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Magnetic nanoparticles are an important asset in many biomedical applications ranging from the local heating of tumours[1] to targeted drug delivery towards diseased sites[2]. Recently, magnetic nanoflowers showed a remarkable heating performance[3,4] in hyperthermia experiments thanks to their complex structure leading to a broad range of magnetic dynamics. To grasp their full potential and to better understand the origin of this unexpected heating performance, we propose the use of Kaczmarz’ algorithm[5] in interpreting magnetic characterisation measurements[6]. It has the advantage that no a priori assumptions need to be made on the particle size distribution, contrasting current magnetic interpretation methods that often assume a lognormal size distribution. Both approaches are compared on DC magnetometry, magnetorelaxometry and AC susceptibility characterisation measurements of the nanoflowers. We report that the lognormal distribution parameters vary significantly between data sets, whereas Kaczmarz’ approach achieves a consistent and accurate characterisation for all measurement sets. Additionally, we introduce a methodology to use Kaczmarz’ approach on distinct measurement data sets simultaneously. It has the advantage that the strengths of the individual characterisation techniques are combined and their weaknesses reduced, further improving characterisation accuracy. Our findings are important for biomedical applications as Kaczmarz’ algorithm allows to pinpoint multiple, smaller peaks in the nanostructure’s size distribution compared to the monomodal lognormal distribution. The smaller peaks permit to fine-tune biomedical applications with respect to these peaks to e.g. boost heating or to reduce blurring effects in images. Furthermore, the Kaczmarz algorithm allows for a standardised data analysis for a broad range of magnetic nanoparticle samples. Thus, our approach can improve the safety and efficiency of biomedical applications of magnetic nanoparticles, paving the way towards their clinical use.

**References:**

&lt;br/&gt;[1] A. Espinosa, R. Di Corato, J. Kolosnjaj-Tabi, et al, ACS Nano, Vol. 10, p. 2436–2446 (2016)&lt;br/&gt;[2] K. T. Al-Jamal, J. Bai, J. T.-W. Wang, et al, Nano Letters, Vol. 16, p. 5652–5660 (2016)&lt;br/&gt;[3] P. Bender, J. Fock, C. Frandsen, et al, J. Phys. Chem. C, Vol.122, p. 3068–3077 (2018)&lt;br/&gt;[4] C. Blanco-Andujar, D. Ortega, P. Southern, et al, Nanoscale, Vol. 7, p. 1768–1775 (2015)&lt;br/&gt;[5] S. Kaczmarz, Bulletin International de l’Academie Polonaise des Sciences et des Lettres, Vol. 35, p. 355–357 (1937)&lt;br/&gt;[6] J. Leliaert, D. Schmidt, O. Posth, et al, Journal of Physics D: Applied Physics, Vol. 50, p.195002 (2017)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/d71e9f523e4a62d6a20fc7c82d90f1d3.jpg)
Best combined fit to a) the magnetorelaxometry measurement data and b) imaginary part of the AC-susceptibility measurement data of suspended nanoflowers using a lognormal distribution and using Kaczmarz’ method. c) The obtained relaxation time distributions corresponding to the fits in panel a) and b).



INTERMAG 2021

 Advanced analysis of magnetic nanoflower measurements to leverage their use in biomedicine

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Magnetic nanoparticles are an important asset in many biomedical applications ranging from the local heating of tumours[1] to targeted drug delivery towards diseased sites[2]. Recently, magnetic nanoflowers showed a remarkable heating performance[3,4] in hyperthermia experiments thanks to their complex structure leading to a broad range of magnetic dynamics. To grasp their full potential and to better understand the origin of this unexpected heating performance, we propose the use of Kaczmarz’ algorithm[5] in interpreting magnetic characterisation measurements[6]. It has the advantage that no a priori assumptions need to be made on the particle size distribution, contrasting current magnetic interpretation methods that often assume a lognormal size distribution. Both approaches are compared on DC magnetometry, magnetorelaxometry and AC susceptibility characterisation measurements of the nanoflowers. We report that the lognormal distribution parameters vary significantly between data sets, whereas Kaczmarz’ approach achieves a consistent and accurate characterisation for all measurement sets. Additionally, we introduce a methodology to use Kaczmarz’ approach on distinct measurement data sets simultaneously. It has the advantage that the strengths of the individual characterisation techniques are combined and their weaknesses reduced, further improving characterisation accuracy. Our findings are important for biomedical applications as Kaczmarz’ algorithm allows to pinpoint multiple, smaller peaks in the nanostructure’s size distribution compared to the monomodal lognormal distribution. The smaller peaks permit to fine-tune biomedical applications with respect to these peaks to e.g. boost heating or to reduce blurring effects in images. Furthermore, the Kaczmarz algorithm allows for a standardised data analysis for a broad range of magnetic nanoparticle samples. Thus, our approach can improve the safety and efficiency of biomedical applications of magnetic nanoparticles, paving the way towards their clinical use.

**References:**

 [1] A. Espinosa, R. Di Corato, J. Kolosnjaj-Tabi, et al, ACS Nano, Vol. 10, p. 2436–2446 (2016) [2] K. T. Al-Jamal, J. Bai, J. T.-W. Wang, et al, Nano Letters, Vol. 16, p. 5652–5660 (2016) [3] P. Bender, J. Fock, C. Frandsen, et al, J. Phys. Chem. C, Vol.122, p. 3068–3077 (2018) [4] C. Blanco-Andujar, D. Ortega, P. Southern, et al, Nanoscale, Vol. 7, p. 1768–1775 (2015) [5] S. Kaczmarz, Bulletin International de l’Academie Polonaise des Sciences et des Lettres, Vol. 35, p. 355–357 (1937) [6] J. Leliaert, D. Schmidt, O. Posth, et al, Journal of Physics D: Applied Physics, Vol. 50, p.195002 (2017)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/d71e9f523e4a62d6a20fc7c82d90f1d3.jpg)
Best combined fit to a) the magnetorelaxometry measurement data and b) imaginary part of the AC-susceptibility measurement data of suspended nanoflowers using a lognormal distribution and using Kaczmarz’ method. c) The obtained relaxation time distributions corresponding to the fits in panel a) and b).

technical paper

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

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

 Approximation Methods for Hybrid Constitutive Model of Magnetostriction

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.

 Prediction of Magnetic Field Strength in Magnetorheological Elastomers using Feedforward Neural Network.

Large magnetostriction and magnetoelastic coupling are properties extensively sought after because of their exploitability in low power consumption devices, energy conversion, and spintronics. Many alloys displaying large magnetostriction coefficients contain rare earths, whereas Fe-Ga systems, being rare earths free, have attracted attention because of their significant magnetostrictive response, coupled with good corrosion resistance and mechanical hardness [1]. Magnetostriction is particularly significant in Fe-rich alloys, with Ga content approximately equal to 19% and 27%. However, in this range of compositions, the Fe-Ga binary system is characterised by the possible presence of multiple phases, whose complex interplay strongly affects the magnetostrictive response. The stabilisation of the A2 phase by means of suitable thermal treatments turns out to be an effective way to obtain remarkable magnetostrictive properties. In this work, Fe-Ga alloys, containing 18, 21 and 23 at.% of Ga, were prepared in bulk form by arc melting. In their as-cast state they display a small magnetostriction, that is strongly improved after annealing at 1000 °C for 24 h, and subsequent rapid cooling. Multiple characterisation techniques, such as X-ray diffraction, Differential scanning calorimetry, Mössbauer spectroscopy, temperature-dependent magnetisation curves, hysteresis loops, magnetic force microscopy and magnetostriction measurements were exploited in synergy to gain a deep understanding of the structure-properties relationships in the studied alloys, before and after annealing. The A2 phase, which is favoured in the lower range of compositions and is promoted at the expense of the D03 one by the annealing, is responsible for characteristic dendritic and maze magnetic domains, shown in Figure 1, reporting domains patterns acquired with a magnetic force microscope on as-cast and annealed samples of the three compositions. Conversely, the lower-contrast regions, especially in the Ga23 as-cast sample, are due to the D03 phase. The transformation of the D03 phase into the A2 one with annealing, followed through extensive microstructure investigations, is responsible for the strong improvement of the magnetostriction, which almost reaches 240 ppm (transverse configuration) in the alloys with 18 at.% of Ga, after annealing (Figure 2).

**References:**

[1] J. Atulasimha, A.B. Flatau, Smart Mater. Sctruct., Vol. 20, p.043001 (2011).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/fb554383092a2de573f3c9c9372c7fef.jpg)
Fig. 1. MFM images of as-cast and annealed Fe-Ga bulk alloys with 18, 21 and 23 at.% of Ga. In the insets, FFT transforms of the same images.

![](https://assets.underline.io/uploads/markdown_image/1/image/f5d001f23cb724c503cddb52ce8186e5.jpg)
Fig. 2. Transverse magnetostriction as a function of Ga at.% for as-cast (blue curve) and annealed (red curve) samples.

 Phase Transformation and Magnetostriction in Fe100-xGax bulk alloys

Magnetoelastic sensors are typically amorphous ferromagnetic ribbons in which mechanical and magnetic properties are coupled, so an acoustic wave can be excited in these materials by the application of an alternating magnetic field and detected by the magnetic changes induced in it. The magnetoelastic ribbon can enter in resonance at certain frequencies of excitation, compatible with the dimensions and elastic properties of the material. These resonance frequencies are sensitive to different external parameters, which can be used to design different types of sensors [1], in particular, differences in ribbon mass loading cause a shift of the resonance frequency of the sensor. This sensitivity and its ability to query and detect remotely, made these devices especially interesting for sensing biological and chemical agents. In the present work, we have used magnetoelastic sensors to monitor the precipitation reaction of calcium oxalate crystals (CaC2O4), one of the most common minerals which form calcifications in the urinary track. The formation of these crystals was measured in real-time by tracking the changes in the resonant frequency of the sensor, which decreases as the precipitate is deposited on its surface (Figure 1), changing its total mass [2]. Previous works by Bouropoulos et al. have used magnetoelastic sensors based on the commercial material Metglas® 2826 for monitoring the kinetics of different precipitation reactions [3]; in this work a different amorphous ferromagnetic alloy ((Fe-Cr)-Based metallic glass of composition Fe73Cr5Si10B12) that has shown to have an improved corrosion resistance has been used [4]. This composition contains a small amount of chromium that allows the formation of a passivation layer on the material, thus favoring its corrosion-resistant behavior, which allows it to be used in biomimetic environments without any pre-treatment of its surface and avoid possible changes in the resonance frequency due to the corrosion of the ribbon. The precipitation reaction of calcium oxalate was carried out by mixing equal parts of solutions at the same concentration of oxalic acid and calcium chloride, and the formation of the insoluble salt crystals was monitored in a 500 s time interval. Solutions of different concentration (1, 5, 10, 30, 50 and 100 mM) were used in order to observe its effect on the rate of reaction (Figure 1). In addition, the monitoring of the reaction was complemented with the numerical fittings of the obtained resonance curves (Figure 2) to an analytical expression that describes the behavior of the resonance [5]. These numerical fittings improve the observation of the temporal evolution of the different parameters that characterize the response of the sensor (such as the resonance frequency, the damping parameters or the quality factor Q), and can be used to monitor the precipitation reaction, since in addition to a decrease in the resonance frequency, the deposition of calcium oxalate crystals on the ribbon causes an increase in the damping experienced by the sensor and therefore, a decrease in its quality factor. The numerical fittings made allowed to obtain this quality factor of the sensor with more accuracy than with traditional methods, which are unreliable when working with damped signals, such as when the sensor is immersed in a liquid and subject to mass loads. In addition, these fittings allow to get rid of the noise associated with the experimental measurements, which facilitates the calibration and use of the sensor. These fittings allow to follow the evolution of the precipitation reaction of calcium oxalate in a more reliable, precise and noiseless way.

**References:**

[1] C. Grimes, C. S. Mungle and K. Zeng, Sensors, Vol. 2, no 7, p. 294-313 (2002). [2] B. Sisniega, A. Sagasti, J. Gutiérrez and A. García-Arribas, Sensors, Vol. 20, no 10, p. 2802 (2020). [3] N. Bouropoulos, D. Kouzoudis and C. Grimes, Sensors and Actuators B: Chemical, Vol. 109, no 2, p. 227-232 (2005). [4] A. Sagasti, V. Palomares, J. M. Porro, Materials, vol. 13, no 1, p. 57 (2020). [5] B. Sisniega, J. Gutiérrez, V. Muto and A. García-Arribas, Materials, vol. 13, no 21, p. 4708 (2020).

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/bb7d61ffc90c7103cae1159d7944bdd0.jpg)
Fig. 1: Temporal evolution of the resonance frequency of the magnetoelastic sensor as the precipitation reaction progresses, for different concentration of reactants in the reaction (30, 50 and 100 mM) and a control measurement (sensor in a vial with distilled water).

![](https://assets.underline.io/uploads/markdown_image/1/image/924bd6ccec73727c2a3c30779f0becc9.jpg)
Fig. 2: Temporal evolution of the resonance curves of the magnetoelastic sensor during the precipitation reaction of calcium oxalate for the case of 30 mM concentration of reagents. As the reaction progresses, the resonant frequency and the quality factor Q of the curves decrease.

 Real Time Monitoring of the Precipitation of Calcium Oxalate by Using Corrosion Resistant Magnetoelastic Resonance Sensors

Magnetostrictive materials are distinguished by an inherent coupling between their mechanical and magnetic processes which is manifested as a direct (actuator) and an inverse (sensor) effect. Iron-gallium alloys (Galfenol, Fe100-xGax with 12&lt;x&lt;30) are a particularly promising magnetostrictive, transducer material that combines highly desirable tensile and compressive mechanical attributes with superior magnetic properties. These characteristics make it possible to design robust structural sensors and energy harvesters. However, the behavior of the mechanical and magnetic phenomena is essentially non-linear, making it difficult to develop analytical models to find an optimal operational condition. Because the magnetostriction strongly depends on both the pre-stress and magnetic field, the characterization of the material is usually conducted under controlled conditions for normalizing the parameters with respect to each other. In this study, large-grained, highly textured, polycrystalline Galfenol samples with 15, 17, 18, and 19 at. % Ga were evaluated. As shown in Figure 1(a), standard magnetic hysteresis loops were measured at various applied compressive stresses. Changes in the flux density, B, as a function of applied stress were determined as the intercept values at specific applied magnetic fields, shown by the red dotted line for the 50 Oe bias magnetic field condition. Figure 1(b) shows an example of the flux density as a function of the mechanical pre-stress at constant magnetic fields of 30, 50, 80, 100, 120, and 150 Oe for 18% Galfenol sample (colored dot symbols). To estimate the response of the magnetic flux density as a function of applied stress, a hyperbolic tangent function was used with 4-nonlinear parameters (a, b, c and d) which are functions of magnetic field, as defined below: B(T,H)=a(H)×tanh[b(H)×(T+c(H))]+d(H) (1) The calculated magnetization based on the fitted parameters is plotted in Figure 1 (b) with solid lines, which fits the experimental data well. The continuous magneto-mechanical sensitivity, defined as the slope of the magnetization versus stress data, was calculated from the fitted equation (1) and the magneto-mechanical sensitivity results are depicted in Figure 2 for various bias magnetic fields. The same data process procedure was applied for the experimental characterization data of 15, 17 and 19% Ga specimens and Figure 3 shows the maximum value of the magneto-mechanical sensitivity as a function of applied bias field for the various compositions of Galfenol. As can be seen in Figure 3, 17% Galfenol shows higher magneto-mechanical sensitivity than 18% Galfenol, which usually shows higher magnetostriction values. Based on the results in Figure 3, 17% Galfenol will give better performance for sensor and energy harvester applications as compared to 18% alloys, which are used for actuation.

**References:**



**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/ee2402680b3e12b17a954ad38c86b980.jpg)
Figure 1. (a) Flux-density (B) versus external applied magnetic field (H) at various pre-stress values (T). (b) Magnetic induction (B) versus external compressive stress (T) at various bias fields (H) and curve-fit results based on hyperbolic tangent function.

![](https://assets.underline.io/uploads/markdown_image/1/image/741346be7a511b6fbdbbcf08a654cc25.jpg)
Figure 2. (a) Magneto-mechanical sensitivity at various bias magnetic fields (30, 50, 80, 100, 120 and 150 Oe). (b) Magneto-mechanical sensitivity at various bias magnetic fields for various composition.

 A Calculation of Magneto-Mechanical Sensitivity for Various Compositions of Galfenol

Magnetic hyperthermia is one of the most promising biomedical applications of magnetic nanoparticles (NP) and is intended to be alternative to cancer therapies based on drug delivery and radiotherapy. It is based on the fact that magnetic NP dissipate heat when an oscillating magnetic field is applied to them in a quantity (specific absorption rate, SAR) that is closely related to the area of the hysteresis loop. The main problem in the field has become to find the suitable range of parameters that maximize SAR for a given material [1], SAR depends of course on the amplitude of the applied magnetic field and its frequency, but also on intrinsic parameters of the NP such as saturation magnetization, anisotropy, shape and size [2]. Although the role of external parameters is somehow well contrasted, there is still ongoing controversy on the role that dipolar interactions (DI) and aggregation state of the assemblies play on SAR. We will present results of Monte Carlo simulations of hysteresis loops of interacting NP assemblies in the macrospin approximation [3]. We will present first results of different regular spatial arrangements of NP, showing the influence of interparticle separation and particle size on SAR. Next, we will study the case of randomly placed NP with varying concentrations mimicking experimentally found situations [4] (inside and at the surface of liposomes/cells, clusters). It is found that formation of chain-like arrangements or assemblies with prolate shapes, lead to considerable increases in SAR due to DI. Acknowledgements: Work supported by Spanish MINECO (PGC2018-097789-B-I00), Catalan DURSI (2017SGR0598) and EU FEDER funds (Una manera de hacer Europa) also CSUC for supercomputer facilities.

**References:**

[1] S.-H. Noh, et al. Nano Today 13, 61 (2017); R. Hergt et al. Nanotechnology 21, 015706 (2010). [2] B. Mehdaoui, et al.Phys. Rev. B 87, 174419 (2013); R. P. Tan et al.Phys. Rev. B 90, 214421 (2014); P. Hasse and U. Nowak Phys. Rev. B 85, 045435 (2012); P. V. Melenev et al. Phys. Rev. B 86, 104423 (2012). [3] C. Martínez-Boubeta et al.Scientific Reports 3, 1652 (2013). [4] Brollo et al., Phys. Chem. Chem. Phys. 20, 17829 (2018); D. Niculaes et al., ACS Nano 11, 12121 (2017)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/e6cf5b763ca228f2ac9da0bc2351d2af.jpg)
Concentration dependence of the hysteresis loop area for different kinds of NP random assemblies.

 Aggregates and Dipolar Interactions in Nanoparticle Assemblies for Magnetic Hyperthermia

Recent wearable ultrasensitive magnetometers opened novel applications of magnetoencephalography (MEG) to brain machine interface. In a recent study, the subject was allowed to move the head in a limited area of approximately 40 x 40 x 40 cm3 having a uniform magnetic field produced by a set of active shimming coils(1). The significant challenge of such wearable MEG systems is to improve the uniformity of magnetic field in the entire space in the magnetic shield room (MSR) and to suppress motion artifacts in MEG signals. A typical approach of realizing high field uniformity has been to make MSRs with highly symmetric shapes such as cylinder and polyhedron. However, these MSRs still require additional adaptive techniques to improve residual non-uniformity. The purpose of this study is to develop a new method of passive shimming for an MSR using ferromagnetic plates attached to its walls. The concept of this study is to optimize the spatial distribution of the magnetic field inside MSR with combination of ferromagnetic plates attached to the surface of MSR. The method mainly consists of two steps. In the forward problem, firstly, we calculated the spatial distributions of magnetic fields with the ferromagnetic plates of 601 mm (1 mm thickness) attached at 16 locations on the surface of a downsized magnetic shield box (MSB) of 200350 mm (1 mm thickness) (Fig. 1(a)). The uniformity was evaluated by obtaining the standard deviation for 9 measurement points inside the MSB. We established the relationship between the individual shimming plates and the resulting magnetic field uniformity inside MSR. Secondly, by solving the inverse problem based on the forward solution, we identified the optimum number of plates to be attached at each location. We used the Tikhonov regularization2) to solve the ill-posed inverse problem, and the L-curve method3) to determine its optimal parameters. In the experiments, we measured and evaluated the spatial distribution of magnetic fields using nine magneto-impedance (MI) sensors (ROHM BM1422AGMV), as shown in Figs. 1(b) and (c). We found one shimming steel plate caused magnetic field variation of 50 ~ 300 nTinside MSB in the forward problem. And then we let the value of magnetic field of 9 measurement points equal to zero and solve the inverse problem to find the number of shimming plates. Figure 2(a) shows the combination of the shimming plates optimized by the inverse problem. Figures 2(b) and (c) show the magnetic field distribution in the area of 120 x 120 mm inside MSB with and without the combination of shimming plates, respectively. The shielding factor increased 131% of MSB, from ~5 to ~8, by the proposed method. We also evaluated magnetic field uniformity by calculating the standard deviation of 9 measurement points inside MSB. The standard deviation is 0.211 and 0.379 with and without the shimming method, indicating the magnetic field uniformity inside MSB has improved 44% by the shimming method. Figures 2(d) and(e) show the magnetic field distribution measured by nine MI sensors in the area of 160 x 160 mm without MSB and with MSB, respectively. The experimental shielding factor is ~7, which is comparable to the simulation results. In conclusion, the proposed methods of passive shimming for an MSR using ferromagnetic plates demonstrates the feasibility of improving the magnetic field uniformity and is applicable to wearable MEG system in the future. This work was supported by MEXT Q-LEAP with the Grant Number JPMXS0118067395.

**References:**

(1) H. Niall, et al., NeuroImage 181: 760-774 (2018). (2) D. Calvetti, et al., Journal of Computational and Applied Mathematics 123.1-2: 423-446 (2000). (3) H. Christian, et al., "The L-curve and its use in the numerical treatment of inverse problems." (1999): 119-142.

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/225e9eac620c2f7bd03e9d474e4e2734.jpg)
(a) Schematic of 16 shimming location and 9 measurement points of shimming design (b) Magnetic shield box assembled by steel. (c) Image of MI sensors setup. We used 9 MI sensors fixed on a resin board to measure the magnetic field distribution, distance of each sensor is 80 cm.

![](https://assets.underline.io/uploads/markdown_image/1/image/0a66a80a354d85830facd3503e6cfc5b.jpg)
Figure.2 (a) Number of plates on each shimming location. 2D Simulation results of (b) no shimming case and (c) with shimming case solved by inverse problem, Experiment results of (d) magnetic field distribution without MSB (Magnetic Shield Box) and (e) magnetic field distribution with MSB (Magnetic Shield Box)

 Shimming Design of Magnetic Shield Room Using Ferromagnetic Plates

Additive manufacturing of smart materials is a novel field in engineering. The ability to produce adaptable materials in varying geometries opens the door to the creation of a tremendous variety of new sensors, actuators and applications in domains including, transportation, energy production and medicine [1][2]. In this work, our objective is to build a ferromagnetic catheter by using additive manufacturing principles employed for the creation of smart materials (Fig. 1). Induction heating phenomena takes place in the ferromagnetic catheter due to the influence of a varying magnetic excitation field. This contactless heat source can be used to heal locally and destroy damaged blood vessels. According to medical specialists, thermal ablation of varicose veins requires a temperature close to 120°C and time constants lower than a few seconds [3]. First investigation results showed great potential for success using the low frequency induction heating (LFIH) method but further surveys and validation steps are still needed before in vivo testing can commence. Fig. 1 – Illustration of the ferromagnetic catheter. In this study, multi-physics simulation results, including ferromagnetic, thermal and fluid mechanics domains were compared to experimental data, then conclusions are proposed to establish the most adapted magnetic waveform (frequency) and its influence on the human tissues. In our processs, heat is transferred through a contactless method. The flexible 3D printed ferromagnetic catheter is subjected to an alternating magnetic excitation. Under the influence of the magnetic excitation, eddy currents are generated through the catheter creating heat. First published results [4][5] describing this technique have experimentally demonstrated the feasibility of heat generation but the conditions tested and simulated were still far from the in vivo environment. In this study, we go deeper into the investigation process, we propose simulation results (magnetic, thermal …) under much realistic experimental conditions. In [4], the authors limit the magnetic excitation signal to a few kHz. The purpose is to avoid undesired charge displacements in the surrounding tissue, but this restriction is strictly assumption based and further investigations must be under-taken to determine the optimal value for ablation while minimizing charge displacement. No further investigations have been proposed to establish a significant cutoff frequency. In this new research, a ferromagnetic model for the simulation of the ferromagnetic composite magnetic behavior is developed first. This model is based on previous work on the same topic [6]-[8] and it investigates the frequency dependence of the composite hysteresis area, i.e. the heat generated through the LFIH influence. Next, comparisons between the model and experimental situations are investigated to set the ferromagnetic model parameters and to establish the most efficient LFIH frequency. Finally, multi-physics simulation results combined with experimental tests are reviewed to predict the ferromagnetic composite’s behavior on living tissues under the influence of the optimized magnetic excitation. Before validating the ferromagnetic catheter as a realistic treatment method for varicose veins, special attention must be given to the undesired charges or ionic displacements in the healthy surrounding tissues causing local temperature elevations and degradations. A compromise between treatment and negative effects on surrounding tissue must be made to establish the best working frequency. Numerical simulations are proposed to establish the optimized configuration of the magnetic source (frequency, amplitude) and of the ferromagnetic catheter behavior, these results constitute a huge step toward a first medical test.

**References:**

[1] Standard A. F2792, Pa, USA: ASTM International (2012) [2] G.N. Levy & al., CIRP Annals-Manuf. Technol., vol. 52 iss.2, pp. 589-609 (2003) [3] S. Nehemiah & al., Cochrane Library, iss. 13 (2013) [4] Z. Xiang & al., polymers 2020, 12(2), 386 (2020) [5] Z. Xiang & al., Mat. & Des., Vol. 174, 107804 (2019) [6] M. A. Raulet & al., IEEE Transactions on Magnetics, vol. 40, n° 2, pp. 872 – 875 (2004) [7] Z. Xiang & al., IEEE Trans. on. Mag, IEEE int. Mag. Conf. (INTERMAG), Singapore (2018) [8] B. Gupta & al., Proc. of the 21st int. conf. on the comp. of electromag. fields COMPUMAG, Daejeon (S. Korea), (2017)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/7cd27e86849e75379de11a4ef8f53341.jpg)

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