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Equilibrium solubility products of molybdenum carbide and tungsten carbide in iron
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Scripta Materialia 66 (2012) 243–246 www.elsevier.com/locate/scriptamat
Equilibrium solubility products of molybdenum carbide and tungsten carbide in iron Erik J. Pavlina,a,⇑ John G. Speerb and Chester J. Van Tyneb a
Pohang University of Science and Technology, Graduate Institute of Ferrous Metallurgy, Pohang, Republic of Korea b Colorado School of Mines, Advanced Steel Processing and Products Research Center, Golden, CO, USA Received 29 August 2011; revised 29 October 2011; accepted 31 October 2011 Available online 7 November 2011
Solubility products for molybdenum carbide and tungsten carbide are absent from the literature despite their importance in secondary hardening steels and in microalloyed steels. Equilibrium solubility products were calculated for MoC, Mo2C, WC and W2C in ferrite and austenite. Molybdenum and tungsten carbides exhibit greater solubility than traditional microalloy carbides in iron and their predicted solution behavior is similar to that of other Group VI transition metal carbides. The calculated solubility products are consistent with precipitation behavior during secondary hardening in steels. Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Carbides; Iron alloys; Steels; Precipitation; Solubility
Precipitation of molybdenum carbide and tungsten carbide is widely utilized to increase the strength at elevated temperatures in high-speed or tool steel applications [1]. Precipitation of M2C-type carbides occurs during tempering of a martensitic microstructure and the associated increase in hardness is designated as secondary hardening [2]. Carbides with an MC stoichiometry are generally found in multicomponent alloy systems, particularly those containing vanadium in addition to molybdenum or tungsten [3]. In molybdenumcontaining high-strength, low-alloy (HSLA) steels the molybdenum is usually added to control the austenite decomposition kinetics and it has been observed in relatively fine complex microalloy carbonitride precipitates that form in ferrite, and to a lesser extent in austenite [4–8]. Molybdenum precipitation in ferrite appears to be promoted by thermomechanical processing schemes that utilize an interrupted accelerated cooling step to produce the desired ferritic or bainitic microstructure followed by slow cooling [6–8]. It has also been suggested that molybdenum inhibits niobium carbonitride coarsening in fire-resistant steels by segregating at the matrix/precipitate interface [9]. Complex precipitate compositions can be predicted by equilibrium considerations based on the Temkin– Hillert and Staffanson model [10–13]. Houghton [4,14]
has calculated complex precipitate compositions based on a general quasi-regular solution model. Each of these model formulations make equilibrium calculations employing solubility products for each binary carbide or nitride compound that comprise the complex precipitate. However, published solubility products for carbides of molybdenum and tungsten in ferrite and austenite are lacking, although the molybdenum carbide solubility product is speculated to be greater than that of vanadium carbide [15]. Ashby and Easterling [16] cite a solubility product for Mo2C in austenite from work by Klumpes [17]. However, examination of the original source did not uncover the stated solubility product and consequently its provenance is uncertain. Solubility products for molybdenum carbide and tungsten carbide would aid in alloy design of secondary hardening steels used in high-speed and tool steel applications as well as for HSLA steels used in linepipe and fire-resistant steel applications. In the present study, solubility products for MoC, Mo2C, WC and W2C in austenite and ferrite are derived using available thermodynamic data. An example of the calculation procedure is shown for the Mo2C compound in austenite, and solubility products for all other compounds are calculated following the same general procedure. Consider the reaction:
⇑ Corresponding author. E-mail: [email protected]
1 1 Moc þ C c () Mo2 C 2 2
1359-6462/$ - see front matter Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2011.10.047
ð1Þ
244
E. J. Pavlina et al. / Scripta Materialia 66 (2012) 243–246
where the superscript indicates solution in austenite, c. At equilibrium, the change in Gibbs free energy is zero and can be expressed as: 10 1 GMo2 C GMo GC ð2Þ 2 2 i is the where 0GMo2C is the free energy of Mo2C and G partial molar free energy of component i. The partial molar free energy for each component can be expressed in the form:
log10 acC ¼
DG ¼ 0 ¼
Gi ¼ 0 Gi þ RT log aci
ð3Þ
where 0Gi is the free energy of component i in its standard state (e.g. carbon in the form of graphite), R is the ideal gas constant, T is absolute temperature, and aci is the activity of component i in austenite. The change in free energy for the carbide precipitation reaction, D0 GMo2 C , can now be expressed as: 10 1 1 DGMo2 C ¼ 0 GMo2 C 0 GMo 0 GC 2 2 2 c1=2 c ¼ RT log aMo aC
ð4Þ
based on the assumption that the activity of the pure Mo2C compound is 1. The activity of each component is related to the mole fraction, Xi, in solution in austenite by the relation: aci ¼ Cci X ci
ð5Þ
where Ci is the activity coefficient of component i. Substituting and rearranging yields: 0 D GMo2 C 1=2 1 1 X Mo X C ¼ ðCcMo Þ ðCcC Þ 2 exp : ð6Þ 2RT Eq. (6) is the solubility product for Mo2C in austenite and defines the locus of molybdenum and carbon solute concentrations in austenite that satisfy equilibrium with the Mo2C precipitate at a given temperature. Solubility products are usually expressed using compositions in weight percent and an approximate relation between the mole fraction and weight percent of component i, Mi, is: Xi
1 AFe Mi 100 Ai
ð7Þ
where Ai is the atomic mass of component i. Substitution yields: 1=2
1
1
K cM02 C ¼ M Mo :M C ¼ ðCcMo Þ ðCcC Þ 2 ! 3=2 D0GMo2 C 100 1=2 exp AMo AC AFe 2RT
ð8Þ
where K cMo2 C is the equilibrium solubility product. Finally, the solubility product is conventionally expressed in the form of: log10 K cMo2 C ¼ log10 ðM Mo :M 1=2 c Þ ¼ P
Q T
ð9Þ
where P and Q can be obtained from regression analysis of log10 K cMo2 C vs. inverse temperature data. Carbon activity in austenite can be calculated using the expression of Ban-Ya et al. [18]:
3770 þ 2:72log10 T 10:525 T 3900 M C þ log10 M C þ 0:43 þ T
ð10Þ
where MC is the weight percent of carbon. Eq. (10) was determined over the temperature range 1173–1673 K. With the carbon activity obtained from Eq. (10), the activity coefficient can be calculated according to Eq. (5). The effects of other alloying elements on carbon activity are neglected in the present analysis. A regular solution model of Fe–Mo is used to calculate the activity coefficient of molybdenum in solution in austenite given by: ð1 X cMo Þ2 c LFe–Mo ð11Þ RT where LcFe–Mo is the regular solution interaction parameter of iron and molybdenum; a value of 9686 J mol1 is used in the present derivation [19]. Table 1 lists the regular solution interaction parameters for the Fe–Mo and Fe– W systems. For dilute solutions, the activity coefficient of a solute element is essentially constant but the use of Eqs. (10) and (11) require the selection of some carbon and molybdenum solute concentration and therefore some flexibility is possible in the selection provided that the solute fractions are low. In the present case, Fe– 0.1C–0.5Mo and Fe–0.1C–0.5W (in wt.%) steel compositions were selected to calculate activity coefficients for each of the two systems. Solubility products calculated for these compositions showed no significant difference with calculations made using leaner alloy compositions. The standard free energy of formation of Mo2C can be calculated according to Seigle et al. [20]. Their expression shows good agreement (2%) with Gleiser and Chipman’s [21] earlier expression that extends to a lower temperature (1200 K) and a linear extrapolation to even lower temperatures is assumed to be reasonable. Table 2 lists expressions for the standard free energy of formation of the MoC, Mo2C, WC and W2C compounds. In all cases, the expressions are assumed valid for temperatures corresponding to the ferrite and austenite phase fields. The expression for D0GMoC is taken from the compilation of Elliot and Gleiser [22], who expressed some reservation about the quality of the thermodynamic data available for MoC. Should improved thermodynamic data for MoC be produced, the present solubility product calculation method is accessible and can be easily modified. With the available calculation methods for CC, CMo and D0 GMo2 C , the solubility product of Mo2C in austenite was computed over the available range of carbon activity data from a linear regression of the calculated values. The resulting solubility product of Mo2C in austenite is: logCcMo ¼
1=2
log10 K cMo2 C ¼ log10 ðM Mo M C Þ ¼ 3:04
2814 : T
ð12Þ
Table 1. Regular solution interaction parameters for binary Fe–Mo and Fe–W systems [19]. Binary system
Iron phase
Interaction parameter (J mol1)
Fe–Mo
Ferrite Austenite Ferrite Austenite
17780 9686 9010 7800–0.45T
Fe–W
E. J. Pavlina et al. / Scripta Materialia 66 (2012) 243–246
245
Table 2. Standard Gibbs free energy of formation for MoC, Mo2C, WC and W2C. Reaction
D0G (J mol1)
Source
Mo + C ! MoC 2Mo + C ! Mo2C W + C ! WC 2 W + C ! W2C
40585–58.58T 47350–9.46T 42300–4.98T 40500–2.34T
[22] [20] [23] [23]
An identical analysis can be conducted to calculate a solubility product for Mo2C in ferrite as well as for MoC, WC and W2C in ferrite and austenite. The outlined calculation procedure only needs to be modified to reflect the stoichiometry of the reaction, the metal species in the carbide compound, and the activity changes due to the precipitation reaction occurring in a ferrite matrix instead of austenite. The activity coefficient of carbon in ferrite is calculated using: 5846 þ 2:687 ð13Þ T which was determined by Lobo and Geiger [24] over the temperature range 953–1118 K. Activity coefficients for molybdenum or tungsten are calculated using the appropriate regular solution interaction parameter listed in Table 1. Table 3 lists the calculated solubility products for MoC, Mo2C, WC and W2C in ferrite and austenite. Figures 1 and 2 compare the calculated results with solubility products for other Group IV (Ti and Zr), Group V (V, Nb and Ta) and Group VI (Cr and Mo) transition metal carbides in ferrite and austenite, respectively, compiled from literature [14,25–34]. A low solubility product indicates low solubility. The questionable Mo2C solubility product in austenite cited by Ashby and Easterling [16] and modified to have a MoC0.5 stoichiometry is also included in the comparison. In general, the relative solubility of transition metal carbides (and nitrides) is lowest for Group IV compounds and greatest for Group VI compounds. The solubility products for MoC, Mo2C, WC and W2C exhibit behavior in line with that of chromium carbides in austenite. The high solubility of these compounds in austenite suggests that incorporation of molybdenum or tungsten into complex carbonitride precipitates formed in austenite may be limited. Rollason [35] examined the secondary hardening response of nominal Fe–0.12C steels containing approximately 0.5–3.0 wt.% Mo. Maximum hardness was obtained after tempering at 873 K for 3600 s resulting from the precipitation of needle-shaped Mo2C particles [2,36,37]. Figure 3a shows the calculated solubility log10 CaC ¼
Table 3. Equilibrium solubility productsa for MoC, Mo2C, WC and W2C in iron. Iron phase
Ferrite
Austenite
a
Compound
v
P
Q
MoC Mo2C WC W2C MoC Mo2C WC W2C
1 0.5 1 0.5 1 0.5 1 0.5
3.19 4.00 6.67 4.47 1.29 3.04 5.00 3.61
4649 5088 8914 5683 523 2814 6058 3566
Solubility products have the form: log10 ðM i M mC Þ ¼ P QT.
Figure 1. Comparison of Mo2C, WC, and W2C solubility products in ferrite (773–1184 K) with other carbide solubility products.
Figure 2. Comparison of Mo2C, WC, and W2C solubility products in austenite (1184–1666 K) with other carbide solubility products.
Figure 3. Calculated solubility product isotherm for (a) Mo2C in ferrite at 873 K with composition range of Rollason [35] indicated and (b) for MoC and Mo2C in ferrite at 923 K with compositions of Chen et al. [8] and Lee et al. [6,7] indicated. Dashed line indicates metastable equilibrium.
product isotherm for Mo2C in ferrite at 873 K with the shaded region indicating the steel composition range investigated by Rollason [35]. In this case, the ferrite is supersaturated with molybdenum at 873 K and precipitation of Mo2C is predicted based on thermodynamic equilibrium. Chen et al. [8] investigated precipitation in ferrite during slow cooling (60.5 K s1) following interrupted accelerated cooling in a titanium-containing Fe–0.1C–0.2Mo steel. The accelerated cooling stop temperature was 923 K in their study. Relatively fine (<7 nm) carbonitride precipitates containing titanium and molybdenum were observed. Figure 3b shows the calculated solubility product isotherms for MoC and Mo2C in ferrite at 923 K and indicates the bulk composition investigated by Chen et al. [8]. Both molybdenum carbide compounds are predicted to be present at 923 K. During the period of slow cooling below 923 K, it is expected that precipitation would continue because the driving force would increase as temperature decreases and solute diffusivity would still be sufficient for precipitate growth in the early stages of cooling (D0T = 100 K), as evidenced by the temperature
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E. J. Pavlina et al. / Scripta Materialia 66 (2012) 243–246
Figure 4. Calculated solubility product isotherm for WC and W2C in ferrite at 923 K with composition of Crafts and Lamont [40] indicated. Dashed line indicates metastable equilibrium.
range for maximum hardness in secondary hardening steels containing molybdenum. Lee et al. [6,7] examined niobium carbonitride precipitation in a series of molybdenum-containing (60.6 wt.% Mo) hot-rolled Fe–0.08C–0.03Nb steels coiled at temperatures between 823 and 923 K. They reported that the concentration of molybdenum in niobium-rich carbide precipitates increases as the precipitate formation temperature decreases, with those that form in ferrite having the highest molybdenum content. The steel compositions studied by Lee et al. [6,7] are indicated in Figure 3b, and molybdenum carbide precipitation would be expected for the range of coiling temperatures examined in their study. Secondary hardening of tungsten-containing steels occurs over a similar temperature range (773–923 K) as for molybdenum-containing steels [3,38–40]. However, the atomic weight of tungsten is about twice that of molybdenum and therefore approximately twice as much tungsten is required (on a weight basis) to achieve the same strengthening effect or potency as molybdenum [3]. For this reason, tungsten-containing secondary hardening steels usually contain a minimum of 2 wt.% W which is far in excess of the predicted solubility limit for either tungsten carbide compound in ferrite. To illustrate this point, Figure 4 shows solubility isotherms for both WC and W2C in ferrite at 923 K. Tungsten carbide precipitation is predicted at any temperature relevant to the tempering of typical tungsten-containing secondary hardening steels because the tungsten concentrations are so high. Figure 4 also indicates a Fe–0.47C–1.13W composition reported by Roberts and Cary [3] from an investigation by Crafts and Lamont [40]. This steel did not produce a true secondary hardening peak but did exhibit a lower rate of softening in comparison to the zero-tungsten base alloy and extensive tungsten carbide precipitation is predicted at 923 K based on the equilibrium solubility products. In summary, equilibrium solubility products were determined for molybdenum carbides and tungsten carbides in ferrite and austenite. High solubility is predicted for these carbide compounds in both ferrite and austenite. Finally, the solubility products predict precipitation behavior that is consistent with experimental studies of secondary hardening steels and microalloyed HSLA steels. [1] M.S. Bhat, W.M. Garrison Jr., V.F. Zackay, Mater. Sci. Eng., A 41 (1979) 1. [2] K.J. Irvine, F.B. Pickering, J. Iron Steel Res. Int. 194 (1960) 137. [3] G.A. Roberts, R.A. Cary, Tool Steels, Fourth ed., American Society for Metals, OH, 1980.
[4] D.C. Houghton, G.C. Weatherly, J.D. Embury, Characterization of carbonitrides in Ti bearing HSLA steels, in: Thermomechanical Processing of Microalloyed Austenite, Pittsburgh, 1982, pp. 267–292. [5] J.D. L’Ecuyer, G. L’Esperance, M.G. Akben, B. Bacroix, Acta Metall. Mater. 35 (1987) 1149. [6] W.B. Lee, S.G. Hong, C.G. Park, K.H. Kim, S.H. Park, Scripta Mater 43 (2000) 319. [7] W.B. Lee, S.G. Hong, C.G. Park, S.H. Park, Metall. Mater. Trans. A 33A (2002) 1689. [8] C.Y. Chen, H.W. Yen, F.H. Kao, W.C. Li, C.Y. Huan, J.R. Yang, S.H. Wang, Mater. Sci. Eng. A 499 (2009) 162. [9] R. Chijiiwa, H. Tamehiro, Y. Yoshida, K. Funato, R. Uemori, Y. Horii, Nippon Steel Technical Report 58 (1993) 47. [10] M. Temkin, Acta Physiochim. USSR 20 (1945) 411. [11] M. Hillert, L.I. Staffanson, Acta Chem. Scand. 24 (1970) 3618. [12] J.G. Speer, J.R. Michael, S.S. Hansen, Metall. Mater. Trans. A 18 (1987) 211. [13] P.R. Rios, Mater. Sci. Eng. A 142 (1991) 87. [14] D.C. Houghton, Acta Metall. Mater. 41 (1993) 2993. [15] B. Aronsson, Steel Strengthening Mechanisms, Climax Molybdenum Symposium, Zurich, 1969, pp. 77–88. [16] M.F. Ashby, K.E. Easterling, Acta Metall. Mater. 30 (1982) 1969. [17] H. Klumpes, DVS Ber. 32 (1974) 71. [18] S. Ban-Ya, J.F. Elliot, J. Chipman, Metall. Mater. Trans. 1 (1970) 1313. [19] G. Kirchner, H. Harvig, B. Uhrenius, Metall. Mater. Trans. 4 (1973) 1059. [20] L.L. Seigle, C.L. Chang, T.P. Sharma, Metall. Mater. Trans. A 10 (1979) 1223. [21] M. Gleiser, J. Chipman, J. Phys. Chem. 66 (1962) 1539. [22] J.F. Elliot, M. Gleiser, Thermochemistry for Steelmaking, Vol. 1, Addison-Wesley, Reading, MA, 1960. [23] D.K. Gupta, L.L. Seigle, Metall. Mater. Trans. A 6 (1975) 1939. [24] L.A. Lobo, G.H. Geiger, Metall. Mater. Trans. A 7 (1976) 1347. [25] B.C. De Cooman, J.G. Speer, I.Y. Pyshmintsev, N. Yoshinaga, Materials Design: The Key to Modern Steel Products, Grips Media GmbH, Germany, 2006. [26] K.A. Taylor, Scripta Metall. Mater. 32 (1995) 7. [27] R.C. Sharma, V.K. Lakshmanan, J.S. Kirkaldy, Metall. Mater. Trans. A 15 (1984) 545. [28] S. Koyama, T. Ishii, K. Narita, J. Jpn. Inst. Met. 37 (1973) 191. [29] J.A. Todd, P. Li, Metall. Mater. Trans. A 17 (1986) 1191. [30] H. Sekine, T. Inoue, M. Ogasawara, J. Iron. Steel Res. Int. Jpn. 8 (1968) 101. [31] K. Narita, J. Iron. Steel Res. Int. Jpn. 15 (1975) 147. [32] K.J. Irvine, F.P. Pickering, T. Gladman, J. Iron. Steel Res. Int. 205 (1967) 161. [33] W. Roberts, A. Sandberg, Swedish Institute for Metals Research, Report No. IM-1489, Stockholm, 1980. [34] H. Chino, K. Wada, Yawata Technical Report 251 (1965) 75. [35] E.C. Rollason, Fundamental Aspects of Molybdenum in Transformation of Steel, Climax Molybdenum Company, London. [36] J. Akre´, F. Danoix, H. Leitner, P. Auger, Ultramicroscopy 109 (2009) 518. [37] W. Rong, G.L. Dunlop, Acta Metall. Mater. 32 (1984) 1591. [38] E. Tekin, M.H. Richman, Metallography 3 (1970) 327. [39] H.F. Fischmeister, S. Karagoz, H.O. Andren, Acta Metall. Mater. 36 (1988) 8917. [40] W. Crafts, J.L. Lamont, T. AIME 180 (1949) 471–512.
Scripta Materialia 66 (2012) 243–246 www.elsevier.com/locate/scriptamat
Equilibrium solubility products of molybdenum carbide and tungsten carbide in iron Erik J. Pavlina,a,⇑ John G. Speerb and Chester J. Van Tyneb a
Pohang University of Science and Technology, Graduate Institute of Ferrous Metallurgy, Pohang, Republic of Korea b Colorado School of Mines, Advanced Steel Processing and Products Research Center, Golden, CO, USA Received 29 August 2011; revised 29 October 2011; accepted 31 October 2011 Available online 7 November 2011
Solubility products for molybdenum carbide and tungsten carbide are absent from the literature despite their importance in secondary hardening steels and in microalloyed steels. Equilibrium solubility products were calculated for MoC, Mo2C, WC and W2C in ferrite and austenite. Molybdenum and tungsten carbides exhibit greater solubility than traditional microalloy carbides in iron and their predicted solution behavior is similar to that of other Group VI transition metal carbides. The calculated solubility products are consistent with precipitation behavior during secondary hardening in steels. Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Carbides; Iron alloys; Steels; Precipitation; Solubility
Precipitation of molybdenum carbide and tungsten carbide is widely utilized to increase the strength at elevated temperatures in high-speed or tool steel applications [1]. Precipitation of M2C-type carbides occurs during tempering of a martensitic microstructure and the associated increase in hardness is designated as secondary hardening [2]. Carbides with an MC stoichiometry are generally found in multicomponent alloy systems, particularly those containing vanadium in addition to molybdenum or tungsten [3]. In molybdenumcontaining high-strength, low-alloy (HSLA) steels the molybdenum is usually added to control the austenite decomposition kinetics and it has been observed in relatively fine complex microalloy carbonitride precipitates that form in ferrite, and to a lesser extent in austenite [4–8]. Molybdenum precipitation in ferrite appears to be promoted by thermomechanical processing schemes that utilize an interrupted accelerated cooling step to produce the desired ferritic or bainitic microstructure followed by slow cooling [6–8]. It has also been suggested that molybdenum inhibits niobium carbonitride coarsening in fire-resistant steels by segregating at the matrix/precipitate interface [9]. Complex precipitate compositions can be predicted by equilibrium considerations based on the Temkin– Hillert and Staffanson model [10–13]. Houghton [4,14]
has calculated complex precipitate compositions based on a general quasi-regular solution model. Each of these model formulations make equilibrium calculations employing solubility products for each binary carbide or nitride compound that comprise the complex precipitate. However, published solubility products for carbides of molybdenum and tungsten in ferrite and austenite are lacking, although the molybdenum carbide solubility product is speculated to be greater than that of vanadium carbide [15]. Ashby and Easterling [16] cite a solubility product for Mo2C in austenite from work by Klumpes [17]. However, examination of the original source did not uncover the stated solubility product and consequently its provenance is uncertain. Solubility products for molybdenum carbide and tungsten carbide would aid in alloy design of secondary hardening steels used in high-speed and tool steel applications as well as for HSLA steels used in linepipe and fire-resistant steel applications. In the present study, solubility products for MoC, Mo2C, WC and W2C in austenite and ferrite are derived using available thermodynamic data. An example of the calculation procedure is shown for the Mo2C compound in austenite, and solubility products for all other compounds are calculated following the same general procedure. Consider the reaction:
⇑ Corresponding author. E-mail: [email protected]
1 1 Moc þ C c () Mo2 C 2 2
1359-6462/$ - see front matter Ó 2011 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2011.10.047
ð1Þ
244
E. J. Pavlina et al. / Scripta Materialia 66 (2012) 243–246
where the superscript indicates solution in austenite, c. At equilibrium, the change in Gibbs free energy is zero and can be expressed as: 10 1 GMo2 C GMo GC ð2Þ 2 2 i is the where 0GMo2C is the free energy of Mo2C and G partial molar free energy of component i. The partial molar free energy for each component can be expressed in the form:
log10 acC ¼
DG ¼ 0 ¼
Gi ¼ 0 Gi þ RT log aci
ð3Þ
where 0Gi is the free energy of component i in its standard state (e.g. carbon in the form of graphite), R is the ideal gas constant, T is absolute temperature, and aci is the activity of component i in austenite. The change in free energy for the carbide precipitation reaction, D0 GMo2 C , can now be expressed as: 10 1 1 DGMo2 C ¼ 0 GMo2 C 0 GMo 0 GC 2 2 2 c1=2 c ¼ RT log aMo aC
ð4Þ
based on the assumption that the activity of the pure Mo2C compound is 1. The activity of each component is related to the mole fraction, Xi, in solution in austenite by the relation: aci ¼ Cci X ci
ð5Þ
where Ci is the activity coefficient of component i. Substituting and rearranging yields: 0 D GMo2 C 1=2 1 1 X Mo X C ¼ ðCcMo Þ ðCcC Þ 2 exp : ð6Þ 2RT Eq. (6) is the solubility product for Mo2C in austenite and defines the locus of molybdenum and carbon solute concentrations in austenite that satisfy equilibrium with the Mo2C precipitate at a given temperature. Solubility products are usually expressed using compositions in weight percent and an approximate relation between the mole fraction and weight percent of component i, Mi, is: Xi
1 AFe Mi 100 Ai
ð7Þ
where Ai is the atomic mass of component i. Substitution yields: 1=2
1
1
K cM02 C ¼ M Mo :M C ¼ ðCcMo Þ ðCcC Þ 2 ! 3=2 D0GMo2 C 100 1=2 exp AMo AC AFe 2RT
ð8Þ
where K cMo2 C is the equilibrium solubility product. Finally, the solubility product is conventionally expressed in the form of: log10 K cMo2 C ¼ log10 ðM Mo :M 1=2 c Þ ¼ P
Q T
ð9Þ
where P and Q can be obtained from regression analysis of log10 K cMo2 C vs. inverse temperature data. Carbon activity in austenite can be calculated using the expression of Ban-Ya et al. [18]:
3770 þ 2:72log10 T 10:525 T 3900 M C þ log10 M C þ 0:43 þ T
ð10Þ
where MC is the weight percent of carbon. Eq. (10) was determined over the temperature range 1173–1673 K. With the carbon activity obtained from Eq. (10), the activity coefficient can be calculated according to Eq. (5). The effects of other alloying elements on carbon activity are neglected in the present analysis. A regular solution model of Fe–Mo is used to calculate the activity coefficient of molybdenum in solution in austenite given by: ð1 X cMo Þ2 c LFe–Mo ð11Þ RT where LcFe–Mo is the regular solution interaction parameter of iron and molybdenum; a value of 9686 J mol1 is used in the present derivation [19]. Table 1 lists the regular solution interaction parameters for the Fe–Mo and Fe– W systems. For dilute solutions, the activity coefficient of a solute element is essentially constant but the use of Eqs. (10) and (11) require the selection of some carbon and molybdenum solute concentration and therefore some flexibility is possible in the selection provided that the solute fractions are low. In the present case, Fe– 0.1C–0.5Mo and Fe–0.1C–0.5W (in wt.%) steel compositions were selected to calculate activity coefficients for each of the two systems. Solubility products calculated for these compositions showed no significant difference with calculations made using leaner alloy compositions. The standard free energy of formation of Mo2C can be calculated according to Seigle et al. [20]. Their expression shows good agreement (2%) with Gleiser and Chipman’s [21] earlier expression that extends to a lower temperature (1200 K) and a linear extrapolation to even lower temperatures is assumed to be reasonable. Table 2 lists expressions for the standard free energy of formation of the MoC, Mo2C, WC and W2C compounds. In all cases, the expressions are assumed valid for temperatures corresponding to the ferrite and austenite phase fields. The expression for D0GMoC is taken from the compilation of Elliot and Gleiser [22], who expressed some reservation about the quality of the thermodynamic data available for MoC. Should improved thermodynamic data for MoC be produced, the present solubility product calculation method is accessible and can be easily modified. With the available calculation methods for CC, CMo and D0 GMo2 C , the solubility product of Mo2C in austenite was computed over the available range of carbon activity data from a linear regression of the calculated values. The resulting solubility product of Mo2C in austenite is: logCcMo ¼
1=2
log10 K cMo2 C ¼ log10 ðM Mo M C Þ ¼ 3:04
2814 : T
ð12Þ
Table 1. Regular solution interaction parameters for binary Fe–Mo and Fe–W systems [19]. Binary system
Iron phase
Interaction parameter (J mol1)
Fe–Mo
Ferrite Austenite Ferrite Austenite
17780 9686 9010 7800–0.45T
Fe–W
E. J. Pavlina et al. / Scripta Materialia 66 (2012) 243–246
245
Table 2. Standard Gibbs free energy of formation for MoC, Mo2C, WC and W2C. Reaction
D0G (J mol1)
Source
Mo + C ! MoC 2Mo + C ! Mo2C W + C ! WC 2 W + C ! W2C
40585–58.58T 47350–9.46T 42300–4.98T 40500–2.34T
[22] [20] [23] [23]
An identical analysis can be conducted to calculate a solubility product for Mo2C in ferrite as well as for MoC, WC and W2C in ferrite and austenite. The outlined calculation procedure only needs to be modified to reflect the stoichiometry of the reaction, the metal species in the carbide compound, and the activity changes due to the precipitation reaction occurring in a ferrite matrix instead of austenite. The activity coefficient of carbon in ferrite is calculated using: 5846 þ 2:687 ð13Þ T which was determined by Lobo and Geiger [24] over the temperature range 953–1118 K. Activity coefficients for molybdenum or tungsten are calculated using the appropriate regular solution interaction parameter listed in Table 1. Table 3 lists the calculated solubility products for MoC, Mo2C, WC and W2C in ferrite and austenite. Figures 1 and 2 compare the calculated results with solubility products for other Group IV (Ti and Zr), Group V (V, Nb and Ta) and Group VI (Cr and Mo) transition metal carbides in ferrite and austenite, respectively, compiled from literature [14,25–34]. A low solubility product indicates low solubility. The questionable Mo2C solubility product in austenite cited by Ashby and Easterling [16] and modified to have a MoC0.5 stoichiometry is also included in the comparison. In general, the relative solubility of transition metal carbides (and nitrides) is lowest for Group IV compounds and greatest for Group VI compounds. The solubility products for MoC, Mo2C, WC and W2C exhibit behavior in line with that of chromium carbides in austenite. The high solubility of these compounds in austenite suggests that incorporation of molybdenum or tungsten into complex carbonitride precipitates formed in austenite may be limited. Rollason [35] examined the secondary hardening response of nominal Fe–0.12C steels containing approximately 0.5–3.0 wt.% Mo. Maximum hardness was obtained after tempering at 873 K for 3600 s resulting from the precipitation of needle-shaped Mo2C particles [2,36,37]. Figure 3a shows the calculated solubility log10 CaC ¼
Table 3. Equilibrium solubility productsa for MoC, Mo2C, WC and W2C in iron. Iron phase
Ferrite
Austenite
a
Compound
v
P
Q
MoC Mo2C WC W2C MoC Mo2C WC W2C
1 0.5 1 0.5 1 0.5 1 0.5
3.19 4.00 6.67 4.47 1.29 3.04 5.00 3.61
4649 5088 8914 5683 523 2814 6058 3566
Solubility products have the form: log10 ðM i M mC Þ ¼ P QT.
Figure 1. Comparison of Mo2C, WC, and W2C solubility products in ferrite (773–1184 K) with other carbide solubility products.
Figure 2. Comparison of Mo2C, WC, and W2C solubility products in austenite (1184–1666 K) with other carbide solubility products.
Figure 3. Calculated solubility product isotherm for (a) Mo2C in ferrite at 873 K with composition range of Rollason [35] indicated and (b) for MoC and Mo2C in ferrite at 923 K with compositions of Chen et al. [8] and Lee et al. [6,7] indicated. Dashed line indicates metastable equilibrium.
product isotherm for Mo2C in ferrite at 873 K with the shaded region indicating the steel composition range investigated by Rollason [35]. In this case, the ferrite is supersaturated with molybdenum at 873 K and precipitation of Mo2C is predicted based on thermodynamic equilibrium. Chen et al. [8] investigated precipitation in ferrite during slow cooling (60.5 K s1) following interrupted accelerated cooling in a titanium-containing Fe–0.1C–0.2Mo steel. The accelerated cooling stop temperature was 923 K in their study. Relatively fine (<7 nm) carbonitride precipitates containing titanium and molybdenum were observed. Figure 3b shows the calculated solubility product isotherms for MoC and Mo2C in ferrite at 923 K and indicates the bulk composition investigated by Chen et al. [8]. Both molybdenum carbide compounds are predicted to be present at 923 K. During the period of slow cooling below 923 K, it is expected that precipitation would continue because the driving force would increase as temperature decreases and solute diffusivity would still be sufficient for precipitate growth in the early stages of cooling (D0T = 100 K), as evidenced by the temperature
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Figure 4. Calculated solubility product isotherm for WC and W2C in ferrite at 923 K with composition of Crafts and Lamont [40] indicated. Dashed line indicates metastable equilibrium.
range for maximum hardness in secondary hardening steels containing molybdenum. Lee et al. [6,7] examined niobium carbonitride precipitation in a series of molybdenum-containing (60.6 wt.% Mo) hot-rolled Fe–0.08C–0.03Nb steels coiled at temperatures between 823 and 923 K. They reported that the concentration of molybdenum in niobium-rich carbide precipitates increases as the precipitate formation temperature decreases, with those that form in ferrite having the highest molybdenum content. The steel compositions studied by Lee et al. [6,7] are indicated in Figure 3b, and molybdenum carbide precipitation would be expected for the range of coiling temperatures examined in their study. Secondary hardening of tungsten-containing steels occurs over a similar temperature range (773–923 K) as for molybdenum-containing steels [3,38–40]. However, the atomic weight of tungsten is about twice that of molybdenum and therefore approximately twice as much tungsten is required (on a weight basis) to achieve the same strengthening effect or potency as molybdenum [3]. For this reason, tungsten-containing secondary hardening steels usually contain a minimum of 2 wt.% W which is far in excess of the predicted solubility limit for either tungsten carbide compound in ferrite. To illustrate this point, Figure 4 shows solubility isotherms for both WC and W2C in ferrite at 923 K. Tungsten carbide precipitation is predicted at any temperature relevant to the tempering of typical tungsten-containing secondary hardening steels because the tungsten concentrations are so high. Figure 4 also indicates a Fe–0.47C–1.13W composition reported by Roberts and Cary [3] from an investigation by Crafts and Lamont [40]. This steel did not produce a true secondary hardening peak but did exhibit a lower rate of softening in comparison to the zero-tungsten base alloy and extensive tungsten carbide precipitation is predicted at 923 K based on the equilibrium solubility products. In summary, equilibrium solubility products were determined for molybdenum carbides and tungsten carbides in ferrite and austenite. High solubility is predicted for these carbide compounds in both ferrite and austenite. Finally, the solubility products predict precipitation behavior that is consistent with experimental studies of secondary hardening steels and microalloyed HSLA steels. [1] M.S. Bhat, W.M. Garrison Jr., V.F. Zackay, Mater. Sci. Eng., A 41 (1979) 1. [2] K.J. Irvine, F.B. Pickering, J. Iron Steel Res. Int. 194 (1960) 137. [3] G.A. Roberts, R.A. Cary, Tool Steels, Fourth ed., American Society for Metals, OH, 1980.
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