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Effect of local elasticity of the matrix on magnetization loops of hybrid magnetic elastomers
Journal of Magnetism and Magnetic Materials 459 (2018) 92–97
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Research articles
Effect of local elasticity of the matrix on magnetization loops of hybrid magnetic elastomers M.V. Vaganov a,⇑, D.Yu. Borin b, S. Odenbach b, Yu.L. Raikher a a b
Institute of Continuous Media Mechanics – The Division of Perm Federal Research Center, Russian Academy of Sciences, Ural Branch, Perm 614013, Russia Technische Universität Dresden, Dresden 01062, Germany
a r t i c l e
i n f o
Article history: Received 9 July 2017 Received in revised form 2 December 2017 Accepted 4 December 2017 Available online 5 December 2017 Keywords: Magnetorheological elastomers Hybrid magnetic elastomers Magnetization curves Magnetic hysteresis FORC diagrams
a b s t r a c t To model magnetization loops of magnetorheological elastomers (MREs) with magnetically hard filler, we consider an assembly of single-domain particles possessing inversion-symmetrical shape and embedded in a soft polymer matrix. To describe the intrinsic behavior of the particle magnetic moments under an applied field, the Stoner-Wohlfarth approach is employed. Unlike the case of solid matrix, the particle in a MRE is able to rotate relative to its elastic environment, so that its equilibrium orientation results from the balance between the magnetic torque (exerted by the applied field at any magnetic moment that does not point along the field) and the elastic torque generated by the matrix. We assume that elastic resistance to the field-induced particle rotation could be presented as comprising of two contributions. The first one is valid for any not perfectly spherical particle and is independent of the particle–matrix adhesion. It reflects the fact that, when trying to rotate, the particle has to ‘‘shoulder its way” by deforming the adjoining regions of the matrix. In that case, for angular deviations up to 90° from the initial position, the resistance torque increases. However, as soon as the rotation angle grows up to the value but infinitesimally exceeding 90°, the elastic torque changes its sign and from now on forces the particle to rotate to 180°, where it attains the geometrical position that coincides with the initial one. Evidently, this process is of the barrier type: both orientations of the particle are equal in elastic energy. The second mechanism stems from the ‘‘memory” that a given particle has of its initial state and may be caused, for example, by some macromolecules grafted to its surface while curing the matrix of the MRE. This restoring force always tends to drag the particle to its 0° (initial) position. Therefore, the observed magnetization of a MRE sample comes out as a result of joint interplay of the Stoner-Wohlfarth and the two elastic mechanisms. In our simulations, we show that the proposed model in a natural way accounts for the two essential features observed (solely or together) in experiment, namely: (i) weak net coercivity of the MREs, whose filler particles as themselves are highly coercive, and (ii) asymmetric positions of the magnetization loops with respect to H ¼ 0 point at the ðM HÞ plane. Ó 2017 Elsevier B.V. All rights reserved.
1. Introduction Composite materials consisting of a soft polymer matrix and embedded into it magnetic particles, can change their physical properties in response to the applied magnetic field. Due to this specific ability, such smart materials are called magnetorheological, magnetoactive or magnetosensitive elastomers. Being synonyms, these three terms are used interchangeably in literature [1–3]. The interest to these field-responsive materials is based on wide prospects of their technical and high-tech [4–6], and biomedical [7,8] applications. In below, we discuss some aspects of ⇑ Corresponding author. E-mail address: [email protected] (M.V. Vaganov). https://doi.org/10.1016/j.jmmm.2017.12.016 0304-8853/Ó 2017 Elsevier B.V. All rights reserved.
internal magnetomechanics of these composites under the action of external fields and hereafter use MRE as a unique acronym designating these composites. Hybrid magnetic elastomers containing a mixture of two distinctly different magnetic fillers – magnetically hard (MH) and magnetically soft (MS) microparticles – make a special class of MREs. As shown in Refs. [9–11], the magnetic properties of hybrid MREs differ significantly both from those of customary MREs with just magnetically soft filler and from those of solid magnetics. In this work we investigate theoretically a model hybrid MRE which contains only MH particles. To justify this choice, we remark that, according to the available experimental data, the MS component, if added, apart from increasing the saturation magnetization of the material, does not bring any new specifics in magnetic behavior
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of a sample. Experimental investigation of the MH particles magnetization has been presented in Refs. [1,9–11]. In what follows, we, first, develop a model describing the response of a magnetically hard particle embedded in a soft elastomer matrix. We then use that model to simulate magnetization loops of the samples and, finally, give an example of application of the theory taking for that the data obtained on MREs filled with NdFeB microparticles.
2. Model When constructing the model, we aim at the experimentally available MREs where the MQP-S-11-9-20001-070 powder (Magnequench, Inc.) was used as a filler. This powder consists of initially isotropic roundish NdFeB-alloy particles with mean linear size of 46.8 lm, according to the manufacturer. Due to production imperfections, a considerable fraction of the particles come out resembling prolate or oblate ellipsoidal shapes than are perfectly spherical. According to the manufacturer, the particles are produced during centrifugal or spinning cup atomization, which is a type of rapid solidification processes [12]. A NdFeB microparticle of that kind consists of metallurgical grains with the size about 20 nm [13] that is below the critical size of single-domainness for NdFeB (0.21 lm [14]). Since in the melt-spun materials the domain walls coincide with the grain borders [15], then during magnetization these single-domain grains should fairly accurately obey the Stoner-Wohlfarth (SW) model [16]. The same should apply to the materials produced by the HDDR (hydrogenation, disproportionation, desorption, and recombination) method, which yield the grains about 0.3 lm [17]. When subjected to a strong field (first magnetization), each grain (domain) of the NdFeB microparticle switches in the direction of the field. On turning off the field, the elementary magnetic moment falls to the closest (in orientational sense) potential well. The resulting net magnetic moment of the particle, in accordance with the SW model, is a half of the maximal value [10,16] Due to that, under a cycled field (its negative values included) the microparticle displays a magnetization loop very similar to the hysteresis obtained with the SW model for an assembly of grains with random distribution of uniaxial anisotropy axes [10]. Hereby, we neglect this detail and approximate magnetization of a NdFeB particle, once subjected to a strong field, with a single SW cycle. As it will be demonstrated in Section 4, the discrepancies resulting from that deliberate simplification are of some quantitative nature but are not relevant for the main line of the work: introducing an extended concept of magnetization for the MREs filled with MH particles. Bender et al. [18,19] modified the conventional SW energy expression to allow for the linear elastic response (Hook resistance) of the matrix to mechanical rotations of the particles. It has been done in order to study polymer-based dispersions with acicular magnetically hard filler. Hereby we modify this magnetoelastic approach in the following way. Consider a typical MRE based on weakly linked silicone rubber plasticized with silicone oil. In such a matrix, an MH particle sits inside a cavity formed around it during the polymerization process. It is tightly enveloped by the environment but is not too strongly ‘‘glued” to it. Because of that, in response to an exerted mechanical torque, the particle would strive to rotate. Moreover, in this motion the plasticizer distributed all over the matrix would serve as a lubricant. If the particle in question is to some extent anisometric, then for its rotation it ought to move apart the surrounding matrix because, when turned from the equilibrium position, the particle is not any longer congruent with the cavity. Apparently, this inference
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equally applies to prolate and oblate objects. Let n be the unit vector of major axis of an axially symmetrical particle, and m the unit vector characterizing the enveloping cavity (Fig. 1) that has the same shape. Those vectors coincide ðn ¼ m Þ in equilibrium, but turn from one another under particle rotation. Evidently, the resistance torque grows with the deviation angle x until the particle attains position n ? m , i.e., x ¼ 90 . As soon as this orientation is traversed, the torque changes its sign: in order to restore its shape, the elastic cavity now generates the stress that urges the particle towards position n ¼ m , i.e., to x ¼ 180 . Whatever the exact form of angular dependence of the torque, one concludes that this function is even with respect to the direction x ¼ 90 , i.e., orthogonal configuration n ? m . This implies that the corresponding term to be added to the particle energy is of the barrier type. With regard to that, we approximate the in-cavity energy barrier term as QVðnm Þ2 ¼ QVðnm Þ2 ¼ cos2 x, where Q is the effective elastic modulus and V the particle volume. To account for a general case, the Hook-like term should be retained to the particle energy expression as well. The basis for that is at least twofold. First, the point-to-point adhesion at the particle/matrix interface is not total zero, so that, when rotating, the particle has to pull and unwind some macromolecules attached to its surface. Second, even if the barrier-type energy prevails, a real particle hardly has perfect inversion-symmetrical shape; then, upon turning it by 180°, its congruence with the cavity would be incomplete. For such a particle the energy wells (minima) positioned at n ¼ m are not equally deep: the one at n ¼ m is preferable. Therefore, to allow for possible coupling of the particle with its initial position, we, as in Ref. [19], add to the energy expression a quasi-Hookean term GVarccos2 ðnm Þ. Let the magnetic moment m of the considered MH particle at the initial state be directed along n. Then the above-described incavity 180°-rotation causes mechanical inversion of the particle magnetic moment without changing its position inside the particle: m rotates together with n. Meanwhile, the SW process renders a different (non-mechanical) type of re-orientation of m that does not require any mobility of the particle axis n. This means that in reality the vectors m and n are independent orientational variables. With allowance for the above-given explanations we present the orientation-dependent energy of an MH particle embedded in an elastic matrix and subjected to a quasistatic external field H ¼ Hh in the form
Fig. 1. MH particle rotation in an elastic matrix.
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U=V ¼ K u ðenÞ2 l0 M s HðehÞ Q ðnm Þ2 þ Garccos2 ðnm Þ;
ð1Þ
where K u is the energy density of the particle internal anisotropy (crystallographic or magnetostatic), which we assume to be uniaxial and which direction is defined by the particle axis n. In formula (1), e ¼ m=m is the unit vector of magnetic moment whose magnitude is m ¼ M s V with Ms being saturation magnetization of the particle material; l0 is the magnetic permeability of vacuum. Setting xOz plane of the coordinate frame in such a way that it passes through vectors m and h, one transforms the energy expression (1) to
U=V ¼ K u sin ðh uÞ l0 M s H cos u þ Q sin x þ Gx2 ; 2
2
ð2Þ
where the choice of angles is presented in Fig. 1, and x ¼ h0 h. Due to the fact that the particles at the beginning of the experiment are magnetically isotropic, after initial magnetization their magnetic moments will be aligned with that direction of the anisotropy axes which make acute angles with the direction of the initially applied field, i.e. h0 2 ½0; 90 . Introducing nondimensional units in (2), one arrives at the expression
1 2
1 2
1 2
g ¼ U=2K u V ¼ sin2 ðh uÞ f cos u þ f sin2 x þ jx2 ;
Fig. 2. Unhysteretic magnetization curve of a particle at h0 ¼ 5 ; f ¼ 0 and
j ¼ 0:2.
ð3Þ
where f ¼ l0 M s H=2K u is normalized magnetic field, f ¼ Q =K u characterizes the elastic energy barrier, and j ¼ G=K u is the linear elasticity parameter. 3. Magnetization curves Eq. (3) provides a direct way of determining the magnetization curve of a particle surrounded by an elastic medium. For that, one has to monitor, by minimization of function gðu; hÞ, the evolution of projection of e on the direction of the field h, i.e., cos u, under cyclic change of parameter f. We note that upon using only first two terms in the rhs of expression (3), one would recover the well-known SW magnetization loop at a given tilt h0 of the particle anisotropy axis. As well known, all those SW loops are centrally-symmetirical in the ðf ; cos uÞ plane. Adding the third term to the rhs of (3), i.e., accounting for the elastic barrier, preserves that symmetry. This is the forth (Hook-like) term that changes the situation. In below, to reveal the role of each of the two elastic contributions, we set in (3) coefficients f and j to zero separately. 3.1. Effect of linear elasticity at f ¼ 0 As indicated above, the forth term in the rhs of expression (3) renders the effect of elastic restoring force that always strives to return the particle to the initial position. Considering a MRE with a very soft matrix, one realizes that it would display unhysteretic behavior because the reorientation of m would go reversibly just by mechanical rotation of the particle. The magnetization curve for this situation is shown in Fig. 2, where the red square marks the region inside which all the possible SW magnetization loops are located. Note the difference between the normalized magnetic moment m=m ¼ h cos u values at f ¼ 1. Manifestation of the same effect in an assembly of particles whose anisotropy axes ðnÞ are distributed at random and whose magnetic and elastic energy factors are comparable ðj 1Þ, is shown in Fig. 3. The essential point is that for a given matrix elasticity j, there exists an angle hjump dividing all the particles on those exhibiting hysteretic magnetization ðh0 < hjump Þ and those changing their magnetization in a non–hysteretic way. The jumps of m occur only for those particles for which internal magnetic
Fig. 3. Magnetic hysteresis of a sample containing 100 particles with random distribution of anisotropy axes.
energy barrier is less than the elastic energy imposed by deforming the surrounding matrix. Therefore, only those particles make the loop width non-zero in Fig. 3 whose anisotropy axes are tilted at ðh0 < hjump Þ. All the other ones contribute only to the shift of the loop along the f axis towards negative values of the applied field. The same difference in remagnetization of the particles explains the non-equivalence between the upper and lower branches with respect to the vertical axis. Indeed, whereas the upper branch accumulates contributions from all the particles, the lower one sums up the contributions only from the particles prone to the magnetic switching, cf. Fig. 2. Therefore, in the case where just the linear elasticity term is taken into account, the occurring hysteresis is of exclusively SW origin (magnetic switching). 3.2. Effect of elastic barrier at
j¼0
The elastic energy barrier corresponding to the third term in the rhs of expression (3) implies a possibility of hysteretic mechanical rotation of an MH particle. Consider a particle whose magnetic moment in the field-free state points along that direction of the easy axis n which makes an acute angle with the direction of the magnetic field that had been imposed in the first magnetization process. When a weak negative field is applied, the elastic barrier would allow the particle only a small mechanical turn, of n and e together, from m . However, under enhancement of the field magnitude, the magnetic torque exerted on the particle would result in the increase of deviation angle. Finally the torque would become strong enough to enable the particle to overcome the energy barrier of height QV. This mechanical jump would entail an abrupt increase of the projection of e on h. Similarly, when going back
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from Hmax to Hmax , the mechanical (and, thus, magnetic) jump towards the positive values would take place. For sufficiently magnetically hard particles, the whole magnetic hysteresis loop (of elastic origin) would be available under the field cycle whose amplitude is lower than that required by the SW scenario; this case is illustrated by Fig. 4. The dependence of the barrier-induced model coercivity on the elastic barrier magnitude is presented in Fig. 5. If f ¼ Q =K u > 1, i.e., the energy of magnetic anisotropy is lower than its elastic analog, then for the particle it is more efficient to change its magnetization not by rotating but by switching the magnetic moment from one direction of the easy axis n to another. However, the softer the matrix the less coercive the particle behavior is. 3.3. Magnetization curve under combined elastic resistance An example of magnetization loop for a particle that experiences both kinds of elastic resistance is shown in Fig. 6. We note that the obtained model curve in a natural way accounts for the two most specific features distinguishing the MRE magnetization loops from those displayed by solid composites, namely, an extremely low coercivity and a pronounced shift along the field axis. 4. Comparison with experiment The MRE samples for experimental characterization were manufactured using polydimethylsiloxane (PDMS) matrix from Wacker
Fig. 4. Comparison of hysteresis loops at h0 ¼ 5 of a SW particle: with infinite (magenta) and finite elastic barrier f ¼ 0:3 (blue). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 5. Dependence of the modelled coercivity f c on the barrier parameter f 2 ½0; 10.
Fig. 6. Hysteresis of a particle with h0 ¼ 5 under combination of two elastic effects at f ¼ 0:3 and j ¼ 0:1.
Elastosil and magnetically hard NdFeB alloy powder MQP-S-11-920001-070 from Magnequench consisting of roundish particles with the mean linear size of 46.8 lm. The elasticity of the matrix was tuned using dilution of the liquid rubber component with silicone oil M100 Baysilone from Bayer. The powder was modified using a mixture of ether and silicone oil in order to provide a better compatibility of the particles with the silicon matrix and to avoid the magnetic filler aggregation. The mixture of the modified powder, liquid matrix and cross-linking agent was mechanically stirred and degassed. Further, it has been poured into a mould made of hard polyethylene and subsequently polymerized at temperature 100 °C for one hour under rotation in the oven at 50 rpm in oreder to prevent sedimentation of the particles. In this way rod-like samples of MRE with overall concentration of magnetic filler 40 vol.% and various elasticity were produced. The elastic modulus was evaluated by means of quasi-static elongation. Magnetization curves were obtained on Lakeshore vibrating sample magnetometer 7407 which provides static magnetic field up to 2000 kA/m and accuracy of magnetization measurement 1 A/m. A scanning electronic microscopy image of the NdFeB particles in the PDMS matrix is presented in Fig. 7 adopted from [10]. The magnetization curve of the magnetically hard particles in epoxy can be found in the same reference. Magnetic measurements on MREs, performed repeatedly, reveal one more essential detail regarding the hysteresis loops. A typical MRE sample displays training effect: when sweeping the field from Hmax to Hmax and back, the coercivity gradually decreases during the first two–three cycles, so that the consecutive loops, like those shown in Fig. 8, become reproducible only from the fourth cycle. The same effect was reported in Refs. [9–11].
Fig. 7. Scanning electron microscopy image of the NdFeB particles in the PDMS matrix.
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Fig. 8. Consecutive magnetization loops of a MRE with Young modulus E ¼ 230 kPa; loop No.1 (points), loop No.4 and all the next ones (solid line); the unit of the field is Hs = 1500 kA/m.
We ascribe the training effect to the ‘‘smoothing” of the cavity/particle interface resulting, for example, from removal of some weak adhesion links and/or from reconfiguring the adjacent polymer regions due to the microdeformations imposed by smallscale surface irregularities of the particle. To model the training effect, one would need a deeper knowledge of mechanochemistry than we have at the moment. Meanwhile in Table 1 some basic properties of the trained experimental samples are presented. Parameters l; w, and h in the Table stand for the length, width and height of a sample, respectively; V is its volume and m its mass. Expectedly, the coercivity of samples grows with the stiffness of the matrix. We remark that the maximum field in our experiments was Hmax ¼ 1500 kA=m whereas the field necessary to reach 95% of magnetization saturation of the NdFeB powder is greater than 1600 kA/m according to the datasheets provided by the manufacturer. Due to that, the precise value of the sample magnetization under true saturation remains unknown. Instead of it, in Table 1 we present the maximum measured values Mmax . The left H # and right H " coercive fields were calculated with allowance for the demagnetizing field of the samples approximated by ellipsoids. In the present form, our model is applicable to the magnetization curves of trained MREs. As an example, we take an assembly of 100 particles with randomly distributed orientations h0 and assume that all the particles possess the same pair of effective parameters j and f. The modelled loop is plotted in Fig. 9 along with the measurement data. The calculation parameters were found by means of the least-square best fitting procedure that involved all the available experimental points; the obtained numerical values are f ¼ 0:13 and j ¼ 0. We note that zeroing out of parameter j was expected a priory since the experimental magnetization loop, as seen, does not display any noticeable shift with respect to the field axis. Using the SW model for describing magnetization of the NdFeB microparticles entails one main inaccuracy: in the fitting curve the line MðHÞ at stronger fields comes out horizontal instead of being
Fig. 9. Magnetization loop No.4 for the tested MRE sample (circles) and its model approximation (solid curve); see the text for the parameter values.
ascending as it should be when the elementary magnetic moments gradually turn to the direction of applied field. Comparison with experiment enables one to estimate the reference height of the elastic energy barrier per particle. From the above-introduced definition one has Q ¼ fK u . The uniaxial anisotropy constant of bulk NdFeB used in the experiment was about 4.3 MJ/m3 [20], but for our microparticles it is anticipated to be considerably lower, in particular, due to the fact that the magnetizing field was not strong enough. Assuming that K u 2 MJ/m3 is a reasonable choice, we get for the effective modulus involved in deformation of the cavity: Q 260 kPa. This comes out very close to the macroscopic elastic characteristic of the MRE whose magnetization is presented in Fig. 9. For that sample, the Young modulus measured mechanically in the absence of external field is E 230 kPa. Given the roughness of the approximations that we use, the similarity of the numbers seems rather occasional than true. However, the agreement by the order of magnitude is, in our view, a hefty argument in favor of the proposed model.
5. Conclusions A model for magnetic behavior of a magnetorheological elastomer filled with magnetically hard particles is developed. It is assumed that under the action of external field the particles strive to rotate with respect to the matrix, and, on doing that, encounter elastic resistance of two types. One is spring-like: it links the particle to the position and orientation that it had in a freshlyprepared composite. Another type of elastic resistance, which we propose, is of the energy barrier type. Presumably, it works when a single-domain particle rotates by a notable angle inside the cavity that it occupies inside the matrix. We show that in a sufficiently soft matrix the jumps of the particle magnetic moments caused by surmounting the elastic barrier are able to compete with the Stoner-Wohlfarth intrinsic magnetic switchings of the particle and, thus, substantially contribute to the observed magnetization curves. Comparison with the experimental magnetization loops of magnetorheological polymers of the above-described kind,
Table 1 Physical properties of the experimental samples.
1 2 3 4
E, kPa
H #, kA/m
H ", kA/m
Mmax , kA/m
V; mm3
l, mm
w, mm
h, mm
m, g
45 120 230 2200
63 78 93 253
60 70 81 258
246 234 232 254
15 12 17 10
4.7 4.7 4.7 4.7
2.7 2.5 4.7 2.5
1.2 1.1 1 0.9
0.05 0.036 0.05 0.033
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shows that the hypothesis of the barrier-like processes, at least in qualitative aspect, fits the evidence quite well, whereas the presence/absence of the spring-like elastic resistance is of low relevance. Acknowledgements Support from the RFBR-DFG grant 16–51-12001 (PAK907) is acknowledged. M.V. and Yu.R. are grateful to O. Stolbov for the discussions on the model formulation. References [1] D.Yu. Borin, G.V. Stepanov, S. Odenbach, Tuning the tensile modulus of magnetorheological elastomers with magnetically hard powder, J. Phys: Conf. Ser. 412 (2013) 012040. [2] D. Ivaneyko, V. Toshchevikov, M. Saphiannikova, G. Heinrich, Mechanical properties of magneto-sensitive elastomers: unification of the continuummechanics and microscopic theoretical approaches, Soft Matter 10 (2014) 2213. [3] K.A. Kalina, J. Brummund, P. Metsch, K.A. Kalina, M. Kas¨tner, D.Yu. Borin, J.M. Linke, S. Odenbach, Modeling of magnetic hystereses in soft MREs filled with NdFeB particles, Smart Mater. Struct. 26 (10) (2017) 105019. [4] K. Zimmermann, D. Böhm, T. Kaufhold, J. ChavezVega, T. Becker, S. Odenbach, T. Gundermann, M. Schilling, M. Martens, Investigations and simulations on the mechanical behaviour of magneto-sensitive elastomers in context with soft robotic gripper applications, Int. Sci. J. IFToMM Prob. Mech. 65 (2016) 13– 25. [5] G.J. Monkman, D. Sindersberger, A. Diermeier, N. Prem, The magnetoactive electret, Smart Mater. Struct. 26 (7) (2017) 075010. [6] T.I. Becker, Yu.L. Raikher, O.V. Stolbov, V. Böhm, K. Zimmermann, Dynamic properties of magneto-sensitive elastomer cantilevers as adaptive sensor elements, Smart Mater. Struct. 26 (2017) 095035. [7] M. Mayer, R. Rabindranath, J. Börner, E. Hörner, A. Bentz, J. Salgado, H. Han, H. Böse, J. Probst, M. Shamonin, G.J. Monkman, G. Schlunck, Ultra-soft PDMSbased magnetoactive elastomers as dynamic cell culture substrata, PLoS One 8 (2013) e76196.
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Research articles
Effect of local elasticity of the matrix on magnetization loops of hybrid magnetic elastomers M.V. Vaganov a,⇑, D.Yu. Borin b, S. Odenbach b, Yu.L. Raikher a a b
Institute of Continuous Media Mechanics – The Division of Perm Federal Research Center, Russian Academy of Sciences, Ural Branch, Perm 614013, Russia Technische Universität Dresden, Dresden 01062, Germany
a r t i c l e
i n f o
Article history: Received 9 July 2017 Received in revised form 2 December 2017 Accepted 4 December 2017 Available online 5 December 2017 Keywords: Magnetorheological elastomers Hybrid magnetic elastomers Magnetization curves Magnetic hysteresis FORC diagrams
a b s t r a c t To model magnetization loops of magnetorheological elastomers (MREs) with magnetically hard filler, we consider an assembly of single-domain particles possessing inversion-symmetrical shape and embedded in a soft polymer matrix. To describe the intrinsic behavior of the particle magnetic moments under an applied field, the Stoner-Wohlfarth approach is employed. Unlike the case of solid matrix, the particle in a MRE is able to rotate relative to its elastic environment, so that its equilibrium orientation results from the balance between the magnetic torque (exerted by the applied field at any magnetic moment that does not point along the field) and the elastic torque generated by the matrix. We assume that elastic resistance to the field-induced particle rotation could be presented as comprising of two contributions. The first one is valid for any not perfectly spherical particle and is independent of the particle–matrix adhesion. It reflects the fact that, when trying to rotate, the particle has to ‘‘shoulder its way” by deforming the adjoining regions of the matrix. In that case, for angular deviations up to 90° from the initial position, the resistance torque increases. However, as soon as the rotation angle grows up to the value but infinitesimally exceeding 90°, the elastic torque changes its sign and from now on forces the particle to rotate to 180°, where it attains the geometrical position that coincides with the initial one. Evidently, this process is of the barrier type: both orientations of the particle are equal in elastic energy. The second mechanism stems from the ‘‘memory” that a given particle has of its initial state and may be caused, for example, by some macromolecules grafted to its surface while curing the matrix of the MRE. This restoring force always tends to drag the particle to its 0° (initial) position. Therefore, the observed magnetization of a MRE sample comes out as a result of joint interplay of the Stoner-Wohlfarth and the two elastic mechanisms. In our simulations, we show that the proposed model in a natural way accounts for the two essential features observed (solely or together) in experiment, namely: (i) weak net coercivity of the MREs, whose filler particles as themselves are highly coercive, and (ii) asymmetric positions of the magnetization loops with respect to H ¼ 0 point at the ðM HÞ plane. Ó 2017 Elsevier B.V. All rights reserved.
1. Introduction Composite materials consisting of a soft polymer matrix and embedded into it magnetic particles, can change their physical properties in response to the applied magnetic field. Due to this specific ability, such smart materials are called magnetorheological, magnetoactive or magnetosensitive elastomers. Being synonyms, these three terms are used interchangeably in literature [1–3]. The interest to these field-responsive materials is based on wide prospects of their technical and high-tech [4–6], and biomedical [7,8] applications. In below, we discuss some aspects of ⇑ Corresponding author. E-mail address: [email protected] (M.V. Vaganov). https://doi.org/10.1016/j.jmmm.2017.12.016 0304-8853/Ó 2017 Elsevier B.V. All rights reserved.
internal magnetomechanics of these composites under the action of external fields and hereafter use MRE as a unique acronym designating these composites. Hybrid magnetic elastomers containing a mixture of two distinctly different magnetic fillers – magnetically hard (MH) and magnetically soft (MS) microparticles – make a special class of MREs. As shown in Refs. [9–11], the magnetic properties of hybrid MREs differ significantly both from those of customary MREs with just magnetically soft filler and from those of solid magnetics. In this work we investigate theoretically a model hybrid MRE which contains only MH particles. To justify this choice, we remark that, according to the available experimental data, the MS component, if added, apart from increasing the saturation magnetization of the material, does not bring any new specifics in magnetic behavior
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of a sample. Experimental investigation of the MH particles magnetization has been presented in Refs. [1,9–11]. In what follows, we, first, develop a model describing the response of a magnetically hard particle embedded in a soft elastomer matrix. We then use that model to simulate magnetization loops of the samples and, finally, give an example of application of the theory taking for that the data obtained on MREs filled with NdFeB microparticles.
2. Model When constructing the model, we aim at the experimentally available MREs where the MQP-S-11-9-20001-070 powder (Magnequench, Inc.) was used as a filler. This powder consists of initially isotropic roundish NdFeB-alloy particles with mean linear size of 46.8 lm, according to the manufacturer. Due to production imperfections, a considerable fraction of the particles come out resembling prolate or oblate ellipsoidal shapes than are perfectly spherical. According to the manufacturer, the particles are produced during centrifugal or spinning cup atomization, which is a type of rapid solidification processes [12]. A NdFeB microparticle of that kind consists of metallurgical grains with the size about 20 nm [13] that is below the critical size of single-domainness for NdFeB (0.21 lm [14]). Since in the melt-spun materials the domain walls coincide with the grain borders [15], then during magnetization these single-domain grains should fairly accurately obey the Stoner-Wohlfarth (SW) model [16]. The same should apply to the materials produced by the HDDR (hydrogenation, disproportionation, desorption, and recombination) method, which yield the grains about 0.3 lm [17]. When subjected to a strong field (first magnetization), each grain (domain) of the NdFeB microparticle switches in the direction of the field. On turning off the field, the elementary magnetic moment falls to the closest (in orientational sense) potential well. The resulting net magnetic moment of the particle, in accordance with the SW model, is a half of the maximal value [10,16] Due to that, under a cycled field (its negative values included) the microparticle displays a magnetization loop very similar to the hysteresis obtained with the SW model for an assembly of grains with random distribution of uniaxial anisotropy axes [10]. Hereby, we neglect this detail and approximate magnetization of a NdFeB particle, once subjected to a strong field, with a single SW cycle. As it will be demonstrated in Section 4, the discrepancies resulting from that deliberate simplification are of some quantitative nature but are not relevant for the main line of the work: introducing an extended concept of magnetization for the MREs filled with MH particles. Bender et al. [18,19] modified the conventional SW energy expression to allow for the linear elastic response (Hook resistance) of the matrix to mechanical rotations of the particles. It has been done in order to study polymer-based dispersions with acicular magnetically hard filler. Hereby we modify this magnetoelastic approach in the following way. Consider a typical MRE based on weakly linked silicone rubber plasticized with silicone oil. In such a matrix, an MH particle sits inside a cavity formed around it during the polymerization process. It is tightly enveloped by the environment but is not too strongly ‘‘glued” to it. Because of that, in response to an exerted mechanical torque, the particle would strive to rotate. Moreover, in this motion the plasticizer distributed all over the matrix would serve as a lubricant. If the particle in question is to some extent anisometric, then for its rotation it ought to move apart the surrounding matrix because, when turned from the equilibrium position, the particle is not any longer congruent with the cavity. Apparently, this inference
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equally applies to prolate and oblate objects. Let n be the unit vector of major axis of an axially symmetrical particle, and m the unit vector characterizing the enveloping cavity (Fig. 1) that has the same shape. Those vectors coincide ðn ¼ m Þ in equilibrium, but turn from one another under particle rotation. Evidently, the resistance torque grows with the deviation angle x until the particle attains position n ? m , i.e., x ¼ 90 . As soon as this orientation is traversed, the torque changes its sign: in order to restore its shape, the elastic cavity now generates the stress that urges the particle towards position n ¼ m , i.e., to x ¼ 180 . Whatever the exact form of angular dependence of the torque, one concludes that this function is even with respect to the direction x ¼ 90 , i.e., orthogonal configuration n ? m . This implies that the corresponding term to be added to the particle energy is of the barrier type. With regard to that, we approximate the in-cavity energy barrier term as QVðnm Þ2 ¼ QVðnm Þ2 ¼ cos2 x, where Q is the effective elastic modulus and V the particle volume. To account for a general case, the Hook-like term should be retained to the particle energy expression as well. The basis for that is at least twofold. First, the point-to-point adhesion at the particle/matrix interface is not total zero, so that, when rotating, the particle has to pull and unwind some macromolecules attached to its surface. Second, even if the barrier-type energy prevails, a real particle hardly has perfect inversion-symmetrical shape; then, upon turning it by 180°, its congruence with the cavity would be incomplete. For such a particle the energy wells (minima) positioned at n ¼ m are not equally deep: the one at n ¼ m is preferable. Therefore, to allow for possible coupling of the particle with its initial position, we, as in Ref. [19], add to the energy expression a quasi-Hookean term GVarccos2 ðnm Þ. Let the magnetic moment m of the considered MH particle at the initial state be directed along n. Then the above-described incavity 180°-rotation causes mechanical inversion of the particle magnetic moment without changing its position inside the particle: m rotates together with n. Meanwhile, the SW process renders a different (non-mechanical) type of re-orientation of m that does not require any mobility of the particle axis n. This means that in reality the vectors m and n are independent orientational variables. With allowance for the above-given explanations we present the orientation-dependent energy of an MH particle embedded in an elastic matrix and subjected to a quasistatic external field H ¼ Hh in the form
Fig. 1. MH particle rotation in an elastic matrix.
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U=V ¼ K u ðenÞ2 l0 M s HðehÞ Q ðnm Þ2 þ Garccos2 ðnm Þ;
ð1Þ
where K u is the energy density of the particle internal anisotropy (crystallographic or magnetostatic), which we assume to be uniaxial and which direction is defined by the particle axis n. In formula (1), e ¼ m=m is the unit vector of magnetic moment whose magnitude is m ¼ M s V with Ms being saturation magnetization of the particle material; l0 is the magnetic permeability of vacuum. Setting xOz plane of the coordinate frame in such a way that it passes through vectors m and h, one transforms the energy expression (1) to
U=V ¼ K u sin ðh uÞ l0 M s H cos u þ Q sin x þ Gx2 ; 2
2
ð2Þ
where the choice of angles is presented in Fig. 1, and x ¼ h0 h. Due to the fact that the particles at the beginning of the experiment are magnetically isotropic, after initial magnetization their magnetic moments will be aligned with that direction of the anisotropy axes which make acute angles with the direction of the initially applied field, i.e. h0 2 ½0; 90 . Introducing nondimensional units in (2), one arrives at the expression
1 2
1 2
1 2
g ¼ U=2K u V ¼ sin2 ðh uÞ f cos u þ f sin2 x þ jx2 ;
Fig. 2. Unhysteretic magnetization curve of a particle at h0 ¼ 5 ; f ¼ 0 and
j ¼ 0:2.
ð3Þ
where f ¼ l0 M s H=2K u is normalized magnetic field, f ¼ Q =K u characterizes the elastic energy barrier, and j ¼ G=K u is the linear elasticity parameter. 3. Magnetization curves Eq. (3) provides a direct way of determining the magnetization curve of a particle surrounded by an elastic medium. For that, one has to monitor, by minimization of function gðu; hÞ, the evolution of projection of e on the direction of the field h, i.e., cos u, under cyclic change of parameter f. We note that upon using only first two terms in the rhs of expression (3), one would recover the well-known SW magnetization loop at a given tilt h0 of the particle anisotropy axis. As well known, all those SW loops are centrally-symmetirical in the ðf ; cos uÞ plane. Adding the third term to the rhs of (3), i.e., accounting for the elastic barrier, preserves that symmetry. This is the forth (Hook-like) term that changes the situation. In below, to reveal the role of each of the two elastic contributions, we set in (3) coefficients f and j to zero separately. 3.1. Effect of linear elasticity at f ¼ 0 As indicated above, the forth term in the rhs of expression (3) renders the effect of elastic restoring force that always strives to return the particle to the initial position. Considering a MRE with a very soft matrix, one realizes that it would display unhysteretic behavior because the reorientation of m would go reversibly just by mechanical rotation of the particle. The magnetization curve for this situation is shown in Fig. 2, where the red square marks the region inside which all the possible SW magnetization loops are located. Note the difference between the normalized magnetic moment m=m ¼ h cos u values at f ¼ 1. Manifestation of the same effect in an assembly of particles whose anisotropy axes ðnÞ are distributed at random and whose magnetic and elastic energy factors are comparable ðj 1Þ, is shown in Fig. 3. The essential point is that for a given matrix elasticity j, there exists an angle hjump dividing all the particles on those exhibiting hysteretic magnetization ðh0 < hjump Þ and those changing their magnetization in a non–hysteretic way. The jumps of m occur only for those particles for which internal magnetic
Fig. 3. Magnetic hysteresis of a sample containing 100 particles with random distribution of anisotropy axes.
energy barrier is less than the elastic energy imposed by deforming the surrounding matrix. Therefore, only those particles make the loop width non-zero in Fig. 3 whose anisotropy axes are tilted at ðh0 < hjump Þ. All the other ones contribute only to the shift of the loop along the f axis towards negative values of the applied field. The same difference in remagnetization of the particles explains the non-equivalence between the upper and lower branches with respect to the vertical axis. Indeed, whereas the upper branch accumulates contributions from all the particles, the lower one sums up the contributions only from the particles prone to the magnetic switching, cf. Fig. 2. Therefore, in the case where just the linear elasticity term is taken into account, the occurring hysteresis is of exclusively SW origin (magnetic switching). 3.2. Effect of elastic barrier at
j¼0
The elastic energy barrier corresponding to the third term in the rhs of expression (3) implies a possibility of hysteretic mechanical rotation of an MH particle. Consider a particle whose magnetic moment in the field-free state points along that direction of the easy axis n which makes an acute angle with the direction of the magnetic field that had been imposed in the first magnetization process. When a weak negative field is applied, the elastic barrier would allow the particle only a small mechanical turn, of n and e together, from m . However, under enhancement of the field magnitude, the magnetic torque exerted on the particle would result in the increase of deviation angle. Finally the torque would become strong enough to enable the particle to overcome the energy barrier of height QV. This mechanical jump would entail an abrupt increase of the projection of e on h. Similarly, when going back
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from Hmax to Hmax , the mechanical (and, thus, magnetic) jump towards the positive values would take place. For sufficiently magnetically hard particles, the whole magnetic hysteresis loop (of elastic origin) would be available under the field cycle whose amplitude is lower than that required by the SW scenario; this case is illustrated by Fig. 4. The dependence of the barrier-induced model coercivity on the elastic barrier magnitude is presented in Fig. 5. If f ¼ Q =K u > 1, i.e., the energy of magnetic anisotropy is lower than its elastic analog, then for the particle it is more efficient to change its magnetization not by rotating but by switching the magnetic moment from one direction of the easy axis n to another. However, the softer the matrix the less coercive the particle behavior is. 3.3. Magnetization curve under combined elastic resistance An example of magnetization loop for a particle that experiences both kinds of elastic resistance is shown in Fig. 6. We note that the obtained model curve in a natural way accounts for the two most specific features distinguishing the MRE magnetization loops from those displayed by solid composites, namely, an extremely low coercivity and a pronounced shift along the field axis. 4. Comparison with experiment The MRE samples for experimental characterization were manufactured using polydimethylsiloxane (PDMS) matrix from Wacker
Fig. 4. Comparison of hysteresis loops at h0 ¼ 5 of a SW particle: with infinite (magenta) and finite elastic barrier f ¼ 0:3 (blue). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Fig. 5. Dependence of the modelled coercivity f c on the barrier parameter f 2 ½0; 10.
Fig. 6. Hysteresis of a particle with h0 ¼ 5 under combination of two elastic effects at f ¼ 0:3 and j ¼ 0:1.
Elastosil and magnetically hard NdFeB alloy powder MQP-S-11-920001-070 from Magnequench consisting of roundish particles with the mean linear size of 46.8 lm. The elasticity of the matrix was tuned using dilution of the liquid rubber component with silicone oil M100 Baysilone from Bayer. The powder was modified using a mixture of ether and silicone oil in order to provide a better compatibility of the particles with the silicon matrix and to avoid the magnetic filler aggregation. The mixture of the modified powder, liquid matrix and cross-linking agent was mechanically stirred and degassed. Further, it has been poured into a mould made of hard polyethylene and subsequently polymerized at temperature 100 °C for one hour under rotation in the oven at 50 rpm in oreder to prevent sedimentation of the particles. In this way rod-like samples of MRE with overall concentration of magnetic filler 40 vol.% and various elasticity were produced. The elastic modulus was evaluated by means of quasi-static elongation. Magnetization curves were obtained on Lakeshore vibrating sample magnetometer 7407 which provides static magnetic field up to 2000 kA/m and accuracy of magnetization measurement 1 A/m. A scanning electronic microscopy image of the NdFeB particles in the PDMS matrix is presented in Fig. 7 adopted from [10]. The magnetization curve of the magnetically hard particles in epoxy can be found in the same reference. Magnetic measurements on MREs, performed repeatedly, reveal one more essential detail regarding the hysteresis loops. A typical MRE sample displays training effect: when sweeping the field from Hmax to Hmax and back, the coercivity gradually decreases during the first two–three cycles, so that the consecutive loops, like those shown in Fig. 8, become reproducible only from the fourth cycle. The same effect was reported in Refs. [9–11].
Fig. 7. Scanning electron microscopy image of the NdFeB particles in the PDMS matrix.
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Fig. 8. Consecutive magnetization loops of a MRE with Young modulus E ¼ 230 kPa; loop No.1 (points), loop No.4 and all the next ones (solid line); the unit of the field is Hs = 1500 kA/m.
We ascribe the training effect to the ‘‘smoothing” of the cavity/particle interface resulting, for example, from removal of some weak adhesion links and/or from reconfiguring the adjacent polymer regions due to the microdeformations imposed by smallscale surface irregularities of the particle. To model the training effect, one would need a deeper knowledge of mechanochemistry than we have at the moment. Meanwhile in Table 1 some basic properties of the trained experimental samples are presented. Parameters l; w, and h in the Table stand for the length, width and height of a sample, respectively; V is its volume and m its mass. Expectedly, the coercivity of samples grows with the stiffness of the matrix. We remark that the maximum field in our experiments was Hmax ¼ 1500 kA=m whereas the field necessary to reach 95% of magnetization saturation of the NdFeB powder is greater than 1600 kA/m according to the datasheets provided by the manufacturer. Due to that, the precise value of the sample magnetization under true saturation remains unknown. Instead of it, in Table 1 we present the maximum measured values Mmax . The left H # and right H " coercive fields were calculated with allowance for the demagnetizing field of the samples approximated by ellipsoids. In the present form, our model is applicable to the magnetization curves of trained MREs. As an example, we take an assembly of 100 particles with randomly distributed orientations h0 and assume that all the particles possess the same pair of effective parameters j and f. The modelled loop is plotted in Fig. 9 along with the measurement data. The calculation parameters were found by means of the least-square best fitting procedure that involved all the available experimental points; the obtained numerical values are f ¼ 0:13 and j ¼ 0. We note that zeroing out of parameter j was expected a priory since the experimental magnetization loop, as seen, does not display any noticeable shift with respect to the field axis. Using the SW model for describing magnetization of the NdFeB microparticles entails one main inaccuracy: in the fitting curve the line MðHÞ at stronger fields comes out horizontal instead of being
Fig. 9. Magnetization loop No.4 for the tested MRE sample (circles) and its model approximation (solid curve); see the text for the parameter values.
ascending as it should be when the elementary magnetic moments gradually turn to the direction of applied field. Comparison with experiment enables one to estimate the reference height of the elastic energy barrier per particle. From the above-introduced definition one has Q ¼ fK u . The uniaxial anisotropy constant of bulk NdFeB used in the experiment was about 4.3 MJ/m3 [20], but for our microparticles it is anticipated to be considerably lower, in particular, due to the fact that the magnetizing field was not strong enough. Assuming that K u 2 MJ/m3 is a reasonable choice, we get for the effective modulus involved in deformation of the cavity: Q 260 kPa. This comes out very close to the macroscopic elastic characteristic of the MRE whose magnetization is presented in Fig. 9. For that sample, the Young modulus measured mechanically in the absence of external field is E 230 kPa. Given the roughness of the approximations that we use, the similarity of the numbers seems rather occasional than true. However, the agreement by the order of magnitude is, in our view, a hefty argument in favor of the proposed model.
5. Conclusions A model for magnetic behavior of a magnetorheological elastomer filled with magnetically hard particles is developed. It is assumed that under the action of external field the particles strive to rotate with respect to the matrix, and, on doing that, encounter elastic resistance of two types. One is spring-like: it links the particle to the position and orientation that it had in a freshlyprepared composite. Another type of elastic resistance, which we propose, is of the energy barrier type. Presumably, it works when a single-domain particle rotates by a notable angle inside the cavity that it occupies inside the matrix. We show that in a sufficiently soft matrix the jumps of the particle magnetic moments caused by surmounting the elastic barrier are able to compete with the Stoner-Wohlfarth intrinsic magnetic switchings of the particle and, thus, substantially contribute to the observed magnetization curves. Comparison with the experimental magnetization loops of magnetorheological polymers of the above-described kind,
Table 1 Physical properties of the experimental samples.
1 2 3 4
E, kPa
H #, kA/m
H ", kA/m
Mmax , kA/m
V; mm3
l, mm
w, mm
h, mm
m, g
45 120 230 2200
63 78 93 253
60 70 81 258
246 234 232 254
15 12 17 10
4.7 4.7 4.7 4.7
2.7 2.5 4.7 2.5
1.2 1.1 1 0.9
0.05 0.036 0.05 0.033
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shows that the hypothesis of the barrier-like processes, at least in qualitative aspect, fits the evidence quite well, whereas the presence/absence of the spring-like elastic resistance is of low relevance. Acknowledgements Support from the RFBR-DFG grant 16–51-12001 (PAK907) is acknowledged. M.V. and Yu.R. are grateful to O. Stolbov for the discussions on the model formulation. References [1] D.Yu. Borin, G.V. Stepanov, S. Odenbach, Tuning the tensile modulus of magnetorheological elastomers with magnetically hard powder, J. Phys: Conf. Ser. 412 (2013) 012040. [2] D. Ivaneyko, V. Toshchevikov, M. Saphiannikova, G. Heinrich, Mechanical properties of magneto-sensitive elastomers: unification of the continuummechanics and microscopic theoretical approaches, Soft Matter 10 (2014) 2213. [3] K.A. Kalina, J. Brummund, P. Metsch, K.A. Kalina, M. Kas¨tner, D.Yu. Borin, J.M. Linke, S. Odenbach, Modeling of magnetic hystereses in soft MREs filled with NdFeB particles, Smart Mater. Struct. 26 (10) (2017) 105019. [4] K. Zimmermann, D. Böhm, T. Kaufhold, J. ChavezVega, T. Becker, S. Odenbach, T. Gundermann, M. Schilling, M. Martens, Investigations and simulations on the mechanical behaviour of magneto-sensitive elastomers in context with soft robotic gripper applications, Int. Sci. J. IFToMM Prob. Mech. 65 (2016) 13– 25. [5] G.J. Monkman, D. Sindersberger, A. Diermeier, N. Prem, The magnetoactive electret, Smart Mater. Struct. 26 (7) (2017) 075010. [6] T.I. Becker, Yu.L. Raikher, O.V. Stolbov, V. Böhm, K. Zimmermann, Dynamic properties of magneto-sensitive elastomer cantilevers as adaptive sensor elements, Smart Mater. Struct. 26 (2017) 095035. [7] M. Mayer, R. Rabindranath, J. Börner, E. Hörner, A. Bentz, J. Salgado, H. Han, H. Böse, J. Probst, M. Shamonin, G.J. Monkman, G. Schlunck, Ultra-soft PDMSbased magnetoactive elastomers as dynamic cell culture substrata, PLoS One 8 (2013) e76196.
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