In-situ Measurements of Solute Partition Coefficients between Solid and Liquid Phases in Fe–Cr–Ni–Mo–Cu Alloys during Solidification

Abstract

The in-situ measurements of the solute partition coefficients, k, between the solid and liquid phases in Fe–Cr–Ni–Mo–Cu alloys were conducted using X-ray transmission imaging and X-ray fluorescence spectroscopy (EDS) in a synchrotron radiation facility, SPring-8.

A nearly planar solid/liquid interface was achieved in a furnace with a temperature gradient (5–10 K/mm) using X-ray transmission imaging. The measurement points in the solid and liquid phases and close to the solid/liquid interface were determined by X-ray imaging. The compositions of the solid and liquid phases were measured by EDS. The solute partition coefficients along the solidification path in the Fe - 19.89–25.82mass% Cr - 24.73–34.81mass% Ni - 4.46–10.28mass% Mo - 1.47–5.79mass% Cu alloy were determined at 56 different liquid compositions. At the beginning of solidification, the partition coefficients of Cr, Ni, Mo and Cu were 0.96, 0.97, 0.70 and 0.86, respectively. The partition coefficients of Cr and Ni were almost constant during the unidirectional solidification. The partition coefficient of Mo gradually changed from 0.7 to 0.6, leading to a severe microsegregation at the end of solidification. In contrast, the partition coefficient of Cu was dispersed in the range from 0.8 to 0.9. This study demonstrated that the in-situ measurement was effective for systematic measurement.

1. Introduction

Common stainless steels including SUS 304 and SUS316 are susceptible to deterioration in very corrosive environments, such as seawater and acidic conditions. Therefore, various austenitic stainless steels consisting of Fe–Cr–Ni–Mo–Cu systems have been developed so far,¹⁾ aiming for increased strength and durability in application to these environments. The Fe–Cr–Ni–Mo–Cu alloys not only contain larger amounts of Cr and Ni than the common grades, but also contain Mo and Cu. It is known that Mo plays a role in the prevention of pitting corrosion, while Cu is an effective element for avoiding corrosion in sulfuric acid environments.^2,3)

The casting process should be controlled to reduce the micro- and macro- segregation and to have the elemental distribution as homogeneous as possible for the alloys to exhibit the properties required for the applications in severe environments. In particular, segregation may result in the deterioration of corrosion resistance, heat resistance and toughness. In addition to its influence on the properties, segregation widens the brittle temperature range at the post solidification temperatures, which leads to cracking during casting^4,5) and hot working.⁵⁾ In general, segregation is significant in the alloys containing the solutes with partition coefficients k, apart from k=1.

It is important to understand the solute partition at the solidification front (S/L interface) to estimate the element distribution because microsegregation takes place because of the solute partition at the S/L interface. However, only a few of the applicable partition coefficients k have been reported for the relatively high alloy of the Fe–Cr–Ni–Mo–Cu system. Yamamoto et al.⁶⁾ determined the k values of Cr, Ni, Mo and Cu for the Fe-20mass% Cr-25mass% Ni-4.5mass% Mo-1.5mass% Cu alloy. Morita et al.⁷⁾ reported the k values of Cr, Ni and Mo in the Fe-20mass% Cr-24mass% Ni-1.3mass% Si-1.0mass% Mn-1.0mass% Mo alloy. So far, the values have been measured for the binary Fe-X and ternary Fe-X-Y alloys,⁸⁾ which should not apply for the k values of the high alloys.

The previous study⁹⁾ has indicated that some problems are associated with the microsegregation model based on the two-dimensional concept to estimate the segregation. The terminal solidification positions corresponding to the interdendritic regions have to be examined in closer detail, implying that the behavior of a solid fraction (fs) higher than 0.9 has to be clarified. Attention should also be paid to the three factors of (1) the diffusion coefficients in the solid phase, (2) the k values and (3) the interfacial area depending on fs.

The k value is usually treated as a constant value, despite the concentration of the solutes in the residual liquid phase with the solidification progress indicating that the k values could vary with fs. The variation of the k value with fs results in a considerable difference at the final stage of solidification in a model. Scheil’s model,¹⁰⁾ widely and simply applied for estimation, gives us a result that the difference of 0.1 in the k value leads to a difference as significant as 1.3 times at fs=0.95. Assuming that 1 mass% solute has the k value of 0.6, the solute concentration in the residual liquid phase is 3.3 mass% at fs=0.95. In comparison, the k value of 0.5 with 1 mass% solute brings 4.5 mass% at fs=0.95, which may affect the properties of the alloy such as corrosion resistance and toughness. The above facts show that the k values for the high alloy systems are still insufficient to accurately estimate the segregation.

Conventionally, the k values are determined by the experiments where the solid phase attains equilibrium with the liquid phase of the aimed alloy at high temperatures. However, it takes a long time to attain equilibrium, after which the sample is taken out of the furnace to quench in a water bath. The cross-section of the sample is then polished to identify the S/L interface. The compositions of the solid and liquid phases are measured by EPMA.^6,7,8) However, only a single dataset of k values of the elements in the alloy is obtained after a long process. The random sampling method is proposed as an alternative method by previous studies.^6,11,12) The k value can be estimated assuming that Scheil’s equation¹⁰⁾ is applicable for the concentration distributions of the solutes. One has to be careful of this method because the obtained k values will include all of the thermal history involving diffusion in the solid phase after the completion of solidification. Thus, it implies that they are not the k values in equilibrium.

Therefore, the k values must be measured more efficiently with an accuracy as high as possible as a function of liquid composition at the solidification front to improve the simulation model of segregation. It is expected that an in-situ measurement would give us more accurate k values, according to the work by Uwabe et al.¹³⁾ The combination of X-ray transmission imaging and X-ray fluorescence spectroscopy enabled us to identify the S/L interface during the experiments on the SUS316 stainless steel at high temperatures. Simultaneously, the solid and liquid phases at the positions close to the S/L interface were directly analyzed by energy dispersive X-ray spectroscopy (EDS).

The objective of this study is to clarify the variation of the k values with liquid compositions along the solidification path. Consequently, the k values as a function of liquid composition at the solidification front are proposed to be applicable for the high alloy system of the Fe–Cr–Ni–Mo–Cu alloys. Therefore, in-situ measurements were attempted for the Fe–Cr–Ni–Mo–Cu alloys using X-ray transmission imaging, X-ray fluorescence spectroscopy and simultaneous EDS analysis.

2. Experimental Procedure

2.1. Experimental Setup

Measurements of solute partition coefficients were conducted at beamline BL20B2 in the synchrotron radiation facility, SPring-8. An experimental apparatus developed in the previous report¹³⁾ was used in this study. The setup as shown in Fig. 1 is configured to observe the S/L interface by X-ray transmission imaging and to measure the X-ray fluorescence spectra. The following items were placed along the incident X-ray beam: beamline slits to adjust the incident beam size, an ion chamber for measuring the intensity of the incident X-ray beam, and a vacuum chamber equipped with a graphite furnace and a beam monitor (visible light conversion type) for observing the transmission images. As explained in the previous report,¹³⁾ the furnace and the specimen were tilted by 30° against the direction of the incident beam. The EDS (Energy dispersive X-ray spectrometer) was also placed against the incident beam with a tilt angle of 45°, as shown in Fig. 1. The setup enabled us to observe the transmission image and to perform the X-ray fluorescence analysis.

Fig. 1.

Schematics of X-ray optics for X-ray transmission imaging and X-ray fluorescence analysis (after ref. 13).

The vacuum pressure in the chamber was kept at <10 Pa during the experiments with a turbomolecular pump and a scroll pump. A container filled with sponge titanium powder was placed above the sample to avoid oxidation. The carbon heater was designed to achieve a vertical temperature gradient of several K/mm in the specimen. Consequently, it was possible to perform unidirectional solidification from the bottom to the top by decreasing the temperature at a constant cooling rate.

The compositions of the Fe–Cr–Ni–Mo–Cu alloys used in this study are listed in Table 1. Alloy No. 1 is the standard composition of this study, while Alloys No. 2–9 are prepared so as to present the composition range of the residual liquid phase entirely during solidification of the standard alloy. Mother alloys (5 kg) were produced using pure metals of Fe, Cr, Ni, Mo and Cu by an induction furnace. The alloys were melted in a MgO crucible. According to the thermodynamic calculations from the Thermo-Calc software,¹⁴⁾ the alloy systems have a fully austenitic solidification mode. The ingots were forged and then partially sliced. Thereafter, they were precisely polished to 0.1 mm in thickness.

Table 1. Chemical compositions of the alloys (mass%).

Alloy No.	Cr	Ni	Mo	Cu	Fe	Solidification path	Single
1	19.89	24.82	4.50	1.47	bal.	○	–
2	22.70	24.73	4.46	1.47	bal.	–	○
3	25.82	24.81	4.52	1.52	bal.	–	○
4	20.06	29.88	4.50	1.48	bal.	–	○
5	19.89	34.81	4.49	1.47	bal.	–	○
6	19.95	25.26	7.36	1.47	bal.	–	○
7	19.90	24.89	10.28	1.47	bal.	○	○
8	20.53	24.78	4.46	3.61	bal.	–	○
9	20.19	24.85	4.50	5.79	bal.	–	○

A specimen holder was composed of a sintered alumina and BN plates as shown in Fig. 2. A thin sample was inserted between two sapphire plates (35 μm in thickness at the incidence side and 150 μm in thickness for the transmission side). The thinner sapphire plate with a thickness of 35 μm facilitated the measurement of the fluorescence X-rays (>5 keV). A monochromatized X-ray beam of 23 keV was used for the observations and measurements.

Fig. 2.

Specimen holder for in-situ measurement of X-ray fluorescence spectra. (Online version in color.)

2.2. Measurement Procedure

Two different procedures were used to measure the partition coefficients. One procedure involves acquiring measurements along the solidification path and is referred to as a solidification path measurement. The other procedure involves a single measurement using a small specimen and is referred to as a single measurement.

Figure 3 shows the procedures for the solidification path measurement. The initial dimensions of the specimen were 1 mm × 5 mm (width × height). Alloys No. 1 and No. 7 in Table 1 were used for the solidification path measurement. These alloys were selected because Alloy No. 1 is the standard composition of the alloy system and Alloy No. 7 has the highest Mo content. As previously clarified,^6,9) Mo is the element receiving the most attention because it has a k value of 0.57, which leads to a significant concentration in the later stage of solidification. To prepare for measurement, the sample was first fully melted and homogenized. An X-ray fluorescence spectrum from the melt was measured by irradiating the sample with an incident X-ray beam at 0.2 mm × 0.1 mm (width × height). Because the melt composition was known, the spectrum was used for the calibration procedure. The measurement time (live time) of EDS was 100 s for each scan.

Fig. 3.

Experimental procedure of solidification path measurement. (Online version in color.)

In the case of Alloy No. 1, the specimen was unidirectionally solidified from the bottom to the top with cooling rates that ranged from 0.5 K/min to 1 K/min. When the S/L interface moved to a certain position, cooling was suspended. It took about 15 min to achieve a smooth S/L interface. The X-ray fluorescence spectra of the solid and liquid phases were measured. The measurement positions were 50–100 μm away from the S/L interface, because the S/L interface could have a convex shape that was perpendicular to the X-ray transmitting direction owing to the wetting behavior. In the case of Alloy No. 7, the specimen was solidified with cooling rates of 0.1–0.3 K/min.

Figure 4 schematically shows the single measurement procedure. Eight solidified specimens (Alloys No. 2 to 9 in Table 1) were inserted into a specimen holder and heated. Once melting was observed in a specimen, a smooth S/L interface was achieved by controlling the furnace temperature. The typical holding time before an EDS measurement was 15 min. The X-ray fluorescence spectra of the solid and liquid phases were measured. The EDS measurements were essentially the same as those for the solidification path measurement. The heating procedures were repeated until the measurements were performed for all specimens. The EDS measurements of the fully melted specimens were also performed for calibration. The calculation manner of the solute concentrations from the intensities of the EDS spectra was described in the previous report.¹³⁾ The solid fraction (f_S) was defined as the area fraction of the solid region. This value was used to show how much solidification had progressed.

Fig. 4.

Experimental procedure of single measurement in various alloys. (Online version in color.)

3. Results and Discussion

3.1. X-ray Transmission Image and X-ray Fluorescence Spectra

Figures 5(a) and 5(b) show the typical X-ray transmission images during the solidification path measurement of Alloy No. 1 and the experiments of Alloy No. 8 with the single measurement, respectively. The density of the liquid phase (γ) is about 4% lower than that of the solid phase.¹⁵⁾ Therefore, the S/L interface can be clearly identified by the contrast attributed to the difference of X-ray transmittance between the solid and liquid phases.

Fig. 5.

Example of X-ray transmission images. Representative images of (a) Solidification path measurement (fs=0.52, Alloy No. 1) and (b) Single measurement (fs=0.44, Alloy No. 8). The S/L interface can be identified by the transmission images.

Figure 6 shows the X-ray fluorescence spectra of the solid and liquid phases close to the S/L interface of the solidification path at fs=0.34 of Alloy No. 1. Figures 6(a), 6(b) and 6(c) show the entire X-ray fluorescence spectra and Gaussian fitting curves to the X-ray fluorescence spectra of Cr, Fe, Ni and Cu, and the Gaussian fitting curves of Mo, respectively. The intensities were normalized by the peak height of the Fe K_α line. As can be seen, the detected peaks are the characteristic X-ray markers for the composed elements, a broad peak of Compton scattering around 21 to 22 keV and the incident X-ray beam at 23 keV. A white light background caused by the scattering of the X-ray beam in the sample is negligibly small in this measurement. Obviously, in Fig. 6(b), the spectral lines of K_α and K_β of the composed elements are clearly seen, except for Cu. The peaks can be fitted with the Gaussian function curves. In Fig. 6(c), the peak intensity of the Mo K_α line is notably much higher than the other elements despite being only several mass% in content. This result proves that the compositions of 4d transition metals can be accurately analyzed by this experiment. Because of the lower content of Cu, the peaks of the Cu K_α (8.04 keV) and Ni K_β (8.26 keV) lines are overlapped with each other, resulting in some analysis error.

Fig. 6.

Typical X-ray fluorescence spectra of the solid and liquid phases near the solid/liquid interface in the solidification path measurement of Alloy No. 1 at fs=0.34. (a) Entire X-ray fluorescence spectra, (b) Gaussian fitting curves to X-ray fluorescence spectra of Cr, Fe, Ni and Cu and (c) fitting curves to X-ray fluorescence spectra of Mo. (Online version in color.)

3.2. Solute Concentration of the Solid and Liquid Phases

The measured compositions of the solid and liquid phases near the S/L interface and the solute partition coefficients are summarized in Table 2. The solidification path measurements of Alloys No. 1 and 7 produced 47 datasets of the solid and liquid compositions. Nine datasets were obtained by the single measurement. In total, 56 datasets of the solute partition coefficients were efficiently obtained, corresponding to their liquid compositions within two days of beamtime.

Table 2. Summary of measured compositions and solute partition coefficients.

Measurement No.	Measurement procedure	Alloy No.	fS	Composition of the solid and liquid phases near S/L interface								Partition coefficient, k (CS/CL)
				Solid composition, CS (mass%)				Liquid composition, CL (mass%)
				Cr	Ni	Mo	Cu	Cr	Ni	Mo	Cu	Cr	Ni	Mo	Cu
1	Solidification path	1	0.082	19.1	23.8	3.3	1.3	19.8	24.7	4.7	1.5	0.96	0.97	0.70	0.86
2			0.174	19.3	23.4	3.3	1.2	19.6	25.1	4.9	1.2	0.99	0.93	0.68	0.93
3			0.258	19.1	23.9	3.4	1.1	19.5	25.3	5.2	1.4	0.98	0.94	0.66	0.80
4			0.344	19.2	24.7	3.5	1.0	19.3	25.4	5.3	1.4	1.00	0.97	0.67	0.77
5			0.389	19.2	23.8	3.6	1.0	20.1	24.8	5.4	1.3	0.96	0.96	0.67	0.80
6			0.515	18.6	25.3	3.9	1.1	19.1	25.9	5.7	1.2	0.97	0.98	0.68	0.91
7			0.584	19.0	24.4	4.1	1.0	19.3	26.2	6.1	1.2	0.98	0.93	0.68	0.83
8			0.664	18.0	25.2	4.4	0.9	18.9	26.5	6.6	1.2	0.95	0.95	0.67	0.81
9			0.779	19.1	25.5	4.9	0.9	18.8	27.0	7.5	1.0	1.02	0.94	0.65	0.88
10	Solidification path	7	0.060	19.9	23.8	7.4	1.3	19.2	24.2	10.6	1.5	1.03	0.98	0.70	0.85
11			0.062	18.5	24.2	7.6	1.2	19.5	24.4	10.7	1.4	0.95	0.99	0.71	0.84
12			0.068	19.5	24.0	7.2	1.2	19.2	24.4	10.6	1.4	1.02	0.98	0.68	0.85
13			0.077	19.9	24.5	7.4	1.2	19.6	24.4	10.9	1.4	1.01	1.00	0.68	0.84
14			0.090	20.1	23.9	7.4	1.2	19.8	24.3	10.9	1.3	1.01	0.98	0.68	0.91
15			0.105	19.0	24.4	7.4	1.2	19.5	24.3	10.9	1.3	0.97	1.00	0.68	0.92
16			0.128	19.3	24.4	7.5	1.2	19.2	25.0	11.3	1.4	1.01	0.98	0.66	0.86
17			0.168	20.0	24.0	7.9	1.2	19.1	24.4	11.8	1.4	1.05	0.99	0.67	0.85
18			0.251	18.6	24.0	7.9	1.1	19.1	24.9	11.6	1.3	0.97	0.96	0.68	0.86
19			0.262	19.2	24.5	7.9	1.1	19.8	24.0	11.9	1.3	0.97	1.02	0.67	0.88
20			0.281	18.9	24.2	7.9	0.9	19.2	24.2	12.1	1.1	0.99	1.00	0.66	0.84
21			0.303	19.1	24.2	8.1	1.0	18.9	25.1	12.1	1.2	1.01	0.96	0.67	0.82
22			0.316	19.3	24.4	8.3	1.0	19.3	24.6	12.5	1.1	1.00	0.99	0.67	0.89
23			0.348	19.4	24.2	8.4	1.0	18.9	24.8	12.5	1.1	1.03	0.98	0.67	0.93
24			0.382	18.9	24.5	8.3	1.0	18.8	24.8	12.5	1.1	1.01	0.99	0.66	0.89
25			0.407	19.6	24.6	8.3	0.9	18.7	24.6	12.8	1.1	1.05	1.00	0.65	0.87
26			0.444	19.2	24.3	8.5	0.9	18.6	24.5	12.6	1.0	1.03	0.99	0.67	0.87
27			0.452	19.3	24.9	8.3	1.0	19.2	24.9	12.8	1.1	1.01	1.00	0.65	0.91
28			0.456	19.0	24.4	8.3	0.9	19.0	24.9	13.0	1.0	1.00	0.98	0.64	0.90
29			0.468	19.6	24.7	8.6	0.8	18.4	24.8	13.1	1.0	1.07	0.99	0.66	0.85
30			0.476	19.7	24.9	8.4	0.8	19.1	24.8	13.2	1.0	1.03	1.01	0.64	0.87
31			0.483	19.2	24.3	8.4	0.8	19.0	25.2	13.3	0.9	1.01	0.96	0.63	0.90
32			0.492	18.8	24.9	8.6	0.9	19.8	24.9	13.2	1.0	0.95	1.00	0.65	0.88
33			0.499	18.8	24.7	8.7	0.9	18.8	24.9	13.3	1.0	1.00	0.99	0.65	0.88
34			0.506	18.7	25.3	8.4	0.8	18.6	25.8	13.5	1.0	1.00	0.98	0.63	0.83
35			0.517	18.1	24.8	8.7	0.7	18.6	24.7	13.7	0.9	0.97	1.00	0.64	0.81
36			0.525	19.0	24.9	8.8	0.7	18.6	25.0	13.4	0.8	1.02	1.00	0.65	0.90
37			0.533	19.8	24.4	8.7	0.8	18.9	24.8	13.7	0.9	1.04	0.99	0.63	0.88
38			0.542	19.2	24.2	8.6	0.8	19.3	24.8	13.9	1.0	0.99	0.98	0.62	0.81
39			0.553	19.1	24.7	8.9	0.8	18.6	24.6	14.2	0.9	1.03	1.00	0.63	0.96
40			0.570	18.4	25.3	9.0	0.7	18.7	25.0	14.0	0.7	0.98	1.01	0.65	0.99
41			0.594	18.8	25.1	9.1	0.7	18.6	25.4	14.2	0.8	1.01	0.99	0.64	0.88
42			0.606	18.7	25.2	9.3	0.7	18.1	25.3	14.3	0.7	1.03	1.00	0.65	1.00
43			0.615	19.3	24.9	9.4	0.7	17.9	25.0	14.8	0.8	1.08	1.00	0.64	0.86
44			0.645	18.4	25.7	9.6	0.7	18.2	25.3	15.0	0.7	1.01	1.02	0.64	1.09
45			0.656	19.3	25.2	9.6	0.7	18.1	24.9	15.2	0.7	1.07	1.01	0.63	1.00
46			0.668	20.2	25.1	9.9	0.7	18.3	25.3	15.2	0.7	1.10	0.99	0.65	0.96
47			0.683	18.7	25.3	10.1	0.7	18.6	24.8	15.7	0.7	1.00	1.02	0.65	1.06
48	Single	2	0.706	23.7	24.1	3.8	2.6	23.7	25.3	5.9	2.4	1.00	0.95	0.63	1.11
49		3	0.235	26.6	23.8	3.3	2.6	27.9	24.2	4.8	3.0	0.96	0.98	0.69	0.86
50		4	0.589	21.0	29.3	3.6	1.7	21.2	30.0	5.5	1.8	0.99	0.98	0.65	0.90
51		5	0.365	19.8	34.2	3.6	1.7	20.0	34.6	5.1	2.0	0.99	0.99	0.70	0.84
52		6	0.704	20.3	25.6	6.6	2.2	19.2	26.1	9.6	2.5	1.05	0.98	0.69	0.88
53		7※	0.191	21.2	23.6	7.3	1.9	19.8	25.0	10.9	2.2	1.07	0.94	0.67	0.87
54		7	0.433	20.3	25.1	7.3	1.6	19.8	25.2	12.2	1.8	1.03	1.00	0.60	0.90
55		8	0.442	20.9	24.0	3.5	3.5	20.7	25.1	5.2	4.0	1.01	0.96	0.66	0.88
56		9	0.308	19.4	24.4	3.4	5.8	20.2	25.1	4.9	7.0	0.96	0.97	0.70	0.84

※  Two interfaces could be measured in sample No. 7.

The measured compositions of Cr, Ni, Mo and Cu along the solidification path are plotted in Fig. 7, which shows the contents of the solid and liquid phases of Alloy No. 1 (Figs. 7(a) and 7(b), respectively) and the solid and liquid phases of Alloy No. 7 (Figs. 7(c) and 7(d), respectively). The compositions were measured near the S/L interface along the solidification path using X-ray fluorescence analysis. The Cr and Cu contents in both the solid and liquid phases slightly decreased with increasing fs, whereas the Ni contents in both of the phases slightly increase with increasing fs. The Mo content increases with increasing fs, especially in the liquid phase, resulting in a significant concentration of Mo in the residual liquid phase.

Fig. 7.

Cr, Ni, Mo and Cu contents of the (a) solid phase and (b) liquid phase of Alloy No. 1 and (c) solid phase and (d) liquid phase of Alloy No. 7, where the compositions were measured near the S/L interface along the solidification path using X-ray fluorescence analysis.

Figure 7 shows that the Cu and Cr contents of both the solid and liquid phases are lower than the initial contents along the solidification path. This reason is attributed to the fact that Cu and Cr evaporated from the liquid phase under vacuum during the experiments owing to their higher vapor pressures. Generally, an inert gas atmosphere can prevent the evaporation. However, it is mostly impossible to detect X-ray fluorescence of the solutes because the gas significantly decreases the detectability.

In particular for Cu, this phenomenon may lead to a reduction in the accuracy of the measurements, in addition to the lower initial content and the overlapped peaks of the Cu K_α and Ni K_β lines.

3.3. Partition Coefficients

Figure 8 shows the partition coefficients obtained with fs from the datasets of the solid and liquid phases near the S/L interface. The measured partition coefficients of Cr, Ni, Mo and Cu between the solid and liquid phases near the S/L interfaces are compared with the calculated coefficients from Eq. (1) for Alloys No. 1 (a) and No. 7 (b) obtained via the solidification path measurement. The measured coefficients are also compared with the calculated data using the single measurement (c). In addition, Fig. 8(d) shows the previously reported partition coefficient values. Notably, the partition coefficients are available within the liquid compositions of the Fe-17.9–27.9mass% Cr-24.0–34.6mass% Ni-4.7–15.7mass% Mo-0.7–7.0mass% Cu alloy. The solid symbols show the values based on the measured compositions provided in Table 2, while the open symbols show the data calculated by Eq. (1) considering the solute concentration dependence on the partition coefficient. The latter concept is described in the following section in detail.

Fig. 8.

Measured partition coefficients of Cr, Ni, Mo and Cu between the solid and liquid phases near the S/L interfaces with the coefficients calculated by Eq. (1) for Alloy No. 1 (a) and Alloy No. 7 (b) by the solidification path measurements with the data using the single measurement (c). Plot of the partition coefficients previously reported (d).

Focusing on the solid symbols in Fig. 8, the k values of Cr and Ni are slightly lower than 1 along the solidification path of Alloy No. 1, while the corresponding values are close to 1 in Alloy No. 7. The plots in Fig. 8(c) obtained from the single measurement consistently show a similar behavior to that of the solidification path measurement. The k values of Cr vary from 0.95 to 1.10, while that of Ni range from 0.93 to 1.02. Therefore, Cr and Ni do not affect the microsegregation much because the k values are close to 1. In contrast, the k values of Mo vary from 0.60 to 0.71, and are much lower than 1. The k values of Mo gradually decrease along the solidification path, by which the microsegregation is facilitated at the final stage of solidification.

The dispersion of the Cu data is larger than those of the other elements. However, the partition coefficients show the values ranging from 0.8 to 0.9, which increases to above 1 at the final stage of the solidification path. However, the k value of 0.85 is considered to be the most accurate, as seen in Fig. 8(c). This assumes that the probable error caused by the evaporation and the overlap with the Ni peak can be compensated by the larger amounts of Cu contained in Alloys No. 8 and 9.

3.4. Solute Concentration Dependence on Partition Coefficient

The dependency of the solute content in the liquid phase on the solute partition coefficients k_i can be comprehensively expressed by applying the first order Taylor series expansion as follows:   

$$k_i=k_i^0+{\sum }_j^N{\gamma }_{i,j}\left(C_j-C_j^0\right)$$(1)

where $k_i^0$ is the partition coefficient in the standard composition, C_j is the liquid concentration of the solute elements j, $C_j^0$ is the standard solute content of elements j, and γ_i,j is the interaction parameter between the solutes. Therefore, the following equations express the k_i values of each solute:   

$$\begin{array}{c}{k}_{Cr}=k_{Cr}^0+{\gamma }_{Cr,Cr}\left(C_{Cr}-C_{Cr}^0\right)+{\gamma }_{Cr,Ni}\left(C_{Ni}-C_{Ni}^0\right)\\ +{\gamma }_{Cr,Mo}\left(C_{Mo}-C_{Mo}^0\right)+{\gamma }_{Cr,Cu}\left(C_{Cu}-C_{Cu}^0\right)\end{array}$$(2)

  

$$\begin{array}{c}{k}_{Ni}=k_{Ni}^0+{\gamma }_{Ni,Cr}\left(C_{Cr}-C_{Cr}^0\right)+{\gamma }_{Ni,Ni}\left(C_{Ni}-C_{Ni}^0\right)\\ +{\gamma }_{Ni,Mo}\left(C_{Mo}-C_{Mo}^0\right)+{\gamma }_{Ni,Cu}\left(C_{Cu}-C_{Cu}^0\right)\end{array}$$(3)

  

$$\begin{array}{c}{k}_{Mo}=k_{Mo}^0+{\gamma }_{Mo,Cr}\left(C_{Cr}-C_{Cr}^0\right)+{\gamma }_{Mo,Ni}\left(C_{Ni}-C_{Ni}^0\right)\\ +{\gamma }_{Mo,Mo}\left(C_{Mo}-C_{Mo}^0\right)+{\gamma }_{Mo,Cu}\left(C_{Cu}-C_{Cu}^0\right)\end{array}$$(4)

  

$$\begin{array}{c}{k}_{Cu}=k_{Cu}^0+{\gamma }_{Cu,Cr}\left(C_{Cr}-C_{Cr}^0\right)+{\gamma }_{Cu,Ni}\left(C_{Ni}-C_{Ni}^0\right)\\ +{\gamma }_{Cu,Mo}\left(C_{Mo}-C_{Mo}^0\right)+{\gamma }_{Cu,Cu}\left(C_{Cu}-C_{Cu}^0\right)\end{array}$$(5)

The composition of Alloy No. 1 in Table 1 was defined as the standard solute contents ($C_j^0$). The k values at the lowest solid fraction, fs = 0.082 of Alloy No. 1 (Measurement No. 1), listed in Table 2 were applied for the k_i⁰ values. Consequently, the γ_i,j values were derived from a multiple regression analysis of all the k values based on the measured compositions listed in Table 2.

The derived interaction parameters are summarized in Table 3 along with the k_i⁰ values. The open symbols in Fig. 8 show the calculated k values using the interaction parameters. The calculated results of Cr, Ni and Mo are in good agreement with the data based on the compositions. The calculated results of Cu are almost consistent with the data based on the compositions owing to the larger dispersion. Here, the relative errors of the calculated values using the k-value interaction parameters were obtained in each solute element as follows:   

$$\mathrm{\Delta }{k}_i=\frac{k_{i\mathrm{\hspace{0.17em} }\text{exp}}-k_{i\mathrm{\hspace{0.17em} }\text{calc}}}{k_{i\mathrm{\hspace{0.17em} } \text{exp}}}\times 100$$(6)

where Δk_i is the relative error of the partition coefficient of the solute element i, and k_{i exp} and k_{i calc} are the partition coefficients of the experimental and the calculated values, respectively. The histograms of the errors in each solute element are shown in Fig. 9. The solid line shows the fitting curve to the Gaussian function. The full width at half maximum of the fitted curves are also shown. For Cr, Ni and Mo, almost all of the estimated values of the partition coefficients are within 5% in relative error. The histogram of Cu is broader than the others. However, 90% of the estimated values are approximately within 10% relative error; this is an acceptable range for the estimation. The estimation of the partition coefficients using the interaction parameters may provide sufficient accuracy to predict any microsegregation in the solidification process.

Table 3. Solute-solute interaction parameters defined by Eq. (1).

Element i	k0i	γi,Cr	γi,Ni	γi,Mo	γi,Cu
Cr	0.96	−0.003	0.004	0.007	0.007
Ni	0.97	0.002	−0.001	0.003	−0.002
Mo	0.70	−0.004	−0.003	−0.006	0.000
Cu	0.86	0.012	0.001	0.006	−0.008

Fig. 9.

Histograms of relative errors of estimated k values using Eq. (6). (Online version in color.)

3.5. Comparison of Partition Coefficients with Previously Reported Values

Figure 8(d) shows the previously reported partition coefficients obtained by the conventional ex-situ methods. Yamamoto et al.⁶⁾ measured the k values by two methods using a Fe–20mass% Cr–25mass% Ni–4.5mass% Mo–1.5mass% Cu alloy, which is equivalent to the composition of Alloy No. 1. One method involved the values of k_E targeting equilibrium coefficients obtained by iso-thermal experiments held at the state where the solid and liquid phases coexisted. The other method involved the k_R values determined by a random sampling method. Morita et al.⁷⁾ measured the k values using a multi-component iron alloy of Fe–20mass% Cr–24mass% Ni–1.3mass% Si–1.0mass% Mn–1.0mass% Mo. The k_M.C. values were determined by a method similar to that used for determining the k_E values.

The data of the solidification path of Alloy No. 1 were compared with the previous results from Yamamoto et al.⁶⁾ and Morita et al.⁷⁾ because of the similar alloy compositions. The k_E values of Cr and Mo tend to be lower than the k values in this study, while the k_E values of Ni tend to be higher. The k_E value of Cu lies within the range of the k values in this study. The k_R values are closer to the k values in this study than the k_E values, except for Mo. Comparing the k_M.C. with the k values of this study, the k_M.C. values of Cr, Ni and Mo lie within the range of the k values of this study.

As stated above, the k values obtained in this study mostly agree with the previous values shown in Fig. 8(d) within 0.1 in the difference. The reason of the difference of this extent is not fully understood at this moment.¹³⁾

3.6. Calculated Concentration Distributions

Using the interaction parameters in Table 3, the k values are applied to calculate the concentrations of the solutes. The highest value of f_S is 0.779, with which the k values are considered available up to 0.8. Therefore, the distributions higher than 0.8 in f_S are calculated with the k values at 0.8 in f_S as obtained by Eq. (1).

Scheil’s model was primarily applied for the calculation to understand how well the k values determined in this study could estimate the segregation without considering the other factors of the S/L interfacial area and diffusion of the solutes. The calculated lines are compared with the previous data measured by EDS,⁶⁾ as shown in Fig. 10, where the measured data are normalized by the initial composition (C_s/C₀) of Alloy No. 1. The measurements are normalized because the data taken by EDS have a measurement error to some extent owing to factors that include the spot size, the way to convert the peaks to the contents and how to treat the background of the spectra. It is obvious in Fig. 10 that the plots agree well with the calculations, particularly in the region of fs below 0.8, which proves the accuracy of the in-situ measurement by fluorescence irradiation. Focusing carefully on the fs > 0.8 region, the plots of Ni, Mo, and Cu pass below the lines, while those of Cr pass above the lines. This reason has to be clarified because segregation is determined at the later stage of solidification. The probabilities are estimated from both sides of the calculation and the measurement by EDS.⁶⁾

Fig. 10.

Calculated concentration distributions of the solutes are compared with the previous data plots experimentally measured by EDS. (Online version in color.)

Scheil’s model has the characteristic of which the concentration of a solute diverges at fs = 1, giving a sharp increase at the high fs region. This is particularly true for the f_S region higher than 0.8. This fact shows the requirement to create a model with the three-dimensional concept considering the evolution of the interfacial area with f_S⁹⁾ taking diffusion of the solutes into account.

From the aspect of the measurement, it is important to understand how accurate the EDS analyses are. The resolution of a point analysis by EDS is typically 1 to 2 μm in diameter as pointed out earlier. This implies that a plot provides averaged compositional information of this area. Focusing on the interdendritic region, a microsegregation of sub-micron order is probable depending on the cooling rate. They do not reflect the actual concentration profiles from the conventional way, despite taking as much data as possible by the random sampling method.⁶⁾ In addition, focusing on the fs = 1 position, the plots marked as “*” contain larger amounts of Cr and Mo, and smaller amounts of Ni and Cu. This implies that intermetallic compounds may be formed at the interdendritic regions. The beam could impinge the position where the matrix and the compound both exist, which can result in an overestimate for the solute contents of Cr and Mo and an underestimate for the Ni and Cu contents. This could plausibly result in the slight difference between the solute distribution profiles and the calculations in the region of fs higher than 0.8. Thus, the measurement should be carried out with the resolution as high as possible.

4. Conclusions

The compositions of the solid and liquid phases of the Fe-17.9–27.9mass% Cr-24.0–34.6mass% Ni-4.7–15.7mass% Mo-0.7–7.0mass% Cu alloys were measured during solidification by in-situ measurements using X-ray transmission imaging and X-ray fluorescence spectroscopy (XRF). The X-ray fluorescence spectra of the composed elements close to the interface were measured by EDS. The following conclusions summarize this study:

(1) The solid/liquid interface was clearly identified by the X-ray transmission image.

(2) The partition coefficients could be efficiently determined in reference to the various liquid compositions.

(3) It was found that the partition coefficient of Mo gradually decreased from 0.7 to 0.6 along the unidirectional solidification path.

(4) The partition coefficients of Cr and Ni were closed to 1, indicating they did not segregate much.

(5) The partition coefficients of Cu were dispersed in the range from 0.8 to 0.9 owing to its lower content amount in the alloy compared with other metals, evaporation that occurred during the experiments and overlap with the Ni peaks.

(6) Interaction parameters between the solutes were obtained by a regression analysis of the partition coefficients that was determined on the basis of the composition measurements.

(7) The calculated results of the k values of Cr, Ni and Mo agreed well with the k values determined on the basis of the composition measurements.

(8) It was suggested that the experiments should be carried out under modified conditions that prevent the vaporization of Cr and Cu to obtain more accurate k values.

Acknowledgments

The synchrotron radiation experiments were performed as a General Proposal for Industrial Application (2017B1581 and 2018A1586) at the beamline BL20B2 in SPring-8 with approvals of the Japan Synchrotron Radiation Research Institute (JASRI). The authors are truly grateful to JASRI for the acceptance to use the beamline.

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