- Email: [email protected]
Modelling of interstitials in the bcc phase
CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 33 (2009) 233–236
Contents lists available at ScienceDirect
CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry journal homepage: www.elsevier.com/locate/calphad
Modelling of interstitials in the bcc phase Bengt Hallstedt ∗ , Dejan Djurovic 1 Materials Chemistry, RWTH Aachen University, Aachen, Germany
article
info
Article history: Received 14 May 2008 Received in revised form 12 August 2008 Accepted 18 September 2008 Available online 7 October 2008 Keywords: Calphad Modelling Interstitials Fe–C Nb–N
a b s t r a c t There are several widespread thermodynamic datasets which produce a spurious bcc interstitial solution at high temperature and high X content (X is an interstitally dissolved element). The reason for this is the standard model for bcc interstitial solutions (M(Va, X)3 ), which requires careful selection of optimising parameters to minimise spurious appearances of the bcc phase. In this work the model M(Va, X)1 is suggested as an alternative. This model is much easier to handle and its parameters can be directly compared with those of the fcc phase. The two models are compared for the Fe–C and Nb–N systems. In the Fe–C system almost identical results are achieved. In Nb–N there are some differences for high N content, but there is no experimental data to clearly support any model. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction The sublattice model M(Va, X)3 is widely used to model interstitial solution in bcc (A2) metals. Here M is a metal, Va represents the normally vacant interstitial site and X is an element that dissolves interstitially, normally C or N, but it can also be O, H or B. This model has a firm crystallographic basis, since there are three distorted octahedral interstitial sites per metal atom in the bcc lattice. There are also tetrahedral interstitial sites (six per metal atom) in the bcc lattice with a similar size as the octahedral sites, but at least for C and N in Fe it is accepted that it is the octahedral sites that are occupied. Ordering among the interstitial sites is in principle possible [1], but has not been observed, except in the case of Fe–C martensite. In this case it is, however, unclear if the ordered state is the stable state, or if it is simply inherited from the parent fcc interstitial solution. The model M(Va, X)3 suggests that a compound MX3 could exist (stable or metastable), and even if it is viewed only as a model compound it will have to be assigned some thermodynamic properties. In most cases it can be expected that this compound is very unstable (not metastable) and, thus, that ab initio methods should not be used to calculate its properties. Since this compound often is the one richest in X it can be difficult to suppress completely, even if it is made relatively unstable. In the literature
∗
Corresponding author. Tel.: +49 241 80 25972; fax: +49 241 80 22295. E-mail addresses: [email protected] (B. Hallstedt), [email protected] (D. Djurovic). 1 Tel.: +49 241 80 25997; fax: +49 241 80 22295. 0364-5916/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.calphad.2008.09.013
several thermodynamic datasets can be found where the bcc phase becomes stable in unexpected locations. One of the most widespread datasets is that of Fe–C from Gustafson [2]. If the stable Fe–C phase diagram is calculated up to 6000 K, the result in Fig. 1 is obtained. The bcc phase has a huge stability region above 4000 K around xC = 0.33. There is also an inverse miscibility gap in the liquid phase, covered by the bcc + liquid two-phase region, with a minimum at 5559 K and xC = 0.53, but this will not be discussed further. The situation is even worse in the Nb–N system. Nb (bcc) shows a fairly high N solubility and the calculated Nb–N phase diagram from the present dataset [3] is shown in Fig. 2. An N-rich bcc becomes stable just above 1 bar of N2 and completely dominates the N-rich part of the phase diagram if the gas phase is not included. Similar problems, though not as serious as in the Nb–N system, can be found in the Nb–C [4], Cr–N [5] and Mo–N [5] (and possibly other) systems. By careful selection of thermodynamic parameters it is possible, at least to some extent, to remove these problems, but it is clear that the M(Va, X)3 model is not helpful in this respect. 2. An alternative model for the bcc phase The interstitial solubility in most bcc metals is small and the choice of model is then uncritical. It is then also relatively simple to change from one model to another. As an alternative model we suggest M(Va, X)1 . This suggestion is for pure convenience and does not suggest that the interstitial solution is ordered. This is the same model as the model used for fcc interstitial solutions. The corresponding fcc MX compound has the NaCl (B1) structure and is stable or metastable in many systems. It is then easy to
234
B. Hallstedt, D. Djurovic / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 33 (2009) 233–236
Fig. 1. The stable Fe–C phase diagram calculated from the standard thermodynamic description [2], showing a stable bcc phase field at high temperature.
Fig. 3. The Nb–N phase diagram calculated with a reoptimised description of the bcc phase. The gas phase was not included in the calculation, and is only shown in the form of an isobar at 1 bar of N2 .
well-known since the scatter in the experimental measurements at high temperature is very large and there are essentially no measurements of the bcc phase above 2500 K. N2 potentials in the bcc phase have been extensively measured. The measurements by Cost and Wert [10] are particularly extensive and were found to be both internally consistent and consistent with other data by Huang [3]. Here the N activity data and to some extent the N solubility data from Cost and Wert [10] was used to optimise the bcc parameters using the Nb(Va, N) model. For all other phases, the parameters from Huang [3] were kept. Even with the Nb(Va, N) model, the bcc phase showed a strong tendency to become stable for high N contents, although there is a stable fcc NbN phase in the system. In order to avoid that, the bcc NbN parameters were related to the fcc NbN parameters and only the interaction parameter (LNb:N,Va ) optimised. The resulting parameters are: Gbcc Nb:N = GHSERNB + GHSERNN − 223 000 + 120T − 4T ln T
(1)
and Lbcc Nb:N,Va = +31 100 − 16.8T . Fig. 2. The Nb–N phase diagram calculated from the thermodynamic description of Huang [3]. The gas phase was not included in the calculation, but is only shown in the form of isobars at 1, 2, 5 and 10 bar of N2 . An N-rich bcc phase becomes stable just above 1 bar N2 .
model the bcc MX compound as less stable than the fcc MX compound, which should nearly always be the case. Once the thermodynamic parameters for the bcc MX compound has been fixed, the interstitial solubility and the properties of the interstitial solution can be fitted using the interaction parameter LM:Va,X . The M(Va, X)1 model has already been used to model the bcc solution in the Ce–O [6], Ti–O [7] and Y–O [8] systems in order to avoid having an MO3 compound in the system. The alternative model M(Va, X)1.5 has been used in the La–O system [9] and some other systems for the same reason. Its use should be abandoned. In the following the M(Va, X)1 model is applied to the Nb–N and Fe–C systems. 3. The Nb–N system The bcc phase in the Nb–N system dissolves up to between 10 and 20 at.% N (see Fig. 2). The maximum solubility is not
(2)
GHSERNB and GHSERNN are the Gibbs energy functions for pure Nb and N2 taken from the SGTE unary database [11]. In comparison, the parameter for the fcc NbN compound is: Gfcc Nb:N = GHSERNB + GHSERNN − 227 779
+ 120.567T − 4T ln T .
(3)
The resulting Nb–N phase diagram is shown in Fig. 3. At high temperature the N solubility is clearly lower than in the original evaluation, but there is no data to really support one or the other. The temperature of the bcc + liquid + Nb2 N invariant equilibrium has changed from 2865 K to 2688 K and its character has shifted from peritectic to eutectic. The solubilities are compared in Fig. 4. In the region where data is available, both evaluations are close. The experimental data shows a higher solubility at the highest temperatures. Extrapolating this to the bcc + liquid + Nb2 N invariant would lead to an excessively high N solubility in the bcc phase. There is more solubility data at lower temperature, not included in Fig. 4, which show a much steeper slope. This change in slope has been discussed by Cost and Wert [10], but a satisfactory explanation was not found. Calculated N2 potentials are compared with experimental data in Figs. 5 and 6. The present evaluation is very similar to the previous one. Only in the region of 10 at.% N (and above) there is some difference.
B. Hallstedt, D. Djurovic / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 33 (2009) 233–236
Fig. 4. The N solubility in bcc-Nb calculated from the thermodynamic description of Huang [3] (dashed line) and the present description (solid line), compared with experimental data from Cost and Wert [10].
235
Fig. 6. N potential as function of N content at different temperatures from the thermodynamic description of Huang [3] (dashed lines) and the present description (solid lines), compared with experimental data from Cost and Wert [10].
part of the phase diagram. Here, the parameters for the Fe(C, Va)1 model were fitted to calculated solubilities and invariant equilibria from the original description [2]. That description represents a thorough evaluation of available experimental data. Although that evaluation is more than 20 years old no further data was found that could change that picture. The resulting parameters are: Gbcc Fe:C = GHSERFE + GHSERCC + 107 390 − 7.335T
(6)
and Lbcc Fe:C,Va = −40T .
Fig. 5. N potential as function of inverse temperature at different N contents in the bcc phase from the thermodynamic description of Huang [3] (dashed lines) and the present description (solid lines), compared with experimental data from Cost and Wert [10].
4. The Fe–C system The original parameters for the bcc phase (Fe(C, Va)3 ) from Gustafson [2] are: Gbcc Fe:C = GHSERFE + 3GHSERCC + 322 050 + 75.667T
(4)
and Lbcc Fe:C,Va
= −190T .
(5)
As for Nb–N, GHSERFE and GHSERCC are taken fron the SGTE unary database [11]. There is also a magnetic contribution, but this was kept unchanged. By increasing the entropy contribution of the interaction parameter and changing the FeC3 compound parameter correspondingly, it is possible to shift the high temperature bcc region to above 6000 K without changing anything in the stable
(7)
In this case the interaction parameter was fixed by trial and error and the FeC compound parameter was optimised. No changes are visible in the phase diagram, except that the high temperature bcc region has disappeared. The bcc(δ -ferrite)+ fcc + liquid equilibrium changed from 1767.76 K to 1767.84 K. The eutectoid reaction changed by less than 0.01 K. Although the C solubility in bcc-Fe is very small, its thermodynamic properties are important also at much higher C content. This can most easily be checked by comparing the present and original bcc–fcc T0 lines. These are compared in Fig. 7 up to about 9 at.% C, corresponding to the maximum C solubility in the fcc phase and there is no significant change. The T0 line is defined by Gbcc = Gfcc . It is important because diffusionless transformations (martensitic or massive) can only take place below this line (when starting with fcc). However, Gustafson [2] did not explicitly include any data on the martensitic transformation in his evaluation, thus, this may require a re-evaluation. At this point we are satisfied with concluding that the change of model does not lead to a significant change by itself. 5. Conclusions The interstitial model M(Va, X)3 for bcc interstitial solutions has a tendency to lead to spurious appearances of the bcc phase at high temperature and high X content, in particular if the modelling parameters are not selected carefully. In most cases experimental data can be reproduced just as well with the model M(Va, X)1 without the problems with the M(Va, X)3 model. An exception is the Zr–H system [12] (and possibly a few other M–H systems), where the H solubility is larger than xH = 0.5.
236
B. Hallstedt, D. Djurovic / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 33 (2009) 233–236
For dilute bcc solutions the conversion can be made easily, using the following equations:
1Gnew =
1 3
1Gold − RT ln 3
(8)
and Lnew =
1 3
Lold .
(9)
Acknowledgement The authors gratefully acknowledge the financial support from the Deutsche Forschungsgemeinschaft (DFG) within the Collaborative Research Centre (SFB) 761 ‘‘Steel ab initio’’. References
Fig. 7. The calculated T0 line between the bcc and fcc phase in the Fe–C system from the thermodynamic description of Gustafson [2] (dashed line) and with the reoptimised bcc phase (solid line).
When using the M(Va, X)1 model for bcc interstial solutions, its modelling parameters can be directly compared with the parameters for the fcc interstitial solution.
[1] B. Hallstedt, N. Dupin, M. Hillert, L. Höglund, H.L. Lukas, J.C. Schuster, N. Solak, CALPHAD 31 (2007) 28–37. [2] P. Gustafson, Scand. J. Metall. 14 (1985) 259–267. [3] W. Huang, Metall. Mater. Trans. A 27A (1996) 3591–3600. [4] B.-J. Lee, Metall. Mater. Trans. A 32A (2001) 2423–2439. [5] K. Frisk, CALPHAD 15 (1991) 79–106. [6] M. Zinkevich, D. Djurovic, F. Aldinger, Solid State Ion. 177 (2006) 989–1001. [7] M. Cancarevic, M. Zinkevich, F. Aldinger, CALPHAD 31 (2007) 330–342. [8] D. Djurovic, M. Zinkevich, F. Aldinger, CALPHAD 31 (2007) 560–566. [9] A.N. Grundy, B. Hallstedt, L.J. Gauckler, J. Phase Equilib. 22 (2001) 105–113. [10] J.R. Cost, C.A. Wert, Acta Metall. 11 (1963) 231–242. [11] A.T. Dinsdale, CALPHAD 15 (1991) 317–425. [12] N. Dupin, I. Ansara, C. Servant, C. Toffolon, C. Lemaignan, J.C. Brachet, J. Nucl. Mater. 275 (1999) 287–295.
Contents lists available at ScienceDirect
CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry journal homepage: www.elsevier.com/locate/calphad
Modelling of interstitials in the bcc phase Bengt Hallstedt ∗ , Dejan Djurovic 1 Materials Chemistry, RWTH Aachen University, Aachen, Germany
article
info
Article history: Received 14 May 2008 Received in revised form 12 August 2008 Accepted 18 September 2008 Available online 7 October 2008 Keywords: Calphad Modelling Interstitials Fe–C Nb–N
a b s t r a c t There are several widespread thermodynamic datasets which produce a spurious bcc interstitial solution at high temperature and high X content (X is an interstitally dissolved element). The reason for this is the standard model for bcc interstitial solutions (M(Va, X)3 ), which requires careful selection of optimising parameters to minimise spurious appearances of the bcc phase. In this work the model M(Va, X)1 is suggested as an alternative. This model is much easier to handle and its parameters can be directly compared with those of the fcc phase. The two models are compared for the Fe–C and Nb–N systems. In the Fe–C system almost identical results are achieved. In Nb–N there are some differences for high N content, but there is no experimental data to clearly support any model. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction The sublattice model M(Va, X)3 is widely used to model interstitial solution in bcc (A2) metals. Here M is a metal, Va represents the normally vacant interstitial site and X is an element that dissolves interstitially, normally C or N, but it can also be O, H or B. This model has a firm crystallographic basis, since there are three distorted octahedral interstitial sites per metal atom in the bcc lattice. There are also tetrahedral interstitial sites (six per metal atom) in the bcc lattice with a similar size as the octahedral sites, but at least for C and N in Fe it is accepted that it is the octahedral sites that are occupied. Ordering among the interstitial sites is in principle possible [1], but has not been observed, except in the case of Fe–C martensite. In this case it is, however, unclear if the ordered state is the stable state, or if it is simply inherited from the parent fcc interstitial solution. The model M(Va, X)3 suggests that a compound MX3 could exist (stable or metastable), and even if it is viewed only as a model compound it will have to be assigned some thermodynamic properties. In most cases it can be expected that this compound is very unstable (not metastable) and, thus, that ab initio methods should not be used to calculate its properties. Since this compound often is the one richest in X it can be difficult to suppress completely, even if it is made relatively unstable. In the literature
∗
Corresponding author. Tel.: +49 241 80 25972; fax: +49 241 80 22295. E-mail addresses: [email protected] (B. Hallstedt), [email protected] (D. Djurovic). 1 Tel.: +49 241 80 25997; fax: +49 241 80 22295. 0364-5916/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.calphad.2008.09.013
several thermodynamic datasets can be found where the bcc phase becomes stable in unexpected locations. One of the most widespread datasets is that of Fe–C from Gustafson [2]. If the stable Fe–C phase diagram is calculated up to 6000 K, the result in Fig. 1 is obtained. The bcc phase has a huge stability region above 4000 K around xC = 0.33. There is also an inverse miscibility gap in the liquid phase, covered by the bcc + liquid two-phase region, with a minimum at 5559 K and xC = 0.53, but this will not be discussed further. The situation is even worse in the Nb–N system. Nb (bcc) shows a fairly high N solubility and the calculated Nb–N phase diagram from the present dataset [3] is shown in Fig. 2. An N-rich bcc becomes stable just above 1 bar of N2 and completely dominates the N-rich part of the phase diagram if the gas phase is not included. Similar problems, though not as serious as in the Nb–N system, can be found in the Nb–C [4], Cr–N [5] and Mo–N [5] (and possibly other) systems. By careful selection of thermodynamic parameters it is possible, at least to some extent, to remove these problems, but it is clear that the M(Va, X)3 model is not helpful in this respect. 2. An alternative model for the bcc phase The interstitial solubility in most bcc metals is small and the choice of model is then uncritical. It is then also relatively simple to change from one model to another. As an alternative model we suggest M(Va, X)1 . This suggestion is for pure convenience and does not suggest that the interstitial solution is ordered. This is the same model as the model used for fcc interstitial solutions. The corresponding fcc MX compound has the NaCl (B1) structure and is stable or metastable in many systems. It is then easy to
234
B. Hallstedt, D. Djurovic / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 33 (2009) 233–236
Fig. 1. The stable Fe–C phase diagram calculated from the standard thermodynamic description [2], showing a stable bcc phase field at high temperature.
Fig. 3. The Nb–N phase diagram calculated with a reoptimised description of the bcc phase. The gas phase was not included in the calculation, and is only shown in the form of an isobar at 1 bar of N2 .
well-known since the scatter in the experimental measurements at high temperature is very large and there are essentially no measurements of the bcc phase above 2500 K. N2 potentials in the bcc phase have been extensively measured. The measurements by Cost and Wert [10] are particularly extensive and were found to be both internally consistent and consistent with other data by Huang [3]. Here the N activity data and to some extent the N solubility data from Cost and Wert [10] was used to optimise the bcc parameters using the Nb(Va, N) model. For all other phases, the parameters from Huang [3] were kept. Even with the Nb(Va, N) model, the bcc phase showed a strong tendency to become stable for high N contents, although there is a stable fcc NbN phase in the system. In order to avoid that, the bcc NbN parameters were related to the fcc NbN parameters and only the interaction parameter (LNb:N,Va ) optimised. The resulting parameters are: Gbcc Nb:N = GHSERNB + GHSERNN − 223 000 + 120T − 4T ln T
(1)
and Lbcc Nb:N,Va = +31 100 − 16.8T . Fig. 2. The Nb–N phase diagram calculated from the thermodynamic description of Huang [3]. The gas phase was not included in the calculation, but is only shown in the form of isobars at 1, 2, 5 and 10 bar of N2 . An N-rich bcc phase becomes stable just above 1 bar N2 .
model the bcc MX compound as less stable than the fcc MX compound, which should nearly always be the case. Once the thermodynamic parameters for the bcc MX compound has been fixed, the interstitial solubility and the properties of the interstitial solution can be fitted using the interaction parameter LM:Va,X . The M(Va, X)1 model has already been used to model the bcc solution in the Ce–O [6], Ti–O [7] and Y–O [8] systems in order to avoid having an MO3 compound in the system. The alternative model M(Va, X)1.5 has been used in the La–O system [9] and some other systems for the same reason. Its use should be abandoned. In the following the M(Va, X)1 model is applied to the Nb–N and Fe–C systems. 3. The Nb–N system The bcc phase in the Nb–N system dissolves up to between 10 and 20 at.% N (see Fig. 2). The maximum solubility is not
(2)
GHSERNB and GHSERNN are the Gibbs energy functions for pure Nb and N2 taken from the SGTE unary database [11]. In comparison, the parameter for the fcc NbN compound is: Gfcc Nb:N = GHSERNB + GHSERNN − 227 779
+ 120.567T − 4T ln T .
(3)
The resulting Nb–N phase diagram is shown in Fig. 3. At high temperature the N solubility is clearly lower than in the original evaluation, but there is no data to really support one or the other. The temperature of the bcc + liquid + Nb2 N invariant equilibrium has changed from 2865 K to 2688 K and its character has shifted from peritectic to eutectic. The solubilities are compared in Fig. 4. In the region where data is available, both evaluations are close. The experimental data shows a higher solubility at the highest temperatures. Extrapolating this to the bcc + liquid + Nb2 N invariant would lead to an excessively high N solubility in the bcc phase. There is more solubility data at lower temperature, not included in Fig. 4, which show a much steeper slope. This change in slope has been discussed by Cost and Wert [10], but a satisfactory explanation was not found. Calculated N2 potentials are compared with experimental data in Figs. 5 and 6. The present evaluation is very similar to the previous one. Only in the region of 10 at.% N (and above) there is some difference.
B. Hallstedt, D. Djurovic / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 33 (2009) 233–236
Fig. 4. The N solubility in bcc-Nb calculated from the thermodynamic description of Huang [3] (dashed line) and the present description (solid line), compared with experimental data from Cost and Wert [10].
235
Fig. 6. N potential as function of N content at different temperatures from the thermodynamic description of Huang [3] (dashed lines) and the present description (solid lines), compared with experimental data from Cost and Wert [10].
part of the phase diagram. Here, the parameters for the Fe(C, Va)1 model were fitted to calculated solubilities and invariant equilibria from the original description [2]. That description represents a thorough evaluation of available experimental data. Although that evaluation is more than 20 years old no further data was found that could change that picture. The resulting parameters are: Gbcc Fe:C = GHSERFE + GHSERCC + 107 390 − 7.335T
(6)
and Lbcc Fe:C,Va = −40T .
Fig. 5. N potential as function of inverse temperature at different N contents in the bcc phase from the thermodynamic description of Huang [3] (dashed lines) and the present description (solid lines), compared with experimental data from Cost and Wert [10].
4. The Fe–C system The original parameters for the bcc phase (Fe(C, Va)3 ) from Gustafson [2] are: Gbcc Fe:C = GHSERFE + 3GHSERCC + 322 050 + 75.667T
(4)
and Lbcc Fe:C,Va
= −190T .
(5)
As for Nb–N, GHSERFE and GHSERCC are taken fron the SGTE unary database [11]. There is also a magnetic contribution, but this was kept unchanged. By increasing the entropy contribution of the interaction parameter and changing the FeC3 compound parameter correspondingly, it is possible to shift the high temperature bcc region to above 6000 K without changing anything in the stable
(7)
In this case the interaction parameter was fixed by trial and error and the FeC compound parameter was optimised. No changes are visible in the phase diagram, except that the high temperature bcc region has disappeared. The bcc(δ -ferrite)+ fcc + liquid equilibrium changed from 1767.76 K to 1767.84 K. The eutectoid reaction changed by less than 0.01 K. Although the C solubility in bcc-Fe is very small, its thermodynamic properties are important also at much higher C content. This can most easily be checked by comparing the present and original bcc–fcc T0 lines. These are compared in Fig. 7 up to about 9 at.% C, corresponding to the maximum C solubility in the fcc phase and there is no significant change. The T0 line is defined by Gbcc = Gfcc . It is important because diffusionless transformations (martensitic or massive) can only take place below this line (when starting with fcc). However, Gustafson [2] did not explicitly include any data on the martensitic transformation in his evaluation, thus, this may require a re-evaluation. At this point we are satisfied with concluding that the change of model does not lead to a significant change by itself. 5. Conclusions The interstitial model M(Va, X)3 for bcc interstitial solutions has a tendency to lead to spurious appearances of the bcc phase at high temperature and high X content, in particular if the modelling parameters are not selected carefully. In most cases experimental data can be reproduced just as well with the model M(Va, X)1 without the problems with the M(Va, X)3 model. An exception is the Zr–H system [12] (and possibly a few other M–H systems), where the H solubility is larger than xH = 0.5.
236
B. Hallstedt, D. Djurovic / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 33 (2009) 233–236
For dilute bcc solutions the conversion can be made easily, using the following equations:
1Gnew =
1 3
1Gold − RT ln 3
(8)
and Lnew =
1 3
Lold .
(9)
Acknowledgement The authors gratefully acknowledge the financial support from the Deutsche Forschungsgemeinschaft (DFG) within the Collaborative Research Centre (SFB) 761 ‘‘Steel ab initio’’. References
Fig. 7. The calculated T0 line between the bcc and fcc phase in the Fe–C system from the thermodynamic description of Gustafson [2] (dashed line) and with the reoptimised bcc phase (solid line).
When using the M(Va, X)1 model for bcc interstial solutions, its modelling parameters can be directly compared with the parameters for the fcc interstitial solution.
[1] B. Hallstedt, N. Dupin, M. Hillert, L. Höglund, H.L. Lukas, J.C. Schuster, N. Solak, CALPHAD 31 (2007) 28–37. [2] P. Gustafson, Scand. J. Metall. 14 (1985) 259–267. [3] W. Huang, Metall. Mater. Trans. A 27A (1996) 3591–3600. [4] B.-J. Lee, Metall. Mater. Trans. A 32A (2001) 2423–2439. [5] K. Frisk, CALPHAD 15 (1991) 79–106. [6] M. Zinkevich, D. Djurovic, F. Aldinger, Solid State Ion. 177 (2006) 989–1001. [7] M. Cancarevic, M. Zinkevich, F. Aldinger, CALPHAD 31 (2007) 330–342. [8] D. Djurovic, M. Zinkevich, F. Aldinger, CALPHAD 31 (2007) 560–566. [9] A.N. Grundy, B. Hallstedt, L.J. Gauckler, J. Phase Equilib. 22 (2001) 105–113. [10] J.R. Cost, C.A. Wert, Acta Metall. 11 (1963) 231–242. [11] A.T. Dinsdale, CALPHAD 15 (1991) 317–425. [12] N. Dupin, I. Ansara, C. Servant, C. Toffolon, C. Lemaignan, J.C. Brachet, J. Nucl. Mater. 275 (1999) 287–295.