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Analytical expression for initial magnetization curve of Fe-based soft magnetic composite material
Journal of Magnetism and Magnetic Materials 423 (2017) 140–144
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Analytical expression for initial magnetization curve of Fe-based soft magnetic composite material
crossmark
⁎
Zuzana Birčákováa, , Peter Kollára, Ján Füzera, Radovan Burešb, Mária Fáberováb a b
Institute of Physics, Faculty of Science, Pavol Jozef Šafárik University, Park Angelinum 9, 04154 Košice, Slovakia Institute of Materials Research, Slovak Academy of Sciences, Watsonova 47, 04001 Košice, Slovakia
A R T I C L E I N F O
A BS T RAC T
Keywords: Analytical expression, Initial magnetization curve, Soft magnetic composite materials Magnetization processes
The analytical expression for the initial magnetization curve for Fe-phenolphormaldehyde resin composite material was derived based on the already proposed ideas of the magnetization vector deviation function and the domain wall annihilation function, characterizing the reversible magnetization processes through the extent of deviation of magnetization vectors from magnetic field direction and the irreversible processes through the effective numbers of movable domain walls, respectively. As for composite materials the specific dependences of these functions were observed, the ideas were extended meeting the composites special features, which are principally the much higher inner demagnetizing fields produced by magnetic poles on ferromagnetic particle surfaces. The proposed analytical expression enables us to find the relative extent of each type of magnetization processes when magnetizing a specimen along the initial curve.
1. Introduction The effort of finding an analytical expression for magnetization curves still has the very important point within the studies in the field of magnetism. Until now various scientific works provided valuable models, e.g. the Jiles-Atherton model of ferromagnetic hysteresis [1,2] or the hysteresis model proposed by J. Takács et al. [3], enabling to predict the curves as well as to reveal the information on magnetic properties, magnetization processes or phenomena in general. These models were experimentally verified on bulk ferromagnetic materials. Recently C.S. Schneider proposed in his studies [4–9] an approach, ideas of it he formulated as “cooperative anisotropic theory of ferromagnetic hysteresis” and applied it to describe the magnetization curves of various, also bulk Fe, Ni and Co based ferromagnets [4–9]. Soft magnetic composite materials (SMCs, consisting of small ferromagnetic particles insulated from each other) belong to the class of ferromagnetic materials, but due to their specificity, we expect, the analysis of the magnetic properties including the magnetization curves of SMCs require modified treatment based on principles developed originally for bulk ferromagnets. SMCs become an interesting kind of soft magnetic materials also from the view of application possibilities [10–18]. In authors previous studies [13–17] dealing with the soft magnetic composite materials, the different and more simple dependences of certain magnetic properties of SMCs were found compared to the majority of bulk ferromagnetic materials (e.g. the dependence of excess energy losses on ⁎
frequency). It is due to the fact that the magnetization process is performed in small ferromagnetic particles (which are not only electrically insulated, but the magnetization process within each particle is more or less independent on the magnetization process of surrounding particles), resulting in some specific properties of SMCs (e.g. low energy losses at medium to higher frequencies, or magnetic and thermal isotropy are valuable for the applications). Particularly these dissimilarities were taken into account in study [19], when B. Zidarič et al. developed a ferromagnetic hysteresis model for the analytical expression of magnetization curves of SMCs based on the Jiles-Atherton model. Similarly we assume the adaptation of other models for composites can bring some new knowledge on these materials. For this reason the soft magnetic composites have been chosen to be the focus of the presented work, with the aim to find the analytical expression of initial magnetization curve for Fe-based SMC, drawing on the selected essential ideas proposed by C.S. Schneider. By means of extending of these ideas to further develop the interpretation of magnetization processes in the view of composite material special features. SMCs were represented by simple Fe-phenolphormaldehyde resin samples. 2. Analytical expression for initial magnetization curve In order to find an expression that would enable to analytically describe the initial magnetization curves of SMCs, we drawn on some
Corresponding author. E-mail addresses: [email protected], [email protected] (Z. Birčáková).
http://dx.doi.org/10.1016/j.jmmm.2016.09.087 Received 29 April 2016; Received in revised form 7 September 2016; Accepted 16 September 2016 Available online 17 September 2016 0304-8853/ © 2016 Elsevier B.V. All rights reserved.
Journal of Magnetism and Magnetic Materials 423 (2017) 140–144
Z. Birčáková et al.
basic ideas proposed by C.S. Schneider [4–9] for selected bulk ferromagnets, based on the analysis of magnetization processes. Schneider described magnetization processes using terms for the hysteresis phenomena, which are not usually used in books such as [10,11,20– 22]. According to the investigations of magnetization processes in SMCs [13–17] we used the selected ideas as an inspiration to extend them meeting the special features of the composites. The ideas are: the possibility to express the initial magnetization curve with a two summand relation and the characterization of the particular magnetization processes by two normalized functions. The initial magnetization curve of ferromagnetic material can be expressed by one relation, from the demagnetized state to saturation, where M is the magnetization, H is the applied magnetic field, HC denotes the coercivity of material, χi is the initial susceptibility and two normalized functions Xr and Xann are introduced:
⎛ M⎞ M = H ⎜Xr χi + Xann ⎟ ⎝ HC ⎠
(1) Fig. 1. Light optical microscopy photographs of composite samples F45-10%, F75-10%, F100-10% and F160-10%.
First normalized function Xr is the magnetization vector deviation function and is defined as a ratio of the reversible to initial susceptibility [6]
Xr =
χrev . χi
Xann = Xr2
for several isotropic bulk ferromagnetic materials based on Fe, Ni, Co [4–6,8] and generalized this observation that the dependence is quadratic for an isotropic ferromagnet. He also predicted the magnetization curves with quite good accuracy.
(2)
This function characterizes the relative amount of reversible magnetization processes (domain wall displacement and/or magnetization vector rotation) through the extent to which all magnetization vectors in material are deviated from the direction of the applied magnetic field. Its maximum is always at the demagnetized state (Xr is equal to 1), further decreasing with the increasing magnetic induction due to the magnetization processes of the magnetization vector rotation towards the field direction. Finally Xr converges to 0 at saturation which means the minimum deviation of magnetization vectors from the magnetic field direction. The reversible susceptibility tells about the proportions of reversible processes in the overall magnetization process at a given magnetic induction. Thus the higher is the value of Xr, the higher is the relative amount of reversible processes, whereas with the decreasing Xr the more the magnetization vector rotation dominates over the reversible domain wall displacement. Second normalized function Xann is the domain wall annihilation function and characterizes the relative amount of irreversible domain wall displacements through the increasing annihilation of domain walls with the increasing magnetic induction (starting from the induction Bμ max , where the total permeability reaches its maximum μmax and domain wall annihilation and nucleation appears [22,24]), or in other words, the decreasing effective number of domain walls participating in magnetization process. As the domains with the magnetization vector oriented favourably to applied magnetic field grow (by the domain wall displacements), some of the domain walls can have the tendency to annihilate. The decreasing effective number of domain walls means the amount of irreversible domain wall displacements is decreasing. Function Xann reaches also values from 1 at the demagnetized state (extrapolated value, physical meaning of the maximum effective number of movable domain walls is at Bμ max ) down to 0 when approaching saturation (minimum effective number of movable domain walls). The proportion of irreversible magnetization processes determines the irreversible susceptibility. The relation (1) becomes in limiting case of low magnetic fields (Rayleigh region), where for H → 0: χrev → χi, Xr → 1, M → H χi, and Xann → 1, the following form:
M = Hχi +
χi 2 H, HC
(4)
3. Experimental The simple Fe-phenolphormaldehyde resin samples were chosen to represent soft magnetic composite material. The samples were prepared by conventional powder metallurgy process, with different resin contents and particle size distributions [15,16]. Iron powder ASC 100.29 (Höganäs AB Sweden [23]) was sieved to obtain particle size distributions showing peaks at 45 μm, 75 μm, 100 μm and 160 μm (granulometric classes labelled F45, F75, F100 and F160) [15,16]. The homogenization of iron powder with 5, 10 and 15 vol% (1, 2 and 3 wt %) of phenolphormaldehyde resin (Bakelite ATM) and acetone was performed. The mixtures were compacted at uniaxial pressure of 800 MPa in the form of a ring (outer diameter 24 mm, inner diameter 18 mm, height from 1.4 to 2.4 mm) and cylinder (diameter 10 mm, height 2 mm). After the compaction a heat treatment at a temperature of 165 °C for 60 min in electric furnace in air atmosphere was performed. Light optical microscopy investigations (Olympus GX71), Fig. 1, confirmed the structure of composite samples to be isotropic, as expected and also observed in studies on similar samples [13,14,17]. Parameters of the samples are in Table 1. Porosity was calculated from dimensions and mass of each sample (density of iron is 7.851 g/cm3 and density of resin is 1.39 g/cm3). The initial magnetization curves were measured by the DC fluxmeter-based hysteresisgraph and the reversible relative permeability was measured using the setup described in [16], both on ring-shaped samples. Magnetic induction was measured referred to the filler content of ferromagnetic material in SMC (subtracting non-ferromagnetic components: resin and pores). The coercivity was measured by Foerster Koerzimat 1.097HCJ on cylinder-shaped samples. 4. Results and discussion 4.1. Analytical expression of initial magnetization curves of SMCs
(3)
The magnetization vector deviation function Xr (B) and the domain wall annihilation function Xann (B) were calculated from the experimental data (Eqs. (1), (2)). The plots Xann vs. Xr for selected samples are in Fig. 2. The dependence Xann vs. Xr was found not to be
determining the Rayleigh coefficient to be χi / HC . C.S. Schneider experimentally observed the dependence Xann vs. Xr to be quadratic 141
Journal of Magnetism and Magnetic Materials 423 (2017) 140–144
Z. Birčáková et al.
Table 1 Parameters of the samples. Sample
Iron powder to insulation ratio (vol.) Density (g/cm3) Porosity (%) Filler content of ferromagnetic material (%)
Table 2 Initial relative permeability µi, coercivity HC, inner demagnetization factor ND and coefficient C1.
F45 − 5%
F45 − 10%
F75 − 10%
F100 − 10%
F160 − 10%
F45 − 15%
95: 5
90: 10
90: 10
90: 10
90: 10
85: 15
6.60 12.2 83.4
6.0 17.0 74.6
5.94 17.5 74.2
6.05 16.4 75.2
6.10 15.0 76.5
5.45 21.0 67.0
Sample
F45 − 5%
F45 − 10%
F75 − 10%
F100 − 10%
F160 − 10%
F45 − 15%
µi (-) HC (A/m) ND (-) C1 (-)
62.5 475 0.006 0.071
42.5 478 0.012 0.033
43.5 440 0.011 0.035
47 415 0.010 0.035
49 377 0.009 0.041
39 476 0.014 0.030
(using µrev = χrev + 1, µi = χi + 1), the values of initial relative permeability µi and coercivity HC are in Table 2 (µi is increasing with increasing filler content of ferromagnetic material in SMC and with increasing ferromagnetic particle size, HC is decreasing with increasing particle size and does not depend on the filler content [15,16]). The coefficient C1 tells about the shift from a quadratic dependence (the lower C1, the more the dependence is shifted, corresponding to the dependence of ND on filler content and particle size) - the obtained values are in Table 2. Coefficients C2 and C3 reached values ~10−7 – 10−5 and ~7.5 – 14, respectively and did not show dependence neither on filler content, nor on particle size. The Eq. (5) obtained from fitting was further implemented into Eq. (1), which was finally used to express analytically the initial magnetization curve, Figs. 3 and 4 show a good agreement with the experimental data. 4.2. Graphical meaning of the dependence Xann vs. Xr The dependence of domain wall annihilation function Xann plotted vs. magnetization vector deviation function Xr suggests the proportions of reversible and irreversible processes in the overall magnetization process. The dependence starts from point [0;0] (saturation) and ends at [1,1] (demagnetized state). From its shape we can deduce which type of magnetization processes dominates at given magnetic state (at given induction), where especially important is the area of magnetic inductions above Bμ max , where the total permeability reaches its maximum μmax and the domain wall annihilation and nucleation appears [22,24] (Bμ max determined for each sample is decreasing from 0.38 to 0.22 T with the increasing resin content [16], which stand for Xr ≈ 0.9, covering thus almost whole graph area). In this area we can directly claim that the magnetization process is the more realized by magnetization vector rotation (mainly reversible) the more is the curve approaching to Xr axis (intersecting Xann = 0), and vice versa, the closer is the curve to Xann axis (intersecting Xr = 0), the higher is the proportion of irreversible domain wall displacements (Barkhausen jumps). In Fig. 2 we can see it is in agreement with the observations in study [16]: In samples with higher resin content and with smaller particle size the effective number of movable domain walls participating in the magnetization process is lower due to higher inner demagnetizing fields produced at magnetic particles surfaces reducing magnetic interaction between particles, resulting in lower proportions of irreversible domain wall displacements [14–16]. To sum up, the comparison of the shapes of Xann vs. Xr dependences (samples with different particle sizes or different resin content) enables to express the tendency of the ratio between the effective number of movable domain walls participating in magnetization process and the extent of deviation of magnetization vectors from the applied magnetic field direction.
Fig. 2. The dependence of domain wall annihilation function Xann on magnetization vector deviation function Xr of selected samples F45-5%, F160-10% and F45-15%, fitted with Eq. (5), quadratic function Xann = Xr2 (dashed) is plotted for comparison.
quadratic for SMC samples. The main feature which differs the SMCs from the bulk materials studied by Schneider is their heterogeneous structure containing non-ferromagnetic components (insulation and pores) which give rise to much higher inner demagnetizing fields produced by magnetic poles on ferromagnetic particle surfaces [15]. Based on this observation we can assume that the quadratic rule (Eq. (4)) covers only isotropic bulk ferromagnets, where he observed the quadratic dependence Xann vs. Xr - function Xann = Xr2 is also depicted in Fig. 2 for comparison. We have observed the dependences are shifting from the quadratic one, the more the value of inner demagnetization factor ND is increasing. ND is increasing with decreasing filler content of ferromagnetic material in SMC and with decreasing ferromagnetic particle size [15] - values are in Table 2. We can assume that Xann = Xr2 might be valid for isotropic ferromagnets only when ND is negligible and in general the relation Xann vs. Xr depends on ND. In order to express the initial magnetization curve with one function it is necessary to know the function Xann = f (Xr). For the investigated SMC samples the function that fits the best the experimental data (for Xr from 0 to ~0.95), with maximum regression coefficients, we propose to use in the following form:
Xann = C1 Xr + (C2 Xr e(C3 Xr ) ).
(5)
with the coefficients C1, C2 and C3 depending on particular material. Eq. (5). We found that for fitting of the data with Eq. (5), in order to get the expression of initial magnetization curve, the required accuracy of fitting is very high at Xr → 0. So the data were fitted for Xr from 0 to 0.95 with the most importance laid on the lower values (the required accuracy is decreasing with the increasing Xr and above 0.95 it is not sensitive at all). The fittings for selected samples are shown in Fig. 2,
5. Conclusion In this work the magnetization process of the specific class of ferromagnetic materials – soft magnetic composite materials (SMCs) was characterized by the analytical expression of the initial magnetization curves. The expression was derived by means of extending the 142
Journal of Magnetism and Magnetic Materials 423 (2017) 140–144
Z. Birčáková et al.
particle size distributions) the form of the function Xann = f (Xr) was found, fitting the best the experimental data. Knowledge of this function enables to express analytically the initial magnetization curves with good accuracy. The tendency of the ratio between the effective number of domain walls participating in magnetization process and the extent of deviation of magnetization vectors from the direction of applied magnetic field can be determined from the comparison of Xann vs. Xr dependence shapes. This corresponds to the tendency of the ratio between reversible and irreversible process proportions within the overall magnetization process at a given induction along the initial curve, found also in previous observations [14,16], which thus can be deduced from the comparison of the Xann vs. Xr shapes. Acknowledgments This work was part of the project “The progressive technology for the preparation of microcomposite materials for electrotechnics“ ITMS:26220220105, and the project NanoCEXmat No. ITMS26220120019, which are supported by the Operational Program “Research and Development” financed through the European Regional Development Fund. This work was also supported by the Slovak Research and Development Agency under contract No. APVV-022210 MAGCOMP and by the Scientific Grant Agency of the Ministry of Education of Slovak Republic and the Slovak Academy of Sciences within the frame of the projects VEGA No. 1/0377/16 and VEGA No. 1/0330/15.
Fig. 3. Initial magnetization curves calculated from analytical expression, of SMC samples with different mean particle sizes (45, 75, 100 a 160 µm), compared with experimental ones.
References [1] D.C. Jiles, D.L. Atherton, Theory of ferromagnetic hysteresis, J. Magn. Magn. Mater. 61 (1986) 48–60. [2] A. Raghunathan, Y. Melikhov, J.E. Snyder, D.C. Jiles, Generalized form of anhysteretic magnetization function for Jiles–Atherton theory of hysteresis, Appl. Phys. Letters 95, 172510-1 - 172510-3, 2009 [3] J. Takács, G. Kovács, L.K. Varga, Internal demagnetizing factor in ferrous metals, Journal of Metallurgy Vol. 2012 Article ID 752871, 1-5, 2012 [4] C.S. Schneider, Cooperative anisotropic theory of ferromagnetic hysteresis, in: B.M. Caruta (Ed.)Trends in Materials Science Research, Nova Science Publishers, Inc., New York, 2006. [5] C.S. Schneider, S.D. Winchell, Hysteresis in conducting ferromagnets, Physica B 372 (2006) 269–272. [6] C.S. Schneider, Role of reversible susceptibility in ferromagnetic hysteresis, J. Appl. Phys. 91 (2002) 7637–7639. [7] C.S. Schneider, Domain cooperation in ferromagnetic hysteresis, J. Appl. Phys. 89 (2001) 1281–1286. [8] C.S. Schneider, Maximum susceptibility of ferromagnetic hysteresis, IEEE Trans. Magn. 48 (2012) 3371–3374. [9] C.S. Schneider, Anisotropic cooperative theory of coaxial ferromagnetoelasticity, Physica B 343 (2004) 65–74. [10] K.H.J. Buschow, Concise encyclopedia of magnetic and superconducting materials, Elsevier, Oxford, UK, 2005. [11] A.J. Moses, Advanced soft magnetic materials for power applications, in: H. Kronmüller, S. Parkin (Eds.), Handbook of magnetism and advanced magnetic materials, Wiley-Interscience, New York, 2007. [12] B.V. Neamtu, O. Geoffroy, I. Chicinas, O. Isnard, AC magnetic properties of the soft magnetic composites based on Supermalloy nanocrystalline powder prepared by mechanical alloying, Mater. Sci. Eng. B 177 (2012) 661–665. [13] P. Kollár, J. Füzer, R. Bureš, M. Fáberová, AC magnetic properties of Fe-based composite materials, IEEE Trans. Magn. 46 (2010) 467. [14] Z. Birčáková, P. Kollár, J. Füzer, M. Lauda, R. Bureš, M. Fáberová, Influence of the resin content on the dynamic energy losses in iron–phenolphormaldehyde resin composites, IEEE Trans. Magn. 50 (2014) 6301507. [15] P. Kollár, Z. Birčáková, V. Vojtek, J. Füzer, R. Bureš, M. Fáberová, Dependence of demagnetizing fields in Fe-based composite materials on magnetic particle size and the resin content, J. Magn. Magn. Mater. 388 (2015) 76–81. [16] Z. Birčáková, P. Kollár, B. Weidenfeller, J. Füzer, M. Fáberová, R. Bureš, Reversible and irreversible DC magnetization processes in the frame of magnetic, thermal and electrical properties of Fe-based composite materials, J. Alloy. Comp. 645 (2015) 283–289. [17] P. Kollár, V. Vojtek, Z. Birčáková, J. Füzer, M. Fáberová, R. Bureš, Steinmetz law in iron - phenolformaldehyde resin soft magnetic composites, J. Magn. Magn. Mater. 353 (2014) 65. [18] Y. Peng, J. Nie, W. Zhang, J. Mab, C. Bao, Y. Cao, Effect of the addition of Al2O3 nanoparticles on the magnetic properties of Fe soft magnetic composites, J. Magn. Magn. Mater. 399 (2016) 88–93.
Fig. 4. Initial magnetization curves calculated from analytical expression, of SMC samples with different insulation content (5, 10, 15 vol%), compared with experimental ones.
selected basic ideas originally proposed for selected ferromagnetic materials, summarized as follows: The expression for the initial magnetization curve consists of two summands with two normalized functions – the magnetization vector deviation function Xr describing the reversible magnetization processes through the extent the magnetization vectors are deviated from the magnetic field direction, and the domain wall annihilation function Xann describing the irreversible processes through the effective numbers of movable domain walls. However, we found out that the dependence of domain wall annihilation function Xann on the magnetization vector deviation function Xr for SMCs showed not the quadratic tendency, which was observed for bulk isotropic homogenous ferromagnets (the microscopy observation of SMCs confirmed their macroscopically isotropic structure). The main feature of SMCs is their heterogeneous structure containing non-ferromagnetic components, which causes much higher inner demagnetizing fields and the magnetization process performed differently as in bulk ferromagnets – within each ferromagnetic particle of SMC more or less independently on the magnetization process in surrounding particles. Correspondingly, the observed non-quadratic dependence Xann vs. Xr was found to significantly depend on the inner demagnetization factor ND, shifting from the quadratic one, the more the value of ND is increasing. Hence Xann = Xr2 can be pronounced to be valid only when ND is negligible. For the investigated SMCs (Fephenolphormaldehyde resin samples with different resin contents and 143
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[22] E. Kneller, Ferromagnetismus, Springer-Verlag, Berlin, 1962. [23] 〈www.hoganas.com〉 [24] F.J.G. Landgraf, J.C. Teixeira, M. Emura, M.F. de Campos, C.S. Muranaka, Separating components of the hysteresis loss of non-oriented electrical steels, Mater. Sci. Forum 302–303 (1999) 440.
[19] B. Zidarič, D. Miljavec, A new ferromagnetic hysteresis model for soft magnetic composite materials, J. Magn. Magn. Mater. 323 (2011) 67. [20] R.M. Bozorth, Ferromagnetism, third edition, IEEE Press, Piscataway, New Jersey, 1993. [21] J.M.D. Coey, Magnetism and magnetic materials, Cambridge University Press, New York, 2009.
144
Contents lists available at ScienceDirect
Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Analytical expression for initial magnetization curve of Fe-based soft magnetic composite material
crossmark
⁎
Zuzana Birčákováa, , Peter Kollára, Ján Füzera, Radovan Burešb, Mária Fáberováb a b
Institute of Physics, Faculty of Science, Pavol Jozef Šafárik University, Park Angelinum 9, 04154 Košice, Slovakia Institute of Materials Research, Slovak Academy of Sciences, Watsonova 47, 04001 Košice, Slovakia
A R T I C L E I N F O
A BS T RAC T
Keywords: Analytical expression, Initial magnetization curve, Soft magnetic composite materials Magnetization processes
The analytical expression for the initial magnetization curve for Fe-phenolphormaldehyde resin composite material was derived based on the already proposed ideas of the magnetization vector deviation function and the domain wall annihilation function, characterizing the reversible magnetization processes through the extent of deviation of magnetization vectors from magnetic field direction and the irreversible processes through the effective numbers of movable domain walls, respectively. As for composite materials the specific dependences of these functions were observed, the ideas were extended meeting the composites special features, which are principally the much higher inner demagnetizing fields produced by magnetic poles on ferromagnetic particle surfaces. The proposed analytical expression enables us to find the relative extent of each type of magnetization processes when magnetizing a specimen along the initial curve.
1. Introduction The effort of finding an analytical expression for magnetization curves still has the very important point within the studies in the field of magnetism. Until now various scientific works provided valuable models, e.g. the Jiles-Atherton model of ferromagnetic hysteresis [1,2] or the hysteresis model proposed by J. Takács et al. [3], enabling to predict the curves as well as to reveal the information on magnetic properties, magnetization processes or phenomena in general. These models were experimentally verified on bulk ferromagnetic materials. Recently C.S. Schneider proposed in his studies [4–9] an approach, ideas of it he formulated as “cooperative anisotropic theory of ferromagnetic hysteresis” and applied it to describe the magnetization curves of various, also bulk Fe, Ni and Co based ferromagnets [4–9]. Soft magnetic composite materials (SMCs, consisting of small ferromagnetic particles insulated from each other) belong to the class of ferromagnetic materials, but due to their specificity, we expect, the analysis of the magnetic properties including the magnetization curves of SMCs require modified treatment based on principles developed originally for bulk ferromagnets. SMCs become an interesting kind of soft magnetic materials also from the view of application possibilities [10–18]. In authors previous studies [13–17] dealing with the soft magnetic composite materials, the different and more simple dependences of certain magnetic properties of SMCs were found compared to the majority of bulk ferromagnetic materials (e.g. the dependence of excess energy losses on ⁎
frequency). It is due to the fact that the magnetization process is performed in small ferromagnetic particles (which are not only electrically insulated, but the magnetization process within each particle is more or less independent on the magnetization process of surrounding particles), resulting in some specific properties of SMCs (e.g. low energy losses at medium to higher frequencies, or magnetic and thermal isotropy are valuable for the applications). Particularly these dissimilarities were taken into account in study [19], when B. Zidarič et al. developed a ferromagnetic hysteresis model for the analytical expression of magnetization curves of SMCs based on the Jiles-Atherton model. Similarly we assume the adaptation of other models for composites can bring some new knowledge on these materials. For this reason the soft magnetic composites have been chosen to be the focus of the presented work, with the aim to find the analytical expression of initial magnetization curve for Fe-based SMC, drawing on the selected essential ideas proposed by C.S. Schneider. By means of extending of these ideas to further develop the interpretation of magnetization processes in the view of composite material special features. SMCs were represented by simple Fe-phenolphormaldehyde resin samples. 2. Analytical expression for initial magnetization curve In order to find an expression that would enable to analytically describe the initial magnetization curves of SMCs, we drawn on some
Corresponding author. E-mail addresses: [email protected], [email protected] (Z. Birčáková).
http://dx.doi.org/10.1016/j.jmmm.2016.09.087 Received 29 April 2016; Received in revised form 7 September 2016; Accepted 16 September 2016 Available online 17 September 2016 0304-8853/ © 2016 Elsevier B.V. All rights reserved.
Journal of Magnetism and Magnetic Materials 423 (2017) 140–144
Z. Birčáková et al.
basic ideas proposed by C.S. Schneider [4–9] for selected bulk ferromagnets, based on the analysis of magnetization processes. Schneider described magnetization processes using terms for the hysteresis phenomena, which are not usually used in books such as [10,11,20– 22]. According to the investigations of magnetization processes in SMCs [13–17] we used the selected ideas as an inspiration to extend them meeting the special features of the composites. The ideas are: the possibility to express the initial magnetization curve with a two summand relation and the characterization of the particular magnetization processes by two normalized functions. The initial magnetization curve of ferromagnetic material can be expressed by one relation, from the demagnetized state to saturation, where M is the magnetization, H is the applied magnetic field, HC denotes the coercivity of material, χi is the initial susceptibility and two normalized functions Xr and Xann are introduced:
⎛ M⎞ M = H ⎜Xr χi + Xann ⎟ ⎝ HC ⎠
(1) Fig. 1. Light optical microscopy photographs of composite samples F45-10%, F75-10%, F100-10% and F160-10%.
First normalized function Xr is the magnetization vector deviation function and is defined as a ratio of the reversible to initial susceptibility [6]
Xr =
χrev . χi
Xann = Xr2
for several isotropic bulk ferromagnetic materials based on Fe, Ni, Co [4–6,8] and generalized this observation that the dependence is quadratic for an isotropic ferromagnet. He also predicted the magnetization curves with quite good accuracy.
(2)
This function characterizes the relative amount of reversible magnetization processes (domain wall displacement and/or magnetization vector rotation) through the extent to which all magnetization vectors in material are deviated from the direction of the applied magnetic field. Its maximum is always at the demagnetized state (Xr is equal to 1), further decreasing with the increasing magnetic induction due to the magnetization processes of the magnetization vector rotation towards the field direction. Finally Xr converges to 0 at saturation which means the minimum deviation of magnetization vectors from the magnetic field direction. The reversible susceptibility tells about the proportions of reversible processes in the overall magnetization process at a given magnetic induction. Thus the higher is the value of Xr, the higher is the relative amount of reversible processes, whereas with the decreasing Xr the more the magnetization vector rotation dominates over the reversible domain wall displacement. Second normalized function Xann is the domain wall annihilation function and characterizes the relative amount of irreversible domain wall displacements through the increasing annihilation of domain walls with the increasing magnetic induction (starting from the induction Bμ max , where the total permeability reaches its maximum μmax and domain wall annihilation and nucleation appears [22,24]), or in other words, the decreasing effective number of domain walls participating in magnetization process. As the domains with the magnetization vector oriented favourably to applied magnetic field grow (by the domain wall displacements), some of the domain walls can have the tendency to annihilate. The decreasing effective number of domain walls means the amount of irreversible domain wall displacements is decreasing. Function Xann reaches also values from 1 at the demagnetized state (extrapolated value, physical meaning of the maximum effective number of movable domain walls is at Bμ max ) down to 0 when approaching saturation (minimum effective number of movable domain walls). The proportion of irreversible magnetization processes determines the irreversible susceptibility. The relation (1) becomes in limiting case of low magnetic fields (Rayleigh region), where for H → 0: χrev → χi, Xr → 1, M → H χi, and Xann → 1, the following form:
M = Hχi +
χi 2 H, HC
(4)
3. Experimental The simple Fe-phenolphormaldehyde resin samples were chosen to represent soft magnetic composite material. The samples were prepared by conventional powder metallurgy process, with different resin contents and particle size distributions [15,16]. Iron powder ASC 100.29 (Höganäs AB Sweden [23]) was sieved to obtain particle size distributions showing peaks at 45 μm, 75 μm, 100 μm and 160 μm (granulometric classes labelled F45, F75, F100 and F160) [15,16]. The homogenization of iron powder with 5, 10 and 15 vol% (1, 2 and 3 wt %) of phenolphormaldehyde resin (Bakelite ATM) and acetone was performed. The mixtures were compacted at uniaxial pressure of 800 MPa in the form of a ring (outer diameter 24 mm, inner diameter 18 mm, height from 1.4 to 2.4 mm) and cylinder (diameter 10 mm, height 2 mm). After the compaction a heat treatment at a temperature of 165 °C for 60 min in electric furnace in air atmosphere was performed. Light optical microscopy investigations (Olympus GX71), Fig. 1, confirmed the structure of composite samples to be isotropic, as expected and also observed in studies on similar samples [13,14,17]. Parameters of the samples are in Table 1. Porosity was calculated from dimensions and mass of each sample (density of iron is 7.851 g/cm3 and density of resin is 1.39 g/cm3). The initial magnetization curves were measured by the DC fluxmeter-based hysteresisgraph and the reversible relative permeability was measured using the setup described in [16], both on ring-shaped samples. Magnetic induction was measured referred to the filler content of ferromagnetic material in SMC (subtracting non-ferromagnetic components: resin and pores). The coercivity was measured by Foerster Koerzimat 1.097HCJ on cylinder-shaped samples. 4. Results and discussion 4.1. Analytical expression of initial magnetization curves of SMCs
(3)
The magnetization vector deviation function Xr (B) and the domain wall annihilation function Xann (B) were calculated from the experimental data (Eqs. (1), (2)). The plots Xann vs. Xr for selected samples are in Fig. 2. The dependence Xann vs. Xr was found not to be
determining the Rayleigh coefficient to be χi / HC . C.S. Schneider experimentally observed the dependence Xann vs. Xr to be quadratic 141
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Table 1 Parameters of the samples. Sample
Iron powder to insulation ratio (vol.) Density (g/cm3) Porosity (%) Filler content of ferromagnetic material (%)
Table 2 Initial relative permeability µi, coercivity HC, inner demagnetization factor ND and coefficient C1.
F45 − 5%
F45 − 10%
F75 − 10%
F100 − 10%
F160 − 10%
F45 − 15%
95: 5
90: 10
90: 10
90: 10
90: 10
85: 15
6.60 12.2 83.4
6.0 17.0 74.6
5.94 17.5 74.2
6.05 16.4 75.2
6.10 15.0 76.5
5.45 21.0 67.0
Sample
F45 − 5%
F45 − 10%
F75 − 10%
F100 − 10%
F160 − 10%
F45 − 15%
µi (-) HC (A/m) ND (-) C1 (-)
62.5 475 0.006 0.071
42.5 478 0.012 0.033
43.5 440 0.011 0.035
47 415 0.010 0.035
49 377 0.009 0.041
39 476 0.014 0.030
(using µrev = χrev + 1, µi = χi + 1), the values of initial relative permeability µi and coercivity HC are in Table 2 (µi is increasing with increasing filler content of ferromagnetic material in SMC and with increasing ferromagnetic particle size, HC is decreasing with increasing particle size and does not depend on the filler content [15,16]). The coefficient C1 tells about the shift from a quadratic dependence (the lower C1, the more the dependence is shifted, corresponding to the dependence of ND on filler content and particle size) - the obtained values are in Table 2. Coefficients C2 and C3 reached values ~10−7 – 10−5 and ~7.5 – 14, respectively and did not show dependence neither on filler content, nor on particle size. The Eq. (5) obtained from fitting was further implemented into Eq. (1), which was finally used to express analytically the initial magnetization curve, Figs. 3 and 4 show a good agreement with the experimental data. 4.2. Graphical meaning of the dependence Xann vs. Xr The dependence of domain wall annihilation function Xann plotted vs. magnetization vector deviation function Xr suggests the proportions of reversible and irreversible processes in the overall magnetization process. The dependence starts from point [0;0] (saturation) and ends at [1,1] (demagnetized state). From its shape we can deduce which type of magnetization processes dominates at given magnetic state (at given induction), where especially important is the area of magnetic inductions above Bμ max , where the total permeability reaches its maximum μmax and the domain wall annihilation and nucleation appears [22,24] (Bμ max determined for each sample is decreasing from 0.38 to 0.22 T with the increasing resin content [16], which stand for Xr ≈ 0.9, covering thus almost whole graph area). In this area we can directly claim that the magnetization process is the more realized by magnetization vector rotation (mainly reversible) the more is the curve approaching to Xr axis (intersecting Xann = 0), and vice versa, the closer is the curve to Xann axis (intersecting Xr = 0), the higher is the proportion of irreversible domain wall displacements (Barkhausen jumps). In Fig. 2 we can see it is in agreement with the observations in study [16]: In samples with higher resin content and with smaller particle size the effective number of movable domain walls participating in the magnetization process is lower due to higher inner demagnetizing fields produced at magnetic particles surfaces reducing magnetic interaction between particles, resulting in lower proportions of irreversible domain wall displacements [14–16]. To sum up, the comparison of the shapes of Xann vs. Xr dependences (samples with different particle sizes or different resin content) enables to express the tendency of the ratio between the effective number of movable domain walls participating in magnetization process and the extent of deviation of magnetization vectors from the applied magnetic field direction.
Fig. 2. The dependence of domain wall annihilation function Xann on magnetization vector deviation function Xr of selected samples F45-5%, F160-10% and F45-15%, fitted with Eq. (5), quadratic function Xann = Xr2 (dashed) is plotted for comparison.
quadratic for SMC samples. The main feature which differs the SMCs from the bulk materials studied by Schneider is their heterogeneous structure containing non-ferromagnetic components (insulation and pores) which give rise to much higher inner demagnetizing fields produced by magnetic poles on ferromagnetic particle surfaces [15]. Based on this observation we can assume that the quadratic rule (Eq. (4)) covers only isotropic bulk ferromagnets, where he observed the quadratic dependence Xann vs. Xr - function Xann = Xr2 is also depicted in Fig. 2 for comparison. We have observed the dependences are shifting from the quadratic one, the more the value of inner demagnetization factor ND is increasing. ND is increasing with decreasing filler content of ferromagnetic material in SMC and with decreasing ferromagnetic particle size [15] - values are in Table 2. We can assume that Xann = Xr2 might be valid for isotropic ferromagnets only when ND is negligible and in general the relation Xann vs. Xr depends on ND. In order to express the initial magnetization curve with one function it is necessary to know the function Xann = f (Xr). For the investigated SMC samples the function that fits the best the experimental data (for Xr from 0 to ~0.95), with maximum regression coefficients, we propose to use in the following form:
Xann = C1 Xr + (C2 Xr e(C3 Xr ) ).
(5)
with the coefficients C1, C2 and C3 depending on particular material. Eq. (5). We found that for fitting of the data with Eq. (5), in order to get the expression of initial magnetization curve, the required accuracy of fitting is very high at Xr → 0. So the data were fitted for Xr from 0 to 0.95 with the most importance laid on the lower values (the required accuracy is decreasing with the increasing Xr and above 0.95 it is not sensitive at all). The fittings for selected samples are shown in Fig. 2,
5. Conclusion In this work the magnetization process of the specific class of ferromagnetic materials – soft magnetic composite materials (SMCs) was characterized by the analytical expression of the initial magnetization curves. The expression was derived by means of extending the 142
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particle size distributions) the form of the function Xann = f (Xr) was found, fitting the best the experimental data. Knowledge of this function enables to express analytically the initial magnetization curves with good accuracy. The tendency of the ratio between the effective number of domain walls participating in magnetization process and the extent of deviation of magnetization vectors from the direction of applied magnetic field can be determined from the comparison of Xann vs. Xr dependence shapes. This corresponds to the tendency of the ratio between reversible and irreversible process proportions within the overall magnetization process at a given induction along the initial curve, found also in previous observations [14,16], which thus can be deduced from the comparison of the Xann vs. Xr shapes. Acknowledgments This work was part of the project “The progressive technology for the preparation of microcomposite materials for electrotechnics“ ITMS:26220220105, and the project NanoCEXmat No. ITMS26220120019, which are supported by the Operational Program “Research and Development” financed through the European Regional Development Fund. This work was also supported by the Slovak Research and Development Agency under contract No. APVV-022210 MAGCOMP and by the Scientific Grant Agency of the Ministry of Education of Slovak Republic and the Slovak Academy of Sciences within the frame of the projects VEGA No. 1/0377/16 and VEGA No. 1/0330/15.
Fig. 3. Initial magnetization curves calculated from analytical expression, of SMC samples with different mean particle sizes (45, 75, 100 a 160 µm), compared with experimental ones.
References [1] D.C. Jiles, D.L. Atherton, Theory of ferromagnetic hysteresis, J. Magn. Magn. Mater. 61 (1986) 48–60. [2] A. Raghunathan, Y. Melikhov, J.E. Snyder, D.C. Jiles, Generalized form of anhysteretic magnetization function for Jiles–Atherton theory of hysteresis, Appl. Phys. Letters 95, 172510-1 - 172510-3, 2009 [3] J. Takács, G. Kovács, L.K. Varga, Internal demagnetizing factor in ferrous metals, Journal of Metallurgy Vol. 2012 Article ID 752871, 1-5, 2012 [4] C.S. Schneider, Cooperative anisotropic theory of ferromagnetic hysteresis, in: B.M. Caruta (Ed.)Trends in Materials Science Research, Nova Science Publishers, Inc., New York, 2006. [5] C.S. Schneider, S.D. Winchell, Hysteresis in conducting ferromagnets, Physica B 372 (2006) 269–272. [6] C.S. Schneider, Role of reversible susceptibility in ferromagnetic hysteresis, J. Appl. Phys. 91 (2002) 7637–7639. [7] C.S. Schneider, Domain cooperation in ferromagnetic hysteresis, J. Appl. Phys. 89 (2001) 1281–1286. [8] C.S. Schneider, Maximum susceptibility of ferromagnetic hysteresis, IEEE Trans. Magn. 48 (2012) 3371–3374. [9] C.S. Schneider, Anisotropic cooperative theory of coaxial ferromagnetoelasticity, Physica B 343 (2004) 65–74. [10] K.H.J. Buschow, Concise encyclopedia of magnetic and superconducting materials, Elsevier, Oxford, UK, 2005. [11] A.J. Moses, Advanced soft magnetic materials for power applications, in: H. Kronmüller, S. Parkin (Eds.), Handbook of magnetism and advanced magnetic materials, Wiley-Interscience, New York, 2007. [12] B.V. Neamtu, O. Geoffroy, I. Chicinas, O. Isnard, AC magnetic properties of the soft magnetic composites based on Supermalloy nanocrystalline powder prepared by mechanical alloying, Mater. Sci. Eng. B 177 (2012) 661–665. [13] P. Kollár, J. Füzer, R. Bureš, M. Fáberová, AC magnetic properties of Fe-based composite materials, IEEE Trans. Magn. 46 (2010) 467. [14] Z. Birčáková, P. Kollár, J. Füzer, M. Lauda, R. Bureš, M. Fáberová, Influence of the resin content on the dynamic energy losses in iron–phenolphormaldehyde resin composites, IEEE Trans. Magn. 50 (2014) 6301507. [15] P. Kollár, Z. Birčáková, V. Vojtek, J. Füzer, R. Bureš, M. Fáberová, Dependence of demagnetizing fields in Fe-based composite materials on magnetic particle size and the resin content, J. Magn. Magn. Mater. 388 (2015) 76–81. [16] Z. Birčáková, P. Kollár, B. Weidenfeller, J. Füzer, M. Fáberová, R. Bureš, Reversible and irreversible DC magnetization processes in the frame of magnetic, thermal and electrical properties of Fe-based composite materials, J. Alloy. Comp. 645 (2015) 283–289. [17] P. Kollár, V. Vojtek, Z. Birčáková, J. Füzer, M. Fáberová, R. Bureš, Steinmetz law in iron - phenolformaldehyde resin soft magnetic composites, J. Magn. Magn. Mater. 353 (2014) 65. [18] Y. Peng, J. Nie, W. Zhang, J. Mab, C. Bao, Y. Cao, Effect of the addition of Al2O3 nanoparticles on the magnetic properties of Fe soft magnetic composites, J. Magn. Magn. Mater. 399 (2016) 88–93.
Fig. 4. Initial magnetization curves calculated from analytical expression, of SMC samples with different insulation content (5, 10, 15 vol%), compared with experimental ones.
selected basic ideas originally proposed for selected ferromagnetic materials, summarized as follows: The expression for the initial magnetization curve consists of two summands with two normalized functions – the magnetization vector deviation function Xr describing the reversible magnetization processes through the extent the magnetization vectors are deviated from the magnetic field direction, and the domain wall annihilation function Xann describing the irreversible processes through the effective numbers of movable domain walls. However, we found out that the dependence of domain wall annihilation function Xann on the magnetization vector deviation function Xr for SMCs showed not the quadratic tendency, which was observed for bulk isotropic homogenous ferromagnets (the microscopy observation of SMCs confirmed their macroscopically isotropic structure). The main feature of SMCs is their heterogeneous structure containing non-ferromagnetic components, which causes much higher inner demagnetizing fields and the magnetization process performed differently as in bulk ferromagnets – within each ferromagnetic particle of SMC more or less independently on the magnetization process in surrounding particles. Correspondingly, the observed non-quadratic dependence Xann vs. Xr was found to significantly depend on the inner demagnetization factor ND, shifting from the quadratic one, the more the value of ND is increasing. Hence Xann = Xr2 can be pronounced to be valid only when ND is negligible. For the investigated SMCs (Fephenolphormaldehyde resin samples with different resin contents and 143
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[22] E. Kneller, Ferromagnetismus, Springer-Verlag, Berlin, 1962. [23] 〈www.hoganas.com〉 [24] F.J.G. Landgraf, J.C. Teixeira, M. Emura, M.F. de Campos, C.S. Muranaka, Separating components of the hysteresis loss of non-oriented electrical steels, Mater. Sci. Forum 302–303 (1999) 440.
[19] B. Zidarič, D. Miljavec, A new ferromagnetic hysteresis model for soft magnetic composite materials, J. Magn. Magn. Mater. 323 (2011) 67. [20] R.M. Bozorth, Ferromagnetism, third edition, IEEE Press, Piscataway, New Jersey, 1993. [21] J.M.D. Coey, Magnetism and magnetic materials, Cambridge University Press, New York, 2009.
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