Potential curves and orientational distributions of magnetic moments of chainlike clusters composed of secondary particles

~ ELSEVIER

Journal of Magnetism and Magnetic Materials 154 (1996) 183- 192

Journalof

magnellc ~H materials

Potential curves and orientational distributions of magnetic moments of chainlike clusters composed of secondary particles A. Satoh a,*, R.W. Chantrell u, S. Kamiyama c, G.N. Coverdale b " Department of Mechanical Engineering, Faculty of Engineering, Chiba University, 1-33, Yayoi-cho, Inage-ku, Chiba 263, Japan b Department of Physics, Keele University, Keele, Staffs ST5 5BG, UK c Institute of Fluid Science, Tohoku University, 1-1, Katahira 2-chome, Aoba-ku, Sendai 980, Japan

Received 25 April 1995; revised 30 August 1995

Abstract The present study investigates the potential curves of linear chainlike clusters which are composed of secondary magnetic particles. The orientational distributions of the magnetic moments of the primary particles within the secondary particles are also clarified by means of the usual Monte Carlo method. The results obtained here can be summarized as follows. For the parallel arrangement, in which one cluster lies just beside the other, repulsive forces act between clusters and attractive forces do not arise for any cluster-cluster separation. On the other hand, for the staggered arrangement, in which one cluster is shifted relative to the other in the saturating field direction by the radius of secondary particles, attractive forces do act between clusters at short range. These forces become stronger as the clusters become longer and as the size of secondary particles increases. Although there is a potential barrier for the staggered arrangement, the height of the barrier is almost constant, irrespective of cluster length.

I. Introduction Magnetic fluids are well known to be colloidal dispersions of ferromagnetic particles in carder liquids such as water, hydrocarbon, ester, or fluorocarbon. The most outstanding feature of magnetic fluids is that they respond to an applied magnetic field. Many researchers and engineers have been attempting to apply these attractive materials in various fields of engineering fields, such as seals, bearings, grinding and polishing technologies, actuators, sensors, etc. It is clear from experimental studies [1,2] that the

* Corresponding author. Present address: Department of Physics, Keele University, Staffs ST5 5BG, UK.

flow characteristics of magnetic fluids are strongly influenced by the chainlike clusters of ferromagnetic fine particles formed in an applied magnetic field. In order to clarify theoretically the flow fields and rheological properties of real magnetic fluids, we first need to develop the governing equations which take into account the cluster formation. The hydrodynamical studies of magnetic fluids, however, have not achieved such sophistication. Before attempting to develop the basic equations, it is desirable to clarify in detail the behaviour of the clusters in a flow field and the influence of cluster formation on rheological properties. In one study [3] such a microscopic approach was adopted to investigate the dependence of the viscosity on cluster formation. In this work we used the simple aggregate model in

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A. Satoh et al. / Journal of Magnetism and Magnetic Materials 154 (1996) 183-192

_lQ+ d

(a) primary particle

D

3>

Co) secondaryparticle

putation time. This simple model is successful in predicting the phenomenon of thick chainlike clusters in magnetic fluids [6]. However, it is possible that subtle interaction phenomena may be obscured by the simplifying approximations in the potential. Thus, future developments rest on a detailed study of the actual potential between secondary particles. The purpose of the present study is to investigate in detail the potential curves of linear chainlike clusters composed of magnetic secondary particles. Furthermore, Monte Carlo simulations were carried out to investigate the influence of the orientational distributions of magnetic moments on the potential curves.

2. Model of secondary particles (c) linear chainlike cluster

Fig. 1. Models of primary and secondary particles, and linear chainlike clusters. (a) Primary particle; (b) secondary particle; (c) linear chainlike cluster.

which the magnetic particles interact with each other to form linear chainlike clusters, and calculated theoretically the orientational distributions of such linear thin clusters in a simple shear flow. Real chainlike clusters formed in an applied magnetic field, however, are clearly much thicker than in such a simple cluster model, since the chainlike clusters can be observed even with an optical microscope [4,5]. The formation of thick chainlike clusters can be explained very well by the concept of secondary particles. Namely, the secondary particles, which are composed of primary particles, are magnetized by an applied magnetic field, and the induced magnetic interactions between secondary particles result in the aggregation of these particles to form thick chainlike clusters along the field direction; Fig. 1 illustrates the concept of primary and secondary particles, and thick chainlike clusters. The usefulness of this concept was clarified in a previous study [6]; Monte Carlo simulations have succeeded in predicting thick chainlike clusters for a two-dimensional model dispersion. The simulations, however, used single particles themselves as a secondary particle model, rather than realistic secondary particles, for reason of com-

The primary particles that form the secondary particles are idealized as spherical particles with central point magnetic dipoles. If the magnitude of the magnetic moment is constant and denoted by m, and the magnetic field strength is H ( H = IHI), the interactions between particle i and the magnetic field, and between particles i and j, respectively, are u i = - kT,~n i • H / H ,

(la)

d3 uij=kTA--3{ni.nj-3(ni.tji)(nj.tji)},

(lb)

r)i

where sc and A are dimensionless parameters representing the strengths of particle-field and particleparticle interactions relative to the thermal energy, respectively. These parameters are written as = IxomH/kT,

A = I~omZ/4"rrd3kT.

(2)

In the above equations, k is Boltzmann's constant, /% is the permeability of free space, T is the liquid temperature, m i is the magnetic moment (m = [miD, and Qi is the magnitude of the vector rji drawn from particles i to j. n~ and tji are unit vectors given by n i =rai/m and t~i = r j i / r j i , and d is the particle diameter. We consider it reasonable to use a three-dimensional model in which secondary particles are formed by primary particles aggregating in a nearly spherical shape. A nearly spherical configuration, however, cannot be obtained unless a secondary particle is

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A. Satoh et aL / Journal of Magnetism and Magnetic Materials 154 (1996) 183-192

composed of many primary particles. Also, we do not have sufficient experimental data to determine the shape of real secondary particles. We therefore consider here an initial study using such a secondary particle model as the above-mentioned primary particles gather in a plane to form an aggregate, or a secondary particle, with a nearly circular shape. Fig. 1 shows schematically the models of primary and secondary particles, and linear chainlike clusters that will be discussed soon. The aggregate mechanism for forming such secondary particles, however, is not considered here [7]. This simple model is sufficient to illustrate the basic micromagnetic processes that govern the interaction between secondary particles, such as non-uniform magnetisation structures. We now consider the magnetic interactions between secondary particles composed of N primary particles; N = 7 for Fig. l(b). Taking the diameter of the circumscribed circle as that of secondary particles and denoting it by D, the interaction between secondary particles a and b is written as Uab

N

N

N

N

d3

E

E uij~=kTA E

E

r3

ia= 1 jb =1

ia=l jb =1

Jbta

×{nio'nj~-3(ni.'tj~io)(nj~'tj~io)} 1

=kTA~

N

N

D3

E

E

rS

i~= I jb=l

Jbta

X {nio.nj -- 3(ni "tjd~)(njb'tjd~)},

(3)

where Eq. (1) has been used, and A is expressed as

A = A(d/D)3N 2 = /x°(mN)2 4,rrD3kT .

(4)

By comparing Eq. (3) with Eq. (1), we see that A is similar to A, i.e. the dimensionless parameter representing the strength of magnetic interactions between secondary particles. Eq. (3) may therefore be regarded as the interaction between single particles with magnetic moment mN. Numerical relationships between A and A are as follows: A/A = 1.8, 2.9, 4.0 and 5.1 for N - 7 , 19, 37 and 61, for example. These data show that it is possible for secondary particles to aggregate to form thick chainlike clusters for the case of larger secondary particles, even if the

interactions between primary particles are weak such as A = 1, which will be clarified later. From Eq. (3), the interaction energy between two clusters (one cluster is composed of Nsp secondary particles) is written as Nsp Nsp U=E

EUab:E a=l b=l

Nsp N~p

N

N

E E EUiaJb " a=l b=l ia=l jb=l

(5)

In the present paper we discuss the behaviour of interacting linear chainlike clusters composed of secondary particles pointing in the magnetic field direction, as shown in Fig. l(c).

3. Monte Carlo calculations of potential curves and orientational distributions of magnetic moments

The magnetic interaction energies between the above-mentioned cluster models and the orientational distributions of magnetic moments of primary particles have been evaluated by means of the usual Monte Carlo method [8]. The calculations were carfled out for various values of the cluster-cluster distance and the cluster length under the conditions A = 1 and ~ = 1 and 5. Note that the value of A = 1 roughly corresponds to that of real magnetic fluids. The main part of the Metropolis Monte Carlo algorithm for a system of two clusters is as follows: (a) specify an initial direction of each primary particle; (b) compute the interaction energy between two clusters, u; (c) select a primary particle (in order or randomly) from two clusters; (d) generate a new direction of magnetic moment of the particle; (e) compute the interaction energy, u', between two clusters for this case; (f) if A u = u' - u < 0, then accept the new direction and return to step (c); (g) if A u > 0, then generate a uniform random number R (0 < R < 1); (h) if exp(-Au/kT)>R, then accept the new direction and return to step (c); (i) if exp(-Au/kT)
186

A. Satoh et al. // Journal of Magnetism and Magnetic Materials 154 (1996) 183-192

The sampling numbers are about 30 000 MC steps for evaluating potential curves and about 300000 MC steps for orientational distributions of magnetic moments.

4. Results and discussion 4.1. P o t e n t i a l c u r v e s o f l i n e a r c h a i n l i k e c l u s t e r s

Figs. 2 - 4 show the potential curves for the two linear chainlike clusters, both containing N~p sec(a)

ondary particles whose axes point in the field direction. Fig. 3 shows the results for a parallel arrangement, in which one cluster is just beside the other. Fig. 2 shows those for a staggered arrangement, in which one cluster is shifted in the saturating field direction by the radius of secondary particles relative to the other, as shown in Fig. 8(d) for example. Fig. 4 shows those for a linear arrangement, in which both clusters lie on a straight line along the field direction. In the figures, the thick solid lines relate to the special case of primary particles themselves as secondary particles ( N = 1), and the other lines are for secondary particles composed of N primary particles described in Section 2. In both cases, the

(b) 1."'

....

' ....

' ....

' ....

N,~=5

0.2 I-..-

~< o. 0.

-0.2

i

-0.4

i

i

i

[

i

,

,

.....

N=7

-----o

N=19 N=91 N=7, ~ =1

-"-iEl

"

N=7, ~ =5

D v

N=19,1~=I N=19, ~ = 5

_2. -._j [

,

I

,

,

I

'

'

'

'

I

'

'

"

'

I

~: =oo.

=

I

,

'

'

"

'

I

....

I

,

,

'

'

[]

I'-

N=91

I

N=7, ~ =1

,

N=7, ~ - 5

[] v

N = 1 9 , ~ =1 N=19, ~ = 5

.

3. '

---N=,,

::~?'i

r/D

--2,

-6.

,

2.

(c)

-4.

.i .f

single

1.

(dl

'

2.

oF Ii

r/D

3.

N,.=1i5

single .

.

.

.....

N=7

-----

N=19

. o

i~

. . . .

I

. . . .

~=oo

N=91 N=7,~=I

,,

N=7, ~ =5

= v

N=19,~=1 N=19, ~e= 5

I

. . . .

2.

I

r/D

L'~i

---

N--19 I ° -

~: ~

-----

N=91 J

:

~1

o

N=7,~=1

:,.

[]

N=19,~=1

:

u=~9.,~--5

-

2.

r/D

3.

~"lll

. . . .

3.

F#I 0 ~ •

1.

.

-

::

Fig. 2. Potential curves for the linear chainlike clusters composed of secondary particles for the case of the staggered arrangement. (a) Nsp = 1; (b) Nsp = 5 ; (c) Nsp = 10; (d) Nsp = 15.

A. Satoh et al. / Journal of Magnetism and Magnetic Materials 154 (1996) 183-192 I

F'4"

'

'

'

'

I

'

'

'

'

I

'

'

'

'

I

'

'

'

"

~

,.~

N~p=lO 4

~

single

..... ---------

-

-

-

N=7 N=19 N=91 N=91

-

2.

o. I

,

,

,

,

I

,

,

,

,

I,

I

,

,

,

2.

,

I

,

,

,

,

riD

s.

Fig, 3, Potential curves for the linear chainlike clusters composed of secondary particles for the case of the parallel arrangement.

results are obtained first for the case ~:= 2, i.e. under the condition that the magnetic moment of each particle points in the field direction. The results obtained by the Monte Carlo method for finite £ are also shown in Fig. 2 using symbols such as circles, triangles, etc. r is the distance between the two cluster axes (see Fig. 8a), but in the case of the linear arrangement, it is the distance between the lower-end particle of the upper cluster and the upper-end particle of the lower cluster. Also note that A and D reduce to A and d for N = 1.

O,

I--

--2,

single

..... N=7 ----- N=19 N=91

-4.

,

,

1.

,

i

=

I

=

,

i

i

I

2.

,

,

,

,

I

r/D

,

,

,

I

3.

Fig. 4. Potential curves for the linear chainlike clusters composed of secondary particles for the case of the linear arrangement.

187

It can be seen from Fig. 3 that, for the parallel arrangement, repulsive forces act between clusters and attractive forces do not occur for any clustercluster distance. These repulsive forces become stronger as the values of N~p increase, i.e. as the clusters increase in length. The curves are almost independent of N, i.e. the dimensions of the secondary particles. However, since the A increases with N, as described in the last part of Section 2, the repulsive forces themselves increase as the clusters become composed of larger secondary particles. It can be seen from Fig. 2 that, for the staggered arrangement, attractive forces come into play between clusters at small separations for all cases of Nsp = l, 5, 10 and 15. These attractive forces increase with increasing length of clusters and the dimensions of secondary particles. Also they act over greater distances with increasing cluster length. Another characteristic is that each potential curve has an energy barrier, the height of which is almost independent of cluster length, except for the case of short clusters (N~p = 1) and is about AkT. This means that larger magnetic moments of primary particles and larger secondary particles lead to higher energy barriers, independent of the cluster length. In this situation, therefore, it is impossible for chainlike clusters to overcome such high energy barriers to form thick chainlike clusters. The results of the Monte Carlo calculations show that a weaker magnetic field leads to a deeper potential minimum and a lower energy barrier. This does not necessarily mean that thick chainlike clusters are formed more easily for a weak magnetic field. Since curved chainlike clusters are preferred to linear clusters in weak magnetic fields [9,10], it is possible that the chain structures assumed in Section 2 are not realistic for very low field situations. We will refer to these results by the Monte Carlo method once again in the next section. The potential curves shown in Fig. 4 for the linear arrangement agree essentially with those for N~p = 5 and 10, which are not shown in the figures. Hence, the mechanism of the cluster-cluster aggregation along the field direction depends mainly on the attractive forces between secondary particles, while the interactions between clusters play a secondary role. Fig. 5 shows the influence of the cutoff radius on

A. Satoh et al. / Journal of Magnetism and Magnetic Materials 154 (1996) 183-192

188

I2.~"

//

,~----,, / _ -

"~ ""'"-,~,.

I: / ,."---_~-~_~:-_"\_~ ....... ---.---.--.-_.___-£

"

@

,

(a)

fl

..... roo:O

L7

---

rco~=5

-z

t~

1.

e.

r/D

3.

(b) (b)

[-- ~

=

...... -Y_.-'-_y---~",-~.-

If/'/"

!//

F

!

6

1.

(~

Fig. 6. Orientational distributions of magnetic moments of primary particles in a single cluster for ~ = 1. (a) N,p = 1; (b) N,p = 2; (c) N~p = 3; (d) N~p = 4.

----"-"--:"

I/l -

(c)

use of rcoff = 5 induces a large deviation. Hence it is seen that we should use a cutoff radius of more than about rcoff = 8 to simulate realistic cluster formation via Monte Carlo simulations.

___-7__ ;:.:? .

~

-----

2.

rcor~=5

r/D

3.

Fig. 5. Influence of the cutoff radius rcoff on potential curves for the case of the staggered arrangement and the cluster length of N,p = 15. (a) N = 1; (b) N = 19.

the potential curves for Nsp = 15. In the Monte Carlo simulations, a potential cutoff radius is usually used to reduce the long computation times. From a Monte Carlo simulation point of view, therefore, it is very important to clarify the dependence of the potential curves on the cutoff radius. Fig. 5(a) and (b) show the results for N = 1 and 19, respectively, for the staggered arrangement, in which rcoff represents the cutoff radius normalized by the secondary particle diameter D. It can be seen from Fig. 5 that a higher energy barrier is predicted by the shorter cutoff radius. The height of the energy barrier for r,orr = 3 is about three times that for rcorr = ~, and also the

@ (a)

(b)

(c)

(d)

Fig. 7. Orientational distributions of magnetic moments of primary particles in a single cluster for £ = 5. (a) Nsp = 1; (b) N~p = 2; (c) N~p = 3; (d) N,p = 4.

A. Satoh et al. / Journal of Magnetism and Magnetic Materials I54 (1996) 183-192

Finally, we would point out that the results obtained by using primary particles themselves as a secondary particle model ( N = 1) capture the essential characteristics of the potential curves for a real secondary particle model, composed of many primary particles, to a first approximation. The reason for this is that the thick chain formation occurs in relatively large fields which are sufficient to saturate the clusters and thereby minimize the effects of non-uniform magnetisation. This means that it is not inappropriate to adopt such a simple model as the secondary particles in Monte Carlo simulations.

(e)

189

4.2. Orientational distributions of magnetic moments of particles Figs. 6 - 9 show the orientational distributions of ensemble-averaged magnetic moments of primary particles. Figs. 6 and 7 are for an isolated single cluster, and Figs. 8 and 9 for the same two clusters arranged in the staggered position. The magnetic field strength is taken as ~ = 1 in Figs. 6 and 8, and = 5 in Figs. 7 and 9. Note that the results on the left- and right-hand sides in Figs. 8 and 9 were obtained for the two cases of the cluster-cluster

(d)

Fig. 8. Influence of cluster-cluster interactions on the orientational distributions of magnetic moments for ~ = 1. The results on the left- and right-hand sides are obtained for r ~ = 0.866 and 1.066, respectively. (a) Nsp = I ; (b) Nso = 2; (c) N~p = 3; (d) N~p = 4.

A. Satoh et al. / Journal of Magnetism and Magnetic Materials 154 (1996) 183-192

190

distance, namely, r * ( = r/D) = 0.866 and 1.066, respectively. It should be noted that the magnetic moments are subject to thermal perturbations concerning orientational distributions, so that the thermal averaged magnetic moments depend on the thermal energy and the magnetic particle-field and particle-particle interactions, although the magnitude of each moment (not averaged) is constant. Consequently, the thermally averaged magnetic moments are drawn as vectors such that the differences in their magnitude are represented by the lengths of the vectors. Fig. 6(a) and Fig. 7(a) clearly show that the orientational distributions of the magnetic moments are not uniform for the case of a single secondary

particle, and that this non-uniformity becomes more pronounced for a weaker magnetic field such as ~:= l. The magnitudes of the averaged values of the magnetic moments for ~ = 5 are almost equal to m, which means that the direction of each magnetic moment is strongly constrained to the field direction for a strong magnetic field. On the other hand, since the magnetic moments are fluctuating around the applied field direction for the case ~ = 1, the magnitudes of the magnetic moments show much smaller values than m. From a comparison of Fig. 6(a), (b), (c) and (d) it can be seen that the interactions between primary particles belonging to different secondary particles suppress the fluctuations in the directions of the magnetic moments around an aver-

(b)

(d)

(c) 2f

)'~

2f

I I

Fig. 9. Influence of cluster-cluster interactions on the orientational distributions of magnetic moments for ~ = 5. The results on the left- and right-hand sides are obtained for r * = 0.866 and 1.066, respectively. (a) N~p = 1; (b) N~p = 2; (c) Nsp = 3; (d) N~p = 4.

A. Satoh et al. / Journal of Magnetism and Magnetic Materials 154 (1996) 183-192

aged direction. Also the directions of primary particles lying at the edge of secondary particles vary significantly from those of single secondary particles. The deviations from uniformity are in such a sense as to reduce the surface charge density of an individual secondary particle. It is also interesting to note that the reduction in the thermally averaged value of the magnetisation for those particles close to the edge of a secondary particle would have a similar effect. The results shown in Fig. 7, however, do not clearly show the above-mentioned features because the magnetic field is too strong. Figs. 8 and 9 clearly show that, if the two clusters contact with each other in the staggered arrangement, the orientational distributions of magnetic moments vary significantly from those of single clusters. Especially for a weak magnetic field such as ~ = 1, this influence extends to almost all primary particles, which is clear comparing Figs. 8 and 6. However, the orientational distributions for the longer chains shown in Fig. 8(b), (c), and (d) are not greatly different from those of the corresponding primary particles. In contrast, for ~ = 5, the interactions between the clusters influence mainly those primary particles close to the edges of the clusters. For the cluster-cluster distance of r * = 1.066, the orientational distributions of the magnetic moments shown in Figs. 8 and 9 are not greatly different from those of a single cluster in Figs. 6 and 7. It has already been shown in Fig. 2 that potential curves with deeper energy minima are obtained for a weaker magnetic field. The reason may be made clear by considering the results for r * = 0.866 in Fig. 8. Namely, the magnetic moments of many primary particles deviate from the field direction so that a lower potential energy of the whole cluster system can be obtained. The fact that the results for ~:--- 1 show lower potential energies than those for ---5 means that the magnetisation deviates further from uniformity with decreasing magnetic field strength. In this situation, the particle-particle interactions are dominant rather than the particle-field interactions for the cluster formation, and therefore chainlike clusters along the field direction are not preferred [9,10]. Hence, thick chainlike clusters are unlikely to be formed for a weak magnetic field, even though the potential curves for chainlike structures have deeper potential minima. Essentially, we

191

expect that calculations assuming closed loop structures would give rise to still lower energies.

5. Conclusions In the present study we have investigated the potential curves of linear chainlike clusters composed of magnetic secondary particles. The orientational distributions of the magnetic moments of primary particles within secondary particles have also been studied by means of the usual Monte Carlo method. The results obtained here can be summarized as follows. For the parallel arrangement, in which one cluster lies just beside the other, repulsive forces act between clusters and attractive forces do not exist for any cluster-cluster separation. On the other hand, for the staggered arrangement, in which one cluster is shifted relative to the other in the saturating field direction by the radius of the secondary particles, attractive forces do act between clusters at small separations. These forces become stronger as the clusters become longer and are composed of larger secondary particles. Although there is a potential barrier for the staggered arrangement, the height of the barrier is almost constant, irrespective of the cluster length.

Acknowledgements The present work was done mainly while the first author was staying at Keele University. The author gratefully acknowledges the financial supports of the British Council and the Ministry of Education, Science and Culture of Japan for the three-month stay in UK. Also one of the authors (R.W. Chantrell) is grateful to Professor J.N. Chapman for handling the editorial duties for this paper, including the anonymous reviews.

References [1] S. Kamiyama, K. Koike and T. Oyama, J. Magn. Magn. Mawr. 39 (1983) 23. [2] S. Kamiyama, K. Koike and Z. Wang, JSME Int. J. 30 (1987) 761.

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A. Satoh et al./ Journal of Magnetism and Magnetic Materials 154 (1996) 183-192

[3] S. Kamiyama and A. Satoh, J. Colloid Interface Sci. 127 (1989) 173. [4] C.F. Hayes, J. Colloid Interface Sci. 52 (1975) 239. [5] S. Wells, K.J. Davies, S.W. Charles and P.C. Fannin, in: Proc. Int. Symp. on Aerospace and Fluid Science (Tohoku University, Sendai, 1993) p. 621. [6] A. Satoh, R.W. Chantrell, S. Kamiyama and G.N. Coverdale, J. Colloid Interface Sci. (1996) in press. [7] A. Satoh and S. Kamiyama, in: Continuum Mechanics and

Its Applications, eds. G.A.C. Graham and S.K. Malik (Hemisphere, New York, 1989) p. 731. [8] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller and E. Teller, J. Chem. Phys. 21 (1953) 1087. [9] R.W. Chantrell, A. Bradbury, J. Popplewell and S.W. Charles, J. Phys. D: Appl. Phys. 13 (1980)Ll19. [10] R.W. Chantrell, A. Bradbury, J. Popplewell and S.W. Charles, J. Appl. Phys. 53 (1982) 2742.