Spinodal Decomposition in Plastically Deformed Fe–Cr–Co Magnet Alloy

Abstract

Fe–Cr–Co alloys are becoming important as half-hard magnet which can be subjected to plastic deformation process for their novel applications including non-contact electromagnetic brake because of its large hysteresis loss. Its magnetic hardness depends on the modulated structure formed by spinodal decomposition. It is important to clarify the effect of plastic deformation on the spinodal decomposition for optimizing the heat treatment after plastic deformation process. In the present study, we examined the spinodal-decomposed structures in Fe–Cr–Co sheets cold-rolled to 25% reduction and that without rolling to clarify the influences of cold rolling. Also, spinodal decomposition under the presence of dislocation structure have been simulated by phase field method for the case with the presence of dislocation cell boundary with a high in-plane solute diffusivity at various migrating speed. It has been found that the spinodal decomposition is accelerated around dislocation owing to the elastic field and higher diffusivity, which results in inhomogeneous microstructure with various wave length of modulation. The existence of dislocation enhances the initiation of phase decomposition and the growth particles. The decomposed structure greatly depends on the in-plane solute diffusivity and migrating speed of the dislocation cell boundary.

1. Introduction

Fe–Cr–Co alloys can be plastically deformed and are inexpensive. Thus, the alloys have been used as permanent magnets in various applications.^1,2,3,4) Recently, demands for semi-hard magnetic materials for non-contact electromagnetic brakes are expected to be increased.^5,6) Fe–Cr–Co alloys have an appropriate coercive force, and the magnetization can be reversed by a small electromagnet. For the use as an electromagnetic brake material, semi-hard magnets are required to be plastically processed into a ring shape before heat treatment and magnetization to fabricate products efficiently. Fe–Cr–Co alloys have excellent workability and can be subjected to hot rolling and cold rolling and are suitable as semi-hard magnetic materials from the viewpoint of efficient manufacturing processes.

In Fe–Cr–Co alloys, an isolated (Fe, Co)-rich ferromagnetic phase (α₁) surrounded by the Cr-rich non-magnetic phase (α₂) corresponds to a single magnetic domain, and magnetic properties depend on the wavelengths of the modulation structure. Thus, a higher performance as a permanent magnet has been obtained by forming a modulated structure with a wavelength of several tens of nm by spinodal decomposition.^1,2,3,4) On the other hand, semi-hard magnetic materials for non-contact brake requires an appropriate coercive force that allows the reversion of magnetization by a small electromagnet while keeping the saturation hard to be reached.^5,6) Therefore, another guideline to control the modulated structure is needed to obtain a suitable magnetic property as a semi-hard magnet.

Magnetic properties of Fe–Cr–Co alloys can be improved by having magnetic shape anisotropy of α₁ ferromagnetic particles as well.^6,7) Chin et al.⁷⁾ found that α₁ particles are mechanically elongated by swaging and cold rolling processes after spinodal decomposition and the magnetic shape anisotropy increases the coercive force. On the other hand, it has been reported^2,7,8) that the Fe–Cr–Co alloy recrystallized under a magnetic field has a highly unidirectionally oriented microstructure with the <100> direction of the easy axis of Fe magnetization oriented to the magnetic field. Li et al.^9,10,11) investigated the effects of edge dislocations on spinodal decomposition in Fe-20 at.%Cr alloys by phase-field simulation, and found that the elastic stress field around the dislocations and the pipe diffusion promote solute atom diffusions and spinodal decomposition.^9,10,11) Thus, it is predicted by simulation that the introduction of dislocations affects the spinodal decomposition behavior. However, the effect of plastic deformation on spinodal decomposition has not been clarified experimentally. Thus, in the present study, we have investigated the differences between the decomposed microstructures formed in the two cases of without rolling (hereafter, denoted by “non-rolled”) and with rolling (hereafter, denoted by “rolled”) to reveal the effects of dislocations introduced by cold-rolling on the spinodal decomposition of Fe–Cr–Co alloys. In addition, we performed phase-field simulations^{12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27)} to discuss the formation mechanisms of the experimentally observed microstructures in terms of defects induced by plastic deformations as a path for rapid diffusion.

2. Methods

2.1. Experimental Procedure

Fe-26 mass% Cr-10 mass% Co-1.5 mass% Ti (Fe-27.4 at.%Cr-9.3 at.%Co-1.7 at.%Ti) alloy ingots were prepared by melting pure iron, chromium, cobalt, and titanium metals. The chemical composition of the ingot was analysed by TEM-EDX, and shown in Table 1. The ingots were subjected to initial solution treatment at 1200°C for 3 h and subsequently subjected to the forced-air cooling to room temperature. A plate with dimension of 2 mm × 4 mm × 60 mm was cut out and cold-rolled in several passes to a thickness of 1.5 mm. To remove the damaged surface, upper and lower surfaces of the samples were polished up to 0.6 mm × 4 mm × 60 mm. Then, the non-rolled and rolled plates were heat treated at 640°C for 1 h under argon atmosphere in the infrared lamp furnace (ULVAC QHC P610CP) and subsequently quenched to water.

Table 1. Chemical composition of the Fe–Cr–Co–Ti alloy ingot.

Element	Fe	Cr	Co	Ti
Concentration [mass%]	Bal.	24.95	10.59	1.56
Concentration [at.%]	Bal.	26.4	9.88	1.79

The phase compositions and textures of non-rolled and rolled samples were analyzed using field emission scanning electron microscope (FE-SEM, Philips XL30S-FEG) equipped with an electron backscatter diffraction (EBSD) detector. The FE-SEM was operated under 20 kV. EBSD datum were analyzed using TSL orientation imaging microscopy (OIM) software. Then, the specimens were cut parallel to the {100} plane using a focused ion beam (FIB) system for scanning transmission electron microscope (STEM) observation. High angular annular dark field diffraction STEM (HAADF-STEM) image spinodal-decomposed microstructures were observed by FEI Titan3^TM G² 60-300 S/TEM Double Cs Corrector/Chemi-STEM. Corresponding element maps was also obtained by the energy-dispersive X-ray spectrometry (EDS) method.

2.2. Computational Method

To reveal the effects of dislocations on spinodal decompositions of Fe–Cr–Co alloy, phase-field (PF) simulations were carried out using the program based on the source code published by Koyama et al.²⁷⁾ Details of the calculation method are described in the appendix. Initially, we used the free energy of the Fe–Cr–Co alloy assessed in Refs. [28, 29]. However, unfortunately, the spinodal decomposition did not occur in the simulation conducted using the free energy function for the composition of Fe-27.4 at.%Cr-9.3 at.%Co which exhibits the spinocal decomposition in experiment. In this study, we have conducted PF simulations using the thermodynamic parameters in Refs. [28, 29] for a different alloy composition at a different temperature to obtain a clue for understanding the formation mechanism of the experimental spinodal microstructures.

The simulation box was 320 nm × 320 nm two-dimensional domain consisting of 256 × 256 meshes. The x and y axes were set to be [100] and [010] directions, respectively. The domain was initially set as α phase with the composition of Fe-40 at.%Cr-40 at.%Co at a temperature of 600°C. The parameters shown in Table 2 were used for the calculation. Using this phase-field simulation model, we have investigated the effect of an array of dislocations forming low angle grain boundary on spinodal decomposition in Fe–Cr–Co alloys. The array of dislocations composing the interface can be the path of rapid diffusion of solute atoms. The interface can migrate owing to the difference in dislocation density in the surrounding matrix on the two sides of the interface and the decrease in the area of interface, which act as the driving force. For simulating the effect of interface migration on the spinodal decomposition, we assumed that diffusions of solute atoms are along the planar interface, i.e. in the direction perpendicular to the x axis, and that interface move from the center of the calculation area to the right end. The mobility of solute i at the interface, $M_i^{\mathrm{I}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}}$, is assumed to be 10 times or 100 times larger than that in the bulk, $M_i^{\mathrm{B}\mathrm{u}\mathrm{l}\mathrm{k}}$, and changes with Gaussian distribution with the variance of 2 meshes (2.5 nm). The migration speed of the interfacial was assumed to be 1.9 nm h⁻¹ (0.52 pm s⁻¹). For the cases of $M_i^{\mathrm{I}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}}$ 100 times larger than $M_i^{\mathrm{B}\mathrm{u}\mathrm{l}\mathrm{k}}$, simulations were performed with the interface migration speed of 0.019 nm h⁻¹ (0.0052 pm s⁻¹) as well.

Table 2. Parameters used in phase-field simulations.

Parameter	Symbol	Value
Gradient-energy coefficient [m2 J mol−1]	κ	1.0 × 10−14
Elastic constants [N m2]	$C_{11}^{\mathrm{F}\mathrm{e}},\mathrm{\hspace{0.17em} }{C}_{11}^{\mathrm{C}\mathrm{o}}$	2.331 × 1011
	$C_{12}^{\mathrm{F}\mathrm{e}},\mathrm{\hspace{0.17em} }{C}_{12}^{\mathrm{C}\mathrm{o}}$	1.3544 × 1011
	$C_{44}^{\mathrm{F}\mathrm{e}},\mathrm{\hspace{0.17em} }{C}_{44}^{\mathrm{C}\mathrm{o}}$	1.1783 × 1011
	$C_{11}^{\mathrm{C}\mathrm{r}}$	3.5 × 1011
	$C_{12}^{\mathrm{C}\mathrm{r}}$	0.678 × 1011
	$C_{44}^{\mathrm{C}\mathrm{r}}$	1.008 × 1011
Atomic mobilities [m2 s−1]	MFe, MCo	1.0 × 10−4 exp (−294000/RT)
	MCr	2.0 × 10−5 exp (−308000/RT)
Interaction parameters of chemical free energy [J mol−1]	LFeCr	20500 − 9.68T
	LFeCo	−23699 + 103.9627T − 12.7886T lnT
	LCrCo	24357 − 19.797T − 2010 (cCo−cCr)
Curie temperatures [K]	TFe	1043
	TCr	−311.5
	TCo	1450
	TFeCr	850
	TFeCo	590
Interaction parameters of atomic magnetic moment [μb]	βFe	2.22
	βCr	−0.01
	βCo	1.35
	βFeCr	0.0247
	βFeCo	2.4127 + 0.2418 (cCo−cCr)

3. Results and Discussion

3.1. Experimental Observation

Figure 1 shows the inverse pole figure (IPF) maps obtained by SEM-EBSD of the non-rolled and the rolled alloy specimens subjected to heat treatment at 640°C for 1 h. Both of the samples were α single phase with a body-centered cubic (BCC) structure. In the rolled sample, there was a crystallographic texture even after subjected to the heat treatment. The crystal grain sizes of the non-rolled and the rolled samples were approximately 0.5 mm and 1.0 mm in diameter, respectively. From these samples, thin foil specimens parallel to the {1 0 0} plane were cut out using FIB, and the microstructures were observed by STEM. Figure 2 shows the bright-field images (Figs. 2(A) and 2(B)) and corresponding EDS element maps of these non-rolled and rolled samples. The EDS maps of the non-rolled sample (Figs. 2(a1)–2(a3)) indicate homogeneous concentration distribution. On the other hand, the EDS maps of the rolled sample (Figs. 2(b1)–2(b3)) show a concentration fluctuation of approximately 10 nm. The Fe-rich regions correspond to the Co-rich regions, and the Cr-rich regions were distributed between them. The selected-area electron diffraction (SAED) patterns of these samples are shown in Fig. 3. The SAED pattern of the non-rolled sample (Fig. 3(A)) shows only 1 1 1 diffraction spots of the BCC structure. On the other hand, the SAED pattern of the rolled sample (Fig. 3(B)) shows the 1 1 1 diffraction spots of the BCC structure and 1 0 0 ordered reflection spots of the B2 structure. These results indicate that the cold-rolling induced strains promote grain growth, spinodal decomposition, and the B2 ordering of (Fe, Co)-rich phase.

Fig. 1.

SEM-EBSD inverse pole figure (IPF) maps of (a) non-rolled and (b) rolled samples subjected to heat treatment at 640°C for 1 h. (Online version in color.)

Fig. 2.

(A, B) STEM bright-field images and corresponding EDS maps of (a1, b1) Fe, (a2, b2) Co, and (a3, b3) Cr of (a1–a4) heat-treated non-rolled and (b1–b4) rolled samples. (Online version in color.)

Fig. 3.

Selected-area electron diffraction patterns of (a) heat-treated non-rolled and (b) rolled samples corresponding to Figs. 2(A) and 2(B), respectively. (Online version in color.)

In the rolled sample, dislocations were introduced inhomogeneity and greatly affect the microstructures. Figure 4 shows the STEM bright-field images and corresponding EDS maps taken from the regions nearby a dislocation, accumulated dislocations, and subgrain boundaries. In Fig. 4(A), dislocations can be seen lying from the upper left corner to the lower right. The (Fe, Co)-rich and Cr-rich regions are along with the dislocation wider than those in the surrounding matrix. Figure 4(B) shows the region including a lot of dislocations in short segments. These dislocations are not parallel to the thin foil and seem to be arranged to form subgrain boundaries. Although slightly Fe-rich regions were observed nearby the subgrain boundaries, there is no significant effect of the subgrain boundaries on the decomposed microstructure. Figure 4(C) shows a two band-shaped dislocation structured region indicated by arrows. An elongated modulated structure appears perpendicular to the band-shaped dislocations. In Fig. 4(D), the dislocation array appears as dots perpendicular to the foil. The dislocation array affects spinodal decomposed behavior: the (Fe, Co)-rich α₁ phase was surrounded by the Cr-rich α₂ phase as in the EDS maps of Figs. 4(d1)–4(d3).

Fig. 4.

(A–D) STEM bright-field images and corresponding EDS mapping of (a1–d4) Fe, (a2–d2) Co, and (a3–d3) Cr of samples heat-treated after rolling. (Online version in color.)

A phase-field simulation¹¹⁾ indicated that a spinodal decomposition can be promoted by an elastic-strain field and/or dislocation-core diffusion. The distribution of defects introduced by cold rolling has some inhomogeneity, and as a result, the geometry and the wave length of modulated structure are not uniform.

Cold rolling⁷⁾ and/or heat-treatment under a magnetic field^2,7,8) can enhance the shape anisotropy and accordingly improve the magnetic properties of Fe–Cr–Co magnets. However, the improvement is limited in only one direction. On the other hand, the modulated structure elongated in various directions will be formed by applying these phenomena, i.e. the elongation of the modulated structure due to the effect of migration of dislocation wall associated with elastic-strain field and rapid diffusions of solute atoms along dislocations. By controlling this effect of dislocations, a Fe–Cr–Co magnet with isotropically enhanced overall magnetic properties, which are enhanced by local shape-anisotropy of magnetic phases, can be fabricated. The isotropic enhancement of magnetic properties by the local shape-anisotropy magnet is expected to gives rise to the good combination of a residual magnetization and a maximum energy product, which are not as high as that of a magnet with a unidirectionally elongated microstructure, but the magnetic properties are enhanced isotropically in various directions. Such an isotropic semi-hard magnet is suitable for applications to products that require isotropically high magnetic performances such as a ring-shaped magnet.

3.2. Phase-field Simulation

Figure 5 shows the simulated spinodal-decomposed microstructures for the case where the interface migrates at a constant velocity of 1.9 nm/h. Blue and red indicate the (Fe, Co)-rich (α₁) and the Cr-rich (α₂) phases, respectively. The interface is initially located at the center of the simulation domain (Fig. 5(a)) and moved to the right end. The diffusivity on the interface ($M_i^{\mathrm{I}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}}$) was assumed to be 10 times larger than that in the surrounding matrix ($M_i^{\mathrm{B}\mathrm{u}\mathrm{l}\mathrm{k}}$) (i.e. $M_i^{\mathrm{I}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}}$ = 10$M_i^{\mathrm{B}\mathrm{u}\mathrm{l}\mathrm{k}}$). As in Fig. 5(b), the spinodal-decomposed microstructure was isotropic in the region where the interface did not pass, while the microstructure was elongated along the direction of the interface migration in the region where the interface has passed. In the elongated region where the interface has passed, the concentration difference between the α₁ and α₂ phases was remarkable compared to that in the matrix region where the interface has not passed. This indicates that the presence of an interface with a high diffusivity of solute atoms promotes the spinodal decomposition in the initial state. When the interface moved a quarter of the calculation domain as shown in Fig. 5(c), the modulated structure continues to grow to be elongated in the direction of interface migrating. However, there is no difference in concentration between the elongated region and the matrix region. When the interface reached the right end of the calculation domain (Fig. 5(d)), the microstructure elongated from the center to the right end was formed. The resulted microstructure is essentially similar to the experimentally observed microstructure shown in Figs. 4(C), 4(c1)–4(c3). This suggests that the experimentally elongated structure was formed as the result of the dislocation array interface. Note that the interface migration velocity was assumed to be as small as 1.9 nm/h. This is much smaller than that expected for the experiment. In this study, the chemical composition and the annealing temperature in the simulation are different from those in the experiment owing to the uncertainty of thermodynamic parameters.^27,28) It is suggested that the rate of the spinodal decomposition in the simulation is slower than that in the experiment because of the smallness of the driving force, and accordingly the interface migration velocity also need to be assumed to be slow to form the experimentally observed microstructure.

Fig. 5.

Concentration profiles of Fe-40Cr-40Co (at.%) alloy model in the spinodal-decomposition simulations with interface along which the solute mobility was ten times larger than that in bulk. The migration rate of the interface was 1.9 nm/h, and the aging temperature and times were 600°C and (a) 0 h, (b) 17.0 h, (c) 170 h, and (d) 1695 h, respectively. (Online version in color.)

Figure 6 shows the simulated microstructure for the case of the solute mobility on the interface $M_i^{\mathrm{I}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}}$ = 100$M_i^{\mathrm{B}\mathrm{u}\mathrm{l}\mathrm{k}}$. The velocity of the interface migration was assumed to be 1.9 nm/h which is the same as in the simulation shown in Fig. 5. In the left-side region of the simulation domain where the interface did not pass, the spinodal-decomposed microstructure with the Cr-rich α₂ phase surrounded by the (Fe, Co) -rich α₁ phase was formed. In contrast, the α₁ phase was surrounded by the α₂ phase in the right-side region where the interface passed. The microstructure formed in the right region is similar to that experimentally observed in the area shown in Figs. 4(D), 4(d1)–4(d4). This similarity suggests that the experimentally observed microstructure in Figs. 4(D), 4(d1)–4(d4) was formed as a result of rapid diffusion along an interface with an extremely high solute mobility which made the region reach an equilibrium state.

Fig. 6.

Concentration profiles of Fe-40Cr-40Co (at.%) alloy model in the spinodal-decomposition simulation with interface along which the solute mobility was 100 times larger than that in bulk. The migration rate of the interface was 1.9 nm/h, and the aging temperature and time were 600°C and 1695 h, respectively. (Online version in color.)

Figure 7 shows the spinodal-decomposed microstructures simulated for the case of $M_i^{\mathrm{I}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}}$ = 100$M_i^{\mathrm{B}\mathrm{u}\mathrm{l}\mathrm{k}}$ and the slower interface migration speed of 0.019 nm/h. In this case, the computation was terminated when the interface moved to the position indicated by the arrow in Fig. 7. In this case, Cr atoms remarkably segregated in the region where interface has passed. On the other hand, the concentrations of Fe and Co segregated on the left side of the original position of interface to form a thin layer of (Fe, Co)-rich region. This modulated structure is similar to the experimental microstructure observed nearby a dislocation line, shown in Figs. 4(A), 4(a1)–4(a4). Thus, the experimental microstructure is suggested to be formed because of rapid diffusion promoted by the slowly migrating interface. As described above, the shape and the wavelength of the simulated modulated structure largely depend on the migration speed of the interface and the diffusivity of solute atoms on the interface.

Fig. 7.

Concentration profiles of Fe-40Cr-40Co (at.%) alloy model in the spinodal-decomposition simulation with interface along which the solute mobility was 100 times larger than that in bulk. The migration rate of the interface was 0.019 nm/h, and the aging temperature and time were 600°C and 1695 h, respectively. (Online version in color.)

The phase-field simulations demonstrate that dislocations can promote spinodal decomposition not only by their elastic stress field but also by fast diffusions of solute atoms via dislocations, and strongly affect the morphology of the modulated structure: the wavelength, the anisotropy, and the arrangement of α₁ and α₂ phases. The simulated microstructures are similar to those observed in the samples heat-treated after rolling. Their formation mechanisms have been proposed based on the simulation results. Their detailed mechanism will be elucidated by additional experiments and simulations, such as experiments using dislocation containing bicrystals^7,30,31) and three-dimensional simulations with dislocation.

As described in Section 2.2, the spinodal decomposition did not occur at 640°C in the simulation for the composition of the alloy used in the experiment. According to the phase-diagram constructed by using the thermodynamic parameters in Refs. [28, 29], the composition of the alloy is out of the range for spinodal decomposition. The thermodynamic parameters are required to be re-evaluated by further assessment. The phase-field simulation using modified thermodynamic parameters is currently underway.

4. Conclusion

In this study, we investigated the spinodal decomposition behavior of Fe–Co–Cr alloys by the TEM observation and the phase-field simulation. TEM observation revealed that spinodal decomposed microstructures in the samples heat treated after cold-rolling have the inhomogeneity in the arrangements of α₁ magnetic phase and α₂ nonmagnetic phase. The phase-field simulation suggests that dislocations act as a rapid diffusion path of solute atoms and promote spinodal decomposition. The simulation results have demonstrated that various microstructures with variety in the wavelength, the anisotropy of the modulated structure, and the arrangement of α₁ and α₂ phases. The simulated microstructures are similar to those observed in the samples heat-treated after rolling, and therefore, the experimentally observed microstructures are suggested to be formed reflecting inhomogeneity of the plastic deformation. The cold-rolled sample has both of microstructures with suitable properties for use under low magnetic field and those suitable for the use under high magnetic field. A semi-hard magnetic material with excellent braking performance in a wide range of magnetic fields is expected to be manufactured efficiently by controlling the inhomogeneity of dislocation structure introduced by plastic deformations.

Acknowledgement

This research is supported by ISIJ Research Promotion Grants from the Iron and Steel Institute of Japan (ISIJ) and Adaptable and Seamless Technology transfer Program through Target-driven R&D (A-STEP) from Japan Science and Technology Agency (JST). We would like to thank Dr. Y. Hayasaka, Dr. Y. Kodama, Dr. K. Suzuki, and Dr. M. Nagasako for technical support for TEM observation.

References

• 1)   M.  Sagawa,  M.  Hamano and  M.  Hirabayashi: Eikyu-Jishaku -Zairyo-Kagaku-to-Oyo (Permanent Magnet -Materials Science and Application), Agne Gijutsu Center, Tokyo, (2007), 163 (in Japanese).
• 2)   H.  Kaneko and  M.  Homma: Jisei-Zairyo (Magnetic Materials), The Japan Institute of Metals and Materials, Sendai, (1977), 186 (in Japanese).
• 3)   H.  Kaneko,  M.  Homma and  K.  Nakamura: AIP Conf. Proc., 5 (1972), 1088.
• 4)   M.  Okada and  M.  Homma: Recent Magnetics for Electronics (JARECT 15), ed. by Y. Sakurai, OHMSHA and North-Holland, Tokyo and Amsterdam, (1984), 231.
• 5)   K.  Tsurumoto and  M.  Imamura: J. Mag. Soc. Jpn., 20 (1996), 545 (in Japanese).
• 6)   K.  Tsurumoto: J. Mag. Soc. Jpn., 11 (1987), 391 (in Japanese).
• 7)   T. S.  Chin,  T. S.  Wu,  C. Y.  Chang,  T. K.  Hsu and  Y. H.  Chang: J. Mater. Sci., 18 (1983), 1681.
• 8)   S.  Sugimoto,  M.  Okada and  M.  Homma: J. Appl. Phys., 63 (1988), 3707.
• 9)   Y.-S.  Li,  Y.-Z.  Yu,  X.-L.  Cheng and  G.  Chen: Mater. Sci. Eng. A, 528 (2011), 8628.
• 10)   Y.-S.  Li,  H.  Zhu,  L.  Zhang and  X.-L.  Cheng: J. Nucl. Mater., 429 (2012), 13.
• 11)   Y.-S.  Li,  S. X.  Li and  T.-Y.  Zhang: J. Nucl. Mater., 395 (2009), 120.
• 12)   T.  Koyama and  H.  Onodera: Nihon Gakujutsushinkokai-Gokinjotaizu-Dai-172-Iinkai-Hokokusho (The Report of the Committee No. 172 [Alloy Phase Diagram]), Vol. 1, Japan Society for the Promotion of Science, Tokyo, (2002), 64.
• 13)   T.  Koyama and  H.  Onodera: Met. Mater. Int., 10 (2004), 321.
• 14)   T.  Koyama and  H.  Onodera: J. Phase Equilib. Diffus., 27 (2006), 22.
• 15)   L. X.  Lv,  L.  Zhen,  C. Y.  Xu and  X. Y.  Sun: J. Magn. Magn. Mater., 322 (2010), 987.
• 16)   Y.  Koizumi,  S. M.  Allen,  M.  Ouchi and  Y.  Minamino: ISIJ Int., 52 (2012), 1678.
• 17)   Y.  Koizumi,  S. M.  Allen,  M.  Ouchi and  Y.  Minamino: Intermetallics, 18 (2010), 1297.
• 18)   Y.  Koizumi,  S. M.  Allen and  Y.  Minamino: Acta Mater., 57 (2009), 3039.
• 19)   Y.  Koizumi,  S. M.  Allen and  Y.  Minamino: Acta Mater., 56 (2008), 5861.
• 20)   Y.  Koizumi,  M.  Ouchi,  Y.  Minamino and  S. M.  Allen: Trans. Mater. Res. Soc. Jpn., 35 (2010), 209.
• 21)   Y.  Koizumi,  T.  Yokoi,  M.  Ouchi,  Y.  Minamino,  M.  Yoshiya and  S. M.  Allen: MRS Symp. Proc., 1295 (2011), 437.
• 22)   Y.  Koizumi,  T.  Yamazaki,  A.  Chiba,  K.  Hagihara,  T.  Nakano,  K.  Yuge,  K.  Kishida and  H.  Inui: MRS Symp. Proc., 1516 (2013), 309.
• 23)   T.  Yamazaki,  Y.  Koizumi,  A.  Chiba,  K.  Hagihara,  T.  Nakano,  K.  Yuge,  K.  Kishida and  H.  Inui: MRS Symp. Proc., 1516 (2013), 145.
• 24)   T.  Yamazaki,  Y.  Koizumi,  K.  Yuge,  A.  Chiba,  K.  Hagihara,  T.  Nakano,  K.  Kishida and  H.  Inui: Comput. Mater. Sci., 108 (2015), 358.
• 25)   T.  Yamazaki,  Y.  Koizumi,  A.  Chiba,  K.  Hagihara,  T.  Nakano,  K.  Yuge,  K.  Kishida and  H.  Inui: Adv. Mater. Res., 922 (2014), 832.
• 26)   T.  Yamazaki,  Y.  Koizumi,  K.  Yuge,  A.  Chiba,  K.  Hagihara,  T.  Nakano,  K.  Kishida and  H.  Inui: Intermetallics, 54 (2014), 232.
• 27)   C. Q.  Zhu,  Y.  Koizumi,  A.  Chiba,  K.  Yuge,  K.  Kishida and  H.  Inui: Intermetallics, 116 (2020), 106590.
• 28)   A.  Fernández Guillermet: Int. J. Thermophys., 8 (1987), 481.
• 29)   K.  Oikawa,  G. W.  Qin,  T.  Ikeshoji,  R.  Kainuma and  K.  Ishida: Acta Mater., 50 (2002), 2223.
• 30)   A.  Nakamura,  T.  Mizoguchi,  K.  Matsunaga,  T.  Yamamoto,  N.  Shibata and  Y.  Ikuhara: ACS Nano, 7 (2013), 6297.
• 31)   A.  Nakamura,  E.  Tochigi,  N.  Shibata,  T.  Yamamoto and  Y.  Ikuhara: Mater. Trans., 50 (2009), 1008.

Appendix

We have simulated spinodal-decomposition using the PF model originally developed by Koyama.^12,13,14) In this model, the Gibbs free energy of the whole system G_sys are assumed to be sum of a chemical free-energy G_chem, a magnetic excess energy G_mag, a gradient energy E_grad and an elastic strain energy E_elast:   

$$G_{\text{sys}}=G_{\text{chem}}+G_{\text{mag}}+E_{\text{grad}}+E_{\text{elast}}.$$(A1)

The G_chem of Fe–Cr–Co ternary systems can be expressed using the regular solution approximation:   

$$\begin{array}{l}{G}_{\text{chem}}=G_{\text{Fe}}{c}_{\text{Fe}}+G_{\text{Cr}}{c}_{\text{Cr}}+G_{\text{Co}}{c}_{\text{Co}}+G_{\text{E}}\\ \hspace{1em}\hspace{1em}\hspace{1em}+RT\left(c_{\text{Fe}}\text{ln}{c}_{\text{Fe}}+c_{\text{Cr}}\text{ln}{c}_{\text{Cr}}+c_{\text{Co}}\text{ln}{c}_{\text{Co}}\right),\end{array}$$(A2)

  

$$G_{\text{E}}=L_{\text{FeCr}}{c}_{\text{Fe}}{c}_{\text{Cr}}+L_{\text{FeCo}}{c}_{\text{Fe}}{c}_{\text{Co}}+L_{\text{CrCo}}{c}_{\text{Cr}}{c}_{\text{Co}}.$$(A3)

Here, T is a temperature, c_i is a concentration of element i, G_i is free energy of a pure substance, G_E is an excess energy, and L_ij is an interaction parameter between elements i and j. The G_mag of the Fe–Cr–Co ternary alloy can be expressed as follows:   

$$G_{\text{mag}}=RT\mathrm{l}\mathrm{n}\left(\beta +1\right)f\left(T\right),$$(A4)

  

$$\begin{array}{c}\begin{array}{l}f\left(T\right)=\\ \{\begin{array}{c}1-\frac{1}{D}\left\{\frac{79{\left(T/T_{\text{c}}\right)}^{-1}}{140p}+\frac{474}{497}\left(\frac{1}{p}-1\right)\left(\frac{{\left(T/T_{\text{c}}\right)}^3}{6}+\frac{{\left(T/T_{\text{c}}\right)}^9}{135}+\frac{{\left(T/T_{\text{c}}\right)}^{15}}{600}\right)\right\},\left(T\le T_{\text{c}}\right)\\ -\frac{1}{D}\left(\frac{{\left(T/T_{\text{c}}\right)}^{-5}}{10}+\frac{{\left(T/T_{\text{c}}\right)}^{-15}}{315}+\frac{{\left(T/T_{\text{c}}\right)}^{-25}}{1\mathrm{\hspace{0.17em} }500}\right),\left(T>T_{\text{c}}\right)\end{array}\end{array}\end{array}$$(A5)

  

$$D=\frac{518}{1\mathrm{\hspace{0.17em} }125}+\frac{11\mathrm{\hspace{0.17em} }692}{15\mathrm{\hspace{0.17em} }975}\left(\frac{1}{p}-1\right).$$(A6)

Here, p is a constant determined by the crystal structure and p = 0.4 for the BCC structure. β and T_c are a magnetic moment per atom and Curie temperature, respectively, and these parameters are calculated from the following equation:   

$$\beta ={\text{c}}_{\text{Fe}}{\beta }_{\text{Fe}}+c_{\text{Cr}}{\beta }_{\text{Cr}}+c_{\text{Co}}{\beta }_{\text{Co}}+c_{\text{Fe}}{c}_{\text{Cr}}{\beta }_{\text{FeCr}}+c_{\text{Fe}}{c}_{\text{Co}}{\beta }_{\text{FeCo}}$$(A7)

  

$$T_{\text{c}}=c_{\text{Fe}}{T}_{\text{Fe}}+c_{\text{Cr}}{T}_{\text{Cr}}+c_{\text{Co}}{T}_{\text{Co}}+c_{\text{Fe}}{c}_{\text{Cr}}{T}_{\text{FeCr}}+c_{\text{Fe}}{c}_{\text{Co}}{T}_{\text{FeCo}}$$(A8)

The E_grad can be expressed by the following equation:   

$$E_{\text{grad}}=\frac{1}{2}\kappa {\left(\nabla c_{\text{Fe}}\right)}^2+\frac{1}{2}\kappa {\left(\nabla c_{\text{Cr}}\right)}^2+\frac{1}{2}\kappa {\left(\nabla c_{\text{Co}}\right)}^2$$(A9)

Here, κ is a concentration gradient energy coefficient and we assumed that κ does not depend on compositions. Because of the difference of lattice constants between Fe and Cr, lattice misfit occurs in a boundary region of α₁ and α₂ phases. This lattice misfit causes the elastic strain energy, which can be expressed by the following equation:   

$$E_{\text{elast}}=\frac{1}{2}{C}_{ijkl}\left\{{\epsilon }_{ij}^c\left(\mathbf{r}\right)-{\epsilon }_{ij}^0\left(\mathbf{r}\right)\right\}\left\{{\epsilon }_{kl}^c\left(\mathbf{r}\right)-{\epsilon }_{kl}^0\left(\mathbf{r}\right)\right\},$$(A10)

  

$$\begin{array}{l}{C}_{ijkl}{\delta }_{\text{ij}}\left\{{\epsilon }_{kl}^c\left(\mathit{r}\right)-{\epsilon }_{kl}^0\left(\mathit{r}\right)\right\}=\\ \left(C_{11}+2C_{12}\right)\left[\left\{{\epsilon }_{11}^0\left(\mathit{r}\right)+{\epsilon }_{22}^0\left(\mathit{r}\right)+{\epsilon }_{33}^0\left(\mathit{r}\right)\right\}-\left\{{\epsilon }_{11}^c\left(\mathit{r}\right)+{\epsilon }_{22}^c\left(\mathit{r}\right)+{\epsilon }_{33}^c\left(\mathit{r}\right)\right\}\right].\end{array}$$(A11)

Here C_ij and ${\epsilon }_{ij}^c$ are an elastic constant and a constraint strain, respectively. ${\epsilon }_{ij}^0$ is eigenstrain which is any mechanical deformation except for an external mechanical stress strain. In this simulation, the eigenstrain is corresponding to the lattice misfit strain and is defined by ${\epsilon }_{ij}^0$ = ξ(c_Cr−c₀)δ_ij. c₀ and δ_ij are an average concentration of Cr and the Kronecker delta, respectively. ξ is a lattice misfit size and is given by ξ = (a_Cr − a_Fe)/a_Fe using the lattice constants of the stable structures of each element, a_i.

The time evolution of solute concentrations was calculated from the Cahn-Hilliard equation:   

$$\frac{\partial c_i}{\partial t}=\nabla \cdot \left(M_i^{\text{Bulk}}\nabla \frac{\delta G_{\text{sys}}}{\delta c_i}+M_{i,j}\nabla \frac{\delta G_{\text{sys}}}{\delta c_j}\right),$$(A12)

  

$$M_{i,\text{Fe}}=\left\{{\left(1-c_i\right)}^2{M}_i^{\text{Bulk}}+c_i{c}_j{M}_j^{\text{Bulk}}+\left(1-c_i-c_j\right)c_i{M}_{\text{Fe}}^{\text{Bulk}}\right\}{c}_i,$$(A13)

  

$$\begin{array}{l}{M}_{\text{Cr},\text{Co}}=M_{\text{Co},\text{Cr}}\\ =\left\{-\left(1-c_{\text{Cr}}\right)M_{\text{Cr}}^{\text{Bulk}}-\left(1-c_{\text{Co}}\right)M_{\text{Co}}^{\text{Bulk}}+\left(1-c_{\text{Cr}}-c_{\text{Co}}\right)M_{\text{Fe}}^{\text{Bulk}}\right\}{c}_{\text{Cr}}{c}_{\text{Co}}.\end{array}$$(A14)

Here, $M_i^{\mathrm{B}\mathrm{u}\mathrm{l}\mathrm{k}}$ is the solute mobility of each element in the bulk, and M_i,j is the Onsager coefficient.