2023 Volume 63 Issue 7 Pages 1245-1250
Fe–Cr–Co alloys are becoming important as a half-hard magnet for their novel applications, including non-contact electromagnetic brakes, because of the controllability of its magnetic hardness depending on the modulated structure formed by spinodal decomposition. However, the experimental optimization of the complicated heat-treatment process to control the microstructure significantly increases the development cost, and microstructure prediction by computational simulation is desired. In this study, we first developed the method of phase-field simulation for spinodal decomposition in Fe–Cr–Co alloy during various heat treatments, including isothermal heat treatment, multistep continuous fast and slow cooling, which allows us to conduct a simulation of spinodal decomposition under conditions close to the condition of practical heat treatment. The simulation results revealed that the morphology of the modulated structure is predominantly determined by the cooling rate and does not change significantly during the subsequent isothermal annealing process, while the difference between the concentrations of the FeCo-rich magnetic phase and the Cr-rich non-magnetic phase increases. Continuous cooling at rates higher than 140 K/h demonstrates the maximum number densities of the ferromagnetic particles of α1-phase seemingly almost reaching saturation, which is expected to give rise to exhibiting the largest coercive force of the Fe–Cr–Co magnet. Moreover, this method can be extended to other materials for designing a modulated structure to show a desired property.
The spinodal decomposition phenomenon has been found in many alloys and polymers. The mechanical and magnetic properties of these materials are strongly enhanced by the optimization of modulated structures.1,2) In particular, Fe–Cr–Co permanent magnetic alloys are attractive because of the combination of good ductility and tunable magnetic properties.3,4,5,6) In the Fe–Cr–Co alloy, the body-centered cubic (BCC) phase (α phase) separates into (Fe, Co)-rich ferromagnetic phase (α1 phase) and Cr-rich nonmagnetic phase (α2 phase). The spinodal decomposition of Fe–Cr–Co is sensitive to heat treatment conditions, and the magnetic properties can be improved by heat treatment in a magnetic field.7,8,9,10) When the ferromagnetic phase precipitates in the nonmagnetic phase of the parent phase and becomes elongated in one direction, each ferromagnetic phase acts like a bar magnet, making it difficult for the reversal of the magnetic domain to occur, thereby improving the coercive force. On the other hand, if the size of the ferromagnetic phase is too small, the magnetic moment is disturbed by thermal oscillations, which reduces the coercive force. Therefore, it is necessary to optimize the heat treatment conditions for controlling precipitation of the ferromagnetic phase with appropriate grain shape. However, it is challenging to investigate them only by experimental research because a heat treatment of Fe–Cr–Co alloys takes dozens of hours, and there are enormous combinations of heat treatment conditions.
Predicting modulated structures formed after heat treatment by computer simulation is expected to significantly reduce the time for optimization. The Phase-Field (PF) method is widely used as an efficient numerical calculation method for investigating the microstructural evolution in materials.11,12,13) Simulation of spinodal decomposition in the Fe–Cr–Co alloys using the PF method has been reported by many researchers.14,15,16) In our previous study,17) the effects of plastic deformation on Fe–Cr–Co spinodal under isothermal conditions were investigated by experiment and phase-field method. Lv16) et al. evaluated the microstructure after step aging treatment by comparing it with the experimental results. However, in actual heat treatment, it is difficult to perform a perfect step heat treatment, and it is thought that there are moments when the temperature changes continuously. Although there have been many studies on the simulation of Fe–Cr–Co alloys using PF methods, microstructure simulation under continuous temperature-changing conditions is still unclear.
In this study, we simulated the spinodal decomposition in Fe–Cr–Co alloy with continuous temperature change by the PF method and compared the microstructures for each heat treatment condition. The influence of cooling rate on the microstructure of spinodal decomposition is investigated.
PF simulations were carried out using the program based on the source code published by Koyama et al.14,15) Details of the calculation method were described in the appendix of the previous paper.17) The simulation box was 320 nm × 320 nm two-dimensional domain with a total mesh number of 256 × 256. The x and y axes were set to be [100] and [010] directions, respectively. The thermodynamic parameters for evaluating the Gibbs energy, such as the interaction parameters, the Curie temperature, and the atomic magnetic moment, were taken from the commercial TCFE6 database,18) and the data proposed by Lv16) were also used. Moreover, numerical values such as elastic constants and gradient energy coefficient for PF simulation were taken from the literature.19) All the parameters used in this study are summarized in Table 1.
| Parameters | Symbols | Values | Ref. |
|---|---|---|---|
| Gradient-energy coefficient [J m2 mol−1] | κ | 1.0 × 10−14 | 17) |
| Elastic constants [N m−2] | 2.331 × 1011 | 19) | |
| 1.3544 × 1011 | 19) | ||
| 1.1783 × 1011 | 19) | ||
| 3.5 × 1011 | 19) | ||
| 0.678 × 1011 | 19) | ||
| 1.008 × 1011 | 19) | ||
| Lattice constants [m] | aFe | 2.8664 × 10−10 | 19) |
| aCr | 2.8840 × 10−10 | 19) | |
| aCo | 2.8460 × 10−10 | 19) | |
| Self-diffusion coefficients [m2 s−1] | D0,Fe,D0,Co | 1.0 × 10−4 exp (−294000/RT) | 19) |
| D0,Cr | 2.0 × 10−5 exp (−308000/RT) | 19) | |
| Interaction parameters of chemical free energy [J mol−1] | LFeCr | 20500 − 9.68T | 18) |
| LFeCo | −23699 + 103.9627T − 12.7886T lnT | 18) | |
| LCrCo | 24357 − 19.797T − 2010 (cCo − cCr) | 18) | |
| Curie temperatures [K] | Tc,Fe | 1043 | 18) |
| Tc,Cr | −311.5 | 18) | |
| Tc,Co | 1450 | 18) | |
| Tc,FeCr | 850 | 16) | |
| Tc,FeCo | 590 | 18) | |
| Interaction parameters of atomic magnetic moment [μb] | βFe | 2.22 | 18) |
| βCr | −0.01 | 18) | |
| βCo | 1.35 | 18) | |
| βFeCr | 0.0247 | 16) | |
| βFeCo | 2.4127 + 0.2418 (cCo − cFe) | 18) | |
| VFe | −81.5280 | 16) | |
| VCr | 51.4884 | 16) | |
| VCo | 85.6565 | 16) |
The initial condition was set as Fe-26.5Cr-11.2Co (at.%. Hereafter, the unit of “at.%” is not indicated) homogeneous α phase with a temperature of 933 K, and then a concentration fluctuation with a mole fraction of less than 0.01 was given to the entire system. The boundary condition used in PF simulation is periodic. In this study, we have investigated microstructure evolution under continuous temperature-changing conditions as follows: (a) equilibria solidification with the cooling rate of 1 K/h, (b) initial aging at 933 K for 2 h and then cooling to 773 K with 20 levels of constant cooling rates from 10 to 200 K/h, and (c) initial aging at 933 K for 2 h followed by continuous cooling with a constant rate of 170 K/h, and then aging at 803 K for different periods. The number and size of α1 and α2 particles in each condition were measured using the ImageJ software, and concentration distributions were analyzed to evaluate the spinodal decomposition.
The simulated equilibrium microstructures of spinodal decomposition in Fe-26.5Cr-11.2Co alloys at different temperatures are shown in Figs. 1(a)–1(d). The colors of red, green, and blue represent the concentrations of Cr, Fe, and Co, respectively. From Fig. 1(a) we can see that the (Fe, Co)-rich α1 phase was surrounded by the Cr-rich α2 phase at 933 K. However, at temperatures lower than 903 K (Figs. 1(b)–1(d)), the α1 phase became larger, and microstructures with the α2 phase surrounded by α1 phase were observed. The reason for the formation of such microstructure can be explained according to the principle of leverage, and phases with a smaller volume fraction would be surrounded by phases with a larger volume fraction. As has been revealed by Nishizawa et al.,20,21) the miscibility gap of the Fe–Cr–Co ternary system has a ridge on the Fe-rich side, the volume fraction of the nonmagnetic α2 phase is larger at high temperatures, and the volume fraction of the ferromagnetic α1 phase is larger at low temperatures. Therefore, the simulated microstructure in Figs. 1(a)–1(d) agrees with the equilibrium phase relationship.
The simulated microstructure of spinodal decomposition in Fe-26.5Cr-11.2Co alloys at (a) 933 K, (b) 903 K, (c) 873 K, (d) 823 K; and (e) the Cr concentration plot evaluated by the PF simulation under the cooling rate of 1 K/h. The solid line is the calculated binodal curve using the thermodynamic parameters in Table 1. (Online version in color.)
In order to reveal the relationships of equilibrium concentration between the thermal equilibrium phase diagram and the simulation results, PF simulations of spinodal decomposition with very slow cooling rates from 933 K were performed in this alloy, and the Cr concentration distribution was evaluated by analyzing the Cr concentration in each mesh of the PF simulation results. Figure 1(e) shows the Cr concentration plot evaluated from the simulation results under the cooling rate of 1 K/h. The semi-transparent red dots indicate the Cr concentration of all the grid points for the simulation at the corresponding temperature. For comparison, the calculated thermal equilibrium binodal curve (the solid line) at the vertical section of Fe-xCr-11.2Co (x: Cr concentration, CCr and hereafter) is also shown in Fig. 1(e) using the thermodynamic parameters listed in Table 1. At the initial period, all the red dots are concentrated at the starting point with the Cr concentration of the alloy composition and the initial temperature of 933 K. The dots spread towards both the Cr-poor and Cr-rich sides as the simulation proceeds with decreasing temperature. The Cr concentration is distributed nearly all over the two-phase region in the thermal equilibrium phase diagram. The simulated Cr concentration at the (Fe, Co)-rich side agrees relatively well with the calculated equilibrium phase diagram. However, a significant concentration gap exists (as indicated by the both-side arrows in Fig. 1(e)) between the highest value of Cr concentration in the simulated results and the concentration limit indicated in the equilibrium phase diagram. Thus, the highest Cr concentration on the Cr-rich side obtained by PF simulation is approximately 5 at.% lower than that of the thermal equilibrium phase diagram.
The inconsistency of Cr concentration may be due to that the tie lines of spinodal decomposition are not located in the calculated vertical phase diagram. Therefore, the Cr concentration of spinodal decomposition in the Fe-26.5Cr-11.2Co alloy along the tie lines at different temperatures was evaluated using one-axis calculation (in this case, the temperature is set as the only variation axis) of the Thermo-Calc software. Figure 2(a) shows the comparison of Cr concentration distribution obtained from PF simulation results (red dot) and the calculated Cr concentration of spinodal decomposition along the tie lines in Fe-26.5Cr-11.2Co alloys at various temperatures (mark), a good agreement between the PF simulation result and the equilibrium phase diagram. The calculated metastable phase diagram at 873 K of the Fe–Cr–Co ternary system is shown in Fig. 2(b) to explain this more definitely, in which the tie lines are denoted in green, and the dashed line demonstrates the Fe-xCr-11.2Co vertical section as shown in Fig. 1(e). In Fig. 2(b), the points a1 and a2 represent the equilibrium compositions of the α1 (FeCo-rich) and α2 (Cr-rich) phase for Fe-26.5Cr-11.2Co alloy, which are on the spinodal decomposition tie line, respectively. On the other hand, the points b1 and b2 represent the compositions at the edges on FeCo-rich side and Cr-rich side of the miscibility gap of (α1 + α2) two-phase region on the Fe-xCr-11.2Co vertical section containing Fe-26.5Cr-11.2Co alloy, respectively. For the α1 phase, the difference in Cr concentration between a1 and b1 is negligible. As for the α2 phase, however, the concentration difference is large, resulting in the concentration gap in Fig. 1(e). To summarize, the PF simulations of spinodal decomposition performed in this study under slow cooling rates agree well with the thermal equilibrium phase diagram. Nevertheless, the cooling rate is expected to affect the microstructure evolution of spinodal decomposition, which will be discussed in the next section.
(a) Comparison of Cr concentration distribution evaluated from PF simulation results under the cooling rate of 1 K/h (red dot) and the calculated Cr concentration of spinodal decomposition along tie lines in Fe-26.5Cr-11.2Co alloys at various temperatures (mark) by Thermo-Calc software. (b) the calculated metastable phase diagram of Fe–Cr–Co ternary system at 873 K using the thermodynamic parameters listed in Table 1. The tie lines are denoted in green, and the dashed line demonstrates the Fe-xCr-11.2Co vertical section as shown in Fig. 1(e). (Online version in color.)
To clarify the influences of cooling rate on the microstructure formed by spinodal decomposition, PF simulations of spinodal decomposition in Fe-26.5Cr-11.2Co alloy were performed under various conditions. The alloy was initially aged at 933 K for 2 h and then subsequently aged with 20 levels of constant cooling rates of 10–200 K/h. Figure 3 shows the simulated microstructures obtained at the moment where the temperature was 773 K after slow-cooling at the cooling rates of 10, 50, 100, and 200 K/h, by way of example (The simulation results under all of the cooling rates can be found in supplemental Fig. S1). The concentrations of Cr, Fe, and Co are also shown for a better understanding of element distribution. At the cooling rate of 10 K/h (as seen in Fig. 3(A)), α2 with the smaller volume fraction was surrounded by α1. In the case of 50 K/h (Fig. 3(B)), α1 was incorporated into α2. Then, as the cooling rate kept increasing, the α1 phase became finer and the α2 became larger. When the cooling rate reached 100 K/h (Fig. 3(C)), α2 became the matrix phase, and α1 fine particles were surrounded by the α2 phase. As for the cooling rates of 100 K/h to 200 K/h, the α2 became thinner while the α1 particles became larger with an increasing cooling rate (Fig. 3(D)). Moreover, the concentrations of α1 and α2 also vary with the cooling rate.
Simulated microstructures at 773 K formed by spinodal decomposition in Fe-26.5Cr-11.2Co alloys subjected to heat treatment with cooling rates of (A) 10, (B) 50, (C) 100, and (D) 200 K/h. The concentrations are shown in (a1, b1, c1, d1) for Cr, (a2, b2, c2, d2) for Fe, and (a3, b3, c3, d3) for Co, respectively. (Online version in color.)
To study quantitatively the simulated spinodal decomposition microstructures under different cooling conditions, the number densities and size of the phase particles were analyzed using ImageJ software and the results are plotted in Fig. 4. To reduce the influence of image boundaries (i.e., the edges of the images) and increase the accuracy of the analysis results, 9 images with the arrangement of 3×3 were used for analysis in each cooling rate utilizing the advantage of periodic boundary condition. The top edge and the bottom edge of the square image of the simulation box can be connected smoothly. Similarly, the left edge and right edge can be smoothly connected. Figures 4(a) and 4(b) show the number density and size of α1 particles as a function of the cooling rate. Moreover, image analysis is performed with various Cr concentrations (19–27 at.%) as the thresholds, and the results are denoted in different colors in Figs. 4(a)–4(b). The number density of α1 particles increases under a higher cooling rate at the range of 10–140 K/h, and then tends to be steady when the cooling rate is 140–200 K/h. In contrast, the size of α1 particles decreases as the cooling rates increase and becomes stable with only slight variation when the cooling rate is higher than 140 K/h. Figure 4(c) shows the area fraction of different particles simulated under various cooling conditions, from which a linear decrease of α1 particles and a linear increase of α2 particles can be observed with the increase in cooling rate.
Quantitative analysis results of the PF-simulated spinodal decomposition microstructures of Fe-26.5Cr-11.2Co alloys at 773 K under cooling rates of 10–200 K/h: (a) the number densities and (b) size of the α1 particles, the color of the mark represents the Cr concentration to be the threshold in atomic percent; (c) area fraction of the α1 and α2 particles. (Online version in color.)
The relationship between the cooling rate and the spinodal decomposition microstructures can be concluded from the simulation results in Figs. 1, 3, and 4. After initial aging at 933 K for 2 h, microstructures with α1 surrounded by α2 are formed (similar to that in Fig. 1(a)). When the subsequent cooling rate is slow (close to the equilibrium solidification state), the α2 phases change their geometry to a more stable one because of long-term diffusion, and α2 was surrounded by α1 for the increment in volume fraction of α1 at a lower temperature (773 K), as seen in Fig. 3(A). When the cooling rate is higher than 50 K/h, the inclusion relationship between α1 and α2 is reversed. At the cooling rate of 50 K/h or higher, the diffusion distance was short, so the shape of each phase particle did not change significantly while maintaining the initial inclusion relationship; thus α1 was surrounded by α2. Along with the elongation of heat treatment time, the α1 particles became coarse; and when the adjacent particles met with each other, they joined to reduce the interface energy of the entire system. Therefore, a relatively slow cooling rate, the heat treatment time of which is long, results in a decrease in number density and an increase in size of α1 particles (Figs. 4(a)–4(b)). On the contrary, the high cooling rate with short heat treatment time can obtain a high number density of α1 particles, although the particle size is small.
It should be noted that, the simulated microstructures in the present work are all performed using two-dimensional (2D) PF models. Hence there are discrepancies between the simulated results and the experimental microstructures due to the interconnection of particles in three-dimensional (3D) space, especially for the results of particle size and number densities. To reveal the relationship between the cooling rates and the microstructure obtained by numerical calculation more quantitatively, 3D-PF simulations, which fit more the experiment, are in progress for our future work.
In order to investigate the change in particle shape during isothermal aging, the PF simulated microstructures were analyzed while aging at 803 K for different periods after initially aged at 933 K for 2 h and then continuously cooled with a constant rate of 170 K/h. The result is shown in Fig. 5(a). No obvious morphology change can be observed, demonstrating the microstructure after spinodal decomposition is determined by the cooling rate under the conditions in this study. However, the particle number density and size vary with the prolonged aging time. After 13 h of aging, the α1 phases are connected and the radius of α1 particles tended to increase with the aging time. The relationship between the number density of particles in each phase and the aging temperature is shown in Fig. 5(b). The number of particles in the α1 phase decreased with time and decreased sharply around 13 h. The number of α2 phase particles increased with time. On the other hand, the concentration of the α1 and α2 particles change with the aging time. The distribution graph of Cr concentration at each time is shown in Fig. 5(c). The Cr concentration did not reach the equilibrium concentration (denoted by dashed lines) after annealing for 12 h, and almost reached the equilibrium concentration after 16 h.
PF simulation results of Fe-26.5Cr-11.2Co alloys aging at 803 K after initially aged at 933 K for 2 h and cooled with a constant cooling rate of 170 K/h: (a) snap shots of the microstructure formed by spinodal decomposition, (b) number of particles, and (c) Cr-concentration plot. (Online version in color.)
Spinodal decomposition in Fe–Cr–Co alloy under continuous multistep heat treatment has been studied by the Phase-Field method. The results obtained are as follows:
• The equilibrium Cr concentration distribution after spinodal decomposition obtained by the PF simulation exhibits a shift to the low Cr side when compared with the calculated vertical thermal equilibrium phase diagram. The inconsistency can be explained that the tie lines of spinodal decomposition are not located in the calculated vertical section, and the PF simulation results agree well with the spinodal decomposition concentration evaluated along the tie lines.
• The morphology of the modulated structure is determined by the cooling rate and does not change during the subsequent isothermal annealing process, while the concentration difference between FeCo-rich α1-phase and Cr-rich α2-phase increases as has been empirically explained. After isothermal aging at a high temperature, a structure with the ferromagnetic phase included in a thin nonmagnetic phase can be obtained by continuously cooling at a higher cooling rate. And when the cooling rate is less than 50 K/h, the inclusion relationship is reversed.
• Continuous cooling at rates higher than 140 K/h demonstrates the maximum number densities of the ferromagnetic particles of α1-phase seemingly almost reaching saturation, which is expected to give rise to exhibiting the largest coercive force of the Fe–Cr–Co magnet.
Simulated microstructures at 773 K formed by spinodal decomposition in Fe-26.5Cr-11.2Co alloys subjected to heat treatment with cooling rates of 10–200 K/h.
This material is available on the Website at https://doi.org/10.2355/isijint-ernational.ISIJINT-2023-044.
This work was partly supported by JSPS KAKENHI Grant Nos. 21H05018 and 21H05193, and by ISIJ Research Promotion Grant from the Iron and Steel Institute of Japan (ISIJ).
