On anisotropic mechanical properties of heterogeneous magnetic polymeric composites

Abstract

This study is devoted to the magneto-mechanical characterization of heterogeneous magnetoactive elastomers based on an elastic polydimethylsiloxane matrix with embedded spherical magnetic soft microparticles and magnetic hard microparticles of irregular shape. An issue of the anisotropic mechanical properties of these smart composites is considered. Non-magnetized and pre-magnetized specimens are characterized using a planar shear and axial loading in an externally applied homogeneous magnetic field. The field direction differs relative to the direction of the field used for the specimens pre-magnetization. Results of the different methods allow comparison of the tensile shear moduli for the samples with an initially identical composition. Obtained results demonstrate a strong correlation between the composite behaviour and orientation of the magnetic field used for the pre-magnetization of the sample relative to the external field applied to a sample during the test. Composites pre-magnetized in the direction parallel to an applied mechanical force and external magnetic field show higher magnetorheological response than composites pre-magnetized transversally to the force and the field. Application of the external field directed opposite to the direction of the pre-magnetization reduces the observed stiffening. Moreover, in this situation a softening of the material can be observed, depending on the magnitude of the external field and the field used for pre-magnetization.

This article is part of the theme issue ‘Heterogeneous materials: metastable and non-ergodic internal structures’.

1. Introduction

Magnetoactive elastomers (MAEs) belong to a class of smart materials with remote controllable physical properties. These elastomers are heterogeneous composites of a polymeric matrix with embedded magnetic particles. There are a couple of designations used in the scientific literature for these composites, such as magnetorheological (MR) elastomers, ferroelastomers, magnetic polymers, magnetic gels, ferrogels, etc. [1–5]. Conspicuous is the fact that there is no unity in the usage of these definitions despite the quite different material compositions and consequently different physical performances. Therefore, it is reasonable to distinguish at least between composites filled with nanoparticles and those filled with microparticles, as well as between composites with soft gel-like matrices and composites with significantly stiffer resin-like matrices. In this context, ferrogel is a suitable designation for a material based on the nanosized particles embedded into a gel-like matrix, while by the term MAE a composite based on an elastomer with magnetic microparticles is to be understood, as is the case with ferrofluids and magnetorheological suspensions [6,7]. It must be additionally noted that materials which are filled with magnetic powder and which are having a very stiff matrix, such as a rubber, vulcanized under pressure, whose rheological properties are not controllable with an externally applied magnetic field, should not be accounted to the MAEs.

As a filler for MAEs, magnetically soft iron microparticles are commonly used. Nevertheless, it is also possible to use a magnetic hard powder, e.g. NdFeB-alloy particles. The magnetic hard component allows the change of the initial state of the MAE tuning the composite remanence magnetization [8,9]. Recently, an approach to use a mixture of magnetic soft and hard particles has been proposed [10–14]. This approach enhances the range of an active control and simultaneously provides a possibility to passively tune the material properties.

A MAE prior to a polymerization process represents a kind of magnetorheological fluid, which contains movable particles. When a MAE is cross-linked under conventional conditions, one deals with a composite with an isotropic particles distribution, neglecting or avoiding any effects of the gravitational sedimentation of the particles. Experimental microstructural investigations [15,16], as well as theoretical studies [17,18], demonstrate an appearance of the particles' structures in initially isotropic MAEs and ferrogels when an external magnetic field is applied to an already cross-linked sample. These structures disappear after the field is switched off for MAEs based on magnetic soft powders. The formation of particle structure as well as particle movement inside an elastic matrix is known to be an essential reason for the magnetorheological effect, i.e. an increase in an elastic modulus and loss modulus of a composite. On the other hand, when an external magnetic field is applied during the sample cross-linking, particles forming elongated structures in the field direction appear, which are preserved in the cross-linked elastomer network. As a result, a material with the structural anisotropy is obtained [19,20]. It has been shown in the past [21] that composites with a magnetic soft filler that were pre-structured in an uniaxial field exhibit much higher MR effects in an externally applied uniform magnetic field than samples with an isotropic distribution of particles. Additionally, an internal structural anisotropy obviously causes macroscopic anisotropic properties of MAEs. Influence of the direction of the applied field, particle alignment and the mechanical stress on the mechanical properties of the composites with magnetic soft filler, was previously studied in [21]. The most significant MR effect was observed for the case when orientations of field, load and particle structures were parallel to each other. In the MAEs containing a magnetic hard component, the particle structures can persist after switching off the external field. For that, the strength of an applied field to the specimen magnetic field must be high enough to provide a sufficiently high remanence magnetization. Consequently, an initially not structured MAE with a magnetic hard component will obtain anisotropic microstructures, as has been proved in [16]. The MR effect in pre-magnetized MAEs solely based on a magnetic hard filler as well as in MAEs with a hybrid composition, i.e. based on a mixture of magnetic hard and soft fillers, was recently studied in [14]. The direction of the magnetic field used for the specimen's magnetization, the external field and mechanical force applied during the characterization in [14], were parallel to each other. In [9], a MAE based on a magnetically hard filler was studied using oscillatory shear. Samples were pre-magnetized perpendicular to the shear strain direction applied in the experiments, while an application of an external field during the shear loading was not considered. Moreover, as shown in [22], results for oscillatory shear can be misleading when performed with a standard rheometric configuration. Thus, to the best of our knowledge, the anisotropy of the MR effect in MAEs with hybrid compositions has still not received any specific attention. However, it is an important issue which should be addressed by a thorough understanding of the complex MAEs' behaviour and their practical applications. The current experimental study is devoted to the anisotropic mechanical testing of a MAE with a complex hybrid composition. Non-magnetized and pre-magnetized samples based on a polydimethylsiloxane (PDMS) matrix, with a mixture of magnetically soft and hard particles embedded, are studied under the shear and axial loading. These different loading conditions have allowed comparison of the elastic modulus E and shear modulus G for samples with initially identical compositions. Due to a different configuration of the measuring set-up in the shear and axial loading tests, the direction of the externally applied magnetic field could be changed relative to the direction of the field used for the pre-magnetization of the specimens.

This paper is organized as follows. First, basic information on the composition and fabrication of samples as well as the set-up and methods used is given. Next, the results of the experiments and a discussion of the results are presented. Finally, conclusions and outlook are given.

2. Samples and methods

(a). Composition and fabrication of samples

Experimental samples of MAE were fabricated using the two-component silicone elastomer compound (SIEL-254, product of the Russian State Scientific Institute for Chemical Technologies of Organoelement Compounds). The SIEL^®-254 compound is initially liquid and can be cross-linked using a hydride-containing agent. Additionally, Baysilone^® oil M100 was mixed with the SIEL-254 in a 1:1 ratio to obtain a sufficiently soft matrix. The magnetic filler is a mixture of carbonyl iron powder (BASF CIP grade CC) and magnetic hard NdFeB-alloy powder. Particles of BASF carbonyl iron powder (CIP) type CC are spherical (average size approximately 5 µm), while particles of NdFeB-alloy powder have irregular geometry with a size in a range between 40 and 100 µm (figure 1). The surface of the powders' particles is modified using the Baysilone^® oil M100 to avoid particle aggregation, making the particles' surfaces hydrophobic to provide a better compatibility with the matrix, as reported in [9].The overall weight concentration of powder in samples is approximately 82%, and the weight ratio between magnetic soft and hard fractions was 1 : 4. The mixed filler is mechanically stirred into the liquid matrix. After thoroughly stirring and degassing of the suspension, it is cast into moulds and cured at approximately 100°C. In this way, cylindrical samples with a diameter of 14 mm and a height of 15–18 mm as well as plate-like samples with a size of 12 × 12 × 2 mm are prepared (figure 2). More details on a multi-step process of the MAE synthesis can be found elsewhere, e.g. [14].

Figure 1.

Microscopic images of (a) magnetically soft BASF CC and (b) magnetically hard NdFeB particles.

Figure 2.

Photography of the cylindrical and plate-like samples.

(b). Magnetization of samples

In experiments, various magnetic situations of samples were studied. First, all specimens were under investigation for various magnetic fields from 0 mT up to 240 mT applied during mechanical loading. Next, specimens were magnetized at various external fields providing different remanence magnetizations, due to the presence of the magnetic hard component (B_M = 700, 900, 1200, 1500 mT). In figure 3, the normalized by weight magnetization curve M(B) for the magnetic hard NdFeB powder, including several minor loops, and the dependence of the remanence magnetization M_r on a magnetization field, are shown. This was measured with a vibrating sample magnetometer (Lake Shore 7407, USA). As seen in figure 3a, the NdFeB powder is magnetically saturated at B = 1500 mT. Magnetic properties of the heterogeneous MAEs reflect a complex interaction between the particles of the mixed powder and the matrix. This is addressed in [10] on the basis of the first-order reversal curve (FORC) analysis and in [23] on the analysis of the static magnetization curves. Initial magnetization curves of the MAEs with the complex composition, as well as dependencies of their remanence magnetization on the external field, are presented in [14].

Figure 3.

(a) Magnetization curve (minor loops are additionally shown) of the NdFeB-alloy powder and (b) dependency of its remanence magnetization M_r (by weight) on the external field B_M which was applied to magnetize the powder. (Online version in colour.)

The magnetization of the specimens was realized using a uniform field provided by the electromagnet B-E25 (Bruker, Germany). It is assumed that magnetization as well as particle rearrangement processes were completed within the used magnetization time of 5 min. Cylindrical samples were magnetized along the vertical axis and plate-like samples were magnetized in the direction perpendicular to the plane, as schematically shown in figure 4. As has been previously observed in [16], the pre-magnetization results in the induced microstructural anisotropy of the composites due to the particle rearrangements. It has to be noted that effective magnetic fields inside the samples for the two different geometries used and various pre-magnetizations are not equal for identically applied external fields, due to the samples own demagnetizing fields. However, it is not possible to take this issue into account within the frame of the current study, since demagnetization factors for the composites with an induced microstructural anisotropy remain unknown. Therefore, we deal with external fields in this work.

Figure 4.

Configurations of the experimental set-up: (a) axial loading of the cylindrical specimen; (b) shear test of a plate-like composite using a layout with two coupled shear gaps; F is a measured mechanical force, s is a controlled displacement, B_M represents a pre-magnetization of the samples, and B represents an external field applied during the test. (Online version in colour.)

(c). Set-up and methods for magneto-mechanical characterization

Mechanical characterization of the fabricated and magnetized samples was conducted with a table-top testing machine (DynaMess TP 5 kN HF, Germany) equipped with a servo-pneumatic drive and a force unit with a nominal load capacity up to 1 kN. The machine provides a resolution of measurements of 0.01 N and 0.01 mm for the force and displacement, respectively. Two different configurations of the experimental set-up are used (figure 4). For the axial loading test (figure 4a), the cylindrical sample is fixed in a movable upper and static lower fixture made of a non-magnetic alloy. In the shear test, a symmetric layout with two coupled shear gaps is realized (figure 4b). The movable upper fixture provides a controlled deformation of the sample (displacement) and the correspondingly induced force (coaxial to the deformation) functions as measuring parameter. Both types of test were conducted at a frequency of 1 Hz. The maximal deformation was set to 5% to ensure linear viscoelastic behaviour as well as to avoid significant deformation of the cylindrical samples in the radial direction. As a result of the cyclic loading, force–displacement diagrams are obtained. To avoid an influence of the inertia of the apparatus on the measuring results as well as to homogenize the specimen's microstructural state, several cycles of loading were performed prior to each test. For a MAE the force–displacement diagrams are hysteresis loops, representing the specimen's viscoelasticity [14]. The slope of the major axis of the force–displacement loop corresponds to the dynamic or complex stiffness of the specimen. Transforming force and displacement into stress and strain, the storage modulus E′ can be determined for axial tests with the cylindrical specimens and the storage shear modulus G′ for shear tests with the plate specimens. The area of the force–displacement loop represents the dissipated energy per cycle of loading. This energy gives access to the loss factor tan δ as the ratio between the loss and storage modulus. In this study we have focused on the passive and active MR effects regarding the moduli G′ and E′. The passive MR effect is a relative change arising from an induced remanence magnetization. The passive effect was quantified for the shear and tensile modulus respectively as

$$R_{P\mathrm{\_}G}=\frac{G^{\mathrm{\prime }}(B_{\mathrm{M}})}{{G^{\mathrm{\prime }}}_0}-1$$ 2.1

and

$$R_{P\mathrm{\_}E}=\frac{E^{\mathrm{\prime }}(B_{\mathrm{M}})}{{E^{\mathrm{\prime }}}_0}-1,$$ 2.2

where $G_0^{\mathrm{\prime }}$ and $E_0^{\mathrm{\prime }}$ are moduli of a non-magnetized sample, $G^{\mathrm{\prime }}(B_{\mathrm{M}})$ and $E^{\mathrm{\prime }}(B_{\mathrm{M}})$ are moduli of a sample pre-magnetized under a magnetic field with a flux density B_M. The active MR effect is a relative change of the moduli occurring during testing due to an applied external magnetic field with a flux density B. The active effect was quantified as

$$R_{A\mathrm{\_}G}=\frac{G^{\mathrm{\prime }}(B)}{{G^{\mathrm{\prime }}}_0}-1$$ 2.3

and

$$R_{A\mathrm{\_}E}=\frac{E^{\mathrm{\prime }}(B)}{{E^{\mathrm{\prime }}}_0}-1,$$ 2.4

where $G_0^{\mathrm{\prime }}$ and $E_0^{\mathrm{\prime }}$ are moduli of a sample without an applied magnetic field, and $G^{\mathrm{\prime }}(B)$ and $E^{\mathrm{\prime }}(B)$ are moduli of a sample under influence of the field B.

A uniform external magnetic field (B up to 240 mT) in experiments was generated by a cylindrical coil. In all tests the external field was oriented along the direction of the mechanical loading, i.e. it was either co-directional or opposite the pre-magnetization orientation for uniaxial tests with cylindrical samples (compare to figure 4a) and was perpendicular to the magnetization of the plate samples for the shear test (compare to figure 4b).

3. Results and discussion

(a). Active MR effect of the non-magnetized composites

Initially, non-magnetized samples (B_M = 0) are characterized. Magnetic particles in the non-magnetized samples are dispersed randomly. The mechanical force F and the direction of the externally applied magnetic field B are parallel. The application of an external field results in an aggregation of magnetic particles in the direction of the applied field, due to the induced magnetic dipole–dipole interaction. As a result, both elastic and shear moduli increase with increasing magnetic field, as is expected for MAEs (figure 5a). The same is true for the active MR effect regarding both moduli (figure 5b). At B = 240 mT the effect reaches approximately 90% for G′ and respectively approximately 130% for E′. The external field applied to the samples (up to 240 mT) is not high enough to significantly influence non-magnetized particles of the hard magnetic NdFeB powder. As has been recently demonstrated within microscopic observations [16], only moderate rotations and movements of magnetic hard particles occur in the elastic matrices of the composites which were not pre-magnetized. Mainly, the soft magnetic CIP particles contribute to the aggregation process. Thus, the active MR effect is not as high as expected from the results recently reported for MAEs filled with CIP only [4,5]. For the incompressible isotropic materials the Poisson ratio is 0.5 and the ratio between moduli should consequently be E/G = 3. For anisotropic materials it is well known that different Poisson ratios need to be taken into account. According to the obtained experimental results, values of the shear modulus G′ are smaller than values of the elastic modulus E′ at all magnetic fields. Microstructural anisotropy induced due to the applied field, i.e. structuring of particles in the field direction, is of the same kind for both the shear test and the axial test. Nevertheless, the increment of the observed MR effect is higher for the elastic modulus E′ than that for the shear modulus G′. This means it is easier to shear the composite with aggregates of particles oriented along the shear than to expose these aggregates to an axial loading. The ratio between the moduli is in the range 2.2–2.7 and the corresponding Poisson ratio would be in the range 0.11–0.36. This is attributed to different relations between the mechanical force and field inducing the structures of particles. The mechanical force F and, accordingly, the stress σ which is required to deform the sample within axial loading or the stress τ within the shear deformation, are opposed to an influence of the magnetic field. Obviously, the mechanisms of the dipole-interaction between the particles and the local non-affine deformation of the polymer matrix are different in these situations. There are various theoretical approaches which consider an axial loading of the MAEs and a shear loading when the shear is applied perpendicular to the field direction, e.g. [18,24–26]. However, predictions for the case when the field and mechanical stress are parallel in the shear test are rather rare. In a very recent theoretical study [27], a softening of the magnetized sample has been observed, which is in contradiction with our experimental results. A possible reason for this disagreement is a difference in the type of structural short-range order in the particle assembly between the theoretical consideration and real samples, as was previously discussed in the context of the magnetodipolar striction in MAEs [28].

Figure 5.

Influence of the external magnetic field B applied during the measurements on (a) the moduli E′ and G′ and (b) active MR effect for the non-magnetized samples. (Online version in colour.)

(b). Passive MR effect of the magnetized composites

Pre-magnetization of studied MAEs evokes a remarkable remanence magnetization for the magnetic hard component. Consequently, this results in a stiffening of the samples and the appearance of a passive MR effect. This influence of the pre-magnetization on the shear modulus G′ and tensile modulus E′ as well as corresponding passive MR effects are demonstrated in figure 6. The tensile modulus can be passively tuned up to approximately 110%, while an increase in the shear modulus reaches up to approximately 36%. Both moduli increase after the pre-magnetization due to an induced microstructural anisotropy and changed magnetic situation. Depending on the field B_M, particles of NdFeB powder are rotating and moving inside the matrix and this particle reorientation partially remains, due to the remanence. At high pre-magnetization fields B_M irreversible structuring is expected, as shown in [16]. Moreover, after a pre-magnetization the magnetic soft particles stay magnetized to a certain degree without the influence of an external magnetic field, due to the remanence magnetization of the NdFeB powder, and it can even be expected that at least at high B_M they are also involved in the pre-structuring. A direct comparison of the absolute values of the moduli E′ and G′ is not meaningful because of a different direction of the pre-magnetization field relative to the direction of the applied mechanical load. However, it can be concluded that the axial stiffness of the particle structures is higher than the transversal stiffness, because the passive MR effect is higher under axial loading.

Figure 6.

(a) Shear modulus G′ and tensile modulus E′, and (b) the passive MR effect for the samples after magnetization at various fields B_M. External field is not applied during the measurements. The mechanical force F is oriented perpendicular to the field B_M in the shear test (G′) and has the same direction as B_M in the axial loading. (Online version in colour.)

(c). Active MR effect of the magnetized composites

When an external magnetic field is applied to the pre-magnetized composites, their observed magnetomechanical response is further changed. Dependencies of the shear modulus G′ for specimens pre-magnetized in the field B_M on the externally applied field B are shown in figure 7a, while figure 7b demonstrates the corresponding active MR effect. In a similar way, results regarding the tensile modulus E′ are shown in figure 8. Qualitatively, dependencies of the moduli on the externally applied field B are similar for all pre-magnetizing fields B_M: moduli increase with increasing flux B and the active MR effect is provided for both experimental configurations. For the highest pre-magnetization (B_M = 1500 mT) and at an external field B = 240 mT, the MR effect reaches up to approximately 54% for the shear modulus and up to approximately 78% for the tensile modulus.

Figure 7.

Influence of the external magnetic field B applied during the measurements on (a) the shear modulus G′ and (b) active MR effect for the samples pre-magnetized under the field B_M. The field B is oriented parallel to the mechanical force F and perpendicular to the field B_M. (Online version in colour.)

Figure 8.

Influence of the external magnetic field B applied during the measurements on (a) the tensile modulus E′ and (b) active MR effect for the samples pre-magnetized under the field B_M. The field B is oriented parallel to the mechanical force F and has the same direction as the field B_M. (Online version in colour.)

The ratio between the moduli E′ and G′ changes with regards to pre-magnetization and external field. This is explained by induced anisotropy of the material and different orientation of the particle structures in the matrix for different experimental situations. The higher the field B_M, the lower the active MR effect at all fields B. Whereby, absolute and relative changes of the modulus are higher for the axial loading, as has been well observed for the non-magnetized samples. The higher the field B_M, the larger the particle structures initially oriented in the same direction and the higher the overall magnetization of the composite in this direction. The application of the external field B obviously either increases the magnetization of the composite, which is the case for axial loading when the fields B and B_M are co-directional, or leads to a complex situation when a prediction of the overall magnetization in a certain direction is no more obvious, which is the case for shear loading when the field B is oriented transversally to B_M. In the first instance, restricted mobility of the pre-magnetized particles inside the elastic matrix and the complex interaction between magnetic hard and soft components must be taken into account. Further discussions on the difference in MR response of the composites at variously oriented fields B and B_M are too speculative without appropriate theoretical approaches and therefore should be avoided.

Moreover, the experimental configuration of the axial loading is capable of applying an external field oriented in the same or in the opposite direction of the pre-magnetization field. Corresponding results for opposing field directions B and B_M are given in figure 9. Because the fields B and B_M are co-axial, the overall magnetization of the composite in the corresponding direction is governed by a difference between the magnetization provided by the field B and remanence magnetization provided by the field B_M. Experimental results show that this leads to a relative softening of the material at low B and high B_M, while an increase of B and decrease of B_M consequently results in an increase of the active MR effect (figure 9b). Furthermore, at lower B_M, in particular at B_M= 700 mT, and B higher than 150 mT, an absolute softening of the composite is observed, i.e. the measured elastic modulus becomes even lower than the modulus of the specimen without pre-magnetization (figure 9a). Thus, a mutual adaptation of the pre-magnetization and the externally applied flux density can provide both softening and stiffening of an MAE containing magnetic soft and hard components. A rather disadvantaging feature of this configuration, which must be mentioned, is a possible demagnetization of the composite. When MAE samples were magnetized in high fields (B_M > 900 mT) and were consequently actively softened applying an opposing external field, its modulus at zero field can also be reduced to 5–7%. However, further application of the co-directional external field leads to a compensation of this softening. This behaviour is related to the mobility of the particles inside the matrix and re-magnetization of the magnetic hard component, as previously discussed in detail in [23].

Figure 9.

Influence of the external magnetic field B applied during the measurements on (a) the tensile modulus E′ and (b) active MR effect for the samples pre-magnetized under the field B_M. The field B is oriented parallel to the mechanical force F and has the opposite direction to the field B_M. (Online version in colour.)

4. Conclusion and outlook

This study was focused on the experimental evaluation of the mechanical properties of heterogeneous magnetic polymeric composites based on a mixture of magnetically hard and soft microparticles. Specimens of such composites were manufactured and characterized using dynamic axial and shear loading under the action of externally applied magnetic fields up to B = 240 mT oriented parallel to the direction of mechanical loading. Composites were pre-magnetized in homogeneous magnetic fields with flux densities up to B_M = 1500 mT to obtain a remanence magnetization. Whereby, cylindrical samples for the axial loading were magnetized in the axial direction, i.e. parallel to the mechanical load and the external field direction applied during the experiment, while plate-like specimens for the shear testing were magnetized perpendicular to the mechanical load and the external field direction. The different pre-magnetizations made it possible to address different issues of the anisotropic material behaviour. To the best of our knowledge this is the first study considering the anisotropy of the MAEs with complex particle compositions (mixtures of magnetically soft and hard particles).

The magnetorheological response of the composites was evaluated based on the dependencies of elastic modulus and shear modulus on the applied field and pre-magnetization. Composites which are pre-magnetized parallel to the applied mechanical load and parallel to the externally applied magnetic field showed higher MR responses than composites pre-magnetized perpendicular to these, wherefore a lower transversal stiffness of the particulate structures can be assumed. Opposing the applied external field to the direction of the pre-magnetization reduces the observed stiffening, and even a softening of the composite was observed for specific ratios of the external field strength to the pre-magnetization field. The behaviour observed is obviously related to complex particle–particle and particle–matrix interactions. The obtained data should serve as a basis for theoretical predictions; however, appropriate approaches for the interaction between magnetic hard and soft particles need to be proposed to shed a light on the complex behaviour of the MAEs with a mixed composition. Corresponding microstructural investigations are also required. On the other hand, the results obtained to date give an input for future technical applications based on MAEs, where composite properties can be tuned passively and controlled actively. This includes a controllable softening of such composites, which could be experimentally demonstrated within this work. Further studies on the anisotropic material behaviour are also required to describe the magneto-mechanics of MAEs, providing their physical parameters for full-featured engineering simulations and successful applications.

Data accessibility

The dataset supporting this article is included in the electronic supplementary material.

Authors' contributions

D.B. designed the experimental set-up, performed experimental measurements and treatment of the samples, evaluated and analysed the obtained data and drafted the article. G.S. manufactured experimental samples and participated in the conducting of the experiments, interpretation of the obtained data, and critically revised the article. E.D. substantially contributed to the interpretation and analysis of the obtained experimental data and also to the drafting as well as critical revision of the article.

Competing interests

We declare we have no competing interests.

Funding

D.B. gratefully acknowledges the financial support by Deutsche Forschungsgemeinschaft (DFG) under Grant Bo 3343/2-1 within SPP1681. G.S. would like to acknowledge the support of RFBR under Grant 16-53-12009.