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Two-dimensional Monte Carlo simulations of structures of a suspension comprised of magnetic and nonmagnetic particles in uniform magnetic fields
ARTICLE IN PRESS Journal of Magnetism and Magnetic Materials 321 (2009) 1221–1226
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Two-dimensional Monte Carlo simulations of structures of a suspension comprised of magnetic and nonmagnetic particles in uniform magnetic fields Xiaoling Peng a,b, Yong Min c, Tianyu Ma a, Wei Luo a, Mi Yan a, a b c
State Key Laboratory of Silicon Materials, Department of Materials Science and Engineering, Zhejiang University, Hangzhou 310027, China College of Material Sciences and Engineering, China Jiliang University, Zhejiang Province Key Laboratory of Magnetism, Hangzhou 310018, China Ningbo Institute of Technology, Zhejiang university, Ningbo 315100, China
a r t i c l e in fo
abstract
Article history: Received 25 June 2008 Received in revised form 30 September 2008 Available online 21 November 2008
The structures of suspensions comprised of magnetic and nonmagnetic particles in magnetic fields are studied using two-dimensional Monte Carlo simulations. The magnetic interaction among magnetic particles, magnetic field strength, and concentrations of both magnetic and nonmagnetic particles are considered as key influencing factors in the present work. The results show that chain-like clusters of magnetic particles are formed along the field direction. The size of the clusters increases with increasing magnetic interaction between magnetic particles, while it keeps nearly unchanged as the field strength increases. As the concentration of magnetic particles increases, both the number and size of the clusters increase. Moreover, nonmagnetic particles are found to hinder the migration of magnetic ones. As the concentration of nonmagnetic particles increases, the hindrance on migration of magnetic particles is enhanced. & 2008 Elsevier B.V. All rights reserved.
Keywords: Monte Carlo simulation Uniform magnetic field Particle distribution Chain-like clusters
1. Introduction When an external magnetic field is applied to a colloidal suspension comprised of magnetic particles (MPs) suspended in a nonmagnetic medium, dipolar interactions are induced within the particles, and various structures are formed in suspension [1–3]. The suspension, which is sensitive to applied magnetic fields, is called magnetorheological (MR) fluid or magnetic fluid (MF) [4,5]. Due to their practical utility, MR and MF have attracted considerable attention in the past several years, and the field-induced structures in the suspension are reasonably well understood [5]. The structure of the suspension comprised of both magnetic and nonmagnetic particles (NPs) is also sensitive to external magnetic fields [6]. Taking advantage of different response to magnetic fields, a new approach for fabricating functionally graded materials (FGMs) have been developed [7]. A detailed understanding of structures and particle distributions of the mixed suspension in magnetic fields is much important. In our previous work, the effects of magnetic field strength and concentration of MPs on distributions of particles in magnetic fields have been studied [7,8]. However, the magnetic interaction and concentration of NPs, which seriously affect the final structure and distribution of particles, have still not been clarified. This is because it is very difficult to observe directly the
Corresponding author. Tel.: +86 57187952730; fax: +86 571 87951403.
E-mail address: [email protected] (M. Yan). 0304-8853/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2008.11.011
distribution of particles in suspension through experiments. Furthermore, not all the factors that may affect the distribution of particles can be reflected by the available experimental conditions, for example, magnetic interaction increased synchronously as magnetic field strength is increased. As an effective method, compute simulation such as Monte Carlo method has been commonly and successfully used in researches of MR and MF [9–13]. Magnetic particles are found to form chain-like structures under uniform magnetic fields, and the particle distribution is affected by magnetic dipolar interaction among particles, magnetic field strength, concentration of magnetic particles, particle size, and shape of particles [14–16]. Thus Monte Carlo simulation is powerful method used to clarify the particle aggregation phenomena, which should also be a suitable method to study the suspension comprised of both MPs and NPs. In this work, the structures of a suspension comprised of both MPs and NPs under magnetic fields are studied using twodimensional Monte Carlo simulations. The effects of interaction between MPs, magnetic field strength and concentration of MPs and NPs on distributions of particles are estimated.
2. Simulation methods 2.1. Physical model of the suspension We suppose that the position of each particle in the suspension remains unchanged during the subsequent drying and sintering
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and tji ¼ rji/rji, and td the ratio of the thickness of the steric layer d to the radius of particle, which is expressed as 2d/d. The dimensionless parameters lH, lm and lv are written as [13,17]
process for fabrication of FGMs. The model for the suspension is given as follows: MPs and NPs with identical diameter d are dispersed homogeneously in the suspension. The number of particles is defined as N ¼ Nm þ Nn
lv ¼
(1)
where Nm is the number of MPs, Nn the number of NPs. The particles are coated with a steric layer with thickness d. When a uniform magnetic field is applied, MPs are magnetized with the moments aligned along the field direction; while NPs are not affected. The motion of a magnetic particle is typically caused by three kinds of energy: the interaction energy miH between particle i and the magnetic field, the interaction energy mijm between magnetic particle i and j, and the interaction energy mijv due to the overlapping of the steric layers. The motion of a nonmagnetic particle is controlled only by the overlapping energy mijv The above-mentioned energies can be expressed as [13,17]
mHi ¼ kT lH ni H=H
mvij ¼ 2lv kT 1
(4)
where k is the Boltzmann’s constant, T the absolute temperature of the suspension, H the field strength of the magnetic field H, rji the magnitude of the vector rji drawn from particles i to j, ni and tji denote the unit vectors given by ni ¼ mi/mi (mi ¼ |mi|)
60
lm ¼ m0 m2 =4pd3 kT
(7)
60
λm = 10
H
(6)
In the present work, two-dimensional Monte Carlo simulations based on Metropolis’s algorithm [18] are conducted to investigate the internal structures and distributions of particles in the suspension comprised of MPs and NPs. The simulation parameters are as follows: The side length of the square simulation box L is 61.78d. The initial configuration of particles is taken randomly. The thickness of the steric layer d and the dimensionless parameter lv are taken as d ¼ 0.01d and lv ¼ 50, respectively. The cut-off radius rcoff for the particle–particle interactions is taken as rcoff ¼ 10d. The maximum displacement of a particle in one trial move is 0.005d. If the distance between the surfaces of the solid particles is smaller than 0.01d, then these particles are assumed to form a cluster.
(3)
r ji r ji d ðt þ 1Þd ln d r ji dt d dtd
lH ¼ m0 mH=kT
2.2. Simulation parameters
(2)
d fni nj 3ðni tji Þðnj tji Þg r 3ji
(5)
2
where ns is the number of surfactant molecules per unit area on the particle surface, m0 the permeability of free space, m the magnitude of the magnetic moment of MPs, which is proportional to the volume pd3/6.
3
mm ij ¼ kT lm
pd2 ns
50
50
40
40
30
30
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20
10
10
H
λm = 40
0
0 0 60
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λm = 160
H
0
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10
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H
50
60
λm = 640
0
0 0
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60
0
10
20
30
40
50
60
Fig. 1. Effects of the interaction between MPs on distributions of particles with lH ¼ 80. (a) lm ¼ 10, (b) lm ¼ 40, (c) lm ¼ 160, and (d) lm ¼ 640.
ARTICLE IN PRESS X. Peng et al. / Journal of Magnetism and Magnetic Materials 321 (2009) 1221–1226
To reduce the computational complexity, the moments of MPs are fixed along the field direction. Periodic boundary conditions are applied in X and Y directions. All simulations are carried out more than 400,000 steps to get a stable structure and distribution of particles. The present simulation results, two-dimensional structures of MP clusters on the XY plane are plotted using MATLAB 7.3.0 (R2006b) software. The magnetic field is horizontal. Red and blue circles represent MPs and NPs, respectively.
3. Results and discussion 3.1. Effects of the interaction between MPs on distributions of particles In this section, the field strength is setting as lH ¼ 80. The effects of the interactions between MPs on the distributions of particles are studied with lm ¼ 10, 40, 160 and 640, respectively. Simulation results are illustrated in Fig. 1. For lm ¼ 10, which means a weak interaction, MPs do not aggregate but move almost independently (Fig. 1a). In the case of lm ¼ 40, MPs become to aggregate and numerous short chain-like clusters are formed (Fig. 1b). On further increasing lm, short clusters merge into long ones and hence the length of clusters increases gradually (Fig. 1c). For a strong interaction, such as lm ¼ 640, nearly all MPs are arranged as long clusters. In order to investigate quantitatively the effects of the magnetic interaction on distribution of particles, the number density distributions of clusters are analyzed and plotted on a logarithmic graph as shown in Fig. 2. The abscissa indicates the cluster size, c, which means the number of particles in a cluster, and the ordinate is the dimensionless number density of each MP cluster. The maximum size of clusters for lm ¼ 40 is a little larger than that for lm ¼ 10. As lm further increases, both the cluster size and the number density of clusters increase drastically. Finally, the maximum cluster size c approach 17 for lm ¼ 640, which correspond to the long clusters along the magnetic field direction, as shown in Fig. 1d. It can be concluded that MPs aggregate to form chain-like clusters under magnetic fields and the clusters are prolonged along the field direction with increasing lm, which is due to the fact that short clusters merge into the longer ones.
Fig. 2. Effects of the interaction between MPs on number density distributions of clusters with lH ¼ 80.
60
60
λm = 10
H
1223
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H
λm = 40
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λm = 160
H
0
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λm = 640
0
0 0
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60
0
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Fig. 3. Effects of magnetic field strength on distributions of particles with lm ¼ 80. (a) lH ¼ 10, (b) lH ¼ 40, (c) lH ¼ 160, and (d) lH ¼ 640.
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the interaction between MPs and magnetic field is changed when the field strength varied. Thus, the simulation results clearly show that the distributions of particles are affected mainly by the interaction between MPs rather than the field strength.
Chain-like magnetic clusters are formed in the suspension comprised of both MPs and NPs. The simulation results are in good agreement with both experimental observations [8] and previous simulation results on magnetic fluids [17,19], which also show that chain-like clusters of MPs can be formed along the field direction. It is suggested that the formation of chain-like clusters is also attributed mainly to the interaction between MPs and larger interaction results in longer chain-like cluster along the field direction.
3.3. Effects of concentration of MPs on distributions of particles To investigate the effects of the concentration of MPs on the distributions of particles in suspension, the field strength lH and
3.2. Effects of magnetic field strength on distributions of particles In this section, the interaction between MPs is setting as
lm ¼ 80. The effects of the magnetic field strength on distributions of particles are studied with lH ¼ 10, 40, 160 and 640, respectively. Simulation results are given in Fig. 3. It can be observed obviously that the distributions of particles for various field strengths are identical. It is suggested that the distributions of particles are not affected by the field strength when magnetic interaction between MPs is fixed. Previous experimental observations [8] give different results, which show that the length of Ni clusters increased with increasing magnetic field strength. In fact, there is no contradiction between experimental observations and simulation results. Once a magnetic field is applied to the suspension comprised of both MPs and NPs, MPs are magnetized with the moments aligned along the field direction. As magnetic field increased, the magnetization intensity of MPs increased, and therefore not only the interaction between MPs and magnetic field but also the interaction between MPs increased. In this section, the value of interaction between MPs is fixed, and only
Fig. 5. Effects of concentration of MPs on number density distributions of clusters with lm ¼ 80 and gH ¼ 80.
60
60
H
Nm = 100
H 50
50
40
40
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Nm = 200
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Nm = 400
0
0 0
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0
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60
Fig. 4. Effects of concentration of MPs on distributions of particles with lm ¼ 80 and gH ¼ 80. (a) Nm ¼ 100, (b) Nm ¼ 200, (c) Nm ¼ 300, and (d) Nm ¼ 400.
ARTICLE IN PRESS X. Peng et al. / Journal of Magnetism and Magnetic Materials 321 (2009) 1221–1226
1225
given in Fig. 6. For Nn ¼ 0, which means a dilute suspension comprised of only MPs, MPs aggregate to form long chain-like clusters (Fig. 6a). In the case of Nn ¼ 400, which reflects a low concentration of NPs, the migration of MPs is partly hindered by NPs, and therefore the length of MP clusters is a little shorter than that for Nn ¼ 0 (Fig. 6b). As Nn further increases, the length of
the interaction between MPs lm are taken as lH ¼ 80 and lm ¼ 80, respectively. The concentration of MPs is setting as Nm ¼ 100, 200, 300 and 400, respectively. Simulation results are given in Fig. 4. It is observed that very few small clusters are discerned for a low concentration of MPs, such as Nm ¼ 100 (Fig. 4a). As Nm increases, more particles aggregate to form small clusters and the size of the clusters increases gradually (Fig. 4b and c). For a relatively high concentration of MPs, such as Nm ¼ 400 (Fig. 4d), numerous chain-like clusters are formed along the magnetic field direction. The number density distributions of clusters are analyzed and plotted on logarithmic graphs as shown in Fig. 5. All curves exhibit a tendency of monotonous decrease. As the concentration of MPs increase, both the cluster size and the number density of clusters increase, whereas the number density of individual MPs decreases. It indicates that MPs tend to aggregate to form clusters as the concentration of MPs increases. As above mentioned, the formation of clusters can be attributed mainly to the interaction between MPs which is extremely sensitive to the distance r between MPs according to Eq. (2). As the concentration of MPs increases, the mean distance between MPs decreases. Thus, the number and size of clusters increase with increasing concentration of MPs.
3.4. Effects of concentration of NPs on distributions of particles To investigate the effects of the concentration of NPs on the distributions of particles in suspension, the field strength lH and the interaction between MPs lm are taken as lH ¼ 80 and lm ¼ 160, respectively. The concentration of NPs is setting as Nn ¼ 0, 400, 1200 and 2516, respectively. Simulation results are
Fig. 7. Effects of concentration of MPs on number density distributions of clusters with lm ¼ 160 and gH ¼ 80.
60
60
H
Nn = 0
H 50
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Nn = 400
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Nn = 1200
50
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H
50
60
Nn = 2516
0
0 0
10
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50
60
0
10
20
30
40
50
60
Fig. 6. Effects of concentration of NPs on distributions of particles with lm ¼ 160 and gH ¼ 80. (a) Nn ¼ 0, (b) Nn ¼ 400, (c) Nn ¼ 1200, and (d) Nn ¼ 2516.
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clusters decreases gradually (Fig. 6c). For a high concentration of NPs, such as Nn ¼ 2516, the migration of MPs are hindered greatly by NPs, and the size of MP clusters has become quite small (Fig. 6d). The number density distributions of clusters are analyzed and plotted on logarithmic graphs as shown in Fig. 7. For Nn ¼ 0, the biggest cluster contains 53 MPs. For Nn ¼ 400, size of clusters become smaller, with 28 MPs in the biggest clusters, suggesting a slightly hindering effect of NPs on the migration of MPs. As Nn increases, both the cluster size and relative density of large curves decrease. For a high concentration of NPs, such as Nn ¼ 2516, the size of clusters is very small and the biggest cluster contains only 9 MPs, indicating a great hindrance of NPs on migration of MPs. It can be concluded that NPs affect structures of suspension by hindering the migration of MPs. When MPs get close to NPs during migration, a repulsive force arises between the contacting particles, which hinders MPs from moving. As the concentration of NPs increases, the hindrance of NPs on migration of MPs is enhanced. Thus, the length of MP clusters decreases with increasing concentration of NPs.
4. Conclusions Through the two-dimensional Monte Carlo simulation, the structures and distributions of particles under magnetic fields in suspension comprised of both magnetic and nonmagnetic particles were studied. Magnetic particles aggregated to form chain-like clusters along the field direction. The size of the clusters increased with increasing magnetic interaction between magnetic particles, while it kept nearly unchanged as the field strength was varied. As the concentration of magnetic particles increased, both the number and size of the clusters increased. Moreover, nonmagnetic particles also affected the distributions of particles by hindering the migration of magnetic particles. As the
concentration of nonmagnetic particles increased, the migration of magnetic particles became more difficult.
Acknowledgment This work was supported by the National Natural Science Foundation of China (No. 50471041), National High Technology Research and Development Program (‘‘863’’ program) of China, Program for New Century Excellent Talents in University (05-0526) and Program for Innovative Research Team in University (IRT-0651). References [1] Y. Zhu, E. Haddadian, T. Mou, M. Gross, J. Liu, Phys. Rev. E 53 (1996) 1753. [2] E.M. Furst, A.P. Gast, Phys. Rev. E 61 (2000) 6732. [3] S.V. Gorobets, I.A. Melnichuk, J. Magn. Magn. Mater. 182 (1998) 61. [4] M. Matsuzaki, H. Kikura, J. Matsushita, M. Aritomi, H. Akatsuka, Sci. Technol. Adv. Mater. 5 (2004) 667. [5] Jing Liu, E.M. Lawrence, A. Wu, M.L. Ivey, G.A. Flores, K. Javier, J. Bibette, J. Richard, Phys. Rev. Lett. 74 (14) (1995) 2828. [6] Y.S. Zhu, N. Umeharab, Y. Idoc, A. Sato, J. Magn. Magn. Mater. 302 (2006) 96. [7] X.L. Peng, M. Yan, W.T. Shi, Scripta Mater. 56 (2007) 907. [8] X.L. Peng, M. Yan, W.T. Shi, T.Y. Ma, J. Inorg. Mater. 23 (4) (2008) 836. [9] A. Satoh, R.W. Chantrell, G.N. Coverdale, S. Kamiyama, J. Colloid Interface Sci. 203 (1998) 233. [10] C. Scherer, J. Magn. Magn. Mater. 289 (2005) 196. [11] P.J. Camp, G.N. Patey, Phys. Rev. E 62 (2000) 5403. [12] M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids, Clarendon Press, Oxford, 1987. [13] A. Satoh, Introduction to Molecular-Microsimulation of Colloidal Dispersions, Elsevier, Amsterdam, 2003. [14] M. Aoshima, A. Satoh, J. Colloid Interface Sci. 288 (2005) 475. [15] M. Aoshima, A. Satoh, J. Colloid Interface Sci. 293 (2006) 77. [16] A. Satoh, J. Colloid Interface Sci. 318 (2008) 68. [17] M. Aoshima, A. Satoh, J. Colloid Interface Sci. 280 (2004) 83. [18] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, E. Teller, J. Chem. Phys. 21 (1953) 1087. [19] M. Yan, X. L. Peng, T. Y. Ma, submitted to J. Alloys. Compd.
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Two-dimensional Monte Carlo simulations of structures of a suspension comprised of magnetic and nonmagnetic particles in uniform magnetic fields Xiaoling Peng a,b, Yong Min c, Tianyu Ma a, Wei Luo a, Mi Yan a, a b c
State Key Laboratory of Silicon Materials, Department of Materials Science and Engineering, Zhejiang University, Hangzhou 310027, China College of Material Sciences and Engineering, China Jiliang University, Zhejiang Province Key Laboratory of Magnetism, Hangzhou 310018, China Ningbo Institute of Technology, Zhejiang university, Ningbo 315100, China
a r t i c l e in fo
abstract
Article history: Received 25 June 2008 Received in revised form 30 September 2008 Available online 21 November 2008
The structures of suspensions comprised of magnetic and nonmagnetic particles in magnetic fields are studied using two-dimensional Monte Carlo simulations. The magnetic interaction among magnetic particles, magnetic field strength, and concentrations of both magnetic and nonmagnetic particles are considered as key influencing factors in the present work. The results show that chain-like clusters of magnetic particles are formed along the field direction. The size of the clusters increases with increasing magnetic interaction between magnetic particles, while it keeps nearly unchanged as the field strength increases. As the concentration of magnetic particles increases, both the number and size of the clusters increase. Moreover, nonmagnetic particles are found to hinder the migration of magnetic ones. As the concentration of nonmagnetic particles increases, the hindrance on migration of magnetic particles is enhanced. & 2008 Elsevier B.V. All rights reserved.
Keywords: Monte Carlo simulation Uniform magnetic field Particle distribution Chain-like clusters
1. Introduction When an external magnetic field is applied to a colloidal suspension comprised of magnetic particles (MPs) suspended in a nonmagnetic medium, dipolar interactions are induced within the particles, and various structures are formed in suspension [1–3]. The suspension, which is sensitive to applied magnetic fields, is called magnetorheological (MR) fluid or magnetic fluid (MF) [4,5]. Due to their practical utility, MR and MF have attracted considerable attention in the past several years, and the field-induced structures in the suspension are reasonably well understood [5]. The structure of the suspension comprised of both magnetic and nonmagnetic particles (NPs) is also sensitive to external magnetic fields [6]. Taking advantage of different response to magnetic fields, a new approach for fabricating functionally graded materials (FGMs) have been developed [7]. A detailed understanding of structures and particle distributions of the mixed suspension in magnetic fields is much important. In our previous work, the effects of magnetic field strength and concentration of MPs on distributions of particles in magnetic fields have been studied [7,8]. However, the magnetic interaction and concentration of NPs, which seriously affect the final structure and distribution of particles, have still not been clarified. This is because it is very difficult to observe directly the
Corresponding author. Tel.: +86 57187952730; fax: +86 571 87951403.
E-mail address: [email protected] (M. Yan). 0304-8853/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2008.11.011
distribution of particles in suspension through experiments. Furthermore, not all the factors that may affect the distribution of particles can be reflected by the available experimental conditions, for example, magnetic interaction increased synchronously as magnetic field strength is increased. As an effective method, compute simulation such as Monte Carlo method has been commonly and successfully used in researches of MR and MF [9–13]. Magnetic particles are found to form chain-like structures under uniform magnetic fields, and the particle distribution is affected by magnetic dipolar interaction among particles, magnetic field strength, concentration of magnetic particles, particle size, and shape of particles [14–16]. Thus Monte Carlo simulation is powerful method used to clarify the particle aggregation phenomena, which should also be a suitable method to study the suspension comprised of both MPs and NPs. In this work, the structures of a suspension comprised of both MPs and NPs under magnetic fields are studied using twodimensional Monte Carlo simulations. The effects of interaction between MPs, magnetic field strength and concentration of MPs and NPs on distributions of particles are estimated.
2. Simulation methods 2.1. Physical model of the suspension We suppose that the position of each particle in the suspension remains unchanged during the subsequent drying and sintering
ARTICLE IN PRESS 1222
X. Peng et al. / Journal of Magnetism and Magnetic Materials 321 (2009) 1221–1226
and tji ¼ rji/rji, and td the ratio of the thickness of the steric layer d to the radius of particle, which is expressed as 2d/d. The dimensionless parameters lH, lm and lv are written as [13,17]
process for fabrication of FGMs. The model for the suspension is given as follows: MPs and NPs with identical diameter d are dispersed homogeneously in the suspension. The number of particles is defined as N ¼ Nm þ Nn
lv ¼
(1)
where Nm is the number of MPs, Nn the number of NPs. The particles are coated with a steric layer with thickness d. When a uniform magnetic field is applied, MPs are magnetized with the moments aligned along the field direction; while NPs are not affected. The motion of a magnetic particle is typically caused by three kinds of energy: the interaction energy miH between particle i and the magnetic field, the interaction energy mijm between magnetic particle i and j, and the interaction energy mijv due to the overlapping of the steric layers. The motion of a nonmagnetic particle is controlled only by the overlapping energy mijv The above-mentioned energies can be expressed as [13,17]
mHi ¼ kT lH ni H=H
mvij ¼ 2lv kT 1
(4)
where k is the Boltzmann’s constant, T the absolute temperature of the suspension, H the field strength of the magnetic field H, rji the magnitude of the vector rji drawn from particles i to j, ni and tji denote the unit vectors given by ni ¼ mi/mi (mi ¼ |mi|)
60
lm ¼ m0 m2 =4pd3 kT
(7)
60
λm = 10
H
(6)
In the present work, two-dimensional Monte Carlo simulations based on Metropolis’s algorithm [18] are conducted to investigate the internal structures and distributions of particles in the suspension comprised of MPs and NPs. The simulation parameters are as follows: The side length of the square simulation box L is 61.78d. The initial configuration of particles is taken randomly. The thickness of the steric layer d and the dimensionless parameter lv are taken as d ¼ 0.01d and lv ¼ 50, respectively. The cut-off radius rcoff for the particle–particle interactions is taken as rcoff ¼ 10d. The maximum displacement of a particle in one trial move is 0.005d. If the distance between the surfaces of the solid particles is smaller than 0.01d, then these particles are assumed to form a cluster.
(3)
r ji r ji d ðt þ 1Þd ln d r ji dt d dtd
lH ¼ m0 mH=kT
2.2. Simulation parameters
(2)
d fni nj 3ðni tji Þðnj tji Þg r 3ji
(5)
2
where ns is the number of surfactant molecules per unit area on the particle surface, m0 the permeability of free space, m the magnitude of the magnetic moment of MPs, which is proportional to the volume pd3/6.
3
mm ij ¼ kT lm
pd2 ns
50
50
40
40
30
30
20
20
10
10
H
λm = 40
0
0 0 60
10
20
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60 60
λm = 160
H
0
50
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H
50
60
λm = 640
0
0 0
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60
0
10
20
30
40
50
60
Fig. 1. Effects of the interaction between MPs on distributions of particles with lH ¼ 80. (a) lm ¼ 10, (b) lm ¼ 40, (c) lm ¼ 160, and (d) lm ¼ 640.
ARTICLE IN PRESS X. Peng et al. / Journal of Magnetism and Magnetic Materials 321 (2009) 1221–1226
To reduce the computational complexity, the moments of MPs are fixed along the field direction. Periodic boundary conditions are applied in X and Y directions. All simulations are carried out more than 400,000 steps to get a stable structure and distribution of particles. The present simulation results, two-dimensional structures of MP clusters on the XY plane are plotted using MATLAB 7.3.0 (R2006b) software. The magnetic field is horizontal. Red and blue circles represent MPs and NPs, respectively.
3. Results and discussion 3.1. Effects of the interaction between MPs on distributions of particles In this section, the field strength is setting as lH ¼ 80. The effects of the interactions between MPs on the distributions of particles are studied with lm ¼ 10, 40, 160 and 640, respectively. Simulation results are illustrated in Fig. 1. For lm ¼ 10, which means a weak interaction, MPs do not aggregate but move almost independently (Fig. 1a). In the case of lm ¼ 40, MPs become to aggregate and numerous short chain-like clusters are formed (Fig. 1b). On further increasing lm, short clusters merge into long ones and hence the length of clusters increases gradually (Fig. 1c). For a strong interaction, such as lm ¼ 640, nearly all MPs are arranged as long clusters. In order to investigate quantitatively the effects of the magnetic interaction on distribution of particles, the number density distributions of clusters are analyzed and plotted on a logarithmic graph as shown in Fig. 2. The abscissa indicates the cluster size, c, which means the number of particles in a cluster, and the ordinate is the dimensionless number density of each MP cluster. The maximum size of clusters for lm ¼ 40 is a little larger than that for lm ¼ 10. As lm further increases, both the cluster size and the number density of clusters increase drastically. Finally, the maximum cluster size c approach 17 for lm ¼ 640, which correspond to the long clusters along the magnetic field direction, as shown in Fig. 1d. It can be concluded that MPs aggregate to form chain-like clusters under magnetic fields and the clusters are prolonged along the field direction with increasing lm, which is due to the fact that short clusters merge into the longer ones.
Fig. 2. Effects of the interaction between MPs on number density distributions of clusters with lH ¼ 80.
60
60
λm = 10
H
1223
50
50
40
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10
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0
H
λm = 40
0 0
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Fig. 3. Effects of magnetic field strength on distributions of particles with lm ¼ 80. (a) lH ¼ 10, (b) lH ¼ 40, (c) lH ¼ 160, and (d) lH ¼ 640.
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the interaction between MPs and magnetic field is changed when the field strength varied. Thus, the simulation results clearly show that the distributions of particles are affected mainly by the interaction between MPs rather than the field strength.
Chain-like magnetic clusters are formed in the suspension comprised of both MPs and NPs. The simulation results are in good agreement with both experimental observations [8] and previous simulation results on magnetic fluids [17,19], which also show that chain-like clusters of MPs can be formed along the field direction. It is suggested that the formation of chain-like clusters is also attributed mainly to the interaction between MPs and larger interaction results in longer chain-like cluster along the field direction.
3.3. Effects of concentration of MPs on distributions of particles To investigate the effects of the concentration of MPs on the distributions of particles in suspension, the field strength lH and
3.2. Effects of magnetic field strength on distributions of particles In this section, the interaction between MPs is setting as
lm ¼ 80. The effects of the magnetic field strength on distributions of particles are studied with lH ¼ 10, 40, 160 and 640, respectively. Simulation results are given in Fig. 3. It can be observed obviously that the distributions of particles for various field strengths are identical. It is suggested that the distributions of particles are not affected by the field strength when magnetic interaction between MPs is fixed. Previous experimental observations [8] give different results, which show that the length of Ni clusters increased with increasing magnetic field strength. In fact, there is no contradiction between experimental observations and simulation results. Once a magnetic field is applied to the suspension comprised of both MPs and NPs, MPs are magnetized with the moments aligned along the field direction. As magnetic field increased, the magnetization intensity of MPs increased, and therefore not only the interaction between MPs and magnetic field but also the interaction between MPs increased. In this section, the value of interaction between MPs is fixed, and only
Fig. 5. Effects of concentration of MPs on number density distributions of clusters with lm ¼ 80 and gH ¼ 80.
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Fig. 4. Effects of concentration of MPs on distributions of particles with lm ¼ 80 and gH ¼ 80. (a) Nm ¼ 100, (b) Nm ¼ 200, (c) Nm ¼ 300, and (d) Nm ¼ 400.
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given in Fig. 6. For Nn ¼ 0, which means a dilute suspension comprised of only MPs, MPs aggregate to form long chain-like clusters (Fig. 6a). In the case of Nn ¼ 400, which reflects a low concentration of NPs, the migration of MPs is partly hindered by NPs, and therefore the length of MP clusters is a little shorter than that for Nn ¼ 0 (Fig. 6b). As Nn further increases, the length of
the interaction between MPs lm are taken as lH ¼ 80 and lm ¼ 80, respectively. The concentration of MPs is setting as Nm ¼ 100, 200, 300 and 400, respectively. Simulation results are given in Fig. 4. It is observed that very few small clusters are discerned for a low concentration of MPs, such as Nm ¼ 100 (Fig. 4a). As Nm increases, more particles aggregate to form small clusters and the size of the clusters increases gradually (Fig. 4b and c). For a relatively high concentration of MPs, such as Nm ¼ 400 (Fig. 4d), numerous chain-like clusters are formed along the magnetic field direction. The number density distributions of clusters are analyzed and plotted on logarithmic graphs as shown in Fig. 5. All curves exhibit a tendency of monotonous decrease. As the concentration of MPs increase, both the cluster size and the number density of clusters increase, whereas the number density of individual MPs decreases. It indicates that MPs tend to aggregate to form clusters as the concentration of MPs increases. As above mentioned, the formation of clusters can be attributed mainly to the interaction between MPs which is extremely sensitive to the distance r between MPs according to Eq. (2). As the concentration of MPs increases, the mean distance between MPs decreases. Thus, the number and size of clusters increase with increasing concentration of MPs.
3.4. Effects of concentration of NPs on distributions of particles To investigate the effects of the concentration of NPs on the distributions of particles in suspension, the field strength lH and the interaction between MPs lm are taken as lH ¼ 80 and lm ¼ 160, respectively. The concentration of NPs is setting as Nn ¼ 0, 400, 1200 and 2516, respectively. Simulation results are
Fig. 7. Effects of concentration of MPs on number density distributions of clusters with lm ¼ 160 and gH ¼ 80.
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Fig. 6. Effects of concentration of NPs on distributions of particles with lm ¼ 160 and gH ¼ 80. (a) Nn ¼ 0, (b) Nn ¼ 400, (c) Nn ¼ 1200, and (d) Nn ¼ 2516.
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clusters decreases gradually (Fig. 6c). For a high concentration of NPs, such as Nn ¼ 2516, the migration of MPs are hindered greatly by NPs, and the size of MP clusters has become quite small (Fig. 6d). The number density distributions of clusters are analyzed and plotted on logarithmic graphs as shown in Fig. 7. For Nn ¼ 0, the biggest cluster contains 53 MPs. For Nn ¼ 400, size of clusters become smaller, with 28 MPs in the biggest clusters, suggesting a slightly hindering effect of NPs on the migration of MPs. As Nn increases, both the cluster size and relative density of large curves decrease. For a high concentration of NPs, such as Nn ¼ 2516, the size of clusters is very small and the biggest cluster contains only 9 MPs, indicating a great hindrance of NPs on migration of MPs. It can be concluded that NPs affect structures of suspension by hindering the migration of MPs. When MPs get close to NPs during migration, a repulsive force arises between the contacting particles, which hinders MPs from moving. As the concentration of NPs increases, the hindrance of NPs on migration of MPs is enhanced. Thus, the length of MP clusters decreases with increasing concentration of NPs.
4. Conclusions Through the two-dimensional Monte Carlo simulation, the structures and distributions of particles under magnetic fields in suspension comprised of both magnetic and nonmagnetic particles were studied. Magnetic particles aggregated to form chain-like clusters along the field direction. The size of the clusters increased with increasing magnetic interaction between magnetic particles, while it kept nearly unchanged as the field strength was varied. As the concentration of magnetic particles increased, both the number and size of the clusters increased. Moreover, nonmagnetic particles also affected the distributions of particles by hindering the migration of magnetic particles. As the
concentration of nonmagnetic particles increased, the migration of magnetic particles became more difficult.
Acknowledgment This work was supported by the National Natural Science Foundation of China (No. 50471041), National High Technology Research and Development Program (‘‘863’’ program) of China, Program for New Century Excellent Talents in University (05-0526) and Program for Innovative Research Team in University (IRT-0651). References [1] Y. Zhu, E. Haddadian, T. Mou, M. Gross, J. Liu, Phys. Rev. E 53 (1996) 1753. [2] E.M. Furst, A.P. Gast, Phys. Rev. E 61 (2000) 6732. [3] S.V. Gorobets, I.A. Melnichuk, J. Magn. Magn. Mater. 182 (1998) 61. [4] M. Matsuzaki, H. Kikura, J. Matsushita, M. Aritomi, H. Akatsuka, Sci. Technol. Adv. Mater. 5 (2004) 667. [5] Jing Liu, E.M. Lawrence, A. Wu, M.L. Ivey, G.A. Flores, K. Javier, J. Bibette, J. Richard, Phys. Rev. Lett. 74 (14) (1995) 2828. [6] Y.S. Zhu, N. Umeharab, Y. Idoc, A. Sato, J. Magn. Magn. Mater. 302 (2006) 96. [7] X.L. Peng, M. Yan, W.T. Shi, Scripta Mater. 56 (2007) 907. [8] X.L. Peng, M. Yan, W.T. Shi, T.Y. Ma, J. Inorg. Mater. 23 (4) (2008) 836. [9] A. Satoh, R.W. Chantrell, G.N. Coverdale, S. Kamiyama, J. Colloid Interface Sci. 203 (1998) 233. [10] C. Scherer, J. Magn. Magn. Mater. 289 (2005) 196. [11] P.J. Camp, G.N. Patey, Phys. Rev. E 62 (2000) 5403. [12] M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids, Clarendon Press, Oxford, 1987. [13] A. Satoh, Introduction to Molecular-Microsimulation of Colloidal Dispersions, Elsevier, Amsterdam, 2003. [14] M. Aoshima, A. Satoh, J. Colloid Interface Sci. 288 (2005) 475. [15] M. Aoshima, A. Satoh, J. Colloid Interface Sci. 293 (2006) 77. [16] A. Satoh, J. Colloid Interface Sci. 318 (2008) 68. [17] M. Aoshima, A. Satoh, J. Colloid Interface Sci. 280 (2004) 83. [18] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, E. Teller, J. Chem. Phys. 21 (1953) 1087. [19] M. Yan, X. L. Peng, T. Y. Ma, submitted to J. Alloys. Compd.