (3)which modifies the Lagrange equations to ddt ∂ L ∂ ˙ q = ∂ L ∂q − ∂ F ∂ ˙ q (4)for all variables q describing the dynamics.The lengths are made dimensionless by defining r = r /(cid:96) where (cid:96) is a characteristic length of therotating magnet (we will choose (cid:96) slightly differentfrom the actual length of the magnet (cid:96) m , as explainedbelow). The magnetic field is also normalized ac-cording to B = B /B ( B a characteristic magneticfield intensity of the rotating magnet). One defines ε = R/(cid:96) and κ = B µ/ [ mR γ ] . This last con-stant can be interpreted as the square of the charac-teristic time of friction times the magnetic pulsation (cid:112) µB /mR . A small value of κ means that frictionwill likely overdamp the oscillations caused by B andits time evolution. Finally the time is normalized byrenaming γt by t . The outcome of these generalizeddissipative and dimensionless Lagrange equations for q ∈ { n , ˙ ψ = γ − ˙ ψ , r, α = α } is Ω = γ − Ω = n × d n dt + ˙ ψ n , (5) V = ( (cid:96)γ ) − V = ˙ r e r + r ˙ α e α − ε Ω × e z , (6) ¨ n = − ( ˙ n ) n − ε ( V × e z ) × n + 5 κ B − n ( n · B )] , (7) ¨ ψ = − ε V · ( e z × n ) , (8) ¨ r = r ˙ α + κε n · ∂ B ∂r − V · e r , (9) d ( r ˙ α ) dt = κε n · ∂ B ∂α − r V · e α . (10)Notice that the derivative of B with respect to α in-cludes the derivative of the unit vectors ( e r , e α ) aswell as that of the coordinates ( B r , B α , B z ) in case ofa representation with cylindrical coordinates.To model the magnetic field created by the rect-angular rotating magnet, we assume it can be de-scribed by five parallel and equidistant lines of mag-netic dipoles characterized by a constant dipolar line-density d M /dx (cid:48) and a length (cid:96) . Their length (cid:96) isclose to (cid:96) m , but is adjusted so that the two maximaof the magnetic field on the plate are separated bythe same distance (41 mm) in the experiment and themodelling. We found (cid:96) = 0 . (cid:96) m = 23 mm. On theother side, the distance between the two extremal linesare fixed to be exactly at w m - the width of the actualmagnet. The actual portrait of the magnetic field ex-perienced by the bead in shown in fig. 5. The fieldcreated at the location r = ( x , y , R ) by a singlemagnetic line of length (cid:96) , directed along the hori-zontal unit vector e (cid:48) x , and symmetrical with respectto the point (0 , , − h w ) is: B ( r , t ) = B (cid:20) r /(cid:96) + ( h /(cid:96) ) e z − s e x (cid:48) ( t ) | r /(cid:96) + ( h /(cid:96) ) e z − s e x (cid:48) ( t ) | (cid:21) s =+1 s = − (11)where B = µ π(cid:96) d M dx (cid:48) and h = h w + R . The nor-malized version B of this field is straightforwardlyobtained by dividing by B and writing the righthand side in terms of r = r /(cid:96) and h = h /(cid:96) .The time dependence of the magnetic field is entirelyborne by the vector e (cid:48) x which rotates counterclock-wise in the laboratory frame ( O, e x , e y ) : e (cid:48) x ( t ) = cos( ωt ) e x +sin( ωt ) e y . Similarly the ( r, α ) dependenceis given by the term r = r e r = r [cos Φ e (cid:48) x + sin Φ e (cid:48) y ] with Φ = α − ωt , see fig. 1. FIG. 5. Magnetic field generated by five magnetic linesparallel to −→ e (cid:48) x , of normalized length 1, regularly placed at y = nλ, n ∈ {− } , with λ = w m / (4 (cid:96) ) = 0 . . A. Numerical simulations results
We simulated the dynamics of the bead embodiedby equations (7-10) using local spherical coordinatesto represent n , namely n = cos θ e r + sin θ (sin φ e α − cos φ e z ) . The dynamical equations in these coordi-nates are given in appendix IX A. To compare quan-titatively experiments and simulations, we have tofix the values of the parameters ε , κ and γ , the lat-ter being involved in dimensionless angular velocities ω = ω /γ and ˙ α = ˙ α /γ for instance . ε is an im-posed geometric parameter : ε = R/(cid:96) = 0 . . Theother two are rather difficult to determine from ex-periments, all the more so the actual friction withinthe experiment is a solid friction, in contrast withour modeling of a linear viscous one. As a result,we chose to adjust κ and γ so as to fit the experi-mental result at best. We found κ = 1 . and γ =312 . s − . It gives a value µB ∼ . − J. For a typ-ical neodymium bead, we have µ ∼ R × BH max /B rem ,where BH max ∼ J · m − is the maximum energyproduct and B rem ∼ T the remanence. We get here µ ∼ − A · m − and B ∼ − T.With these values the agreement between experi-ment and simulations is quite quantitative, as can beseen in fig. 6.
FIG. 6. Comparison between experiments (symbols) andsimulation (solid lines). Left ordinate and blue circles: Mean normalized radius (cid:104) r (cid:105) . Right ordinate and redsquares : Mean normalized bead angular velocity (cid:104) ˙ α (cid:105) . In-set : Standard deviation δr = [ (cid:104) r (cid:105) − (cid:104) r (cid:105) ] / vs. ω . V. ANALYSIS OF MOTION
FIG. 7. Simulation for the frequency ω = 0 . corre-sponding to the experiment of fig. 2. The color codes forthe intensity of n z and the arrows show the direction andthe relative length of the horizontal component of n . A. 4-fold counter-rotation
The advantage of numerical simulations is to pro-vide easily the rotation of the bead. We first anal-yse the case represented in fig. 2 corresponding to ω = 0 . . In fig. 7, the simulated trajectory is repre-sented with a color code corresponding to the values of n z ∈ [ − , (A movie of the simulation can be foundin Supplemental Material [ film_0215.avi ], where themotion of both the bead and the rotating magnet areshown). Small gray arrows are also added to see thedirection and relative length of n − n z e z . It is worthnoting that the mean value of n z is not zero, whichindicates that a different, conjugate solution at thisfrequency exists where polarities of n and B are si-multaneously reversed. In this case the corners of thesquare-like shape of the trajectories would correspondto minima n z (cid:39) − . It is worth mentioning that in FIG. 8. Same as fig. 7 (upper right corner only), thecolor codes showing in the top plot the angle between themagnetic axis of the bead and the local magnetic field (indegrees), and in the bottom plot the magnitude of the co-incidental point velocity, proportional to the friction forceexperienced by the bead. this motion, the bead’s moment stays remarkably par-allel to the magnet field, as can be seen in the fig. 8(top) : Their relative angle does not exceed 4 degrees.The bottom plot shows the magnitude of the coinci-dental velocity. It can be seen that the high frictionzones are tightly correlated to those (rare) momentswhere the bead axis cannot follow the rate of varia-tion of the magnet’s field. It is also interesting to notethat the magnitude of the magnetic field experiencedby the bead during its revolution does not vary morethan 13% with respect to its mean value (not shown).Figure 8 is interesting also because it allows a dis-cussion on the linear viscous friction hypothesis. Why indeed does the model with a viscous friction describeso well the experiments ? We see from the figure 8(bottom) that the friction force acts on quite localizedmoments of the trajectory, close to the turning points.So one can expect that some dynamical details of thetrajectories in the vicinity of the turning points maybe only approximately accounted for (for instance, theturning points in the real ω = 0 . case (see for in-stance fig. 2 where the turning points look sharperthan those of fig. 7). On the other hand, the restof the trajectories should be correctly described, pro-vided the characteristic times associated to the energydissipation of both friction mechanisms be compara-ble. For a dry friction with parameter µ dyn , the char-acteristic time of dissipation is τ ∼ I ˙ φ / ( µ dyn Rmg ) where ˙ φ ∼ ω . For a viscous dissipation, one has in-stead τ ∼ I/ [ mR γ ] . On equating both expressions,one finds γ (cid:39) µ dyn g/ [ Rω ] . So, in principle γ de-pends on ω , but in the range ω (cid:39) Hz where thefestoons are well developed and therefore the frictionis not negligible, it gives γ (cid:39) Hz, i.e. the or-der of magnitude we got for γ by direct fit (we found γ = 312 . s − ). B. Many-fold counter-rotation
FIG. 9. Trajectory of the magnetic bead for ω = 0 . .The color modulates according to the magnitude of thefriction force. The inset (solid blue line) shows the closedtrajectory of the vector n in the local coordinate frame ( e r , e α , e z ) . For increasing values of ω , the counter-rotationtends to adopt a more circular shape, as can be seenfrom fig. 9 where the trajectory of the bead for ω = 0 . is shown (a movie of the simulation can befound in Supplemental Material [ film_04.avi ]). Thefestoons are numerous (15 for ω = 0 . ) and have asmall amplitude. As before, the magnetic bead staysremarkably parallel to the local magnetic field (theangle is never larger than 3 ◦ ), and in the local frame ( e r , e α , e z ) , the trajectory is closed and almost cir-cular, with the magnetic moment vector displaying amild tilt ( (cid:39) ◦ ) with respect to the vertical. C. Order of the patterns and period halving
As can be seen in figs 7 and 9, (i) the patternshave different orders corresponding to the number offestoons they are made of and (ii) in general, thepatterns are slowly shifting and do not superposewhen the bead has completed a revolution. The cri-terion to have a truly periodic motion in the labo-ratory frame, with N festoons reads π/ ( N | ˙ α | ) =(2 π/ω )(1 − /N ) ⇔ N = a (1 + ω / | ˙ α | ) with a = 1 ifthe bead has a periodic motion in the rotating mag-net frame requiring a full turn, i.e. r (Φ + 2 π ) = r (Φ) (symmetry S1). The possibility of a period halv-ing exists if the trajectory has the finer symmetry r (Φ + π ) = − r (Φ) (symmetry S2, with the rota-tional reversal n (Φ + π ) = − n (Φ) ). In this casethe number of festoons is related to the frequency by N = a (1 + ω / | ˙ α | ) with a = 2 . By trial and er-rors we were able to find in the numerical simulationsthe frequencies ω for which the pattern is approxi-mately periodic in the laboratory frame. The resultsare shown in the table V C. One sees that on enhanc- frequency ω a a ( ω/ | ˙ α | + 1) ing ω , a period halving transition occurs in the range ω ∈ [0 . , . , going from S1-invariant trajecto-ries to S2-invariant ones. The precise location of thetransition has been found for ω (cid:39) . as can beseen in figure 10, where the order parameter p hasbeen chosen as p = ∆ + / ∆ − − , namely the larger-than-one ratio of the amplitudes of two successive os-cillations of the parameter r (minus 1). One sees thatthe transition is supercritical, which is confirmed bythe diverging relaxation time associated with the con-vergence of p near the transition. FIG. 10. Supercritical transition of period halving near ω = 0 . . The inset shows r ( t ) in the p (cid:54) = 0 region inorder to highlight the definition of p as the ordered ratioof two successive amplitudes in r ( t ) . VI. TIME AVERAGINGS
The full dynamical behaviour of the bead is com-plicated and certainly non integrable.One can nevertheless try to make some predic-tions concerning the mean radius of counter-rotationand the associated rotation frequency, at least in theregime where r stays reasonably constant.The map of the field shows that close to the maxi-mum, it has essentially an (slightly tilted) orthoradialstructure. It is thus reasonable to assume that in thedynamical regimes where r (cid:39) , one can neglect theradial dependence of n , assume θ (cid:39) π/ and write n (cid:39) sin φ e α − cos φ e z . Likewise, we neglect also thefluctuations in r and ˙ α . Writing again r and ˙ α for thetemporal averages (cid:104) r (cid:105) and (cid:104) ˙ α (cid:105) , we have r ˙ α + κε ∂ r (cid:104) B α sin φ − B z cos φ (cid:105) − (cid:104)V r (cid:105) , (12) κε (cid:104) [ ∂ α B α + B r ] sin φ − ∂ α B z cos φ (cid:105) − r (cid:104)V α (cid:105) , (13) κε (cid:104) B α cos φ + B z sin φ (cid:105) − (cid:104)V α (cid:105) . (14)In the approximation considered, one has, from theformula (20) of the Appendix IX A, V α (cid:39) r ˙ α + ε ˙ φ .The equation (12) shows that the centrifugal force iscounterbalanced by a magnetic force only if φ oscil-lates with the same frequency as Φ = α − ωt . Thisleads us to assume φ = − Φ + χ where χ is a constantphase. One can show (but the calculation is cumber-some) that the averages implying the magnetic fieldin eqs (13) and (14) are all ∝ sin( χ ) for symmetryreasons, whereas that of (12) is ∝ cos( χ ) . The solu-tion of these equations is therefore somewhat simpli- FIG. 11. Test of the formulas (15) : The solid lines show (cid:104) ˙ φ (cid:105) (blue, left ordinate) and (cid:104) ˙ α (cid:105) (red, right ordinate) andthe dashed show the result of (15). fied, since they reduce to (i) χ ≡ modulo π and (ii) (cid:104)V α (cid:105) = 0 and (iii) eq. (12). Actually, one can guessin advance that the phase locks to χ = π , becauseit corresponds to the most stable situation where thebead visits the region of maximum magnetic field inthe orientation which minimizes the magnetic energyinteraction. Combining ˙ φ = ω − ˙ α and r ˙ α + ε ˙ φ = 0 ,we obtain ˙ α = − εωr − ε and ˙ φ = ωrr − ε . (15)The comparison of these formulas with the actual av-erages of ˙ φ and ˙ α are shown in fig. 11 and the result isconvincing for ω (cid:62) . , that is for frequencies higher than those obtained in the experiments of fig. 6. Thismeans that the hypothesis of free rolling correspond-ing to the relation r ˙ α + ε ˙ φ = 0 is quantitatively correctonly at quite large frequencies. Regarding the predic-tion for the mean value of r , one would use eq. (12),but this equation would be tractable only if (cid:104)V r (cid:105) isnegligible with respect to the other terms, since theexpression (19) for V r contains a term − ε ˙ ψ sin θ sin φ addressing directly the rotation of the bead around itsmagnetic axis, a motion that is coupled to all degreesof freedom. As can be seen from the inspection offig. 12, the friction term −(cid:104)V r (cid:105) is not at all negligiblein the regime ω < . and becomes negligible with re-spect to the other two only at quite higher frequencies.As a result, one concludes that the quantitative fea-tures of the counter-rotating regime cannot be simplyobtained in the moderate driving frequencies wherethe festooning of the trajectories is marked.A final comment can be made about an implicitchoice made in assuming φ = − Φ + χ , a relation dic-tated by the requirement that φ and Φ must leads to FIG. 12. Evolution with ω of the three terms of eq. (12).“centrifugal” refers to r ˙ α , “magnetic” to κε ∂ r (cid:104) B α sin φ − B z cos φ (cid:105) and “friction” to −(cid:104)V r (cid:105) . The dotted yellow curveshows ε (cid:104) ˙ ψ sin θ sin φ (cid:105) , the term of −(cid:104)V r (cid:105) depending on therotation of the bead around its magnetic axis. All curvesare divided by ε . resonant terms in the magnetic force. There is herean implicit because φ = Φ + χ would have been alsoa valid choice. In Appendix IX B it is shown why thisAnsatz, which would give a co-rotative regime, is ac-tually never observed. A. Physical origin of the festoons
FIG. 13. Fluctuations of r (solid blue) and | B | (dashedred) during half a revolution of the magnet. The abscissais the angle between the bead and the rotating magnet.Notice that when the magnet and the bead are on top ofeach other, the value of r is minimal and is . , i.e. thelocation of the maximum of the field. On the qualitative level, the origin of the festoonscan be understood if one realizes that the magneticaxis of the bead stays always nearly colinear to the lo-cal magnetic field. As a result, the effective magneticforce for the bead’s center of mass is high near theends of the magnet where the field varies substantiallyover a short distance. As shown in fig. 13, one seesthat the radius is minimal, around . (the locationof the absolute maximum of field), when the magnetcrosses the bead angular position. When the magnetaxis goes away from the bead angular position, thefield variations weaken, the centrifugal force “wins”and drives the bead away from r = 0 . , whence themaximum of r at precisely α − ωt = π/ . However,the detailed shape of the festoons cannot be accountedfor by such a simple force balance argument, becausein the vicinity of the maxima of r , the friction force isno longer negligible in the budget, as can be seen infig. 8 (bottom). B. Paramagnetic bead
As correctly noticed in [11], the mechanism for thecounter-rotation proposed by [7] relies on the pres-ence of a remanent magnetization in the beads, andthe counter-rotation observed with steel beads wouldbe entirely due to it. With the theory presented inthis work and summarized by equations (5-10), it ispossible to test an ideal case where the magnetic inter-action would be solely paramagnetic. It amounts toreplacing the interaction potential in the Lagrangianby V = − α m B . The most important consequenceof this new interaction is that the rotational dynam-ics of the bead is now decoupled from the magneticfield by direct interaction, that is the term ∝ κ in (7)disappears. All the arguments put forth previously toaccount for the counter-rotation are no longer valid,and we indeed never observed counter-rotation in sim-ulations of the purely paramagnetic bead.To have a theoretical indication (not a proof) ofwhy the counter-rotating stationary is generally sup-pressed, we consider the time evolution of the energyfunction [14] h = ˙ q∂ L /∂ ˙ q −L is dh/dt = − F − ∂ L /∂t which yields after time averaging and noticing that ∂ t B = − ω∂ α B (cid:104)V r + V α (cid:105) = α m ω (cid:28) ∂B ∂α (cid:29) (16)On the other hand, the time averaging of eq. (10)(with the first term of the right hand side replaced by α m ∂ α B ) yields α m (cid:104) ∂B /∂α (cid:105) = (cid:104) r V α (cid:105) . So that wehave from (16) ω (cid:104) r V α (cid:105) > (17)Physically, it means that the friction force removesangular momentum from the particle with respect to the rotating magnet. Writing that (cid:104) d ( m v G ) /dt (cid:105) = 0 ,we have also ωα m (cid:104) ∂B /∂α (cid:105) = (cid:104) ˙ r V r + r ˙ α V α (cid:105) > (18)which shows that the work of the friction force on thesphere center of mass is resistive. in average. So, ifone assumes, for high enough frequencies, a motionwhich is very close to a free rolling at a fixed distancefrom the magnet’s center, it means that r and ˙ α arealmost constant and that ˙ r (cid:39) , whence the righthand side of the preceding equation is asymptotically ∼ (cid:104) ˙ α (cid:105)(cid:104) r V α (cid:105) . If this is correct, we have both (17) and (cid:104) ˙ α (cid:105)(cid:104) r V α (cid:105) > yielding (cid:104) ˙ α (cid:105) ω > , i.e. the counter-rotation is impossible. Although not a mathematicalproof, the argument is qualitatively correct, providedthat the radial fluctuations are negligible in the highfrequency regime, as well as those of ˙ α . It is worthnoting however that the argument relies on the pos-itivity of (16), which is of thermodynamical origin,since its corresponds to the dissipative work of thefriction force. As a result, the argument should there-fore apply to a pure paramagnetic bead experiencinga dry friction as well. VII. CONCLUSION
In this paper, we have presented a coherent studyof the counter-rotation of a ferromagnetic bead, con-strained to move on a magnetic stirrer’s surface, andexcited by the rotation at constant angular velocityof the magnet installed beneath the slab of the stir-rer. By fitting two parameters, we were able to re-produce quantitatively by numerical simulations theexperimental observations, first observed by [11] in avery similar experiment, in spite of the different typeof friction of the bead on the slab utilized (viscous vs.dry friction). The expression of the dynamical equa-tions of the ten degrees of freedom of the problem (twofor the bead’s center of mass, three for the orienta-tion of the bead, plus the same number for their timederivatives), allowed us to make also some theoreticalanalysis in the regime of high frequencies. We show inparticular that the corotative regime is never stable athigh frequencies (whereas it would be observable for asystem of a magnetic disk holonomically constrainedto roll at a fixed distance from the center) and further-more that a purely paramagnetic and isotropic beadcan never display counter-rotation. Therefore the slowcounter-rotation observed by [11] with steel spheresare entirely attributable to the slight remanent mag-netization or magnetic anisotropy of the spheres. Wefurther analyzed the dynamical behavior of the beadwhen festoons are present and showed that the asso-ciated modulation of the radial distance of the beadis tightly correlated to —and therefore mainly dueto— the modulation of the radial component of the0magnetic force : When this component weakens, thecentrifugal force moves the bead away from the ro-tation center, and conversely. This simple argumentis asymptotically true only for ω → ∞ . At lower fre-quencies, the friction force is not negligible, cf. fig. 12.We noted that the proper asymptotic regime would beexperimentally difficult to obtain since it correspondsto frequencies ∼ Hz.This study can be pursued in several interestingways : What happens when the bead is constrainedto move in a fluid or on a fluid surface, and is likely tobe sensitive to the waves generated by itself ? Recentstudies have shown the extraordinary behaviors whichhappen in such composite systems of a bead or dropletand an interacting fluid, when the latter has a longrelaxation time [8, 10, 15]. Another interesting ques-tion would be to probe the behaviour of non sphericalmagnets : As the asymptotic ( ω → ∞ ) is a free rollingfor the sphere, how does a non spherical magnet ac-commodate to high excitation frequencies ? Finally, athird class of follow-ups would be to inquire the col-lective behaviour of several beads excited together bythe magnetic stirrer. Such systems may display emer-gent properties, typical of dissipative-active systems,where a continuous flux of energy drives assemblies ofparticles far from equilibrium into a unexpected sta-tionary and complex dynamical regimes. VIII. ACKNOWLEDGEMENTS
We thank numerous students who worked duringtheir internship on this experiment or on variants ofit : Elodie Adam, Vanessa Bach, Jérémie Geoffre,Vincent Hardel, Alexandre Ohier, Yona Schell andCamille Vandersteen Mauduit-Larive.
IX. APPENDICESA. Dynamical equations in spherical coordinates
For sake of completeness, we provide here the dy-namical equations (Eq. (7)) for the magnet axis n in the spherical coordinates defined by n = cos θ e r +sin θ (sin φ e α − cos φ e z ) : V r = ˙ r − ε (cid:16) ˙ θ cos φ + ˙ α sin θ sin φ cos φ − ˙ φ sin θ cos θ sin φ + ˙ ψ sin θ sin φ (cid:17) , (19) V α = r ˙ α + ε ( ˙ φ sin θ + ˙ α sin θ cos θ cos φ + ˙ ψ cos θ ) . (20) ddt (cid:20) ˙ θ + ˙ α sin φ (cid:21) = ˙ φ sin θ cos θ + ˙ α sin θ cos θ sin φ + ˙ α ˙ φ cos(2 θ ) cos φ + 5 κ ∂ n ∂θ · B + 52 ε V r cos φ, (21) ddt (cid:104) ˙ φ sin θ + ˙ α sin θ cos θ cos φ (cid:105) = ˙ α sin θ sin φ cos φ + ˙ α ˙ θ cos φ − ˙ α ˙ φ sin θ cos θ sin φ + 5 κ ∂ n ∂φ · B − κ ε [ V r sin θ cos θ sin φ + V α sin θ ] . (22) ¨ ψ = 52 ε [ V r sin θ sin φ − V α cos θ ] . (23) B. Why is rapid corotation never observed ?
In the preceding analysis, we found only a counter-rotating regime (i.e. ω ˙ α < ) because we have as-sumed φ = − Φ+ χ = ωt − α + χ . Another possibility tohave a nonzero radial magnetic force resisting the cen-trifugal force would have been to write φ = Φ + χ . Inthis case, we would find a corotative regime, with ˙ α = εω/ ( r + ε ) and ˙ φ = − ωr/ ( r + ε ) , and χ = π becausethe stability criterion assumed above is obviously stillvalid. To understand why this corotative regime isobserved neither in the experiments nor in the simula-tions, we assume for sake of simplicity that the drivingfrequency is so high that we can disregard the frictionterm (cid:104)V r (cid:105) in (12). We also model the magnetic fieldexperienced by the bead by the orthoradial structure B (cid:39) (cid:98) B ( r )[sin(Φ) e α +cos(Φ) e z ] where (cid:98) B ( r ) is the typi-cal field amplitude along the trajectory at mean radius r . Notice that the trigonometric factors in this for-mula are dictated by the geometrical structure of thefield, see fig. 5. The mean magnetic force resisting thecentrifugal force is F mag = κε ∂ r (cid:104) B α sin φ − B z cos φ (cid:105) .With the counter-rotative Ansatz φ = − Φ+ π , we have F mag = κε ∂ r (cid:98) B ( r ) , which is negative (as required) fortypical values of r larger than the value where (cid:98) B ( r ) is maximum. With the corotative Ansatz φ = Φ + π ,we would have F mag = − κε ∂ r (cid:98) B ( r ) (cid:104) cos(2Φ + π ) (cid:105) = 0 .In fact, this value is not strictly zero, since we madeapproximations concerning the structure of the field.However, it is small and therefore precludes the sta-bilization of a corotative motion.1.In fact, this value is not strictly zero, since we madeapproximations concerning the structure of the field.However, it is small and therefore precludes the sta-bilization of a corotative motion.1