Wave-induced motion of magnetic spheres

0.37 MB·

by  Christophe Gissinger

epl draft
Wave-induced motion of magnetic spheres
Christophe Gissinger1
1 Laboratoire de Physique Statistique, Ecole Normale Suprieure, CNRS, Universit P. et M. Curie, Universit Paris
Diderot - Paris, France
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1
PACS 05.45.-a–Nonlinear dynamics and chaos
0
PACS 45.40.-f –Dynamics and kinematics of rigid bodies
2
PACS 47.63.-b–Biological fluid dynamics
n PACS 85.70.Rp–Magnetic levitation, propulsion and control devices
a
J Abstract –We report an experimental study of the motion of magnetized beads driven by a

6 travelling wave magnetic field. For sufficiently large wave speed, we report the existence of a

backwardmotion,inwhichthespherecanmoveinthedirectionoppositetothedrivingwave. We
]
t show that the transition to this new state is strongly subcritical and can lead to chaotic motion
f of the bead. For some parameters, this counter-propagation of the sphere can be one order of
o
s magnitudefaster than thedriving wavespeed. These results are understood in theframework of
. a model based on the interplay among solid friction, air resistance and magnetic torque.
t
a
m
-
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o Introduction. – Many situations, in laboratory or
c
in Nature, involve the locomotion of spherical particles
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on a solid plane through a viscous fluid [1]. Despite the
,-"+’./)"0
1apparent simplicity of the problem, there are still many
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unsolvedquestions concerningthe exactmechanisms con-
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trollingthemotioninthesesystems,suchasthetransition
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1fromstatic torollingfriction,the roleofsurfaceelasticity,
1or the hydrodynamical interaction between the fluid and
0the moving object. "
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1 Duringthe lastdecade,this type ofproblemhasgained !"#$#%&’()%*+
0somerenewedinterestin the fieldof microfluidics. For in- ,"#"!
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stance, biomedical applications involve microrobots such
1
as artificial micro-swimmers made of super-paramagnetic
:
vbeads subject to an externalmagnetic field [2,3]. In some Fig. 1: Experimental set-up. A bead moves on a flat surface
Xicases (so-called ’surface walkers’), it has been reported under the effect of a traveling magnetic wave created by 16
that actuation of the microrobotcan be controlledby the Neodymium magnets located on a rotating disc.
r
apresence of a surface wall [4,5]. On the other hand, the
problemofparticlestransportedbyelectromagneticwaves
iswidelystudiedintheframeworkoftheplasma-electrons ofwidth ap =5mm. Thebead, ofradiusa,is madeeither
interaction. Surprisingly,only few studies havebeen done of weakly ferromagneticmild steel or Neodymium NdFeB
on electromagnetic control of particles on macroscopic (permanent magnet) depending on the experiments and
scales [6]. In this letter, we focus on the motion of can freely move on the surface. At a distance h below the
macroscopic magnetic beads on a flat horizontal surface plane is locateda rotatingdisc containing16 Neodymium
driven by the magnetic force due to a travelling wave. magnets disposed with a regular spacing along a circle of
radius R = 83mm. These magnets are cylinders of di-
ameter d = 20mm and height h = 10mm, generating
m m
Experiment. – Fig.1-left shows a schematic picture a magnetic field of B0 = 0.45T at their surface. The
m
of the experiment: a spherical bead is placed on a fixed magnets are arranged such that two adjacent magnets,
horizontalplanemadeofmethyl-methacrylate(plexiglass) separated by a distance d =2πR/16=32.5mm, are ori-
m
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Christophe Gissinger1
ented with opposite polarity. This rotating disc therefore 1 V1.2
generates a sinusoidal magnetic field propagating in the 1
V
twahzaeivmdeunistucam.lTbdehirreekcbtei=oandπtw/hdietmrhe,foawrephuemrlseoavfteidsonwisiωtthh=earv2oeπtloafctdii/oty1n6Vraa(ntnedoora-f d speed 00..68 0000....2468 BSaynckcwhraorndo us
a Reversing state
malized by the wave speed c) under the influence of this be0.4 0
d -0.2
travelling magnetic field along circular trajectories which ze0.2 -0.4
are recorded by a fast camera located on the top of the mali -0.6
set-up. By changing the distance h between the rotating Nor 0 5 6 7 8 Time 9[s]
disc and the fixed plate, one can also experimentally vary -0.2
themagnitudeofthe magneticfieldappliedtothesphere. -0.4
Figure 2 shows the evolutionof the normalizedvelocity 0 2 4 6 8 10
Wave speed c [m.s-1]
V of a mild steel bead ofradius a=3mmas a function of
thewavespeedc. Atsmalldiscfrequency,thebeadmoves
Fig. 2: Bifurcation diagram of the velocity V of a mild steel
insynchronismwiththewave(V =1,upperbranch). This
bead of radius a = 3mm and disc/plate distance h = 4mm.
synchronous translation is relatively intuitive: under the
Note the coexistence of two solutions, corresponding to ei-
effect of the applied field, the magnetizedbead is trapped
ther synchronism with the wave or counter-propagating mo-
inside the potential well of one of the magnet of the disc.
tion. Typical time series are shown in inset (see text).
As the disc rotates, the sphere therefore slides above the
magnet without rolling and keeps its magnetic moment
alignedwith the localfield (time seriesin the inset figure, Such magnetic spheres show the same type of behavior
black curve, for c=3m.s−1). discussed above, with a bistability between forward and
However, for larger wave speed (c > 5m.s−1), the sys- backwardsolutions. Note however in Fig.3 that the max-
tem suddenly bifurcates to a very different solution, in imum negative velocity can be very large: for a=9.5mm
which the bead now propagates in the backward direction for instance, the sphere propagates backward 10 times
related to the driving wave (typical time series in green, faster than the forward magnetic wave associated to the
for c = 6.3m.s−1). Note that this backward motion is in rotating (See Supplemental Material at URL for videos)
this case slower than the wave speed (V 0.4), and
∼ −
slightly decreases with c.
Theoretical model. – In order to clarify the mech-
If the wave speed is decreased from this new state, the
anism by which such a solution occurs, a simple one-
transition back to synchronous positive translation is ob-
dimensional model is now derived, in which the sphere
tained for much smaller velocity, thus showing that this
now moves along some horizontal x coordinate at a dis-
bifurcationissubcriticalandassociatedwithastronghys-
tance z from the bottom magnetic wave. Let us suppose
teresis.
that the magnetic field due to the rotating disc (black
For some parameters,the bistability between these two
linesinFig.4-bottom)canbemodeledbyamagneticwave
solutions can yield complex non-linear behavior. The red B=B0e−kz[cos(ω0t kX)ex+sin(ω0t kX)ez],where
curve in the inset of Fig.2 shows a time series obtained − −
X represents the local position along the trajectory, k is
for h = 8mm, a = 2.5mm, c = 0.68m.s−1. For these
thewavenumberofthemagneticwaveandzisthedistance
parameters, both positive and negative V solutions are
fromthedisc. The beadofmassM isalsosupposedtobe
accessible and the system undergoes chaotic switches
uniformly magnetized (Magnetization density M0) with
between the two branches.
a dipole moment m = m0[cosθ(t)ex+sinθ(t)ez] of con-
stant amplitude m0 but with a direction which depends
In fact, the evolution of the counter-propagative solu-
on the orientation of the sphere (white arrow in Fig.4-
tionstronglydependsontheparametersusedintheexper-
bottom). Both B and m are supposed to lie in the (x,z)
iment. Fig.3showstheevolutionof V (onlythenegative
− plane. ThehorizontalcomponentFx ofthemagneticforce
solutionisshown)asafunctionofcforh=10mmandfor
F = (m. )B and the magnetic torque Γ = m B act-
B
differentvaluesofthesphereradius. Forthesmallestvalue ∇ ×
ing on the particle are given by:
of a, the velocity V is constant, whereas it systematically
decreaseswith c for largera. Note that the maximum ve-
locity reached by the sphere first increases with a, before Fx =km0B0sin(ϕ θ) ΓB =m0B0sin(ϕ θ)
− −
decreasing for the largest values of a. For a = 6mm, this (1)
maximum velocity reaches V = 1.2 (for c = 0.5m.s−1), where ϕ(t) = ω0t kX(t) is the phase of the magnetic
− −
meaning that the bead is travelling backward faster than fieldatpointX andtime t. Duringitsmotion,the sphere
the magnetic wave driving its motion. is also subject to the solid friction r from the surface
x
IfmildsteelbeadsarereplacedbysphericalNeodymium on which it is moving and to the fluid friction from the
NdFeB magnets, the magnetic moment of the sphere air F = 1C πa2ρ U2sign(U), where U = X˙ is the ve-
a 2 D air
is now fixed, rather than induced by the applied field. locity of the bead and C is the drag coefficient for a
D
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Wave-induced motion of magnetic spheres
101 max h=4 mm-Steel
eed -V aaaa====1346...m255mmmm mmm- S--- tSSSettteeeeleeelll velocity-V 101 hhhh====9181 0 2mm mmmmmm--SN--SNtdetFdeeFeleBelB
d sp aa==76m.5mm m- S- tNeedlFeB bead h=15 mm-NdFeB
d bea100 a=9.5mm - NdFeB alized 100 hT=h2e0o rmetmica-Nl sdoFleuBtion (see text)
e m
aliz nor
m m
Nor mu
10-1 Maxi10-1
10-1 ka 100
0 1 2 3 4 5
Wave speed c [m.s-1]
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Fig. 3: Velocity of counter-propagative beads as a function of
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the wave speed c, for h = 10mm. For some parameters, the
backward velocity can be10 times higher than c. " # !!" " # !!" " # !!"
! " ! " ! "
Fig. 4: Top: maximum velocity of thecounter-propagativeso-
sphere rolling on a plane. The motion of the sphere is
lutionasafunctionofka. Dashedlinecorrespondstosolution
then governed by Newton’s law for both the velocity V (5). Bottom: simple picture of the mechanism leading to this
of the center of mass and the angular velocity ω with re- solution. As the bottom magnets move to the right at speed
specttothesamecenterofmass. Byusingthewavespeed c,thesphererotatescounter-clockwiseinordertostayaligned
c = ω0/k as a typical velocity scale and l0 = a as a typ- with the field, producing a translation to the left at a speed
ical length scale, the equations of motion reduced to the v>c .
simple dimensionless system :
In Fig.4-top, this theoretical backward solution V0 is
V˙ = kaτsinψ+R KV2sign(V) (2) plotted as a function of ka (dashed line) and compared
− to the maximumamplitude ofthe experimentalbackward
5
ω˙ = τsinψ+R (3) solution obtained for different values of h (i.e. different
2
values of the magnetic field magnitude). It appears that
ψ˙ = ka kaV ω (4)
− − most of the points collapse on the theoretical prediction
where R = r a/Mc2 is the dimensionless (unknown) V0, independently of the values of a, h and the bead ma-
x terials.
solid friction, ψ = ϕ θ is the difference of phase be-
− Solution (5) has a simple physical meaning: In the ’lo-
tween the magnetic field at point X and the magnetic
3 cal’centerofmassreferenceframe,thesphereexperiences
moment of the sphere. K is given by K = C ρ /ρ,
8 D f
where ρf and ρ are respectively the density of the sur- a magnetic field rotating at a pulsation ϕ˙ = ω0 − kU,
i.e. the local rotation frequency of the field is doppler-
rounding fluid and of the bead. For simplicity, the drag
shifted by the displacement of the bead. Under the effect
torquehasbeenneglectedineq.(3),sinceitisexpectedto
of this (locally) rotating field, the magnetic moment of
be much smaller than the translationaldrag. In addition,
the sphere locks to the field and a synchronous rotation
we assume no deformation of the plane during the bead
θ˙ = ϕ˙ is achieved. Since the sphere rolls without slid-
displacement, so rolling friction does no work. Three di-
ing, this positive angular rotation enhances the transla-
mensionless numbers therefore control the problem: ka,
K and τ = M0B0/ρc2, which compares the magnitude of tional counter-motion of the sphere, increasing in return
the doppler-shiftofthe wavespeed(seepicturesofFig.4).
the magnetic driving to inertial effects.
We now search for for stationary solutions, ψ˙ = ω˙ = An important point is that this motion is not bounded
V˙ = 0. First, we are interested in the solution for which by the velocity of the travelling field: as ka tends to
1 (i.e. when the half-perimeter of the bead equals the
thesphererollswithoutsliding,correspondingtoω = V
− distance between two adjacent magnetic poles of the
inour dimensionless variables. Eq.(4) andcombinationof
wave), only an infinitesimal displacement of the magnetic
eqs. (2) and (3) leads to the solution:
waveis necessaryforthe sphereto travelfromonepoleto
the other while keeping constant its angle with the local
2 2
ka Kka Kka magnetic field. In other words, the sphere propagates
V0 =−(1 ka), sinψ0 = τ(1 ka)3, R0 = (1− ka)3 along the field lines several times faster than c in order
− − −
(5) to conserve synchronous rotation with the local magnetic
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Christophe Gissinger1
h=6 mm , a=9.5 mm
h=4 mm , a=3 mm h=8 mm , a=2.5 mm
0 h=9 mm , a=4 mm 0 hh==88 mmmm ,, aa==91..55 mmmm
h=10 mm , a=1.5 mm h=8 mm , a=3. mm
h=8 mm , a=5 mm
h=10 mm , a=3 mm h=8 mm , a=6.5 mm
h=10 mm , a=3.5 mm -0.5 hh==1125 mmmm ,, aa==95. 5m mmm
h=10 mm , a=4 mm h=20 mm , a=1.5 mm
-0.5 h=10 mm , a=4.5 mm
/V0 hh==1104 mmmm ,, aa==53 mmmm /V0 -1 1100
V V 1000
900
-1.5
800
-1 700
600
-2 500
400
300
200
-1.5 -2.5 100200 400 600 800 1000 1200 1400
10-3 10-2 10-1 100 101 102 10-3 10-2 10-1 100 101 102
N = µKMρ0cB20(1−kkaa)3 N = µKMρ0cB20(1−kkaa)3
Fig. 5: Evolution of V/V0 as a function of N for mild steel Fig.6: EvolutionofV/V0asafunctionofN forNdFeBspheres.
beads. Most experimental data collapse on the dashed line Atsmall N,abistability isobservedbetweentherolling back-
correspondingtomodelequations(5)and(7),i.e. purerolling ward solution V0 and a faster motion. Inset: corresponding
for N <1, and rolling-sliding for N >1. bidimensionnal oscillating trajectories.
field.
isproportionaltotheambientmagneticfield. Withtypical
values of the experiment (B0 300G and c 3m.s−1),
Note that the aboveargumentis only valid fora sphere τ ranges between 10−2 and 1∼00. Some expe∼riments [7,
rolling without sliding, the corresponding rolling friction
8] suggest that the air resistance coefficient for a sphere
R0 being smaller than the sliding friction. For a sphere
rollingon a plane is largerthan its usualvalue C =0.45
D
which slides during its motion, the friction R on the bot-
inunboundedfluid. Forsimplicity,weusedC =1inthe
D
tomsurfaceis rathergivenby R = µW, whereµ is the
S − following. Since it is hard to know precisely the value of
frictioncoefficientandW =kaτcosψ isthedimensionless
the kinetic frictioncoefficientin ourexperiment, µ is kept
normalforce due to the verticalmagnetic attractionforce
asa fitting parameterbutservesonlytoscalethe value of
(gravity being neglected).
ourexperimentalparameterN. Inthefollowing,µ=0.02
A transition from pure rolling to sliding is expected
is used, which is slightly smaller than the value expected
when the rolling resistance R0 exceeds the sliding friction
for a lubricated metal-plastic interface.
RS. By using the expression (5) for R0, this leads to the
Figure 5 shows our experimental data for mild steel
definition of a new dimensionless number controlling the
beads compared to the model described above. In
motion of the bead:
this figure, bead velocities have been rescaled by the
R0 Kτ ka backward solution V0 (eq.5) and plotted as a function
N = | | = (6)
R µ (1 ka)3 of our dimensionless number N. Although it involves a
S
| | − variety of values for h and a, our experimental data show
wherethelimitψ 1(smallphaselag)hasbeentaken.
≪ a very good agreement with both solutions (5) and (7),
When N > 1, the fast solution V = V0 disappears and representedby the dashedlines. The bifurcationfromthe
the sphere starts sliding during its backward motion. By
rolling solution to the sliding state occurring at N = 1
injecting the sliding friction R into equations (2-4), one
S is relatively sharp, and very weak departure from theory
find the following solution:
is observed, although 4 orders of magnitude for N are
V0 explored.
V = , sinψ =µka (7)
S S
√N
For Neodymium magnets (Fig.6), the rescaling also
This second solution still corresponds to a bead propa-
shows a fairly good agreement with the theory at large
gating backwardand a synchronousangular rotation,but
N, but exhibits clear departures from the model at small
duetosomesliding,thespherenowmovesslowerthanthe
N. First, the transition from rolling to sliding is strongly
pure rolling solution V0.
imperfect,involvingasmoothertransitionandsmallerve-
Comparison between theory and experiment. – locities than expected. Moreover, the data are spread
Atthispointitisinterestingtocomparethemodelderived around the backward-rolling solution. In particular, note
above with our experimental setup. When the mild steel thatsomeexperimentsinvolvebackwardvelocitiesseveral
bead is first put on the table, it is magnetized by the times faster than V0 as N is reduced. The inset of Fig.6,
magnetic field below, such that its magnetization density which focus on the trajectory obtained for h = 8mm and
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Wave-induced motion of magnetic spheres
a = 3mm, shows that this departure is due to a more
∗∗∗
complextrajectorythanthe one-dimensionalmotionused
during the model derivation: the bead starts oscillating The author is thankful to S. Fauve, F. P´etr´elis, G.
radially around its circular trajectory. It is interesting to Michel and B. Semin for their comments and criticisms,
note that this type of motion which involves a complex which helped to improve the manuscript. This work was
interaction between the magnetic restoring force and the supported by funding from the French program ”Retour
inertiaofthe sphere,canincreasethe velocityofthe bead Postdoc” managed by Agence Nationale de la Recherche
even further . Clearly, this motion would require a mod- (Grant ANR-398031/1B1INP)
ification of our 1-D model, which is beyond the scope of
this paper.
REFERENCES
Additional experiments were done using beads made
of copper: in this non-magnetic case, a similar counter- [1] Michaelides E.E, J. Fluids Eng. , 119 (1997) 233-247
motion is observed, although one order of magnitude [2] Peyer K.E. and Zhang L. and Nielson B., Nanoscale,
5(2013) 1259
smaller than the one reported here for magnetic spheres.
[3] Pamme N.,Lab Chip, 6 (2006) 24-38
In this case, eddy currents in the sphere produce weaker
[4] Sing C.E. et al.,PNAS USA, 107 (2010) 535-540
but similar forces than the magnetic ones described
[5] KarleM.etal., MicrofluidNanofluid,939(2011)10:935
here, thus leading to identical behavior. Finally, note
[6] Bolcato R. and Etay R. and Fautrelle Y. and Mof-
that by adding random noise in equations (2)-(3), these
fatt H.K, Phys. Fluids A, 519931852
equationseasily explainthe originofthe chaotic behavior [7] Cart J.J. Jr, B.S. Thesis,M.I.T.1957
reported in Fig.2: it originates from imperfections of [8] Garde, R. J. and S. Sethuraman, La Houille Blanche,
the bottom surface, generating noise in the friction and 71969727-732
allowing random transitions between the two bistable [9] MahonetA.andAbbott J.,App.Phys. Lett.,99(2011)
forward/backwardsolutions. 134103
[10] Yan j. and Skoko D. and Marko J.F., Phys. Rev. E,
70 (2004) 011905
Conclusion. – The motion of a magnetized sphere
driven by an external travelling magnetic field has been
studied. When the speed of the driving wave is large
enough, we observe a transition from a state involving
a forward synchronous translation with no rotation, to a
backward translation associated to synchronous angular
rotation. Under several assumptions, a simple model was
proposed to explain this backward motion, which can be
severaltimes faster than the driving wave. A good agree-
ment is obtained between the theory and experimental
data.
The behavior reported in this paper may have several
applications. First, the present experiment shares some
similarities with recent studies on the controlof magnetic
beads in microfluidic channel [5], [9] and may therefore
be regarded as a new method for manipulating nano-
microscale objects using the tumbling motion of particles
induced by a travelling magnetic wave. Rolling of a small
magnetic particle can also be relevant to micromanipula-
tion of magnetic beads for magnetic twizzer [10] or pol-
ishing techniques. It is also interesting that the present
mechanism offers a simple explanation to counter-flows
observednear boundariesin ferrofluidssubmitted to a ro-
tating magnetic field. On the other hand, the mechanism
of locomotion described here is very general, and may
be applied to controlled transport of objects at macro-
scopic scales. In this perspective, several questions may
be addressed in future work, such as the role of electri-
calconductivity orthe factorslimiting the stability ofthe
counter-propagativesolution.
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EPS file