Influence of microstructure on magnetic properties of anisotropic magnetic elastomers

Abstract

The magnetic properties of elastomers based on silicone matrix and iron microparticles assembled in linear chain-like aggregates of different length are experimentally investigated. For this purpose, elastomer samples with a low concentration of magnetic filler are structured in a magnetic field of various strength. The influence of the particle aggregates morphological characteristics on the macroscopic magnetic response of the samples is revealed. Moreover, the influence of the angle between the direction of particle aggregates, that is, the field applied in the process of crosslinking the polymer matrix, and the direction of the field applied during magnetic measurements on the macroscopic magnetic properties of the composite is taken into account. Magnetic measurements are supported by the evaluation of the real microstructure of the samples using X-ray computed microtomography and corresponding digital image processing methods.

Keywords

magnetic elastomer anisotropy field-structuring microstructure

Introduction

Composites based on magnetic particles and an elastic polymer matrix are called magnetic elastomers. Depending on the size and concentration of magnetic particles and the elasticity of the matrix, such materials can have a significant magnetorheological (MR) effect and are traditionally referred to as MR elastomers (Wereley, 2014). Regardless of the above parameters and the magnitude of the MR effect, the presence of a magnetic component makes such composites magnetically sensitive. The application of an external homogeneous magnetic field during the polymerization process makes it possible to obtain anisotropic material due to the structuring of particles in the initially liquid polymer matrix (see Figure 1).

Figure 1.

Enlarged, cropped isometric images of the magnetic elastomer microstructure, representing an isotropic distribution of magnetic microparticles (a), as well as particles that formed linear aggregates under the influence of a 1500 mT magnetic field applied during the cross-linking of the matrix (b).

It is obvious that the induced anisotropy significantly changes the magneto-mechanical response of composites to an external magnetic field. Recently, the influence of the mutual orientation of particle aggregates and the external magnetic field applied during magneto-mechanical tests on the macroscopic mechanical properties of MR elastomers was systematically investigated, see for example, Yao et al. (2019), Wang et al. (2023). At the same time, the known results of obtaining magnetization curves and determining the corresponding magnetic properties of magnetic elastomers as a function of the mutual orientation of the particle structures and the external field are limited to theoretical predictions, see for example, Romeis and Saphiannikova (2023). The known experimental publications consider parallel and perpendicular orientations only (Borin and Odenbach, 2017). On the other hand, varying the concentration of magnetic filler and the magnitude of the magnetic field applied during material crosslinking makes it possible to obtain a composite with different morphology of particle aggregates, see for example, Borin (2020). However, the relationship between the morphology of particle aggregates and the macroscopic properties of the material has not been sufficiently investigated to date. The question of the influence of the spatial orientation of different particle structures relative to the direction of the magnetic field applied in the experiment remains open. Accordingly, this study examines the influence of the angle between the direction of particle aggregates of different morphologies and the direction of the field applied during magnetic measurements on the macroscopic material response. Measurements of the magnetic characteristics of various composite samples are accompanied by microstructural studies.

Materials and methods

Composition and preparation of samples

The study examines elastomeric composite samples based on two-component polydimethylsiloxane matrix (silicone and crosslinker Neukasiel RTV 230 and A149). The shear modulus of the cross-linked matrix is $G =$ 700 kPa, so the particles in the cured matrix should be immobilized regardless of the magnetization process as shown in Borin et al. (2024). Sigma-Aldrich 209309 iron powder with a particle average diameter of 44 $μ$ m are used as magnetic filler (Figure 2).

Figure 2.

Optical microscopic image of the used Sigma-Aldrich iron powder particles.

The specimens are fabricated by mechanically mixing all the components, using manual stirring and a vortex mixer. The suspension of iron particles in polydimethylsiloxane is degassed under vacuum, poured into the mold, and then cross-linked at room temperature (23°C). The concentration of powder for all samples is 1 wt.% (approx. 0.14 vol.%) and is ensured by accurate weighing of the components prior to mixing. The reference isotropic samples were cross-linked in a zero field. To prevent the sedimentation of microparticles during matrix cross-linking, the isotropic samples were rotated on a laboratory Loopster. An external uniform magnetic field was applied to cross-link the anisotropic samples, as described below. The issue of particle sedimentation for anisotropic samples was irrelevant given the short time required to transfer the samples to the structuring setup.

Structuring and magnetic characterization

The structuring magnetic field is provided by the electromagnet of a vibrating sample magnetometer (Lake Shore VSM 7407), which is further used for magnetic measurements. Accordingly, a standard Lake Shore VSM 730935 sample holders are used as the molds for the specimens. The basic principles for the preparation of composites with different particle aggregate morphologies, visualized using computed tomography methods, are presented in Borin (2020). In the current study, a very low structuring field of about 1 mT is used to obtain a composite with short chains. Whereas a high field of 1500 mT is used to obtain a composite with long chains. The spatial orientation of particle structures relative to the direction of the external field in magnetic measurements is changed by rotating the VSM rod with the sample holder around the vertical axis. The magnetization curve is measured for different values of angle $α$ , which is the angle between the direction of the external magnetic field during measurement (magnetization field) and the direction in which the magnetic particle aggregates are oriented during matrix cross-linking (structuring field), see Figure 3. The analyzed parameter is an apparent initial magnetic susceptibility $χ_{ini}$ , that is, the slope of the initial linear section of the measured magnetization $M$ versus effective magnetic field $H$ dependence. A magnetometer measures the magnetic moment in the direction of the applied external field, and it is assumed that for low fields in which magnetization is linear, the global macroscopic magnetization of the sample under investigation coincides in direction with the external field $H_{ext}$ (see e.g. Abbott et al., 2007) Effective field $H$ is defined as $H = H_{ext} - N_{g} \cdot M$ , where $H_{ext}$ is measured by a VSM Tesla meter and $N_{g}$ is the global geometric demagnetization factor, which is taken as 0.44 for the sample’s shape approximated as ellipsoidal in the specified holder.

Figure 3.

Schematic view of the sample holder (left) and the orientation of the sample during the experiment (right).

Microstructural characterization

Microstructural investigations are conducted using the own laboratory X-Ray microtomography setup TomoTU (Borin and Odenbach, 2023). To avoid damaging the elastomer composite structured directly in the VSM sample holder, tomography was performed without removing the specimen from it. The reconstruction process is performed calculating a three-dimensional model from the individual radiographs using a Feldkamp–Davis–Kress algorithm (Feldkamp et al., 1984) based software package developed in house. The open source platform Fiji (Schindelin et al., 2012)) was used for digital processing and analysis of the reconstructed cross-sectional images obtained.

Results and discussion

Visualization of the different internal microstructure of studied magnetic elastomer samples is shown in Figure 4. A composite with short chains formed by structuring of microparticles in a low ( $1$ mT) magnetic field (a), a composite with long chains formed by structuring of microparticles in a high magnetic field ( $1500$ mT; b) are shown next to the isotropic specimens (c).

Figure 4.

Obtained using X-Ray computed microtomography three-dimensional renderings of exemplary microstructures of the specimens: (a) short chain composite, (b) long chain composite, and (c) isotropic composite.

Figure 5 shows examples of reconstructed and processed images of cross sections of samples with short and long chain aggregates and isotropic particle distribution. At the same time, Figure 6 shows examples of longitudinal section images of the samples under study, with the section plane parallel to the direction of the structuring field.

Figure 5.

Processed reconstructured X-Ray tomographic images of exemplary cross sections of specimens in VSM specimen holder: (a) short chain composite, (b) long chain composite, and (c) isotropic composite.

Figure 6.

Processed reconstructured X-Ray tomograpnhic images of exemplary longitudinal sections of specimens in VSM specimen holder (section plane parallel to the direction of the structuring field): (a) short chain composite, (b) long chain composite, and (c) isotropic composite.

Using a median filter and binarization of the cross-sectional and longitudinal section images, a quantitative assessment of the length of chain aggregates was performed.

Chain aggregates in a composite structured in a field of $B$ = 1500 mT completely bridge the sample holder. Accordingly, their maximal length corresponds to the sample holder diameter and is up to approximately 3400 $μ$ m. The shortest chains in this composite are located closer to the edges of the sample holder and are approximately 1500 $μ$ m long. The average chain length in this composite, determined from cross-sectional images, is $2550 \pm 820 μ$ m.

In a composite structured in a field of $B$ = 1 mT, significantly shorter chains are distributed throughout the sample holder. The maximum length of the chains, as calculated from both cross-sectional and longitudinal section images, is up to approximately $~ 1140 μ$ m, while the shortest chains have a length of $~ 100 μ$ m. The average chain length in this composite, determined from cross-sectional and longitudinal section images, is $360 \pm 250 μ$ m and $~ 500 \pm 320 μ$ m correspondingly.

The thickness of the observed chain aggregates corresponds to the diameter of individual powder particles. Thus, considering the average chain lengths, the length-to-thickness ratio of the aggregates is roughly 60 for a composite structured at $B$ = 1500 mT and 10 for a composite structured at $B$ = 1 mT.

Figures 7 to 9 show the magnetization curves of differently aligned samples with chains of particles of varying lengths and an isotropic microstructure. The direction of the sample structuring field in Figure 7 coincides with the direction of the magnetometer magnetic field (angle $α = 0^{°}$ ), that is, the chains of particles are aligned with $H$ . The macroscopic magnetic susceptibility of an isotropic sample is lower than that of a structured sample, and the sample with the longest average particle length has the highest susceptibility. As the angle $α$ increases, the difference in susceptibility decreases (see, e.g. Figure 8 for $α = 45^{°}$ ). In the case of perpendicular orientation of chains relative to the field $H$ , the susceptibility for short and long aggregates is the same and less than for a composite with isotropic particle distribution. (Figure 9). The dependence of the apparent initial magnetic susceptibility $χ_{ini}$ , as extracted from magnetization curves, on the angle $α$ between the field $H$ and the orientation of particle aggregates is shown in Figure 10.

Figure 7.

Magnetization curves of samples with long and short particle chains oriented parallel ( $α = 0^{°}$ ) to the field applied by VSM, and isotropic particle distribution.

Figure 8.

Magnetization curves of samples with long and short particle chains oriented at $α = 45^{°}$ to the field applied by VSM, and isotropic particle distribution.

Figure 9.

Magnetization curves of samples with long and short particle chains oriented perpendicular ( $α = 90^{°}$ ) to the field applied by VSM, and isotropic particle distribution.

Figure 10.

Apparent initial magnetic susceptibility of the samples with long and short particle chains oriented at various angle $α$ to the field applied by VSM.

The obtained dependencies clearly demonstrate the relation between macroscopic magnetic properties and microstructure anisotropy, which determines the effective magnetic field affecting the composite specimen. When evaluating the effective magnetic field, the influence of particle aggregates must also be taken into account. The demagnetizing factor of specimens with complex magnetic particle mixtures depends not only on the global demagnetizing factor $N_{g}$ , but also on the geometric shape of particles and particle aggregates that differ from the global shape. That is, on the demagnetizing factors of particles $N_{p}$ and aggregates $N_{a}$ . In accordance with Skomski et al. (2007), accepting certain simplifications and assumptions, the demagnetizing factor of a particulate composite with a nonmagnetic matrix can be estimated as

N = N_{p} (1 - f_{p}) + N_{a} f_{p} (1 - f_{a}) + N_{g} f_{a} f_{p},

(1)

where $f_{a}$ is the volume fraction of the aggregates in the global matrix and $f_{p}$ is the volume fraction of the particles in the aggregates. For an isotropic sample with a very low particle concentration ( $f_{a} = 1$ and $f_{p} \to 0$ ), the demagnetizing factor of the particles, $N_{p}$ , is decisive.

Generally, for the samples considered in this study, the variable parameters are the shape of the particle aggregates, that is, the parameter $N_{a}$ , and the volume fraction of the particles in the aggregates $f_{p}$ . Both are determined by the structuring field. Moreover, assuming that the chain aggregates are linear and consist of individual microparticles, then the only variable characteristic is $N_{a}$ . Approximating the chain aggregate as a long rod, a simple expression obtained by Sato and Ishii (1989) can be used to determine $N_{a}$ for $α = 0^{°}$ , that is, the chain of particles is parallel to the field $H$ , and $α = 90^{°}$ , that is, the chain of particles is perpendicular to the field $H$ :

N_{a} (α = 0^{°}) = \frac{1}{2 (2 \frac{l}{d} / \sqrt{π}) + 1},

(2)

N_{a} (α = 90^{°}) = \frac{2 \frac{l}{d} / \sqrt{π}}{2 (2 \frac{l}{d} / \sqrt{π}) + 1},

(3)

where $l / d$ is the ratio between the length and diameter of the rod. For the accuracy, it is important to note that these expressions are simplifications. For instance, they do not take into account the non-uniformity of the demagnetizing field in the rod.

The ratio $l / d$ , estimated above from microstructural analysis for average aggregate sizes, is approximately 60 and 10 for long and short chains, respectively. The demagnetizing factor $N_{a} (α = 0^{°})$ is inversely proportional to the ratio $\frac{l}{d}$ , therefore, it is several times smaller for longer chains. In particular, using equation (2) $N_{a} (α = 0^{°}) \approx$ 0.007 for $\frac{l}{d}$ = 60 and $N_{a} (α = 0^{°}) \approx$ 0.042 for $\frac{l}{d}$ = 10. The apparent initial magnetic susceptibility is inversely proportional to the demagnetizing factor. Consequently, at $α = 0^{°}$ , the composite with longer chains has a higher $χ_{ini}$ than the composite with shorter chains, as it is also observed in the experimental measurements (see Figure 10). When the chain of particles is oriented perpendicular to the magnetic field, the difference between the demagnetizing factors of the long ( $\frac{l}{d}$ = 60) and short ( $\frac{l}{d}$ = 10) chains is relatively small. In particular, using equation (3) $N_{a} (α = 90^{°}) \approx 0.496$ and $N_{a} (α = 90^{°}) \approx 0.478$ for the long and short chain respectively. This reasons why, in the experiment, the apparent initial magnetic susceptibility of both types of composite at $α = 90^{°}$ is relatively the same.

Changing the orientation angle of the sample in the range $0^{°} < α < 90^{°}$ changes the surface area of particle aggregates facing in the direction of the applied field and consequently the parameter $N_{a}$ in the direction of the field. The surface projection of the aggregate onto a plane perpendicular to the applied magnetic field is determined by the cosine function. Accordingly, the factor $N_{a}$ increases and $χ_{ini}$ decreases with $α$ increasing from $0^{°}$ to $90^{°}$ . The experimental dependencies $χ_{ini} (α)$ for structured samples also show the decrease of $χ_{ini}$ with increasing $α$ . Moreover, these dependencies are fairly well fitted by a cosine squared function of the form $χ_{ini} = a \cos^{2} α + b$ , where $a$ and $b$ are free parameters (see dotted curves in Figure 10). The asymptotic standard error for all free parameters in both fits does not exceed 1.5%. This function is chosen because of the bidirectional symmetry characteristic of soft magnetic materials. Thus, considerations regarding the changing demagnetizing factor of particle aggregates depending on the angle of application of the field to the composite are confirmed.

The fact that $χ_{ini}$ at $α \to 90^{°}$ for structured samples is lower than $χ_{ini}$ for isotropic samples is also related to the difference in demagnetizing factors. The shape of individual particles can be approximated by a sphere or ellipsoid with a moderate difference in the lengths of the principal axes (see Figure 4(c)), that is, the maximum demagnetizing factor for particles of this shape is in the range $N_{p} ~ 0.33 - 0.41$ (see e.g. Osborn, 1945). That is, $N_{p} < N_{a} (α \to 90^{°})$ and, accordingly, $χ_{ini}$ of the isotropic sample is higher than that of samples with transverse alignment of chains to the applied field. Thus, the experimentally obtained dependencies $χ_{ini} (α)$ for composites with different internal anisotropy are attributed involving quantitative analysis of the actual microstructure.

Summary and outlook

Through experimentation, we demonstrated differences in the magnetization of magnetic elastomers with various microstructural anisotropies. We focused on elastomers with linear chain aggregates. Quantitative microstructure analysis is key to understanding the macroscopic magnetic behavior of the studied composites. The magnetic properties of these elastomers depend strongly on the morphology of the aggregates and their relative alignment to the external magnetic field applied during magnetic characterization. Structured samples exhibit a strong dependence of the apparent initial susceptibility on the tilt angle. Samples with longer chains exhibit a greater change in susceptibility compared to samples with shorter chains. This behavior is due to the tilt-angle-dependent demagnetizing factor of the single particle chain aggregates. For composites with more complex microstructures further research is needed. The study examined composite samples containing a low concentration of magnetic particles in a sufficiently rigid matrix. The magnetorheological effect, that is, the change in viscoelastic properties induced by a magnetic field, is absent in such materials; consequently, investigations into their mechanical properties were of no interest. Nevertheless, from an application perspective, such composites possess precisely tunable magnetic anisotropy, which is induced during manufacture by controlled structuring using an applied field. Therefore, they are of interest for sensing devices similar to advanced, flexible, anisotropic magnetic films. See, for example, the corresponding review in Wu et al. (2026). The results obtained in the current study can serve as a basis for further development toward sensor applications.

Footnotes

ORCID iD

Dmitry Borin

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: N.M. and S.O. are gratefully acknowledging the financial support by Deutsche Forschungsgemeinschaft (DFG) under Grant OD 18/35-1 within Research Unit FOR5599. D.R. would like to acknowledge the financial support by DFG under Grant RO 6756/3-1.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data availability statement

The data that support the findings of this study are available upon reasonable request from the authors.

References

Abbott

Ergeneman

Kummer

, et al. (2007) Modeling magnetic torque and force for controlled manipulation of soft-magnetic bodies. IEEE Transactions on Robotics 23: 1247–1252. https://doi.org/10.1109/tro.2007.910775

Borin

(2020) Targeted patterning of magnetic microparticles in a polymer composite. Philosophical Transactions of the Royal Society A 378: 20190256. https://doi.org/10.1098/rsta.2019.0256

Borin

Odenbach

(2023) On the use of X-ray microtomography to investigate the field-driven structure and rheology in magnetorheological elastomers. Chapter 8. In: de Vicente

(ed.) Magnetic Soft Matter.The Royal Society of Chemistry, pp.213–228. https://doi.org/10.1039/BK9781839169755-00213

Borin

Vaganov

Odenbach

(2024) Magnetic training of the soft magnetorheological elastomers. Journal of Magnetism and Magnetic Materials 589: 171499. https://doi.org/10.1016/j.jmmm.2023.171499

Borin

Odenbach

(2017) Initial magnetic susceptibility of the diluted magneto polymer elastic composites. Journal of Magnetism and Magnetic Materials 431: 115–119. https://doi.org/10.1016/j.jmmm.2016.07.055

Feldkamp

Davis

Kress

(1984) Practical cone-beam algorithm. Journal of the Optical Society of America A 1: 612.

Osborn

(1945) Demagnetizing factors of the general ellipsoid. Physical Review 67: 351–357. https://doi.org/10.1103/physrev.67.351

Romeis

Saphiannikova

(2023) Effective magnetic susceptibility in magnetoactive composites. Journal of Magnetism and Magnetic Materials 565: 170197. https://doi.org/10.1016/j.jmmm.2022.170197

Sato

Ishii

(1989) Simple and approximate expressions of demagnetizing factors of uniformly magnetized rectangular rod and cylinder. Journal of Applied Physics 66: 983–985. https://doi.org/10.1063/1.343481

10.

Schindelin

Arganda-Carreras

Frise

, et al. (2012) Fiji: An open-source platform for biological-image analysis. Nature Methods 9: 676–682. https://doi.org/10.1038/nmeth.2019

11.

Skomski

Hadjipanayis

Sellmyer

(2007) Effective demagnetizing factors of complicated particle mixtures. IEEE Transactions on Magnetics 43: 2956–2958. https://doi.org/10.1109/tmag.2007.893798

12.

Wang

Chen

Jiang

, et al. (2023) Magneto-mechanical properties of anisotropic magnetorheological elastomers with tilt angle of magnetic chain under compression mode. Journal of Magnetism and Magnetic Materials 570: 170441. https://doi.org/10.1016/j.jmmm.2023.170441

13.

Wereley

(2014) Magnetorheology. RSC Publishing.

14.

Liu

, et al. (2026) A review of magnetic elastomers from material design and fabrication methods to applications in soft robots. Langmuir 42: 5330–5366. https://doi.org/10.1021/acs.langmuir.5c05890

15.

Yao

Yang

Gao

, et al. (2019) Magnetorheological elastomers with particle chain orientation: Modelling and experiments. Smart Materials and Structures 28: 095008. https://doi.org/10.1088/1361-665x/ab2e21