Self-diffusion in iron-based Fe–Mo alloys

Acta Materialia 54 (2006) 2833–2847 www.actamat-journals.com

Self-diffusion in iron-based Fe–Mo alloys Hiroyuki Nitta *, Kensuke Miura 1, Yoshiaki Iijima

2

Department of Materials Science, Graduate School of Engineering, Tohoku University, Aoba-yama 6-6-11, Sendai 980-8579, Japan Received 23 November 2005; received in revised form 16 February 2006; accepted 16 February 2006 Available online 18 April 2006

Abstract Tracer diffusion coefficients of 59Fe and 99Mo in high-purity iron–molybdenum alloys containing up to 1.8 at.% molybdenum have been determined in the temperature range 823–1173 K by use of a serial sputter-microsectioning technique. The diffusion coefficient of iron (solvent) increases with increasing molybdenum content, whereas the diffusion coefficient of molybdenum (solute) decreases. The influence of magnetic transformation on the diffusion of iron in the iron–molybdenum alloys decreases with increasing molybdenum content, while that on the diffusion of molybdenum in the alloys is a maximum at about 0.8 at.% molybdenum. Analysis of jump frequency ratios shows that a vacancy is weakly bound by a molybdenum atom.  2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Iron alloys; Bulk diffusion; Magnetic properties

1. Introduction It is very important to use fossil fuels with as high an efficiency as possible. Thermal power stations are required to operate at higher temperatures and higher pressures. Excellent thermal fatigue properties are essential for the materials used in thermal power stations because of the severe heat cycles during operations. Improvements in the structural materials for these applications have been attempted for many years. Ferritic iron–molybdenum alloys are some of the promising candidates because a small addition of molybdenum to iron is very effective in improving the creep strength [1] and because the thermal expansion of body-centered cubic (bcc) ferritic alloys is small, being an advantage in terms of thermal fatigue.

* Corresponding author. Present address: Institute for Materials Research, Tohoku University, Sponsored Division, Katahira 2-1-1, Sendai 980-8577, Japan. Tel.: +81 22 215 3463; fax: +81 22 215 3465. E-mail address: [email protected] (H. Nitta). 1 Present address: Nippon Yakin Kogyo Co. Ltd., Kawasaki 210-0861, Japan. 2 Present address: Department of Materials Science and Engineering, Faculty of Engineering, Iwate University, Morioka 020-8551, Japan.

Atomic movement plays a key role in creep deformation at high temperatures because the creep rate is basically proportional to the diffusion coefficient in metals. Thus, precise experimental data for the diffusion of components in the alloys are essential for evaluating the suitability of the alloys at high temperatures. In particular, knowledge of the solute dependence of the diffusivities is very important for improving the performance of the alloys. However, the appropriate diffusion data for the alloys are quite sparse [2–4]. The tracer diffusion coefficients in binary Fe–Mo alloys have been measured by Million and Kucˇera [5] and by Nohara and Hirano [6] in the temperature range 1000– 1600 K using the residual activity method. To use ferritic iron alloys in high-efficiency thermal power station applications, diffusion data determined experimentally around 900 K are necessary. Furthermore, diffusion data for dilute alloys containing less than 2 at.% solute are necessary to evaluate the dislocation velocity in the solid solution strengthening mechanism. To eliminate the influence of impurities on these evaluations, high-purity alloys should be used for the diffusion measurements. The precise determination of the diffusion coefficient of an atom in a solid is commonly made using the radioactive

1359-6454/$30.00  2006 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2006.02.020

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H. Nitta et al. / Acta Materialia 54 (2006) 2833–2847

tracer method in combination with an appropriate serial sectioning technique to establish the penetration profiles of the diffusing tracer in the specimen [7]. It is also essential to use, as far as possible, high-purity single-crystal specimens with stress free surfaces. Alternatively, specimens with large grain size should be employed. Otherwise, apparent diffusion coefficients affected by impurities and short-circuiting diffusion paths such as dislocations and grain boundaries are recorded. The influence of shortcircuit paths on diffusion is more pronounced at lower temperatures [4,8,9]. Therefore, the diffusion coefficient calculated from the diffusion parameters measured at high temperatures should not be used to estimate the characteristics of iron alloys such as the creep lifetime at intermediate temperatures. To analyze submicrometer penetration profiles for diffusion at low temperatures, a serial sputtermicrosectioning technique is indispensable [7–10]. It is well known that the temperature dependence of the diffusion coefficient in iron and its alloys below the Curie temperature deviates downwards from the Arrhenius relationship extrapolated from the paramagnetic state [4,6,7]. The temperature dependence of the diffusion coefficient, D, in the whole temperature range across the Curie temperature is well described by [11] D ¼ Dp0 exp½Qp ð1 þ as2 Þ=RT  Dp0

ð1Þ

p

and Q are, respectively, the preexponential facwhere tor and the activation energy for the diffusion in the paramagnetic state. The value of s, the ratio of the spontaneous magnetization at T K to that of 0 K, for pure iron has been experimentally determined by Potter [12] and Crangle and Goodman [13]. The constant a, expressing the extent of the influence of the magnetic transformation on diffusion, is given by a ¼ ðaf þ am Þ=Qp

ð2Þ

where af and am are the increments of formation energy and migration energy, respectively, of a vacancy by the magnetic transformation [11]. Determination of the value of a is useful for estimating the reduction of diffusivity at a temperature in the ferromagnetic region. For substitutional solute diffusion in a bcc metal via the monovacancy mechanism, two different jump frequency models I and II, have been proposed [14,15]. The jump behavior of a vacancy adjacent to a solute atom is characterized by several jump frequencies. In model I, the interactions between a solute atom and a vacancy adjacent to it in first and second neighbor positions are taken into account, while in model II only the nearest-neighbor interaction between the solute and the vacancy is taken into account. At the present time, the individual jump frequencies cannot be determined experimentally, but in each model two jump frequency ratios, x2 =x03 and x3 =x03 in model I and x2/x3 and x4/x0 in model II, can be estimated using a set of any two of the four experimentally determinable values D2/D0, b1, f2, and G, where D2/D0 is the ratio of the solute diffusion coefficient to the solvent self-diffusion coefficient,

b1 is the linear enhancement factor for solvent diffusion by solute addition, f2 is the correlation factor for solute diffusion, and G is the vacancy flow factor. Using experimental data for D2/D0 and b1 obtained in the present work, jump frequency ratios can be estimated, which are important information as regards solute behavior in the solvent metal. The purpose of this work is to study the diffusion behavior of iron and molybdenum in high-purity dilute Fe–Mo alloys over a wide range of temperature across the Curie temperature by use of a sputter-microsectioning technique and 59Fe and 99Mo radioactive tracers. 2. Experimental High-purity Fe–Mo alloys were prepared as follows. (i) High-purity electrolytic iron (4 N) and high-purity electrolytic molybdenum (4 N) were induction-melted in a cold copper crucible at 2 · 104 Pa after evacuation of the chamber to a base pressure less than 107 Pa. The concentrations of molybdenum in the alloys were 0.4 and 1.5 at.%, designated as MM0.4 and MM1.5. (ii) Although the melting method and the source of electrolytic molybdenum were the same as those of the MM alloys, a different source of high-purity electrolytic iron (4 N) was used to produce alloys AM0.4 and AM1.0, containing 0.4 and 1.0 at.% molybdenum, respectively. (iii) High-purity electrolytic iron (4 N) and electrolytic molybdenum (2N5) were made by vacuum induction melting and the concentrations of molybdenum in the alloys were 0.7 and 1.8 at.%. These alloys were designated as KM0.7 and KM1.8. The ingots were hot-forged and machined in the form of rods of 12 mm in diameter. The results of the chemical analyses of these materials are given in Table 1. The rods were electropolished at 273 K in an aqueous solution containing 77 vol.% acetic acid and 18 vol.% perchloric acid. To induce grain growth and reduce the contents of carbon, nitrogen, and oxygen, the rods were annealed at 1723 K for 36 h in a stream of hydrogen gas purified by permeation through a palladium tube at 700 K [16]. The resultant grain size was about 3–5 mm. After grain growth, the rods were cut to make disk specimens 1 mm in thickness. The specimens were ground using abrasive papers and polished with fine alumina paste and electropolished under the same conditions as mentioned above. To obtain a stress-free surface, the specimens were further annealed at 1173 K for 7.2 ks in a stream of the purified hydrogen gas. Table 1 Chemical analysis of Fe–Mo alloy (ppm by mass) C

N

O

P

MM alloy 16 5

18

3

4

<1

6

3

AM alloy <30 <8

62

3

<5

<100

<100

<100

KM alloy 6

18

3

1

2

6

2

6

S

Cr

Ni

Si

H. Nitta et al. / Acta Materialia 54 (2006) 2833–2847

The radioisotope 59Fe (c-rays, 1.095 and 1.292 MeV; half-life, 45.6 d) in the form of FeCl3 in 0.5 kmol/m3 HCl solution was supplied by New England Nuclear Corporation, USA. The radioisotope 99Mo (b-rays, 17% 0.448 MeV and 82% 1.234 MeV; half-life, 65.84 h) was supplied in the form of Na2MoO4 in 0.11 kmol/m3 NaOH solution by the Japan Atomic Energy Research Institute. Taking care to avoid oxidation of the specimens, the radioisotopes were electroplated on the surface of the specimens with a current density of 15–25 mA/cm2 for 0.5–4.0 min. The specimens were then diffused in quartz tubes at 104 Pa at a temperature in the range 823–1173 K for 1.80–508.80 ks in furnaces preset at the desired temperatures and controlled to within ±0.5 K. To measure the penetration profiles of the diffusing species into the specimen, three types of serial sputter-microsectioning apparatus were used: an ion beam type with a sputtering rate of 7–10 nm/min and a radio-frequency type with a sputtering rate of 30– 60 nm/min were used to measure short penetration profiles and a magnetron type with a sputtering rate of 150– 300 nm/min was used to measure long penetration profiles. The details of the methods are described elsewhere [10,17,18]. For each specimen, 15–35 successive sections were sputtered. A constant fraction of the sputtered-off material was collected on an aluminum foil. The intensity of the radioactivity of the isotope in each section was measured using a well-type Tl-activated NaI detector with a 1024-channel pulse height analyzer. The radioactive diffusant 99Mo is a b emitter which decays through the reaction b ;2:1105 y 99 b ;66h 99m c;6h 99 99 ! 44 Ru 42 Mo ! 43 Tc ! 43 Tc 

via the radioactive daughter 99mTc (c-rays, 80% 0.142 MeV; half-life, 6.01 h). For reasons of efficiency the 0.142 MeV c peak of 99mTc was counted. After a sufficiently long time gap after the diffusion anneal (at least 10 half-lives of 99mTc), 99mTc activity is proportional to the concentration of 99Mo [19,20]. 99Ru decay product on the final count is negligible because the NaI detector used for counting is programmed to record the c-rays coming from 99mTc in a selective manner. The b-rays coming from all other isotopes are easily absorbed and do not influence the counts.

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before the diffusion. For very shallow sectioning, distortion of penetration profiles due to the measuring process is sometimes observed. The sputter-sectioning condition of the present experiments was examined for the concentration–depth profile of 99Mo from undiffused a-Fe. As shown in Fig. 1A, the intensity of 99Mo decreases drastically at a very near-surface region in an undiffused specimen. Penetration profiles for 99Mo in a-Fe at 828 and 863 K from previous work [20] are also shown in Fig. 1B. Figs. 1A and B show that the influence of distortion due to the measuring process is negligible. Figs. 2–4 show representative plots of ln I(X, t) vs. X2 for the diffusion of 59Fe and 99Mo in MM0.4, AM1.0, and KM1.8 alloys, respectively. In accordance with Eq. (3), the main slopes of all the plots in Figs. 2–4 are linear, indicating that the concentration profiles are governed by volume diffusion. The diffusion coefficients calculated from the slopes of the plots are listed in Tables 2–4. The experimental error in the diffusion coefficients is less than 3%. Figs. 5 and 6 show the molybdenum concentration dependence of the solvent (59Fe) and solute (99Mo) diffusion coefficients, respectively. As shown, the values of both the solvent and solute diffusion coefficients are different among MM, AM, and KM alloys. The diffusion coefficients of both 59Fe and 99Mo in MM alloys are about 50% smaller than those in AM and KM alloys at 1073 K. The differences in diffusion coefficients are significant even if one takes the experimental error into consideration. As shown in Fig. 5, the diffusion coefficients of 59Fe in pure iron and MM alloys increase linearly with the concentration of molybdenum in the temperature region of the paramagnetic state. In contrast to Fig. 5, the diffusion coefficients of 99Mo in pure iron and MM alloys shown in Fig. 6 decrease linearly with the concentration of molybdenum for the same temperatures. However, the concentration dependence of the diffusion coefficients of both 59 Fe and 99Mo in AM and KM alloys are not obvious.

(B)

(A)

873K 1.5ks

3. Results For one-dimensional volume diffusion of a tracer from an infinitesimally thin surface layer into a sufficiently long rod, the solution of Fick’s second law is given by h pffiffiffiffiffiffiffiffii IðX ; tÞ / CðX ; tÞ ¼ M= pDt expðX 2 =4DtÞ ð3Þ where I(X, t) and C(X, t) are, respectively, the intensity of the radioactivity and the concentration of the tracer at a distance X from the original surface after a diffusion time t. D is the volume diffusion coefficient of the tracer and M is the total amount of tracer deposited on the surface

838K 85.5ks

0

2

4

6

X / 10 -8m

8

0

20

40

60

80

100

120

140

X 2 / 10 -16m2

Fig. 1. (A) Concentration–depth profile of 99Mo from an undiffused a-iron. (B) Concentration–depth profiles of 99Mo diffused into a-iron.

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H. Nitta et al. / Acta Materialia 54 (2006) 2833–2847

Fig. 2. Examples of penetration profiles for diffusion of (A)

59

Fig. 3. Examples of penetration profiles for diffusion of (A)

59

Fe and (B)

99

Fig. 4. Examples of penetration profiles for diffusion of (A)

59

Fe and (B)

99

Fe and (B)

99

Mo in Fe–0.4 at.% Mo alloy (MM0.4).

Mo in Fe–1.0 at.% Mo alloy (AM1.0).

Mo in Fe–1.8 at.% Mo alloy (KM1.8).

H. Nitta et al. / Acta Materialia 54 (2006) 2833–2847

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Table 2 Diffusion coefficients of iron and molybdenum in MM alloys Composition (at.%) Iron diffusion Fe–0.4Mo

Fe–1.5Mo

Molybdenum diffusion Fe–0.4Mo

Fe–1.5Mo

Temperature, T (K)

Diffusion time, t (ks)

Diffusion coefficient, D (m2/s)

Techniquea

1173 1143 1123 1088 1073 1053 1038 1023 973 953 923 873 823

1.92 2.10 2.10 1.80 1.80 3.30 61.20 65.40 57.30 326.28 11.46 252.00 427.80

2.40 · 1015 1.23 · 1015 7.78 · 1016 3.56 · 1016 1.96 · 1016 1.22 · 1016 4.81 · 1017 2.65 · 1017 3.64 · 1018 9.73 · 1019 2.45 · 1019 1.38 · 1020 9.49 · 1022

MS RF RF RF RF RF RF RF RF MS IBS IBS IBS

1173 1153 1120 1113 1103 1088 1073 1063 1053 1033 1023 993 953 923 903 873 858 823

3.60 2.70 1.80 2.40 2.40 4.98 2.28 3.00 5.40 54.60 5.46 1.80 1.80 72.60 83.94 246.00 328.20 402.90

2.70 · 1015 2.03 · 1015 9.16 · 1016 7.92 · 1016 6.48 · 1016 3.78 · 1016 2.56 · 1016 1.90 · 1016 1.61 · 1016 6.05 · 1017 3.70 · 1017 8.56 · 1018 1.15 · 1018 2.22 · 1019 1.07 · 1019 1.92 · 1020 1.22 · 1020 1.19 · 1021

MS MS RF RF RF MS RF RF MS RF RF IBS IBS IBS IBS IBS IBS IBS

1143 1123 1104 1089 1073 1063 1053 1044 1034 1023 1003 983 973 963 953 923 903 887 873 863 843 834

2.10 2.10 2.70 1.80 1.80 5.40 3.30 3.72 61.20 65.40 63.00 57.30 13.20 1.80 8.10 7.80 65.10 63.00 12.60 67.20 77.10 82.20

1.80 · 1015 1.14 · 1015 6.62 · 1016 4.52 · 1016 2.65 · 1016 1.99 · 1016 1.49 · 1016 1.07 · 1016 6.69 · 1017 3.05 · 1017 1.22 · 1017 5.50 · 1018 3.43 · 1018 2.23 · 1018 1.43 · 1018 3.13 · 1019 1.31 · 1019 3.93 · 1020 2.07 · 1020 1.26 · 1020 4.75 · 1021 2.79 · 1021

RF RF RF RF RF RF RF RF RF RF RF RF IBS IBS IBS IBS IBS IBS IBS IBS IBS IBS

1120 1103 1088 1073 1063

1.62 1.80 1.50 1.80 3.60

7.34 · 1016 5.19 · 1016 2.91 · 1016 2.19 · 1016 1.48 · 1016

RF RF RF RF RF (continued on next page)

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H. Nitta et al. / Acta Materialia 54 (2006) 2833–2847

Table 2 (continued) Composition (at.%)

Temperature, T (K)

Diffusion time, t (ks)

Diffusion coefficient, D (m2/s)

Techniquea

1053 1040 1033 1004 984 970 953 923 903 873 858 838

1.80 6.60 3.00 69.00 1.80 4.50 65.40 10.80 14.40 65.88 70.80 85.50

1.13 · 1016 6.52 · 1017 4.84 · 1017 1.09 · 1017 4.60 · 1018 2.81 · 1018 1.28 · 1018 2.97 · 1019 1.02 · 1019 1.95 · 1020 8.97 · 1021 3.62 · 1021

RF RF RF RF IBS IBS RF IBS IBS IBS IBS IBS

Furthermore, the diffusion coefficients of 99Mo in AM and KM alloys are larger than that of 99Mo in pure iron, although those in MM alloys are smaller than that in pure iron. Figs. 7 and 8 show Arrhenius plots of the diffusion coefficients of 59Fe and 99Mo in MM0.4, AM1.0, and KM1.8 alloys. The Arrhenius lines of the self-diffusion coefficient [8,9,21] and of the diffusion coefficient of Mo [20] in a-Fe are also shown. The Arrhenius plots of both components in the alloys curve at the Curie temperature because the diffusion is suppressed in the ferromagnetic region. However, above the Curie temperature the plots show linearity without any sign of the influence of short-range magnetic spin ordering on the diffusion. The activation energy, Qp, and the preexponential factor, Dp0 , for diffusion in the paramagnetic state were calculated and are listed in Table 5. Fig. 9 shows the solute concentration dependence of Qp and Dp0 for the diffusion of iron in Fe–Mo alloys. Both the activation energy and the preexponential factor increase linearly with increasing molybdenum content. A similar tendency is also observed for the diffusion of molybdenum in the alloys, as shown in Fig. 10, although the values of both Qp and Dp0 for the diffusion of molybdenum are larger than those for the diffusion of iron in the alloys. It is noted that, in both Figs. 9 and 10, the data points for AM and KM alloys for Qp lie on the linear line drawn for MM alloys, but those for Dp0 are above the linear line. The gap between the data points is shown by a dotted line. This suggests that some impurities contained in AM and KM alloys enhance the diffusion but the influence on the value of the activation energy is negligible. In Figs. 7 and 8, the Arrhenius plots of the diffusion coefficients of 59Fe and 99Mo below the Curie temperature deviate downwards owing to magnetic spin ordering. The temperature dependence of the diffusion coefficient for the whole temperature range across the Curie temperature is well described by Eq. (1). To evaluate the value of a, Eq. (1) can be rewritten as follows:  p DðT Þ Qpcal aQ T ln P ¼   ð4Þ s2 R R D0

Substituting the values of D(T) and Dp0 obtained in the present work and the empirical value of s(T) for pure iron obtained by Potter [12] and Crangle and Goodman [13] into Eq. (4), we can obtain a plot of T ln½DðT Þ=Dp0  as a function of s2. Examples for MM0.4, AM1.0, and KM1.8 alloys are shown in Fig. 11. From the slope and intercept of the linear line, a and Qpcal can be obtained. The values of a and Qpcal thus determined are listed in Table 5. The recalculated value of Qpcal is in good agreement with Qp determined directly from the Arrhenius plot of D(T) in the paramagnetic state. The activation energy Qf(=(1 + a)Qp) for the ferromagnetic state is also listed in Table 5. Using the values of a, Dp0 , and Qp obtained in the present work, D(T) for MM, AM, and KM alloys were calculated, and their Arrhenius lines are shown as the solid lines in Figs. 7 and 8. They are in good agreement with the experimental points. The molybdenum concentration dependence of a on the diffusion of 59Fe and 99Mo is shown in Fig. 12. The value of a for the diffusion of iron decreases linearly with increasing molybdenum content, whereas the value of a for the diffusion of molybdenum is about half of that for the diffusion of iron and shows a maximum at about 0.8 at.% molybdenum. 4. Discussion 4.1. Diffusional feature of Fe–Mo alloys Diffusion coefficients in AM and KM alloys are higher than in MM alloys, as seen in Figs. 5 and 6. Furthermore, it seems that the diffusion coefficients of iron and molybdenum in AM and KM alloys do not show obvious molybdenum concentration dependence as compared with MM alloys. This result is unexpected, because both MM and AM alloys were made of high-purity electrolytic iron and high-purity electrolytic molybdenum using the same melting method, although two different sources of electrolytic iron were used. Furthermore, no particularly high concentration of impurity in the rods

H. Nitta et al. / Acta Materialia 54 (2006) 2833–2847

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Table 3 Diffusion coefficients of iron and molybdenum in AM alloys Composition (at.%) Iron diffusion Fe–0.4Mo

Fe–1.0Mo

Molybdenum diffusion Fe–0.4Mo

Fe–1.0Mo

a

Temperature, T (K)

Diffusion time, t (ks)

Diffusion coefficient, D (m2/s)

Techniquea

1173 1138 1123 1103 1073 1052 1023 993 973 923 898 873 823

2.46 2.46 2.70 3.60 3.00 7.20 14.52 89.40 355.06 10.80 61.38 257.10 257.10

3.44 · 1015 1.55 · 1015 1.02 · 1015 6.90 · 1016 3.00 · 1016 1.73 · 1016 3.42 · 1017 8.08 · 1018 2.98 · 1018 3.36 · 1019 7.61 · 1020 2.12 · 1020 2.20 · 1021

MS MS MS MS MS MS MS MS MS IBS IBS IBS IBS

1168 1143 1123 1103 1085 1073 1035 1023 973 956 903 873 823

2.70 1.80 1.80 1.80 3.00 2.70 2.70 16.20 3.36 2.40 58.8 65.40 508.18

3.25 · 1015 2.10 · 1015 1.12 · 1015 6.50 · 1016 4.68 · 1016 3.62 · 1016 7.95 · 1017 4.40 · 1017 3.62 · 1018 1.59 · 1018 1.87 · 1019 2.33 · 1020 1.95 · 1021

MS RF RF RF RF RF RF MS RF IBS IBS IBS IBS

1155 1133 1113 1093 1073 1053 1033 923 903 853

1.80 2.10 2.40 2.70 2.58 1.80 2.46 3.60 5.40 69.00

4.42 · 1015 2.01 · 1015 1.292 · 1015 7.64 · 1016 4.32 · 1016 2.21 · 1016 9.73 · 1017 4.26 · 1019 2.86 · 1019 1.21 · 1020

MS MS MS MS MS MS MS IBS IBS IBS

1143 1123 1111 1103 1086 1073 1063 1053 1034 1023 1012 995 953 923 913 903 873 853 833

1.80 1.80 1.62 1.80 1.80 1.80 1.80 2.40 4.20 12.60 57.60 65.40 66.00 7.02 14.40 14.40 77.40 86.40 85.02

3.07 · 1015 1.67 · 1015 1.27 · 1015 1.05 · 1015 6.02 · 1016 4.55 · 1016 3.26 · 1016 2.02 · 1016 1.05 · 1016 5.72 · 1017 2.73 · 1017 1.63 · 1017 2.06 · 1018 4.77 · 1019 2.82 · 1019 1.77 · 1019 3.60 · 1020 1.08 · 1020 3.71 · 1021

RF RF RF RF RF RF RF RF RF RF RF RF RF IBS IBS IBS IBS IBS IBS

MS, magnetron sputter-microsectioning; RF, radio-frequency sputter-microsectioning; IBS, ion-beam sputter-microsectioning.

of MM and AM alloys was found in the chemical analysis, as shown in Table 1, although the method of analysis for AM alloy was less quantitative than that for

MM alloy because some substitutional impurities are estimated to be less than the minimum limit of analysis (<100 ppm). Additionally, the concentration of oxygen

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H. Nitta et al. / Acta Materialia 54 (2006) 2833–2847

Table 4 Diffusion coefficients of iron and molybdenum in KM alloys Composition (at.%) Iron diffusion Fe–0.7Mo

Fe–1.8Mo

Molybdenum diffusion Fe–0.7Mo

Fe–1.8Mo

a

Temperature, T (K)

Diffusion time, t (ks)

Diffusion coefficient, D (m2/s)

Techniquea

1173 1143 1123 1103 1073 1053 1033 1023 993 973 953 923 873 848 823

1.92 2.40 3.30 3.72 4.80 7.20 3.60 14.40 259.20 73.44 326.28 20.10 252.00 508.80 501.18

3.10 · 1015 1.53 · 1015 8.02 · 1016 5.12 · 1016 2.65 · 1016 1.54 · 1016 8.23 · 1017 3.38 · 1017 1.04· 1017 4.20 · 1018 1.56 · 1018 3.10 · 1019 2.12 · 1020 7.33 · 1021 1.67 · 1021

MS MS MS MS MS MS MS MS MS RF MS IBS IBS IBS IBS

1173 1153 1112 1103 1086 1071 1063 1051 1033 1023 993 973 956 903 873 848

2.10 2.70 1.50 1.80 1.80 1.80 4.80 3.60 3.60 14.40 259.20 230.70 2.40 96.42 68.40 508.80

5.32 · 1015 2.58 · 1015 1.18 · 1015 9.15 · 1016 6.09 · 1016 3.62 · 1016 2.77 · 1016 2.09 · 1016 8.69 · 1017 5.90 · 1017 1.35 · 1017 4.49 · 1018 1.50 · 1018 1.81 · 1019 2.82 · 1020 9.12 · 1021

MS MS RF RF RF RF MS RF MS MS MS MS IBS MS IBS IBS

1153 1133 1113 1093 1073 1053 1043 1033 1003 973 923 873 833

1.86 2.40 2.16 2.46 2.40 2.40 56.82 2.70 57.60 59.40 5.40 56.82 64.98

2.82 · 1015 2.54 · 1015 1.23 · 1015 8.03 · 1016 3.71 · 1016 1.59 · 1016 1.50 · 1016 8.41 · 1017 1.87 · 1017 4.73 · 1018 3.60 · 1019 4.63 · 1020 2.50 · 1021

MS MS MS MS MS MS RF RF RF RF IBS IBS IBS

1155 1133 1113 1093 1073 1043 1033 973 923 873 853

1.86 2.40 2.16 2.46 2.40 3.30 2.70 59.40 5.40 14.40 64.98

2.87 · 1015 1.64 · 1015 7.74 · 1016 4.20 · 1016 2.62 · 1016 1.21 · 1016 6.18 · 1017 3.70 · 1018 3.85 · 1019 3.04 · 1020 1.02 · 1020

MS MS MS RF MS RF RF RF IBS IBS IBS

MS, magnetron sputter-microsectioning; RF, radio-frequency sputter-microsectioning; IBS, ion-beam sputter-microsectioning.

in AM alloy is 62 ppm which is higher than the 18 ppm in MM alloy. However, further examination of the specimens by energy dispersive X-ray analysis did not show

any peak except for iron and molybdenum. Furthermore, the method of preparation of KM alloys has two differences as compared with that of MM alloys. One is the

H. Nitta et al. / Acta Materialia 54 (2006) 2833–2847

-15

DMo / m2s-1

10

2841

1103K

1073K

1053K 10

-16

0.0

0.5

1.0

1.5

2.0

nMo / at% Fig. 5. Molybdenum concentration dependence of DFe in high-purity iron–molybdenum alloys.

Fig. 6. Molybdenum concentration dependence of DMo in high-purity iron–molybdenum alloys.

use of less pure molybdenum, and the other is the method of melting under a low vacuum at a base pressure. However, no particular impurity in three alloys was found in the chemical analysis as shown in Table 1. In conclusion, the difference in the diffusion coefficients among the three alloys is most likely caused by different sources of materials and melting methods. Therefore, it seems that the diffusion coefficients obtained for AM

and KM alloys are enhanced by some impurities, although the impurities have not been identified. The activation energy and the preexponential factor for solvent iron diffusion in the paramagnetic state increase linearly with increasing molybdenum, as shown in Fig. 9. A similar tendency is also observed for the diffusion of molybdenum in Fig. 10. Furthermore, the values of both Qp and Dp0 for the diffusion of molybdenum are larger than

Fig. 7. Examples of Arrhenius plots of diffusion coefficient of iron in (A) MM0.4, (B) AM1.0, and (C) KM1.8 alloys. Arrhenius line for self-diffusion in a-iron is also shown.

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Fig. 8. Examples of Arrhenius plots of diffusion coefficient of molybdenum in (A) MM0.4, (B) AM1.0, and (C) KM1.8 alloys. Arrhenius line for diffusion of molybdenum in a-iron is also shown.

Table 5 Preexponential factor Dp0 , activation energies, Qp, and parameters a for diffusion of iron and molybdenum in iron–dilute molybdenum alloys Alloy

Dp0 (m2/s)

Qp (kJ/mol)

Qpcal ðkJ=molÞ

a

Qf = Qp(1 + a) (kJ/mol)

Iron diffusion MM0.4

4 4:56þ6:34 2:65  10

253.3 ± 8.0

253.7

0.153 ± 0.008

292.1

4 5:74þ6:53 3:05  10 þ4:55 6:322:64  104 4 7:79þ11:4 4:63  10 þ8:49 4:713:03  104 4 11:49þ26:3 8:00  10

253.3 ± 7.0

253.8

0.151 ± 0.008

291.5

253.0 ± 5.0

253.4

0.158 ± 0.008

293.0

254.1 ± 8.3

254.1

0.148 ± 0.013

291.6

251.8 ± 9.6

251.5

0.145 ± 0.01

288.5

256.1 ± 10.0

256.3

0.146 ± 0.01

293.7

MM1.5 AM0.4 AM1.0 KM0.7 KM1.8

Molybdenum diffusion 2 MM0.4 2:02þ1:26 0:78  10

285.1 ± 4.4

285.8

0.090 ± 0.002

310.7

MM1.5

2 2:67þ8:70 2:04  10

290.2 ± 11.3

290.3

0.076 ± 0.005

312.3

AM0.4

2 3:67þ6:71 2:37  10

286.5 ± 9.5

286.8

0.083 ± 0.021

310.5

AM1.0

2 4:98þ3:70 2:12  10

289.1 ± 5.5

289.2

0.089 ± 0.002

314.8

KM0.7

3:18þ15:6 2:93 4:62þ13:0 3:41

2

286.7 ± 19.5

286.6

0.091 ± 0.01

312.7

 104

292.6 ± 12.2

293.0

0.068 ± 0.02

312.4

KM1.8

 10

those for the diffusion of iron in the alloys. A large solute atom requires high enthalpy for migration in the matrix. However, the vacancy concentration increases around the large solute atom [22]. For a molybdenum atom in the iron matrix, the former effect is larger than the latter one, because molybdenum in iron requires a large activation energy for diffusion. In addition, for the diffusion of transition metal solutes in paramagnetic iron a linear relationship between the activation energy and the atomic radius has been observed [22]. Molybdenum atoms in pure iron diffuse faster than iron atoms [20]. However, this relation reverses with an increase

of molybdenum content, as seen in Figs. 5 and 6. This feature can be explained on the basis of the above discussion, that is, the diffusion coefficient of iron in MM alloy increases with increasing molybdenum content owing to the increase of vacancy concentration, whereas the diffusion coefficient of molybdenum in the alloy decreases owing to the increase of migration energy. As shown in Fig. 12, the value of a for diffusion of iron decreases with increasing molybdenum content, whereas the value of a for the diffusion of molybdenum is about half of that for the diffusion of iron and shows a maximum at about 0.8 at.% molybdenum. The magnitude of a is most

H. Nitta et al. / Acta Materialia 54 (2006) 2833–2847

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Fig. 9. Solute concentration dependence of Qp and Dp0 for the diffusion of iron.

Fig. 10. Solute concentration dependence of Qp and Dp0 for the diffusion of molybdenum.

likely related to the magnetic field around the diffusing atom. The large a is due to the positive change in the magnetization of iron on alloying with the solute, while the small a is due to the negative change with the presence of

the solute atom. Since the change in magnetization of iron on diluting with molybdenum is 2.3 lB [23], as a result, the value of a for solvent diffusion decreases with increasing molybdenum content, as shown in Fig. 12. In contrast,

Fig. 11. Examples of plots of T ln½DðT Þ=Dp0  vs. s2 for diffusion of iron and molybdenum in (A) MM0.4, (B) AM1.0, and (C) KM1.8 alloys of the ferromagnetic state.

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and the concentration of solute is less than 2 at.%, Eqs. (5) and (6) are experimentally approximated by the linear expression of b1 and B1 in many bcc alloys [25–32]. The enhancement factors for both the solvent and solute diffusion in MM alloys shown in Figs. 5 and 6 were determined using Eqs. (5) and (6). In the present work, only two concentrations (0.4 and 1.5 at.% Mo) are examined in MM alloys. However, the concentration dependence of the diffusion coefficients of both components can be regarded as linear against the molybdenum concentration including the value at 0 at.%. The higher order terms can then be neglected and only the first terms b1 (=b) and B1 (=B) are taken into consideration. The values of b and B are listed in Table 6. As shown in Table 6, the enhancement factor of solvent diffusion, b, is 34–35 in the paramagnetic state and 58–68 in the ferromagnetic state. Fig. 13 shows the temperature Fig. 12. Molybdenum concentration dependence of a in iron–molybdenum alloys of the ferromagnetic state.

for the diffusion of molybdenum in the alloys, the magnetic field around the diffusing atom is influenced by the interaction between a diffusing molybdenum atom and other molybdenum atoms. The interaction acts between a diffusing molybdenum atom and a solute molybdenum for a low concentration of solute molybdenum. The number of solute molybdenum atoms that interact with a diffusing molybdenum atom increases from one to two in the range 0.7–1.0 at.% molybdenum. When the molybdenum concentration exceeds 1 at.%, the interaction arises among three molybdenum atoms, which are one diffusing atom and two solute atoms. Consequently, the value of a for the diffusion of molybdenum takes a maximum at about 0.8 at.% solute.

Table 6 Enhancement factors b and B for diffusion of iron and molybdenum in MM alloys Temperature (K)

b

B

1173 1153 1133 1093 1073 1053 1033 1003 973 923 873 823

35 35 35 34 34 34 68 58 58 62 64 67

14 14 15 16 17 18 14 18 20 24 27 30

4.2. Analysis of jump frequencies It is well known that the vacancy concentration varies in the vicinity of the solute atom. This makes the jump frequencies of both the solvent and solute atoms in the neighborhood of a solute atom perturb. The change in the jump frequencies of the diffusing atom depends on the distance from the solute atom. It relates to the dependence of the solvent and solute diffusion on the atomic fraction c of the solute. These diffusion coefficients of Dsolvent(c) and Dsolute(c) are represented by [24] Dsolvent ðcÞ ¼ Dsolvent ð0Þð1 þ b1 c þ b2 c2 þ   Þ 2

Dsolute ðcÞ ¼ Dsolute ð0Þð1 þ B1 c þ B2 c þ   Þ

ð5Þ ð6Þ

where Dsolvent(0) and Dsolute(0) are the solvent and solute diffusion coefficients, respectively, in a pure solvent. The constants bi (i = 1, 2, . . .) and Bi (i = 1, 2, . . .) are the solvent and solute enhancement factors, respectively. For the case that the interaction between solute atoms is weak

Fig. 13. Temperature dependence of enhancement factor b in high-purity iron–molybdenum alloy.

H. Nitta et al. / Acta Materialia 54 (2006) 2833–2847

dependence of b, which is linear in the paramagnetic state, while it is complicated in the ferromagnetic state. The relation between the enhancement factor of solvent diffusion, jump frequencies, and the correlation factor in a bcc metal are formulated by LeClaire and Jones [14,15] into two models, as mentioned earlier. The jump frequencies xi (i = 0, 1, . . .) and x0i ði ¼ 0; 1; . . .Þ are defined in the literature [14], as shown in Fig. 14. Model I. The model assumes that all potential energies between a vacancy and solvent atoms in the position distant from the third neighbor of a solute atom are the same. All jumps to the first and second neighbors from more distant sites are supposed to occur with the pure solvent jump frequency x0. The jump frequencies different from x0 and x2 are x3, x03 , x4, x04 , and x5. Model II. The interactions up to the first nearest neighbors of a solute atom only are taken into consideration. All the dissociative jumps from the first nearest-neighbor site of the solute atom occur with the same jump frequency x3. The jump frequencies different from x0 and x2 are x3, x4, and x5. Recently, from a study of the diffusion of transition elements in a-Fe, the present authors have found that the value of a in Eq. (1) can be expressed by a linear function of the change in magnetization of the first and second nearest-neighbor shells of a solute atom in a-Fe in the ferromagnetic state [33]. This is also true for the diffusion in iron-based dilute alloys. Therefore, model I is more suitable for the diffusion in iron-based Fe–Mo alloys than model II, because the former distinguishes potential energies of a vacancy and solvent atoms in the first and second nearest neighbors of a solute atom, whereas the latter does not take this into consideration. Model I is then applied to the present analysis. In model I, the jump frequencies are expressed as follows:

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x03 ¼ x003 ; x04 ¼ x004 ¼ x6 ¼ x0 ; xx30 ¼ xx45 3    Dg  f1 f2 x03 exp  Dg exp  ¼ x ¼ x 5 0 RT RT

ð7Þ

where Dgf1 and Dgf2 are the changes of free energy to form a vacancy to the first and second neighbors of a solute atom, respectively. Furthermore, the enhancement factor of solvent diffusion b1, jump frequency ratio x2 =x03 , and correlation factor fi are represented by the following equations:      l1 x3 m1 b1 ¼ 20 þ 14 þ6 0 ð8Þ f0 x3 f0 x2 f0 uðDsolute ð0Þ=Dsolvent ð0ÞÞ ð9Þ ¼ x03 u  2f 0 ðDsolute ð0Þ=Dsolvent ð0ÞÞ   x3 ðx3 =x03 Þ u ¼ 3 þ 2 0 þ 0:51 ð10Þ ð0:51 þ x3 =x03 Þ x3 u ð11Þ fi ¼ 2ðx2 =x03 Þ þ u where f0 is the correlation factor (=0.727) for self-diffusion in a bcc metal, and l1 and m1 are the mean partial correlation factors with functions of x2 =x03 and x3 =x03 , respectively. In this model, b depends on x3 =x03 according to Eq. (8). Jones and LeClaire have calculated mean partial correlation factors, l1 and m1, for some conditions of x2 =x03 and x3 =x03 [15]. Putting the value of b of the present experiments, DFe [8] and DMo [20] in a-Fe, and mean partial correlation factors [15] into Eqs. (8)–(10), the ratios of jump frequencies, x3 =x03 and x2 =x03 , at each temperature were calculated. The correlation factor for diffusion of molybdenum, fMo, was also obtained by applying the values of the ratio of jump frequencies and the partial correlation factor in Eq. (11). The values thus obtained are given in Table 7. Fig. 15 shows the temperature dependence of x3 =x03 and x2 =x03 , where x3 =x03 takes a value from 8.9 to 9.4 in the paramagnetic state and from 16 to 19 in the ferromagnetic state. The large ratio indicates that the vacancy in the first nearest neighbor of a molybdenum atom tends to jump to the second nearest neighbor much rather than to the third nearest neighbor. Moreover, since x2 =x03 takes a ratio from 1.1 to 1.7 in the paramagnetic state and from 1.2 to 2.2 in Table 7 Jump frequency ratios and correlation factor, fMo, in MM alloys

Fig. 14. A schematic illustration of two unit cells in the bcc structure showing the possible jumps of a vacancy to various sites and their respective frequencies.

Temperature (K)

DMo/DFe

fMo

ðx3 =x03 Þ

ðx2 =x03 Þ

(x2/x3)

1173 1153 1133 1093 1073 1053 1033 1003 973 923 873 823

2.02 1.90 1.79 1.58 1.48 1.39 1.58 1.98 2.20 2.57 2.70 2.77

0.87 0.88 0.88 0.89 0.90 0.91 0.94 0.92 0.91 0.90 0.90 0.90

9.4 9.4 9.4 9.2 9.2 8.9 19 16 16 17 17 18

1.7 1.6 1.5 1.3 1.2 1.1 1.2 1.6 1.8 2.1 2.2 2.2

0.2 0.2 0.2 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

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H. Nitta et al. / Acta Materialia 54 (2006) 2833–2847

Fig. 15. Temperature dependence of (a) x3 =x03 , (b) x2 =x03 , and (c) x2/x3.

the ferromagnetic state, the vacancy in the first nearest neighbor of a solute molybdenum atom exchanges the position of the solute molybdenum atom rather than an iron atom in the third nearest neighbor. Using the values of x3 =x03 and x2 =x03 , the ratio x2/x3 can be obtained, as listed in Table 7. Since x2/x3 takes a value from 0.1 to 0.2, the jumping probability of a vacancy in the nearest neighbor of a molybdenum atom to the second nearest neighbor is 5–10 times larger than the probability of exchanging the positions for the solute molybdenum atom. This means that the vacancy in the vicinity of a solute molybdenum atom jumps more frequently to an iron atom which locates in the first and second nearest neighbors of the solute molybdenum atom rather than other sites. The large variation of x3 =x03 at the Curie temperature shown in Fig. 15(a) can be explained by the magnetic property of a molybdenum atom in a-Fe. According to the theoretical calculation by Dritter et al. [34], the change in magnetization of the first nearest-neighbor shell around a solute molybdenum atom in the a-Fe matrix is 1.44 lB and that of the second nearest-neighbor shell around it is 0.198 lB. In contrast, the value for the third nearestneighbor shell is +0.492 lB. Since x3 is the jump frequency of the vacancy from the first nearest neighbor of a solute molybdenum atom to the second nearest neighbor, it is a jump from the shell (1.44 lB) to the shell (0.198 lB). Since x03 is the jump frequency of the vacancy from the first nearest neighbor of a solute molybdenum atom to the third nearest neighbor, it is a jump from the shell (1.44 lB) to the shell (+0.492 lB). Therefore, x03 is strongly influenced by the magnetic transformation in comparison with x3, and x03 decreases with increasing magnetization. For a similar reason, the ratio x2 =x03 shown in Fig. 14(b) increases below the Curie temperature.

The correlation factor fMo for solute molybdenum diffusion is 0.87–0.91 in the paramagnetic state and 0.90–0.94 in the ferromagnetic state. These values are larger than the correlation factor f (=0.727) for self-diffusion in a bcc metal. The ratios of jump frequencies x3 =x03 , x2 =x03 , and x2/x3 listed in Table 7 show a tendency of the attractive interaction between a molybdenum atom and a vacancy. Since molybdenum has an atomic radius about 10% larger than iron [35], the distortion of the iron lattice due to the oversized solute atom results in attractive interactions between the solute atom and vacancies. According to recent theoretical investigations on the enhancement factor B in a bcc dilute alloy by Segel et al. [25,26], B is represented by a complex function of seven jump frequencies of a vacancy adjacent to a solute atom, two additional free energies to form a vacancy at a specific neighboring site, and two binding energies between two solute atoms. The enhancement factor B for a dilute bcc alloy containing less than 2 at.% solute takes a value within the range between 20 and 50 [25,26]. The value of B for MM alloys was estimated to be 16 to 15 above the Curie temperature and 31 to 25 below the Curie temperature, as shown in Table 5. This means that the solute diffusion in dilute Fe–Mo alloys belongs to the framework of weak interaction between a solute atom and a vacancy, although below the Curie temperature interactions between solute atoms make a small contribution. 5. Conclusions Using the serial sputter-microsectioning technique with Fe and 99Mo radioactive tracers, the solvent and solute diffusion coefficients in high-purity Fe–Mo alloys have been determined in the temperature range 823–1173 K.

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Solvent diffusion of iron is enhanced with increasing molybdenum content, while the solute diffusion coefficient of molybdenum decreases. The activation energies and the preexponential factors of both components increase as the molybdenum concentration increases. Arrhenius plots of the diffusion coefficients of iron and molybdenum below the Curie temperature deviate from the linear Arrhenius relationships in the paramagnetic state owing to magnetic spin ordering. Analysis of jump frequency ratios shows that a vacancy is weakly bound by a molybdenum atom. Acknowledgments The authors thank Messrs. T. Iida and R. Kanno and Dr. Y. Yamazaki for their assistance in experiments. The authors are grateful to Dr. S. Ogu of Mitsubishi Heavy Industries Ltd. for supply of the iron–molybdenum alloys. This work was partly supported by an ISIJ Research Promotion Grant. References [1] Austin CR, St. John CR, Lindsay RW. Trans Metall Soc AIME 1945;162:84. [2] Bakker H. In: Mehrer H. editor. Diffusion in solid metals and alloys, Landort-Bo¨rnstein, New Series, Group 3, vol. 26. Berlin: Springer; 1990. p. 213. [3] Kucˇera J, Stra˘nskyˆ K. Mater Sci Eng 1982;52:1. [4] Iijima Y, Hirano K. In: Agarwala RP, editor. Diffusion processes in nuclear materials. Amsterdam: Elsevier Science; 1992. p. 169. [5] Million B, Kucˇera J. Kov Mater 1984;22:372. [6] Nohara K, Hirano K. J Jpn Inst Metals 1976;40:407. [7] Rothman SJ. In: Murch GE, Nowick AS, editors. Diffusion in crystalline solids. New York (NY): Academic Press; 1984. p. 1.

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