- Email: [email protected]
Break-up of dipolar rings under an external magnetic field
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
288
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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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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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.
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[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.
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
288
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
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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.
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