Precipitation Mode and Kinetics of Fe₂X (X: Ta, Hf) Laves Phase on a Eutectoid Type Reaction in High Chromium Ferritic Alloys

Abstract

We recently found the formation of periodically arrayed rows of very fine Fe₂Hf Laves phase by interphase precipitation on a eutectoid type reaction path: δ-Fe → γ-Fe+Laves in 9Cr ferritic alloys with a small addition of Hf. In the present work, the precipitate morphology and precipitation mode of Laves phase were investigated on the eutectoid path in Hf or Ta doped high Cr ferritic alloys. The precipitation mode was found to change from fibrous precipitation to interphase precipitation with raising the δ→ γ transformation kinetics. The transition would be related to the time availability for solute diffusion to grow the fibrous precipitates through the advancing interface boundary diffusion. The nucleation of interphase precipitation of Fe₂Hf phase was measured to be ~2 orders of magnitude faster than that of Fe₂Ta. A thermodynamical consideration suggests that the faster kinetics of the Fe₂Hf phase mainly derived from the higher chemical driving force for the nucleation.

1. Introduction

Heat resistant 9Cr ferritic steels are an important class of materals for high temperature components such as pipes, tubing, and turbines in thermal power generation systems due to their low thermal expansion coefficient, high thermal conductivity, and relatively low cost, compared with austenitic heat resistant steels and nickel based superalloys. The creep strength of high Cr ferritic steels has been improved over the years, which has led to a continuous increase in the steam temperature and pressure and thereby raising the thermal efficiency of steam turbine power plants.^1,2) Using fossil fuels is currently against the worldwide trend towards carbon neutralization, but thermal power will remain as an important power source to keep the resistance to the fluctuation of the alternating current in case of an emergency as well as to back-up the fluctuation of output from the variable renewable energy. Also, thermal power plant systems will be decarbonized by using clean fuels such as H₂ and NH₃ and/or carbon capture technologies. It is, therefore, important to continue to make efforts to further increase the efficiency and the durability of thermal power generation systems, and thus improving the high temperature durability of heat-resistant steels and alloys for the systems will remain a key subject.³⁾

The high Cr ferritic steels for power plant applications are typically heat-treated to form a tempered lath martensitic structure strengthened with M₂₃C₆ type carbide (M: Cr, Fe, Mo) and (V, Nb)(C, N) carbonitride precipitate particles against dislocation motion, recovery, and recrystallization. The precipitate particles are, however, susceptible to particle coarsening and decomposition, which are believed to drive the degradation of long-term creep strength of the materials.^1,2)

Laves phase, A₂B type intermetallic compound, exists as a thermodynamically stable phase in many Fe-based binary alloy systems.⁴⁾ It is reported that Fe₂W and Fe₂Mo type Laves phases are formed in the ferrite matrix during high-temperature exposures in conventional high Cr ferritic steels [5]. The Laves phase precipitates are observed as fine particles along the lath/block boundaries and/or as coarse globular ones, depending on the type and the content of the Laves phase forming elements. There have been efforts to modify the distribution of Laves phase precipitates and to determine the creep properties of high Cr ferritic heat resistant steels with modified distribution of Laves phase. As a result, it was found that an addition of a large amount of W led to the formation of fine Laves phase in the ferritic matrix and improved the creep resistance.^5,6)

One of the present authors found the formation of periodically arrayed rows of very fine Fe₂Hf Laves phase particles in a 9 wt.% Cr based ferrite matrix in a diffusion couple sample.⁷⁾ The subsequent studies^{7,8,9,10,11,12)} revealed that the fine precipitates were formed through interphase precipitation (hereafter designated as IPP) along a eutectoid type reaction path: δ-ferrite → γ-austenite + Fe₂Hf-Laves, followed by a γ to α-ferrite phase transformation. This finding allows us to expect to develop a new type of heat resistant ferritic steel strengthened with this new eutectoid-type precipitation route. The IPP was, however, found to occur in a narrow heat treatment window with a temperature/time range of 1000–1050°C/1 s ~1 min. The precipitation mode competes with the precipitation of relatively coarse Laves phase within the δ-ferrite matrix at the longer time side of TTT diagram.⁹⁾ This suggests that the faster the precipitation start time is, the wider the window for the IPP becomes. It is therefore important to understand effects of Laves phase forming elements and influencing factors on the upper critical bound of the IPP kinetics to enable the distribution of fine Laves phase particles in high Cr ferritic steels.

The same eutectoid type reaction path: δ→ γ+Laves is available in the Fe–Ta and the Fe–Nb binary alloy systems⁴⁾ and in their Cr doped ternary alloys. Our preliminary experiments revealed that Fe₂Ta type Laves phase was formed with fibrous morphology along the eutectoid reaction path in Cr doped ternary Fe–Cr–Ta alloys. The formation of the fibrous Laves phase, hereafter is called “fibrous precipitation”,¹³⁾ designated as FP, was found to occur by more than three orders of magnitude slower than IPP observed in Fe–Cr–Hf ternary alloys with similar Cr contents. We also found that a transition from FP to IPP took place when the δ→ γ transformation kinetics was raised by lowering the Cr content in Fe–Cr–Ta alloys in our recent study.

The aim of this work is two-fold: the first is to show how the precipitate morphology and precipitation mode of Laves phase change with raising the δ→ γ transformation kinetics and to discuss the transition between the two precipitation modes (FP vs. IPP) on the eutectoid reaction path: δ→ γ+Laves; the second is to investigate alloying effects on the upper bound of the IPP kinetics on the eutectoid reaction and to grasp influencing metallurgical factors on the kinetics.

2. Experimental Procedure

The chemical compositions of the studied alloys are summarized in Table 1. The alloys are divided into two groups. The first group is a series of Fe–Cr–Ta alloys with different Cr contents and a fixed eutectoid Ta content. Changing Cr content allows to make variations in the δ→ γ transformation kinetics (i.e. the velocity of the δ/γ interphase boundary migration) while keeping a fixed precipitation kinetics of Laves phase. The second group is a series of Fe–Cr–Hf ternary alloys with different Hf contents and a fixed Cr content. In contradistinction with the first group, the changes in Hf contents allows to observe variations in the driving force for the precipitation of Laves phase with little change in the velocity of δ/γ boundary migration.

Table 1. Chemical compositions and designations of the studied alloys.

Series	Alloy designation	Chemical composition (at.%)
		Fe	Cr	Hf	Ta
Series I	Fe-7.5Cr-0.6Ta	bal.	7.5	–	0.65
	Fe-7.7Cr-0.6Ta	bal.	7.7	–	0.65
	Fe-8.1Cr-0.6Ta	bal.	8.1	–	0.65
	Fe-8.5Cr-0.6Ta	bal.	8.5	–	0.65
	Fe-9.5Cr-0.6Ta	bal.	9.5	–	0.62
Series II	Fe-9.5Cr-0.06Hf	bal.	9.5	0.058	–
	Fe-9.5Cr-0.09Hf	bal.	9.5	0.085	–
	Fe-9.4Cr-0.12Hf	bal.	9.4	0.116	–

All the alloys were prepared by arc melting in an argon atmosphere. The chemical compositions of the alloys were analyzed using inductively coupled plasma optical emission spectrometer (ICP-OES) method, the results of which are listed in Table 1. The chemical compositions are given in atomic percent through this paper.

The arc-melted alloys were fabricated to cylinders of 3 mm in diameter and 10 mm in length with a hole of 5 mm in depth drilled along the longitudinal direction of the cylinder. The cylindrical specimens were solutionized at 1250°C for the first series of alloys or 1300°C for the second series of alloys in their δ single-phase region for a period of 5 min, then directly cooled to isothermally aged temperatures in the γ+Fe₂X two-phase region, hold for certain periods of time, and finally gas cooled. During the heat treatments, the temperature profiles were tracked using an R-type thermocouple which was inserted in the hole of the specimen. The cooling rate from the solutionizing temperature to the isothermal aging temperature was about 35°C/s. Figure 1 shows an example of temperature profile obtained from a heat treatment in which solutionizing was performed at 1300°C for 5 min and isothermal aging at 1100°C for 30 s.

Fig. 1.

The temperature profile for a isothemal heat treatment in which the sample was hold at 1300°C for 5 min, cooled down to 1100°C, hold for 30 s and then cooled down to room temperature.

Heat-treated specimens were sectioned parallel to the basal planes of the cylindrical specimens at 1 mm inside from the planes. The cross sections were prepared by grinding, mechanically polishing down to 1 μm Al₂O₃ polishing suspension, and chemical polishing with SiO₂ polishing suspension. Microstructures were observed with an optical microscope (OM) and a field emission scanning electron microscope (FE-SEM) with backscattered electron (BSE) detector. A free image analysis software, ImageJ,¹⁴⁾ was used to characterize the microstructures of the specimens.

3. Results

3.1. Measurement of δ/γ Interface Velocity

The δ→γ transformation occurred at the surface of the specimens and δ grain boundaries during heat treatments. Figure 2 shows low magnification micrographs taken from the polished cross sections of the Fe-9.5Cr-0.6Ta samples isothermally aged at 1050°C for different periods of time. Large equiaxed grains with a size of 200–400 μm, as observed in the center area of the sections, are interpreted as δ-ferrite grains which were retained during the isothermal aging and cooling processes to room temperature. The fine-grained areas, as observed in the vicinity of the surface of the cross sections and δ grain boundaries, are the areas where the δ→γ transformation (or eutectoid reaction) took place, and the γ phase was subsequently transformed to the α phase during cooling from the isothermal aging temperature. An increase in the fraction of the fine-grained areas with isothermal aging (cf differences between Figs. 2(a) and 2(b)) demonstrates that the δ/γ interphase migrated towards the center of the sample from the surface or into grain interiors from the grain boundaries.

Fig. 2.

Backscattered electron images of the Fe-9.5Cr-0.6Ta alloy aged at 1050°C directly after a δ single-phase heat treatment at 1300°C aged for: (a) 5 min, (b) 30 min.

The velocity of the δ/γ interface migration was estimated at both γ nucleating sites by measuring the width of the fine-grained areas. Half of the width value was judged as the migration distance of the δ/γ interface in the case of grain boundary nucleation site. Figure 3 shows the values of the migration distance measured at both nucleation sites as a function of isothermal aging time in the Fe-9.5Cr-0.6Ta samples. It can be seen that the distances increase almost linearly with aging time under the experimental conditions, and the interface velocities at both sites are similar. The velocity was thus estimated by linear approximation of the measured distances.

Fig. 3.

Growth distance of the γ phase at the sample surface and δ grain boundaries on aging at 1050°C as a function of aging time in the Fe-9.5Cr-0.6Ta alloy.

Figure 4 shows the estimated interface velocity as a function of temperature for Ta doped alloys with different Cr contents and for the second series of Hf doped alloys. The estimated velocity was found to show an inverse C shape in the aging condition. The velocity for the Fe-9.5Cr-0.6Ta alloy is ~3 orders of magnitude as slow as that for the Hf doped alloys with similar Cr contents. The velocity in the Ta doped alloys increased with decreasing the Cr content from 9.5% to 8.5% by one order of magnitude, and from 8.5% to 8.0% by another one order of magnitude. The velocity for the Fe-7.5Cr-0.6Ta alloy was too fast to be measured. The velocity in the Hf doped alloys was found to increase with decreasing the Hf content and was to high to be measured at the lower Hf contents in the present study.

Fig. 4.

The δ/γ interface velocity measured at aged temperatures in Ta doped and Hf doped alloys. The broken lines for the Hf doped alloys indicate they are qualitatively estimated.

3.2. Precipitation Mode and Morphology vs. δ/γ Interface Velocity

The precipitation mode and precipitate morphology changed with raising the δ/γ interface velocity. Figure 5 shows some examples of micrographs taken from Ta doped alloys isothermally aged. The precipitation mode was identified as FP at a relatively slow δ/γ interface velocity range according to the facts that the precipitates are fibrous, and their longitudinal direction is oriented perpendicular to the δ/γ interface (Figs. 5(a), 5(b)). It is also seen that the fibers are fragmented in some places and continuous in other areas even in one eutectoid colony. At an intermediate δ/γ interface velocity range, fibrous precipitates coexisted with fine precipitate arrays in between the fibers (Fig. 5(c)). The orientation of the fine precipitate arrays suggests that they are formed by IPP. Coexistence of FP and IPP in different eutectoid colonies was also observed in this range, as reported in previous studies.^15,16) Typical microstructures formed by IPP were found at a relatively higher δ/γ interface velocity, where periodical arrays of fine particles are arranged parallel to the advancing δ/γ interface (Fig. 5(d)). It is noted that the size and distribution of IPP particles observed in Ta doped alloys are coarser by a factor of ~3 for the inter-sheet particle spacing and of ~5 for the intra-sheet particle spacing than those obtained in Hf doped alloys.^{7,8,9,10,11,12)} At an even faster δ/γ interface velocity range, the formation of Laves phase via the eutectoid reaction was rarely recognized. Only IPP mode was, on the other hand, recognized in the Hf doped alloys. The observed precipitation mode and morphological features in Ta doped and Hf doped alloys are summarized in Fig. 6, and Tables 2 and 3.

Fig. 5.

Backscattered electron images of Ta doped alloys heat-treated at d phase field and subsequently aged at 950°C for: (a) 30 min in Fe-9.5Cr-0.6Ta alloy, (b) 5 min in Fe-8.5Cr-0.6Ta alloy, (c) 10 min in Fe-8.1Cr-0.6Ta alloy, (d) 100 s in Fe-7.7Cr-0.6Ta alloy. The phase constituents identified are designated in the images. δ (γ) indicates that the region was δ phase at the aging temperature and it transformed to γ phase on cooling after the aging.

Fig. 6.

The precipitation mode (FP or IPP or No precipitation) identified in Ta and Hf doped alloys with different δ/γ interface velocities. The data for the Hf doped alloys are plotted on the qualitatively estimated lines.

Table 2. Morphological features of the precipitates observed in Ta doped alloys.

Aging temp. /°C	Alloys studied
	Fe-9.5Cr-0.6Ta	Fe-8.5Cr-0.6Ta	Fe-8.1Cr-0.6Ta	Fe-7.7Cr-0.6Ta
1100	FP (f)	–	No precipitate	–
1050	FP (f+c)	FP (f+c)	No precipitate	–
1000	FP (f+c)	FP (f+c)	No precipitate	–
950	FP (f)	FP (f+c)+IPP	FP (f+c)+IPP	IPP

f: fragmental fibre, c: continuous fibre, – not observed

Table 3. Morphological features of the precipitates observed in Hf doped alloys.

Aging temp. /°C	Alloys studied
	Fe-9.4Cr-0.12Hf	Fe-9.5Cr-0.09Hf	Fe-9.5Cr-0.06Hf
1200	IPP	–	–
1100	IPP	–	–
1050	IPP	IPP	–
1000	IPP	IPP	IPP

– Not observed

3.3. Alloying Effects on the Upper Bound of δ/γ Interface Velocity Which Allows IPP

Upper bound of the velocity of δ/γ interface migration which allows IPP was investigated by identification of fine IPP particles on samples with intentionally changed δ/γ interface velocities. The δ/γ interface velocity was estimated for Hf doped alloys in the same way as done for Ta doped alloys. Figure 6 summarizes the identified δ/γ interface velocity conditions to allow or not to allow IPP in Ta and Hf doped alloys. The upper bound of the velocity was estimated to be ~200 μm/s in the Hf doped alloys. In Ta doped alloys, the upper bound was estimated as 5–8 μm/s, which was about two orders of magnitude slower than in the Hf doped alloys.

4. Discussion

4.1. Transition between the Precipitation Modes (FP vs. IPP)

The precipitation mode on the eutectoid type reaction path: δ→ γ+Fe₂Ta was observed to change from FP to IPP through a coexisting state with increasing the δ/γ interface velocity at a critical velocity of 0.2–1 μm/s. The transition between the precipitation modes is discussed below by analyzing the obtained results in terms of available theoretical treatments on the growth of eutectoid product phases.

It is generally considered that fibrous precipitation mode occurs on eutectoid reaction when the precipitate phase can grow in a cooperative manner with the other product phase in the reaction. The precipitate phase corresponds to the Laves phase and the other product phase to the γ phase in the present case. Treatments on the growth rate through volume diffusion or boundary diffusion of solute elements were adopted for the fibrous aggregates observed in the present study. By referring,^17,18) the volume diffusion-based growth rate v_v of the present case is given by the Eq. (1):   

$$v_{\mathrm{v}}=D_{\mathrm{V}}{}^{\delta }/(f^{\gamma }{f}^{\mathrm{F}{\mathrm{e}}_2\mathrm{T}\mathrm{a}}S)\mathrm{\hspace{0.17em} }(x^{\delta /\gamma }-x^{\delta /\mathrm{F}{\mathrm{e}}_2\mathrm{T}\mathrm{a}})/(x_{\mathrm{e}}{}^{\mathrm{F}{\mathrm{e}}_2\mathrm{T}\mathrm{a}/\gamma }-x_{\mathrm{e}}{}^{\gamma /\mathrm{F}{\mathrm{e}}_2\mathrm{T}\mathrm{a}})$$(1)

where D_V^δ is volume diffusion coefficient of solute element (Ta) in the δ phase, f is the volume fraction of each phase, S is interlamellar (inter-fiber) spacing. x^δ/γ and $x^{\delta {\text{/Fe}}_{\text{2}}\text{Ta}}$ is the solute content (in mole fraction) in the δ phase in front of the eutectoid γ phase and Fe₂Ta phase, respectively. $x_{\text{e}}{}^{{\text{Fe}}_{\text{2}}\text{Ta/}\gamma }$ and $x_{\text{e}}{}^{\gamma {\text{/Fe}}_{\text{2}}\text{Ta}}$ is the equilibrium solute content in the Fe₂Ta and the γ phase, respectively. The boundary diffusion-based growth rate v_B is given by the Eq. (2):   

$$\begin{array}{l}{v}_{\mathrm{B}}=8k\mathrm{\hspace{0.17em} }w{D}_{\mathrm{B}}/(3f^{\gamma }{f}^{\mathrm{F}{\mathrm{e}}_2\mathrm{T}\mathrm{a}}{S}^2)\mathrm{\hspace{0.17em} }(x^{\delta /\gamma }-x^{\delta /\mathrm{F}{\mathrm{e}}_2\mathrm{T}\mathrm{a}})/\\ \hspace{1em}\hspace{1em}(x_{\mathrm{e}}{}^{\mathrm{F}{\mathrm{e}}_2\mathrm{T}\mathrm{a}/\gamma }-x_{\mathrm{e}}{}^{\gamma /\mathrm{F}{\mathrm{e}}_2\mathrm{T}\mathrm{a}})\end{array}$$(2)

where w is the width of advancing δ/γ interface, D_B is boundary diffusion coefficient, and k is defined as the distribution coefficient for the solute element between the δ/γ interface and the initial parent δ phase. Figure 7 shows a schematic illustration of a FP growth and some parameters used in the treatment. Both types of growth rates were calculated using the measured inter-fiber spacing, reported diffusion coefficient values,^19,20) and the volume fractions and the solute contents which were estimated from our phase diagram study on the Fe–Cr–Ta ternary system,¹²⁾ ignoring the molar volume difference among the constituent phases. The specific δD_B value for Ta at ferrite/austenite boundary was not available, so that the reported value for grain boundary diffusion of Fe in pure iron was taken, which would be reasonable since the value was reported to be independent of the diffusive species (Fe, Ni, Co, and Cr) and of the matrix phases (ferrite or austenite).²⁰⁾ The calculated v values and some parameters used for calculation are summarized in Table 4. The value of 1 was taken for k since the values of x^δ/γ and $x^{\delta {\text{/Fe}}_{\text{2}}\text{Ta}}$ are almost equally away from the bulk Ta content. The calculated growth rate values demonstrate that the growth rate which is achievable by boundary diffusion is 5 orders of magnitude higher than the rate by the volume diffusion.

Fig. 7.

Schematic illustration of FP growth and some parameters used to discuss the growth mechanism.

Table 4. Thermodynamic, diffusional and microstrucural parameters used to calculate the velocity of eutectoid fibrous growth in the Fe-8.5Cr-0.6Ta alloy based on the Eqs. (1) and (2). x stands for the Ta content in mol fraction.

Temp (°C)	xδ/γ (at.%)	$x^{\delta {\text{/Fe}}_{\text{2}}\text{Ta}}$ (at.%)	$x^{{\text{Fe}}_{\text{2}}\text{Ta/}\gamma }$ (at.%)	$x^{\gamma {\text{/Fe}}_{\text{2}}\text{Ta}}$ (at.%)	DVδ (μm2/s)	wDB (μm3/s)	S (μm)	vV (μm/s)	vB (μm/s)
1100	0.78	0.56	26.9	0.21	1.0×10−6	7.0×10−2	2.1	2.7×10−7	2.4×10−2
1050	0.83	0.49	27.0	0.16	3.5×10−7	4.0×10−2	1.5	1.9×10−7	3.8×10−2
1000	0.81	0.42	27.2	0.12	1.0×10−7	2.5×10−2	0.9	9.3×10−8	6.9×10−2
950	0.69	0.36	27.4	0.09	3.5×10−8	8.0×10−2	0.6	3.9×10−8	3.9×10−2

As explained in the section 3.2., fibrous precipitates are often fragmented. The fragmented fibrous feature suggests that the advancing δ/γ interface moves at a rate just above the rate at which the precipitate can collect sufficient solute element to be continuously grown. Closer values between the measured δ/γ interface velocity range for FP and the calculated v_B rather than v_v indicate that the growth of fibrous precipitates is controlled by boundary diffusion of solute element through the advancing δ/γ interface. As the interface velocity increases, the time for the boundary diffusion becomes insufficient to supply the solute element to grow the fibers continuously, and then the fibers get more fragmented. The increased interface velocity may also allow the solute content to remain high at the δ/γ interface, and the supersaturated solute is available for the nucleation of IPP at the moving δ/γ interface, which is a possible scenario of the transition from FP to IPP which was observed in the present study.

4.2. The Factors to Influence the Nucleation Rate of IPP

Upper bound of the velocity of δ/γ interface migration which allows IPP was found to be about two orders of magnitude slower in Ta doped alloys than in Hf doped alloys. This result indicates that Fe₂Hf particles can nucleate at the migrating interface much faster than Fe₂Ta particles, by considering the moving ledge mechanism for the nucleation of IPP.¹⁴⁾ A schematic of the nucleation of IPP is shown in Fig. 8(a). When a ledge moves towards the right, the interface moves upward in the schematic. The period t^int for which the δ/γ interface is available for nucleation of IPP starts when the interface appears by the arrival of a ledge and ends when the interface disappears by the arrival of a ledge on the successive sheets. This time range can be estimated as   

$$t^{\mathrm{i}\mathrm{n}\mathrm{t}}=\lambda /u=h/v$$(3)

where λ is moving ledge spacing, h is the inter-sheet spacing, u is the velocity of lateral ledge motion, and v is the upward interface velocity. A similar estimation was recently made by a previous work on the formation of VC by IPP in steels.²¹⁾ Taking observed h (150 μm for Hf doped alloys, 450 μm for Ta doped alloys) and v values, t^int was calculated in different samples. The results are displayed in Fig. 9, which shows that the least time that is required for the nucleation of IPP t^IPP, is 0.5–1 millisecond for Fe₂Hf and 50–100 milliseconds for Fe₂Ta.

Fig. 8.

Schematic drawing of the nucleation of IPP: (a) the nucleation at the terrace of δ/γ interface by ledge mechanism, (b) solute content profile along the interface.

Fig. 9.

The time required for the nucleation of IPP. In the figure, the presence or absence of IPP is shown as a function of the estimated period for which the δ/γ interface is available for nucleation of IPP t^int. x in Fe-xCr-0.6Ta stands for the Cr contents in the alloys studied.

The calculated difference in t^IPP for the Fe₂Hf and Fe₂Ta phases is discussed below. The calculated t^IPP was used to estimate the radius of Laves phase particle which is attained by assuming the classical Zener type growth model¹³⁾ and a supply of the solute element to Laves phase particle through boundary diffusion along the δ/γ interface. The flow of solute element and concentration profile at the interface is schematically drawn in Fig. 8(b). The amount (in mol) of the solute element which is extracted from the diffusion field m can be given by the Eq. (4):   

$$m=1/2\pi L^{2}w(x_0-x_e{}^{\gamma /\mathrm{F}{\mathrm{e}}_2\mathrm{X}})/V_{\mathrm{m}}$$(4)

where L is the diffusion distance of the solute element, x₀ and $x_{\text{e}}{}^{\gamma {\text{/Fe}}_{\text{2}}\text{X}}$ is initial and equilibrium solute content in the γ phase, respectively, and V_m is the molar volume. According to the Zener approximation,   

$$L=2\surd (Dt)$$(5)

If the particle is assumed to be a double spherical cap with an aspect ratio of 3, which is closer to the observed shape of the precipitates rather than sphere, the amount (in mol) of the solute element in the Laves phase particle m’ can be given by:   

$$m’=4\pi r^3(x_{\mathrm{e}}{}^{\mathrm{F}{\mathrm{e}}_2\mathrm{X}/\gamma }-x_0)/V_{\mathrm{m}}$$(6)

where $x_{\text{e}}{}^{{\text{Fe}}_{\text{2}}\text{X/}\gamma }$ is equilibrium solute content in the Laves phase. Equating (4), (5), (6) gives the long radius of the particle 3r as:   

$$3r=3^3\surd (1/8{(Dt)}^{2}w(x_0-x_e{}^{\gamma /\mathrm{F}{\mathrm{e}}_2\mathrm{X}})/(x_{\mathrm{e}}{}^{\mathrm{F}{\mathrm{e}}_2\mathrm{X}/\gamma }-x_0))$$(7)

The 3r values, by taking the values of 0.5 nm for w,²²⁾ the boundary diffusion coefficient D_B for D, and the calculated t^IPP for t, were estimated as 5–11 nm for Fe₂Hf and 23–44 nm for Fe₂Ta. This estimation might imply that Fe₂Hf can nucleate at a smaller size than Fe₂Ta can by a factor of about one fourth. It should be noted that the L values calculated were 250–750 nm for Hf and 2300–6800 nm for Ta, which difference is in a reasonable agreement with the difference in the intra-sheet particle spacing observed for the two Laves phase particles.

In order to understand factors to cause the difference in the nucleation kinetics between the two Laves phases, a classical nucleation theory was considered. The Gibbs free energy change during heterogeneous nucleation at the interphase interface, ΔG_het, can be given by the Eq. (8):   

$$\begin{array}{l}\mathrm{\Delta }{G}_{\mathrm{h}\mathrm{e}\mathrm{t}}=-V^{\mathrm{F}{\mathrm{e}}_2\mathrm{X}}(\mathrm{\Delta }{G}_{\mathrm{V}}-\mathrm{\Delta }{G}_{\mathrm{S}})+A^{\mathrm{F}{\mathrm{e}}_2\mathrm{X}/\delta }{\sigma }^{\mathrm{F}{\mathrm{e}}_2\mathrm{X}/\delta }\\ \hspace{1em}\hspace{1em}\hspace{1em}+A^{\mathrm{F}{\mathrm{e}}_2\mathrm{X}/\gamma }{\sigma }^{\mathrm{F}{\mathrm{e}}_2\mathrm{X}/\gamma }-A^{\delta /\gamma }{\sigma }^{\delta /\gamma }\end{array}$$(8)

where $V^{{\text{Fe}}_{\text{2}}\text{X}}$ is the volume of precipitate nucleus with a double spherical cap shape, ΔG_V is the chemical driving force for nucleation of precipitate phase, ΔG_S is the elastic strain energy change associated with nucleation, A is the interfacial area between each different phase, and σ is the interfacial energy. The Eq. (8) is written in terms of wetting angle θ and cap radius r under an assumption that: ${\sigma }^{{\text{Fe}}_{\text{2}}\text{X/}\delta }$ = ${\sigma }^{{\text{Fe}}_{\text{2}}\text{X/}\gamma }$ as.   

$$\begin{array}{l}\mathrm{\Delta }{G}_{\mathrm{h}\mathrm{e}\mathrm{t}}=-\{4/3\pi r^3(\mathrm{\Delta }{G}_{\mathrm{V}}-\mathrm{\Delta }{G}_{\mathrm{S}})-4\pi r^2{\sigma }^{\mathrm{F}{\mathrm{e}}_2\mathrm{X}/\delta }\}\\ \hspace{1em}\hspace{1em}\hspace{1em}(2-3\mathrm{c}\mathrm{o}\mathrm{s}\theta +\mathrm{c}\mathrm{o}{\mathrm{s}}^3\theta )/2\end{array}$$(9)

The critical radius of nucleus r* is given by the Eq. (10):   

$$r*=2{\sigma }^{\mathrm{F}{\mathrm{e}}_2\mathrm{X}/\delta }/(\mathrm{\Delta }{G}_{\mathrm{V}}-\mathrm{\Delta }{G}_{\mathrm{S}})$$(10)

Interfacial energy ${\sigma }^{{\text{Fe}}_{\text{2}}\text{X/}\delta }$ and strain energy ΔG_S values are currently unavailable to be able to compare the difference between the nucleation events of the two Laves phases, as far as the authors are concerned. On the other hand, the chemical driving force for the nucleation of Fe₂Hf and Fe₂Ta phases are available from a thermodynamic data. The chemical driving force ΔG_V values were then calculated under an assumption that a local equilibrium is achieved at δ/γ interface using the Gibbs free energy values which fulfils the phase equilibrium determined by experiments in both Fe–Cr–X ternary systems.¹²⁾ Figure 10 shows the ΔG_V values estimated in a Hf doped alloy and a Ta doped alloy as a function of temperature. It can be seen that the ΔG_V values exhibit inverse C curves in both alloys and the values at their nose temperatures are about 4 times higher in the Hf doped case than in the Ta doped case. This difference in the ΔG_V values may give one forth critical radius of nucleus of Fe₂Hf compared with that of Fe₂Ta, which is in a good agreement with the nucleus size estimated by the calculated t^IPP and boundary diffusion. This agreement might suggest that the IPP nucleation kinetics would be mainly governed by the chemical driving force for the precipitating phase among the three energy terms although it is not easy to conclude since the other energy terms just compromised with each other.

Fig. 10.

The chemical driving force for the nucleation of Laves phase calculated with Gibbs free energy values which reproduce experimentally determined phase diagrams in the Fe–Cr–X ternary systems.

5. Summary

In the present work, the precipitate morphology and precipitation mode of Laves phase on a eutectoid type reaction path: δ-Fe → γ-Fe+Laves were investigated in Ta or Hf doped 9Cr ferritic alloys. The main results are:

(1) The precipitation mode was found to change from fibrous precipitation to interphase precipitation with raising the δ→ γ transformation kinetics.

(2) Semi-quantitative analysis suggests that the growth of fibrous precipitates is controlled by boundary diffusion of solute element through the advancing δ/γ interface rather than the volume diffusion in the parent phase. The transition would thus be related to the time availability for solute diffusion through the advancing interface boundary diffusion to grow the fibrous precipitates.

(3) The nucleation of IPP of Fe₂Hf was found to be about 2 orders of magnitude faster than that of Fe₂Ta. A driving force consideration suggests that the faster kinetics of the Fe₂Hf phase mainly derived from the higher chemical driving force for the nucleation.

Acknowledgement

This study was financed by the 31st ISIJ Research Promotion Grand (2020), university research aid of JFE 21st Century Fundation, and JSPS KAKENHI Grand Number 15K06496.

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