4 December 2000
Physics Letters A 277 (2000) 287–293
www.elsevier.nl/locate/pla
Break-up of dipolar rings under an external magnetic field
F. Kun a,b,∗ , K.F. Pál c , Weijia Wen d,e , K.N. Tu e
a Department of Theoretical Physics, University of Debrecen, P.O. Box 5, H-4010 Debrecen, Hungary
b Institute for Computer Applications (ICA1), University of Stuttgart, D-70569 Stuttgart, Germany
c Institute of Nuclear Research (ATOMKI), P.O. Box 51, H-4001 Debrecen, Hungary
d Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
e Department of Materials Sciences and Engineering, UCLA, Los Angeles, CA 90095-1595, USA
Received 28 June 2000; accepted 16 October 2000
Communicated by J. Flouquet
Abstract
An experimental and theoretical study of the deformation and break-up process of rings, formed by magnetic microspheres,
under the application of an external magnetic field is reported in this Letter. When the external magnetic field is applied parallel
to the plane of the rings, we found that the break-up process has three different outcomes depending on the way of application
and time history of the external field: (a) deformation into a compact set of dipoles with a triangular lattice structure, (b) opening
into a single chain, and (c) break-up into two chains with various relative sizes. A thorough theoretical investigation of the breakup process has been carried out based on computer simulations, taking into account solely the dipole–dipole and dipole–external
field interactions, without thermal noise. The experimental results and the simulations are in good agreement.  2000 Elsevier
Science B.V. All rights reserved.
PACS: 83.10.Pp; 82.70.Dd; 41.20.-q; 61.46.+w
Keywords: Magnetic microspheres; Aggregation; Rings; Break-up
1. Introduction
Magnetorheological (MR) fluids are generally composed of micrometer sized magnetic particles suspended in a non-magnetic viscous liquid. In the absence of an external magnetic field the particles with
permanent magnetic moment aggregate due to the interplay of the dipole–dipole interaction and of the
Brownian motion of the particles, and build up complex structures. The intriguing effect of long range
dipolar forces on the dynamics of growth processes
* Corresponding author.
E-mail address: [email protected] (F. Kun).
and on the structure of growing aggregates in colloids
have attracted much scientific and industrial interest
during the past years [1–4]. In these studies the two
dimensional structures formed by dipolar particles on
the bottom plate of a container can serve as a starting
point due to their simplicity. Recently, we reported an
experimental and theoretical investigation of the formation of circularly shaped rings of dipoles in MR
fluids in the absence of an external magnetic field, and
that of the competition of rings with randomly oriented
open chains and labyrinthine structures when changing the volume fraction of particles [4].
The micromechanical properties of structures
formed by magnetic microspheres, their stability and
disintegration under various kinds of external pertur-
0375-9601/00/$ – see front matter  2000 Elsevier Science B.V. All rights reserved.
PII: S 0 3 7 5 - 9 6 0 1 ( 0 0 ) 0 0 6 7 7 - 0
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F. Kun et al. / Physics Letters A 277 (2000) 287–293
bations is also a very interesting issue and it has initiated an intensive research. A theoretical study of the
break-up of rings under the application of an external
magnetic field perpendicular to the plane of the ring,
and under thermal excitations has been carried out in
Ref. [5]. Very recently, the micromechanical properties, deformation and rupturing of dipolar chains and
columns, formed under the application of an external
magnetic field, was studied [6]. We have analyzed the
stability of rings of dipoles with respect to external
mechanical perturbations by analytic means, and we
also tested experimentally the stability of structures
formed in the absence of an external field against vibrations [4].
In this Letter we present for the first time a thorough
experimental and theoretical investigation of the deformation and break-up process of dipolar rings subjected
to an external magnetic field parallel to the plane of
the ring. Based on experiments and computer simulations we have revealed that the break-up process of
dipolar rings has three different outcomes depending
on the way of application and time history of the external field: (a) deformation into a compact set of dipoles
with a triangular lattice structure, (b) opening into a
single chain, and (c) break-up into two chains with
various relative sizes. This behaviour is much richer
than what we may expect in the case when the field
is applied perpendicular to the ring [5]. The analysis
of Ref. [5] does not predict qualitatively different outcomes of the break-up process depending on the way
the field is applied. Our study can serve as a basis for
the understanding of the stability and disintegration of
more complex structures occurring in MR fluids.
2. Experiments
For the experimental study of the break-up process
the circularly shaped rings of dipolar particles have
been produced as described in Ref. [4]. The momentcontrollable magnetic particles were fabricated by
selecting uniform glass microspheres with average
diameter d of 47 µm as an initial core, and coating
a layer of nickel of thickness about 3.3 µm using a
chemical coating process [8]. The magnetization M
of this nickel layer was 480 emu/cm3. We argued in
Ref. [4] that in our experimental setup the formation
of two dimensional rings of magnetic particles on the
Fig. 1. The opening of a dipolar ring into a single chain. The experiments (a)–(e) and the simulations (f)–( j) are in good agreement.
bottom plate of the vessel is due to the relatively large
particle size, which hinders the Brownian motion and
it also results in a larger value of dipole moment µ,
leading to a magnetic coupling the strength of which
is much larger than the thermal energy.
In the experiments the optical side of a two-inch
Si wafer was used as the bottom plate on which
four plastic barriers are mounted to form a container
filled with silicone oil of viscosity η = 517.68 mPas.
The container was placed in the center region of a
pair of Helmholz coils, where the magnetic strength
of coils was controlled by a current amplifier. The
pattern evolution of microspheres in the container was
monitored in situ by a CCD camera and a video
recorder. First, the nickel-coated microspheres were
magnetized and then dispersed randomly onto the
container in the absence of an external magnetic
field. The particles with relatively large size settled
down onto the bottom plate of the container and they
formed two-dimensional aggregates due to the dipole–
dipole interaction. After the formation of the circularly
shaped rings of magnetic microspheres, those rings
were selected for the further studies which occurred
far from the other structures in the MR fluid, in order
to minimize the disturbing effect of the surroundings.
In the next stage of the experiments, the external
magnetic field was switched on such that its direction
was fixed to be parallel to the plane of the rings.
The magnitude B of the field was increased linearly
with time up to Bmax = 260–320 G varying the rate
of increase dB/dt in a broad interval between 1 and
320 G/s. Hence, the time needed to reach Bmax ranged
from 4 min to 1 s. Fig. 1 presents the experimental
results for the time evolution of a dipolar ring when B
is increased slowly at rate dB/dt = 1 G/s. One can
F. Kun et al. / Physics Letters A 277 (2000) 287–293
Fig. 2. Asymmetric break-up of a ring into two chains with different
sizes. One can observe the good agreement of the experiments
((a)–(e)) and simulations ((f)–( j)).
observe that when the ring tries to minimize its total
potential energy (the sum of the energies due to the
dipole–dipole and dipole–external field interactions)
first it gets deformed asymmetrically in the direction
perpendicular to the field (Figs. 1(a) and (b)). At a
critical value of B the ring suddenly opens into a single
chain (Fig. 1(c)), which then gradually aligns itself
parallel to the field (Figs. 1(d) and (e)). The opening
into a single chain was observed for slowly increasing
external fields, however, the outcome of the process
drastically changed at larger values of dB/dt as it
is shown in Fig. 2 for dB/dt = 10 G/s. In this case
the ring gets less deformed before the break-up and
it breaks into two chains of different sizes. Further
increase of the value of dB/dt (including the case of
suddenly switching on the field Bmax ) changes only
the relative size of the two resulting chains but the
process remains qualitatively the same.
3. Simulation of the break-up process
The problem of the break-up of rings when the external field is applied parallel to the plane of the ring is
not tractable analytically (contrary to the case when
the field is perpendicular to the ring), hence, to get
a deeper insight we have performed extensive computer simulations of the process. The two-dimensional
model implemented here is the same as the one used
in our recent work [4], where the construction of the
model has been presented in detail. Here we give only
a short overview of the main ideas. In the simulation, the system under consideration is modeled as a
289
monodispersive suspension of non-Brownian soft particles of number N with radius R and magnetic moment µ. The particles are represented by spheres having three continuous degrees of freedom in two dimensions, i.e., the two coordinates of the center of mass
and the rotation angle around the axis perpendicular to
the plane of the motion.
The time evolution of the system is followed by
solving the equations of motion for the translational
degrees of freedom of the particles (molecular dynamics [9]), and by applying a self-consistent relaxation
technique to capture the rotational motion of the particles. The particles are subjected to the dipole–dipole
and dipole–external field interaction, to hydrodynamic
resistance due to their motion relative to the liquid
phase, and to an elastic restoring force (soft particle dynamics) in order to take into account the finite
size of the particles. The magnetic force FEijm acting
between two dipoles µEei and µEej separated by distance rij is supposed to have the form FEijm = µ2 fEijm
with
3
nij
fEijm = 4 5 cos βi cos βj nEij − cos(βi − βj )E
rij
− eEi cos βj − eEj cos βi ,
(1)
where nE ij denotes the unit vector pointing from dipole i to dipole j , and βi , βj are the angles of
the direction of the dipoles with respect to nEij . Although this formula implies that point-like dipoles
are assigned to the center of the spherical particles, it is also exact for a homogeneous distribution of dipole moments in a spherical shell of uniform thickness. In our approximation we neglect
any spatial inhomogeneities arising either from the
manufacturing of the particles or the polarization
effects of the neighboring dipoles. The hydrodyhyd
namic force FEi on a sphere is treated as Stockes’s
hyd
drag FEi = −α vEi , where vEi denotes the velocity
of particle i and α = 6Rπη. The elastic restoring
force, arising between contacting particles due to their
overlap, is introduced according to the Herz contact
law [10]
FEijcont = −k(d − rij )3/2 · nE ij = −k fEijcont ,
where k is a material dependent constant.
(2)
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F. Kun et al. / Physics Letters A 277 (2000) 287–293
For simplicity, in the simulations the system is
supposed to be fully dissipative, non-inertial, hence,
µ2 X Em k X Econt
d rEi
=
fij −
fij ,
dt
α
α
j
rij <2R
i = 1, . . . , N,
(3)
first-order differential equation system is solved numerically to obtain the trajectories of the particles [9].
In our simulations we took R, µ and α to be unity,
and we chose k such that no significant overlap occurs. This implies that the physical quantities for the
present experimental situation are measured in the
following units: distance R = 27 µm, magnetic field
B0 = µ/R 3 = 650 G, force F0 = µ2 /R 4 = 3.1 dyn,
velocity v0 = F0 /α = 12 cm/s (where α = 0.26 g/s),
energy E0 = µ2 /R 3 = 8 × 10−3 erg, and time t0 =
R/v0 = 2.3 × 10−4 s. Since the external magnetic field
BE is assumed to be homogeneous, it affects the translational motion only through its influence on the dipole
orientation.
For each spatial configuration a self-consistent relaxation algorithm is applied to find the equilibrium
orientation of the dipoles, where each dipole points towards the local (dipolar plus external) magnetic field.
The method corresponds to the limiting case of infinite
rotational mobility, when the dipole orientations are in
equilibrium in any moment. This simulation technique
has been successfully used in Ref. [4] to study the formation of rings of dipoles in the absence of an external
magnetic field.
4. Comparison of experiments and simulations
In the simulations, the particles were placed initially
on a circle such that the particle system is in equilibrium in the absence of an external field. Then the
external magnetic field was increased linearly as in
the experiments, i.e., B(t) ∼ t was imposed parallel
to the plane of the ring. dB/dt is measured in unit of
B0 /t0 = 2.877 × 106 G/s. The simulated results obtained at dB/dt = 5 × 10−4 are compared to the experimental ones in Fig. 1, where in the snapshots generated by simulations the direction of the dipoles is
also indicated. One can observe in Fig. 1 the deformation of the ring, the opening into a single chain at a
critical value of the field, and the final alignment of the
chain parallel to the external field, in reasonable qualitative agreement with the experimental results. These
results also imply that from the location of the opening of a ring with respect to the direction of the external magnetic field, one can determine experimentally the orientation of dipoles around the ring. It was
found that varying dB/dt in the interval 10−4 –10−3
the ring opens always into a single chain, however,
when dB/dt is larger than 10−3 , in the simulations
the ring breaks up into two chains with practically the
same size. The qualitative explanation of these observations is the following. The dipole moments of a ring,
with fixed spatial coordinates of the particles, have two
distinct stable arrangements in a wide range of the
external magnetic field. In the configuration optimal
at zero field the dipole moments point into the direction of the local tangent, and they are oriented in the
same way around the circumference of the ring (see
Fig. 1(f )). An external field distorts this arrangement,
but up to a certain field strength it remains stable. Although one half of the ring is oriented wrongly with respect to the external field, the dipoles are kept that way
by each other’s magnetic field. In the other stable configuration — optimal from some field value — the dipole moments in this half of the ring are reversed. Both
halves are oriented according to the external field, but
the orientation of the dipoles suddenly changes at two
opposite parts of the ring. The stability of the arrangement of the dipole moment directions does not mean
that the ring itself is stable if translational motion is
allowed. In the first configuration the ring will tend to
become more and more deformed perpendicular to the
external field, even in weak fields. However, the forces
deforming the ring are weak, so the deformation proceeds slowly. At a certain field value (the stronger the
deformation the lower this value is) it will become favorable for the ring to split at the middle of the half
oriented against the external field. In this case the ring
may open up into a single chain. However, if the field
changes fast enough, it may reach the value where this
arrangement of dipole moments becomes unstable before the ring has time to open up, and suddenly the
dipole directions get rearranged. At the new configuration the ring splits at the two opposite parts of the ring
where the dipole orientation is discontinuous. Therefore, the split is always symmetric. However, as it is
shown in Fig. 2, experimentally the ring may break
up into two chains quite asymmetrically, and this is
F. Kun et al. / Physics Letters A 277 (2000) 287–293
Fig. 3. The deformation of a ring in a very slowly increasing external
magnetic field for dB/dt = 5 × 10−5 .
what most often happens for a rapidly increasing field.
Clearly, asymmetric split can only be reproduced by
models in which the rearrangement of the dipole moment directions does not happen instantly. The simplest extension of the model to take into account the finite rotational mobility of the particles is the dynamic
treatment of the rotational degree of freedom analogously to the translational motion. This means that the
angular velocity of each particle is taken to be proportional to the torque of the local magnetic field. Unfortunately, the proportionality factor characterizing the
rotational mobility of the dipoles appears as an additional parameter whose value cannot be determined
a priori. However, if we choose its value such that significant translational and rotational movements due to
the forces and torques typical in the system occur on
similar time scales, the asymmetric split of rings can
be reproduced. A representative example is presented
in Fig. 2 where the good agreement of the simulations
and experiments can be observed.
To study the limiting case of a very slowly increasing external magnetic field simulations were performed varying dB/dt in the range 10−5 –10−4 . We
note that in this case there is no difference between
the results of the two different treatments of the rotational degree of freedom. In Fig. 3 snapshots of the
291
time evolution of a ring obtained by simulations are
presented for dB/dt = 5 × 10−5 . One can observe
that first the ring gets strongly deformed asymmetrically, then the deformed ring gradually closes to form
two parallel chains of dipoles perpendicular to the exE Increasing BE further this conformation
ternal field B.
of dipoles becomes unstable and the system suddenly
reorganizes itself into a triangular structure minimizing its surface energy. According to the argument in
the previous paragraph this extreme deformation for
a very slowly increasing field that prevents the ring
completely from opening up is not surprising. However, in the experiments it was impossible to observe
this regime of the break-up process, instead, even in
the case of the smallest dB/dt available for our equipment, the ring opened into a single chain, similarly to
Fig. 1. One possible reason is that besides the hydrodynamic force, also the friction between the particles and
the bottom plate of the vessel hinders the motion of the
particles, which is not taken into account in our model.
Another limitation of our model, which can play an
important role in the break-up process, is the assumption of a homogeneous distribution of dipole moments
in a spherical shell of uniform thickness (equivalent to
a point-dipole) neglecting any spatial inhomogeneities
like in the case of Ref. [7]. Further tests are necessary
to clarify how the distribution of dipole moments affects the relevant interactions governing the structural
evolution.
To give a quantitative characterization of the change
of conformation and break-up of rings, in the simulations the energy of the dipole–dipole Ed–d and the
dipole–external field Ed–ext interactions were monitored. Examples from each regime of the process revealed by simulations are presented in Fig. 4, where
Fig. 4. Ed–d /N and (Ed–d + Ed–ext )/N as a function of the external magnetic field B at several values of dB/dt for N = 20.
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F. Kun et al. / Physics Letters A 277 (2000) 287–293
regime the maximum of Ed–d occurs after the opening of the ring. The curves of Bm and Bd differ significantly only in the first regime. Finally, substituting the units of the corresponding quantities, beside
the qualitative agreement, we obtained that the typical
field values B, where the break-up occurs, fall in the
range of 200–300 G, which agrees well with the experimental values. However, for dB/dt there is a significant difference between experiments and simulations,
which might be due to the effect of friction and the
non-trivial distribution of dipole moments inside the
particles, which have been neglected here.
5. Conclusions
Fig. 5. Bm and Bd as a function of the coefficient dB/dt. The
ranges of dB/dt corresponding to the three different outcomes of
the break-up process are separated by arrows.
Ed–d and the total energy Ed–d + Ed–ext divided by the
number of particles N are plotted as a function of the
external field B. It can be seen that in all cases the energy of the dipole–dipole interaction Ed–d has a sharp
maximum at a specific value of the external field Bm .
However, the total energy of the system monotonically
decreases, since the decrease of Ed–ext compensates
the increase of Ed–d . In Fig. 4 it can also be seen that
the value of Bm depends on the rate of increase dB/dt
of the external field B. Another characteristic quantity is the field strength Bd corresponding to the decisive conformation of the ring, which determines the
outcome of the process. In the simulations it can be
defined as the conformation where at least one particle has displacement larger than the diameter 2R, or
two particles, which were not neighbors in the initial configuration, touch each other (see Fig. 3(b) for
the case of a slowly increasing field). The dependence
of Bm and Bd on the rate of increase dB/dt is presented in Fig. 5, where the ranges of dB/dt corresponding to the three different outcomes of the breakup process are also indicated. One can see that Bd is a
monotonically increasing, smooth function of dB/dt,
while Bm has three distinct regimes corresponding to
the three different outcomes of the break-up process.
Simulations revealed that in the first and third regimes
Bm coincides with the final rearrangement and breakup of the ring, respectively, however, in the middle
We presented an experimental and theoretical investigation of the deformation and break-up process of
dipolar rings subjected to an external magnetic field
parallel to the plane of the ring. We demonstrated that
in this case the break-up process is much richer than
in the case when the field was applied perpendicular
to the ring. Based on experiments and computer simulations we have revealed that the break-up process
of dipolar rings has three different outcomes depending on the way of application and time history of the
external field: (a) deformation into a compact set of
dipoles with a triangular lattice structure, (b) opening
into a single chain, and (c) break-up into two chains
with various relative sizes. The simulated results are in
reasonable agreement with the experimental findings.
Our study can serve as a basis for the understanding of
the stability and disintegration of more complex structures occurring in MR fluids.
Acknowledgement
F. Kun acknowledges financial support of the Alexander von Humboldt Stiftung (Roman Herzog Fellowship). F. Kun was also supported by the Bólyai János
fellowship of the Hungarian Academy of Sciences.
References
[1] P.G. De Gennes, P.A. Pincus, Phys. Kondens. Mater. 11 (1970)
189.
F. Kun et al. / Physics Letters A 277 (2000) 287–293
[2] A.T. Skjeltorp, Phys. Rev. Lett. 51 (1983) 2306;
G. Helgesen, A.T. Skjeltorp, P.M. Mors, R. Botet, R. Jullien,
Phys. Rev. Lett. 61 (1988) 1736;
G. Helgesen, A.T. Skjeltorp, J. Appl. Phys. 69 (1991) 8277.
[3] R. Pastor-Satorras, J.M. Rubi, Phys. Rev. E 51 (1995) 5994;
N. Vandewalle, M. Ausloos, Phys. Rev. E 51 (1995) 597.
[4] W. Wen, F. Kun, K.F. Pál, D.W. Zheng, K.N. Tu, Phys. Rev.
E 59 (1999) R4758.
[5] P. Jund, S.G. Kim, D. Tománek, J. Hetherington, Phys. Rev.
Lett. 74 (1995) 3049.
[6] E.M. Furst, A.P. Gast, Phys. Rev. Lett. 82 (1999) 4130.
293
[7] E.M. Furst, A.P. Gast, Phys. Rev. E 61 (2000) 6732.
[8] W.Y. Tam, G. Yi, W. Wen, H. Ma, M.M.Y. Loy, P. Sheng, Phys.
Rev. Lett. 78 (1997) 2987;
W. Wen, K. Lu, Phys. Fluids 9 (1997) 1826.
[9] M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids,
Clarendon Press, Oxford, 1994.
[10] S. Luding, in: H.J. Herrmann, J.-P. Hovi, S. Luding (Eds.),
Physics of Dry Granular Media, NATO-ASI Series, Kluwer
Academic Publishers, Dordrecht, 1998.