High Magnetic Field Effects on the Solid-liquid Reaction of Fe–Ga System

Abstract

The effect of magnetic fields on solid-liquid reactions in the Fe–Ga binary system was investigated using Fe/Ga diffusion couples. The reaction and phase growth proceeded by diffusion-controlled process for both 0 and 10 T. It is found that the growth of the intermetallic phases was suppressed by magnetic fields of 10 T regardless of whether α-Fe was in a ferromagnetic or paramagnetic state. The pre-exponential factors in the parabolic coefficient of the Fe₃Ga + eutectic region at 0 and 10 T were 2.50 × 10⁴ m²/s and 1.32 × 10³ m²/s, respectively. Meanwhile, activation energies at 0 and 10 T were 336 kJ/mol and 318 kJ/mol, respectively, which indicated the ineffectiveness of magnetic fields. That is, reduction of pre-exponential factor of the parabolic coefficient leads the magnetic-field-induced suppression of the reaction in Fe/Ga.

1. Introduction

Diffusion in metals under magnetic fields has been studied, such as magnetic-field-induced motion of the grain boundary^1,2,3) and suppression of the intermediate phase growth of diffusion couples.^4,5,6) The field-induced suppression of crystal growth was explained by the ambipolar model⁵⁾ or atomic movement by Lorentz forces.⁶⁾ For ferromagnetic metals, the diffusion of carbon in iron under a magnetic field of 6 T was reported.⁷⁾ The diffusion constants of carbon to ferromagnetic α-Fe were suppressed and enhanced by a static magnetic field and magnetic field gradient, respectively. In these systems, the previous studies pointed out that the reason for field-induced suppression of the diffusion was a reduction in the pre-exponential factor.^4,5,6,7) Although there are many studies of solid-phase diffusion under high magnetic fields, magnetic field effects (MFEs) on the liquid-solid systems have not been clarified in detail.

It is known that reactions, transformations, and equilibria of the ferromagnetic phase are influenced by application of a magnetic field because of the contribution of Zeeman energy.^{8,9,10,11,12,13)} Differences of Zeeman energy between ferromagnetic and non-ferromagnetic phases leads to the rise of the A₁ and A₃ lines in the Fe–C system, resulting in a change in the phase diagram.⁸⁾ Magnetic-field-induced enhancement of the solid-phase reaction of the Bi–Mn system was found due to the increase of formation rate at grain boundaries and a reduction of activation energy.^12,13) Crystallization of an ferromagnetic phase from an amorphous form under magnetic fields has been also studied.^14,15) Field-induced enhancement of nucleation of Fe–Si–B was explained by the magnetic field reducing the critical radius of the crystallized nuclei and the activation energy. Therefore, the reaction of a ferromagnetic system under magnetic field tends to decrease the activation energy. Meanwhile, the magnetic state also influences the self-diffusion constant.^16.17) The diffusion constant of pure ferromagnetic metal fell below the Curie temperature T_C. Therefore, the magnetic state, as well as an external magnetic field, was expected to influence the inter-diffusion of the metals. For systematic investigation of the MFE on a liquid/solid reactive system, the contribution of Zeeman energy and the system’s magnetic state should be considered.

The Fe–Ga binary system has solid/liquid reactions that include a ferromagnetic state.¹⁸⁾ The equilibrium phases of the Fe–Ga system have a wide variety of magnetic properties.^{19,20,21,22,23)} For example, Fe-15at.%Ga was reported to exhibit a large magnetostriction and is expected to be a candidate for vibration energy harvesting using inverse magnetostriction effects.²³⁾ In this study, to investigate the effects of external magnetic fields and magnetism on the solid/liquid diffusion, in-field annealing of Fe/Ga diffusion couples was performed.

2. Experimental Procedure

Fe/Ga diffusion couples were prepared by the following process. First, 4N-Fe flakes were arc-melted under an Ar atmosphere. After flattening of the surface of the button-shaped ingot, Fe ingots were sealed in a quartz tube in an Ar atmosphere and subsequently annealed at 773 K for 24 h. The surface of the ingot was re-polished and confirmed to be a single α-Fe phase by X-ray diffraction measurement at room temperature. After that, Fe and 4N-Ga were sealed in quartz tube with Ar gas. Figure 1 shows an Fe/Ga sample sealed in a quartz tube. Ga was on the surface of the Fe, and the Fe/Ga interface was set at the magnetic field center and perpendicular to the magnetic field direction.

Fig. 1.

Overview of Fe/Ga diffusion couple sealed in a quartz tube. The arrow indicates the magnetic field direction. (Online version in color.)

In-field heat treatments were carried out in magnetic fields up to μ₀H = 15 T and at T = 973, 1023, and 1073 K, which were selected to be below, near, and above the T_C of α-Fe, respectively. Magnetic field was also applied during heating process to the target temperature and cooling process to room temperature. The magnetic field was generated by a cryogen-free superconducting magnet. The sample was heated by an electric furnace, which was subjected to magnetic field at 10 and 15 T.²⁴⁾

After heat treatment, the Fe/Ga couples were cut perpendicular to the Fe/Ga interface. The microstructural analysis was performed by an electron probe micro-analyzer.

3. Experimental Results

Figure 2 shows the backscattered electron (BSE) images for Fe/Ga interfaces annealed at 1023 K for 1 day at 0 T (a) and 10 T (b). The obtained morphology can be divided into five regions. Point composition analysis indicated that the obtained phases are Fe₃Ga, an eutectic region of Fe₃Ga and Fe₆Ga₅, Fe₆Ga₅, Fe₃Ga₄, and FeGa₃, as read from the left side of the image. It was found that all phases annealed at 10 T were thinner than those annealed at 0 T. Figure 3 shows the thickness of the reactant phases, w, as a function of the square root of the annealing time, t^1/2, in a zero field (a) and at 10 T (b). A parabolic relationship was reported for diffusion-controlled systems,^4,25) which can be expressed as   

$$w^2= k(T)t,$$(1)

where k(T) and t are the parabolic coefficient and the annealing time, respectively. A linear relation between w and t^1/2 was observed for all phases, suggesting that the solid-liquid reaction of Fe–Ga was diffusion controlled. Table 1 shows the evaluated k(T) for each phase, where k (1023 K) for each phase was suppressed by 55% for the Fe₃Ga phase, 64% for the eutectic region, 62% for the Fe₆Ga₅ phase, and 68% for the Fe₃Ga₄ phase formed at 10 T.

Fig. 2.

BSE images of Fe/Ga interface annealed at 1023 K for 24 h in a zero field (a) and at 10 T (b).

Fig. 3.

Thickness of the formed phases w as a function of square root of annealing time t^1/2 at 1023 K in a zero field (a) and at 10 T (b). (Online version in color.)

Table 1. Parabolic coefficient k (1023 K) for each phase in a zero field and at 10 T.

Magnetic field (T)	kFe3Ga	keutectic	kFe6Ga5	kFe3Ga4
	(10−14 m2/s)	(10−14 m2/s)	(10−16 m2/s)	(10−15 m2/s)
0	4.29	3.94	1.89	5.33
10	1.92	1.41	0.727	2.23

The effects of the annealing temperature on the in-field phase formation were examined. Figure 4 provides the BSE images for the sample annealed at 1073 K for 24 h in a zero field (a) and at 10 T (b), respectively. This annealing temperature was greater than the T_C of α-Fe; that is, α-Fe was in a paramagnetic state. The phase formation at 1073 K was faster than that at 1023 K. Although Fe₃Ga and the eutectic region were observed, the thickness of Fe₆Ga₅ and Fe₃Ga₄ could not be evaluated because of the brittleness of the Ga-rich phases. As seen in Fe₃Ga and the eutectic region, it was found that phase formation at 1073 K, as well as at 1023 K, was suppressed by application of a magnetic field.

Fig. 4.

BSE images of Fe/Ga interface annealed at 1073 K for 24 h in a zero field (a) and at 10 T (b).

Figure 5 displays the w-t^1/2 relation at 973 K (a) and 1073 K (b). The growth of all phases showed a linear relation to t^1/2 as well as T = 1023 K. The growth of both the Fe₃Ga phase and Fe₃Ga + Fe₆Ga₅ eutectic region was suppressed by application of 10 T at 1073 K. Because the MFE for the eutectic region was not clearly observed for μ₀H = 10 T at 973 K, in-field annealing at 15 T was performed, and the result is also plotted in Fig. 5(a). The application of 15 T clearly reduced the thickness of the Fe₃Ga phase and eutectic region. Thus, the present study clearly showed that phase formation in a solid-liquid Fe–Ga system was suppressed by magnetic fields regardless of the magnetic state of α-Fe.

Fig. 5.

Thickness of the formed phases w as a function of square root of annealing time t^1/2 at 973 K (a) and 1073 K (b). (Online version in color.)

4. Discussion

In this section, MFEs of liquid-solid reactions are discussed according to the magnetic state of α-Fe and viscosity of liquid Ga.

The parabolic coefficient k(T) was expressed as   

$$k(T)=k_0\text{exp}\left(\frac{-Q}{RT}\right),$$(2)

where k₀, Q, and R are the pre-exponential factor, activation energy, and gas constant, respectively.

For μ₀H = 0 and 10 T, k₀ and Q were evaluated by Eq. (2). Figure 6 shows the log k versus T⁻¹ for Fe₃Ga and the eutectic region at μ₀H = 0 and 10 T. According to the Fe–Ga phase diagram, Fe₃Ga, Fe₆Ga₅, and the eutectic region were formed from α and α’/α’’ ordered phases during the cooling process.¹⁸⁾ The log k - T⁻¹ relation shows good linearity. Table 2 shows the obtained pre-exponential factor and activation energy for Fe₃Ga and the eutectic phase. It is found that k₀ at 10 T remarkably decreased, whereas Q was almost not changed by application of a magnetic field. These behaviors were the same as those in other systems, such as Fe–C, Ni–Al, and Mg–Al diffusion couples.^4,5,6,7) Reductions of k₀ for the interstitial solid solution were previously discussed with regard to the relationship between magnetostriction and interstitial sites.⁷⁾ Meanwhile, for substitutional solid solutions, an ambipolar diffusion process model was proposed by Youdelis et al.²⁶⁾ After that, Nakajima et al. reported the ineffectiveness of a magnetic field of 4 T on diffusion in the Ni/Ti system, and pointed out that the ambipolar diffusion process model was not enough to explain the MFE on the diffusion well.²⁷⁾ In contrast, Li et al. proposed that the spiral motion of the atoms during an atomic-jump process was disturbed by the Lorentz force.⁶⁾

Fig. 6.

Inverse temperature 1/T dependence of the parabolic coefficient k for the Fe₃Ga and eutectic region with and without a magnetic field of 10 T. Solid curves were obtained by least-squares calculations.

Table 2. Pre-exponential factor k₀ and the activation energy Q in a zero field and at 10 T.

Phase	Magnetic field (T)	k0 (× 103 m2/s)	Q (kJ/mol)
Fe3Ga + eutectic region	0	25.0	336
	10	1.32	318

On the other hand, magnetic-field-induced reduction of atomic collision exhibits at initial Fe/Ga solid-liquid interface. The contribution of induction current influenced the viscosity, η, of liquid Ga and suppressed diffusion. To date, a parabolic relationship between η and μ₀H has been reported for the Sb–Bi system²⁸⁾ and a Ga-based liquid.²⁹⁾ Parabolic enhancement of η in liquid Ga under magnetic fields up to 0.05 T was explained by an induction force.²⁹⁾ The viscosity of liquid metal under magnetic fields can be expressed by the following equation:   

$$\eta (H)=\eta (0)+\beta H^2,$$(3)

where η(H) is viscosity under a magnetic field and β is a constant related to the sample size, angular velocity, and electrical resistivity of the melt. When η(H) was extrapolated to μ₀H = 10 T, η(10 T) reached ~10⁴ mPa·s, which was 10⁴ times larger than that at μ₀H = 0 T. Enhancement of the viscosity leads to a reduction of the frequency of atomic collisions at Fe/Ga interfaces. However, as the phase growth was diffusion-controlled process (see Figs. 3 and 5), magnetic-field-induced suppression of the diffusion of Ga in α-Fe is effective rather than above MFE on the reaction at the initial Fe/Ga interface.

Next, effects of magnetic state of α-Fe on the in-field reaction were discussed. The gain of Zeeman energy influences the reactions in ferromagnetic systems.^12,13) Previously, we reported the magnetic-field enhancement of the reaction of ferromagnetic MnBi, pointing out that the contribution of a small Zeeman energy of less than 0.25 kJ/mol reduced the activation energy by about 4 kJ/mol.¹³⁾ In addition, the ineffectiveness of magnetic fields on liquid-solid reactions of a Mn–Ga system was found, which was probably due to the competition between reductions of activation energy and the pre-exponential factor.³⁰⁾ These reductions of activation energy depend on the difference in free energy gain between non-ferromagnetic raw materials and ferromagnetic reactants.

According to the Fe–Ga binary phase diagram,¹⁸⁾ reaction of an Fe/Ga diffusion couple proceeds by the formation of a α’/α’’ ordered phase and an Fe₆Ga₅/Fe₃Ga₄ phase at the annealing temperatures. It is reported that the T_C of the α-Fe and α’/α’’ phase were reduced with increasing Ga composition from 1047 K in pure iron to ~920 K at 20at.%Ga. The gain of Zeeman energy in pure Fe was larger than that in other ordered phases and intermediate phases. Therefore, the activation energy of reactions in a magnetic field is larger than that in a zero field regardless of the magnetic state of α-Fe. As seen in Table 2, Q at 10 T was not changed clearly from that at 0 T, which is inconsistent with the above explanation. Therefore, at the present stage, the magnetic state of α-Fe is supposed to be not dominantly for in-field phase growth. In addition, the reduction ratio of k by applying 10 T, k(10 T)/k(0 T), at 973, 1023, and 1073 K were evaluated to be 0.53, 0.40, and 0.43, respectively. Considering measurement error (see error in Fig. 6), although k(10 T)/k(0 T) at 973 K was larger than others, difference was not found clearly between ferromagnetic and paramagnetic state. If the magnetic state influences the intensity of MFE, one reason is probably magnetostriction as with the case of interstitial solution because magnetostriction of α-phase was much larger than pure Fe.³¹⁾

Consequently, the present results suggested that magnetic-field-induced suppression of the reaction was due to the reduction of k₀. In addition, the suppression under magnetic fields exhibited regardless of the magnetic state of α-Fe.

5. Conclusions

Magnetic field effects on solid-liquid reactions between Fe/Ga interfaces were suppressed under magnetic fields. The phase growth proceeded by diffusion-controlled process for both 0 T and 10 T. The pre-exponential factor of the parabolic coefficient remarkably decreased, whereas the activation energy did not change clearly in a magnetic field of 10 T, as previously reported for solid-solid diffusion in intermediate phase growth. Solid-liquid reaction of Fe/Ga system was suppressed by application of magnetic field regardless of the magnetic state of α-Fe because of the reduction of pre-exponential factor in parabolic coefficient.

Acknowledgments

This works partly supported by 26th ISIJ research promotion grant, and KAKENHI grant no. 16K14374. This work was performed at High Field Laboratory for Superconducting Materials, Institute for Materials Research, Tohoku University (Project No. 18H0044 and 17H0050).

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