Stacking fault energies of Mn, Co and Nb alloyed austenitic stainless steels

Stacking fault energies of Mn, Co and Nb alloyed austenitic stainless steels

Available online at Acta Materialia 59 (2011) 5728–5734 Stacking fault energies of Mn, Co and ...

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Acta Materialia 59 (2011) 5728–5734

Stacking fault energies of Mn, Co and Nb alloyed austenitic stainless steels Song Lu a,⇑, Qing-Miao Hu b, Bo¨rje Johansson a,c, Levente Vitos a,c,d a

Applied Materials Physics, Department of Materials Science and Engineering, Royal Institute of Technology, Stockholm SE-100 44, Sweden b Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, 72 Wenhua Road, Shenyang 110016, China c Condensed Matter Theory Group, Physics Department, Uppsala University, Uppsala SE-75120, Sweden d Research Institute for Solid State Physics and Optics, P.O. Box 49, Budapest H-1525, Hungary Received 15 April 2011; accepted 24 May 2011 Available online 27 June 2011

Abstract The alloying effects of Mn, Co and Nb on the stacking fault energy (SFE) of austenitic stainless steels, Fe–Cr–Ni with various Ni contents, are investigated via quantum–mechanical first-principles calculations. In the composition range (cCr = 20%, 8 6 cNi 6 20%, 0 6 cMn, cCo, cNb 6 8%, balance Fe) studied here, it is found that Mn always decreases the SFE at 0 K but increases it at room temperature in high-Ni (cNi J 16%) alloys. The SFE always decreases with increasing Co content. Niobium increases the SFE significantly in low-Ni alloys; however, this effect is strongly diminished in high-Ni alloys. The SFE-enhancing effect of Ni usually observed in Fe–Cr–Ni alloys is inverted to an SFE-decreasing effect by Nb for cNb J 3%. The revealed nonlinear composition dependencies are explained in terms of the peculiar magnetic contributions to the total SFE. Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Stacking fault energy; First-principles electron theory; Austenitic stainless steels

1. Introduction The stacking fault energy (cSFE) is a parameter of significant importance on the mechanical properties of closepacked face centered cubic (fcc) alloys, such as strength, toughness and fracture. In austenitic stainless steels, it has been demonstrated that cSFE correlates closely with the plastic deformation mechanisms [1,2]. It has been recognized that plastic deformation is mainly realized by martensitic transformations (c (fcc) ! a0 (bcc/bct) or c !  (hcp)) at low cSFE values and by twinning at intermediate cSFE (18 [ cSFE [ 45 mJ m2). At even higher cSFE, plasticity and strain hardening are controlled solely by the slide of dislocations [3–5].

⇑ Corresponding author. Tel.: +46 8 7906215; fax: +46 8 100411.

E-mail address: [email protected] (S. Lu).

In order to optimize the mechanical properties of austenitic steels as desired, cSFE has to be adjusted to an appropriate value. Several variables, such as composition [3,6,7], temperature [8–12], grain size [13,14] and deformation ratio [2,4], have been shown to affect cSFE. The alloying effects on cSFE are quite complicated and sometimes contradict each other from different experimental measurements. Additionally, the experimental cSFE values are not accurate and are usually associated with large error bars [6–8,15,16]. Based on the existing databases, several empirical relationships between cSFE and chemical compositions have been proposed [6,7,17]. However, the application of these empirical relationships is limited and in most cases they are unable to reproduce the complex nonlinear dependence of cSFE on the composition [18–20]. Moreover, these empirical relationships hardly address the interactions between the different alloying elements.

1359-6454/$36.00 Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2011.05.049

S. Lu et al. / Acta Materialia 59 (2011) 5728–5734

Manganese, cobalt and niobium are commonly used alloying elements to improve the specific mechanical and/ or corrosion-resistive properties of stainless steels. For Mn-containing austenitic stainless steels, a large number of experiments have been performed to measure the composition-dependent properties, including the SFE [6,7,21]. In the cSFE vs. composition relations developed by those authors for the calculation of cSFE, there are serious discrepancies on the effect of Mn (increase or decrease). For Co and Nb, there are no adequate data with which to judge their effects on cSFE. Though it has been noted that significant interactions among alloying elements occur and extra care should be taken when extending the empirical equations beyond the composition ranges used to derive the equations [22], there is a clear need to set up the interaction profiles between alloying elements and to understand the fundamental mechanisms controlling the alloying effects on cSFE. Using a quantum–mechanical first-principles method, Vitos et al. [18–20] established the composition dependence map of cSFE in random paramagnetic Fe–Cr–Ni alloys. The nonlinear composition dependence and temperature effect on cSFE have been well reproduced. They demonstrated that a fourth alloying element may have a totally opposite effect on cSFE, depending on the initial composition of the matrix [19]. This work implies the importance of the volume effect, magnetism and the interactions between the alloying elements on cSFE. Following this pioneering work, we study here the effect of additional alloying elements and their interactions with Ni on cSFE of Fe80nmCr20NinXm (X = Mn, Co and Nb) alloys with 8 6 n 6 20 and 0 6 m 6 8. The rest of paper is arranged as follows. In Section 2, we briefly introduce the model used to calculate cSFE and describe the calculation details. In Section 3, we first discuss the magnetic transition in hexagonal close-packed (hcp) Fe–Cr–Ni-X alloys driven by alloying elements and during the local structure relaxation process, then present the calculated cSFE as a function of composition in Mn-, Co- and Nb-containing austenitic alloys, respectively. The paper ends with a summary and conclusions. 2. Methodology and calculation details The intrinsic SFE was calculated within the axial interaction model, taking into account the interactions between layers up to the third nearest neighbors. According to that, cSFE can be expressed as [19,23,24]   cSFE ¼ F hcp þ 2F dhcp  3F fcc A2D ð1Þ where Fhcp, Fdhcp and Ffcc are the free energies of hexagonal close-packed, double hexagonal close-packed (dhcp) and fcc structures, respectively. A2D denotes the area of the stacking fault. Details of the model and its accuracy when applied to the Fe–Cr–Ni alloys can be found in Ref. [19]. In the previous work by Vitos et al. [18–20], all energies were calculated for relaxed volumes with rigid lattices (i.e.


ideal c/a for hcp and dhcp structures). However, in real alloys, the in-plane lattice constants of the faulted area are constrained by the fcc (1 1 1) plane. Because of this, relaxation can only be realized along the direction perpendicular to the stacking fault plane. To mimic this situation, in the present calculations we adopt a more realistic model by relaxing the c lattice constants of the hcp and dhcp lattices while keeping the in-plane lattice constants fixed to ð111Þ afcc . The latter is calculated by minimizing the total energy of the fcc lattice with respect to the volume. The Fe-rich Fe–Cr–Ni solid solutions have relatively low magnetic transition temperatures [25]. At room temperature they are normally paramagnetic, and the magnetism state can be described by the disordered local magnetic moments (DLMs) approximation [26–28]. Within this approximation, at 0 K no local magnetic moments develop on Cr and Ni sites, except on Fe sites. Thermal spin fluctuations, however, can induce non-zero local magnetic moments on the Cr and Ni sites as well. At room temperture, the SFE in Fe–Cr–Ni alloys can be roughly divided into two contributions: the chemical part c0SFE and the magnetic part cmag. c0SFE corresponds to the SFE calculated at 0 K. The contribution of electronic entropy and lattice vibrational entropy at room temperature were verified to be relatively insignificant [18–20] and thus are neglected in the present work. Therefore, the dominant temperature dependent part of cSFE is due to the magnetic entropy of the local magnetic moments, namely  . mag mag cmag ¼ T S mag ðlÞ þ 2S ðlÞ  3S ðlÞ A2D ð2Þ hcp dhcp fcc mag mag where S mag hcp ðlÞ; S dhcp ðlÞ and S fcc ðlÞ denote the magnetic entropies in the hcp, dhcp and fcc phases, respectively. In a completely disordered paramagnetic alloy the magnetic entropy may be estimated using the mean-field expression Smag = RikBcilog (li + 1) (where li denotes the local magnetic moments of atom i, kB is the Boltzmann constant and ci is the concentration of atom i) [29]. The definition of the cmag implies that a large cmag is expected if the difference in the local magnetic moments from the hcp ðlhcp i Þ or fcc dhcp ðldhcp Þ and fcc ðl Þ phases is significant. i i According to Eq. (2), the nonlinearity in the temperature factor of the SFE may enter only via the temperature dependence of the local magnetic moments li. Low-carbon Fe–Cr–Ni alloys, examined by Lecroisey and co-workers [8,9] in the temperature range 150–400 K, showed a linear correspondence between the SFE and temperature. Similarly, the characteristic of SFE vs. temperature in other literature may be captured by a linear fitting at low temperature [10–12]. Because of this, in the present study we ignore the thermally induced spin fluctuations and assume that li is constant at low temperature. This approximation means that cSFE depends linearly on the temperature. In order to verify our assumption, in Fig. 1 we compare the present temperature dependence of cSFE calculated for Fe70Cr18Ni12 with the available experimental and theoretical data. The almost perfect correspondence


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Fig. 1. Theoretic temperature dependence of the SFE for Fe70Cr18Ni12 (solid line) compared to the available experimental [3] (black square) and previous theoretical data [20] (red dashed line). (For interpretation of the references to colours in this figure legend, the reader is referred to the web version of this paper.)

between the three sets of data confirms the validity of our approximation. The total energies of all phases were calculated using the exact muffin-tin orbitals (EMTO) method [30,31] combined with the coherent potential approximation (CPA) [32,33]. The EMTO–CPA approach is an appropriate tool to describe a system with chemical and magnetic disorder. In the self-consistent EMTO–CPA calculations, the oneelectron equations were solved within the scalar-relativistic and frozen-core approximation, and the Green function for the valence states was calculated for 16 complex energy points. In the muffin-tin basis set we included s, p, d and f orbitals. The k-point numbers were carefully test and we used 1500–2000 uniformly distributed k-points in the irreducible wedge of the fcc, hcp and dhcp Brillouin zones. For the exchange–correlation functional the generalized gradient approximation of Perdew, Burke and Ernzerhof was applied [34]. 3. Results and discussion 3.1. Magnetic transition In Fig. 2a we show the total energies of hcp Fe80nCr20Nin as functions of hexagonal lattice parameters ahcp and c/ ahcp for various Ni contents at 0 K. The corresponding local magnetic moments of Fe atoms are shown in Fig. 2b. The solid and dashed lines mark two different ways of relaxing the hcp structures: the prior corresponds to a ð111Þ path when ahcp is kept fixed to afcc and c is relaxed (route I), and the latter relaxes the volume while keeping c/ahcp at the ideal value of 1.633 (route II). These two routes represent two typical ways when applying the axial interaction model to calculate the SFE in the literature. A third route often applied in full-potential calculations (route III, not shown) is the fully relaxed hcp lattice, which corresponds to the global minimum in Fig. 2a.

The dominant feature of the energy contours in Fig. 2a is that the global minimum state gradually moves from the low-volume region (small c/ahcp and small ahcp) to the highvolume region (large c/ahcp and large ahcp) with increasing Ni content. These two regions correspond to the low-spin (LS) and high-spin (HS) states of Fe atoms, respectively, as seen in Fig. 2b. At the same time, the HS area spreads from the high ahcp, high c/ahcp corner to the low ahcp, low c/ahcp corner with increasing Ni. This means that the local magnetic moment of Fe in hcp phase experiences a transition (LS ! HS) with Ni at around cNi = 16 at.% [19]. For alloys with intermediate content of Ni, e.g. Fe64Cr20Ni16 (middle panel from Fig. 2), following either of the relaxation routes I and II we observe two metastable states, corresponding to the LS and HS states, respectively. The difference is that these two routes (and also route III) will not show the magnetic transition at exactly the same composition. Usually, the local magnetic moments of Fe in fcc and dhcp phases change smoothly with the concentration of Ni or Cr [19]. Then the transition of the magnetic state of hcp phase implies a transition in cmag (large cmag! small cmag). Therefore, one should be very careful when choosing the state of the hcp phase used to calculate the SFE, especially for the compositions close to the magnetic transition. The difference in the energies of these two states may be very small, giving a negligible difference in the SFE at 0 K c0SFE , but the difference in the local magnetic moment is relatively large and leads to a big difference in cmag. Lecroisey and Pineau [9] measured the lattice parameters of the  phase (hcp structure) in Fe–Cr–Ni alloys and found that the fcc ! hcp transformation is accompanied by a contraction along the hcp c axis of about 1%. In situ experiments performed by Brooks et al. [35] showed a 2% decrease in the interplanar spacing of the close-packed planes at the intrinsic stacking fault area in austenitic stainless steels. Following route I, the lattice contraction along c direction is 1–3%, as shown in Fig. 2a, depending on the composition. It should be noted that in real case of dislocation-terminated stacking faults the contraction perpendicular to the stacking fault may introduce coherent strain and add a positive contribution (cstrain) to the ideal SFE which corresponds to the infinitely extended stacking fault [36,3]. Usually in thermodynamical calculations, the coherent strain energy is assumed to be composition independent and negligible [3,37], whereas our result suggests that the strain contribution to the SFE (when taken into account) should not be treated as being composition independent. The smaller equilibrium volume of the hcp phase than that of the fcc phase is due to the loss of local magnetic moment. We can see that for the LS hcp phase, e.g. Fe72Cr20Ni8, the calculated equilibrium c/ahcp  1.57 on route I is much smaller than the ideal one (1.633), while for the HS Fe60Cr20Ni20, the equilibrium c/ahcp  1.62 is very close to the ideal one and the cstrain is expected to be relatively small in this case.

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Fig. 2. The total energy (a) (in units of mRy) and the Fe magnetic moment lFe (b) (in units of lB) of hcp Fe–Cr–Ni as a function of ahcp and c/ahcp for 8%, ð111Þ 16% and 20% Ni at 0 K. Upon changing the c/ahcp ratio the ahcp is fixed to afcc . Solid and dashed lines mark two typical ways of relaxing hcp structure, see details in text.

The details of the magnetic transition in the hcp phase are expected to depend strongly on the additional alloying elements. The total energies of hcp Fe–Cr–Ni-X (X = Mn, Nb and Co) alloys as a function of c/ahcp for various compositions are plotted in Fig. 3. Generally, we can observe two local minima as a function of c/ahcp in line with Fig. 2a, and their relative stability changes with the amount of the additional alloying element X. The local minima at small c/ahcp are associated with nearly vanishing local magnetic moments of Fe (LS), and consequently contribute




Fig. 3. Total energies of the hcp Fe–Cr–Ni-X alloys as a function of hexagonal lattice parameter c/ahcp. The energies are plotted relative to the respective equilibrium values. X represents Mn (a), Co (b) and Nb (c), respectively, and the actual alloy compositions are shown in the legends. For notations, see the caption for Fig. 2.

with large cmag to the total SFE, whereas the local minima at higher c/ahcp have larger local magnetic moments (HS) and relatively small cmag. No local magnetic moment develops on the sites of Mn, Co and Nb in the hcp structure. From Fig. 3, Co and Nb are found to favor the HS state as they lower the energy of the HS state relative to that of the LS state. Manganese tends to stabilize the LS state as Cr [19]. Niobium is very efficient in shifting the hcp phase from the LS to the HS state. As seen in Fig. 3c, with m = 2, the stable state of hcp Fe72mCr20Ni8Nbm is LS, whereas it becomes HS with m = 4. The reason is that Nb strongly increases the equilibrium volume. Manganese and cobalt have negligible effects on the equilibrium volume and consequently change the spin state of the hcp phase very weakly. By manipulating the local magnetic state, alloying additions to Fe–Cr–Ni show a greater capability in altering the formation energy of the stacking fault besides their intrinsic chemical effects at 0 K. This will be discussed in detail in the following sections. 3.2. The SFE of Fe–Cr–Ni alloys In Fig. 4, the calculated cSFE for Fe80nCr20Nin at 300 K is compared to previous theoretical and experimental results. Taking into account that the experimental data were measured for samples containing small amounts of other impurities, such as N, C interstitials and Si, Mn substitutes, the agreement between theory and experiment can be considered reasonable. Nickel was found to increase cSFE quite significantly in alloys with less than 20 at.% Cr. Namely, an almost linear dependence of the SFE on Ni concentration was reported with the gradient of 2 < ocSFE/on < 3 mJ m2 per at.% Ni, depending on the experimental data used for the fit [6,7,17]. In the present


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Fig. 4. Comparison between the theoretical and experimental SFEs for Fe80nCr20Nin. References in the legend can be found in Refs. [6,11,15,20,38–40].

work, for alloys with 20 at.% Cr, the linear relationship between cSFE and Ni concentration is reproduced for n [ 16 with a mean slope of ocSFE/ on  3 mJ m2 per at.% Ni. At higher Ni, cSFE slightly decreases, which is in line with some experimental or theoretical results [7,10,15], but is not supported by others [19– 21]. In particular, the discrepancy between the present result for high-Ni alloys and those from Refs. [19,20] is due to the two different routes adopted when relaxing the hcp structure. While here high-Ni hcp alloys reach the HS state, in Refs. [19,20] the hcp Fe–21.5Cr–20Ni was taken to be in an intermediate magnetic state (yielding a relatively significant magnetic and thus temperature contribution to the SFE). At high Ni content, we find that the calculated SFE at 0 K remains nearly constant with Ni, but since the magnetic contribution cmag decreases with Ni, the total cSFE also decreases with Ni at 300 K. However, cmag for the Fe–Cr–Ni alloy with composition in the HS state is relatively small ( 5 mJ m2), and thus the decreasing effect may be easily screened by other factors in experiments, e.g. other micro-amount alloying elements or the temperature dependence of the magnetic moment, which are not considered here. We note that in the isoSFE contour plots in the Fe-rich corner of the Fe–Cr–Ni ternary diagram, at around 20 wt.% Cr, the SFE exhibits a saturation/declining trend with increasing Ni exceeding 15 wt.% in Refs. [7,10,41]. 3.3. The SFE of Fe–Cr–Ni-X alloys Fig. 5a, c and e displays the calculated cSFE maps as a function of the chemical composition for alloying elements Mn, Co and Nb at 300 K while Fig. 5b, d and f shows the corresponding magnetic contribution to the SFEs (cmag), respectively. For a fixed concentration of Ni, from Fig. 5a, we observe that cSFE decreases with Mn concentration in alloys with n [ 16 and slightly increases with Mn at higher

Ni. The magnetic entropy contribution (cmag) to the SFE always increases with increasing Mn content, as shown in Fig. 5b. At 0 K, Mn is found to decrease the SFE, but this decreasing effect is gradually weakened with increasing Ni content, which is the reason why Mn increases the SFE at high Ni content at 300 K. In Fe–Cr–Ni ternary alloys, Cr was also found to stabilize the hcp phase and reduce the SFE in low-Ni alloys (n [ 16), but in high-Ni alloys a change (negative to positive) was observed in the slope of cSFE vs. Cr content at about 20 at.% Cr [20]. Manganese behaves very similarly to Cr on the SFE because they are both LS stabilizers and tend to decrease the SFE chemically at 0 K. Taking into account the interactions among alloying elements, Dai et al. [21] demonstrated a positive slope of the SFE vs. Mn content beyond 10 at.% Mn in their empirical equation, in line with what we predict. The dependence of cSFE on Ni is also altered upon adding Mn, as shown in Fig. 5a. The magnetic transition from LS to HS in the hcp phase with increasing Ni is found to be postponed at high Mn. In particular, the critical point cNi where cSFE begins to decrease is 18 at.% at 8 at.% Mn, compared to cNi  16 at.% obtained for the case without Mn. Cobalt is an hcp-stabilizing alloying element and always tends to decrease cSFE in the whole concentration range studied, as shown in Fig. 5c. The slope ocSFE/om is estimated to be about 0.5 mJ m2 per at.% Co in low-Ni (n [ 16) alloys and about 2 mJ m2 per at.% Co in high-Ni (n J 16) alloys. The Ni-enhanced decreasing effect of Co on the SFE is also present at 0 K, meaning that this effect has a chemical origin. In Fig. 5d, we can see that cmag is only slightly decreased by Co in the whole concentration range of Ni (8 6 n 6 20). The behavior pattern of Ni on cSFE is not acutely altered by Co. Cobalt is known as a useful alloying element that improves the steel resistance against galling [42,43]. An enhanced galling effect, in turn, is thought to be associated with enhanced ductility. According to our study, Co decreases the SFE and thus decreases the ductility of austenitic stainless steels. This might explain why Co acts as an efficient anti-galling alloying ingredient. In low-Nb alloys (m [ 3), at 300 K, cSFE increases with Ni for n [ 16, while in high-Nb alloys cSFE decreases with Ni for the whole concentration range of 8 6 n 6 20 (Fig. 5e). This phenomenon is also present at 0 K and thus stems from the intrinsic chemical effect of Nb. Niobium strongly increases cSFE in the low-Ni alloys, but in highNi alloys this increasing effect is weakened. On the other hand, Nb always decreases the magnetic contribution cmag to the SFE, as shown in Fig. 5f. Because Ni and Nb are both HS-stabilizing elements in the hcp phase, cmag has large values only in the low-Ni, low-Nb region (n [ 16, m [ 3), beyond which cmag is rather small. A commonly referred to experimental work on the role of Nb on the SFE in Fe–Cr–Ni alloys is that of Martinez et al. [44]. The SFEs of Fe–15%Cr–15%Ni alloys comprising 0, 0.5, 1 and 2 wt.% Nb, measured by X-ray diffraction

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Fig. 5. The calculated SFE (cSFE) and the magnetic contribution to the SFE (cmag) maps of Fe–Cr–Ni-X alloys plotted as functions of composition for T = 300 K. X represents Mn in (a) and (b), Co in (c) and (d), and Nb in (e) and (f), respectively.

analysis, indicated that Nb strongly decreases the SFE of austenitic stainless steels. However, this decreasing effect was mainly observed for Nb concentration from 0 to 0.5 wt.%, whereas for alloys with 0.5, 1 and 2 wt.% Nb the SFEs are almost the same (the difference is smaller than the error bar associated with the method [45]). Addition-

ally, the authors noted that the measured SFE for the Nb-free composition was unusually high. They interpreted the decreasing effect of Nb by using the variation of the effective numbers of d electrons (Nd) upon alloying. However, this simple model applies only to systems where the magnetic contribution to the SFE is small and the alloying


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elements do not change the lattice parameters of the host significantly [19]. In the present work, we obtained the totally opposite result, namely that Nb increases the SFE at temperatures up to 300 K. Our finding is in fact fully supported by the thermodynamic analysis performed by Ishida [46], who calculated the change of the SFE of Fe– 18% Ni–10% Cr austenitic stainless steels by alloying 1% M (M = Nb, W, V, Mo, etc.) and found positive effects for Nb, W, V and Mo. The above results are also in line with earlier experimental results measured by Dulieu et al. using an extended dislocation nodes method [40]. We would like to add that, since the magnetic SFE is to a large extent cancelled by Nb, at very high temperature Nb could lead to decreasing SFE. 4. Summary Using a quantum–mechanical all-electron first-principles method, we have calculated the SFE in austenitic stainless steel alloys as a function of composition. The effect of the additional alloying elements Mn, Co and Nb on the SFE of Fe–Cr–Ni alloys with various Ni contents has been examined. From the SFE maps with respect to composition we have shown the interesting effect of the interactions between different alloy components on the SFE. Manganese is found to decrease the SFE in alloys with less than 16 at.% Ni, beyond which the SFE rises slightly with Mn. Cobalt always tends to decrease the SFE, and this decreasing effect is enhanced in high-Ni alloys. Niobium strongly increases the SFE in low-Ni alloys, but the increasing effect is weakened by Ni. Niobium is found to overwrite the effect of Ni on the SFE of Fe–Cr–Ni ternary alloys. Acknowledgements The authors acknowledge the Swedish Research Council, the Swedish Foundation for International Cooperation in Research and High Education, the European Research Council, the Swedish Steel Producers’ Association, the Carl Tryggers Foundation, the Hungarian Scientific Research Fund (research project OTKA 84078) and the China Scholarship Council for financial support. Q.-M. H. acknowledges the financial support from the MoST of China under Grant No. 2011CB606404. References [1] Fujita M, Kaneko Y, Nohara A, Saka H, Zauter R, Mughrabi H. ISIJ Inter 1994;34:697.

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