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Application of a modified Ostwald ripening theory in coarsening of γ′ phases in Ni based single crystal superalloys
Journal of Alloys and Compounds 632 (2015) 558–562
Contents lists available at ScienceDirect
Journal of Alloys and Compounds journal homepage: www.elsevier.com/locate/jalcom
Letter
Application of a modified Ostwald ripening theory in coarsening of c0 phases in Ni based single crystal superalloys Cheng Ai a, Xinbao Zhao b, Jian Zhou a,⇑, Heng Zhang a, Lei Liu a, Yanling Pei a, Shusuo Li a,⇑, Shengkai Gong a a b
School of Materials Science and Engineering, Beihang University, No. 37 Xueyuan Road, Beijing 100191, China Xi’an Thermal Power Research Institute Co., Ltd., Xi’an, Shaanxi 710032, China
a r t i c l e
i n f o
a b s t r a c t A modified Ostwald ripening theory was used to evaluate the coarsening behavior of c0 phases in Ni based single crystal superalloys during thermal exposure. After introducing a dimensionless factor correlated with volume fraction of c0 phase into the classical Lifshitz–Slyozov–Wagner (LSW) coarsening theory, this model was thus developed by coupling formulaic interfacial energy and diffusion coefficient (including activation energy and pre-exponential factor) as function of alloy composition. The coarsening rate coefficients and radius sizes of c0 phases were then presented, which show good agreement with the corresponding experimental results. Ó 2015 Elsevier B.V. All rights reserved.
Article history: Received 23 July 2014 Received in revised form 19 January 2015 Accepted 25 January 2015 Available online 2 February 2015 Keywords: Ni based single crystal superalloy c0 phase Diffusion Coarsening Ostwald ripening
Ni based single crystal (SC) superalloys have been most widely applied in modern turbine blades due to their excellent mechanical properties, which largely depend on the strengthening phase (c0 phases) with an ordered structure (L12) [1–4]. However, SC superalloys must serve at high temperature for comparatively long time, resulting in significant increase of the particle sizes of c0 phases, which yet leads to the decrement of stress rupture properties [5– 7]. As a result, due to the importance of the particle sizes of c0 phases for the mechanical properties of superalloys, it is necessary to evaluate the evolution of particle sizes (coarsening) with time during thermal exposure. In fact, considering that a great deal of time and cost would be required to perform the experimental determination of particle sizes, a theoretical evaluation of coarsening behavior should be fairly helpful and deserved to be performed. Consequently, theoretical researches on the coarsening behavior of c0 phase in superalloys have been widely conducted based on classical LSW coarsening theory of Ostwald ripening [7–17]: 3
r 3 r 30 ¼ k t
ð1Þ 0
where r and r0 are the average radius of c phases at time t and t = 0 respectively, and k the coarsening rate coefficient determined by [14–17]:
⇑ Corresponding authors. Tel.: +86 10 82314488; fax: +86 10 82338200. E-mail addresses: [email protected] (J. Zhou), [email protected] (S. Li). http://dx.doi.org/10.1016/j.jallcom.2015.01.215 0925-8388/Ó 2015 Elsevier B.V. All rights reserved.
k¼
8Deff rNa V m 9RT
1=3 ð2Þ
where Deff is the effective diffusion coefficient, r the interfacial energy, Na the equilibrium mole fraction of alloying elements in c phases, Vm the molar volume of c0 phases, R the gas constant and T the coarsening temperature. Actually, the LSW coarsening theory assumes infinitesimally small volume fraction of secondary phases [15,16], which is obviously not suitable for the cases of Ni based SC superalloys due to their extremely high volume fractions of secondary phases (50–80%) [1,17]. The researches on Ni–Al binary (at high volume fraction region (50–90%)) [18] and Ni–Al–Mo ternary systems [19] both show the coarsening rate coefficients are significantly increased with increasing volume fractions of c0 phases (U). Therefore, in order to account for the deviation of theoretical calculations from experimental values, a dimensionless factor A(U) correlated with U was introduced, which has various mathematical forms based on different theories, such as Modified LSW (MLSW), Tsumaraya–Miyata (TM), Marqusee–Ross (MR) and Voorhees–Glicksman (VG) theories [17]. Consequently, Eq. (2) is thus turned into:
k¼
8Að/ÞDeff rNa V m 9RT
1=3 ð3Þ
All of these models indicate that A(U) is independent on alloy systems but largely relies on the volume fraction (U). Generally, a positive correlation of A(U) with U was found, which actually
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C. Ai et al. / Journal of Alloys and Compounds 632 (2015) 558–562
resulted from the fact that higher U leads to shorter diffusion distances, thus promoting the diffusion of elements and then the coarsening of c0 phases [17–19]. Furthermore, it has been shown by Wang et al. [19] that A(U) for Ni–Al–Mo ternary alloys rapidly increased at high U region (U = 0.3–0.6), which can be preferably described by MLSW/TM/VG theory [17]. Besides, A(U) were also found to be sensitive to the morphology of c0 phases, e.g. A(U) for cubic c0 phases was lower than that for spherical c0 phases at the same volume fraction of c0 phases [19]. Fortunately, the morphologies of c0 phases in Ni based SC superalloys are cubic or near cubic, which means that A(U) only relies on U during the process of coarsening. Therefore, a predefined A(U) in accordance with cubic/near cubic morphology can be applied in modeling the coarsening behavior of c0 phases in Ni based SC superalloys. It was found that through the comparison between the calculated results of different models (not shown in this paper), the model IV of TM theory, which assumed that the particles of secondary phase are dispersed throughout the matrix in the unoccupied region available to them within the constraint of no particle interpenetration, presented the best agreement between theoretical calculation and experimental data, and was thus chosen to revise the LSW theory [20]. Actually, as for this model, the differential equation incorporating A(U) cannot be analytically solved so that the analytical formula of A(U) is incapable of being achieved and thus replaced by the fitted polynomial (i.e. A(U) = 2.22 + 25.55 ⁄ U 1.60 ⁄ U2 + 0.65 ⁄ U3). It should be noted that in terms of Eq. (3), the parameters Deff and r are closely correlated with alloy compositions, resulting in a composition-dependent rate coarsening coefficient. Considering the complexity of multicomponent alloys, there are a few researches focusing on the quantitative relationship between coarsening rate coefficients and alloy compositions for superalloys. Based on the philosophy of LSW theory, Li calculated coarsening rate coefficients of c0 phases in polycrystalline Ni based superalloys, and good agreement with experimental results has been achieved [14]. Furthermore, using Li0 s methodology, Phase Transformation module in ‘‘Single Crystal’’ database of software JMatPro 7.0 (dynamics database) can be used to determine coarsening rate coefficients of c0 phases in Ni based SC superalloys [21]. Nonetheless, the accuracy of which still remains to be verified. The key to quantitatively evaluate coarsening behavior is to establish the function-relation between the physical parameters in Eq. (3) (e.g. effective diffusion coefficient and interfacial energy, etc.) and the corresponding alloy composition. Therefore, in the current work, we tried to adopt reasonable physical models of parameters and then incorporated them into the modified Oswald equation to derive the alloy-composition dependent coarsening behavior. Normally, as for Deff, considering its Arrhenius form:
Deff ¼ D0;eff eQ eff =RT
ð4Þ
where D0,eff and Qeff are the pre-exponential factor and activation energy of effective diffusion coefficient respectively, and both of D0,eff and Qeff are functions of alloy composition. Generally, two types of function-relation between Qeff and alloy composition exist. P One is Qeff = QNi,Ni + mCmQNi,m, where QNi,Ni is the intrinsic diffusion activation energy for pure Ni, QNi,m the diffusion activation energy of element m in Ni, and Cm the atomic concentration of element m. Using this relation, Zhu et al. [22] calculated the activation energies during creep of SC superalloys, which exceeds 300 kJ/mol and are obviously higher than the experimental values during unstressed thermal exposure of SC superalloys (260.09 kJ/mol1 [8] and 272.40 kJ/mol [10]). As for another relation, i.e. 1
Average value of three different heat treatment regimes.
Q eff
X ¼ C m Q Ni;m
ð5Þ
m
good agreements between theoretical calculations and experimental results have been achieved, which can be considered to effectively reflect the contributions of all elements in SC superalloys into the diffusion behavior during thermal exposure. For instance, with regard to superalloys CMSX-2 [8], based on the corresponding alloy compositions, Qeff was calculated by Eq. (5) as 271.11 kJ/mol, which is close to Qexperiment = 260.09 kJ/mol [8]. Besides, as for superalloys CMSX-4, the corresponding Qeff was calculated to be 271.73 kJ/mol, in agreement with Qexperiment = 272.40 kJ/mol [10]. It should be noted that, as for the above calculation, the data of QNi,m was chosen based on Ni-m binary interdiffusion. Actually, the addition of ternary element would have influence on the value of QNi,m [23]. Nevertheless, considering the above calculated results by Eq. (5) are pretty close to the experimental data, for the sake of simplicity, the effect of other elements on Ni-m binary interdiffusion (i.e. the interaction of alloying elements) was not taken into account in this work. As for the pre-exponential factor D0,eff, the harmonic mean method was adopted to consider the contribution of alloy elements in this work [22]:
1 D0;eff ¼ X C m
ð6Þ
m
DNi;m 0
In fact, pre-exponential factors can also be determined by arithP metic mean method, i.e. D0,eff = mCmDNi,m [14,22]. The reason for 0 choosing harmonic mean method was depicted as follows. Normally, the addition of refractory elements can decrease diffusion coefficients [22]. Therefore, when adopting the function-relation Eq. (5), Qeff is calculated to be smaller than QNi,Ni (284 kJ/mol), thus leading to a larger Deff. In order to match the calculated values and experimental results, the theoretically determined D0,eff of SC superalloys should be significantly lower than DNi–Ni 0 (1.90 ⁄ 104 m2/s). Zhu0 s study showed that the order of magnitude for D0,eff calculated using harmonic mean method is 105 m2/s, which is lower than that for DNi–Ni (104 m2/s) by one order of 0 magnitude, as well as that for D0,eff (104 m2/s) calculated using arithmetic mean method [22]. Moreover, previous research shows the effective diffusion coefficients significantly decrease with the addition of a spot of refractory elements (such as Re) [22]. By comparing these two methods, only the harmonic mean method (Eq. (6)) can present D0,eff, which is notably affected by the addition of a small amount of elements when they possess low diffusion coefficient, and is thus more suitable for the theoretical evaluation of D0,eff [22]. In order to calculate Qeff and D0,eff, the pre-exponential factors (DNi,m ) and activation energies (QNi,m) for the interdiffusion of 0 element m in Ni is shown in Table 1.
Table 1 DNi,m and QNi,m for the interdiffusion of element m in Ni. 0 Element
DNi,m (m2/s) 0
QNi,m (kJ/mol)
Refs.
Ni Al Co Cr Hf Mo Re Ru Ta Ti W
1.90 ⁄ 104 1.00 ⁄ 103 7.50 ⁄ 105 3.00 ⁄ 106 1.62 ⁄ 104 1.15 ⁄ 104 8.20 ⁄ 107 2.48 ⁄ 104 2.19 ⁄ 105 4.10 ⁄ 104 8.00 ⁄ 106
284.00 272.09 285.10 170.70 250.10 281.30 255.00 304.40 251.00 275.00 264.00
[22] [24] [22] [22] [25] [26] [27] [28] [27] [29] [27]
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Table 2 The composition of Ni based SC superalloys and thermal exposure temperature (T) used in this work (wt.%). Classification
Alloy
Ni
Al
Co
Cr
Hf
Mo
Re
Ru
Ta
Ti
W
C
T (°C)
Refs.
1st generation
CMSX-2 M09A
Bal. Bal.
5.62 4.50
4.60 9.00
7.80 12.00
– –
0.60 2.00
– –
– –
6.00 5.00
1.00 2.50
7.90 4.00
– 0.01
800–1050 900
[8] [9]
2st generation
CMSX-4 Alloy 2Re12Co
Bal. Bal.
5.60 6.00
9.00 12.00
7.00 3.00
0.10 0.10
0.60 1.00
3.00 2.00
– –
7.00 8.00
1.00 –
6.00 6.00
– –
850–1000 950–1050
[10] [11]
3rd generation
Alloy 4Re0Co Alloy 4Re3Co Alloy 4Re12Co CMSX-10
Bal. Bal. Bal. Bal.
6.00 6.00 6.00 5.70
– 3.00 12.00 3.30
3.00 3.00 3.00 2.20
0.10 0.10 0.10 –
1.00 1.00 1.00 0.40
4.00 4.00 4.00 6.30
– – – –
8.00 8.00 8.00 8.30
– – – 0.23
6.00 6.00 6.00 5.50
– – – –
950–1050 950–1050 950–1050 950–1050
[11] [11] [11] [7]
Ru-containing Al-rich
Alloy 15.8Co IC6SX
Bal. Bal.
6.00 7.55
15.80 –
3.40 –
– –
2.60 13.96
2.10 –
2.00 –
8.00 –
– –
6.10 –
– –
1100 1100
[12] [13]
In addition to Deff, r also plays an important role in determining k. As for Ni based superalloys, considering the interfaces of c (matrix phase) and c0 (secondary phase) phases are coherent [1], it can thus be calculated by Li according to the following relation [14]:
r¼
zk N DH m zk N0
ð7Þ
where zk⁄ and zk are the number of broken bonds and coordination number in the kth shell respectively. Actually, zk⁄/zk = 0.3526 is derived based on the nearest-neighbor broken-bond analyze [30]. Besides, N⁄ is the number of cross bonds per atom at the interface and can be determined using the relation of N⁄ = 41.5/a2 with a = (ac0 + ac)/2, where ac0 and ac are the lattice constants of c0 and c phases at coarsening temperature respectively. Moreover, N0 is the Avogadro’s constant and DHm the enthalpy of solution of 1 mol c0 phase in c phase at coarsening temperature. Clearly, with regard to Eq. (7), the values of ac0 , ac and DHm are unknown, which, herein, were determined using Thermo-Physical Properties module in ‘‘Single Crystal’’ database of JMatPro 7.0 (thermodynamics
database). In order to show the feasibility of this methodology, a comparison was presented between the reference value and calculated result by Eq. (7) for r. According to the composition and temperature (CMSX-4, 1000 °C) in Ref. [31], r was evaluated by Eq. (7) as 79.51 mJ/m2, which was pretty close to the reference value 80 mJ/m2 [31], implying Eq. (7) is suitable for the calculation of interfacial energy. Considering the effective diffusion coefficient and interfacial energy are tightly correlated with alloy composition, it can be inferred that alloy composition plays an import role in affecting the coarsening rate coefficient. Based on the above analysis, a methodology that applies the models of effective diffusion coefficient and interfacial energy for multi-component alloys in a modified Ostwald ripening theory was proposed to describe the coarsening behavior of c0 phases in Ni based SC superalloys. Consequently, an application of the present model in some typical Ni based SC superalloys is given as follows. Table 2 lists the compositions and thermal exposure temperatures (T) of some SC superalloys. As for CMSX-4, the model parameters (U, A(U), Deff, r, Na and Vm) have been determined as a
Fig. 1. Influence of temperature on (a) U and A(U), (b) Deff and r, (c) Na and Vm, and (d) comparison between experimental (dot, from [10]) and calculated (dashed line, by Eq. (3)) radius sizes of c0 phases in CMSX-4 as a function of time.
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Table 3 Calculated U, A(U), D0,eff, Qeff, Deff, r, Na, Vm and k (calculated by Eq.(3)) of IC6SX, CMSX-2 and CMSX-10 at 950 °C, it is noted that D0,eff of CMSX-10 is the lowest for these three alloys. Alloy IC6SX CMSX-2 CMSX-10
U 0.738 0.627 0.638
A(U) 20.47 17.78 18.04
D0,eff (m2/s) 4
2.05 ⁄ 10 2.61 ⁄ 105 2.32 ⁄ 105
Qeff (kJ/mol) 281.88 271.11 277.46
Deff (m2/s) 16
1.88 ⁄ 10 6.87 ⁄ 1017 3.28 ⁄ 1017
r (mJ/m2)
Na
Vm (mol/cm3)
k (nm/s1/3)
Characteristic
82.79 79.44 75.52
0.22 0.37 0.26
7.22 7.26 7.22
3.52 2.85 1.93
7.6Al0Re 5.6Al0Re 5.7Al6.3Re
Fig. 2. (a) Comparison between experimental (dot, from [7,8,13]) and calculated (dashed line, by Eq. (3)) k of IC6SX [13], CMSX-2 [8] and CMSX-10 [7] as a function of temperature; (b) comparison between calculated (by Eq.(3)) and experimental k [7–13]; (c) comparison between calculated (by JMatPro 7.0) and experimental k [7–13].
function of temperature2, as seen in Fig. 1a–c. It is easy to find that with increasing temperature, U, A(U), r and Na decreased while Deff and Vm increased. Especially for Deff, the corresponding order of magnitude increased from 1017 m2/s to 1016 m2/s at 8501000 °C suggesting the change of Deff is the main reason of significant increment in k with increasing temperature [10]. Furthermore, good agreement was achieved between the experimental data (dot, from [10]) and calculated results (dashed line, by Eq.(3)) for the radius sizes of c0 phases as a function of time (as seen in Fig. 1d). Table 3 gives the calculated values of model parameters for some Ni based SC superalloys at 950 °C. According to Table 1, Al element had the highest pre-exponential factor (one magnitude higher than that of Ni) and the activation energy lower than that of Ni, while Re element had the lowest pre-exponential factor (three magnitude higher than that of Ni) and the activation energy close to that of Ni. Accordingly, at the same element content, Al and Re respectively have the greatest influence on D0,eff and Qeff, further playing the most important role in affecting Deff. As shown in Tables 2 and 3, Al-rich Ni based SC superalloys have higher Al content (7.55 wt.% Al) and thus possess larger Deff than those of Ni based SC superalloys (4–6 wt.% Al). Meanwhile, compared with CMSX-2 (0Re), the Re content of CMSX-10 (6.3 wt.%) is higher, resulting in lower Deff. As shown in Table 3, through comparing the calculated values of model parameters for these three alloys, it was found that Deff had the greatest impact on k, in accordance with the fact that the coarsening of c0 phases during thermal exposure are mainly caused by the diffusion of elements. Furthermore, a comparison of experimentally obtained (dot, from [7,8,13]) and theoretically calculated (dashed line, by Eq. (3)) k for IC6SX [13], CMSX-23 [8] and CMSX-10 [7] at different temperatures is shown in Fig. 2a. In order to verify the rationality of the present methodology, Fig. 2b presents the experimental and theoretical values of k [7–13] for Ni based SC superalloys. Clearly, good agreement between experimental data and calculated results was achieved.
2 In the present work, U, Na and Vm were calculated by Thermodynamic Properties module and Thermo-Physical Properties module in ‘‘Single Crystal’’ database of JMatPro 7.0 (thermodynamics database). 3 Average values of three different heat treatment regimes.
Furthermore, a comparison between the current methodology and the Phase Transformation module of ‘‘Single Crystal’’ database of JMatPro 7.0 (dynamics database) was performed, as seen in Fig. 2b and c. The calculation method of coarsening rate coefficient in JMatPro 7.0 is also based on Ostwald ripening theory [14]. However, the calculated results of JMatPro 7.0 were mostly obviously higher than the experimental data, which might be caused by the different calculation method of D0,eff. JMatPro 7.0 used the arithmetic mean method to calculate D0,eff (one magnitude higher than that of harmonic mean method) and led to the overestimate of the coarsening rate coefficient. Applying the models of diffusion coefficient (including activation energy and pre-exponential factor) and interfacial energy dependent on alloy composition combined with the dimensionless factor (correlated with volume fraction of c0 phase) of coarsening rate coefficient, a modified Ostwald ripening theory was proposed to describe the coarsening behavior of c0 phases in Ni based SC superalloys during thermal exposure. Good agreement was also achieved between theoretical calculation and experimental results for the coarsening rate coefficients and radius sizes of c0 phases. Besides, as compared with JMatPro 7.0, the current model presents a better description for the coarsening behavior of c0 phases in Ni based SC superalloys and may play certain role in the optimization of alloy composition for Ni based SC superalloys. Acknowledgements The authors would like to acknowledge many useful discussions with Dr. Mingming Gong (State Key Laboratory of Solidification Processing, Northwestern Polytechnical University). This research is sponsored by National Nature Science Foundations of China under Grant Nos. 50871005 and 51371014, China Postdoctoral Science Foundation under Grant No. 2013M540037. References [1] R.C. Reed, The superalloys: fundamentals and applications, Cambridge University Press, Cambridge, 2008. Chapters 2 and 3. [2] R. Gilles, D. Mukherji, H. Eckerlebe, L. Karge, P. Staron, P. Strunz, Th. Lippmann, J. Alloys Comp. 612 (2014) 90. [3] F. Wang, D. Ma, J. Zhang, L. Liu, S. Bogner, A. Bührig-Polaczek, J. Alloys Comp. 616 (2014) 102. [4] S. Gao, Y.Z. Zhou, C.F. Li, J.P. Cui, Z.Q. Liu, T. Jin, J. Alloys Comp. 610 (2014) 589.
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Contents lists available at ScienceDirect
Journal of Alloys and Compounds journal homepage: www.elsevier.com/locate/jalcom
Letter
Application of a modified Ostwald ripening theory in coarsening of c0 phases in Ni based single crystal superalloys Cheng Ai a, Xinbao Zhao b, Jian Zhou a,⇑, Heng Zhang a, Lei Liu a, Yanling Pei a, Shusuo Li a,⇑, Shengkai Gong a a b
School of Materials Science and Engineering, Beihang University, No. 37 Xueyuan Road, Beijing 100191, China Xi’an Thermal Power Research Institute Co., Ltd., Xi’an, Shaanxi 710032, China
a r t i c l e
i n f o
a b s t r a c t A modified Ostwald ripening theory was used to evaluate the coarsening behavior of c0 phases in Ni based single crystal superalloys during thermal exposure. After introducing a dimensionless factor correlated with volume fraction of c0 phase into the classical Lifshitz–Slyozov–Wagner (LSW) coarsening theory, this model was thus developed by coupling formulaic interfacial energy and diffusion coefficient (including activation energy and pre-exponential factor) as function of alloy composition. The coarsening rate coefficients and radius sizes of c0 phases were then presented, which show good agreement with the corresponding experimental results. Ó 2015 Elsevier B.V. All rights reserved.
Article history: Received 23 July 2014 Received in revised form 19 January 2015 Accepted 25 January 2015 Available online 2 February 2015 Keywords: Ni based single crystal superalloy c0 phase Diffusion Coarsening Ostwald ripening
Ni based single crystal (SC) superalloys have been most widely applied in modern turbine blades due to their excellent mechanical properties, which largely depend on the strengthening phase (c0 phases) with an ordered structure (L12) [1–4]. However, SC superalloys must serve at high temperature for comparatively long time, resulting in significant increase of the particle sizes of c0 phases, which yet leads to the decrement of stress rupture properties [5– 7]. As a result, due to the importance of the particle sizes of c0 phases for the mechanical properties of superalloys, it is necessary to evaluate the evolution of particle sizes (coarsening) with time during thermal exposure. In fact, considering that a great deal of time and cost would be required to perform the experimental determination of particle sizes, a theoretical evaluation of coarsening behavior should be fairly helpful and deserved to be performed. Consequently, theoretical researches on the coarsening behavior of c0 phase in superalloys have been widely conducted based on classical LSW coarsening theory of Ostwald ripening [7–17]: 3
r 3 r 30 ¼ k t
ð1Þ 0
where r and r0 are the average radius of c phases at time t and t = 0 respectively, and k the coarsening rate coefficient determined by [14–17]:
⇑ Corresponding authors. Tel.: +86 10 82314488; fax: +86 10 82338200. E-mail addresses: [email protected] (J. Zhou), [email protected] (S. Li). http://dx.doi.org/10.1016/j.jallcom.2015.01.215 0925-8388/Ó 2015 Elsevier B.V. All rights reserved.
k¼
8Deff rNa V m 9RT
1=3 ð2Þ
where Deff is the effective diffusion coefficient, r the interfacial energy, Na the equilibrium mole fraction of alloying elements in c phases, Vm the molar volume of c0 phases, R the gas constant and T the coarsening temperature. Actually, the LSW coarsening theory assumes infinitesimally small volume fraction of secondary phases [15,16], which is obviously not suitable for the cases of Ni based SC superalloys due to their extremely high volume fractions of secondary phases (50–80%) [1,17]. The researches on Ni–Al binary (at high volume fraction region (50–90%)) [18] and Ni–Al–Mo ternary systems [19] both show the coarsening rate coefficients are significantly increased with increasing volume fractions of c0 phases (U). Therefore, in order to account for the deviation of theoretical calculations from experimental values, a dimensionless factor A(U) correlated with U was introduced, which has various mathematical forms based on different theories, such as Modified LSW (MLSW), Tsumaraya–Miyata (TM), Marqusee–Ross (MR) and Voorhees–Glicksman (VG) theories [17]. Consequently, Eq. (2) is thus turned into:
k¼
8Að/ÞDeff rNa V m 9RT
1=3 ð3Þ
All of these models indicate that A(U) is independent on alloy systems but largely relies on the volume fraction (U). Generally, a positive correlation of A(U) with U was found, which actually
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C. Ai et al. / Journal of Alloys and Compounds 632 (2015) 558–562
resulted from the fact that higher U leads to shorter diffusion distances, thus promoting the diffusion of elements and then the coarsening of c0 phases [17–19]. Furthermore, it has been shown by Wang et al. [19] that A(U) for Ni–Al–Mo ternary alloys rapidly increased at high U region (U = 0.3–0.6), which can be preferably described by MLSW/TM/VG theory [17]. Besides, A(U) were also found to be sensitive to the morphology of c0 phases, e.g. A(U) for cubic c0 phases was lower than that for spherical c0 phases at the same volume fraction of c0 phases [19]. Fortunately, the morphologies of c0 phases in Ni based SC superalloys are cubic or near cubic, which means that A(U) only relies on U during the process of coarsening. Therefore, a predefined A(U) in accordance with cubic/near cubic morphology can be applied in modeling the coarsening behavior of c0 phases in Ni based SC superalloys. It was found that through the comparison between the calculated results of different models (not shown in this paper), the model IV of TM theory, which assumed that the particles of secondary phase are dispersed throughout the matrix in the unoccupied region available to them within the constraint of no particle interpenetration, presented the best agreement between theoretical calculation and experimental data, and was thus chosen to revise the LSW theory [20]. Actually, as for this model, the differential equation incorporating A(U) cannot be analytically solved so that the analytical formula of A(U) is incapable of being achieved and thus replaced by the fitted polynomial (i.e. A(U) = 2.22 + 25.55 ⁄ U 1.60 ⁄ U2 + 0.65 ⁄ U3). It should be noted that in terms of Eq. (3), the parameters Deff and r are closely correlated with alloy compositions, resulting in a composition-dependent rate coarsening coefficient. Considering the complexity of multicomponent alloys, there are a few researches focusing on the quantitative relationship between coarsening rate coefficients and alloy compositions for superalloys. Based on the philosophy of LSW theory, Li calculated coarsening rate coefficients of c0 phases in polycrystalline Ni based superalloys, and good agreement with experimental results has been achieved [14]. Furthermore, using Li0 s methodology, Phase Transformation module in ‘‘Single Crystal’’ database of software JMatPro 7.0 (dynamics database) can be used to determine coarsening rate coefficients of c0 phases in Ni based SC superalloys [21]. Nonetheless, the accuracy of which still remains to be verified. The key to quantitatively evaluate coarsening behavior is to establish the function-relation between the physical parameters in Eq. (3) (e.g. effective diffusion coefficient and interfacial energy, etc.) and the corresponding alloy composition. Therefore, in the current work, we tried to adopt reasonable physical models of parameters and then incorporated them into the modified Oswald equation to derive the alloy-composition dependent coarsening behavior. Normally, as for Deff, considering its Arrhenius form:
Deff ¼ D0;eff eQ eff =RT
ð4Þ
where D0,eff and Qeff are the pre-exponential factor and activation energy of effective diffusion coefficient respectively, and both of D0,eff and Qeff are functions of alloy composition. Generally, two types of function-relation between Qeff and alloy composition exist. P One is Qeff = QNi,Ni + mCmQNi,m, where QNi,Ni is the intrinsic diffusion activation energy for pure Ni, QNi,m the diffusion activation energy of element m in Ni, and Cm the atomic concentration of element m. Using this relation, Zhu et al. [22] calculated the activation energies during creep of SC superalloys, which exceeds 300 kJ/mol and are obviously higher than the experimental values during unstressed thermal exposure of SC superalloys (260.09 kJ/mol1 [8] and 272.40 kJ/mol [10]). As for another relation, i.e. 1
Average value of three different heat treatment regimes.
Q eff
X ¼ C m Q Ni;m
ð5Þ
m
good agreements between theoretical calculations and experimental results have been achieved, which can be considered to effectively reflect the contributions of all elements in SC superalloys into the diffusion behavior during thermal exposure. For instance, with regard to superalloys CMSX-2 [8], based on the corresponding alloy compositions, Qeff was calculated by Eq. (5) as 271.11 kJ/mol, which is close to Qexperiment = 260.09 kJ/mol [8]. Besides, as for superalloys CMSX-4, the corresponding Qeff was calculated to be 271.73 kJ/mol, in agreement with Qexperiment = 272.40 kJ/mol [10]. It should be noted that, as for the above calculation, the data of QNi,m was chosen based on Ni-m binary interdiffusion. Actually, the addition of ternary element would have influence on the value of QNi,m [23]. Nevertheless, considering the above calculated results by Eq. (5) are pretty close to the experimental data, for the sake of simplicity, the effect of other elements on Ni-m binary interdiffusion (i.e. the interaction of alloying elements) was not taken into account in this work. As for the pre-exponential factor D0,eff, the harmonic mean method was adopted to consider the contribution of alloy elements in this work [22]:
1 D0;eff ¼ X C m
ð6Þ
m
DNi;m 0
In fact, pre-exponential factors can also be determined by arithP metic mean method, i.e. D0,eff = mCmDNi,m [14,22]. The reason for 0 choosing harmonic mean method was depicted as follows. Normally, the addition of refractory elements can decrease diffusion coefficients [22]. Therefore, when adopting the function-relation Eq. (5), Qeff is calculated to be smaller than QNi,Ni (284 kJ/mol), thus leading to a larger Deff. In order to match the calculated values and experimental results, the theoretically determined D0,eff of SC superalloys should be significantly lower than DNi–Ni 0 (1.90 ⁄ 104 m2/s). Zhu0 s study showed that the order of magnitude for D0,eff calculated using harmonic mean method is 105 m2/s, which is lower than that for DNi–Ni (104 m2/s) by one order of 0 magnitude, as well as that for D0,eff (104 m2/s) calculated using arithmetic mean method [22]. Moreover, previous research shows the effective diffusion coefficients significantly decrease with the addition of a spot of refractory elements (such as Re) [22]. By comparing these two methods, only the harmonic mean method (Eq. (6)) can present D0,eff, which is notably affected by the addition of a small amount of elements when they possess low diffusion coefficient, and is thus more suitable for the theoretical evaluation of D0,eff [22]. In order to calculate Qeff and D0,eff, the pre-exponential factors (DNi,m ) and activation energies (QNi,m) for the interdiffusion of 0 element m in Ni is shown in Table 1.
Table 1 DNi,m and QNi,m for the interdiffusion of element m in Ni. 0 Element
DNi,m (m2/s) 0
QNi,m (kJ/mol)
Refs.
Ni Al Co Cr Hf Mo Re Ru Ta Ti W
1.90 ⁄ 104 1.00 ⁄ 103 7.50 ⁄ 105 3.00 ⁄ 106 1.62 ⁄ 104 1.15 ⁄ 104 8.20 ⁄ 107 2.48 ⁄ 104 2.19 ⁄ 105 4.10 ⁄ 104 8.00 ⁄ 106
284.00 272.09 285.10 170.70 250.10 281.30 255.00 304.40 251.00 275.00 264.00
[22] [24] [22] [22] [25] [26] [27] [28] [27] [29] [27]
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Table 2 The composition of Ni based SC superalloys and thermal exposure temperature (T) used in this work (wt.%). Classification
Alloy
Ni
Al
Co
Cr
Hf
Mo
Re
Ru
Ta
Ti
W
C
T (°C)
Refs.
1st generation
CMSX-2 M09A
Bal. Bal.
5.62 4.50
4.60 9.00
7.80 12.00
– –
0.60 2.00
– –
– –
6.00 5.00
1.00 2.50
7.90 4.00
– 0.01
800–1050 900
[8] [9]
2st generation
CMSX-4 Alloy 2Re12Co
Bal. Bal.
5.60 6.00
9.00 12.00
7.00 3.00
0.10 0.10
0.60 1.00
3.00 2.00
– –
7.00 8.00
1.00 –
6.00 6.00
– –
850–1000 950–1050
[10] [11]
3rd generation
Alloy 4Re0Co Alloy 4Re3Co Alloy 4Re12Co CMSX-10
Bal. Bal. Bal. Bal.
6.00 6.00 6.00 5.70
– 3.00 12.00 3.30
3.00 3.00 3.00 2.20
0.10 0.10 0.10 –
1.00 1.00 1.00 0.40
4.00 4.00 4.00 6.30
– – – –
8.00 8.00 8.00 8.30
– – – 0.23
6.00 6.00 6.00 5.50
– – – –
950–1050 950–1050 950–1050 950–1050
[11] [11] [11] [7]
Ru-containing Al-rich
Alloy 15.8Co IC6SX
Bal. Bal.
6.00 7.55
15.80 –
3.40 –
– –
2.60 13.96
2.10 –
2.00 –
8.00 –
– –
6.10 –
– –
1100 1100
[12] [13]
In addition to Deff, r also plays an important role in determining k. As for Ni based superalloys, considering the interfaces of c (matrix phase) and c0 (secondary phase) phases are coherent [1], it can thus be calculated by Li according to the following relation [14]:
r¼
zk N DH m zk N0
ð7Þ
where zk⁄ and zk are the number of broken bonds and coordination number in the kth shell respectively. Actually, zk⁄/zk = 0.3526 is derived based on the nearest-neighbor broken-bond analyze [30]. Besides, N⁄ is the number of cross bonds per atom at the interface and can be determined using the relation of N⁄ = 41.5/a2 with a = (ac0 + ac)/2, where ac0 and ac are the lattice constants of c0 and c phases at coarsening temperature respectively. Moreover, N0 is the Avogadro’s constant and DHm the enthalpy of solution of 1 mol c0 phase in c phase at coarsening temperature. Clearly, with regard to Eq. (7), the values of ac0 , ac and DHm are unknown, which, herein, were determined using Thermo-Physical Properties module in ‘‘Single Crystal’’ database of JMatPro 7.0 (thermodynamics
database). In order to show the feasibility of this methodology, a comparison was presented between the reference value and calculated result by Eq. (7) for r. According to the composition and temperature (CMSX-4, 1000 °C) in Ref. [31], r was evaluated by Eq. (7) as 79.51 mJ/m2, which was pretty close to the reference value 80 mJ/m2 [31], implying Eq. (7) is suitable for the calculation of interfacial energy. Considering the effective diffusion coefficient and interfacial energy are tightly correlated with alloy composition, it can be inferred that alloy composition plays an import role in affecting the coarsening rate coefficient. Based on the above analysis, a methodology that applies the models of effective diffusion coefficient and interfacial energy for multi-component alloys in a modified Ostwald ripening theory was proposed to describe the coarsening behavior of c0 phases in Ni based SC superalloys. Consequently, an application of the present model in some typical Ni based SC superalloys is given as follows. Table 2 lists the compositions and thermal exposure temperatures (T) of some SC superalloys. As for CMSX-4, the model parameters (U, A(U), Deff, r, Na and Vm) have been determined as a
Fig. 1. Influence of temperature on (a) U and A(U), (b) Deff and r, (c) Na and Vm, and (d) comparison between experimental (dot, from [10]) and calculated (dashed line, by Eq. (3)) radius sizes of c0 phases in CMSX-4 as a function of time.
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Table 3 Calculated U, A(U), D0,eff, Qeff, Deff, r, Na, Vm and k (calculated by Eq.(3)) of IC6SX, CMSX-2 and CMSX-10 at 950 °C, it is noted that D0,eff of CMSX-10 is the lowest for these three alloys. Alloy IC6SX CMSX-2 CMSX-10
U 0.738 0.627 0.638
A(U) 20.47 17.78 18.04
D0,eff (m2/s) 4
2.05 ⁄ 10 2.61 ⁄ 105 2.32 ⁄ 105
Qeff (kJ/mol) 281.88 271.11 277.46
Deff (m2/s) 16
1.88 ⁄ 10 6.87 ⁄ 1017 3.28 ⁄ 1017
r (mJ/m2)
Na
Vm (mol/cm3)
k (nm/s1/3)
Characteristic
82.79 79.44 75.52
0.22 0.37 0.26
7.22 7.26 7.22
3.52 2.85 1.93
7.6Al0Re 5.6Al0Re 5.7Al6.3Re
Fig. 2. (a) Comparison between experimental (dot, from [7,8,13]) and calculated (dashed line, by Eq. (3)) k of IC6SX [13], CMSX-2 [8] and CMSX-10 [7] as a function of temperature; (b) comparison between calculated (by Eq.(3)) and experimental k [7–13]; (c) comparison between calculated (by JMatPro 7.0) and experimental k [7–13].
function of temperature2, as seen in Fig. 1a–c. It is easy to find that with increasing temperature, U, A(U), r and Na decreased while Deff and Vm increased. Especially for Deff, the corresponding order of magnitude increased from 1017 m2/s to 1016 m2/s at 8501000 °C suggesting the change of Deff is the main reason of significant increment in k with increasing temperature [10]. Furthermore, good agreement was achieved between the experimental data (dot, from [10]) and calculated results (dashed line, by Eq.(3)) for the radius sizes of c0 phases as a function of time (as seen in Fig. 1d). Table 3 gives the calculated values of model parameters for some Ni based SC superalloys at 950 °C. According to Table 1, Al element had the highest pre-exponential factor (one magnitude higher than that of Ni) and the activation energy lower than that of Ni, while Re element had the lowest pre-exponential factor (three magnitude higher than that of Ni) and the activation energy close to that of Ni. Accordingly, at the same element content, Al and Re respectively have the greatest influence on D0,eff and Qeff, further playing the most important role in affecting Deff. As shown in Tables 2 and 3, Al-rich Ni based SC superalloys have higher Al content (7.55 wt.% Al) and thus possess larger Deff than those of Ni based SC superalloys (4–6 wt.% Al). Meanwhile, compared with CMSX-2 (0Re), the Re content of CMSX-10 (6.3 wt.%) is higher, resulting in lower Deff. As shown in Table 3, through comparing the calculated values of model parameters for these three alloys, it was found that Deff had the greatest impact on k, in accordance with the fact that the coarsening of c0 phases during thermal exposure are mainly caused by the diffusion of elements. Furthermore, a comparison of experimentally obtained (dot, from [7,8,13]) and theoretically calculated (dashed line, by Eq. (3)) k for IC6SX [13], CMSX-23 [8] and CMSX-10 [7] at different temperatures is shown in Fig. 2a. In order to verify the rationality of the present methodology, Fig. 2b presents the experimental and theoretical values of k [7–13] for Ni based SC superalloys. Clearly, good agreement between experimental data and calculated results was achieved.
2 In the present work, U, Na and Vm were calculated by Thermodynamic Properties module and Thermo-Physical Properties module in ‘‘Single Crystal’’ database of JMatPro 7.0 (thermodynamics database). 3 Average values of three different heat treatment regimes.
Furthermore, a comparison between the current methodology and the Phase Transformation module of ‘‘Single Crystal’’ database of JMatPro 7.0 (dynamics database) was performed, as seen in Fig. 2b and c. The calculation method of coarsening rate coefficient in JMatPro 7.0 is also based on Ostwald ripening theory [14]. However, the calculated results of JMatPro 7.0 were mostly obviously higher than the experimental data, which might be caused by the different calculation method of D0,eff. JMatPro 7.0 used the arithmetic mean method to calculate D0,eff (one magnitude higher than that of harmonic mean method) and led to the overestimate of the coarsening rate coefficient. Applying the models of diffusion coefficient (including activation energy and pre-exponential factor) and interfacial energy dependent on alloy composition combined with the dimensionless factor (correlated with volume fraction of c0 phase) of coarsening rate coefficient, a modified Ostwald ripening theory was proposed to describe the coarsening behavior of c0 phases in Ni based SC superalloys during thermal exposure. Good agreement was also achieved between theoretical calculation and experimental results for the coarsening rate coefficients and radius sizes of c0 phases. Besides, as compared with JMatPro 7.0, the current model presents a better description for the coarsening behavior of c0 phases in Ni based SC superalloys and may play certain role in the optimization of alloy composition for Ni based SC superalloys. Acknowledgements The authors would like to acknowledge many useful discussions with Dr. Mingming Gong (State Key Laboratory of Solidification Processing, Northwestern Polytechnical University). This research is sponsored by National Nature Science Foundations of China under Grant Nos. 50871005 and 51371014, China Postdoctoral Science Foundation under Grant No. 2013M540037. References [1] R.C. Reed, The superalloys: fundamentals and applications, Cambridge University Press, Cambridge, 2008. Chapters 2 and 3. [2] R. Gilles, D. Mukherji, H. Eckerlebe, L. Karge, P. Staron, P. Strunz, Th. Lippmann, J. Alloys Comp. 612 (2014) 90. [3] F. Wang, D. Ma, J. Zhang, L. Liu, S. Bogner, A. Bührig-Polaczek, J. Alloys Comp. 616 (2014) 102. [4] S. Gao, Y.Z. Zhou, C.F. Li, J.P. Cui, Z.Q. Liu, T. Jin, J. Alloys Comp. 610 (2014) 589.
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