First-order reversal curve analysis of magnetoactive elastomers †

The ﬁ rst magnetization loop and the ﬁ rst stress – strain cycle of magnetoactive elastomers (MAEs) in a magnetic ﬁ eld di ﬀ er considerably from the following loops and cycles, possibly due to the internal restructuring of the magnetic ﬁ ller particles and the matrix polymer chains. In the present study, the irreversible magnetization processes during the ﬁ rst magnetization of MAEs with di ﬀ erent ﬁ ller compositions and tensile moduli of the matrix are studied by ﬁ rst-order reversal curve (FORC) measurements. For MAEs with mixed magnetic NdFeB/Fe ﬁ llers the FORC distributions and magnetization distributions of the ﬁ rst major loop reveal a complex irreversible magnetization behavior at interaction ﬁ elds H u < (cid:1) 50 kA m (cid:1) 1 due to the magnetostatic coupling between the magnetically hard NdFeB and the magnetically soft Fe particles. This coupling is enhanced either if the interparticle distance is reduced by particle motion and restructuring or by an increase in the particle densities. If the sti ﬀ ness of the matrix is increased, the structuring and thus the interparticle interactions are suppressed and the magnetization reversal is dominated by domain processes in the NdFeB particles at high coercive ﬁ elds of H c > 600 kA m (cid:1) 1 .


Introduction
Magnetoactive elastomers (MAE) combine the properties of nonmagnetic elastic polymer matrixes with those of solid magnetic llers, offering a number of fascinating magnetically controllable effects such as magnetostriction, magnetodeformation, magnetorheological and shape memory effects. 1 In both isotropic and anisotropic MAE, the structure and mobility of the magnetic ller within the elastic matrix inuences the macroscopic behavior of the MAE. 2,3 The magnetic ller usually consists of magnetically so particles such as magnetite or carbonyl iron, or magnetically hard particles such as CoFe 2 O 4 or NdFeB. Combinations of magnetically so and hard llers have rarely been studied, although MAE with these mixed magnetic llers are interesting candidates for technical applications since they allow active and passive control of the mechanical properties 4,5the mixed magnetic ller can either be structured actively by applying an external magnetic eld, or passively by inducing a specic inner remanent magnetization of the magnetically hard phase which keeps the so phase partially or fully magnetized and structured when the external magnetic eld is removed.
A standard method for the magnetic characterization of magnetic materials is the measurement of major magnetization loops. The parameters derived from these measurements present bulk averages of the magnetic properties of all particles in a sample. To distinguish the magnetic response of different components of ne particle samples a more sophisticated approach is necessary, for example isothermal remanent magnetization unmixing or rst-order reversal curve (FORC) measurements. The FORC method has the advantage that it provides not only a qualitative ngerprint of the magnetization processes but also quantitative information of the switching or coercive eld distribution and the local interaction eld. 6 Initially, the method was applied in geo-and paleomagnetic studies for mineral and domain state discrimination. [7][8][9] With recent advancements in measurement automation and data processing the FORC method has become a characterization tool for many hysteretic systems, including magnetic recording media, 10-12 multi-layered nanowires 13 and hard/so exchangespring composites. [14][15][16][17] For the latter Panagiotopoulos 18 derived a theoretical model based on a mean-eld approach to explain the twin interference features in the FORC diagrams. Schre et al. simulated the FORC diagrams of sintered Nd 2 Fe 14 B with regions of magnetically so defects by LLGmicromagnetics simulations and compared the diagrams to raw and desheared experimental data, 19 while Chen et al. studied the difference between a common sintered Nd 2 Fe 14 B sample and Nd 2 Fe 14 B particles of a rapid solidication process embedded in a nonmagnetic varnish. 20 In this work we apply for the rst time the FORC method to magnetoactive elastomers with magnetically hard and hard/so mixed magnetic llers and compare the results with non-elastic systems to investigate the effect of the composition of the magnetic ller and the matrix elasticity on the irreversible magnetization processes and their local switching and interaction elds.

Sample preparation
The MAE samples manufactured for the present study were based on a polydimethylsiloxane (PDMS) matrix which is identical to the SIEL® 3,5 and Wacker Elastosil® material. An organometallic crosslinking agent has been used to initiate the polymerization process. To modify the elasticity of the matrix the liquid rubber component has been diluted with silicone oil (M100 Baysilone from Bayer) prior use.
The magnetic ller in the samples were magnetically hard NdFeB alloy particles (MQP-S-11-9-20001-070 from Magnequench) or mixtures of NdFeB particles and magnetically so carbonyl iron particles (precoated, grade CC from BASF). The NdFeB and the iron particles were spherical with mean diameters of 46.8 mm and 4.4 mm, i.e. magnetic multidomain particles. The SEM image in Fig. 1 shows as an example the spherical NdFeB particles embedded in the polymerized matrix. To prevent particle aggregation and to enhance the adhesion to the polymeric matrix the NdFeB particles were pretreated with a mixture of ether and silicone oil 100 : 10 : 1 by mass, which was evaporated before the particles were added to the rubber component of the matrix. The mixture of rubber and magnetic ller was degassed and subsequently mixed with the crosslinking agent of the matrix to yield a ller volume concentration of $28%. Finally the mixture of matrix chemicals and ller was degassed a second time and cured at 80 C for three hours. In addition to the MAE samples, reference samples with an inelastic matrix of epoxy resin and an aliphatic amine hardener (plus endfest 300 from UHU) were prepared, one with 100% NdFeB and one with $75% NdFeB in the ller. The sample compositions and mechanical properties are summarized in Table 1 in Section 3.1.

Mechanical characterization
The elastic moduli of the manufactured MAE samples were measured within a quasi-static axial compression. In the test, the cylindrical sample is xed in a movable upper and static lower xture. The measurements were performed at a speed of 0.05 mm s À1 controlling the displacement, which was limited to approximately 5% of the initial sample length in order to avoid signicant inuence of the sample deformation in the radial direction. To determine the elastic modulus, the measured forces and displacements were transformed into stress (s) and strain (3) values, respectively, taking into account the crosssection and the height of the sample. The tensile modulus E is dened as E ¼ ds/d3, i.e. as the slope of the stress-strain curve which remains linear in the range of deformation used.

Magnetic characterization
The magnetic properties were measured in a Lakeshore vibrating sample magnetometer 7400, which offers the advantage of high eld intensities across a large sample space. The FORC measurement procedure was implemented according to the algorithm of the MicroMag soware of the Princeton Measurements Corporation. This algorithm extends the chosen measurement range by a certain number of points, in the present work by ve points, to avoid rst point artefacts and to compensate for range losses caused by the smoothing in the subsequent data processing.
Prior to measuring each FORC the sample was saturated in a magnetic eld of 1750 kA m À1 . Next a calibration measurement was recorded to monitor eld dris. Then the eld was decreased to the reversal eld H r . Starting from this reversal eld an ascending FORC was recorded as the magnetization M(H a ,H r ) when the applied magnetic eld H a was swept back to saturation with an average measuring time of two seconds per point. For the subsequent FORCs the reversal eld H r was decreased to obtain a set of FORCs, which covered the range of positive to negative coercive elds.
The FORC distribution of ascending FORCs is dened as the mixed derivative of the magnetization M(H r ,H a ) with respect to the reversal eld H r and the applied eld H a : 6 The distribution is calculated from a second order polynomial surface t to the local magnetization of the FORCs. In the present work the analysis package FORCinel 2.05 with  a LOESS smoothing factor of 10 has been used for the calculations. 21 The resulting distributions were plotted rotated by 45 so that the horizontal axis relates to the coercive or switching eld H c and the vertical axis to the interaction eld H u . To preserve the information of the plots for black and white printing we have used a cubehelix color map which converts to a greyscale with a continuous decrease in brightness. 22 In the FORC plots, distribution values of zero indicate reversible magnetization processes and non-zero values indicate irreversible magnetization processes. If the measured FORCs are free of rst point artefacts, the initial slopes of consecutive FORCs can be used to calculate the initial irreversible and reversible magnetization changes, DM irr and DM rev , as illustrated in Fig. 2. From these differences the irreversible and reversible coercivity distributions, f irr and f rev , are calculated. 20,23,24 Due to the nite size of the eld increments in any experiments these experimentally determined distributions are, strictly spoken, only the distributions near the upper branch of the hysteresis loop but not of the upper branch (H / H r ) as dened in eqn (2). 23,24 Compared to typical FORC measurements with 100 to 200 FORCs, 6 we have recorded 699 FORCs with an averaging time of 2 s per point and eld increments between 3.0 and 3.7 kA m À1 to assure the relaxation of the samples and a sufficiently high eld resolution for the evaluation of the distributions.
Since Schre et al. have shown that the demagnetizing eld related to the outer shape of the sample can introduce artefacts in FORC distributions, 19 the FORCs in the present work have been desheared to present the magnetization in an effective internal eld M(H r ,H a ) with and the corresponding coercive and interaction eld axes of the rotated FORC plots H c ¼H a ÀH r 2 and H u ¼H a þH r 2 : The demagnetizing factors N shape of the samples have been approximated by the demagnetizing factors of ellipsoids of revolution. 25 For particle samples with a known internal structure a linear interpolation between the shape demagnetizing factor of the internal structure and the demagnetizing factor due to the outer shape of the sample has been proposed. 26,27 Assuming that the internal structure are either spherical particles with N ¼ 1/3 or elongated structures of particles aligned parallel to the direction of the magnetic eld with N < 1/3, such interpolated demagnetizing factors would be lower than the demagnetizing factor of a continuous medium as assumed in this work. Thus, the demagnetizing factors N shape used for the FORCs represent an upper limit in order to check for any artefacts arising from the outer sample shape.

Mechanical characterization
The tensile modulus E has been evaluated for all manufactured MAE samples in a non-magnetized state using a quasi-static compression test as described above. The parameters of the samples are collected in Table 1. Samples S1 to S4 differ in the concentration ratio of magnetically so iron (f so ) and magnetically hard NdFeB particles (f hard ) and have tensile moduli, which are in the same order of magnitude, while samples S5 to S7 have the same particle concentration ratio but they differ signicantly in their stiffness.

Standard magnetic characterization
For the basic characterization the initial magnetization M ini from zero eld to positive saturation, the rst major magnetization loop and repeated major magnetization loops have been measured. The susceptibility of the initial magnetization curve dM ini /dH of the MAEs with mixed magnetic ller (Fig. 3) displays two maximaone in weak magnetic elds related to the magnetization of the iron particles and one in strong magnetic elds due to the magnetization of the NdFeB particles. In the samples S1 to S4 with similar tensile moduli and varied ller composition, shown in Fig. 3a, the magnetization of the iron particles occurs at a eld of $60 kA m À1 and grows in intensity if the iron concentration of the magnetic ller is increased. The maximum in the susceptibility appears shied to higher elds, i.e. 60 kA m À1 , compared to the maximum in the susceptibility of compressed iron powder (<7 kA m À1 ) or in the mixed magnetic epoxy samples ($21 kA m À1 ).
A similar shi of the maximum susceptibility has already been observed in samples with solely so magnetic llers and has been attributed to the movement and restructuring of the ller particles. 28 The formation of structures aligned parallel to the applied eld reduces the local demagnetizing eld and thus increases the susceptibility in a certain eld interval more than the simultaneous decrease by the magnetic saturation of the particles. Inversely, the rst maximum in the susceptibility attens and shis to lower elds, as seen in Fig. 3b, when the stiffness of the matrix is increased and the restructuring of the iron particles is suppressed. The second maximum in the susceptibility, related to the NdFeB particles, appears shied to lower elds in MAEs with lower tensile moduli since the reduction of the local demagnetizing eld due to the structuring of the iron particles aids the magnetization of the NdFeB particles.
Compared to the major magnetization loops of epoxy samples, in which the magnetization of the immobilized ller particles reverses by nucleation and expansion of domains, 19 the bulk coercivity of the MAE with mobile ller particles is signicantly reduced (Fig. 4a). In the MAE samples with mixed magnetic llers the saturation remanence decreases with increasing amount of the iron in the ller and the saturation magnetization increases as summarized in Table 2 in Section 3.3. Furthermore, the major hysteresis loops of the MAEs with mixed magnetic llers display a continuous magnetization reversal in contrast to the stepped two-phase magnetization in the epoxy sample shown in Fig. 4b.
Most remarkably, however, is that the rst major magnetization loops of the MAEs with mixed magnetic llers differ considerably from their repeated loops ( Fig. 4c and ESI †) analogous to the rst and repeated stress-strain cycles observed in MAEs in a magnetic eld. 29 In the descending branch of the rst loop the magnetic particles have previously seen only a positive saturation eld, hence there is only a positive remanence in the sample. In the repeated magnetisation curves the particles have already seen a positive and a negative saturation eld. Therefore, the statistical distribution of the magnetic moments and thus also the inner structure are different compared to the rst loop. Due to the preferential orientation and magnetization of  the particles in the direction of the positive saturation eld the rst loop is asymmetric to both the eld and the magnetization axis. Similar asymmetric loops are observed in ferroelectrics with preferential polarization and strain congurations. 30 When the magnetic eld is reduced from positive saturation the preferential orientation is removed and the ller particles and polymer chains progressively arrange themselves to the structures present in the repeated loops. These principal magnetization and restructuring processes are probed by FORC measurements with gradually reduced reversal elds.

FORCs of MAEs with a variation of the ller composition
The deshearing of the FORCs according to eqn (3) did not remove or introduce any additional features in the majority of the FORC distributions compared to the distributions calculated from the raw FORCs. In these cases only the desheared data are presented, otherwise the changes in the distributions will be discussed.
Similar to sintered permanent magnets or MQP-B NdFeB particles 15,20 the MQP-S NdFeB particles exhibit two distinct positive regions of the FORC distribution in an epoxy resin matrix (Fig. 5a). The rst irreversible magnetization region, related to the magnetically hard phase in the NdFeB particles, occurs at a high coercive eld of H c ¼ 702 kA m À1 and the second region at a low coercive eld H c < 0 kA m À1 in the desheared FORC distribution, which corresponds to H c ¼ 0 kA m À1 in the raw FORC distribution. The positive regions are accompanied by negative twin features and an additional weak interference region at H c ¼ 403 kA m À1 and H u ¼ 392 kA m À1 . In contrast to the FORC distribution of MQP-B particles the low coercivity region of the MQP-S particles is shied to a strong negative interaction eld of H u ¼ À776 kA m À1 . According to Panagiotopoulos 18 such a shi will occur in systems with a magnetically hard and a magnetically so component, if these components interact with each other but have a negligible overlap of their coercivity distributions. NdFeB particles can contain an intrinsic so phase, such as a-Fe or FeB, 20 which exchange couples to the magnetically hard phase in the particles. However, the low coercivity region of the MQP-S particles is rather narrow compared to the wide spread regions observed in sintered NdFeB or MQP-B NdFeB particles and the magnetization distribution of the major loop (Fig. 6a) is dominated by irreversible magnetization, in contrast to the distribution of MQP-B particles, 20 in which the reversible and irreversible contributions are balanced. This indicates that the MQP-S NdFeB particles of the present study do not contain excessive concentrations of a so phase but mainly reverse their magnetization as a single hard magnetic phase.
If the NdFeB particles are embedded in the elastic matrix of a MAE the FORC distribution (Fig. 5b) changes drastically. The distribution exhibits only one major irreversible magnetization region at a weak interaction eld of H u > À50 kA m À1 and a coercive eld of H c ¼ 30 kA m À1 since the particles reverse their magnetization by rotating and moving within the matrix rather than energetically unfavorable domain processes within the particles. The magnetization distribution of the rst major loop (Fig. 6a) is therefore dominated by irreversible magnetization, which is stronger than in the epoxy sample, although both samples contain similar volume concentrations of NdFeB.
With the addition of magnetically so iron particles to the ller of the MAE the FORC distributions (Fig. 5c-f) become more complex. An asymmetric reversible ridge along the H u axis at H c ¼ 0 kA m À1 appears, which is caused by the coupling of the reversible magnetization of the added iron particles to the irreversible magnetization states of the system. 31 The overall intensity of the irreversible FORC features decreases with increasing amount of iron particles and the reversible contributions to the magnetization of the major loop (Fig. 6b) increase. Furthermore, several regions of irreversible magnetization, listed in Table 2, occur at interaction elds H u < À50 kA m À1 due to the magnetostatic coupling and the motion of the particles within the matrix. These features also occur in the FORC distributions of the unsheared FORCs, as exemplarily shown in Fig. 5f.
Based on the strength of the reversal eld, we would assign the irreversible magnetization features at weakly negative reversal elds, labelled 1 and 2 in Fig. 5f, to the magnetization and structuring of the so iron phase, and the irreversible magnetization at a strong negative reversal eld, labelled 3 in Fig. 5f, to the hard NdFeB phase. Interestingly, the FORC features of the so phase occur at a negative and a positive applied eld. The feature at a negative applied eld is likely to be related to the magnetization reversal of iron particles in the close vicinity to positively magnetized NdFeB particles. To reverse the magnetization of these particles the applied eld has to be negative to compensate the positive remanent eld of the neighboring NdFeB particles. The feature at a positive    applied eld, on the other hand, is likely to originate from the motion and restructuring of the iron particles in the matrix, similar to the structuring observed in the susceptibility measurements described in the previous section.

FORCs of MAEs with a variation of the matrix elasticity
To investigate the inuence of the matrix elasticity on the magnetization and structuring of the ller particles, the FORCs of MAEs with a constant sample composition but increasing tensile moduli E were measured. Fig. 7 shows the respective FORC distributions of the unsheared and desheared FORCs and a projection of the distributions onto the FORCs. All samples display a reversible ridge at H c ¼ 0 kA m À1 due to the so iron particles in the ller. In contrast to the distribution of sample S3 the irreversible magnetization of the so phase at negative applied elds in sample S5 and S6 is only observed in the distributions of the unsheared FORCs. In the distributions of the desheared FORCs it seems to be covered by the negative twin feature of the strong irreversible magnetization in positive applied elds. This demonstrates that the FORC method is able to detect even subtle differences between MAE samples, such as the rate of cross-linking of the matrix surrounding the particles or the adhesion of the polymer molecules to the particles, which are not apparent from major hysteresis loops. If the stiffness of the matrix is increased an additional FORC feature at high coercive elds H c > 600 kA m À1 occurs (Fig. 7 E z 120 kPa and E z 440 kPa) and the low coercivity features gradually disappear. The irreversible contribution to the magnetization of the rst major loop (Fig. 8) decreases accordingly and shis to lower elds. These changes in the magnetization processes can be explained by the progressive suppression of the particle motion. On the one hand, the reduced mobility reduces the interparticle interaction since the particles cannot rearrange themselves in structures close to each other. On the other hand, the magnetization reversal by domain processes is enhanced within the immobilized NdFeB particles. Hence, the FORC distributions of sample S6 with an intermediate tensile modulus display both, features of mobile coupled particles as well as immobile decoupled particles. Sample S7 with a stiff matrix resembles the FORC distribution of the epoxy sample S1 with an additional reversible ridge due to the iron particles and shows a stepped contour of the FORCs, which is typical for the magnetization switching of two decoupled phases.

Conclusion
The rst stress-strain cycle 29 and the rst magnetization loop of magnetoactive elastomers in a magnetic eld differ considerably from the following cycles and loops. The presented study provides an insight into the irreversible magnetization processes during the rst magnetization of MAEs with magnetically hard and hard/so mixed magnetic llers. The magnetization processes were analyzed by high resolution rstorder reversal curve measurements, which proved to be a powerful and sensitive tool to detect the local coercive and interaction elds in dependence of the magnetization history of the MAE samples. By a variation of the hard/so ller composition and the matrix elasticity it has qualitatively been shown that the irreversible magnetization processes are the result of interparticle interactions and of a complex restructuring of the ller particles and the matrix polymer chains in an applied magnetic eld.
Future work could, on the one hand, investigate how other factors like the eld ramp rate or the ratio of the particle sizes of the magnetically so and hard component, which were kept constant in the present study, inuence the reversible and irreversible magnetization processes in MAEs. On the other hand, 3D imaging techniques such as X-ray m-computertomography 32,33 should be enhanced, with respect to their resolution but also to the available magnetic eld intensities, in order to provide microstructural information of MAEs which allows a precise correlation between the changes of the internal structure and the corresponding changes in the local demagnetizing eld and thus in the FORC distributions. If we gain a deeper knowledge of the physics of MAEs in a magnetic eld and the corresponding magnetomechanical properties we could ideally use the FORC method as a monitoring tool for the sample quality of MAEs for future technical applications.