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Coupling piezoelectric and magnetoelastic materials together enables the electrical control of magnetism. However, the lack of an accurate non-linear magnetoelastic constitutive model complicates designing macroscale devices with these multiferroic composites. While micromagnetic simulations are the gold standard for mesoscale magnetic modeling, their use is limited to very small systems (i.e., typically below 10s of cubic microns). The focus of this research is on generating an accurate constitutive model for macroscale magnetoelastic materials that is consistent with micromagnetics.&lt;br/&gt;Multiple different constitutive models are currently used to describe macroscale magnetoelastic materials [1,2,3]. Two common approaches, polynomial and statistical energy models, each have their own strengths and limitations. For example, series expansion models that use polynomial expansions are straightforward to setup, however they are not capable of capturing the saturating behavior of magnetic materials. Additionally, to obtain nonlinear material properties well-controlled experiments to curve fit the models are required [4], which can be prohibitively challenging, time consuming, and expensive for anisotropic materials. Alternatively, statistical models can overcome the inherent limitations of polynomial expansions. These models utilize Boltzmann statistics to calculate the average magnetization and magnetostriction. These calculations account for the Zeeman, magnetoelastic, and magnetocrystalline anisotropy energies. A challenge of this statistical approach is that the requisite integral expressions do not possess general closed form solutions and therefore require numerical solutions. The first of these integrals is the partition function, which is dependent on the micromagnetic energies of the system &lt;i&gt;u,&lt;/i&gt; and the thermal energy. In the title of Figure 1 the partition function includes beta which is used as a nondimensional term representing the magnitude of the micromagnetic energies in &lt;i&gt;u&lt;/i&gt;. The other integral to consider is the average magnetization, which is derived from the partition function. While standard numerical integration methods can be utilized, they are computationally expensive, limiting their use in finite element models. In this work a constitutive model utilizing statistical physics to bridge micromagnetics to the macroscale is developed, and the accuracy and computational complexity of several integration methods are analyzed to produce a model suitable for use in high fidelity numerical models.&lt;br/&gt;The two methods for approximating the partition function that were examined are a partial series expansion and Laplace’s method. Conceptually a series expansion is asymptotically exact at the point its expanded about, conversely Laplace’s method is asymptotically exact at higher energy values (i.e., the methods are complimentary). The accuracy of each method was compared to an accurate, but computationally expensive numerical integral with relative error set at machine precision. The average magnetization, and therefore the partition function, has a closed form solution when only the Zeeman energy is included. If the additional micromagnetic energies are small compared to the Zeeman energy, they can be expanded in a Maclaurin series, which maintains integrals with exact solutions. Figure 1(a) shows the absolute error of this partial series expansion method as the magnitude of the applied energies (beta) increase. The inset graph also plots the absolute error, but beta 1 represents when only Zeeman energy is applied, and beta 2 shows when only the other micromagnetic energies are applied. Figure 1(b) shows the absolute error of Laplace’s method as the magnitude of the applied energies increase. The inset graph again considers the main contributor to the energy of the system. These two figures show that the two methods are indeed complimentary and so a hybrid model would maintain low relative errors for the entire range of beta values. Additionally, Laplace’s method only requires a small number of points (e.g., between two and six) to approximate integrals, making it more computationally efficient compared to standard integration methods. To conclude, we will use this model to simulate both the magnetization and magnetostriction of Galfenol subjected to magnetic fields up to 20 mT and mechanical stresses between -50 and 50 MPa. The results will be compared to experimental data from the literature [5].

**References:**

[1] P. G. Evans and M. J. Dapino, Journal of Applied Physics., Vol. 107, (2010)&lt;br/&gt;[2] Yining Li et al. Journal of Magnetism and Magnetic Materials., Vol. 472, p. 59–65 (2019)&lt;br/&gt;[3] Z. Xiao et al. ACS Applied Materials &amp; Interfaces., Vol 12.5, p. 6752–6760 (2020)&lt;br/&gt;[4] Z. Xiao and S. Le, Journal of Applied Physics., Vol 100.6, (2006)&lt;br/&gt;[5] A. Mahadevan, P.G. Evans, and M.J. Dapino, Applied Physics Letters., Vol. 96, (2010)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/2ace4b46782f3a53b9f2bb7ecae3629c.jpg)
Figure 1: Absolute error of both approximation methods. (a) Results for partial series expansion (b) Results for Laplace&#39;s method



INTERMAG 2021

 Approximation Methods for Hybrid Constitutive Model of Magnetostriction

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Coupling piezoelectric and magnetoelastic materials together enables the electrical control of magnetism. However, the lack of an accurate non-linear magnetoelastic constitutive model complicates designing macroscale devices with these multiferroic composites. While micromagnetic simulations are the gold standard for mesoscale magnetic modeling, their use is limited to very small systems (i.e., typically below 10s of cubic microns). The focus of this research is on generating an accurate constitutive model for macroscale magnetoelastic materials that is consistent with micromagnetics. Multiple different constitutive models are currently used to describe macroscale magnetoelastic materials [1,2,3]. Two common approaches, polynomial and statistical energy models, each have their own strengths and limitations. For example, series expansion models that use polynomial expansions are straightforward to setup, however they are not capable of capturing the saturating behavior of magnetic materials. Additionally, to obtain nonlinear material properties well-controlled experiments to curve fit the models are required [4], which can be prohibitively challenging, time consuming, and expensive for anisotropic materials. Alternatively, statistical models can overcome the inherent limitations of polynomial expansions. These models utilize Boltzmann statistics to calculate the average magnetization and magnetostriction. These calculations account for the Zeeman, magnetoelastic, and magnetocrystalline anisotropy energies. A challenge of this statistical approach is that the requisite integral expressions do not possess general closed form solutions and therefore require numerical solutions. The first of these integrals is the partition function, which is dependent on the micromagnetic energies of the system u, and the thermal energy. In the title of Figure 1 the partition function includes beta which is used as a nondimensional term representing the magnitude of the micromagnetic energies in u. The other integral to consider is the average magnetization, which is derived from the partition function. While standard numerical integration methods can be utilized, they are computationally expensive, limiting their use in finite element models. In this work a constitutive model utilizing statistical physics to bridge micromagnetics to the macroscale is developed, and the accuracy and computational complexity of several integration methods are analyzed to produce a model suitable for use in high fidelity numerical models. The two methods for approximating the partition function that were examined are a partial series expansion and Laplace’s method. Conceptually a series expansion is asymptotically exact at the point its expanded about, conversely Laplace’s method is asymptotically exact at higher energy values (i.e., the methods are complimentary). The accuracy of each method was compared to an accurate, but computationally expensive numerical integral with relative error set at machine precision. The average magnetization, and therefore the partition function, has a closed form solution when only the Zeeman energy is included. If the additional micromagnetic energies are small compared to the Zeeman energy, they can be expanded in a Maclaurin series, which maintains integrals with exact solutions. Figure 1(a) shows the absolute error of this partial series expansion method as the magnitude of the applied energies (beta) increase. The inset graph also plots the absolute error, but beta 1 represents when only Zeeman energy is applied, and beta 2 shows when only the other micromagnetic energies are applied. Figure 1(b) shows the absolute error of Laplace’s method as the magnitude of the applied energies increase. The inset graph again considers the main contributor to the energy of the system. These two figures show that the two methods are indeed complimentary and so a hybrid model would maintain low relative errors for the entire range of beta values. Additionally, Laplace’s method only requires a small number of points (e.g., between two and six) to approximate integrals, making it more computationally efficient compared to standard integration methods. To conclude, we will use this model to simulate both the magnetization and magnetostriction of Galfenol subjected to magnetic fields up to 20 mT and mechanical stresses between -50 and 50 MPa. The results will be compared to experimental data from the literature [5].

**References:**

[1] P. G. Evans and M. J. Dapino, Journal of Applied Physics., Vol. 107, (2010) [2] Yining Li et al. Journal of Magnetism and Magnetic Materials., Vol. 472, p. 59–65 (2019) [3] Z. Xiao et al. ACS Applied Materials & Interfaces., Vol 12.5, p. 6752–6760 (2020) [4] Z. Xiao and S. Le, Journal of Applied Physics., Vol 100.6, (2006) [5] A. Mahadevan, P.G. Evans, and M.J. Dapino, Applied Physics Letters., Vol. 96, (2010)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/2ace4b46782f3a53b9f2bb7ecae3629c.jpg)
Figure 1: Absolute error of both approximation methods. (a) Results for partial series expansion (b) Results for Laplace's method

technical paper

Magnetocaloric effect (MCE) receives an increasing interest for its promising applicability in energy efficient refrigerators. Among the different magnetocaloric materials having potential applications for cooling devices that operate near room temperature, those exhibiting a first-order phase transition (FOPT) usually show larger responses than those exhibiting a second-order phase transition (SOPT), but the intrinsic hysteresis could be deleterious for cyclic operations [1]. Some FOPT materials such as Heusler alloys can also exhibit the Curie transition of their magnetic phases, thus leading to alloys exhibiting both direct (positive isothermal entropy change, ΔSiso) and inverse MCE (negative ΔSiso). However, the close proximity of the two types of transitions can produce a convoluted and compensated global response due to the different nature, direct and inverse, of their MCEs [2]. In order to evaluate the potential of the FOPT for magnetocaloric applications and, therefore, ascertain the usefulness of tuning the Curie temperature of the alloy by compositional modifications, it is necessary to be able to deconvolute its responses. It has previously been shown that concurrent SOPTs can be deconvoluted by applying the scaling laws of the MCE [3]. In this work, we propose an extension of the methodology to deconvolute concurrent FOPT and SOPT. For this purpose, a Heusler alloy with stoichiometry Ni48.1Mn36.5In15.4 has been used as a model case. Upon heating, this sample exhibits a FOPT from a low magnetization martensite to an austenite with higher magnetization. This magneto-structural transition produces an inverse MCE that is immediately followed by the Curie transition of the austenite, with its corresponding conventional MCE. With increasing the magnetic field, two different behaviors are observed: 1) the onset of the FOPT shifts to lower temperatures, with a peak ΔSiso value that increases abruptly at the beginning, reaches a maximum for ~5 T and subsequently decreases, and 2) a Curie transition whose response increases gradually in height and width, with a peak position that remains almost field independent. Due to the proximity of these two competing transitions, the overlapping of their responses increases with increasing field. The application of the scaling laws for the SOPT magnetocaloric response allows us to deconvolute the contributions of the two transitions. Top panel of Fig. 1 shows the experimental ΔSiso of the sample together with its reconstructed response. The separated contributions of the two phase transitions are presented in the bottom panel of Fig. 1. The good agreement between experimental and reconstructed curves allows us to reach the following conclusions: - Universal scaling can be used to deconvolute overlapping first and second order phase transitions. - The broadening of the SOPT with increasing field causes a larger overlap between transitions despite the shift of the FOPT to lower temperatures. - The experimental decrease of the inverse MCE peak above ~5T is due to the competing contribution of the SOPT and is not intrinsic to the FOPT. This deconvolution methodology allows to extract the information of the different phase transitions in the material without having to experimentally synthesize new alloys with different compositions. Work supported by AEI/FEDER-UE (grant PID2019-105720RB-I00), US/JUNTA/FEDER-UE (grant US-1260179), Consejería de Economía, Conocimiento, Empresas y Universidad de la Junta de Andalucía (grant P18-RT-746), and Army Research Laboratory under Cooperative Agreement Number W911NF-19-2-0212.

**References:**

[1] V. Franco, J.S. Blázquez, J.J. Ipus, J.Y. Law, L.M. Moreno-Ramírez, A. Conde, Prog. Mater. Sci. 93 (2018) 112–232. [2] J. Y. Law, Á. Díaz-García, L. M. Moreno-Ramírez, V. Franco, A. Conde, A. K. Giri, Acta Mater. 166 (2019) 459. [3] Á. Díaz-García, J. Y. Law, P. Gebara, V. Franco, JOM 72 (2020) 2845–2852.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/181f71a01791952162fe6704a962d0ff.jpg)
Fig. 1 (Top panel) Experimental temperature dependence of ΔSiso of Ni48.1Mn36.5In15.4 Heusler alloy for different magnetic fields (open symbols) and its reconstructed ΔSiso response (stars symbols). (Bottom panel) Temperature dependence of ΔSiso for different fields of the deconvoluted responses of the SOPT (open symbols) and the FOPT (solid symbols).

 Deconvolution of First and Second Order Phase Transitions using the Scaling Laws of the Magnetocaloric Effect

Since the discovery of the Giant Magnetocaloric Effect (MCE) on Gd5Si2Ge2 by Pecharsky and Gschneidner [1], the Gd5(Si1-xGex)4 family compounds have been intensively studied for applications in magnetic refrigeration (MR) at room temperature. The vast investigation at the macroscale on these alloys up to this time revealed many other important properties, such as giant magnetoresistance and colossal magnetostriction (CMS) [2]. Such behaviors arise from a strong magneto-volume coupling extreme sensitivity to variation of external (e.g., temperature, magnetic field, and pressure) and internal (e. g., stoichiometry, dimensionality, and doping) parameters. For this reason, this family compound presents a rich phase diagram that, for example, gives a wide range of temperatures for device operations. However, the low mechanical stability and oxidations issues are some drawbacks for long periods of appliance operation. Reducing it to the micro and nanoscales can improve the alloys' properties and also find novel properties. In the last years, some experimental approaches have been used to produce micro/nanostructures of Gd5(Si,Ge)4 [3]. It is possible to mention high-energy ball milling, RF magnetron sputtering, and laser ablation techniques, being the most promising. Using a femtosecond pulsed laser deposition, our group successfully obtained Gd5Si1.3Ge2.7 nanogranular film composed of 80 nm particles presenting a magnetostructural transition [4] and a negative thermal expansion [5]. Although this evidence revealed the great potential of ultrashort laser ablation, a more conventional and low-cost technique should be pursued aiming practical applications of these compounds. For example, Tarasenka et al. were able to produce colloidal suspensions of Gd-Si-Ge nanoparticles in liquid with a more accessible Nd:YaG nanosecond pulsed laser [6]. Although the NPs presented fine crystalline features, the formation of silicon carbide and particle segregation after an additional laser ablation occurs. Given this, in addition to finding the most appropriate experimental technique, reasonable control of the intrinsic structures is essential to optimize the material production on a large scale. Here, we demonstrate that a more simple gas-phase synthesis using a Nd:YaG and KrF Excimer nanosecond pulsed lasers can be used to produce Gd5(Si1-xGex)4 nanostructures with x = 0, 0.45 and 0.60. The stoichiometries were selected to cover all the possible structures adopted by this family: orthorhombic-II [O(II)] for x = 0; monoclinic (M) of x = 0.45; and orthorhombic-I [O(I)] for x = 0.60. It is worth mentioning that x = 0.45 and 0.60 are in the vicinity of the phase-mixed region to show the feasibility of this method along with the family phase-diagram. In particular, the KrF Excimer was chosen due to the laser wavelength that presents higher penetration depth and reduced reflectance in metals. The nanostructures were obtained by ablating the polished surface of ingots produced by tri-arc melting with 10 Hz of frequency and maximum energy of 550 mJ and 200 mJ, for Nd:YaG and KrF Excimer lasers, respectively. The chamber was pumped several times down to a base pressure of 10-6 Torr to prevent oxidation. During the deposition, Argon inert gas was used in a continuous flux in constant pressure of 1 Torr inside the chamber. From this process, Gd5(Si,Ge)4 grains with an average particle size between 10-30 nm were achieved with good crystalline features, as can be seen in Fig.1(a). Synchrotron X-ray Diffraction confirmed the great level of the structures crystallinity and revealed a change in symmetry for x = 0 and 0.45 respectively from O(II) and M to O(I). The structural shrinkage was found to be around 2% for the Si-rich compositions and less than 1% for Gd5Ge4 (x=0). Given the lower unit cell volume for the O(I) structure, a shortening between the Si(Ge)-Si(Ge) distances is responsible for a change in the transition order from a first to a second one for x = 0.45. Also, the magnetic transition occurs at 290 K for this 0.45 composition nanostructure, see Fig.1(b), confirming the XRD analysis. Distinctly, for x = 0, the reduction to the nanometric scale leads to a superparamagnetic-like behavior that leads to a maximum of 8 J/kgK at 5 K that drastically reduces to null values as the temperature increases. Furthermore, the reduction of the unit cell volume was found to be directly related to the bulk target volumetric compressibility that is due to intrinsic surface tension (st) rising from the nanostructuring process. Thereby, we can conclude that the nanosecond PLD successfully produces dispersed crystalline nanoparticles of Gd5(Si1-xGex)4 family compounds. This technique allows the formation of NPs with particles size under 30 nm without loss on the stoichiometry of the targets, as observed through Synchrotron XRD [7]. Notwithstanding, the direct relation between bulk compressibility and nanoparticle size due to an intrinsic surface pressure might be the path to understand the particles' growth from pulsed laser deposition techniques under inert gas in the chamber.

**References:**

[1] V. K. Pecharsky and K. A: Gschneidner Jr., Phys. Rev. Lett., vol. 78, p. 4494 (1997); [2] L. Morellon, et al., Phys. Rev. B, vol. 58, p. R14721(R) (1998); [3] J. H. Belo, et al., J. of Mater. Res., vol. 34, p. 134 (2019); [4] R. L. Hadimani, et. al., Appl. Phys. Lett., v. 106, p. 032402 (2015); [5] J. H. Belo, et al., PRB, vol. 100, p. 134303 (2019); [6] N. Tarasenka, et al., J. Nanosci. Nanotechnol., vol. 16, p. 7451 (2016); [7] V. M. Andrade, et al., J. Alloys Compd., vol. 849, p. 156384 (2020[AP1] ).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/afe5422997f395a3795a6766687f1f04.jpg)
Figure 1: (a) TEM images for x = 0.45 nanoparticles with 27 nm of the mean size where the SAED reveals the good crystalline features of the structures. (b) Magnetic entropy change curves for all produced samples for an applied field of 5 T.

 Unit cell volume reduction of Gd5(Si,Ge)4 nanoparticles controlled by bulk compressibility

Faraday effect was a first demonstration of magnet influence on the light. Since then many generations of scientists tried to strengthen it. A decade ago, a breakthrough based on plasmonics was proposed for a significant enhancement of the magneto-optical interaction. However, all metals, even noble ones, introduce rather notable absorption which worsen quality factor of the optical resonances and constrains their role in the magneto-optical enhancement. Here we demonstrate that low lossy all-dielectric magnetic structures are more advantageous in this respect. We propose a specially designed all-dielectric magnetic meta-surface allowing a novel mechanism of the magneto-optical intensity effect [1]. It consists of a 2D array of iron-garnet nanopillars on an ultrathin iron-garnet slab. The nanostructure exhibits high transparency and superior quality-factor resonances followed by multifold increase of the magnetooptical intensity modulation. Unlike the magneto-optical intensity effects in smooth films and plasmonic gratings the photonic crystal metasurface provides the magnetically induced modulation of light intensity not only in a single light polarization, e.g. p-polarization, but also in the orthogonal one (s-polarization). It is due the wide variety of the excited mode types in the all-dielectric nanostructure. It is very promising for sensing, magnetometry and light modulation applications. This work was financially supported by the Russian Ministry of Education and Science, Megagrant project N 075-15-2019-1934.

**References:**

[1] D.O. Ignatyeva et al., Nature Communications 11, 5487 (2020).

**Images:**

 All-dielectric magnetophotonics INVITED

We present analysis of microscopic origin of magneto-optical spectra from the electronic structure of bcc Fe. The magneto-optical (MO) permittivity spectra are obtained by the WIEN2k code [1] and compared to the corresponding experimental MO Kerr effect spectra. The ab-initio spectra are given by the Kubo formula, i.e., as a result of summation over all pairs of electronic bands and integration over the Brillouin zone that all sum up to the single MO spectrum. We investigate what features of electronic structure contribute to the outgoing total MO spectrum. We have also developed several novel ways of visualizing MO-related phenomena in the Brillouin zone. It turns out that strong MO signal comes from several isolated k-points in the Brillouin zone, where the band degeneracy is avoided by the spin-orbit interaction (Fig.1). There are two major types of MO contributions determined by the nature of the avoided degeneracy (given by topology of the approaching bands), that contribute differently to the total MO spectrum. Both types are visualized and their contributions to the total spectrum are demonstrated in the form of local MO spectra. K-points, where the strong MO signal is originated, also give rise to strong Berry curvature. We show the flow of the Berry curvature in the vicinity of those k-points (Fig.2) and in the full Brillouin zone.

**References:**

[1] P. Blaha, K. Schwarz, G. Madsen, D. Kvasnicka and J. Luitz, WIEN2k

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/89049a3a7102cf0d095032f0f6b7b2d8.jpg)
(a) Band structure of bcc Fe with colors corresponding to the d-character of bands with spin-up being positive and spin-dwon negative. (b) Joint band structure with colors representing the strength of the magneto-optical response. Strong magneto-optical transitions are marked by arrows and labelled by the corresponding band indices.

![](https://assets.underline.io/uploads/markdown_image/1/image/dc6681d80eadbec89b015c676343ea08.jpg)
Flow of the Berry curvature in the vicinity of k-point with strong magneto-optical response.

 Magneto-Optical Effect in bcc Fe: Microscopic Origin and Topological Aspects

In the last decade, a huge amount of work has been devoted to constructing a composite thin film where magnetic anisotropy becomes tunable by coupling ferromagnetic layer to a functionalized layer. Electrical control of interfacial anisotropy with an insulator and interfacial strain transferred from an electrically controlled piezoelectric or ferroelectric material are still heavily pursued. In 2016, Feng et al. [1] proposed a novel approach to manipulate magnetic anisotropy by interfacial strain transferred from a thermally controlled NiTiNb shape memory alloy (SMA). They obtained pioneering results in showing small variation of interfacial anisotropy in NiTiNb (0.5mm)/FePt (10–20 nm) with in-plane anisotropy. This innovative architecture benefits from the shape memory effect, in particular, from the two-way shape memory effect (TWSME). The TWSME consists in a reproducible hysteretic transformation of shape memory alloys from a strained B19’ structure (at low temperature) to a differently strained B2 structure (at high temperature) during the thermal cycle. NiTi-based alloys are among the most used shape memory alloys within a wide range of applications in industry, military technologies, health care, etc. [2]. They are metallic and may be grown by regular microelectronics industry deposition techniques. NiTi-based SMAs can produce as much as 4% of TWSME strain and recover up to 10% of the initial deformation [3]. The transformation temperatures can be easily tuned from 250 K to 1000 K [2]. In our talk, we demonstrate that the NiTi alloy can be used to reproducibly switch by 90° the in-plane uniaxial anisotropy of a 20 nm thick Ni thin film during a thermal cycle between room temperature and 100°C (as shown in Figure 1) [4]. We first properly quantify NiTi sample longitudinal strain and transverse strain produced by an external uniaxial tensile stress followed by a thermal cycle, required to setup the TWSME. We demonstrate that NiTi is a Pauli paramagnetic material and explain temperature variations of its magnetic susceptibility by comparing experimental data with ab-initio calculations [5]. We show that both martensitic transition temperature and magnetic susceptibility depend on NiTi lattice structure, which we tuned through strain and characterized by TEM and XRD techniques. Secondly, we characterize magnetic anisotropy of the Ni layer, deposited by PVD on top of NiTi after tensile deformation. As shown in Fig. 1, when NiTi is in its martensite (respectively, austenite) phase, Ni anisotropy points along (respectively, perpendicular to) the initial tensile axis. Variation of Ni anisotropy up to 1.105 J.m-3 was achieved. It is a promising result since the strain transfer at NiTi/Ni interface is not yet optimized. Indeed during the TWSME strain in NiTi varies from 0% to 2.7% whereas Ni strain varies only from 0% to 1.3. We performed transmission electron microscopy and reveal that a thin oxidized interface layer is the main origin of strain loss at the NiTi/Ni interface.

**References:**

[1] C. Feng et al., Appl. Phys. Lett. 109, 212401 (2016) [2] J. Mohd Jani, et al., Mater. Des. (1980–2015) 56, 1078 (2014) [3] O. Benafan, et al., J. Appl. Phys. 112, 093510 (2012) [4] A. Kyianytsia et al., Appl. Phys. Lett. 115, 222402 (2019) [5] A. Kyianytsia et al., Appl. Phys. Lett. 117, 122411 (2020)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/c19037b2f749a5e8b58c235308b7d528.jpg)
(a–b) Magnetization vs magnetic field when the field is applied along the x-axis (0°), and y-axis (90°), when NiTi is in martensite state (E state) and austenite state (F state). (c) Remanent magnetization of NiTi/Ni as a function of magnetic field angle for E and F state.

 Magnetic anisotropy switching induced by shape memory effect in NiTi/Ni bilayer

High-temperature shape memory alloys (HT-SMAs) are required in specific automotive, aerospace or energy applications [1], [2]. Common binary Ni-Ti SMAs are characterized by a limited transformation temperature and, thus, cannot be used for applications operating at 100°C or above. Ternary elements (Pt, Au, Hf, Zr) can be added to increase the transformation temperature [3]. The ternary Ni-Ti-Hf alloy is currently the most promising candidate material, however, pronounced brittleness and high costs due to the significant amount of Hf hinders its technological breakthrough. Alternative HT-SMAs are Co-Ni-Ga Heusler alloys. These materials undergo a first-order magnetostructural transformation (FOMST) from high-temperature B2-ordered austenite to tetragonal L10 low-temperature martensite [4]. A fully reversible superelastic response up to 500°C as well as excellent cyclic stability up to temperatures of 100°C have been reported for single-crystals [5]. However, polycrystalline Co-Ni-Ga suffers from intergranular cracking and a premature failure after several transformation cycles due to the anisotropic volume change of randomly crystallographic orientated grains. Many efforts in the fields of grain boundary and microstructure engineering have been done to synthesize HT-SMAs with favorable grain boundary configuration in order to fully prevent intergranular cracking and premature failure. It was reported that a columnar-grained microstructure with strong &lt;001&gt; texture and straight low-angle grain boundaries can overcome the structural and functional limitations in Cu-based SMAs [6]. Additive manufacturing (AM) is a very promising technique to synthesize HT-SMA since it allows a direct microstructure design. Selective laser melting (SLM) and directed energy deposition (DED) are two common AM techniques for metallic alloys. During the SLM process, pre-alloyed powder is molten layer-by-layer using a laser system operating under inert gas atmosphere to prevent oxidation. Microstructure can be directly tailored by different processing parameters like laser power, scanning velocity and scanning path. In contrast to the powder-bed based SLM, DED is a powder-stream based process using nozzles to directly transfer the powder material into a focused laser beam. Similar to the SLM process the microstructure can be directly tailored by varying the processing parameters such as laser power and scanning speed. In our present work, we compare the microstructure and magnetic properties of Co49Ni21Ga30 alloys processed by SLM and DED with single crystals as well as polycrystalline material prepared by conventional casting. The preferred columnar-grained microstructure could be obtained by proper choice of processing parameters for the different AM techniques [7], [8]. Fig. 1 shows the microstructure of Co-Ni-Ga fabricated by SLM. The electron backscatter diffraction (EBSD) analysis of the cross-section shows a polycrystalline columnar-grained microstructure along the building direction (BD) [7]. A similar columnar-grained microstructure is also observed in the DED processed material [8]. In addition, the DED sample is characterized by a strong &lt;001&gt; texture along BD. Due to these highly anisotropic microstructures, AM processed Co-Ni-Ga obtained by both, SLM and DED, shows excellent HT-SMA properties without any post-processing [7], [8]. To study the FOMST in more detail, temperature and field-dependent magnetization measurements were performed. Fig. 2 shows the temperature-dependent magnetization of additively processed samples, polycrystalline material in the as-cast condition and in a single crystalline state for different orientations. The SLM sample reveals the largest transition width and thermal hysteresis while the DED sample provides for a transformation behavior being very similar to the cast and single crystalline counterparts. Since the transformation behavior depends on parameters such as grain size, residual stresses, defects and precipitates [9] the influence of the processing parameters on these parameters is essential to understand the FOMST and functional properties in Co-Ni-Ga in more detail. This work was supported by the ERC Advanced Grant "CoolInnov" (No 743116), the CRC/TRR 270 “HoMMage” (DFG). TN acknowledges funding by DFG (No 398899207).

**References:**

[1] J. Ma, I. Karaman, R. D. Noebe, International Materials Reviews, vol. 55, pp. 257-315, (2010) [2] K. Otsuka, Ed., Shape memory materials, Cambridge Univ. Press, (1999) [3] M. Elahinia et al., Scripta Materialia, vol. 145, pp. 90-94, (2018) [4] A. Reul et al., Shap. Mem. Superelasticity, vol. 4, pp. 61-69, (2018) [5] P. Krooß et al., Shap. Mem. Superelasticity, vol. 1, pp. 6-17, (2015) [6] J.-L. Liu, H.-Y. Huang, J.-X. Xie, Materials & Design, vol. 64, pp. 427-433, (2014) [7] C. Lauhoff et al., Metall and Mat Trans A, vol. 51, pp. 1056-1061, (2020) [8] C. Lauhoff et al., Materials Research Letters, vol. 8, pp. 314-320, (2020) [9] F. Scheibel et al., Energy Technol., vol. 6, pp. 1397-1428, (2018)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/237965b772da9e5c7d95872cd06b33fe.jpg)
Fig. 1: Microstructure of Co-Ni-Ga processed by SLM: (a) optical micrograph, the building direction is perpendicular to the surface, (b) EBSD inverse pole figure map of the cross-section revealing a polycrystalline columnar-grained microstructure along the building direction (BD)

![](https://assets.underline.io/uploads/markdown_image/1/image/ca9914c1650bb2f6243d137454ae4bf8.jpg)
Fig. 2: Temperature dependent magnetization of (a) Co-Ni-Ga single crystals, (b) cast polycrystalline Co-Ni-Ga and Co-Ni-Ga alloy processed by SLM and DED.

 Influence of additive manufacturing processes on the microstructure and magnetic properties of Co-Ni-Ga shape memory Heusler alloy

Recent progress in multiferroic antenna shows high promise for communication in rf-denied environment for transmitting signals in lossy electrically conductive media. The antenna utilizes the ultra-low (ULF/VLF) frequency band (1 – 30 kHz) while also maintaining a compact design [1]. In such multiferroic antennas, containing magnetostictive materials that are mechanically actuated, becoming a near-field magnetic field transmitter. This transduction mechanism is based on magnetization changes inside the magnetostrictive material, due to the application of time-variable stresses generated by an electrically-driven piezoelectric material (which is elastically coupled with the magnetostrictor). Magnetoelectric coupling between the piezoelectric driving element and magnetostrictive material allows for modulation of a large magnetic signal form the multiferroic antenna [2]. Of particular importance is the material with a high piezomagnetic coefficient coupled with a large change in the internal flux density. Iron-gallium alloys, Fe1-xGax (Galfenol) with 0.12 &lt; x &lt; 0.30, are promising magnetostrictive, transducer materials that combine highly desirable tensile and compressive mechanical excitation with superior magnetic properties [3]. Doping studies of Galfenol show a maximum in the magnetostrictive response when the Ga composition is x = 0 16 – 0.19; while there is an additional peak at higher gallium compositions, this range maintains a large internal flux response [3,4]. This work investigates the mechanically-driven magnetic response of several Galfenol compositions as a function of driving frequency. A solid and laminated rod of similar size are also compared to determine the effect of eddy current losses on the flux response of the rod. The laminated rod is composed of 12 μm, epoxy bonded Galfenol sheets with x = 0.186. Magnetostrictive transducer elements under test were placed in a nonmagnetic frame with an applied, axial static prestress and permanent magnets placed above the Galfenol rods. When placed in an optimal magnetic bias field and compressively actuated through driving with a Pb[Zr,Ti]O3 (PZT) stack the magnetostrictive Galfenol changes magnetic states depending upon the applied stress. The bias magnetic field was optimized by determining the distance where the permanent magnets would allow for the highest Galfenol flux density response. A fluxgate magnetometer at 1 m and search coil surrounding the Galfenol rods determined the radiated magnetic field and changing internal flux, respectively; both showed similar frequency responses. Figure 1 shows the near-field (1 m) response of 3 samples (2 solid rods and 1 laminated) revealing an improved response at the lower Ga composition. Interestingly, a larger signal was measured from the laminated sample at all frequencies despite being a higher composition (18.4 at. % Ga); it also showed little variation with increasing frequency over the measured range. The laminated sample showed better operational properties compared with the solid samples that decreased with increasing frequency. Mechanically driven laminated Galfenol samples showed an improved dynamic response when compared with solid rods. Additional optimizations to the device configuration are possible as variations in production quality, heat treatments, and microstructural properties such as grain size and domain boundaries can affect the magnetostrictive and magnetic response of the material. Further investigations are needed to verify if any additional improvements are possible through annealing.

**References:**

[1] Joseph D. Schneider, John P. Domann, and Gregory P. Carman, Journal of Applied Physics 126, 224104 (2019) [2] Junran Xu, Chung Ming Leung, and Dwight Viehland, Sensors, 19(4), 853 (2019) [3] G. Petculescu, K. B. Hathaway, and A. E. Clark, Journal of Applied Physics 97, 10M315 (2005) [4] A. Mahadevan, P. G. Evans, and M. J. Dapinoa, Appl. Phys. Lett. 96, 012502 (2010)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/edccf778d2f230d28c475ab79dc55641.jpg)
Figure 1: Frequency response of the fluxgate magnetometer 1 meter from the actuated Galfenol rods (solid: 17 and 18 at. % Ga; laminated: 18.4 at. % Ga), showing a nearly flat frequency response for the laminated rod from 300Hz to 2kHz. Both solid rods decrease with increasing frequency. The 18% solid rod has been annealed under tension.

 Dynamic response of Fe-Ga for multiferroic magnetic field transmitters

There has been a recent interest in the use of magnetoelastic materials to develop new magnetic antenna technologies [1,2,3]. Magnetic antennas show promise for surpassing shortcomings in current technologies, such as communicating in lossy environments (e.g., underground or underwater). One commonly proposed design is a resonant composite multiferroic beam consisting of magnetoelastic, elastic, and piezoelectric layers. Appling a voltage to the piezoelectric layer causes the beam to bend, straining the magnetoelastic layer, modulating the magnetic field around the antenna, thereby transmitting information. Current analytical and finite element magnetoelastic beam models [4,5,6] appear to work well for sufficiently long and thin structures, however the analytical models often neglect or overly simplify the magnetostatic field (i.e. demag) or spatially distributed material properties such as Young’s modulus or piezomagnetic coupling coefficients. Therefore, an analysis of under what conditions reduced-order analytical models are valid is needed before they can be used as a tool for analyzing general geometries where complex magnetic field distributions are expected. This work presents a detailed comparison of a finite element model and an analytical reduced order Euler-Bernoulli beam model with the goal of determining under what conditions the reduced order model is valid, and which foundational assumptions are responsible for any discrepancies. Both the finite element model and the analytical reduced order Euler-Bernoulli beam are set up to solve the same geometry, governing equations, and boundary conditions. The antenna has a simple geometry composed of three rectangular plates; with a linear elastic layer sandwiched between piezoelectric and magnetoelastic layers. The governing equations consist of the equations of magnetostatics, electrostatics, and linear elasticity (i.e., suitable for a near-field antenna operating at low kHz frequencies). The mechanical boundary conditions are setup to model a free-free Euler Bernoulli beam. The finite element analysis is conducted in a series of steps. First, a uniform magnetic field is applied, and the bias state of the antenna is solved for using a coupled, nonlinear, magnetoelastic constitutive model (along with linear elastic, and piezoelectric behaviors). A linearized modal analysis is then performed around this bias state to determine the dynamic eigenmodes of the mechanical vibrations. Once found, the first eigenmodes is subsequently used to model the mechanical deformation, which is then fed into the nonlinear magnetic material model to obtain the dynamic magnetization changes (i.e., a one-way coupled model, with mechanics driving the magnetics). The reduced order Euler-Bernoulli beam model is constructed using a similar approach. A bias strain distribution for a composite Euler-Bernoulli beam is first computed. In contrast to the FEA model, this assumes a spatially uniform demag factor, and spatially uniform material properties. A linearized eigenmode analysis is then performed that utilizes a linear piezomagnetic approximation to obtain a fully-coupled linearized solution. Both the finite element model and the analytical model have their strengths and weaknesses. The finite element model produces more accurate results utilizing spatially non-uniform and nonlinear material properties, and an accurate description of demag. However, the speed of the analytical model greatly outperforms the finite element model making it more suitable for understanding design alternatives and optimization. This work will examine how the assumption of uniform material properties and demag fields affects the accuracy of the analytical model and determine under what conditions the approximations are valid. Fig. 1 shows an example, from the finite element model, of the spatially distributed piezomagnetic coupling in the magnetoelastic material where the maximum coupling is found to be at the end of the beam. A specific focus will be made on examining the validity of using an analytical model to optimize the geometry of a magnetic antenna.

**References:**

J. Domann, and G. Carman. Journal of Applied Physics Vol. 121(4), p.044905 (2017) J. Xu, C. Leung, X. Zhuang et al. Sensors Vol.19(4), p.853 (2019) C. Dong, Y. He, M. Li et al. IEEE Antennas and Wireless Propagation Letters Vol. 19(3), p.398–402 (2020) L. Shu, M. Dapino, P. Evans, et al. Journal of Intelligent Material Systems and Structures Vol. 22(8), p.781–93 (2011) H. Zhou, and Y. Zhou. Smart Materials and Structures Vol. 16(1), p.198–206 (2007) V. Guerrero, and R. Wetherhold. Journal of Applied Physics Vol. 94(10), p.6659–66 (2003)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/ca0752a517da73cb8517cd348fe69b9c.jpg)
Fig. 1. Spatially distributed piezomagnetic coupling coefficient in the magnetoelastic layer of the beam. Note that the image is from the finite element model where quarter symmetry is utilized (center of the beam is the bottom right corner).

 Validation of a Reduced Order Magnetoelastic Beam Model

The paper presents the steps that brought the authors to design and implement the proof of principle of a fuel injector based on a custom Terfenol-D actuator. Magnetostrictive actuators play a relevant role in the perception of usefulness of smart materials and devices. Their applications are potentially wider than that of piezoelectric actuators because of the higher energy density and intrinsic robustness [1]. This activity is aimed to investigate the basic elements related to the fuel injection by fully controlling the injection rate. This would contribute to the NOx emission reduction that is a crucial goal for internal combustion engines. The study provides the design, development and a feasibility analysis of a magnetostrictive actuator for fuel injection, by providing both magneto-static numerical analysis of the actuator, the adaptation of a commercial fuel injector and its final experimental testing. Commercial fuel injectors work under on/off conditions and the modulation of the fuel rate is made over time. Recent developments make use of piezo-stacks to control the fuel injector needle [2] but the system is rather complex because the piezo-stacks cannot directly withstand the fuel high pressures. Indeed, the use of magnetostrictive materials would allow to have a simpler device with a full direct control of the injection and with higher achievable injection pressures [3]. The first step in the proof of principle has been the accurate design and analysis of the actuator employing a Terfenol-D magnetostrictive alloy aimed for the needle driving of a commercial fuel diesel injector with hydraulic drive. To this aim, a custom actuator employing a magnetostrictive alloy and fulfilling reasonable constraints for fuel injection (needle Lift of 50 - 200 μm, injection Pressure of 250-1000 bar, rise Time of 100 μs and a duty Cicle of 1/20 ms/ms) has been designed. The scheme of the magnetostrictive actuator, mechanically coupled with the modified injector, is shown in Fig. 1-left. The mechanical frame is acting also as a magnetic circuit and can accommodate different magnetostrictive rods. It is equipped with embedded force and displacement force sensors. The Fig. 1-right shows the realized prototype ready for injection tests. The magnetic circuit includes permanent magnets aimed to provide a bias magnetic field able to force the MS rods to the maximum stroke at zero current in the coil. In this condition the injector’s holes are closed at zero current. The coil is designed to reduce magnetization in the rod driving back the needle and allowing the fuel injection. The system has been assembled and successfully tested with different driving signals. The Fig.2 shows the trigger signal (A), needle displacement (B), flow rate (C) and coil current (D) in case of square (red curves) and triangular (black curves) references and it is apparent that the flow rate sufficiently follows the reference signal shape. In the complete paper more details about the design, the numerical analysis and the experimental tests will be given.

**References:**

[1] Apicella, V.; Clemente, C.S.; Davino, D.; Leone, D.; Visone, C. Review of Modeling and Control of Magnetostrictive Actuators. Actuators 2019, 8, 45. [2] H. Nouraei, R. Ben-Mrad and A. N. Sinclair, Development of a Piezoelectric Fuel Injector, IEEE Transactions on Vehicular Technology, 65(3),1162-1170, 2016, doi: 10.1109/TVT.2015.2410136. [3] G. Xue et al, A review of giant magnetostrictive injector (GMI), Sensors and Actuators A: Physical, 273 (2018), 159-181, https://doi.org/10.1016/j.sna.2018.02.001.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/3c13ab81c3773b95c2eb5a9ba9ebdcdf.jpg)
Fig.1: Scheme of the custom actuator (left) and of the assembled prototype (right).

![](https://assets.underline.io/uploads/markdown_image/1/image/d0cdb6c26116660b98d6714072534796.jpg)
Fig.2: Reference signal (A), needle displacement (B), flow rate (C) and coil current (D) in case of square (red curves) and triangular (black curves) references at a 500-bar injection pressure.

 Design and analysis of a fuel injector based on a magnetostrictive actuator

Introduction A magnetorheological elastomer (MRE) is a composite material with magnetic particles suspended or arranged within a non-magnetic elastomer matrix [1]. On the application of the magnetic field, it exhibits controllable stiffness and damping properties. When compared to a magnetorheological fluid, MRE has added advantage that it is leak-proof, deposition resistant, has a wide range of controllable properties, compatibility to mechanical components, and has a fast response and low power requirements. These advantages make it a very good option in various engineering applications. Changes in composition, shape and size of the MRE has led to a wide range of applications in vibration absorber, adaptive base isolator, and sandwich beams [2]. The magnetic field generated inside the MRE plays an important role in evaluating its properties based on the external load. However, due to non-availability of B-H curve data for MRE, it becomes cumbersome to numerically model it. If this data is available, the MRE and its magnetic circuit can be modelled in finite element (FE) software. B-H curve data for MRE can be found experimentally at a particular composition and size of MRE [3]. Li and Li [4] did FE analysis of adaptive base isolators utilizing laminated base MRE layers proposed by Li et al [5] and studied the magnetic field distribution in MRE layers in the deformed position. Therefore, FE modelling of MRE device can be an effective way to determine its optimal geometry and shape to draw the maximum magnetic field through it. However, all possible geometric combinations cannot be made every time to find out the maximum magnetic field through it. Data-driven modelling paves the way for it. In literature, most of the authors have applied the data-driven technique to model the dynamic behaviour of MRE [6][7]. However, predicting the magnetic field inside MRE using neural network data-driven technique has not been reported till now. In this study, a single-hidden layered feedforward neural network (FNN) is fitted for the magnetic flux data generated from FE model for different dimensions of the magnetic circuit and the study proposes an optimal dimension of it that will generate maximum magnetic flux density inside the MRE region of the magnetic circuit. The data fitting has been done using Neural Net Fitting toolbox and the optimization has been carried out using SQP method in optimization toolbox of MATLAB® 2017a. Methodology: FE modelling and FNN architecture An effective magnetic circuit is very important to maintain a strong magnetic in the MRE region. An axisymmetric FE model of the magnetic circuit is developed in ANSYS-APDL software which is shown in Fig. 1. For the FE analysis, the magnetic field is calculated in terms of magnetic flux density generated in the mid-section of the MRE and this has been set as an output parameter for a given geometry and applied current. Nodal solution for magnetic flux density at 3 A of current is fetched. After the FE model is developed, a log file is generated, which is later modified to include defined geometric variables. These geometric parameters are the same as the design variables (DVs) for the optimization problem. In the present work, the FNN is fitted to predict the magnetic flux density in the MRE region [8]. Training data for the FNN are generated from the results of the FE model encoded in a log file for different values of geometric variables taken between the bounds. The total training data have been divided as 80% for training, 10% for validation and 10% for testing. Results The number of activation units (or nodes) in the hidden layer is decided through error analysis. It is found that 8 activation units would be most suitable as it gives minimal mean square error (MSE) of all data sets as compared to other activation units. At this architecture, MSE for training data, validation data and testing data are 9.73×10-5, 4.3×10-5, and 8.21×10-5 respectively. The optimal weights obtained can be fed in MATLAB® software for a given number of inputs and activation units and the magnetic flux density can be predicted for any set of geometric input. Fig. 2 shows the regression analysis for the same where training data achieve the fitness value, R=0.9967, validation as R=0.9984 and testing as R=0.9971. Optimization problem aims at maximising magnetic flux density in the MRE region of the magnetic circuit for the given constraints on the geometric variables of the magnetic circuit. From the optimal results it is observed that the height of the steel block, Hsb, coil thickness, tc and annular yoke thickness, tay tend to take on their upper bound values, whereas the radius of the steel block, Rsb takes on the lower bound, for the magnetic flux density in the MRE region to be maximum. Conclusion The study presents the way to analyse the distribution of magnetic field in the MRE for different geometric dimensions of the magnetic circuit using the data-driven model and 30-50% of what it predicts that would be considered as approximate for real-time application. The application of data-driven modelling in MRE can be extended to decide the optimization of the geometric configuration of the magnetic circuit with the constraint on magnetic saturation limit and also to be able to predict the magnetic field inside the MRE when it is in a deformed shape. This would help in evaluating the performance of MRE at its different shapes and configurations of the magnetic circuit.

**References:**

[1] Carlson J D and Jolly M R, "MR fluid, foam and elastomer devices", Mechatronics, 10(4-5), 555-69, 2000. [2] Li Y, Li J, Li W and Du H, "A state-of-the-art review on magnetorheological elastomer devices", Smart Mater. Struct., 23, 123001, 2014. [3] Zeng J, Guo Y, Li Y, et al., "Two-dimensional magnetic property measurement for magneto-rheological elastomer", Journal of Applied Physics, 113(17), 919, 2013. [4] Li Y and Li J, " Finite element design and analysis of adaptive base isolator utilizing laminated multiple magnetorheological elastomer layers" in 14th Int. Conf. on Electrorheological Fluids and Magnetorheological Suspensions, Granada, July 2014, 26(14), 1861-70, 2014. [5] Li Y, Li J, Li W, et al., "Development and characterization of a magnetorheological elastomer based adaptive seismic isolator", Smart Mater. Struct., 22(3), 035005, 2013. [6] Jie Fu, Guanyao Liao, Miao Yu, Peidong Li and Junjie Lai, "NARX neural network modeling and robustness analysis of magnetorheological elastomer isolator", Smart Mater. Struct., 25, 125019, 2016. [7] Dingxin Leng, Kai Xu, Yong Ma, Guijie Liu and Lingyu Sun, "Modeling the behaviors of magnetorheological elastomer isolator in shear-compression mixed mode utilizing artificial neural network optimized by fuzzy algorithm (ANNOFA)", Smart Mater. Struct., 27, 115026, 2018. [8] Manjeet Keshav and Sujatha Chandramohan, "Geometric optimization of magnetorheological valve using feedforward neural networks for distribution of magnetic flux density inside the valve", Smart Mater. Struct., 28, 105018, 2019.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/6e00f83e27ff2b41ec45edbbe12691d3.jpg)
1. Axisymmetric 2D FE model of magnetic circuit of MRE.

![](https://assets.underline.io/uploads/markdown_image/1/image/5b4905de10e9fc056c2fd68d1cddb9f4.jpg)
2. Regression analysis of the fitted FNN model.

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