A. Microstructure

There is an intrinsic difficulty on distinguishing between phases with structures A2 and D0₃, using X-ray diffraction, since the most intense reflection peaks of these structures are coincident. Although, ordered structure superlattices have several extra peaks, the intensities of these are usually very small. This occurs because this intensity is proportional to the difference between the atoms’ scattering factors, particularly for XRD with Cu K-α radiation.^{Reference Loureiro, Batista, Khomchenko, Costa and Le Caër12} For the phases Fe(Ga)-α (A2) and Fe₃Ga (D0₃), this problem occurs, but using Mo K-α radiation, it was possible to detect the presence of the D0₃ structure as depicted in Fig. 2.

FIG. 2.

X-ray diffraction patterns of the Fe–Ga and Fe–Ga–B alloys.

The reflection corresponding to the plane (111) of the D0₃ structure and occurring at 2θ ∼ 12.5° is definitely present for both alloys and only attributed to the phase Fe₃Ga. In addition, the highest relative intensity of this peak for the Fe–Ga–B alloy suggests a higher volume fraction of the ordered Fe₃Ga (D0₃) phase in this alloy. In the alloy with B, the presence of Fe₂B phase was also detected and identified by 4 extra peaks on the XRD pattern compared to the sample without B. Using SEM micrographs of the Fe–Ga–B alloy as the one presented in Fig. 3, together with the image analysis freeware program Image J,¹³ it was possible to estimate a volume fraction of the Fe₂B phase of 12.5%. This result was used as a guide for a qualitative estimation, made by simulation using PowderCell®^{Reference Kraus and Nolze14} and the DRX patterns, of the volume fractions of structures A2 and D0₃ and the Fe₂B phase in the Fe–Ga–B alloy. The results of phase volume fraction on the Fe–Ga–B alloy are 11.5% of Fe₂B and 88.5% of D0₃, which means that the A2 structure (α-phase) is not present. Using the same type of simulation for the Fe–Ga alloy, the volume fractions found were 52.9% of A2 and 47.1% of D0₃. These results of simulations are qualitative, but show a tendency. For example, the absence of A2 in the sample with B could not be absolutely true, but the simulation suggests that the volume of Fe₃Ga (D0₃) in the Fe–Ga–B alloy is probably much higher than in the binary alloy Fe–Ga. For a quantitative result, neutron diffraction or high resolution X-ray diffraction could be used.^{Reference Zhao, Mellors, Kilcoyne and Lord15,Reference Cao, Bai, Li, Viehland, Lograsso and Gehring16}

FIG. 3.

Optical and SEM micrographs of Fe–Ga and Fe–Ga–B alloys.

Figure 3 shows optical micrographs of Fe–Ga and Fe–Ga–B alloys. It is not possible to distinguish between the phases with structures A2 and D0₃ because they have coherent interface. The binary alloy resembles single phase and the ternary alloy resembles a two phase alloy. The second phase in Fe–Ga–B is Fe₂B.^{Reference Gao, Li, Zhu, Li and Zhang8} The grain structure was refined due to the addition of B. However, from X-ray analysis, both alloys are two-phases alloys; Fe–Ga with A2 and D0₃ structures and Fe–Ga–B with D0₃ and Fe₂B phase. The Fe₂B particles have spheroid shape and are mainly localized at grain boundaries but there are also particles inside the grains.

The chemical composition measured by EDS resulted in 18.6% of Ga (at.%) for both Fe–Ga and Fe–Ga–B alloys (matrix), with an error lower than 2%. This result also suggests that the excess of Fe added in the Fe–Ga–B alloy before melting was indeed used to generate the Fe₂B phase, therefore corroborating with the prior assumption that B did not go into the lattice. It is worth to say that the mass losses during melting and heat treatment were negligible.

Material thermal history is also an important variable in the ordering/disordering behavior of Fe–Ga alloys as already explained.^{Reference Rahman, Gou, Liu, Ma and Yan3,Reference Ikeda, Kainuma, Ohnuma, Fukamichi and Ishida4} For both magnetic materials, Fe–Ga and Fe–Ga–B, the temperature of annealing used 1200 °C is well above the Curie temperature (650 °C) and both materials present D0₃ structure ordering as discussed by Borrego et al.^{Reference Borrego, Blazquez, Conde, Conde a and Roth17} However, the alloys with B presented even higher quantities of Fe₃Ga (D0₃) compared to the one without B. The main reason for this is that the sample is massive (∼45 g) and even with the quenching after the annealing, the cooling process is not fast enough to prevent ordering.^{Reference Borrego, Blazquez, Conde, Conde a and Roth17,Reference Emdadi18} But as the cooling process of both Fe–Ga and Fe–Ga–B alloys was the same, some additional cause happened to the Fe–Ga–B alloy to have much more D0₃. We suggest the possibility that during the cooling of the alloys, the boride Fe₂B (T _C = 742 °C,^{Reference Iga, Tawara and Yanase19}) in the Fe–Ga–B alloy turn out to be ferromagnetic before the Fe–Ga matrix, therefore Fe₂B deforms due to spontaneous magnetostriction just close to 742 °C. This volumetric deformation may cause local stresses around Fe₂B and therefore in the matrix. From thermodynamics point of view, the phase diagram of solids depends on temperature, concentration of components, pressure, and volume. Normally, pressure and volume are kept constant. However, if the pressure is not constant, and that would be the case of stresses caused on the matrix by the phase Fe₂B, the phase equilibria and stability between the phases α and Fe₃Ga (D0₃) in the Fe–Ga–B alloy phase diagram could change.

In Fe–Al–B alloys,^{Reference Bormio-Nunes and Hubert20} the addition of 1.6% of B to the alloy with 20% of Al caused the formation of Fe₂B too and the matrix of the ternary alloy enriched in Al. In that case, the appearance of the Fe₃Al (D0₃) phase was associated with the change of the composition of the matrix. However, in the present case, the composition of the Fe–Ga–B matrix was kept 18.6%, the same of the Fe–Ga binary alloy.

B. Static magnetic measurements

B versus H curves are shown in Fig. 4(a) for diverse fixed values of compression stresses up to |σ| = 70.7 MPa. In Fig. 4(b), the relative permeability μ_r versus H curves obtained by the derivative of B versus H curves and normalized by the value of the vacuum magnetic permeability are depicted.

FIG. 4.

Fe–Ga and Fe–Ga–B alloys: (a) B and (b) μ_r versus H, for compressive stress values of |σ| = 0, 17.7, 28.3, 38.9, 49.5, 60.1, and 70.7 MPa.

From B versus H curves, it is possible to notice that with the increasing of the applied stress, the inclination of the curves diminishes for both alloys. As a consequence, the relative permeability also decreases with increasing of the applied stress. This occurs because the compression stresses oblige the magnetic moments to align in the plane perpendicular to the force direction. Then, the compression stress induces an extrinsic anisotropy to the material, which is added to the intrinsic anisotropy. Consequently, more energy is necessary to overcome the additional extrinsic anisotropy due to the compression forces and magnetization saturation occurs in a higher field compared to σ = 0 condition. Relative permeability varies significantly with the applied stress in the alloy Fe–Ga, but varies less in Fe–Ga–B.

Although the permeability of Fe–Ga is higher than that of the Fe–Ga–B alloy, this happens only for compression stresses values of 0 and −17.7 MPa. For |σ| ≥ 28.3 MPa, μ_r is practically the same for both alloys. For Fe–Ga alloys with 18.4 and 20.9% of Ga, μ_r decreases significantly with application of compression stress,^{Reference Mahadevan, Evans and Dapino21} the same result observed in the polycrystalline Fe–Ga alloy of the present work. In addition, the alloy with 20.9% has lower permeability than the one with 18.4% of Ga, and the difference is attributed to the low crystal anisotropy of the alloy with highest Ga content. This decrease of the anisotropy is due to the partial ordering of the structure A2 → D0₃ for higher Ga contents than 18.4%. There is not an important contribution of the Fe₂B phase to the magnetization mechanisms.

In this phase at the field level explored in the present work, the rotation of magnetic moments occurs only in the easy directions,^{Reference Bormio-Nunes and Hubert20} which result in a high initial susceptibility. Therefore, Fe₂B does not contribute to the decrease of the initial susceptibility for σ = 0.

Consequently, we attribute the noteworthy drop observed in μ_r for σ = 0 of Fe–Ga–B also to the A2 → D0₃ induced by the slow cooling rate and the presence of the Fe₂B phase during the solidification process.

In addition, with respect to the hysteresis width, values of coercive field of 200 A/m and 249 A/m were found for Fe–Ga and Fe–Ga–B alloys, respectively. These values of H _c reveal that the increase of grain boundaries in Fe–Ga–B, due to the refined grain structure compared to Fe–Ga, is not providing strong domain wall pinning since the increase on H _c is not too strong.

The longitudinal magnetostriction λ_long and the magnetomechanical coefficient d ₃₃ = dλ_long/dH curves as a function of the applied field H are depicted in Figs. 5(a) and 5(b), respectively. From these curves, the detrimental effect of B addition to the magnetomechanical properties of the Fe–Ga alloy is clear. The alloy Fe–Ga–B is almost not influenced by the application of prestress in the range 0 ≤ |σ| ≤ 70.7 MPa. The saturation value of λ_long is constant ∼50 ppm. Therefore, the jump effect, which is the enhancement of the magnetostriction due to application of stress, is almost not present in Fe–Ga–B. This is coherent because this effect is much more pronounced in materials that have high magnetic anisotropy. In the case of Fe–Ga–B, the phase Fe₃Ga with D0₃ structure has a much smaller anisotropy compared to the α-phase.^{Reference Kumagai, Fujita, Fukamichi, Oikawa, Kainuma and Ishida22}

FIG. 5.

Magnetostriction and sensitivity of alloys Fe–Ga and Fe–Ga–B: (a) λ_long and (b) sensitivity d ₃₃ = dλ_long/dH versus H for compressive stress values in the range 0 ≤ |σ| ≤ 70.7 MPa.

While, Fe–Ga has saturation λ_long increased from 78 to ∼120 ppm for |σ| ≥ 28.3 MPa. In fact, the magnetomechanical coefficient depends on the inclination of the curves λ_long versus H before saturation, and from Fig. 5(a), it is possible to see the higher variation of the slope of the curves of the Fe–Ga alloy compared to Fe–Ga–B.

The domain wall movement controls magnetostriction increase up to saturation and the width of the wall is important on this mechanism. The phase α has higher anisotropy than Fe₃Ga (D0₃)^{Reference Rafique, Cullen, Wuttig and Cui23} and the domain wall width is $\delta = \pi {\left( {{A / K}} \right)^{1{\rm{/}}2}}$, where A is the exchange stiffness constant and K is the anisotropy constant.^{Reference O’Handley24} Therefore, considering that Fe atoms have the same exchange energy in both structures, it is expected that the domain walls in the D0₃ structure are much wider than in A2. Indeed, larger domain walls was found for slow cooled Fe–Ga alloys compared to quenched ones^{Reference Golovin25} in a maze-like domain pattern, a characteristic of the ordered structure.^{Reference Emdadi18} As a result, in the D0₃ structure, the magnetostriction is controlled basically by magnetic moment rotation, while in A2 is effectively domain wall movement by the rotation of a smaller quantity of magnetic moments. In addition, the random distribution of magnetic moments in Fe₂B results in smaller magnetostriction curve inclinations.

For both materials, the maximum d ₃₃ (sensitivity) values occur for σ = 0, being 18 nm/A and 5.7 nm/A for Fe–Ga and Fe–Ga–B, respectively. The lowest values occur for σ = −70.7 MPa and are 8.10 nm/A (Fe–Ga) and 4.1 nm/A (Fe–Ga–B). The rate of decrease on sensitivity is more pronounced in Fe–Ga.

C. Quasi static magnetic measurements

The curves of magnetic induction as a function of the continuously applied compression stresses at fixed values of the magnetic field H are shown in Figs. 6(a) and 6(b). In Fig. 7, the corresponding derivatives of the curves of Fig. 6 are depicted and represent the PZM coefficient (sensitivity) of the materials to be used as a force sensor, d ₃₃*.

FIG. 6.

Magnetic induction B versus |σ| curves of alloys (a) Fe–Ga and (b) Fe–Ga–B, for fixed magnetic field values of 1, 2, 4, 6, 8, and 10 kA/m.

FIG. 7.

d ₃₃* = dB/d|σ| versus |σ| of the alloys Fe–Ga (a) and Fe–Ga–B (b) in the range of 0 ≤ |σ| ≤ 80 MPa and fixed magnetic field values of 1, 2, 4, 6, 8, and 10 kA/m.

Once more, the sensitivity is higher for Fe–Ga than the Fe–Ga–B alloy and consequently, the PZM coefficient. This coefficient representing the capacity of transforming mechanical energy in electric (magnetic) energy is also higher.

For Fe–Ga, maximum values of d ₃₃* varies from 6 to 16 mT/MPa for applied fields between 2 and 8 kA/m. As for the Fe–Ga–B, varies from 2 to 5 mT/MPa for an applied field of 8 kA/m.

The Fe–Ga d ₃₃* versus σ curves have a parabolic shape with negative concavity. To work with the maximum sensibility, a field of 2 kA/m should be used up to 11.7 MPa, 4 kA/m for 11.7 < |σ| < 35 MPa, 6 kA/m for 35 < |σ| < 51 MPa, and 8 kA/m for 51< |σ| < 80 MPa. While, Fe–Ga–B d ₃₃* versus σ curves have a linear shape with negative inclination and for 8 kA/m, the maximum sensibility is achieved in the hole σ interval.

A constant behavior of the sensitivity should be ideal in a defined σ window, to have the same accuracy for all values of σ measurements. In addition, a maximum sensibility, at a fixed field, is better for the engineering of a sensor than in a varying field. In a dispositive, the field can be applied by a constant current or even a permanent magnet, simplifying the electronics and mounting of the sensor.

Therefore, although Fe–Ga–B has a lower sensitivity than Fe–Ga–B, the fact that it can be used in constant field is a good feature. Moreover, the Fe–Ga–B alloy mechanical processing is unproblematic. The presence of the intergranular phase Fe₂B promotes more cohesion of grain boundaries and fracture of transgranular type occur instead of intergranular, which makes the material better machinable than Fe–Ga.^{Reference Gao, Li, Zhu, Li and Zhang8}