Training effect in magnetoactive elastomers due to undermagnetization of magnetically hard filler

Journal Pre-proof Training effect in magnetoactive elastomers due to undermagnetization of magnetically hard filler M.V. Vaganov, D. Yu. Borin, S. Odenbach, Yu. L. Raikher

PII: DOI: Reference:

S0921-4526(19)30751-3 https://doi.org/10.1016/j.physb.2019.411866 PHYSB 411866

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Physica B: Physics of Condensed Matter

Received date : 29 June 2019 Revised date : 2 October 2019 Accepted date : 4 November 2019 Please cite this article as: M.V. Vaganov, D.Y. Borin, S. Odenbach et al., Training effect in magnetoactive elastomers due to undermagnetization of magnetically hard filler, Physica B: Physics of Condensed Matter (2019), doi: https://doi.org/10.1016/j.physb.2019.411866. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.

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Training Effect in Magnetoactive Elastomers due to Undermagnetization of Magnetically Hard Filler

of Continuous Media Mechanics, Russian Academy of Sciences, Ural Branch, Perm, 614013, Russia b Institute of Mechatronic Engineering, TU Dresden, Dresden, 01069, Germany

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a Institute

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M. V. Vaganova,b,∗, D. Yu. Borinb , S. Odenbachb , Yu. L. Raikhera

Abstract

Hysteresis loops of magnetoactive elastomers (MAE) exhibit several peculiar features, including low coercivity in comparison with mechanically hard materials and a striking mismatch between first consecutive magnetization loops. The difference between loops diminishes during cycling of the applied magnetic field, thus displaying a training

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effect. The article presents the results of our research on a hypothesis explaining this phenomenon. Magnetization of a multigrain ferromagnetic particle is modeled at fields not exceeding the anisotropy field of the constituent grain material. It is shown that because of rotation of particles in an elastomer matrix, different branches of their hysteresis loops stem from the magnetization switching of different grain groups. This process is illustrated in detail by use of

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simple systems with a small number of grains. The results of simulation are supported with magnetic measurements of both MAE samples and epoxied powder. Coercivity of the modeled trained loops is in the order-of-magnitude agreement with the experiment.

Keywords: magnetoactive elastomer, magnetorheological elastomer, magnetic hysteresis, multigrain magnetic particle, asymmetric loop, training effect

1. Introduction

The specifics of magnetoactive elastomers (MAE) of mixed content, i.e., the composites, where the filler (at least partially) contains magnetically hard (MH) powder, is that the magnetization of the particles depends not only on their intrinsic magnetic properties but on the level of their mobility in a polymeric matrix as well [1–3]. The most known type of MH fillers is rapidly solidified NdFeB powders, which possess large coercivity, magnetization, and

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relatively high Curie temperature [4, 5]. The magnetization process of such MAEs exhibits four major features. First, the coercivity of a sample with soft matrix is much lower than of that with a solid matrix. Second, all the loops display negative bias: they are shifted to the negative area of the field axis. Third, the magnetization of MAEs at the points of ∗ Corresponding

author: M. V. Vaganov: [email protected]

Preprint submitted to JMMM

October 3, 2019

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maximal field grows with the sequence number of measured loops. Fourth, under cyclic magnetizing field, the width of the obtained hysteresis loops gradually decreases, attaining its stationary value only after a few turns of the field. From experiment [6, 7, 1], it is well established that the coercivity of a MAE declines as the matrix compliance

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increases, i.e., as the rotational mobility of the particle inside the matrix enhances. However, to our knowledge, the difference between consecutive loops and their concomitant negative bias have not received enough attention and have not been investigated thoroughly yet.

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In this work we put aside the issue whether or not at its first field-induced rotation an MH particle irreversibly disturbs its adhesion to the surrounding polymer. Although such an assumption seems quite reasonable and definitely needs to be investigated, nevertheless, we do not have at our disposal sufficient experimental evidence to that effect insofar. Therefore, we focus on the above-mentioned features, assuming that they are imparted to MAE magnetization by: (i) complex fine-grain structure of NdFeB particles and (ii) their ability to rotate under the action of the applied

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field in an elastic matrix whose rheology is, however, unaffected by the particle motion.

We base our consideration on the following essential issue. The main sources of high coercivity in the MH microparticles of the type under consideration are single domain nanograins of Nd2 Fe14 B [5]. Since the internal magnetocrystalline anisotropy field of this material is about 7.3 T, an applied during hysteresis measurements field is

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not always high enough to switch the magnetization of every grain in the sample. This fact combined with possible particle rotation and collective reorientations of the grain easy axes causes the situation when at each ramping of the field the magnetic structure of each microparticle becomes more trained gradually, but is not determined only by its state at the first maximal field.

The article comprises six sections. In section 2, we present results of magnetic and mechanical measurements on MAEs and highlight physical properties of the employed NdFeB powder. Section 3 briefly describes the mathematical model developed for simulating MAE magnetization process. In section 4, simple cases are considered to show how the training effect emerges in such materials. The results of computer simulations and their comparison with experimentally measured coercivity of the trained loops are presented in section 5. Section 6 summarizes the work done.

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2. Magnetization curves of MAEs

The samples were prepared according to a standard procedure, which was identical to that very recently reported in [8]. Matrices of the composites were made of a two-component polydimethylsiloxane (PDMS) compound Elastosil RT623 (Wacker Chemie AG, Germany). Tuning of the matrix elasticity was accomplished through diluting Wacker Elastosil with silicone oil M1000 Baysilone from Bayer at different ratios D of Elastosil to oil, see Table 1. 2

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Table 1: Mechanical properties of the MAE samples

D (matrix:oil) 4:0 4:1 4:2 4:3 4:4

G1 , kPa 319 88.5 27.2 12.3 5.5

G2 , kPa 412 115 40.0 19.2 8.4

G∗2 , kPa 461 128 40.0 17.8 8.0

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Sample s1 s2 s3 s4 s5

The MQP-S-11-9-20001 powder (Magnequench) of isotropic spherical NdFeB microparticles was used as a mag-

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netic filler of the MAEs at concentration φp = 11 vol.%. More details on the powder treatment procedure, as well as on the sample manufacturing technique, can be found, for instance, in [9, 10].

The shear moduli of the rod-like samples of pure matrices (G1 ) and of MAEs (G2 ) were obtained via a quasi-static torsion test on Anton Paar MCR301 rheometer according to the procedure described in [11]. The measured moduli

Guth-Smallwood equation [12–15]:

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G2 agree well (see Table 1) with the estimates G∗2 obtained for composites of spherical particles, using the classical   G∗2 = G1 1 + 2.5φ p + 14.1φ2p

(1)

Magnetization curves were measured by means of a vibrating sample magnetometer (Lake Shore 7407) able to

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generate magnetic fields up to 2.7 T. For this purpose, thin disc-shaped specimens were used, which were accurately cut off from rod-like samples. In each experiment at the VSM, the field was first ramped from zero to a certain maximal value H0max and after that was cycled several times between H0max and −H0max , so that at the end we obtained the data for the initial magnetization curve and for up to nine first consecutive hysteresis loops. To investigate magnetization of the immovable powder itself and exclude the influence of soft polymeric matrix, several specimens of epoxied MQP-S particles were prepared at the same concentration φp = 11 vol.%. Each such sample formed a disk of diameter 4.7 mm and height 1.3 mm.

Due to low volume concentration of the MQP-S powder in our samples, the demagnetizing correction is unnecessary [16], therefore, in presenting the experimental data, we further identify an applied field H0 with that acting inside a sample H.

Typical magnetization curves of an MAE sample are shown in Fig. 1. The most notable features of the hysteresis loops are: (i) significantly lower coercivity in comparison to mechanically hard materials, (ii) the shift of the loops

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to the negative values of the field axis, (iii) their ”reptation”, that is, the change in magnetization with the number of field cycles, and (iv) the presence of a training effect: gradual diminution of discrepancy between consecutive magnetization loops. As seen, the training effect manifests itself most strongly between the first and second loops, and quickly fades out after that. 3

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Figure 1: Magnetization loops of the sample s3 with G1 = 27.2 kPa and G2 = 40.0 kPa at Hmax = 15 kOe or ≈1200 kA/m

Figure 2: Optical microscopy image of an MQP-S-11-9-20001 particle.

An image of an MQP-S particle of the magnetic filler is shown in Fig. 2. Those particles are produced by means of the spinning-cup atomization technology and, after rapid solidification, acquire spherical shape with the mean radius 46.8 µm

According to the technical specifications [17] and the data obtained from transmission electron microscopy [18], the particles of rapidly solidified MQP-S-11-9-20001 powder consist of spherical Nd2 Fe14 B nanograins separated by Nd-rich phase. The grain size ranges from 20 to 400 nm and, thus, is less or about the critical size of single-

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domainness for Nd2 Fe14 B [4, 19]. Although the theoretical estimate for the anisotropy field of this material is 7.3 T [5], the measured value of this quantity for a pressed or somehow else immobilized powder is about 4 times less than that number. In the literature, this phenomenon is known as Brown’s paradox [20], and its origin could be attributed to several sources. First place, these are ubiquitous imperfections and impurities that provoke nucleation. Second, the 4

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magnetic state of the interstitial Nd-rich phase, which surrounds the highly coercive nanograins in a microparticle, is still debatable among researches [21, 22]. In general, it is believed to be paramagnetic, so its presence might affect the overall coercivity of the particle to a poorly determined extent. Third, in order to make the grains as small as possible,

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the manufacturer adds to the MQP-S powder the so-called grain growth inhibitors, thus reducing the concentration of pure Nd2 Fe14 B grains further.

The magnetization curves of the MAEs were compared with those of MQP-S powder in epoxy. The hysteresis

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loops measured at H0max = 400 kA/m (or 5000 Oe) are visually identical for all the samples: an example is given in Fig. 3 for our hardest and softest samples. The observed training effect of the NdFeB powder in epoxy can be associated with the presence of exchange-bias anisotropy that should exist at the interfaces between the Nd2 Fe14 B grains and interstitial Nd-rich phase. At the initial stage of magnetization, the exchange-bias is “tuned up”, and this unidirectional anisotropy provides the magnetization loops with negative shift. A surprising match between

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the loops (obtained at H0max ≤ 400 kA/m) of materials with different mechanical properties might be explained by the insufficiency of the applied field, making it impossible for the reversal (magnetic switching) processes in the

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Nd2 Fe14 B grains to be enabled, and for the microparticles to start rotating even inside our softest matrix.

(a) Epoxy resin

(b) MAE sample S5 with G1 =5.5 kPa and G2 =8.4 kPa

Figure 3: Hysteresis loops of the MQP-S powder in different matrices at H0max = 400 kA/m

To characterize the discrepancy between the first two consecutive magnetization loops, we introduce the normal-

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ized difference between their respective negative coercivities:

 δ = (H−1 − H−2 ) H−2 ;

(2)

here H−1 and H−2 are the field points at which the descending branches of the first and second hysteresis curves cross the field axis, see Fig. 1. In Fig. 4, the dependence of δ on the maximal value of the cycled field is shown for powder in epoxy. As seen, for such material the difference between the loops almost disappears at H0max ≥1200 kA/m, but 5

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it does not happen for mechanically soft MAEs, see, for instance, Fig. 1. This points out that in that range of H0max the exchange-bias effect may be neglected and the mechanical rotation of a microparticle significantly affects the

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magnetization reversal of the nanograins constituting it.

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Figure 4: Normalized difference between negative coercivity of the first and second hysteresis loops of MQP-S powder in epoxy at various H0max

Figure 5: Hysteresis loop of the MQP-S-11-9-20001 powder in epoxy at H0max = 2000 kA/m.

By fitting magnetization loops of the powder in epoxy with the well-known Stoner-Wohlfarth model of an assembly of non-interacting randomly oriented single domain particles, one can calculate the effective anisotropy field of the powder as Ha = Hc /0.479 [23]. For that purpose we took the hysteresis data of the MQP-S powder in epoxy at H0max ≈ 2000 kA/m, see Fig. 5. Given Hc ≈ 750 kA/m, one obtains Ha = 1.57 MA/m, and – assuming uniaxial

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anisotropy – from expression Ha = 2K/µ0 Ms one gets for the effective anisotropy energy density K = 1.08 MJ/m3 . As mentioned, the MQP-S powder could be brought to the true saturation Ms only at a field exceeding the anisotropy field of Nd2 Fe14 B nanograins. Nevertheless, according to the specification [17], the field allowing one to reach 95% of the powder magnetic saturation is H0max = 1.6 MA/m. Extrapolating our measurements undertaken at this field, we have estimated Ms at about 1.10 MA/m. 6

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The found value of the effective Ha should be considered as average. Evidently, in the MQP-S powder there are always some grains with extremely high and low coercivity, and ultimately magnetization processes of all the grains strongly depend on the orientation of their anisotropy axes. Since in MAE the microparticles can move, their rotation

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can considerably change the conditions at which its grains are able to switch. In the following sections, we propose a model for the magnetization of MQP-S particles in an elastic matrix and discuss the origin of difference between hysteresis loops of MAEs magnetized by the fields greater or equal than 1200 kA/m but significantly less than the

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field of uniaxial anisotropy of particle grains.

3. Magnetization model

Taking into account the aforementioned properties of the MQP-S particles, it is reasonable to model each of them as a spherical close-packed structure built of MH nanograins separated by non-magnetic medium [8]. The Nd2 Fe14 B

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grains cannot move relative to each other, but the microparticle that they constitute, being positioned inside an elastic medium, is able to move with respect to the coordinate frame of a measuring device. For simplification, in below we consider all the grains to be identical with volume V that is far smaller than the reference volume of single-domainness. On the other hand, in the systems being considered, the nanograin are on average far greater than those that would be

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significantly affected by superparamagnetic effects. Then, it is correct to derive equilibrium magnetic characteristics (magnetization, etc.) of such particles from minimization of the pertinent potential energy. Each i-th single domain grain of Nd2 Fe14 B possesses tetragonal crystal structure and, therefore, exhibits strong uniaxial magnetocrystalline anisotropy with energy:

  Ui, anis = KV 1 − (ei ni )2 ,

(3)

where ei is the unit vector of the grain magnetic moment µi , and ni is unit vector denoting one of the directions of its easy axis.

The field-induced contribution to the i-th grain energy is

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Ui, m = −µ0 M s V(ei Hi, loc )

(4)

The local field Hi, loc is a sum of the macroscopic field H imposed on a microparticle inside an MAE sample and the field Hi,dd produced by all the other grains of the same microparticle. In macroscopic approach, field H is presented as vector sum of the applied field H0 and the demagnetizing field Hd proportional to the overall magnetization of the 7

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sample and generated by other neighboring microparticles. Hi,loc = H + Hi,dd = (H0 + Hd ) + Hi,dd

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(5)

As noted, for a MAE with low particle concentration, the demagnetizing field is very low and might be neglected. To evaluate Hdd , we assume that in magnetic aspect all the grains can be treated as point magnetic dipoles, so that

Hi,dd =

N i Ms V X 1 h 3(e j rˆ i j ) − e j , 3 4π j,i ri j

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this field is expressed by the sum

(6)

where N is the number of grains in a microparticle, ri j denotes radius-vector connecting the i-th and j-th grains, and rˆ i j is its unit vector.

Given a microparticle placed in an isotropic incompressible linearly elastic medium [24], the mechanical energy

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of the deformed matrix is given by:

Uelast = 3G1 Vp γ2 ,

(7)

where G1 is the shear modulus of the elastomer matrix, Vp stands for the volume of the particle; γ denotes the angular

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displacement of the particle from its initial position.

Normalizing the above-presented energies by 2KV, one obtains convenient expressions for computer simulations: N

u=

ui,m

X  U ui,anis + ui,m + uelast , = 2KV i

ui,anis = 21 (1 − (ei ni )2 ), !3 N i λ X a h = −h(ei q) − 3(ei rˆ i j )(e j rˆ i j ) − (ei e j ) , 12 i, j ri j uelast = κNγ2 /2φ,

(8) (9) (10) (11)

where a is the grain radius, and the magnetic field strength is normalized with respect to the anisotropy field

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h = µ0 Ms H/2K.

(12)

The model has two parameters: λ represents the level of intergrain magnetodipole interaction in a microparticle: λ = µ0 Ms2 /K;

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(13)

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and κ renders the ratio of elastic to magnetic anisotropy energy densities and serves as a measure of the matrix stiffness:  κ = 3G1 K.

N N 1 X Ms V X µi = ei . Vp i Vp i

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M=

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The microparticle magnetization is defined as

(14)

(15)

In the simulations, we calculate its projection of M on the field direction and normalize it with respect to the saturation value Msat = Ms NV/Vp : m = (Mq) /Msat =

N 1 X (ei q). N i

(16)

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At a given field h, the stable state of a microparticle is found via tracing local minimum of its total potential energy 8, starting from h = 0. Coordinates of the minimum include angle components of all the unit vectors {ei } and two angles describing declination of the microparticle from its initial position in 3D space. Minimization of the energy function was performed by means of fminunc function from the MATLAB software, which implements the

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Quasi-Newton BFGS method [25–28].

4. Training effect

In order to elucidate the mechanism of MAE magnetization and to explain the nature of the concomitant training effect, let us begin with scrutinizing magnetization of the particles comprised of a small number of grains.

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4.1. Single-grain microparticle

Figure 6: Single-grain particle a) at the absence of the field and b) at a non-zero positive field

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In this case, where the microparticle is identical to its only grain, depending on the maximal field strength and the matrix stiffness, the magnetization vs. field curve can be either hysteretic or non-hysteretic. However, such a curve never changes its type while cycling the field between ±hmax . Schematic representation of

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a single-grain particle is shown in Fig. 6. Here and further in similar images, the color (blue or red) of the arrow denoting magnetic moment µ indicates the particular direction of the anisotropy axis on the part of which µ experiences maximal attraction. Markers p0 and p denote initial and current orientations of the particle in space.

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Basically, the magnetic behavior of the considered system is entirely determined by the angle θ10 between the initial direction of the magnetic moment – at h = 0 it lies along one of the directions of the anisotropy axis – and the field. At κ → ∞ (solid matrix) the model reduces to a standard Stoner-Wohlfarth scheme [23]. If κ is finite and hmax is sufficiently strong, the magnetization curves become loop-shaped, their width shrinks with the decrease in κ (see Fig. 7) [29].

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For any initial orientation of the grain axis n1 , there exists a certain value κ∗ , such that for κ < κ∗ the magnetization curve becomes single-valued. This implies extremely soft polymer matrix where the particle can easily rotate without switching its magnetic moment, see black curve in Fig. 7. Note that such a curve can be ascribed a non-zero negative “coercivity” although being not loop-shaped.

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1

0.5

-2

-1.5

-1

-0.5

0.5

1

1.5

2

-0.5

-1

Figure 7: Magnetization loops of a single-grain particle with θ10 = 60◦ at different values of matrix stiffness parameter κ

4.2. Double-grain particle

The situation becomes more complicated if the particle hosts two grains. The relative position of the grains

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and the orientations of their anisotropy axes highly affect the magnetization curves. However, pursuing the goal of qualitative understanding the intrinsic mechanism of magnetization in a multigrain particle, we make certain additional assumptions. Let the easy axes of both grains lie in the same plane that includes also the field vector, see Fig. 8. Also, we do not take into consideration the magnetostatic interaction between grains setting λ = 0; in that case the distance between the grains inside the particle is irrelevant. 10

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Figure 8: Double-grain particle a) at the absence of the field and b) at a non-zero positive field

For that model, ramping of the field from 0 to hmax and then two cycles in the range ±hmax were simulated. In

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result, a number of different variants for the magnetization vs. field dependencies were obtained. They are: 1. Switching of the magnetic moments never occurs. Such a scenario might take place in a very soft matrix or when hmax is rather low, see Fig. 9a; consequently, no hysteresis is encountered. 2. Neither of the grain switches under initial magnetization, however, when the field is ramped from hmax to −hmax

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the first grain switches. After that no switching occurs, no matter how many times the field is cycled; this situation is illustrated in Fig. 9b. As seen, the field hmax = 0.65 did suffice for switching of one grain; after that the particle pair attained a magnetic configuration whose magnetization is a single-valued function of the field at all the further cycles. The turning of the first loop into a non-hysteretic curve can be considered as a very first (“toy”) example of the training effect in a multigrain particle 3. Neither of the grains switches in the course of initial magnetization. Only one grain switches each time when the field changes between ±hmax ; Fig. 9c illustrates the case. There is no difference between consecutive hysteresis loops, but their negative bias, caused by a non-hysteretic magnetization of the second grain, is well visible. 4. If the field is high enough to switch both grains of the microparticle, neither negative bias nor difference between consecutive loops can be obtained, see Fig. 9d. The case corresponds to a standard situation of a full magnetization of a magnetic particle yielding identical symmetric hysteresis loops.

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A microparticle comprised of two grains cannot possess two different magnetization loops, though it is still prone to the training effect: the difference is possible only if one of the curves is single-valued. Review of the following hypothetical cases can be helpful in comprehending the reasons for that. 1. Suppose, neither of the grains switches during initial magnetization, but as the field is cycled from hmax to 11

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1 0.5

-0.2

0.2 -0.5

0.4

0.6

-0.8

0.8

Initial magnetization First loop Second loop

0.4

0.6

0.8

-1.5

(c) Magnetization at hmax = 0.79 and κ = 0.35

-1

-0.5

0.5

-0.5

Initial magnetization First loop Second loop

-1

1

1.5

Initial magnetization First loop Second loop

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-1

0.6

0.5

0.2 -0.5

0.4

1

0.5

-0.2

0.2

(b) Magnetization at hmax = 0.65 and κ = 0.35

1

-0.4

-0.2 -0.5

(a) Magnetization at hmax = 0.65 and κ = 0.10

-0.6

-0.4

Initial magnetization First loop Second loop

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-0.6

0.5

(d) Magnetization at hmax = 1.50 and κ = 0.35

Figure 9: Magnetization loops of a double-grain particle with θ10 = −60◦ , θ20 = 45◦ at κ = 0.4

−hmax and back, one of the grains switches. Then, the second loop may differ from the first one only if the

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former is generated by jumps in magnetization of the second or both the grains, but it is impossible if neither the orientation of the easy axes nor the value of hmax is changed. 2. The loops may differ from each other if their branches are formed by switching of different grains. Let us assume that again neither of the grains switches during initial magnetization. Next, as the field goes from hmax to negative −hmax the first grain switches; and as the field goes back to hmax , the second grain switches, so that at h = 0 after one cycle, the magnetic moments of both the grains have jumped between the directions of the corresponding easy axes once, and the current state of the particle is a mirror reflection of its initial state. It means, that at such a mirror situation, the first grain also had to switch during the last ramping of the field from 0 to positive hmax (part of the ascending branch of the first loop). Consequently, if it happens, the situation corresponds to the case 4 from the previous list and both of the grains must switch at every ramping of the field from 0 to ±hmax ; as for the whole particle, it would be fully magnetized without being affected by the training effect.

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This case covers all the other hypothetical situations when one of the grains switches only during ramping the field to hmax (−hmax ) and the second grain switches right after that during ramping the field to −hmax (hmax ), in other words, if one grain switches only when the field is applied in positive direction and the other grain switches only in negative fields. Such situations are impossible in the developed model. 12

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Any other case can be reduced to one presented on these lists. 4.3. Triple-grain microparticle

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The simplest system, whose first consecutive hysteresis loops might differ from each other, is a particle containing three grains. As in previous cases, here the shape of magneization curve strongly depends on initial orientations of the grain easy axes ni , hence, one needs to introduce some assumptions about their mutual orientation. From the technical

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specification it follows that MQP-S particles are structurally and magnetically isotropic, so one has to assume that the grain easy axes directions are distributed at random inside the particles. To comply with that assumption in a triplegrain particle, we put the easy axes and initial magnetic moments in the same plane (this is simplification) at 120 degrees with respect to each other, thus making the initial state of the particle to have zero net magnetic moment. Preventing computational errors, we deviate the anisotropy axes from the coordinate axes; as an example, we set

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θ10 = 5◦ , θ20 = 125◦ and θ30 = −115◦ , see Fig. 10b. The initial magnetization curve and two first hysteresis curves of such a system are presented in Fig. 10a, where the numbers denote different stages of the magnetization process. Additionally, in Figs. 10b–10n, the orientation of the microparticle in space and the directions of its grain magnetic moments are shown in details for each of the stages.

During initial magnetization, the grain magnetic moments deviate from the respective easy axes striving to align

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with the direction of the field, see Fig. 10c. At point 3, it becomes energetically more favorable for the second and third grains to switch their magnetic moments (the arrows transform from blue to red). At this part of the cycle and up to the maximal field, the particle just barely tilts with respect to the field direction, see Fig. 10e). After removal of the field (point 5), the particle returns to its initial position possessing high remnant net magnetic moment µp . Due to it, when the field is applied in negative direction, the particle experiences a significant torque µp × H that forces it to turn. Simultaneously, the grain magnetic moments deviate from the anisotropy axes, see Fig. 10g. As the field evolves in negative direction, the magnetic moment of the second grain finds itself being bound to an unfavorable (red) easy axis direction. That is why, when the field strength grows up to about -0.56 (point 7), the second grain switches. By that, the particle acquires a considerable energy gain that enables it to abruptly change its orientation bringing it much closer to the initial position: reducing the elastic part of its energy. Now the first grain is set under similar energetically unfavorable conditions, and when the field strength increases yet more, µ1 jumps to another direction

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(the arrow transforms from red to blue), see Fig. 10i. Note that under an even stronger field (if available) the third grain also finally switches its magnetic moment. The states of the system at points 4 and 8 differ from each other, cf. Fig. 10e and Fig. 10i; because of that the ascending and descending branches of the first loop are not symmetrical and the whole hysteresis curve is fieldbiased. When the field is ramped from −hmax to point 9, the particle, at first, returns to its initial state at h = 0 and then 13

(d) Stage 3.

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(c) Stage 2.

(g) Stage 6.

(h) Stage 7.

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(k) Stage 10.

(b) Triple-grain particle at the absence of the field. Stage 1.

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(a) Magnetization curves at κ = 0.035 and hmax = 0.80

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(l) Stage 11.

(e) Stage 4.

(f) Stage 5.

(i) Stage 8.

(m) Stage 12.

Figure 10: Magnetization stages of a triple-grain particle with θ10 = 5◦ , θ20 = 125◦ , θ30 = −115◦

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(j) Stage 9.

(n) Stage 13.

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begins to rotate clockwise, thus putting the first grain to energetically unfavorable position. To avoid that, magnetic moment µ1 switches at about h = 0.7, cf. Figs. 10j and 10k. While the field is ramped further to hmax , no other grain switches. By the field reduction to h = 0, the particle returns to its initial orientation, see Fig. 10l. Since the state of

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the microparticle by the moment of revisiting the positive maximal field is not the same as when the field attained hmax value for the first time, the upper branches of the first and second curves do not coincide. When the field is applied in negative direction second time, the only grain that finds itself in unfavorable orientation is the first one (see Fig. 10m).

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As the states of the microparticle at points 8 and 13 are identical, the ascending branches of the first and second loops coincide: the first grain switches back to the red direction of the easy axis at a point that is equivalent to the point 10. Apparently, all the consecutive loops would be the same as the second one: the system has been trained. Thus, one sees that a triple-grain scheme, in principle, can explain the difference between the first two consecutive hysteresis loops of a MAE. The system exhibits a training effect because at points, where the field strength once

follows different paths. 4.4. Multigrain microparticle

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and again comes to maximum, the states of the microparticle are not the same, and as a consequence, function m(h)

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The acquired qualitative knowledge facilitates addressing the magnetization process in a multigrain microparticle that results from cycled application of the field whose maximal strength is below the anisotropy field of the grain material.

The conceivable scenario of the process is as follows.

1. During initial magnetization, the particle turns by angle γ0 , and some number of its grains Γ0 switch their magnetic moments between the “hemispheres” associated with the two directions of the respective anisotropy axes.

2. As the descending branch of the first hysteresis loop is measured (or modeled), the particle tilts by angle γ−1 and another group of grains (Γ−1 ) switch their magnetic moments. 3. When the ascending branch of the first hysteresis loop is measured, the particle tilts by angle γ+1 and the group of grains Γ+1 switches. The states of the particle at the first and second maximal fields can differ from each other in the general case.

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4. As the descending branch of the second hysteresis loop is measured, the particle tilts by γ−2 and the group of grains Γ−2 switches.

5. When the ascending branch of the second hysteresis loop is measured, the particle tilts at γ+2 and grains Γ+2 switch their magnetization. 15

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Figure 11: Dependencies of the normalized coercivity difference δ on the maximal field hmax for different values of the matrix stiffness parameter κ

6. And so on.

In the general case, intersection of any two groups of grains Γi and Γ j (i , j) is not empty, but at the same time these groups do not coincide. Moreover, the declinations of the particle γi from its initial position are new at every

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new cycle. The more times the field is cycled, the more stable response to the field is worked out by the system. In result, the difference between consecutive magnetization loops decreases. Ultimately, after passing through several training loops, the particle under the action of the field tilts by the same angles γ, and the same grain groups Γ switch their magnetization. The number of the loops to be executed in order to train a sample depends on the number of the

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grains inside the microparticles. 5. Results of MAE modelling

5.1. Hysteresis of a custom multigrain particle

As it follows from the foregoing, the magnitude of the training effect in MAE highly depends on the stiffness of a polymeric matrix and on the maximal field applied during hysteresis measurements. A set of simulations was run on particles consisting of 115 grains at various hmax and κ. Pursuing the consistency of discussion with respect to the previous section, the level of intergrain interaction λ was set to zero for now, nevertheless its influence on the system coercivity had been highlighted in [8].

Dependency of the normalized difference in coercivity δ of the first two consecutive hysteresis loops on the maximal applied field hmax for different elastic environments characterized by κ is shown in Fig. 11. As expected, the

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training effect of MAE due to particles undermagnetization is only revealed at hmax less that anisotropy field of the particles h = 1. The effect also comes to naught at stiff matrices. The presented dependencies prove that the difference between consecutive magnetization loops of an MAE can be brought to existence by satisfying two conditions simultaneously: (i) application of low or moderate magnetic fields in comparison with the anisotropy of an MAE filler, and (ii) use of soft matrices that enable rotational mobility of the particles. 16

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Figure 12: Half-width of the second minor magnetization loops for NdFeB microparticle with 115 grains at λ = 1.41 in different matrices at hmax = 0.7.

Under the formulated conditions, the first and second model magnetization loops differ from each other as the measured curves do. The discrepancy between the model and experiment appears when comparing the third and following loops. In the model they coincide exactly with the second one, whereas experimentally the difference, al-

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though minuscule, exists up to the tenth curve. It should be noted, however, that while the model particle contains just about hundred identical grains, real particles consist of much larger number of grains, which are not at all necessarily monodisperse.

By operating conditions under a regularly changing field, one of the most important parameter of an MAE would

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be coercivity of the trained loops. Taking into account the possible negative field-bias of magnetization curves, it is more reasonable to use the half-width of the loops, that is, half the difference between field values, where magnetization equals zero. To reflect the main tendency, the magnetization simulation of one model NdFeB multigrain microparticle containing 115 closely packed grains with λ = 1.41 has been performed at various κ and maximal field value hmax = 0.7. In Fig. 12 the dependence of the half-width hw of the second hysteresis loops on the matrix stiffness κ is presented.

In very soft matrices (κ < 0.021) the embedded microparticle exhibits no hysteresis after measuring the first magnetization curve, since mechanical rotation requires less energy than intrinsic grain switching. The magnetization curve corresponding to this situation is shown in Fig. 13. The initial magnetization endows the microparticle with a magnetic moment that points within the “hemisphere” defined by the positive direction of the field. When a negative field is applied, the particle starts to rotate tending to align its net magnetic moment with the field. The deformed matrix generates a resistance torque; that is why the particle displays positive magnetization at negative fields and its

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hysteresis loops have negative field-bias.

In the matrices with κ > 0.021, magnetization of the microparticle exhibits hysteretic behavior. If the matrix is still soft (κ ∈ [0.021, 0.15]), not all the grains switch their magnetic moments and, consequently, the loop is also shifted to the negative values of the field; we term the direction of the field first applied to the sample as positive. The occurring 17

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Figure 13: Magnetization curves of a microparticle with N=115 in a matrix with κ=0.019.

shift (bias) decreases with the increase in matrix stiffness; model magnetization curves for that case are shown in Fig. 14.

A small difference in magnetization curves at maximal positive field is caused by rearrangements in the magnetic

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structure of the grain assembly at moderate fields. If the applied field exceeds the anisotropy of the grain material, the state extremely close to the magnetic saturation can be reached at hmax and the value of magnetization at these points belonging to different consecutive curves will not differ by definition. If one applies a field less than h = 1, the

maximal fields.

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already discussed training effect occurs, that implies not only change in coercivity, but also in magnetization values at

In soft matrices, a considerable number of the grain magnetic moments up to the end of the positive half-cycle are oriented within the positive “hemisphere”. As soon as the first negative half-cycle begins, the particle starts to rotate tilting this “hemisphere”. As the field comes to its maximal negative value, some more grains switch their magnetic moments to the anisotropy axes directions lying in that “hemisphere”, now strongly tilted. The return of the field to its positive maximum restores the positive orientation of the “hemisphere” that now contains an increased number of magnetic moments, i.e., the net magnetic moment of the particle becomes larger. In other words, in a hard matrix there are always grains that cannot be switched by fields h < 1 no matter how many cycles of hysteresis measurements we perform. At zero field, these grains constantly return to their initial location defined at the beginning of experiment of simulation. The situation changes if a particle is situated in a soft matrix and can rotate inside it. The angles between easy axes of the grains and the field change also, thus altering the switching field of the grains. Therefore, the grains, which in a hard matrix would not be involved in the switching

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process, now one by one change their host hemisphere and increase the value of magnetization at maximal field at late hysteresis loops.

In yet more stiff systems with κ > 0.15, the matrix elasticity but slightly affects the microparticle coercivity, and so the discrepancy between the first magnetization loops is hardly observable. This is also supported by the experiment: 18

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Figure 14: Magnetization curves of a MQP-S microparticle with N=115 in a matrix with κ=0.025.

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Figure 15: Model magnetization curves at κ = 0.078 corresponding to the measured curves of sample s3 in Fig. 1

coercivity of the MAE sample S1 (G1 = 319, G2 = 412 kPa) is quite close to that of the powder in epoxy (750 kA/m). 5.2. MAE with MQP-S-11-9 powder

To make quantitative comparison with experiment, several calculations have been carried out for a set of particles consisting of 206 grains and assigned various magnetic properties. The resulting magnetization curves are shown in Fig. 15, they correspond to the dependence presented in Fig. 1. All the particles under simulation possessed the same saturation magnetization Ms = 1.10 MA/m, but the value of anisotropy constant K was selected at random from the normal distribution with the mean value 1.08 MJ/m3 and standard deviation 0.23 MJ/m3 to make the computer simulation more realistic.

Results of the calculation for H0max ≈ 1.2 MA/m are presented in Table 2. Each sample has been simulated by an ensemble of 100 particles. The model is in the order-of-magnitude agreement with the experiment in estimating the

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∗ coercivity H2c of the trained second loop. The relative difference δ of negative coercivities for the first and second

loops does not show good agreement, however. A possible reason for that, above all, is the unaccounted irreversible mechanical processes inside the matrix and at the particle-elastomer interfaces.

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Table 2: Comparison of the measured values of coercivity with the modelled data.

H2c , kA/m 657 207 92 57 48

∗ H2c , kA/m 601 586 256 69.4 10.2

δ 0.02 1.44 1.36 1.81 1.16

δ∗ 0.01 0.02 0.31 0.38 0.26

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6. Conclusions

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Sample s1 s2 s3 s4 s5

In this work, appearance of the training effect in a magnetoactive elastomer (MAE) composite with magnetically hard filler is studied by means of experiment and computer simulations. Magnetic measurements on MAEs were preceded by investigation of the intrinsic magnetic properties of MQP-S-11-9 powder, using epoxied samples. In computer simulations, the microparticles of the powder were presented as complex entities consisting of a large

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number of single domain nanograins that magnetize according to the Stoner-Wohlfarth scheme. The magnetization curves of the considered highly coercive multigrain spherical microparticles were obtained via energy minimization (i.e., in quasi-static regime) under condition that the maximal strength of the applied field is lower than the anisotropy field of the grain material.

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The obtained simulation results enable one to explain appearance of significant difference between first consecutive magnetization loops of MAEs consistently. The decrease in coercivity with the growth of matrix compliance is attributed to enhanced mobility of the microparticles inside the elastomer. The model hysteresis loops are in full qualitative agreement with the experiment, although the quantitative differences are significant. This points out the necessity of further work on the problem; in particular, the effect of irreversible processes in the elastomer matrix ought to be accounted for. Notably, the developed model is readily extendable for multigrain particles of arbitrary shapes and of diverse grain materials.

Acknowledgement. Development of the theoretical models was funded by RFBR according to the research project 18-32-00817. Computer simulations and experimental measurements were accomplished with the financial support by RFBR-DFG projects 19-52-12045, Bo 3343/2-1 and Od 18/24-1 within SPP1681 and PAK907. M.V. is additionally grateful to the DAAD for their generous support during the research. [1] G. V. Stepanov, D. Yu. Borin, A. V. Bakhtiiarov, P. A. Storozhenko, Magnetic properties of hybrid elastomers with magnetically hard fillers:

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rotation of particles, Smart Materials and Structures 26 (2017) 035060.

[2] M. Sch¨umann, D. Yu. Borin, S. Huang, G. K. Auernhammer, R. M¨uller, S. Odenbach, A characterisation of the magnetically induced movement of ndfeb-particles in magnetorheological elastomers, Smart Materials and Structures 26 (2017) 095018. [3] P. Andriushchenko, K. Nefedev, G. Stepanov, Calculations of magnetoactive elastomer reactions in a uniform external magnetic field, The European Physical Journal B 87. doi:10.1140/epjb/e2013-31097-1.

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[4] J. J. Croat, Rapidly Solidified Neodymium-Iron-Boron Permanent Magnets, Elsevier, 2018. [5] J. Lucas, P. Lucas, T. Le Mercier, A. Rollat, W. G. Davenport, Rare Earths: Science, Technology, Production and Use, Elsevier, 2015. [6] J. M. Linke, D. Yu. Borin, S. Odenbach, First-order reversal curve analysis of magnetoactive elastomers, RSC Advances 6 (2016) 100407– 100416.

a hard magnetic filler, Journal of Magnetism and Magnetic Materials 431 (2017) 138–140.

of

[7] G. V. Stepanov, D. Yu. Borin, P. A. Storozhenko, Rotation of magnetic particles inside the polymer matrix of magnetoactive elastomers with

[8] M. V. Vaganov, D. Yu. Borin, S. Odenbach, Yu. L. Raikher, Modeling magnetomechanical behavior of a multigrain magnetic particle in an

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elastic environment, Soft Matter 15 (2019) 4947–4960.

[9] G. V. Stepanov, D. Yu. Borin, E. Yu. Kramarenko, V. V. Bogdanov, D. A. Semerenko, , P. A. Storozhenko, Magnetoactive elastomer based on magnetically hard filler: Synthesis and study of viscoelastic and damping properties, Polymer Science, Ser. A 56 (5) (2014) 603–613. [10] D. Yu. Borin, G. V. Stepanov, E. Dohmen, Hybrid magnetoactive elastomer with a soft matrix and mixed powder, Archive of Applied Mechanics (2018) doi:10.1007/s00419–018–1456–9 – to appear.

[11] D. Yu. Borin, N. Kolsch, G. V. Stepanov, S. Odenbach, On the oscillating shear rheometry of magnetorheological elastomers, Rheologica Acta 57 (3) (2018) 217–227.

re-

[12] G. Kraus, Reinforcement of elastomers, Ed. Interscience, New York, 1965.

[13] J. A. Manson, L. H. Sperling, Polymer Blends and Composites, Plenum Press, 1976.

[14] H. M. Smallwood, Limiting law of the reinforcement of rubber, Journal of Applied Physics 15 (1944) 758. [15] E. Guth, Theory of filler reinforcement, Journal of Applied Physics 16 (1945) 20.

[16] E. A. P´erigo, B. Weidenfeller, P. Koll´ar, J. F¨uzer, Past, present, and future of soft magnetic composites, Applied Physics Reviews 5 (2018)

urn al P

031301.

[17] Magnequench, MQP-S-11-9-20001-070 isotropic powder, material description (2009). [18] R. K. Mishra, Microstructure of melt-spun Nd-Fe-B magnequench magnets, Journal of Magnetism and Magnetic Materials 54-57 (1) (1986) 450–456.

[19] A. P. Guimar˜aes, Principles of Nanomagnetism, Springer-Verlag, Berlin-Heidelberg, 2009. [20] W. F. Brown, Micromagnetics, J. Wiley, New York–London, 1963.

[21] Y. Murakami, T. Tanigaki, T. T. Sasaki, Y. Takeno, H. S. Park, T. Matsuda, T. Ohkubo, K. Hono, D. Shindo, Magnetism of ultrathin intergranular boundary regions in Nd-Fe-B permanent magnets, Acta Materialia 71 (2014) 370–379. [22] J. Liu, H. Sepehri-Amin, T. Ohkubo, K. Hioki, A. Hattori, K. Hono, Microstructure evolution of hot-deformed Nd-Fe-B anisotropic magnets, Journal of Applied Physics 115 (2014) 17A744.

[23] E. C. Stoner, E. P. Wohlfarth, A mechanism of magnetic hysteresis in heterogeneous alloys, Philosphical Transactions of Royal Society (London) 240 (1948) 599–642.

[24] J. Ohayon, P. Tracqui, Computation of adherent cell elasticity for critical cell-bead geometry in magnetic twisting experiments, Annals of Biomedical Engineering 33 (2) (2005) 131–141.

[25] C. G. Broyden, The convergence of a class of double-fank minimization algorithms, Journal of the Institute of Mathematics and its Applica-

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tions 6 (1970) 76–90.

[26] R. Fletcher, A new approach to variable metric algorithms, Computer Journal 13 (1970) 317–322. [27] D. Goldfarb, A family of variable metric updates derived by variational means, Mathematics of Computing 24 (1970) 23–26. [28] D. F. Shanno, Conditioning of quasi-Newton methods for function minimization, Mathematics of Computing 24 (1970) 647–656. [29] P. Bender, A. G¨unter, A. Tsch¨ope, R. Birringer, Synthesis and characterization of uniaxial ferrogels with Ni nanorods as magnetic phase,

21

Journal Pre-proof

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urn al P

re-

pro

of

Journal of Magnetism and Magnetic Materials 323 (2011) 2055–2063.

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*Conflict of Interest form

Declaration of interests

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☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: