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Influence of demagnetizing field on the permeability of soft magnetic composites
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 305 (2006) 291–295 www.elsevier.com/locate/jmmm
Influence of demagnetizing field on the permeability of soft magnetic composites G.Q. Lina,, Z.W. Lia, Linfeng Chena, Y.P. Wub, C.K. Ongb a
Temasek Laboratories, National University of Singapore, 5, Sports Drive 2, Singapore 117508, Singapore Centre for Superconducting and Magnetic Materials, Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542, Singapore
b
Received 22 June 2005 Available online 3 February 2006
Abstract The influence of demagnetizing field on the effective permeability of magnetic composites has been investigated. A theoretical expression of the effective permeability has been obtained and discussed according to four typical composites with spheres, needles, flakes, and aligned prolate ellipsoidal particles. The results indicate that the demagnetizing field within the particles can reduce the effective permeability significantly. In order to increase the effective permeability, it is necessary to decrease the demagnetizing field within the particles. A linear relationship between effective permeability and volume fraction is also observed for composites filled with spherical particles at low volume fraction. r 2006 Elsevier B.V. All rights reserved. PACS: 75.30.Cr; 75.50.y Keywords: Demagnetizing factor; Demagnetizing field; Soft magnetic composites; Effective permeability
1. Introduction Soft magnetic composites are materials comprising particles with soft magnetism (such as soft magnetic ferrites or metals) embedded in a non-magnetic matrix, and have extensive applications in defense and industry, such as electromagnetic (EM) materials which usually consist of ferrite fillers and epoxy matrix. Recently, composites with insulated ferrous powders were also investigated in detail to obtain suitable materials for high-frequency applications. However, experimental results showed that the effective permeability of composites is usually much lower than the permeability of the corresponding bulk materials, especially when the volume concentration of magnetic particles in the composite is low [1–3]. The decrease is mainly attributed to the demagnetizing field, generated by the magnetic poles on the surface of the particles [4,5]. Corresponding author. Tel.: +65 65161930; fax: +65 68726840.
E-mail address: [email protected] (G.Q. Lin). 0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.01.014
A number of theoretical studies were conducted to calculate the effective permeability in soft magnetic composites. The Maxwell-Garnett (MG) [6,7] and Bruggeman [8] effective medium theories (EMT) have been investigated experimentally and theoretically for many years. However, most of their conclusions were based on spherical particles with dipolar interactions. In 1990, Visser and Johnson proposed a simple magnetic circuit model to explain the dependence of permeabilities on the grain sizes for spinel ferrites [9,10]. The model was later employed to study the effective permeability of composites with ferrite inclusions [4,5]. Recently, Mattei et al. derived another theoretical equation to explain and predict the resonance frequency of composites [11–14]. However, all attempts in these studies were made to find numerical expressions for the effective permeability of composites. No attention has been paid to the much lower permeability found in composites. In this work, the influence of the demagnetizing field on the effective permeability of composites is investigated using the basic theories of ferromagnetism. A theoretical
ARTICLE IN PRESS G.Q. Lin et al. / Journal of Magnetism and Magnetic Materials 305 (2006) 291–295
292
expression is given and its applications in some typical cases are discussed in detail. Some experimental results are also presented to support our theoretical expression.
meff ¼ 1 þ
2. Model For composites with n magnetic particles embedded in a non-magnetic matrix, the effective magnetization M eff in the direction of an external magnetic field H e can be described as M eff ¼
n X
4pðm 1Þ p, 4p þ Nðm 1Þ
(6b)
where m is the average permeability of particles and p is the total volume fraction of magnetic particles in the composites. 3. Numerical results and discussions
M i pi ,
(1)
i
where M i and pi are the magnetization along the direction of the external field and the volume fraction of the ith particle, respectively. Eq. (1) can be rewritten as n X
Beff H e ¼
ðBi H i Þpi ,
(2)
i
in CGS units. Here, Beff and Bi are the effective induction for the composites and the ith particle, respectively. H i is the internal field strength of the ith particle, which can be expressed as H i ¼ H e H d;i , where H d;i is the demagnetizing field of the ith particles along the direction of the external field. It is known that the permeability can also be defined as the ratio of induction to field strength, i.e., meff ¼ Beff =H e and mi ¼ Bi =H i under the influence of a small magnetic field. Substituting the expressions of meff and mi into Eq. (2) gives meff ¼ 1 þ
With the assumption that all particles are identical, Eq. (6a) can be simplified to
n X i
ðmi 1Þ
H e H d;i pi . He
Eqs. (6a) and (6b) are the expressions of the effective permeability of composites. According to the formulae, the effective permeability of composites meff depends on the permeability of the particles m, the volume fraction p, and the demagnetizing factor N of the particles. The following discussions are based on Eq. (6b). First, consider a simple case of the composites with single-domain spherical particles embedded in a non-magnetic matrix at low volume concentration. In CGS units, N ¼ N a ¼ N b ¼ N c ¼ 4p=3. Hence, Eq. (6b) can be rewritten as meff ¼ 1 þ
(7a)
The effective permeability meff is shown in Figs. 1 and 2 as a function of the particle’s permeability m and its volume fraction p, respectively. For a comprehensive understanding, the theoretical results from Eq. (7a) are also extrapolated to high-volume fraction with the assumption that the inter-particle interactions are very weak in the composites with high-volume fraction of particles. There-
(3)
max=4.0 4.0
For each particle, the field strength can be calculated using the following equation: Bi ¼ 4pM i þ H i ¼ 4pM i þ ðH e H d;i Þ ¼ mi H i ¼ mi ðH e H d;i Þ.
3ðm 1Þ p. 3 þ ðm 1Þ
3.5
p=0.90
3.0
p= 0. 70
ð4Þ 2.5
H e H d;i 4p . ¼ 4p þ N i ðmi 1Þ He
(5)
Substituting Eq. (5) into Eq. (3), we can obtain the theoretical effective permeability of the composites, which is meff ¼ 1 þ
n X i
4pðmi 1Þ p. 4p þ N i ðmi 1Þ i
(6a)
p=0.50
eff
For composites with single-domain particles at low volume fraction, the distance between the particles is so large that the interactions between the magnetic poles on the surface of nearby particles are very weak. Hence, H d;i ¼ N i M i , where N i is the demagnetizing factor of the ith particles. Eq. (4) now gives the results of
2.0 p=0.30 1.5 p=0.10 1.0 0.5 1
50
100 i
150
200
Fig. 1. The effective permeability vs the permeability of particles, for composites with single-domain spherical particles at various volume fractions. Dashed lines are the results corresponding to composites with high-volume fraction, neglecting the interactions between particles. The theoretical maximum permeability, mmax at p ¼ 1:0, is also indicated.
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293
permeability of the composites is given by 4.0
A : i=10,000
A
B : i =50 3.0
meff ¼ 1 þ ðm 1Þp.
C
C : i=15
eff
D : i =5
meff ¼ 1 þ
2.5
D
1.5
1.0
0.2
0.4
0.6
0.8
1.0
pi Fig. 2. The effective permeability for composite with single-domain spherical particles as a function of the volume fraction of magnetic particles in the composites. Dashed lines are the results extrapolated from low- to high-volume fraction, neglecting the interactions between particles. The theoretical maximum permeability, mmax at p ¼ 1:0, is also indicated.
fore, the demagnetizing field in the particle is considered to be the same as that at low-volume fraction. The results are shown as dashed lines in Figs. 1 and 2. Basically, the effective permeability meff increases for composites with particles of high permeability m or highvolume fraction p. A linear relationship is also observed between the effective permeability and volume fraction when the volume fraction p in the composites is small. When the particle’s permeability m is above 50, its contributions to the effective permeability of the composites meff can be neglected (see Curves A and B in Fig. 2). Meanwhile, there is a maximum effective permeability found in composites which can be calculated from Eq. (7a). meff ¼ 1 þ
3ðm 1Þ pp1 þ 3p 3 þ ðm 1Þ
m1 p m
(7c)
for N ¼ N c ¼ 4p. For the aligned prolate ellipsoidal particles, let c=a ¼ c=b ¼ 3:70 where a; b, and c are the semi-axes of the ellipsoid, with the c-axis of the particles parallel to the external field. Then, the demagnetizing factor along the caxis [15] is N c ¼ p=3. Hence, N ¼ N c ¼ p=3. The effective permeability of the composites is
2.0
0.0
(7b)
For the flakes, the demagnetizing factors are N a ¼ N b ¼ 0 and N c ¼ 4p. Let c be parallel to the external field. Hence, the effective permeability of the composites is
B
meff ¼ 1 þ
(7d)
N=0
N=4π /3 100
B (A)
where pp1:0,
as the particle’s permeability m tends to infinity. For composites with volume fraction of p ¼ 0:30, the obtained maximum permeability meff is 1.90, even though the permeability of the particles m is infinite. It is much lower than the permeability of most soft magnetic materials, such as ferrous alloy and Ni–Zn ferrites. The maximum effective permeability, mmax , with p ¼ 1:0 is also shown in Figs. 1 and 2. The results indicate that a demagnetizing field immediately gives rise to a low effective permeability in composites, regardless of how high the particle’s permeability is. Besides the composites with spherical particles, there are three other types of simple cases, namely, composites with needles, flakes, and aligned prolate ellipsoidal particles. For needles, N a ¼ N b ¼ 2p and N c ¼ 0. Let c be parallel to the external field; hence, N ¼ N c ¼ 0. The effective
12ðm 1Þ p. 12 þ ðm 1Þ
All theoretical effective permeabilities from Eqs. (7a)–(7d) are shown in Fig. 3. The relationship between the demagnetizing field and the effective permeability in the composites is clearly shown in Fig. 3. For needles, there is no demagnetizing field along the direction studied. For the other three cases, there is a demagnetizing field within each particle, with the largest found in flakes and the smallest found in prolate ellipsoids. It is observed in Fig. 3 that higher demagnetizing field often leads to smaller effective permeability in composites. Nevertheless, the effective permeability for composites with demagnetizing field is much lower than the composites without demagnetizing field. For example, although there
H
(B)
N=4π eff
3.5
max=4.0
(C)
10
H N=4π /12
H
(D)
H
D A C
1
p=0.30 1
500
1000 i
1500
2000
Fig. 3. The effective permeability for composites with (a) spheres, (b) needles, (c) flakes, and (d) aligned prolate ellipsoidal particles. The Curves A, B, C, and D are plotted according to Eqs. (7a), (7b), (7c), and (7d) with the volume fraction of 0.30, respectively.
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complex permeabilities of the toroidal samples with an outer diameter of 6.9 mm, inner diameter of 3.0 mm, and thickness of 2.0 mm approximately were measured using HP4291B impedance analyzer over 0.01–1.8 GHz and HP8722D vector network analyzer (VNA) over 0.5–16.5 GHz. All real parts of permeability are shown in Fig. 4. The effective permeabilities of the composites were obtained by averaging the real permeabilities at low frequency. All results are presented in Table 1 and plotted in Fig. 5. The theoretical results from Eq. (7a) are also shown as the solid lines in Fig. 5, where the particle’s permeabilities of bulk Co1:3 Zn0:7 W and Co2 Z are about 8.0 [16] and 12.0 [16–19], respectively. Usually, in ferrites, two types of resonance can be observed, namely domain-wall resonance at low frequency and natural resonance at high frequency. Therefore, there are always two absorption peaks in the imaginary permeability spectra, especially for W-type ferrites [20]. However, for composites with single-domain particles, because of a lack of domain wall in particle, the domainwall resonance disappears and only one peak can be observed in the imaginary permeability spectra. The complex permeability spectra for composites with ballmilling particles at the volume fraction of 50% are shown in Fig. 6. For comparison, another two coaxial samples at the volume fraction of 50% were prepared using Co1:3 Zn0:7 W and Co2 Z powder without ball milling. The particle size is estimated to be about 10:0 mm from the SEM images. Their complex permeability spectra are also presented in Fig. 6. Fig. 6 shows that in the composites with the ball-milling particles, the contributions from domain-wall movement to the permeability become very weak. As a result, these ball-milling Co1:3 Zn0:7 W and Co2 Z particles can be regarded as single-domain particles, which indicates a high demagnetizing field in each particle. Immediately, it is observed that in their corresponding composites, the measured effective permeabilities are in excellent agreement with the theoretical prediction when the volume fraction is lower than 50%, as shown in Fig. 5, which supports our derivation accordingly. Since the interactions between particles at high-volume fraction are not negligible, the demagnetizing field within the particles decreases greatly. The assumption of H d ¼ NM is no longer valid. To take the inter-particle interactions into account, a correction function f ðpÞ maybe defined in terms of H d ¼ f ðpÞNM. The value of f ðpÞ depends on the shape of the particles, domain structures in
is only a small demagnetizing field in the prolate ellipsoid, the effective permeability is still much smaller than that of the composites with needles. The results imply that a demagnetizing field in composites can decrease the permeability significantly. To increase the effective permeability in composites, one of the possible methods is to remove the demagnetizing field within the particles. 4. Experimental results The theoretical results for composites with spherical particles are investigated experimentally. The ferrites of Co1:3 Zn0:7 W (BaCo1.3Zn0.7Fe16O27) and Co2 Z (Ba3Co2 Fe24O41) were synthesized using the conventional double sintering method. Mixed raw powders were first sintered at 1180 1C and second sintered at 1270 1C. All double-sintered samples were then crushed into powder and followed by a high-energy ball milling for 4 h. The particle size are estimated to be about 1:0 mm from the SEM images. All powders were conducted with a post-oxidation treatment at 900 1C for 24 h. The composites were prepared by mixing the ferrites powder and epoxy at various volume fractions. The
(a)
2.0
Permeability (µ)
1.5
1.0
0.5 (b) 2.0 1.5 1.0 0.5
0.1
1 Frequency (GHz)
10
Fig. 4. The real permeability spectra for composites with the inclusions of (a) Co1:3 Zn0:7 W and (b) Co2 Z at various volume fractions. The volume fraction of ferrites increases from 5% to 50% at the intervals of 5%.
Table 1 The experimental permeabilities for the composites with the inclusions of Co1:3 Zn0:7 W and Co2 Z at various volume fractions p p
5%
10%
15%
20%
25%
30%
35%
40%
45%
50%
Co1:3 Zn0:7 W Co2 Z
1.13 1.16
1.24 1.26
1.34 1.35
1.44 1.47
1.53 1.60
1.63 1.69
1.76 1.81
1.87 1.94
1.98 2.02
2.07, 2.47a 2.14, 3.31a
a
Particles without high-energy ball milling.
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295
and when p ! 1:0, f ðpÞ ! 0. Then, Eq. (6a) becomes 2.2 Experimental results:
meff ¼ 1 þ
Co1.3Zn0.7W Co2Z
2.0
i
eff
1.8
5. Conclusions
0.2
A model that describes the relationship between the effective permeability and the demagnetizing field in composites was proposed. Due to the demagnetizing field, the effective permeability meff of composites is much smaller than the permeability m of bulk materials or particles. The effective permeability meff is strongly related to the shape of the particle. It is large for needles and small for flakes. Moreover, a linear relationship is observed between the effective permeability and the volume fraction, for low-volume fraction of magnetic particles. A good agreement is also found between our theoretical predictions and experimental results for the composites with soft magnetic ferrites of spherical shape. The results are useful to the design of soft magnetic composites with high permeability.
1.2
1.0 0.0
0.1
0.3
0.4
0.5
p Fig. 5. The experimental permeability vs volume fractions for composites with the inclusions of Co1:3 Zn0:7 W and Co2 Z. Theoretical results are also shown as the solid lines with the particle’s permeability of 8.0 and 12.0 for Co1:3 Zn0:7 W and Co2 Z, respectively.
A
(a) B
2.0
Permeability (µ)
1.0
References
A : Without high-energy ball milling; B : With high-energy ball milling. A
0.5
B 0.0 3.6 3.0
(b) A
2.4 B 1.8 1.2
A
0.6 0.0 0.01
(8)
=8.0
1.4
1.5
4pðmi 1Þ p. 4p þ f ðpÞN i ðmi 1Þ i
=12.0 1.6
2.5
N X
B 0.1
1
10
Frequency(GHz)
Fig. 6. The complex permeability spectra for the composites with the inclusions of (a) Co1:3 Zn0:7 W and (b) Co2 Z at the volume fraction of 50%. Curve A is the results corresponding to the powders without ball milling and Curve B, with ball milling for 4 h.
the particles, and the volume fraction of the particles in the composites. It indicates the contribution of demagnetizing field to the properties in composites and should be subject to certain limiting conditions, i.e. when p ! 0, f ðpÞ ! 1:0
[1] L.P. Lefebvre, S. Pelletier, C. Gelinas, J. Magn. Magn. Mater. 176 (1997) L93. [2] Y. Konishi, H. Komori, Microwave Opt. Technol. Lett. 16 (1997) 156. [3] J.A. Bas, J.A. Calero, M.J. Dougan, J. Magn. Magn. Mater. 254–255 (2003) 391. [4] T. Tsutaoka, M. Ueshima, T. Tokunaga, T. Nakamura, K. Hatakeyama, J. Appl. Phys. 78 (1995) 3983. [5] T. Tsutaoka, J. Appl. Phys. 93 (2003) 2789. [6] C.M. Garnett, Philos. Trans. Roy. Soc. London 203 (1904) 385. [7] C.M. Garnett, Philos. Trans. Roy. Soc. London 205 (1906) 237. [8] D.A.G. Bruggeman, Ann. Phys. (Leipzig) 24 (1935) 636. [9] M.T. Johnson, E.G. Visser, IEEE Trans. Magn. 26 (1990) 1987. [10] E.G. Visser, M.T. Johnson, J. Magn. Magn. Mater. 101 (1991) 143. [11] J.L. Mattei, O. Minot, M. Le Floc’h, J. Magn. Magn. Mater. 140–144 (1995) 2189. [12] M. Le Floc’h, J.L. Mattei, P. Laurent, O. Minot, A.M. Konn, J. Magn. Magn. Mater. 140–144 (1995) 2191. [13] J.L. Mattei, P. Laurent, O. Minot, M. Le Floc’h, J. Magn. Magn. Mater. 160 (1996) 23. [14] P. Laurent, G. Viau, A.M. Konn, Ph. Gelin, M. Le Floc’h, J. Magn. Magn. Mater. 160 (1996) 63. [15] J.A. Osborne, Phys. Rev. 67 (1945) 351. [16] T. Nakamura, K. Hatakeyama, IEEE Trans. Magn. 36 (2000) 3415. [17] J. Smit, H.P.J. Wijn, Ferrites, Philips Technical Library, Eindhoven, 1959. [18] X.H. Wang, T.L. Ren, L.T. Li, Z.L. Gui, S.Y. Su, Z.X. Yue, J. Zhou, J. Magn. Magn. Mater. 234 (2001) 255. [19] H.I. Hsiang, H.H. Duh, J. Mater. Sci. 36 (2001) 2081. [20] Y.P. Wu, C.K. Ong, Z.W. Li, L.F. Chen, G.Q. Lin, S.J. Wang, J. Appl. Phys. 97 (2005) 063909.
Journal of Magnetism and Magnetic Materials 305 (2006) 291–295 www.elsevier.com/locate/jmmm
Influence of demagnetizing field on the permeability of soft magnetic composites G.Q. Lina,, Z.W. Lia, Linfeng Chena, Y.P. Wub, C.K. Ongb a
Temasek Laboratories, National University of Singapore, 5, Sports Drive 2, Singapore 117508, Singapore Centre for Superconducting and Magnetic Materials, Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore 117542, Singapore
b
Received 22 June 2005 Available online 3 February 2006
Abstract The influence of demagnetizing field on the effective permeability of magnetic composites has been investigated. A theoretical expression of the effective permeability has been obtained and discussed according to four typical composites with spheres, needles, flakes, and aligned prolate ellipsoidal particles. The results indicate that the demagnetizing field within the particles can reduce the effective permeability significantly. In order to increase the effective permeability, it is necessary to decrease the demagnetizing field within the particles. A linear relationship between effective permeability and volume fraction is also observed for composites filled with spherical particles at low volume fraction. r 2006 Elsevier B.V. All rights reserved. PACS: 75.30.Cr; 75.50.y Keywords: Demagnetizing factor; Demagnetizing field; Soft magnetic composites; Effective permeability
1. Introduction Soft magnetic composites are materials comprising particles with soft magnetism (such as soft magnetic ferrites or metals) embedded in a non-magnetic matrix, and have extensive applications in defense and industry, such as electromagnetic (EM) materials which usually consist of ferrite fillers and epoxy matrix. Recently, composites with insulated ferrous powders were also investigated in detail to obtain suitable materials for high-frequency applications. However, experimental results showed that the effective permeability of composites is usually much lower than the permeability of the corresponding bulk materials, especially when the volume concentration of magnetic particles in the composite is low [1–3]. The decrease is mainly attributed to the demagnetizing field, generated by the magnetic poles on the surface of the particles [4,5]. Corresponding author. Tel.: +65 65161930; fax: +65 68726840.
E-mail address: [email protected] (G.Q. Lin). 0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.01.014
A number of theoretical studies were conducted to calculate the effective permeability in soft magnetic composites. The Maxwell-Garnett (MG) [6,7] and Bruggeman [8] effective medium theories (EMT) have been investigated experimentally and theoretically for many years. However, most of their conclusions were based on spherical particles with dipolar interactions. In 1990, Visser and Johnson proposed a simple magnetic circuit model to explain the dependence of permeabilities on the grain sizes for spinel ferrites [9,10]. The model was later employed to study the effective permeability of composites with ferrite inclusions [4,5]. Recently, Mattei et al. derived another theoretical equation to explain and predict the resonance frequency of composites [11–14]. However, all attempts in these studies were made to find numerical expressions for the effective permeability of composites. No attention has been paid to the much lower permeability found in composites. In this work, the influence of the demagnetizing field on the effective permeability of composites is investigated using the basic theories of ferromagnetism. A theoretical
ARTICLE IN PRESS G.Q. Lin et al. / Journal of Magnetism and Magnetic Materials 305 (2006) 291–295
292
expression is given and its applications in some typical cases are discussed in detail. Some experimental results are also presented to support our theoretical expression.
meff ¼ 1 þ
2. Model For composites with n magnetic particles embedded in a non-magnetic matrix, the effective magnetization M eff in the direction of an external magnetic field H e can be described as M eff ¼
n X
4pðm 1Þ p, 4p þ Nðm 1Þ
(6b)
where m is the average permeability of particles and p is the total volume fraction of magnetic particles in the composites. 3. Numerical results and discussions
M i pi ,
(1)
i
where M i and pi are the magnetization along the direction of the external field and the volume fraction of the ith particle, respectively. Eq. (1) can be rewritten as n X
Beff H e ¼
ðBi H i Þpi ,
(2)
i
in CGS units. Here, Beff and Bi are the effective induction for the composites and the ith particle, respectively. H i is the internal field strength of the ith particle, which can be expressed as H i ¼ H e H d;i , where H d;i is the demagnetizing field of the ith particles along the direction of the external field. It is known that the permeability can also be defined as the ratio of induction to field strength, i.e., meff ¼ Beff =H e and mi ¼ Bi =H i under the influence of a small magnetic field. Substituting the expressions of meff and mi into Eq. (2) gives meff ¼ 1 þ
With the assumption that all particles are identical, Eq. (6a) can be simplified to
n X i
ðmi 1Þ
H e H d;i pi . He
Eqs. (6a) and (6b) are the expressions of the effective permeability of composites. According to the formulae, the effective permeability of composites meff depends on the permeability of the particles m, the volume fraction p, and the demagnetizing factor N of the particles. The following discussions are based on Eq. (6b). First, consider a simple case of the composites with single-domain spherical particles embedded in a non-magnetic matrix at low volume concentration. In CGS units, N ¼ N a ¼ N b ¼ N c ¼ 4p=3. Hence, Eq. (6b) can be rewritten as meff ¼ 1 þ
(7a)
The effective permeability meff is shown in Figs. 1 and 2 as a function of the particle’s permeability m and its volume fraction p, respectively. For a comprehensive understanding, the theoretical results from Eq. (7a) are also extrapolated to high-volume fraction with the assumption that the inter-particle interactions are very weak in the composites with high-volume fraction of particles. There-
(3)
max=4.0 4.0
For each particle, the field strength can be calculated using the following equation: Bi ¼ 4pM i þ H i ¼ 4pM i þ ðH e H d;i Þ ¼ mi H i ¼ mi ðH e H d;i Þ.
3ðm 1Þ p. 3 þ ðm 1Þ
3.5
p=0.90
3.0
p= 0. 70
ð4Þ 2.5
H e H d;i 4p . ¼ 4p þ N i ðmi 1Þ He
(5)
Substituting Eq. (5) into Eq. (3), we can obtain the theoretical effective permeability of the composites, which is meff ¼ 1 þ
n X i
4pðmi 1Þ p. 4p þ N i ðmi 1Þ i
(6a)
p=0.50
eff
For composites with single-domain particles at low volume fraction, the distance between the particles is so large that the interactions between the magnetic poles on the surface of nearby particles are very weak. Hence, H d;i ¼ N i M i , where N i is the demagnetizing factor of the ith particles. Eq. (4) now gives the results of
2.0 p=0.30 1.5 p=0.10 1.0 0.5 1
50
100 i
150
200
Fig. 1. The effective permeability vs the permeability of particles, for composites with single-domain spherical particles at various volume fractions. Dashed lines are the results corresponding to composites with high-volume fraction, neglecting the interactions between particles. The theoretical maximum permeability, mmax at p ¼ 1:0, is also indicated.
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293
permeability of the composites is given by 4.0
A : i=10,000
A
B : i =50 3.0
meff ¼ 1 þ ðm 1Þp.
C
C : i=15
eff
D : i =5
meff ¼ 1 þ
2.5
D
1.5
1.0
0.2
0.4
0.6
0.8
1.0
pi Fig. 2. The effective permeability for composite with single-domain spherical particles as a function of the volume fraction of magnetic particles in the composites. Dashed lines are the results extrapolated from low- to high-volume fraction, neglecting the interactions between particles. The theoretical maximum permeability, mmax at p ¼ 1:0, is also indicated.
fore, the demagnetizing field in the particle is considered to be the same as that at low-volume fraction. The results are shown as dashed lines in Figs. 1 and 2. Basically, the effective permeability meff increases for composites with particles of high permeability m or highvolume fraction p. A linear relationship is also observed between the effective permeability and volume fraction when the volume fraction p in the composites is small. When the particle’s permeability m is above 50, its contributions to the effective permeability of the composites meff can be neglected (see Curves A and B in Fig. 2). Meanwhile, there is a maximum effective permeability found in composites which can be calculated from Eq. (7a). meff ¼ 1 þ
3ðm 1Þ pp1 þ 3p 3 þ ðm 1Þ
m1 p m
(7c)
for N ¼ N c ¼ 4p. For the aligned prolate ellipsoidal particles, let c=a ¼ c=b ¼ 3:70 where a; b, and c are the semi-axes of the ellipsoid, with the c-axis of the particles parallel to the external field. Then, the demagnetizing factor along the caxis [15] is N c ¼ p=3. Hence, N ¼ N c ¼ p=3. The effective permeability of the composites is
2.0
0.0
(7b)
For the flakes, the demagnetizing factors are N a ¼ N b ¼ 0 and N c ¼ 4p. Let c be parallel to the external field. Hence, the effective permeability of the composites is
B
meff ¼ 1 þ
(7d)
N=0
N=4π /3 100
B (A)
where pp1:0,
as the particle’s permeability m tends to infinity. For composites with volume fraction of p ¼ 0:30, the obtained maximum permeability meff is 1.90, even though the permeability of the particles m is infinite. It is much lower than the permeability of most soft magnetic materials, such as ferrous alloy and Ni–Zn ferrites. The maximum effective permeability, mmax , with p ¼ 1:0 is also shown in Figs. 1 and 2. The results indicate that a demagnetizing field immediately gives rise to a low effective permeability in composites, regardless of how high the particle’s permeability is. Besides the composites with spherical particles, there are three other types of simple cases, namely, composites with needles, flakes, and aligned prolate ellipsoidal particles. For needles, N a ¼ N b ¼ 2p and N c ¼ 0. Let c be parallel to the external field; hence, N ¼ N c ¼ 0. The effective
12ðm 1Þ p. 12 þ ðm 1Þ
All theoretical effective permeabilities from Eqs. (7a)–(7d) are shown in Fig. 3. The relationship between the demagnetizing field and the effective permeability in the composites is clearly shown in Fig. 3. For needles, there is no demagnetizing field along the direction studied. For the other three cases, there is a demagnetizing field within each particle, with the largest found in flakes and the smallest found in prolate ellipsoids. It is observed in Fig. 3 that higher demagnetizing field often leads to smaller effective permeability in composites. Nevertheless, the effective permeability for composites with demagnetizing field is much lower than the composites without demagnetizing field. For example, although there
H
(B)
N=4π eff
3.5
max=4.0
(C)
10
H N=4π /12
H
(D)
H
D A C
1
p=0.30 1
500
1000 i
1500
2000
Fig. 3. The effective permeability for composites with (a) spheres, (b) needles, (c) flakes, and (d) aligned prolate ellipsoidal particles. The Curves A, B, C, and D are plotted according to Eqs. (7a), (7b), (7c), and (7d) with the volume fraction of 0.30, respectively.
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complex permeabilities of the toroidal samples with an outer diameter of 6.9 mm, inner diameter of 3.0 mm, and thickness of 2.0 mm approximately were measured using HP4291B impedance analyzer over 0.01–1.8 GHz and HP8722D vector network analyzer (VNA) over 0.5–16.5 GHz. All real parts of permeability are shown in Fig. 4. The effective permeabilities of the composites were obtained by averaging the real permeabilities at low frequency. All results are presented in Table 1 and plotted in Fig. 5. The theoretical results from Eq. (7a) are also shown as the solid lines in Fig. 5, where the particle’s permeabilities of bulk Co1:3 Zn0:7 W and Co2 Z are about 8.0 [16] and 12.0 [16–19], respectively. Usually, in ferrites, two types of resonance can be observed, namely domain-wall resonance at low frequency and natural resonance at high frequency. Therefore, there are always two absorption peaks in the imaginary permeability spectra, especially for W-type ferrites [20]. However, for composites with single-domain particles, because of a lack of domain wall in particle, the domainwall resonance disappears and only one peak can be observed in the imaginary permeability spectra. The complex permeability spectra for composites with ballmilling particles at the volume fraction of 50% are shown in Fig. 6. For comparison, another two coaxial samples at the volume fraction of 50% were prepared using Co1:3 Zn0:7 W and Co2 Z powder without ball milling. The particle size is estimated to be about 10:0 mm from the SEM images. Their complex permeability spectra are also presented in Fig. 6. Fig. 6 shows that in the composites with the ball-milling particles, the contributions from domain-wall movement to the permeability become very weak. As a result, these ball-milling Co1:3 Zn0:7 W and Co2 Z particles can be regarded as single-domain particles, which indicates a high demagnetizing field in each particle. Immediately, it is observed that in their corresponding composites, the measured effective permeabilities are in excellent agreement with the theoretical prediction when the volume fraction is lower than 50%, as shown in Fig. 5, which supports our derivation accordingly. Since the interactions between particles at high-volume fraction are not negligible, the demagnetizing field within the particles decreases greatly. The assumption of H d ¼ NM is no longer valid. To take the inter-particle interactions into account, a correction function f ðpÞ maybe defined in terms of H d ¼ f ðpÞNM. The value of f ðpÞ depends on the shape of the particles, domain structures in
is only a small demagnetizing field in the prolate ellipsoid, the effective permeability is still much smaller than that of the composites with needles. The results imply that a demagnetizing field in composites can decrease the permeability significantly. To increase the effective permeability in composites, one of the possible methods is to remove the demagnetizing field within the particles. 4. Experimental results The theoretical results for composites with spherical particles are investigated experimentally. The ferrites of Co1:3 Zn0:7 W (BaCo1.3Zn0.7Fe16O27) and Co2 Z (Ba3Co2 Fe24O41) were synthesized using the conventional double sintering method. Mixed raw powders were first sintered at 1180 1C and second sintered at 1270 1C. All double-sintered samples were then crushed into powder and followed by a high-energy ball milling for 4 h. The particle size are estimated to be about 1:0 mm from the SEM images. All powders were conducted with a post-oxidation treatment at 900 1C for 24 h. The composites were prepared by mixing the ferrites powder and epoxy at various volume fractions. The
(a)
2.0
Permeability (µ)
1.5
1.0
0.5 (b) 2.0 1.5 1.0 0.5
0.1
1 Frequency (GHz)
10
Fig. 4. The real permeability spectra for composites with the inclusions of (a) Co1:3 Zn0:7 W and (b) Co2 Z at various volume fractions. The volume fraction of ferrites increases from 5% to 50% at the intervals of 5%.
Table 1 The experimental permeabilities for the composites with the inclusions of Co1:3 Zn0:7 W and Co2 Z at various volume fractions p p
5%
10%
15%
20%
25%
30%
35%
40%
45%
50%
Co1:3 Zn0:7 W Co2 Z
1.13 1.16
1.24 1.26
1.34 1.35
1.44 1.47
1.53 1.60
1.63 1.69
1.76 1.81
1.87 1.94
1.98 2.02
2.07, 2.47a 2.14, 3.31a
a
Particles without high-energy ball milling.
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and when p ! 1:0, f ðpÞ ! 0. Then, Eq. (6a) becomes 2.2 Experimental results:
meff ¼ 1 þ
Co1.3Zn0.7W Co2Z
2.0
i
eff
1.8
5. Conclusions
0.2
A model that describes the relationship between the effective permeability and the demagnetizing field in composites was proposed. Due to the demagnetizing field, the effective permeability meff of composites is much smaller than the permeability m of bulk materials or particles. The effective permeability meff is strongly related to the shape of the particle. It is large for needles and small for flakes. Moreover, a linear relationship is observed between the effective permeability and the volume fraction, for low-volume fraction of magnetic particles. A good agreement is also found between our theoretical predictions and experimental results for the composites with soft magnetic ferrites of spherical shape. The results are useful to the design of soft magnetic composites with high permeability.
1.2
1.0 0.0
0.1
0.3
0.4
0.5
p Fig. 5. The experimental permeability vs volume fractions for composites with the inclusions of Co1:3 Zn0:7 W and Co2 Z. Theoretical results are also shown as the solid lines with the particle’s permeability of 8.0 and 12.0 for Co1:3 Zn0:7 W and Co2 Z, respectively.
A
(a) B
2.0
Permeability (µ)
1.0
References
A : Without high-energy ball milling; B : With high-energy ball milling. A
0.5
B 0.0 3.6 3.0
(b) A
2.4 B 1.8 1.2
A
0.6 0.0 0.01
(8)
=8.0
1.4
1.5
4pðmi 1Þ p. 4p þ f ðpÞN i ðmi 1Þ i
=12.0 1.6
2.5
N X
B 0.1
1
10
Frequency(GHz)
Fig. 6. The complex permeability spectra for the composites with the inclusions of (a) Co1:3 Zn0:7 W and (b) Co2 Z at the volume fraction of 50%. Curve A is the results corresponding to the powders without ball milling and Curve B, with ball milling for 4 h.
the particles, and the volume fraction of the particles in the composites. It indicates the contribution of demagnetizing field to the properties in composites and should be subject to certain limiting conditions, i.e. when p ! 0, f ðpÞ ! 1:0
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