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Effect of particle size and size distribution on the permeability of soft magnetic liquid silicone rubber composites
Composites Science and Technology 177 (2019) 26–33
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Composites Science and Technology journal homepage: www.elsevier.com/locate/compscitech
Effect of particle size and size distribution on the permeability of soft magnetic liquid silicone rubber composites
T
Dirk W. Schuberta,∗, Siegfried Wernera, Ingo Hahnb, Veronika Solovievab a b
Institute of Polymer Materials, University of Erlangen-Nuremberg, Martensstrasse 7, 91058 Erlangen, Germany Institute of Electrical Drives and Machines, University of Erlangen-Nuremberg, Konrad-Zuse-Str. 3/5, 91052 Erlangen, Germany
A R T I C LE I N FO
A B S T R A C T
Keywords: A: polymer matrix composite B: magnetic properties C: modelling E: powder processing
Magnetic composites made of LSR (Liquid Silicone Rubber) and a magnetic filler have been investigated with respect to magnetic field guidance for transcutaneous energy transport. Effects of particle size and size distribution on the effective initial magnetic permeability are analyzed. A simple model is presented to describe the permeability as a function of the degree of filling, particle size and size distribution. From this, a calculus for the relevant size average of the filler was developed at 20 vol% of magnetic filler. The calculus showed self-consistency and can be used to tailor optimized powders for better field guidance.
1. Introduction Soft magnetic materials are widely used in electronics as transformers, e.g. in high frequency applications for impedance matching and the balancing of antenna systems [1]. Another important application of soft magnetic materials is the wireless transport of energy [2,3]. Therefore, a pair of coils is used to transfer energy via induction and resonance coupling from a primary coil connected to the charging station to a secondary coil connected to the device [4]. The efficiency of this setup can be enhanced by deliberately guiding the magnetic flux between the coils using a soft magnetic encapsulation typically made of ferrite or other rigid metal based materials [4]. This technology is well known and already implemented into consumer electronics such as cell phones or electric tooth brushes. But the transfer to medical applications is still to be made, although wirelessly charging built-in supporting systems like pace makers or a ventricular assist device (VAD) offers extreme benefits for the patient as recurring surgeries or a permanent percutaneous drive line can be avoided [5–7]. One limiting factor for conventional rigid setups is that the irregular surface and the 3D shape of the human body increase the distance between the power transmitting coils and, thereby, reduce their efficiency. This problem can be solved by using a flexible coil system which can adapt to the body surface and minimizes the distance between the coils. Another limiting factor is an efficient guidance of the field by a deliberately designed material permeability as encapsulation for the coils. These two requirements can be met by using a composite made from flexible liquid silicone rubber (LSR) and rigid soft magnetic filler.
∗
In literature soft magnetic materials derived from a soft magnetic powder, surrounded by an insulating layer are referred to as soft magnetic composites (SMC) [2]. In general an increase in permeability with larger particles has been observed for SMCs [8,9]. However, larger particles also lead to a reduced flexibility of the part. So the impact of particle size on the permeability has to be understood thoroughly. But conventional effective medium theories as e.g. the Böttcher [10], De Loor [11] or Bruggeman [12] equation, nicely summarized by Anhalt and Weidenfeller, only focus on the volume fraction of the magnetic filler and do not include the effect of particle size [13]. However, it is known that for the same material an increase of the particle size yields a steady increase in the permeability [3,14–16]. Anhalt found a linear relation between particle size and permeability looking at particles with an arithmetic mean diameter of 30 μm–170 μm [17]. But the investigated diameter range was small and the arithmetic mean is sensitive to a skewed particle size distribution. Recently, Li et al. modified the Bruggeman equation to account for the particle size distribution. They obtained the effective composite permeability by integrating the contributions from particles of different size and, consecutively, of different permeability [18]. They assumed a log-normal distributed particle size and achieved a good prediction of the permeability for particles with a median diameter of 7 μm–84 μm. However, the size range is again small and knowledge of the size distribution function is required. Yet, no work is known to the authors that takes into account that a tailor made mean value for the particle size distribution, that is not the arithmetic mean or the median, can give a better description of the
Corresponding author. E-mail address: [email protected] (D.W. Schubert).
https://doi.org/10.1016/j.compscitech.2019.04.005 Received 6 October 2018; Received in revised form 29 March 2019; Accepted 6 April 2019 Available online 08 April 2019 0266-3538/ © 2019 Elsevier Ltd. All rights reserved.
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nearly constant with a thickness of 20 μm due to the processing routine [21]. Fig. 2 shows that the minimum and maximum Feret diameter were proportional to each other with the aspect ratio of the particles as proportionality constant. All size fractions feature an aspect ratio of about 1.65 and therefore have a very similar shape. Due to the proportionality, it is also sufficient to consider only one Feret diameter typ. In this work the minimum Feret diameter was arbitrarily chosen.
particle-size-permeability relation. Therefore, it is the purpose of this paper to identify a calculus for a more accurate mean particle size to predict the permeability, which works from the micrometer to the millimeter range. In particular, we apply a strategy which was originally developed to reveal the relevant molar mass average of a polymer mixture for the film thickness of a spin coated film [19]. But instead of mixing polymers with different bimodal molar mass distribution, deliberately prepared bimodal particle size distributions and their influence on magnetic permeability of the composite is analyzed (In fact the effective initial magnetic permeability is determined, but for simplicity it is just called magnetic permeability throughout the paper). Therefore, toroidal cores of LSR filled with different size fractions of magnetic particles and combinations of those were prepared. An inductance is generated by winding the cores with a certain number of turns of an electric wire, which makes the permeability of the core accessible. Furthermore, the hysteresis behavior and the particle dispersion are investigated using a magnetometer and SEM, respectively. As recently suggested by Schubert et al. [20] a simple model is used to describe the permeability as a function of the degree of filling with only one adjustable parameter k. To isolate the effects of particle size and size distribution we focus only on a volume fraction of 20% for the composite, which permitted an easy processing and had a sufficiently large effect on the permeability.
2.3. Manufacturing toroidal cores Toroidal cores consisting of LSR and Vitroperm 250 powder were prepared at a constant powder volume fraction ϕp of 20%. The toroids had an outer diameter do of 50 mm, an inner diameter di of 20 mm and a height h of 15 mm. As matrix, the high temperature vulcanizing twocomponent silicone rubber Elastosil LR 3003/30 (Wacker Chemie AG, Munich, Germany) was used, which is typically applied in high volume technical products [23]. To prepare a toroidal core, the two silicone components were weighed out in a ratio of 1:1 and mixed in a speed mixer for 5 min at 1000 rpm. Subsequently the desired amount of filler was added to the silicone and dispersed by mixing for 3 min at 2000 rpm. After mixing, incorporated air was removed in a desiccator. This procedure was repeated three times to ensure proper dispersion of the particles. The prepared compounds were poured into toroidal polytetrafluoroethylene molds and newly incorporated air was again removed in a desiccator. Consecutively, the compound was cross-linked for 4 h at 120 °C. First, toroids filled with the six pure size fractions were produced to obtain the relationship between particle size and permeability. Second, combinations of two size fractions in different ratios were used to determine a mean that considers the size distribution in the best way. Therefore, the fractions 500-250 μm and 100-63 μm as well as the fractions 500-250 μm and 150-100 μm were mixed in different ratios. The ratio of the fractions was characterized by the content n of the fraction 500-250 μm in the mixture as shown in Equation (1). A content n of 0.8, 0.7, 0.6, 0.5, 0.4, 0.3 and 0.2 was used for both mixtures.
2. Experimental section 2.1. Powder preparation Powders with different degrees of fineness were generated from the soft magnetic material Vitroperm 250 (VACUUMSCHMELZE GmbH & Co. KG, Hanau, Germany). Due to its nanocrystalline structure it showed a superior increase in permeability than other commercial soft magnetic materials such as Neosid G and Neosid F [20]. Table 1 shows the parameters of the filler material, where μr is the magnetic permeability, Hc the coercive magnetic field strength and ρ the density. The permeability of the bulk material is constant up to a frequency of 300 kHz [21]. The processing of nanocrystalline soft magnetic alloys with a rapid solidification process yields thin ribbons, which are wound to toroidal cores. Therefore, the variable rotor speed mill Pulverisette 14 (Fritsch GmbH, Idar Oberstein, Germany) was used to mill the cores down to primary powders of certain fineness. A coarse primary powder was produced using a rotational speed of 6000 rpm together with a sieve ring with mesh openings of 2.0 mm and a fine primary powder using a speed of 18000 rpm and a 0.5 mm sieve ring. From the coarse and fine primary powder, the size fractions > 500 μm, 500-250 μm, 250- 150 μm and 150-100 μm, 100-63 μm and < 63 μm were isolated, respectively, using a test sieve shaker according to DIN 66165-1 (2016-08) [22]. Two primary powders of different fineness were generated to achieve an effective production of the large and small size fractions.
n=
The toroidal cores were wound with 40 turns of electric wire to form an inductance as shown in Fig. 3. The corresponding inductivity was measured using an AE20204 LC meter (Ascel Electronics, Hohwacht, Germany). The relative magnetic permeability μr of the core was obtained by Equation (2) describing the inductance L of a toroidal coil:
L=
Table 1 Characteristic parameters of the magnetic filler [21]. Hc [A*m−1]
ϱ [g*cm−3]
Vitroperm 250
4000
<3
7.2 ± 0.1
μr Aμ0 2 N le
(2)
where N is the number of turns, A the cross section of the toroid, μ 0 the magnetic field constant and le the mean magnetic length calculated from the outer diameter do and the inner diameter di of the toroid with π (do + di )/2 . A detailed description is given by Schubert et al. [20], which also demonstrates that the imaginary part of the permeability is negligibly small in the investigated frequency region up to at least 300 kHz. After determining the permeability the hysteresis behavior of the composites was analyzed using the ring-core method following the industry standard IEC 60404-6:2003 [24]. This requires a toroid with a ratio of da /di smaller than 1.4. The initial toroid had a diameter ratio of 2.5. So a smaller toroid with an outer diameter of 40 mm was punched out from the initial toroids yielding a remaining toroid with da of 50 mm and a ratio of only 1.25. These remaining toroids with reduced width were covered with electric insulation and equipped with two windings in order to build a ring core transformer. A sinusoidal current supply in the primary coil leads to a sinusoidal magnetic field strength in the toroidal sample which can be then determined with Equation (3).
The actual particle size was determined using a video microscope (type: VHX-1000D, Keyence Corp., Osaka, Japan) and image processing software Fiji. The particle size was described by the minimum Feret diameter d and the maximum Feret diameter D of 73–100 particles per fraction as shown in Fig. 1. The size of the particles varied from the micrometer to the millimeter range, but their platelet shape remained
μ r [−]
(1)
2.4. Characterizing magnetic properties and particle dispersion
2.2. Particle characterization
Filler
m500 − 250μm m500 − 250μm + m100 − 63μm
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Fig. 1. Example of the determination of the maximum Feret diameter D (left) and the minimum Feret diameter d (right).
B (t ) =
1 Ns A
T
∫ Us (t ) dt 0
(4)
The measurements were conducted at a frequency of 50 Hz and 100 kHz, respectively. The commutation curves of the samples are constructed by connecting the intersection points of the peak values of the magnetic field density and the magnetic field strength of the hysteresis. curves [25]. The punched out toroids (outer diameter 40 mm) were used for analyzing the particle dispersion. A kryo fracture was done to access the particle distribution without plastic deformation of the matrix. They were frozen in liquid nitrogen and fractured with hammer and clippers. The samples were sputtered with gold and SEM images were taken of the toroid cross section. 3. Results and discussion In a first step, the magnetic behavior and particle dispersion inside the toroids was characterized to ensure the applicability of the used models. Second, the arithmetic mean of the minimum Feret diameter was used to develop a relation between the permeability and the particle size of the pure size fraction. Consecutively, an equation for different means was deduced and the governing mean was revealed by analyzing mixtures of two fractions in different ratios. Due to the inevitable errors of this procedure, the developed mean was tested for self-consistency and verified with another mixture.
Fig. 2. Mean and standard deviation for maximum vs. minimum Feret diameter for the pure size fractions together with the particles used in the previous work of Schubert et al. [20] The solid line demonstrates the proportional relation between both Feret diameters and similar aspect ratio for all size fractions.
3.1. Characterization of magnetic behavior The SMCs show a soft magnetic response to an applied magnetic field. The achieved flux density is rather low compared to compact ferrites made by conventional processing technologies. Representative hysteresis curves are shown in Fig. 4 for various particle size fractions. From Fig. 4 a linear relation (in particular a direct proportionality) between the magnetic flux density B and the applied magnetic field strength H is evident for applied magnetic fields up to 2 kA/m. Additionally, the hysteresis is almost perfectly negligible. The commutation curves of the samples are shown in Fig. 5. For small values of the magnetic field strength, a linear correlation between H and B was revealed. Due to experimental limitations, higher values of the magnetic field strength H could not be adjusted at high frequency (100 kHz) for all samples as a consequence of the increasing impedance of the primary coil. Fig. 5 also shows that the commutation curves are almost frequency independent because the graphs of the corresponding curves, for 50 Hz and 100 kHz merge into each other and therefore have the
Fig. 3. Toroidal core (diameter 50 mm) with 20 vol-% of the fraction 250150 μm without (left) and with 40 turns of an electric wire to measure the inductance with an LC meter (right).
H (t ) =
Np I (t ) le
(3)
In this case Np is the number of primary turns and I is the current in the primary coil. The magnetic flux density B can be calculated according to Equation (4) with the induced voltage Us on the secondary side of the ring transformer and the number of secondary turns Ns .
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Fig. 4. Hysteresis curves of toroids with the pure size fractions 500-250 μm, 150-100 μm and 100-63 μm measured at a frequency of 50 Hz with three different amplitudes of the H-field up to 9 kA/m show very low losses and an almost perfect linear relation until a magnetic field of 2 kA/m.
magnetic field strength, the correction of Pak Ming Hui et al. has to be applied [27]. Nevertheless, the strategy suggested in this paper is universal and still applicable even if the initial equation for describing the permeability is modified.
same slope in the linear range. However, the material will reach a regime of saturation at very high magnetic fields. Thus, the permeability can be taken as constant for magnetic field strength values under 2 kA/m. The permeability values determined at a magnetic field of 0.12 mA/m in section 3.2 and 3.3 were thus located in the linear regime. This is an important requirement to apply the Clausius-Mossotti relation later on. Additionally, it must be pointed out, that the intended application of this material is typically at a magnetic field strength of approximately 1 kA/m as a VAD requires a maximum power of up to 25.5 W [26]. If the application requires to use the SMC in a regime where the permeability is depending on the
3.2. Characterization of particle dispersion Fig. 6 shows SEM images of the flanks of a cryo fracture through the toroid cross section for the pure size fractions 500-250 μm and 10063 μm as well as for mixtures of both. The pure size fractions show a homogeneous distribution of the particles. For the mixtures, small 29
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Fig. 5. Commutation curves of toroid samples at 50 Hz (left) and 100 kHz (right).
[20]. This parameter contains the effects of particle size and particle size distribution on the polarizability [28,29]. As parameter k is present in the nominator and denominator of Equation (5.1) the influence of the particle size is expected to be dampened. Therefore, the expression is rearranged to Equation (5.2) and the parameter k is analyzed in the following at a constant degree of filler:
particles are found in the matrix, in close proximity to large particles and between them indicating that they are homogenously distributed and no segregation has occurred. 3.3. Relationship between particle size and permeability Fig. 7 shows the achieved permeability for the different size fractions, which are described by the arithmetic mean of the minimum Feret diameter 〈d〉a . The permeability shows a steady increase with particle size from the micrometer to the millimeter range. The permeability increases by more than a tenfold as the particle size rises by a twentyfold at a volume fraction of 20%. In previous work, a simple equation was derived from the ClausiusMosotti relation to describe the dependency of the magnetic permeability on the volume fraction of a magnetic filler using only one adjustable parameter k [20]. It was shown empirically that although the Clausius-Mosotti relation is derived for spherical particles the equation also describes the behavior of platelet particles in a reasonable range
μr =
1 + 2kϕP 1 − kϕP
(5.1)
μr − 1 μr + 2
(5.2)
kϕP =
With the volume fraction fixed at 20% the k parameter was determined from the permeability. It should be pointed out that due to the percolation-like behavior of Equation (5.1) it should only be applied where the product of k and λ and Φ is below 0.9. The results also showed a steady increase with the arithmetic mean of the minimum Feret diameter as shown in Fig. 8. To describe these results we suggest a Fig. 6. SEM images of the cross section of the toroids for the pure size fraction 500-250 μm (a) and 10063 μm (b) as well as for mixtures of both with a content n of the fraction 500-250 μm of 70% (c) and 30% (d). Pictures show a good dispersion of the fillers even for mixtures as the small particles are located between the flat and large particles and in the matrix (arrows).
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function, which means an infinitely small width and infinite height of the corresponding size distribution. Therefore, our particle fractions should be called quasi-monomodal because there widths are finite. 3.4. Revealing the governing mean Equation (6) considers the boundary condition of saturation inherently as described in section 3.3. Nevertheless, a power law can be applied to describe the data within the experimental error as well. Alternatively, one can roughly estimate a power law after Taylor expansion of Equation (6) for rather small particles d< < d 0 :
d k = k0 ⎛ ⎞ d ⎝ 0⎠ ⎜
λ
⎟
(7)
The reader may note that the approximation is insufficient as the distribution under consideration and the value for d 0 (Fig. 8) does not satisfy d< < d 0 entirely. However, the Tailor expansion yielding Equation (7) from Equation (6) was only given to indicate the strong similarity of both equations, at least in a certain parameter range. Furthermore, the average value of the parameter k shall be described by the average particle size 〈d〉 as shown in Equation (8a). With a certain size distribution ni and an average of i particles the average value of the parameter k has to be calculated according to Equation (8b).
Fig. 7. Steady increase of permeability with increasing minimum Feret diameter of the six pure size fractions at 20 vol-%.
〈d〉 ⎞ 〈k〉 = k 0 ⎛ ⎝ d0 ⎠ ⎜
⟨k ⟩ =
λ
⎟
∑i ni k 0
(8a) d λ d0
( )
∑i ni
(8b)
Relating Equation (8a) and (8b) and rearranging for a relevant average size < d> gives: 1
λ ⎛ ∑ ni di ⎞ ⟨d⟩λ = ⎜ i ∑i ni ⎟ ⎠ ⎝
λ
(9)
Where the index λ on 〈d〉 denotes a certain average. For example λ = 1 yields the arithmetic mean formerly described as 〈d〉a . Mixing two size fractions can be depicted by combining the two quasi-monomodal functions as shown in Equation (10). Where n represents the ratio of the two quasi-monomodal functions and d1 and d2 their characteristic particle sizes 1
⟨d⟩λ = [(1 − n) d1λ + nd 2λ ] Fig. 8. Parameter k as a function of the arithmetic mean of the minimum Feret diameter 〈d〉a . The solid line shows the fit according to Equation (6).
It should be noted that also on the base of Equation (6) an average can be calculated with an adjustable parameter λ . However, the resulting equation is a bit more complicated and less instructive for explaining the strategy, but yields the same result with respect to λ within the experimental errors.
stretched exponential function as given in Equation (6) having an asymptote k 0 for infinite particle size and d 0 and λ being adjustable parameters. The asymptote is considered to be a material limit as the LSR has a permeability of approximately unity and addition of 20 vol-% Vitroperm cannot surpass 20% of the permeability of the neat Vitroperm material:
1
⎜
λ di λ ⎞⎞ ⎞ d0
( ) ⎠ ⎟⎟
⎛ ⎛ ∑ ni exp⎛− i ⎜ ⎝ ⟨d⟩λ = d 0 −ln⎜ ⎜ ∑i ni ⎜ ⎜ ⎜ ⎝ ⎝
λ
⎛ ⎡ 〈d〉 ⎞ ⎤ ⎞ k = k 0 ⎜1 − exp ⎢−⎛ ⎥⎟ d ⎣ ⎝ 0 ⎠ ⎦⎠ ⎝
(10)
λ
⎟⎟ ⎟⎟ ⎠⎠
(11)
⎟
To summarize the equations and statements above: Equation (6) is only utilized to correlate the parameter k to the Feret diameter, as shown in Fig. 8. The power law in Equation (8a) and the corresponding equations for the averages in Equation (9) and (10) are of the general character of a trial function to demonstrate the approach to reveal the relevant averages, even on the base of non-mono-modal calibration system as in this work. In other words the λ in Equation (9) and (10) must not necessarily be identical to the λ in Equation (6). To reveal the relevant averaging procedure, bimodal mixtures were prepared from two different monomodal size fractions, varying the ratio of the two fractions, but keeping the total powder volume fraction
(6)
Nevertheless, the respective particle size fractions have a distribution of the minimum Feret diameter. Thus, an averaging procedure must be considered, as starting point the arithmetic mean is arbitrarily chosen. In the following a calculus for a mean particle size will be developed which accounts for the actual size distribution in the best way. The averaged value will be used representatively for the size distribution during the calculations. This will be indicated by the usage of the term quasi-monomodal. For clarity, a distribution only consisting of the mode is called monomodal and is only given by a Dirac delta 31
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Fig. 10. Minimum Feret diameters determined in Fig. 9 plotted as a function of the composition n of the mixture of the two quasi-monomodal fractions 500250 μm and 100-63 μm.
Fig. 9. Determining the average minimum Feret diameter for the bimodal mixtures from the value of their k parameter using the fit of the quasi-monomodal size fractions determined in Fig. 8.
constant at 20%. With the parameter k determined for each composition the corresponding particle size (average of minimum Feret diameter) was determined as shown in Fig. 9. As starting point the arithmetic mean was used. These minimum Feret diameters were subsequently plotted as a function of the composition n of the two combined quasi-monomodal size fractions and fitted utilizing Equation (10). The fractions are treated as quasi-monomodal, but in fact they have a distribution with finite size, an appropriate average must be chosen with the true λ value for averaging. The true lambda value λtrue must further be self-consistent. So the λ used for determining the particle size of the quasimonomodal fractions must reappear in the λ exponent of the fit in Equation (10) for the bimodal mixtures. Therefore, different initial values for the exponent λin are tested for the size average of the quasimonomodal fractions and compared to the corresponding output exponent λ out obtained from the fit of the quasi-bimodal mixtures. A plot of λin − λ out as a function of λin reveals the self-consistent λtrue as λin − λ out = 0 . This was done for a mixture of 500-250 μm and 10063 μm as well as 500-250 μm and 150-100 μm. The initial fit shown in Fig. 10 gives a reasonable range for λin values, i.e. the mean λ value plus/minus the half and full error. As shown in Fig. 11 a value of λtrue of 0.4666 was obtained for both mixtures. We suggest that this λtrue value yields the most representative size average to represent the particle size and its distribution. Together with Equation (6) describing the relationship between particle size and the k parameter, it is possible to calculate the permeability of a soft magnetic composite from the particle size and size distribution of the filler particles. Nevertheless, it must be kept in mind that the considered particles have a platelet shape with a constant thickness due to the processing technique. It can be expected that the value of λtrue may be different for other filler types. However, the strategy to reveal the appropriate calculus for averaging is generally applicable whenever the calibrating quantity is distributed by analyzing bimodal mixtures or even multimodal blends followed by a search for self-consistency. Tuning the permeability even further could be achieved by controlled orientation of the particles by crosslinking under a magnetic field. The silicon rubber matrix in this work is quite stiff with a shore hardness type A of 30 and the initial components are highly viscous. This prevents particles form orientation in an applied field. But recently, new progress has been made in using an ultra-soft silicone
Fig. 11. The difference λ in − λ out plotted against λ in to reveal the self-consistent value of λtrue = 0.4666 for mixtures of the two fractions 500-250 μm and 10063 μm as well as 500-250 μm and 150-100 μm.
matrices with good controllability of mechanical properties and a strong magnetorheological effect [30,31]. Combining this with the platelet shape of the particles makes it possible to access the influence of orientation next to the influence of particle size and size distribution to develop a profound understanding of the permeability of SMCs. 4. Conclusion Considering the effect of particle size of magnetic fillers on the permeability, we have demonstrated a strategy to reveal the appropriate average value and calculus with respect to the particle size distribution. The data enables permeability of LSR filled with soft magnetic particles to be calculated, making it possible to determine optimization potential for processing. Declarations of interest None. 32
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Funding
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Composites Science and Technology journal homepage: www.elsevier.com/locate/compscitech
Effect of particle size and size distribution on the permeability of soft magnetic liquid silicone rubber composites
T
Dirk W. Schuberta,∗, Siegfried Wernera, Ingo Hahnb, Veronika Solovievab a b
Institute of Polymer Materials, University of Erlangen-Nuremberg, Martensstrasse 7, 91058 Erlangen, Germany Institute of Electrical Drives and Machines, University of Erlangen-Nuremberg, Konrad-Zuse-Str. 3/5, 91052 Erlangen, Germany
A R T I C LE I N FO
A B S T R A C T
Keywords: A: polymer matrix composite B: magnetic properties C: modelling E: powder processing
Magnetic composites made of LSR (Liquid Silicone Rubber) and a magnetic filler have been investigated with respect to magnetic field guidance for transcutaneous energy transport. Effects of particle size and size distribution on the effective initial magnetic permeability are analyzed. A simple model is presented to describe the permeability as a function of the degree of filling, particle size and size distribution. From this, a calculus for the relevant size average of the filler was developed at 20 vol% of magnetic filler. The calculus showed self-consistency and can be used to tailor optimized powders for better field guidance.
1. Introduction Soft magnetic materials are widely used in electronics as transformers, e.g. in high frequency applications for impedance matching and the balancing of antenna systems [1]. Another important application of soft magnetic materials is the wireless transport of energy [2,3]. Therefore, a pair of coils is used to transfer energy via induction and resonance coupling from a primary coil connected to the charging station to a secondary coil connected to the device [4]. The efficiency of this setup can be enhanced by deliberately guiding the magnetic flux between the coils using a soft magnetic encapsulation typically made of ferrite or other rigid metal based materials [4]. This technology is well known and already implemented into consumer electronics such as cell phones or electric tooth brushes. But the transfer to medical applications is still to be made, although wirelessly charging built-in supporting systems like pace makers or a ventricular assist device (VAD) offers extreme benefits for the patient as recurring surgeries or a permanent percutaneous drive line can be avoided [5–7]. One limiting factor for conventional rigid setups is that the irregular surface and the 3D shape of the human body increase the distance between the power transmitting coils and, thereby, reduce their efficiency. This problem can be solved by using a flexible coil system which can adapt to the body surface and minimizes the distance between the coils. Another limiting factor is an efficient guidance of the field by a deliberately designed material permeability as encapsulation for the coils. These two requirements can be met by using a composite made from flexible liquid silicone rubber (LSR) and rigid soft magnetic filler.
∗
In literature soft magnetic materials derived from a soft magnetic powder, surrounded by an insulating layer are referred to as soft magnetic composites (SMC) [2]. In general an increase in permeability with larger particles has been observed for SMCs [8,9]. However, larger particles also lead to a reduced flexibility of the part. So the impact of particle size on the permeability has to be understood thoroughly. But conventional effective medium theories as e.g. the Böttcher [10], De Loor [11] or Bruggeman [12] equation, nicely summarized by Anhalt and Weidenfeller, only focus on the volume fraction of the magnetic filler and do not include the effect of particle size [13]. However, it is known that for the same material an increase of the particle size yields a steady increase in the permeability [3,14–16]. Anhalt found a linear relation between particle size and permeability looking at particles with an arithmetic mean diameter of 30 μm–170 μm [17]. But the investigated diameter range was small and the arithmetic mean is sensitive to a skewed particle size distribution. Recently, Li et al. modified the Bruggeman equation to account for the particle size distribution. They obtained the effective composite permeability by integrating the contributions from particles of different size and, consecutively, of different permeability [18]. They assumed a log-normal distributed particle size and achieved a good prediction of the permeability for particles with a median diameter of 7 μm–84 μm. However, the size range is again small and knowledge of the size distribution function is required. Yet, no work is known to the authors that takes into account that a tailor made mean value for the particle size distribution, that is not the arithmetic mean or the median, can give a better description of the
Corresponding author. E-mail address: [email protected] (D.W. Schubert).
https://doi.org/10.1016/j.compscitech.2019.04.005 Received 6 October 2018; Received in revised form 29 March 2019; Accepted 6 April 2019 Available online 08 April 2019 0266-3538/ © 2019 Elsevier Ltd. All rights reserved.
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nearly constant with a thickness of 20 μm due to the processing routine [21]. Fig. 2 shows that the minimum and maximum Feret diameter were proportional to each other with the aspect ratio of the particles as proportionality constant. All size fractions feature an aspect ratio of about 1.65 and therefore have a very similar shape. Due to the proportionality, it is also sufficient to consider only one Feret diameter typ. In this work the minimum Feret diameter was arbitrarily chosen.
particle-size-permeability relation. Therefore, it is the purpose of this paper to identify a calculus for a more accurate mean particle size to predict the permeability, which works from the micrometer to the millimeter range. In particular, we apply a strategy which was originally developed to reveal the relevant molar mass average of a polymer mixture for the film thickness of a spin coated film [19]. But instead of mixing polymers with different bimodal molar mass distribution, deliberately prepared bimodal particle size distributions and their influence on magnetic permeability of the composite is analyzed (In fact the effective initial magnetic permeability is determined, but for simplicity it is just called magnetic permeability throughout the paper). Therefore, toroidal cores of LSR filled with different size fractions of magnetic particles and combinations of those were prepared. An inductance is generated by winding the cores with a certain number of turns of an electric wire, which makes the permeability of the core accessible. Furthermore, the hysteresis behavior and the particle dispersion are investigated using a magnetometer and SEM, respectively. As recently suggested by Schubert et al. [20] a simple model is used to describe the permeability as a function of the degree of filling with only one adjustable parameter k. To isolate the effects of particle size and size distribution we focus only on a volume fraction of 20% for the composite, which permitted an easy processing and had a sufficiently large effect on the permeability.
2.3. Manufacturing toroidal cores Toroidal cores consisting of LSR and Vitroperm 250 powder were prepared at a constant powder volume fraction ϕp of 20%. The toroids had an outer diameter do of 50 mm, an inner diameter di of 20 mm and a height h of 15 mm. As matrix, the high temperature vulcanizing twocomponent silicone rubber Elastosil LR 3003/30 (Wacker Chemie AG, Munich, Germany) was used, which is typically applied in high volume technical products [23]. To prepare a toroidal core, the two silicone components were weighed out in a ratio of 1:1 and mixed in a speed mixer for 5 min at 1000 rpm. Subsequently the desired amount of filler was added to the silicone and dispersed by mixing for 3 min at 2000 rpm. After mixing, incorporated air was removed in a desiccator. This procedure was repeated three times to ensure proper dispersion of the particles. The prepared compounds were poured into toroidal polytetrafluoroethylene molds and newly incorporated air was again removed in a desiccator. Consecutively, the compound was cross-linked for 4 h at 120 °C. First, toroids filled with the six pure size fractions were produced to obtain the relationship between particle size and permeability. Second, combinations of two size fractions in different ratios were used to determine a mean that considers the size distribution in the best way. Therefore, the fractions 500-250 μm and 100-63 μm as well as the fractions 500-250 μm and 150-100 μm were mixed in different ratios. The ratio of the fractions was characterized by the content n of the fraction 500-250 μm in the mixture as shown in Equation (1). A content n of 0.8, 0.7, 0.6, 0.5, 0.4, 0.3 and 0.2 was used for both mixtures.
2. Experimental section 2.1. Powder preparation Powders with different degrees of fineness were generated from the soft magnetic material Vitroperm 250 (VACUUMSCHMELZE GmbH & Co. KG, Hanau, Germany). Due to its nanocrystalline structure it showed a superior increase in permeability than other commercial soft magnetic materials such as Neosid G and Neosid F [20]. Table 1 shows the parameters of the filler material, where μr is the magnetic permeability, Hc the coercive magnetic field strength and ρ the density. The permeability of the bulk material is constant up to a frequency of 300 kHz [21]. The processing of nanocrystalline soft magnetic alloys with a rapid solidification process yields thin ribbons, which are wound to toroidal cores. Therefore, the variable rotor speed mill Pulverisette 14 (Fritsch GmbH, Idar Oberstein, Germany) was used to mill the cores down to primary powders of certain fineness. A coarse primary powder was produced using a rotational speed of 6000 rpm together with a sieve ring with mesh openings of 2.0 mm and a fine primary powder using a speed of 18000 rpm and a 0.5 mm sieve ring. From the coarse and fine primary powder, the size fractions > 500 μm, 500-250 μm, 250- 150 μm and 150-100 μm, 100-63 μm and < 63 μm were isolated, respectively, using a test sieve shaker according to DIN 66165-1 (2016-08) [22]. Two primary powders of different fineness were generated to achieve an effective production of the large and small size fractions.
n=
The toroidal cores were wound with 40 turns of electric wire to form an inductance as shown in Fig. 3. The corresponding inductivity was measured using an AE20204 LC meter (Ascel Electronics, Hohwacht, Germany). The relative magnetic permeability μr of the core was obtained by Equation (2) describing the inductance L of a toroidal coil:
L=
Table 1 Characteristic parameters of the magnetic filler [21]. Hc [A*m−1]
ϱ [g*cm−3]
Vitroperm 250
4000
<3
7.2 ± 0.1
μr Aμ0 2 N le
(2)
where N is the number of turns, A the cross section of the toroid, μ 0 the magnetic field constant and le the mean magnetic length calculated from the outer diameter do and the inner diameter di of the toroid with π (do + di )/2 . A detailed description is given by Schubert et al. [20], which also demonstrates that the imaginary part of the permeability is negligibly small in the investigated frequency region up to at least 300 kHz. After determining the permeability the hysteresis behavior of the composites was analyzed using the ring-core method following the industry standard IEC 60404-6:2003 [24]. This requires a toroid with a ratio of da /di smaller than 1.4. The initial toroid had a diameter ratio of 2.5. So a smaller toroid with an outer diameter of 40 mm was punched out from the initial toroids yielding a remaining toroid with da of 50 mm and a ratio of only 1.25. These remaining toroids with reduced width were covered with electric insulation and equipped with two windings in order to build a ring core transformer. A sinusoidal current supply in the primary coil leads to a sinusoidal magnetic field strength in the toroidal sample which can be then determined with Equation (3).
The actual particle size was determined using a video microscope (type: VHX-1000D, Keyence Corp., Osaka, Japan) and image processing software Fiji. The particle size was described by the minimum Feret diameter d and the maximum Feret diameter D of 73–100 particles per fraction as shown in Fig. 1. The size of the particles varied from the micrometer to the millimeter range, but their platelet shape remained
μ r [−]
(1)
2.4. Characterizing magnetic properties and particle dispersion
2.2. Particle characterization
Filler
m500 − 250μm m500 − 250μm + m100 − 63μm
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Fig. 1. Example of the determination of the maximum Feret diameter D (left) and the minimum Feret diameter d (right).
B (t ) =
1 Ns A
T
∫ Us (t ) dt 0
(4)
The measurements were conducted at a frequency of 50 Hz and 100 kHz, respectively. The commutation curves of the samples are constructed by connecting the intersection points of the peak values of the magnetic field density and the magnetic field strength of the hysteresis. curves [25]. The punched out toroids (outer diameter 40 mm) were used for analyzing the particle dispersion. A kryo fracture was done to access the particle distribution without plastic deformation of the matrix. They were frozen in liquid nitrogen and fractured with hammer and clippers. The samples were sputtered with gold and SEM images were taken of the toroid cross section. 3. Results and discussion In a first step, the magnetic behavior and particle dispersion inside the toroids was characterized to ensure the applicability of the used models. Second, the arithmetic mean of the minimum Feret diameter was used to develop a relation between the permeability and the particle size of the pure size fraction. Consecutively, an equation for different means was deduced and the governing mean was revealed by analyzing mixtures of two fractions in different ratios. Due to the inevitable errors of this procedure, the developed mean was tested for self-consistency and verified with another mixture.
Fig. 2. Mean and standard deviation for maximum vs. minimum Feret diameter for the pure size fractions together with the particles used in the previous work of Schubert et al. [20] The solid line demonstrates the proportional relation between both Feret diameters and similar aspect ratio for all size fractions.
3.1. Characterization of magnetic behavior The SMCs show a soft magnetic response to an applied magnetic field. The achieved flux density is rather low compared to compact ferrites made by conventional processing technologies. Representative hysteresis curves are shown in Fig. 4 for various particle size fractions. From Fig. 4 a linear relation (in particular a direct proportionality) between the magnetic flux density B and the applied magnetic field strength H is evident for applied magnetic fields up to 2 kA/m. Additionally, the hysteresis is almost perfectly negligible. The commutation curves of the samples are shown in Fig. 5. For small values of the magnetic field strength, a linear correlation between H and B was revealed. Due to experimental limitations, higher values of the magnetic field strength H could not be adjusted at high frequency (100 kHz) for all samples as a consequence of the increasing impedance of the primary coil. Fig. 5 also shows that the commutation curves are almost frequency independent because the graphs of the corresponding curves, for 50 Hz and 100 kHz merge into each other and therefore have the
Fig. 3. Toroidal core (diameter 50 mm) with 20 vol-% of the fraction 250150 μm without (left) and with 40 turns of an electric wire to measure the inductance with an LC meter (right).
H (t ) =
Np I (t ) le
(3)
In this case Np is the number of primary turns and I is the current in the primary coil. The magnetic flux density B can be calculated according to Equation (4) with the induced voltage Us on the secondary side of the ring transformer and the number of secondary turns Ns .
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Fig. 4. Hysteresis curves of toroids with the pure size fractions 500-250 μm, 150-100 μm and 100-63 μm measured at a frequency of 50 Hz with three different amplitudes of the H-field up to 9 kA/m show very low losses and an almost perfect linear relation until a magnetic field of 2 kA/m.
magnetic field strength, the correction of Pak Ming Hui et al. has to be applied [27]. Nevertheless, the strategy suggested in this paper is universal and still applicable even if the initial equation for describing the permeability is modified.
same slope in the linear range. However, the material will reach a regime of saturation at very high magnetic fields. Thus, the permeability can be taken as constant for magnetic field strength values under 2 kA/m. The permeability values determined at a magnetic field of 0.12 mA/m in section 3.2 and 3.3 were thus located in the linear regime. This is an important requirement to apply the Clausius-Mossotti relation later on. Additionally, it must be pointed out, that the intended application of this material is typically at a magnetic field strength of approximately 1 kA/m as a VAD requires a maximum power of up to 25.5 W [26]. If the application requires to use the SMC in a regime where the permeability is depending on the
3.2. Characterization of particle dispersion Fig. 6 shows SEM images of the flanks of a cryo fracture through the toroid cross section for the pure size fractions 500-250 μm and 10063 μm as well as for mixtures of both. The pure size fractions show a homogeneous distribution of the particles. For the mixtures, small 29
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Fig. 5. Commutation curves of toroid samples at 50 Hz (left) and 100 kHz (right).
[20]. This parameter contains the effects of particle size and particle size distribution on the polarizability [28,29]. As parameter k is present in the nominator and denominator of Equation (5.1) the influence of the particle size is expected to be dampened. Therefore, the expression is rearranged to Equation (5.2) and the parameter k is analyzed in the following at a constant degree of filler:
particles are found in the matrix, in close proximity to large particles and between them indicating that they are homogenously distributed and no segregation has occurred. 3.3. Relationship between particle size and permeability Fig. 7 shows the achieved permeability for the different size fractions, which are described by the arithmetic mean of the minimum Feret diameter 〈d〉a . The permeability shows a steady increase with particle size from the micrometer to the millimeter range. The permeability increases by more than a tenfold as the particle size rises by a twentyfold at a volume fraction of 20%. In previous work, a simple equation was derived from the ClausiusMosotti relation to describe the dependency of the magnetic permeability on the volume fraction of a magnetic filler using only one adjustable parameter k [20]. It was shown empirically that although the Clausius-Mosotti relation is derived for spherical particles the equation also describes the behavior of platelet particles in a reasonable range
μr =
1 + 2kϕP 1 − kϕP
(5.1)
μr − 1 μr + 2
(5.2)
kϕP =
With the volume fraction fixed at 20% the k parameter was determined from the permeability. It should be pointed out that due to the percolation-like behavior of Equation (5.1) it should only be applied where the product of k and λ and Φ is below 0.9. The results also showed a steady increase with the arithmetic mean of the minimum Feret diameter as shown in Fig. 8. To describe these results we suggest a Fig. 6. SEM images of the cross section of the toroids for the pure size fraction 500-250 μm (a) and 10063 μm (b) as well as for mixtures of both with a content n of the fraction 500-250 μm of 70% (c) and 30% (d). Pictures show a good dispersion of the fillers even for mixtures as the small particles are located between the flat and large particles and in the matrix (arrows).
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function, which means an infinitely small width and infinite height of the corresponding size distribution. Therefore, our particle fractions should be called quasi-monomodal because there widths are finite. 3.4. Revealing the governing mean Equation (6) considers the boundary condition of saturation inherently as described in section 3.3. Nevertheless, a power law can be applied to describe the data within the experimental error as well. Alternatively, one can roughly estimate a power law after Taylor expansion of Equation (6) for rather small particles d< < d 0 :
d k = k0 ⎛ ⎞ d ⎝ 0⎠ ⎜
λ
⎟
(7)
The reader may note that the approximation is insufficient as the distribution under consideration and the value for d 0 (Fig. 8) does not satisfy d< < d 0 entirely. However, the Tailor expansion yielding Equation (7) from Equation (6) was only given to indicate the strong similarity of both equations, at least in a certain parameter range. Furthermore, the average value of the parameter k shall be described by the average particle size 〈d〉 as shown in Equation (8a). With a certain size distribution ni and an average of i particles the average value of the parameter k has to be calculated according to Equation (8b).
Fig. 7. Steady increase of permeability with increasing minimum Feret diameter of the six pure size fractions at 20 vol-%.
〈d〉 ⎞ 〈k〉 = k 0 ⎛ ⎝ d0 ⎠ ⎜
⟨k ⟩ =
λ
⎟
∑i ni k 0
(8a) d λ d0
( )
∑i ni
(8b)
Relating Equation (8a) and (8b) and rearranging for a relevant average size < d> gives: 1
λ ⎛ ∑ ni di ⎞ ⟨d⟩λ = ⎜ i ∑i ni ⎟ ⎠ ⎝
λ
(9)
Where the index λ on 〈d〉 denotes a certain average. For example λ = 1 yields the arithmetic mean formerly described as 〈d〉a . Mixing two size fractions can be depicted by combining the two quasi-monomodal functions as shown in Equation (10). Where n represents the ratio of the two quasi-monomodal functions and d1 and d2 their characteristic particle sizes 1
⟨d⟩λ = [(1 − n) d1λ + nd 2λ ] Fig. 8. Parameter k as a function of the arithmetic mean of the minimum Feret diameter 〈d〉a . The solid line shows the fit according to Equation (6).
It should be noted that also on the base of Equation (6) an average can be calculated with an adjustable parameter λ . However, the resulting equation is a bit more complicated and less instructive for explaining the strategy, but yields the same result with respect to λ within the experimental errors.
stretched exponential function as given in Equation (6) having an asymptote k 0 for infinite particle size and d 0 and λ being adjustable parameters. The asymptote is considered to be a material limit as the LSR has a permeability of approximately unity and addition of 20 vol-% Vitroperm cannot surpass 20% of the permeability of the neat Vitroperm material:
1
⎜
λ di λ ⎞⎞ ⎞ d0
( ) ⎠ ⎟⎟
⎛ ⎛ ∑ ni exp⎛− i ⎜ ⎝ ⟨d⟩λ = d 0 −ln⎜ ⎜ ∑i ni ⎜ ⎜ ⎜ ⎝ ⎝
λ
⎛ ⎡ 〈d〉 ⎞ ⎤ ⎞ k = k 0 ⎜1 − exp ⎢−⎛ ⎥⎟ d ⎣ ⎝ 0 ⎠ ⎦⎠ ⎝
(10)
λ
⎟⎟ ⎟⎟ ⎠⎠
(11)
⎟
To summarize the equations and statements above: Equation (6) is only utilized to correlate the parameter k to the Feret diameter, as shown in Fig. 8. The power law in Equation (8a) and the corresponding equations for the averages in Equation (9) and (10) are of the general character of a trial function to demonstrate the approach to reveal the relevant averages, even on the base of non-mono-modal calibration system as in this work. In other words the λ in Equation (9) and (10) must not necessarily be identical to the λ in Equation (6). To reveal the relevant averaging procedure, bimodal mixtures were prepared from two different monomodal size fractions, varying the ratio of the two fractions, but keeping the total powder volume fraction
(6)
Nevertheless, the respective particle size fractions have a distribution of the minimum Feret diameter. Thus, an averaging procedure must be considered, as starting point the arithmetic mean is arbitrarily chosen. In the following a calculus for a mean particle size will be developed which accounts for the actual size distribution in the best way. The averaged value will be used representatively for the size distribution during the calculations. This will be indicated by the usage of the term quasi-monomodal. For clarity, a distribution only consisting of the mode is called monomodal and is only given by a Dirac delta 31
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Fig. 10. Minimum Feret diameters determined in Fig. 9 plotted as a function of the composition n of the mixture of the two quasi-monomodal fractions 500250 μm and 100-63 μm.
Fig. 9. Determining the average minimum Feret diameter for the bimodal mixtures from the value of their k parameter using the fit of the quasi-monomodal size fractions determined in Fig. 8.
constant at 20%. With the parameter k determined for each composition the corresponding particle size (average of minimum Feret diameter) was determined as shown in Fig. 9. As starting point the arithmetic mean was used. These minimum Feret diameters were subsequently plotted as a function of the composition n of the two combined quasi-monomodal size fractions and fitted utilizing Equation (10). The fractions are treated as quasi-monomodal, but in fact they have a distribution with finite size, an appropriate average must be chosen with the true λ value for averaging. The true lambda value λtrue must further be self-consistent. So the λ used for determining the particle size of the quasimonomodal fractions must reappear in the λ exponent of the fit in Equation (10) for the bimodal mixtures. Therefore, different initial values for the exponent λin are tested for the size average of the quasimonomodal fractions and compared to the corresponding output exponent λ out obtained from the fit of the quasi-bimodal mixtures. A plot of λin − λ out as a function of λin reveals the self-consistent λtrue as λin − λ out = 0 . This was done for a mixture of 500-250 μm and 10063 μm as well as 500-250 μm and 150-100 μm. The initial fit shown in Fig. 10 gives a reasonable range for λin values, i.e. the mean λ value plus/minus the half and full error. As shown in Fig. 11 a value of λtrue of 0.4666 was obtained for both mixtures. We suggest that this λtrue value yields the most representative size average to represent the particle size and its distribution. Together with Equation (6) describing the relationship between particle size and the k parameter, it is possible to calculate the permeability of a soft magnetic composite from the particle size and size distribution of the filler particles. Nevertheless, it must be kept in mind that the considered particles have a platelet shape with a constant thickness due to the processing technique. It can be expected that the value of λtrue may be different for other filler types. However, the strategy to reveal the appropriate calculus for averaging is generally applicable whenever the calibrating quantity is distributed by analyzing bimodal mixtures or even multimodal blends followed by a search for self-consistency. Tuning the permeability even further could be achieved by controlled orientation of the particles by crosslinking under a magnetic field. The silicon rubber matrix in this work is quite stiff with a shore hardness type A of 30 and the initial components are highly viscous. This prevents particles form orientation in an applied field. But recently, new progress has been made in using an ultra-soft silicone
Fig. 11. The difference λ in − λ out plotted against λ in to reveal the self-consistent value of λtrue = 0.4666 for mixtures of the two fractions 500-250 μm and 10063 μm as well as 500-250 μm and 150-100 μm.
matrices with good controllability of mechanical properties and a strong magnetorheological effect [30,31]. Combining this with the platelet shape of the particles makes it possible to access the influence of orientation next to the influence of particle size and size distribution to develop a profound understanding of the permeability of SMCs. 4. Conclusion Considering the effect of particle size of magnetic fillers on the permeability, we have demonstrated a strategy to reveal the appropriate average value and calculus with respect to the particle size distribution. The data enables permeability of LSR filled with soft magnetic particles to be calculated, making it possible to determine optimization potential for processing. Declarations of interest None. 32
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Funding
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