# Size Dependence of the Magnetoelastic Properties of Metallic Glasses for Actuation Applications

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{®}2826MB3. The strips have a length-to-width ratio R = L/w ranging from 2 to over 20. Significant variations of the apparent saturation Young’s modulus and the ΔE effect with strip geometry, changing from 160 GPa and 4% for L = 10 mm, w = 5 mm and R = 2, to 164 GPa and 9.6% for L = 35 mm, w = 1.7 mm and R = 20.6, have been observed. In order to obtain the highest values of the ΔE effect, the magnetomechanical coupling coefficient, k, and the quality factor of the resonance, Q, a value R > 14 is needed. The effective anisotropy field H

_{k}

^{*}, taken as the minimum of the E(H) curve, and its width ΔH, are not as strongly influenced by the R value, and a value of R > 7 is enough to reach the lowest value. From our measurements we infer that the formerly predicted value of R > 5 needed for a good magnetic and magnetoelastic response of the material must be actually regarded as the lowest limit for this parameter. In fact, we show that the demagnetizing factor N, rather than the length-to-width ratio R, is the parameter that governs the magnetoelastic performance of these strips.

## 1. Introduction

_{max}, and corresponding quality factor of the resonance, Q. For this study, strips of different length-to-width ratio R = L/w (ranging from 2 to 20.6) were cut from a single long ribbon of Metglas

^{®}2826MB3 to ensure they shared the same basic properties. The results indicate large differences in the measured main magnetoelastic parameters, differences that will be qualitatively analyzed as a function of the length-to-width ratio R value of each strip.

## 2. Materials and Methods

^{®}2826MB3 (Fe

_{37}Ni

_{42}Mo

_{4}B

_{17}, see [15]), was chosen for the experiments. It is a commercial material often used in magnetoelastic sensor applications due to its good magnetic and magnetoelastic properties, and resistance to corrosion [16,17]. Room-temperature hysteresis loops of the material were measured by a classical induction method, obtaining a saturation magnetization μ

_{0}M

_{S}= 0.88 T and initial susceptibility $\chi $ = 15,000. Saturation magnetostriction was determined by using strain gages connected to an electronically balanced bridge, obtaining a value of λ

_{S}= 12 ppm.

^{®}2826MB3 of 30 µm thickness in as-cast state, strips were laser cut with perfect rectangular shape of length varying from L = 35 mm to L = 10 mm in steps of 5 mm, and in four different widths: w = 5.0 mm, 3.33 mm, 2.5 mm and 1.66 mm. That is, the initial value of w = 5 mm of the long ribbon was reduced in a factor of 2/3, 1/2 and 1/3, respectively. Following this procedure, we obtained a set of 24 different rectangular strips and subsequently a set of magnetoelastic resonators with different length-to-width ratios R = L/w, ranging from 2 to 20.6.

_{r}) and anti-resonant (f

_{a}) frequencies, at an external applied magnetic field (or bias) H. Precise determination of frequencies was performed using a Hewlett Packard 3589A Spectrum Analyzer, working in the 50–250 kHz range. A detailed description of the set-up can be found in [9,18]. Due to magnetoelasticity, the measured resonant frequency (f

_{r}) will vary with the bias field H, and so too will the Young’s modulus, determined as $E(H)={\left[2L{f}_{r}(H)\right]}^{2}\rho $ [19], where L and ρ are the length and density of the sample. This field-dependence of the elastic modulus is known as ΔE effect and is usually given in % variations:$\Delta E(\%)=(1-E(H)/{E}_{S})\times 100$, E

_{S}being the Young’s modulus measured at magnetic saturation. Other important magnetoelastic parameters that can be determined from these measurements are the magnetomechanical coupling coefficient (${k}^{2}=({\pi}^{2}/8)(1-{({f}_{r}/{f}_{a})}^{2})$) [20] and quality factor of the resonance $(Q={f}_{r}/\Delta f)$, all quantities being function of the applied external magnetic field.

## 3. Results

_{S}and ΔE effect magnitude decrease from 164.5 GPa and 9.6% to 159.9 GPa and 4%. Similarly, maximum magnetoelastic coupling reduces from 0.3 to 0.14. All these changes are accompanied by an evident increase in the bias external field necessary to reach those minima in the E(H) and maxima in the k(H) behaviors, from 400 to 1435 A/m.

^{®}2826MB3 strips with different lengths L and different widths w are summarized in Table 1.

_{max}, the quality factor of the corresponding resonance, Q(k

_{max}), and the width ΔH of the measured E(H) curves, as a function of the length-to-width ratio R.

_{max}reaches its maximum value (about 0.3) at R > 12, while at R = 5, it remains at about 0.2–0.25. Corresponding Q(k

_{max}) at those same R values are the lowest measured, with a value about 20, while strips with value of R = 5 had resonance quality factors ranging from about 30 to 35.

## 4. Discussion

_{eff}through the expression:

^{®}2826MB3 was found, so a uniform applied bias H field along the longitudinal axis of the ribbon was assumed.

_{min}, Es and ΔE effect (about 1% for all of them) is accompanied by a monotonous decrease of ${H}_{k}^{*}$ and ΔH, and also an increase in the magnetoelastic coupling, k

_{max}, mainly due to the different L and w values.

_{f}of a general rectangular prism of dimensions 2a × 2b × 2c along x, y, z axis respectively, so that L = 2c, w = 2a and 2b = 30 μm, the thickness of the ribbon of Metglas

^{®}2826MB3. Obtained values of N

_{f}appear in Table 3.

_{f}values have been obtained assuming $\chi $ = 0, but the error generated by Chen et al’s. [25] calculations when assuming $\chi ={10}^{9}$ is within 1%. From Table 3 it is clear that despite the almost constant value of R = 6, the fluxmetric demagnetizing factor changes by almost one order of magnitude, from $1.42\xb7{10}^{-4}$ (for the longest strip) to $1.01\xb7{10}^{-3}$ (for the shortest one). Hence, despite the strong influence of the length-to-width ratio R on the magnetoelastic properties of Metglas

^{®}2826MB3 strips, the most important parameter is actually the demagnetizing factor of each strip, N.

_{min}, Es) and ΔE effect, while the best magnetoelastic parameters are only guaranteed for the strips with the lowest demagnetizing factors.

## 5. Conclusions

^{®}2826MB3 magnetoelastic strips in the magnetoelastic properties they exhibit has been extensively analyzed. A significant variation in Young’s modulus and the ΔE effect with changing geometry has been observed. Saturation Young’s modulus value E

_{S}and ΔE effect magnitude change from 159.9 GPa and ΔE = 4% (worst magnetoelastic case, L = 10 mm, w = 5 mm and R = 2), to 164.5 GPa and 9.6% (best magnetoelastic case, L = 35 mm, w = 1.7 mm and R = 20.6). In order to obtain the highest values of the ΔE effect, k and Q magnetoelastic parameters, we have observed that a value of R > 14 is needed. However, the effective anisotropy field ${H}_{k}^{*}$ from the minimum of the E(H) curve and its width ΔH, seems not to be so strongly influenced by the length-to-width ratio parameter, and a value R > 7 is enough to reach their lowest value. This is due to the strong influence—even higher than the influence of the R parameter—of the demagnetizing factor, N. Thus, a rectangular strip with moderate length-to-width ratio, but with a low demagnetizing factor value, can show an acceptable magnetoelastic response.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**A schematic proposal for: (

**a**) a simple open/close gas valve, or (

**b**) light flux shutter controller.

**Figure 2.**(

**a**) ΔE effect and (

**b**) magnetoelastic coupling coefficient measured for strips of Metglas

^{®}2826MB3 with the highest (R = 20.6) and the lowest (R = 2) R value.

**Figure 4.**Measured (

**a**) ΔE(%) magnitude and (

**b**) ${H}_{k}^{*}$ value as a function of the length-to-width ratio R, for all the measured strips.

**Figure 5.**Measured (

**a**) k

_{max}value and (

**b**) Q(k

_{max}) value as a function of the length-to-width ratio R, for all the measured strips.

**Figure 6.**Measured ΔH width as a function of the length-to-width ratio R for all the measured strips.

**Table 1.**Magnetoelastic characterization values obtained from the resonance/anti-resonance frequency measurements.

L (mm) | w (mm) | R = L/w | f_{r} (Hz) | E_{min} (GPa) | E_{S} (GPa) | ΔE (%) | ${\mathit{H}}_{\mathit{k}}^{*}\text{}(\mathit{A}/\mathit{m})$ | k_{max} | Q (k_{max}) | ΔH (A/m) |
---|---|---|---|---|---|---|---|---|---|---|

35 | 5 | 7 | 63,145 | 154.3 | 164.5 | 6.2 | 582.4 | 0.25 | 30.7 | 622.2 |

35 | 3.33 | 10.6 | 63,100 | 154.1 | 166.9 | 7.7 | 536.9 | 0.27 | 24.6 | 505.0 |

35 | 2.5 | 14 | 62,240 | 149.9 | 164.2 | 8.7 | 526.5 | 0.29 | 21.5 | 447.5 |

35 | 1.66 | 20.6 | 61,965 | 148.6 | 164.5 | 9.6 | 413.2 | 0.30 | 19.2 | 518.5 |

30 | 5 | 6 | 73,655 | 154.3 | 163.6 | 5.7 | 619.8 | 0.23 | 33.4 | 682.9 |

30 | 3.33 | 9.1 | 73,160 | 152.2 | 164.5 | 7.5 | 572.8 | 0.26 | 25.3 | 514.5 |

30 | 2.5 | 12 | 73,397 | 153.2 | 166.8 | 8.1 | 549.6 | 0.27 | 23 | 487.4 |

30 | 1.66 | 17.6 | 72,400 | 149.1 | 164.9 | 9.6 | 432.4 | 0.29 | 19.4 | 524.9 |

25 | 5 | 5 | 88,782 | 155.7 | 164.9 | 5.6 | 649.4 | 0.23 | 34 | 688.5 |

25 | 3.33 | 7.6 | 87,645 | 151.7 | 163.9 | 7.5 | 583.9 | 0.26 | 25.3 | 494.6 |

25 | 2.5 | 10 | 88,180 | 153.6 | 164.3 | 6.5 | 542.5 | 0.23 | 29.2 | 662.1 |

25 | 1.66 | 14.7 | 87,700 | 151,9 | 165.1 | 8 | 482.6 | 0.25 | 23.4 | 525.7 |

20 | 5 | 4 | 110,993 | 155.7 | 163.6 | 4.8 | 789.0 | 0.2 | 39.9 | 814.5 |

20 | 3.33 | 6.1 | 109,942 | 152.8 | 163.5 | 6.6 | 675.7 | 0.23 | 28.9 | 583.9 |

20 | 2.5 | 8 | 109,760 | 152.3 | 165.5 | 8 | 669.3 | 0.24 | 25.3 | 521.7 |

20 | 1.66 | 11.8 | 110,060 | 153.1 | 165.1 | 7.3 | 522.5 | 0.22 | 25.9 | 572.0 |

15 | 5 | 3 | 148,012 | 155.7 | 162.9 | 4.4 | 990.8 | 0.17 | 44.3 | 988.4 |

15 | 3.33 | 4.5 | 153,475 | 167.5 | 177.8 | 5.8 | 899.1 | 0.19 | 32.8 | 709.2 |

15 | 2.5 | 6 | 147,500 | 154.7 | 164.7 | 6.1 | 776.0 | 0.19 | 31.2 | 708.4 |

15 | 1.66 | 8.8 | 153,050 | 166.5 | 178.6 | 6.8 | 702.8 | 0.19 | 28.1 | 641.4 |

10 | 5 | 2 | 220,400 | 153.5 | 159.9 | 4 | 1418.4 | 0.14 | 48.5 | 1185.5 |

10 | 3.33 | 3 | 221,295 | 154.7 | 162.7 | 4.9 | 1286.0 | 0.14 | 39.2 | 895.1 |

10 | 2.5 | 4 | 222,902 | 157 | 164.1 | 4.4 | 1152.7 | 0.12 | 44.4 | 1197.4 |

10 | 1.66 | 5.9 | 221,175 | 154.6 | 164.2 | 5.9 | 1005.2 | 0.14 | 32.5 | 819.3 |

**Table 2.**Magnetoelastic characterization values obtained from the resonance/anti-resonance frequency measurements for strips with R = 6 equal or close value.

L (mm) | w (mm) | R = L/w | E_{min} (GPa) | E_{S} (GPa) | ΔE (%) | ${\mathit{H}}_{\mathit{k}}^{*}\text{}(\mathbf{A}/\mathbf{m})$ | k_{max} | Q (k_{max}) | ΔH (A/m) |
---|---|---|---|---|---|---|---|---|---|

30 | 5 | 6 | 154.3 | 163.6 | 5.7 | 619.8 | 0.23 | 33.4 | 682.9 |

20 | 3.33 | 6.1 | 152.8 | 163.5 | 6.6 | 675.7 | 0.23 | 28.9 | 583.9 |

15 | 2.5 | 6 | 154.7 | 164.7 | 6.1 | 776.2 | 0.19 | 31.2 | 708.4 |

10 | 1.66 | 5.9 | 154.6 | 164.2 | 5.9 | 1005.2 | 0.14 | 32.5 | 819.3 |

L = 2c (mm) | w = 2a (mm) | R = c/a | a/b ^{1} | c/(ab)^{1/2} | N_{f} ^{2} |
---|---|---|---|---|---|

30 | 5 | 6 | 166.7 | 77.5 | 0.00014248 |

20 | 3.33 | 6.1 | 111 | 63.3 | 0.00019609 |

15 | 2.5 | 6 | 83.3 | 54.8 | 0.00022911 |

10 | 1.66 | 5.9 | 55.3 | 44.8 | 0.00101335 |

^{1}2b = 30 µm, thickness of the ribbon of Metglas

^{®}2826MB3;

^{2}Fluxmetric demagnetizing factor value, extrapolated by using Table II in [25].

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Sagasti, A.; Gutiérrez, J.; Lasheras, A.; Barandiarán, J.M. Size Dependence of the Magnetoelastic Properties of Metallic Glasses for Actuation Applications. *Sensors* **2019**, *19*, 4296.
https://doi.org/10.3390/s19194296

**AMA Style**

Sagasti A, Gutiérrez J, Lasheras A, Barandiarán JM. Size Dependence of the Magnetoelastic Properties of Metallic Glasses for Actuation Applications. *Sensors*. 2019; 19(19):4296.
https://doi.org/10.3390/s19194296

**Chicago/Turabian Style**

Sagasti, Ariane, Jon Gutiérrez, Andoni Lasheras, and José Manuel Barandiarán. 2019. "Size Dependence of the Magnetoelastic Properties of Metallic Glasses for Actuation Applications" *Sensors* 19, no. 19: 4296.
https://doi.org/10.3390/s19194296