Magnetodielectric effect in magnetoactive elastomers: Transient response and hysteresis

Polymer 127 (2017) 119e128

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Magnetodielectric effect in magnetoactive elastomers: Transient response and hysteresis Inna A. Belyaeva a, Elena Yu. Kramarenko b, c, Mikhail Shamonin a, * a

East Bavarian Centre for Intelligent Materials (EBACIM), Ostbayerische Technische Hochschule Regensburg, Seybothstr. 2, D-93053 Regensburg, Germany Faculty of Physics, Lomonosov Moscow State University, Moscow 119991, Russia c A.N. Nesmeyanov Institute of Organoelement Compounds of Russian Academy of Sciences, Moscow 119991, Russia b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 28 July 2017 Received in revised form 19 August 2017 Accepted 26 August 2017 Available online 28 August 2017

Magnetodielectric properties of magnetoactive elastomers comprising micrometer-sized iron particles dispersed in compliant elastomer matrices are experimentally studied in stepwise time-varying dc magnetic fields. It is found that imposition of magnetic field significantly increases both the effective lossless permittivity of these composite materials as well as their effective conductivity. These magnetodielectric effects are more pronounced for larger concentrations of soft-magnetic filler particles and softer elastomer matrices. The largest observed relative change of the effective dielectric constant in the maximum magnetic field of 0.57 T is of the order of 1000%. The largest observed absolute change of the loss tangent is approximately 0.8. The transient response of the magnetodielectric effect to a step magnetic-field excitation can be rather complex. It changes from a simple monotonic growth with time for small magnetic-field steps (<0.1 T) to a non-monotonic behavior with a significant rapidly appearing overshoot for large magnetic-field steps (>0.3 T). The settling time to the magnetic-field step excitation can reach roughly 1000 s and it depends on the applied magnetic field and sample composition. There is also significant hysteresis of the magnetodielectric effect on the externally applied magnetic field. These findings are attributed to the rearrangement of ferromagnetic filler particles in external magnetic fields. The results will be useful for understanding and predicting the transient behavior of magnetoactive elastomers in applications where the control magnetic field is time dependent. © 2017 Elsevier Ltd. All rights reserved.

Keywords: Magnetoactive elastomer Magnetodielectric effect Smart material Magnetorhelogical elastomer Transient response Hysteresis

1. Introduction Magnetoactive elastomers (MAEs) can be classified as magnetic composite materials, i.e. polymer matrices filled with magnetic micro-particles [1]. The growing interest in MAEs is determined by the possibility of controlling their mechanical and other physical properties (e.g. dielectric permittivity) by application of technically easy realizable magnetic fields [2e6]. The physical reason for observed changes of MAE properties are field-induced reconfigurations of the microstructure formed by the magnetic particles [1,2,7e9]. These rearrangements of filling particles in the externally applied magnetic field have been observed directly by optical [10] and X-Ray methods [11]. This restructuring of the filler can also be observed indirectly in the so-called flocculation experiment, where an external stimulus is applied as a stepwise excitation, and

* Corresponding author. E-mail address: [email protected] (M. Shamonin). http://dx.doi.org/10.1016/j.polymer.2017.08.056 0032-3861/© 2017 Elsevier Ltd. All rights reserved.

the transient behavior of physical properties (usually the shear storage modulus) is measured [12]. Time-resolved rheology is also used for studying polymer degradation in complex systems such as polymer nanocomposites, whose rheological response stems from the combination of polymer and nanoparticle contributions [13,14]. In general, polymer nanocomposites represent a promising alternative to conventional dielectric materials for fabrication of electronic devices [15e17], complemented by a specific sustainability feature when both the polymer matrix and the filler are obtained from sustainable sources [16,17]. An et al. [18] investigated in detail the long time response of soft magnetorheological (MR) gels. The resulting transient rheological response indirectly characterizes the microstructural evolution of the filler and predicts the time response of the MR device, where a MR material should be employed. It has been shown that the transient MR response of MR gels is clearly different from the response of MR fluids: two time constants were necessary for MR gels instead of one time constant for MR fluids. Very recently it has been shown that the transient MR response of MAEs can be even

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more complicated [19]: three time constants, different approximately by one order of magnitude, came into play. Square-wave (i.e., periodically on-off switching) magnetic fields have been used for characterizing the degradation of the MR response in ageing magnetic carrageenan gels and magnetic polyurethane elastomers [20]. Using the same experimental technique it has been found that the time for reaching the saturation of the shear storage modulus for magnetic elastomers comprising both magnetic and non-magnetic particles is longer than that for elastomers containing only magnetic particles [21]. The transient MR response of such bimodal MAEs to a step magnetic field excitation has been also investigated and it has been found that the time profiles of the storage modulus can be fitted by a linear combination of two exponential functions with two characteristic times showing the alignment of magnetic particles [22]. Xu et al. systematically investigated the creep and recovery behaviors of the so-called MR plastomers (a class of solid-like gels) in order to understand their deformation mechanism under constant stress [23]. Analysis of the transient response with the suddenly imposed magnetic field of 930 mT has led to the conclusion that the small strain amplitude of 0.1% is unlikely to destroy the microstructure of MR gels during oscillator shear tests [24]. The magnetically induced flocculation of filler particles in magnetic elastomer composites comprising two different magnetic filler particles at fixed overall concentration have been successfully studied by square-wave magnetic fields [25]. Obviously, the transient rearrangement of filler particles should also be observable in other physical properties related to the microstructure of the filler. It is well known that dielectric and conductive properties of MAEs depend on external magnetic fields [26e29]. Very recently Wang et al. [30] proposed that real-time monitoring of MAE impedance can be used as a method for detecting the fatigue of MAE elements in devices relying on their MR properties. However, only scarce information on transient behavior of magnetodielectric properties in MAEs can be found in the literature [31]. In the past, the time dependence of piezoresistance in conductor-filled polymer composites has been intensively investigated (see, e.g. Ref. [32] and references cited therein). The application of stress led to a change in the inter-particle separation in these composites. The piezoresistance varies with time during the test because the inter-particle separation is also a function of time, what can be explained by the creep of polymer [32]. In the present paper, the emphasis is on the magnetodielectric effect. Magnetodielectric effect (MDE) is commonly defined as the variation of the dielectric permittivity due to externally applied magnetic fields [33]. Recently, the interest to obtaining materials with a large MDE is greatly increased due to their potential applications in novel devices such as tunable filters, four-state memories, magnetic sensors, and spin-charge transducers [34]. Conventional strategies used to enhance the MDE include control of the interphase physical effects in multilayer systems and amplification of MDE near to (ferro-) magnetic transitions in (ferro)magnetic dielectrics [33]. The effect is usually very weak at room temperature (approximately 4% in the external magnetic field of 500 mT [33]) and it is much larger at low temperatures (below 80 K), where the MDE exceeding 100% is designated as the “colossal effect” [34]. In Ref. [27] an MDE of approximately 150% has been observed in MAEs filled with hard-magnetic FeNdB particles. It will be demonstrated below that the MDE up to approximately 1500% at room temperature is achievable in MAEs, where the highly compliant elastomer matrix is filled with soft-magnetic microparticles. However, the origin of the MDE in investigated composite materials is different from that observed in conventional polymer based composites. It is caused by the rearrangement of magnetic

filler particles in external magnetic fields. The highly compliant polymer matrix, where the particles are embedded allows such rearrangements of the filler [35]. The purpose of this paper is first to investigate the transient behavior and the hysteresis of the magneto-dielectric effect in MAE samples. Investigations of the transient behavior of rheological properties may provide complimentary information about the processes of formationedestructionereformation of the internal filler structure under the simultaneously applied mechanical force and magnetic field [19]. Presented investigations of MDE in MAEs allow one to exclude the influence of mechanical deformations on structuring of the magnetic filler. In this sense, they are similar to investigations of effective magnetic properties of MAEs. In particular, the measurements of magnetodielectric properties can be easily performed with much better time resolution Dt than MR measurements (where Dt ~ 101 s). A significant magnetic hysteresis has been previously reported in MAEs where soft-magnetic inclusions with negligible hysteresis loops were embedded into mechanically soft elastomer matrices (cf. Fig. 7 of Ref. [35]). A simple explanation of such an effect has been offered in Ref. [36]. Hysteresis behavior of the magnetostriction of MAE samples on the external magnetic field has been observed in Refs. [37,38]. Therefore, it can be expected that there will be hysteresis of magnetodielectric properties as well. Furthermore, we demonstrate the largest, to the best of our knowledge, MDE in polymer based composite materials at room temperature. 2. Experimental 2.1. Materials The base polymer VS 100000 (vinyl-functional polydimethylsiloxane) for addition-curing silicones, the chain extender modifier 715 (SiH-terminated polydimethylsiloxane), the reactive diluent polymer MV 2000 (monovinyl functional polydimethylsiloxane), the crosslinker 210 (dimethylsiloxane-methyl hydrogen siloxane copolymer), the Pt-Catalyst 510 and the Inhibitor DVS were provided by Evonik Hanse GmbH, Geesthacht, Germany. The silicone oil WACKER® AK 10 (linear, non-reactive polydimethylsiloxane) was purchased from Wacker Chemie AG, Burghausen, Germany. The carbonyl iron powder (CIP) type SQ (mean particle size of 4.5 mm), provided by BASF SE Carbonyl Iron Powder & Metal Systems, (Ludwigshafen, Germany) was used as the ferromagnetic filler. These particles are soft-magnetic. The fabrication of the polydimethilsiloxane (PDMS) samples was performed accordingly to the known recommendations [39]. The polymer VS 100000, the polymer MV 2000, the modifier 715 and the silicone oil AK 10 were put together and blended with an electric mixer (Roti®-Speed-stirrer, Carl Roth GmbH, Germany) to form an initial compound. The mixture consists of 14 g VS 100000 combined with 2.5 g of MV 2000, 0.05 g of Modifier 715 and 33 g of AK 10. In the next step, the initial compound (6 g) was mixed with CIP (60%, 70% or 80% by mass) and crosslinker 210 (0.03 g). These concentrations of CIP particles correspond approximately to 16%, 23% and 34% by volume. The crosslinking reaction was activated by the Pt-Catalyst 510 (0.02 g). For the activity control of the Ptcatalyst, the inhibitor DVS was employed. Accordingly to the literature, the recommended quantity of the inhibitor is between 0.01 und 1% [40e42]. The necessary amount of the inhibitor for this MAE composition is 0.01 g. The Petri dishes (35 mm high, Greiner Bio-One GmbH, Germany) were filled with the finished, but uncured MAE (the thickness of the samples is about 1 mm). The air bubbles in the MAE samples were removed using a vacuum desiccator for about 10 min.

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Finally, the MAE samples were cured in the universal oven Memmert UF30 (Memmert GmbH, Schwabach, Germany) at 80
C for 1 h and then cured at 60
C 24 h with air circulation. The resulting disc-shaped specimens with a diameter of 20 mm were mechanically cut out of the obtained MAE layer. Such a MAE layer sandwiched between two thin copper plates formed the parallel plate capacitor. The distance between the electrodes was maintained constant during the measurements. The chemical composition described above was further modified in order to fabricate additional samples with the same concentration of iron particles of 70% but different storage moduli of the composite material. Such an alteration can be easily achieved by varying the proportions of the crosslinker and the silicone oil in the elastomer mixture. The MAE samples and the elastomer matrices were characterized mechanically in the absence of magnetic field. Table 1 lists the synthesized samples and their storage modulus G00 without external magnetic field. The shear storage modulus of the softer MAE sample (denoted MAE70_Soft) is an order of magnitude smaller than the one of the reference MAE70 sample, while the modulus of the harder sample (denoted MAE70_Hard) is approximately 2 times larger than that of the reference sample. One can expect that the matrix rigidity will affect the filler restructuring processes in magnetic field and, thus, the magnitude of MDE. The number in the notation of the samples stands for the corresponding mass concentration of iron particles. It is seen that the modulus of the samples slightly increases with the filling degree, however, the rigidities of the samples are rather similar. Rheological measurements were made using a commercially available rheometer (Anton Paar, model Physica MCR 301) the magnetic cell MRD 170/1 T. The angular oscillation frequency u was maintained constant at 10 s1. The rheological characterization of the materials followed the methodology described in Ref. [43]. To avoid slippage the normal force of approximately 1 N was applied. The moduli were measured at constant strain amplitude g ¼ 0.01% which corresponds to the linear viscoelastic regime. For each sample, ten consecutive measurements of the moduli were performed (20 s per point). These average values of G00 are presented in Table 1.

2.2. Measurements Fig. 1 shows schematically the experimental setup for measuring the dielectric properties of synthesized MAE samples. The samples have been placed between the pole shoes of the electromagnet EM2 (MAGMESS Magnet-Messtechnik Jürgen Ballanyi e.K., Bochum, Germany). The diameter of the pole shoe (10 cm) is much larger than the diameter of the sample, therefore, the external magnetic field is homogeneous over the entire sample surface. The distance between the poles was about 35 mm, where the total height of the sample (capacitor) holder was about 31 mm. The flat capacitor consists of copper plates with diameter of 20.0 ± 0.1 mm and the fixed distance of 1.00 ± 0.01 mm allowed for Table 1 Average values of the storage moduli of MAE and pure elastomer samples in zero magnetic field. Sample

G00 [kPa]

MAE60 MAE70_Soft MAE70 MAE70_Hard MAE80 Elastomer Soft elastomer Hard elastomer

12.27 1.93 24.87 49.96 29.16 5.51 1.08 7.29

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the test sample between conducting plates. The magnetic field was directed perpendicular to the capacitor's plane. The magnetic field was measured using the Lakeshore gaussmeter455 DSP with the Hall sensor head HMMT-6J04-VR (Lake Shore Cryotronics, Inc., Westerville, OH). The drive current I was generated by the power supply EA-PSI 8160-60 2U (EA Elektro-Automatik, Viersen, Germany). The time constants for on/off switching of the current I were determined experimentally. The corresponding time dependences of the current I were well approximated by exponential functions. The time constant ton for switching the current on was equal to 0.22 s, and the time constant toff for switching the current off was measured to be 0.64 s. The corresponding rise time between 10% and 90% of the step height was tr;on z2; 197ton ¼ 0:48 s and the similarly defined fall time was tf ;off z1:41 s. These characteristic times are experimental limits for determining the time constants due to step magnetic field excitations. The steady-state excitation current I of 0, 1.5, 4 or 9 A corresponded to the magnetic flux density of 0.0, 0.10, 0.27 or 0.57 T, respectively. The LCR Meter Hioki IM 3533-01 (supplied by ASM GmbH, Moosinning, Germany) was employed to measure the parameters of the capacitor. The measurements of the ac capacitance and the loss tangent (dissipation factor) tand were performed at frequency f ¼ 1 kHz with the ac voltage amplitude of 1.414 V (rms voltage of 1 V). The nominal measurement time for a single measurement was 20 ms. The device settings, the input of the measurement parameters and the automated data acquisition were performed using the LabVIEW® software. The relative lossless effective permittivity εr (referred to simply as the dielectric constant [44]) was calculated using the reference value for an air-filled (empty) flat capacitor with the same distance between the conducting plates. In the discrete-circuit perspective (lumped element model of a capacitor), the loss tangent tan d ¼ XR , where R is the resistance of the sample and X is its reactance. 3. Results and discussion Following the methodology presented in Ref. [19] we shall first investigate the response to the step magnetic-field excitation and then consider the response to repeated stimuli in the form a pyramid excitation and periodic on-off switching of magnetic field. In the pyramid excitation, the external magnetic field was first stepwise increased and then decreased in the staircase manner. 3.1. Step excitation Fig. 2 presents the time dependences of the dielectric constant εr and the loss factor tand. The magnetic field was first absent, then switched on for 30 min and finally switched off. The measurements in each time interval, when the magnetic field was switched off, lasted 10 min. Fig. 2 (e) and (f) show for comparison the results for the unfilled elastomer (matrix), where it can be observed that there is no influence of magnetic field on dielectric properties of a nonmagnetic material. The curves for specific samples have been shown with the same color through the entire paper. For example, the green color has been assigned to the MAE80-sample, while the MAE70-sample is associated with the red color. In Figs. 2e5, the time interval between measurement points is 1 s. Table 2 presents the dielectric properties of synthesized samples in the absence of magnetic field. It is seen that at zero magnetic field all fabricated materials are good dielectrics (tand0 z 0.001e0.02), tand0 of the elastomer matrix is so small that it is of comparable with the uncertainty of measurement. This explains the “noisy” behavior of tand in Fig. 2 b. However, these fluctuations of tand are not significant at large magnetic fields B  0.27 T (Fig. 2 d, f).

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Fig. 1. Schematic diagram of the experimental setup.

Fig. 2. Time dependences of the effective permittivity εr (a,c,e) and the loss tangent tand (b,d,f) of the samples MAE60, MAE70 and MAE80 for different step magnetic-field excitations with B ¼ 0.1 T (a,b), 0.27 T (c,d) and 0.57 T (e,f).

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The values of εr for elastomer matrices (unfilled elastomers) agree well with the known data at frequencies below 1 MHz [44,45]. According to Ref. [44] the lossless permittivity of PDMS is nearly constant in this frequency range. Large increase of εr and tand has been observed in a magnetic field. The gain of εr and tand is bigger for larger magnetic fields. The absolute value of the dielectric constant grows with the filler concentration both in the absence and the presence of magnetic fields (cf. Fig. 2aec). It is well known that εr increases with the growing concentration of conducting particles [46,47]. This can be explained by the effective medium or percolation theories [48]. The growth of εr upon application of a magnetic field can be related to the rearrangement of magnetizable filler particles along the magnetic field lines [27,49]. In Ref. [27] a simple model elucidating the effect of spatial distribution of conducting particles within a dielectric medium has been proposed. This model has qualitatively explained the role of filler

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particle re-organization under the action of a magnetic field. Numerical modeling in Ref. [49] has demonstrated that formation of chainlike structures could enhance the bulk permittivity by 85% in comparison with that for a composite with randomly distributed filler particles. In our material we observe a much higher increase of permittivity reaching 1000% in the field of 600 mT. This huge difference in the absolute values of magnetodielectric effect can be explained by the model limitations. In Ref. [49], calculations were performed for materials based on magnetic particles which are an order of magnitude smaller in size, the filling degrees did not exceed 18 vol%. The simplified model presented in Ref. [49] assumes formation of ideal chains oriented perpendicular to capacitor electrodes. This model is idealized and seems to be not applicable to the cases with high concentration of particles, where large displacements of particles are restricted by polymer elasticity but particle aggregates can be formed and rearranged in external magnetic fields. Nevertheless,

Fig. 3. Time dependences of the dielectric constant εr (a,c,e) and the loss tangent tand (b,d,f) of the samples containing 70 mass% of magnetic filler but different elastomer matrices for various step magnetic-field excitations with B ¼ 0.1 T (a,b), 0.27 T (c,d) and 0.57 T (e,f).

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Fig. 4. (a) Time dependence of the effective relative permittivity εr for the samples with 60, 70 and 80 mass% of magnetic filler during the magnetic-field pyramid excitation. (b) Corresponding time dependence of magnetic flux density B.

Fig. 5. Time dependence of the dielectric permittivity εr for the samples with 60, 70 and 80 mass% of magnetic filler during the periodic on/off-switching of the magnetic field with different amplitudes: B ¼ 0.1 T (a), 0.27 T (b), 0.57 T (c). The corresponding time dependences of the magnetic flux density (d).

the physical reason for growth of εr is believed to come from restructuring of the filler. It is assumed that the loss tangent of composite dielectric ma00 ε’’ terials can be expressed as tan d ¼ εrr , where εr is the imaginary part Table 2 The effective dielectric permittivity εr0 and the loss tangent tand0 of the synthesized samples in the absence of magnetic field. Sample

εr0

tand0

MAE60 MAE70_Soft MAE70 MAE70_Hard MAE80 Elastomer Elastomer_Soft Elastomer_Hard

5.0 5.8 7.0 6.8 10.4 2.7 2.8 2.6

0.005 0.002 0.010 0.022 0.006 0.001 0.002 0.001

00

of the complex dielectric constant [44]. εr represents the losses due to both conduction and polarization phenomena. The effective conductivity se depends on the imaginary part of the dielectric 00 constant as se ¼ 2pf ε0 εr , where ε0 is the vacuum permittivity [44]. An increase of tand means that se (which is proportional to the ac conductance of a sample) becomes more significant in comparison to the effective dielectric constant. For example, in the maximum

Table 3 Settling time TS for different MAE samples and various magnetic fields. Sample

TS (0.10 T)[s]

TS (0.27 T)[s]

TS (0.57 T)[s]

MAE60 MAE70_Soft MAE70 MAE70_Hard MAE80

581 158 464 910 633

21 31 280 489 544

266 1085 927 400 1436

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magnetic field of 0.57 T, the effective dielectric constant of all samples in Fig. 2 grows approximately threefold in comparison to zero magnetic field, while the gain of tand is roughly 50. The corresponding increase of the effective conductivity is approximately 150-fold. The magnetic-field-induced restructuring of the filler particles has more influence on the effective conductivity than on the effective permittivity. It is also seen that in the maximum magnetic field tand is the largest for the MAE70 sample, it is intermediate for the MAE80 sample and it is the smallest for the MAE60 sample. However, it can be estimated that in the maximum magnetic field the effective conductivity of the MAE80 sample is larger than the conductivity of the MAE70 sample. Both εr and se grew with the increasing magnetic flux density, but the corresponding rates of their growth were different for different filler concentrations. This led to the decrease of tand in the MAE80 sample in comparison to the MAE70 sample. Remarkably, after the magnetic field has been switched off, tand declines as a monotonous function of the filler concentration. Recall that the samples shown in Fig. 2 have the same elastomer matrix but different concentrations of the filler. Fig. 3 presents the results similar to those depicted in Fig. 2 but for the samples with the same filler concentration and different elastomer matrices. It is seen that in the presence of magnetic field both εr and tand increase with the decreasing storage modulus of the elastomer matrix. Obviously, a softer matrix promotes more restructuring of the magnetizable filler particles. Similar conclusion has also been achieved from the analysis of the MR effect and magnetization curves in MAEs with the same filler but different storage moduli of the polymer matrix [50e52]. Notice that the time dependences of εr and tand in Figs. 2 and 3 are qualitatively similar. The MAE70_Soft-sample exhibits the highest MDE effect (cf. Fig. 3 e). The relative change of the effective relative permittivity εr reaches approximately 1500% at the overshoot and it remains around 900% in the steady state. To the best of the author's knowledge, this is the largest MDE observed in polymer based composite materials at room temperature. The corresponding maximum change of the loss tangent is approximately 0.8 at the overshoot and it is roughly 0.7 in the steady state. In a small magnetic field (B ¼ 0.1 T), the transient response of εr to a magnetic-field step excitation can be described by a series of exponential functions. Such a behavior in small magnetic fields has been previously reported by Bica et al. [26]. However, in the highest magnetic field the behavior is clearly different. An overshoot appears shortly after the magnetic field is switched on. After the maximum of εr is reached at overshoot, the value of εr does not necessarily decline monotonically to its final (steady state) value, but, after a local minimum is reached, some slight growth can be observed. For example, in Fig. 3 (e) (largest external magnetic field) there exists a local minimum of εr in the time dependences. This minimum occurs at time moment t z 1000e1400 s. The corresponding value of εr is smaller than the value of εr at the end of measurements in the magnetic field (t ¼ 2400 s). Such a complex behavior cannot be easily described by a series of exponential functions of time. The switching time of a mechanical device from the off-state to the activated state is an important property in practical use [53]. Therefore, we resort to the characterization of the transient response by the settling time TS (see Table 3). The settling time TS is the time elapsed from the application of an ideal instantaneous step input to the time at which the output has entered and remained within a specified error band. In our case, this is time when the change of εr, Dεr(t) ¼ εr(t ¼ 0) - εr(t ¼ 2400 s), remains within the specified band Dεr(t ¼ 2400s)$(1 ± 0,05) (i.e. ± 5% of the value at the end of measurements). It is seen that, for a given sample composition, the smallest settling time is typically observed at an intermediate magnetic flux

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density of 0.27 T. This is explained that between 0.1 T and 0.57 T the transient response changes from the monotonic growth to the nonmonotonic behavior with an overshoot. The MAE70_Hard-sample represents an exception from this rule what can be explained by the mechanical resistance of the harder matrix to the rearrangement of ferromagnetic inclusions. Indeed, it is seen in Fig. 3 e that the overshoot is most strongly pronounced in the sample with the softest matrix. For the MAE70_Hard-sample the transition between two regimes occurs at a higher value of magnetic induction B z 0.57 T. A threshold between different transient response regimes is expected to be defined by an interplay between elastic and magnetic interactions, thus, being dependent on both matrix elasticity and magnetic filling degree. Large retardation times in the response of MAEs have been observed in our previous work on the transient behavior of the shear storage modulus in the magnetic field [19]. We have shown that at least three exponential functions are required to reasonably describe the time behavior of the storage shear modulus on a large time scale and the largest relaxation time could reach several thousand seconds. In the previous work, the measurements of the shear modulus were performed at oscillatory shear deformations. In the present paper, there are no such oscillations. However, the measurements of MDE confirm the existence of retardation processes with long characteristic times which order of magnitude can reach 1000 s. We attribute this retardation to the rearrangement of filler particles in externally applied magnetic fields. 3.2. Pyramid excitation As in Ref. [19], the following set of experiments has been designed in the view of the prospective applications where MAEs are used as adaptive elements. For example, a MAE vibration isolator, possessing controllable stiffness, enables it to adjust its isolation frequency in real time [3]. The magnetic field was first increased in a staircase manner (the magnetic flux density was set equal to 0 T, 0.1 T, 0.27 T and 0.57 T). At each current the dielectric constant was measured during 10 min. Then the field was decreased in the same stepwise manner (the magnetic field was decreased down to 0.27 T, 0.1 T, 0 T). Fig. 4 shows the results of measurements. For the current steps corresponding to the increasing magnetic field, the behavior of εr qualitatively repeats the course of the curves observed in Fig. 2 for the step excitation. After a magnetic-field step from a larger to a smaller magnetic field, the dielectric permittivity monotonically declines with time. However, if the magnetic field jumps from a smaller to the larger values, a non-monotonous transient behavior of εr (exhibiting an overshoot) can be observed. In its appearance, this effect is somewhat similar to the ”cross-over” effect observed in Ref. [19], except that it appears for increasing and not for decreasing magnetic-field steps. Remarkably, a pronounced asymmetric behavior can be observed in Fig. 4. The value of εr strongly depends on the history of the magnetic-field variations. For example, the steady-state value of εr with the magnetic flux density B ¼ 0.27 T in the MAE70 sample is approximately equal to 9.5 for the ascending part of the pyramid excitation while it is about 13 for the descending part of the pyramid excitation. It has been previously suggested that hysteresis of physical properties is an intrinsic property of MAEs [36,43,54,55]. The physical reason is presumably due to the dependence of filler restructuring on the history of magnetic and/or mechanical loading. It is important that the hysteresis behavior cannot be attributed to the hysteresis of the magnetization of soft-magnetic filler particles in external magnetic fields, because these particles are practically free from the magnetic hysteresis. The hysteresis of the dielectric permittivity εr on the external magnetic field will be investigated in more detail below.

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3.3. Repeated on-off switching of the magnetic field As in Ref. [19], the next set of experiments was to simulate the periodic loading of MRE in a typical application (e.g. as a vibration isolator), where mechanical stiffness of an adaptive element is controlled by externally applied magnetic fields. The external magnetic field has been periodically switched on and off several times between the maximum (B ¼ 0.1, 0.27 or 0.57 T) and zero values. The on and off intervals have been taken to be equal to 10 min. Fig. 6 presents the measurement results. It is seen that in small magnetic fields (B ¼ 0.1 and 0.27 T), the cycles are reproduced very well. However, in the maximum field, a significant variation (drift) of the final and maximum values of εr over the subsequent “on” stages can be observed. The largest variations occur between the first and second cycle. The largest changes of the maximum value of εr (an increase of þ3.71) and of the final value of εr (a decrease of 1.37) has been observed for the MAE80 sample between the first and the second cycle. The variations between the second and the following cycles are more moderate. The drift of the final value of εr is approximately þ0.45 between the cycles.

3.4. Hysteresis Fig. 6 presents the dependences of the relativity permittivity εr on external magnetic flux density B. The time interval between the subsequent measuring points is 20 s. This corresponds to the usual measurement time interval in MR measurements. The drive current of the electromagnet was first stepwise increased to its maximum value and then stepwise decreased. Ten such cycles have been recorded. The experimental results are presented in Fig. 6. Dεrn denotes the normalized variation of the relative

permittivity εr during a single cycle of magnetic loading:

Dεrn ¼

εr ðIÞ  εr ðI ¼ 0AÞ : εr ðI ¼ 9AÞ  εr ðI ¼ 0AÞ

(1)

The following conclusions can be drawn from Fig. 6: It is seen that the initial (increasing magnetic field) curve normally significantly differs from the subsequent cycles. Similar effects have been observed for dynamic modulus [43], normal force behaviour [43,56] and magnetic hysteresis [52]. It can be expected that during the first cycle principal restructuring of the filler takes place. Initially this results in major changes, whereas further changes are minor. The observed hysteretic behaviour is attributed to the rearrangement of the filler. The following theoretical considerations support such qualitative explanation. Biller et al. [54] analyzed the magnetostatic interaction of two soft-magnetic particles embedded into an elastomer matrix. It was shown that pair clusters of multidomain ferromagnetic particles may form or break by a hysteresis scenario. Zubarev et al. [36] explained the hysteresis of magnetic properties by the hysteresis of the consolidation of filler particles into chain-like aggregates. Since the dielectric properties of a composite material depend on the internal structuring of the filler, a hysteresis of dielectric properties can be expected as well. The solid arrows designate the branches with ascending/ descending magnetic field. It is observed that for the same magnetic flux density the relative permittivity at the descending branch is significantly larger than at the ascending branch. The dotted arrows designate the moving direction of the points corresponding to the largest/smallest magnetic field of the cycle. It is observed that the hysteresis loops rotate clockwise with subsequent magnetic-loading cycles. The absolute value of the hysteresis (the area of the loop) is

Fig. 6. Dependence of the effective relative permittivity εr (a,b) and the normalized change of the relative permittivity Dεrn (c,d) on the externally applied magnetic flux density B for samples with the same matrix but various filler concentrations (a,c) and for samples with the same filler concentration but various storage moduli of the matrix (b,d). The lines connecting experimental points serve as a guide to the eye. The Figures c and d show the results for the 10th magnetic/loading cycle. The solid arrows indicate the direction of magnetic-field change.

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more pronounced in samples with the larger filler concentration (Fig. 6 a) and softer elastomer matrices (Fig. 6 b), where the losses to the rearrangement of the filler are more important. The effect of the sample composition is especially clearly seen in Fig. 6 c, d, where the tenth hysteresis loop is presented in a normalized way. 4. Conclusion Magnetoactive elastomers comprising micrometer-sized iron particles dispersed in soft silicone matrices have been synthesized and their magnetodielectric properties depending on the concentration of magnetic filler and matrix compliance have been experimentally studied in stepwise time-varying magnetic fields. It has been show that: ⁃ Relative effective permittivity εr and dielectric loss tand in MAEs can be strongly influenced by external magnetic fields. These magnetodielectric effects can easily reach several hundred percent. ⁃ Both εr and tand increase with the growing external magnetic field. The corresponding gains of εr and tand depend on the concentration of conducting magnetizable filler particles and the magnitude of the shear storage modulus of the matrix. ⁃ If the concentration of the filler particles is fixed, the magnetodielectric effect is more pronounced in mechanically softer matrix. ⁃ The transient response of εr and tand to magnetic-field step excitation can be a complex function of time. The corresponding settling time may reach the order of magnitude of 1000 s and it depends on the composition of the sample and applied magnetic field. ⁃ There is a hysteresis of the magnetodielectric effect in MAEs on the externally applied magnetic field. This hysteresis is pronounced more clearly for mechanically soft elastomer matrices and large concentrations of filler particles.

5. Outlook Our experiments confirm that hysteresis of physical properties of MAEs in dependence on external magnetic fields is an intrinsic property of these materials. Therefore, it is tempting to transfer the known analysis methods and theories of hysteresis to these materials. Recently, the first-order-reversal-curves (FORC) diagrams [52,57] have been employed to analyze magnetic hysteresis in MAEs. In general, the method of FORC diagrams is not limited to magnetic properties and can be extended to other physical properties (dynamic modulus, normal force, electric permittivity etc.) as well. However, in conventional FORC diagram analysis of magnetic properties the material response is assumed to be instantaneous. In MAEs, it seems that the material response can be retarded due to the rearrangement of filler particles in the viscoelastic matrix. Clearly, further experimental and theoretical studies are required in order to understand and analyze hysteretic behavior of MAEs in magnetic field. Acknowledgements The authors thank Mr. Dominik Stadler, Mr. Tobias Probst and Mr. Anton Udalzow for their valuable help with the experimental setup. E.K. gratefully acknowledges the financial support of the Russian Foundation for Basic Research (project No. 16-29-05276). I.A.B and M.S. thank OTH Regensburg for financial support within the framework of cluster funding.

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