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A thermodynamic model proposed for calculating the standard formation enthalpies of ternary alloy systems
Scripta Materialia 56 (2007) 975–978 www.actamat-journals.com
A thermodynamic model proposed for calculating the standard formation enthalpies of ternary alloy systems W.C. Wang, J.H. Li, H.F. Yan and B.X. Liu* Advanced Materials Laboratory, Department of Materials Science and Engineering, Tsinghua University, Beijing 100084, China Received 21 December 2006; revised 30 January 2007; accepted 31 January 2007 Available online 6 March 2007
Based on Miedema’s method for binary alloys, a thermodynamic model is proposed for calculating the standard formation enthalpies of ternary alloy systems. In the proposed model, both the atomic size difference of the constituent metals and the interaction between the third metal and the other two are taken into account. Compared with the Miedema’s original model, the newly proposed model can considerably improve the precision of calculation. Ó 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Thermodynamics model; Alloys; Formation enthalpy
It is of vital importance to develop relevant theories to provide thermodynamic and kinetic insight into basic issues in the field of physical metallurgy, especially issues related to the formation and transformation of equilibrium and nonequilibrium phases. Since the 1950s, great efforts have been made to build theoretical models, among which Miedema’s theory, which combines concepts from the thermodynamics of solids and energy band theory, is one of the most successful [1–4]. Miedema’s theory has been widely used to calculate the formation enthalpies of liquids, solid solutions and compounds, as well as to construct Gibbs free energy diagrams of the alloy systems which could provide thermodynamic insight into the relative stability of various alloy phases in the respective systems [5,6]. Meanwhile, some research groups have performed extensive experimental studies to obtain the thermodynamic properties of binary as well as ternary alloy systems. For instance, Kleppa et al. measured some 260 standard formation enthalpies of binary intermetallic compounds [7,8]. Some researchers have found that there is a systemic deviation between the measured experimental values and the results calculated by Miedema’s theory for binary alloy systems. In addition, results calculated by Miedema’s theory are even worse for ternary alloys. Consequently, the proposal of new theories to solve these problems is of importance. In the present work,
* Corresponding author. Tel.: +86 10 6278 1255; e-mail: dmslbx@ tsinghua.edu.cn
we propose a model which can calculate standard formation enthalpy for ternary as well as binary alloys. In Miedema’s theory, the enthalpy effect upon alloying is due to the electron redistribution generated at the contact surfaces between dissimilar atomic cells. In order to calculate the standard formation enthalpies of the binary alloy systems, Miedema’s theory [4] presented a method by calculating microscope interfacial energy, which can be described as DH cA
in B
¼ xA fBA DH int A in B ;
ð1Þ fBA
is a funcwhere xA is the mole fraction of metal A. tion which accounts for the degree to which atoms of type A are surrounded by atoms of type B, and can be given by 2
fBA ¼ xSB ½1 þ cðxSA xSB Þ ; xSA ¼ xSB ¼
2=3 A
xA V ; 2=3 2=3 xA V A þ xB V B xB V B2=3 : 2=3 2=3 xA V A þ xB V B
ð2Þ ð3Þ ð4Þ
In Eq. (2), c is an empirical parameter which is used to describe the short-range-order difference of the liquid and the ordered compound, for which it is usually taken to be 0 and 8, respectively. xSA and xSB are the fraction of surface area of metal A and B, respectively. DH int A in B is the amplitude concerning the magnitude of the electron redistribution interaction and is a constant for a specific binary alloy system, i.e. the interfacial
1359-6462/$ - see front matter Ó 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2007.01.044
976
W. C. Wang et al. / Scripta Materialia 56 (2007) 975–978
energy for 1 mol metal A ideally surrounded by metal B. DH int A in B can be obtained by DH int A in
B
¼
V 2=:3 A
2
1=3
ðnWS Þav
1=3 2
½P ðD/ Þ þ QðDnWS Þ R ; ð5Þ
where VA is the volume of one mole metal A; P, Q and 1=3 R* are three empirical constants; ðnWS Þav is the mean value of the electron density at the boundary of the Wigner–Seitz cell as derived for the pure elements in 1=3 1=3 the metallic state ðnWS ÞA and ðnWS ÞB ; D/* describes the electronegativity difference between D/A and D/B ; 1=3 and DnWS describes the electron density difference between the two constituent metals. It has been mentioned above that there is a systemic deviation between the formation enthalpies calculated by Miedema’s theory and experimental values of binary systems. Based on Miedema’s theory, the contact between two dissimilar Wigner–Seitz unit cells is ideally matched. However, this is not the case because of the atomic size difference between the constituent metals. Therefore, a pre-factor S(x), which is proposed to take into account the significant effect of the atomic size difference, is introduced into Miedema’s theory. Two forms of S(x) have been proposed, by Zhang et al. and Chen et al. [9–11], respectively. However, the formula of formation enthalpy is unsymmetrical by using Zhang’s S(x), while it is symmetric in Miedema’s theory. Though the formula is symmetric in form when introducing Chen’s complex S(x), the physical meaning of Miedema’s theory is lost. We revise the method of calculating the interfacial energy DH int A in B by introducing a new factor S(x), which is more reasonable than that published before, and is described as 2=:3
DH int A in
B
¼
SðxÞ V A 1=3
ðnWS Þav
1=3
½P ðD/ Þ2 þ QðDnWS Þ2 R ; ð6Þ
SðxÞ ¼ 1
C xA xB jV A V B j ; xA xA V A þ xB xB V B
ð7Þ
where C is an empirical parameter that describes the effect of the atomic size difference in a semi-quantitative manner and is taken as 0.5 and 2.0 for the liquid alloy and the ordered compound, respectively. By introducing the new S(x), it turns out that the modified Miedema’s model accords well with the experimental values for binary alloy systems, as shown in Figure 1b. As a comparison, one can find calculated results by Miedema’s original model in Figure 1a. For calculating formation enthalpies of ternary systems, Miedema’s theory presents a simple method based on the method used to calculate binary systems [4,12– 14]. A predigestion is employed which turns a ternary problem into three binary problems and can be described as DH ABC ¼ DH cA in B þ DH cB in C þ DH cA in C ;
ð8Þ
where DHABC is the standard formation enthalpy of the ternary intermetallic compound or ternary liquid alloy.
Figure 1. Comparison of the experimental values with the calculated results of standard formation enthalpies by (a) the original Miedema’s model [9] and (b) the modified Miedema’s model with the new prefactor S(x), for binary compounds. The linear line y = x represents 100% agreement between the calculated and experimental values, and two dashed and dotted lines set the defined data zone, with an error bar to be ±23 kJ mol1. AE: average relative error, AE ¼ P set jy i xi j 1 N jxi j ; R: correlation factor. N
DH cA in B is the heat of solution of metal A in metal B, regardless of the effect of metal C, i.e. the method of calculating DH cA in B , DH cB in C and DH cA in C for ternary systems is similar to that for binary systems, respectively. Using the original Miedema model (OMM), formation enthalpies of about 90 ternary intermetallic compounds are calculated and compared with corresponding experimental values. Formation enthalpies for corresponding systems are also calculated by the revised Miedema model (RMM), in which the pre-factor S(x) is introduced in calculating DH cA in B ; DH cB in C and DH cA in C . Figure 2a and b illustrates the calculated results compared with measured experimental values [15,16]. From the figure, one can see that the results in Figure 2b are a little better than those in Figure 2a, which shows that the pre-factor S(x) has an effect in calculating ternary alloy systems. However, even the RMM cannot produce a precise result. There is one point that should be paid attention: by using Eq. (8), the formation enthalpy of ternary alloys can be split into three items corresponding to three binary alloys. The predigestion itself assumes that the third metal has no effect on the other two metals in calculating the interfacial energy. In reality, introducing a third 1=3 metal can affect D/* and DnWS in Eq. (5). Therefore, the predigestion may be responsible for Miedema’s model, even when modified by S(x), losing precision in predicting the thermodynamic properties of ternary alloy systems. Considering the interaction of the elements, we propose a two-step calculation method (TSCM) for calculating DHABC, which could solve the fatal problem. Under the TSCM hypothesis, the alloying process is split into two steps. The first step is the alloying of metals
W. C. Wang et al. / Scripta Materialia 56 (2007) 975–978
977
As the above three processes are about equal, taking an average from the above three equations, i.e. Eqs. (11)– (13), will achieve a best approximation as follows: 1 c DH A in B þ DH cC in AB DH ABC ¼ 3 þ DH cB in C þ DH cA in BC þ DH cA in C þ DH cB in AC : ð14Þ
Figure 2. Comparison of the experimental values with the calculated results of standard formation enthalpies by (a) the OMM; (b) the RMM; and (c) the PM, respectively, for 90 Al-based and Mg-based ternary intermetallic compounds.
A and B according to the alloy composition, and the second step is the introducing the third metal, metal C, into the mixture of metals A and B. The heat of solution of the two steps can be described as DH cA in B and DH cC in AB , respectively, and can be calculated by ð9Þ DH cA in B ¼ xA fBA DH int A in B ; C DH cC in AB ¼ xC fAB DH int C in AB :
ð10Þ
Employing the proposed model (PM) of Eq. (14), the standard formation enthalpies of the ternary intermetallic compounds and the mixing enthalpies of the ternary liquid alloys can be calculated. Accordingly, 90 Al-based and Mg-based ternary intermetallic compounds have been calculated by the PM and the results compared with those obtained in experiments reported in the literature [15,16]. Figure 2c displays the calculated results obtained by the PM, and can be compared with Figure 2a and b, which represents the results calculated by the OMM and RMM, respectively. One can see that the PM gives results with better precision than the OMM statistically, with over 85% of the calculated results improved. For instance, Table 1 lists the data for the Al–Ni–Y system, clearly showing that the PM can improve the precision compared with the OMM. Also, from the average relative error and the correlation factors in Figure 2, one can conclude that the PM is better than both the OMM and RMM. Some 420 ternary liquid alloys have also been calculated by the PM and OMM, respectively, and then compared with the experimental values obtained from the literature [17–22]. Figure 3 shows the results calculated by the PM and OMM. It can be seen that, as the heat of formation becomes more negative, the improvement by the PM becomes more obvious. The figure can thus be divided into two regions, I and II. In region I, the calculated results by the PM or OMM are almost the same as the experimental values, and the degree of improvement by the PM is not obvious. For the ternary alloy systems in this region, we find that the difference of the property (V, /, nws) among the three elements is Table 1. The standard formation enthalpies (kJ mol1) of Al–Ni–Y ternary intermetallic compounds measured in experiments and calculated by the PM and OMM, respectively
DH cC in AB ,
In calculating the properties of the mixture of metals A and B comply with the following reasonable assumptions: xAB = xA + xB; VAB = (xAVA + xBVB)/ xAB; /AB = (xA/A + xB/B)/xAB; nwsAB = (xA nwsA + xB nwsB)/xAB. By considering the entire process of alloying, the formation enthalpy can be calculated by ð11Þ DH ABC ¼ DH cA in B þ DH cC in AB : There are another two alternative processes that could achieve the same ternary alloy, i.e. by first alloying metals B and C and metals A and C as the first step, respectively, and then introducing metals A and B, respectively. Accordingly, the following two equations are for calculating DHABC: DH ABC ¼ DH cB in C þ DH cA in BC ; DH ABC ¼
DH cA in C
þ
DH cB in AC :
ð12Þ a
ð13Þ
Compounds
Experimental PM values (EXP)
OMM
Improvement by the PM (%)a
Al0.08Ni0.67Y0.25 Al0.15Ni0.68Y0.17 Al0.18Ni0.55Y0.27 Al0.25Ni0.58Y0.17 Al0.33Ni0.33Y0.33 Al0.50Ni0.25Y0.25 Al0.50Ni0.33Y0.17 Al0.58Ni0.25Y0.17 Al0.60Ni0.07Y0.33 Al0.60Ni0.20Y0.20 Al0.67Ni0.17Y0.16 Al0.69Ni0.23Y0.08 Al0.70Ni0.18Y0.12
37.90 47.33 48.50 48.30 54.10 62.80 62.80 61.30 56.70 59.80 54.00 47.00 50.60
54.42 62.01 83.58 81.05 107.41 97.73 90.60 81.82 74.57 80.68 66.70 53.09 58.51
42.96 20.57 71.48 60.26 93.94 46.75 35.86 20.32 21.85 22.00 9.13 4.39 0.00
Improvement ð%Þ ¼
38.14 42.38 48.08 51.94 56.59 57.23 57.52 53.24 51.21 52.07 46.23 42.98 42.69
EXP jDH PM ABC DH ABC j 100%. OMM jDH ABC DH EXP ABC j
978
W. C. Wang et al. / Scripta Materialia 56 (2007) 975–978
The authors are grateful for the financial support from the National Natural Science Foundation of China (50531040), The Ministry of Science and Technology of China (2006CB605201) and the Administration of Tsinghua University.
Figure 3. Comparison of the experimental values with the calculated results of mixing enthalpies by the OMM and the PM, respectively, for 420 ternary liquid alloys.
not distinct. Therefore, the influence of atomic size and of the interaction among elements on the results calculated by the OMM can be neglected. However, for the ternary alloy systems in region II, the difference of the property (V, /, nws) among the three elements is distinct and cannot be neglected. Therefore the calculated error of the OMM is larger. This is also the reason why there is more scatter in this region, and why the improvement by the PM in this region is more obvious. In Figure 3, 100% of the circles fall in the data zone in region II, whereas only about 85% of squares fall in the same data zone, indicating that the PM could achieve more precise results than the OMM in calculating the mixing enthalpies of ternary liquid alloys. The improvement of the PM for ternary systems originates from two important factors. Firstly, the atomic size difference of the constituent metals is taken into consideration by introducing a new pre-factor, S(x). Secondly, the interactions between the third metal and the other two are taken into account by employing a TSCM method. The ternary model is verified by calculating formation enthalpies of about 90 intermetallic compounds and by calculating mixing enthalpies of about 420 liquid alloys. The continuity of the ternary model is also analyzed, and it is confirmed that the model can be applied to binary alloys. In addition, the PM is equal to the RMM for binary alloy systems.
[1] W. Hume-Rothery, G.W. Mabbot, K.M. Channel-Evans, Philos Trans. Roy. Soc. 233 (1934) 1. [2] A.R. Miedema, P.F. de Chatel, in: Proceedings of the Symposium Theory of Alloy Phase Formation, New Orleans, 1979. [3] A.R. Miedema, P.F. de Chatel, F.R. de Boer, Physica B 100 (1980) 1. [4] F.R. de Boer, R. Boom, W.C.M. Mattens, A.R. Miedema, A.K. Niessen, Cohesion in Metals: Transition Metal Alloys, North-Holland, Amsterdam, 1988. [5] B.X. Liu, W.S. Lai, Q. Zhang, Mater. Sci. Eng. R 29 (2000) 1. [6] H. Bakker, Enthalpies in Alloys: Miedema’s Semi-Empirical Model, Trans Tech Publications, Zurich, 1998. [7] J.W. Wang, Q.T. Guo, O.J. Kleppa, J. Alloy. Compd. 313 (2000) 77. [8] Q.T. Guo, O.J. Kleppa, J. Alloy. Compd. 321 (2001) 169. [9] R.F. Zhang, B.X. Liu, Appl. Phys. Lett. 81 (2002) 7. [10] X.Q. Chen, R. Podioucky, P. Rogl, Appl. Phys. Lett. 86 (2005) 216103. [11] R.F. Zhang, B.X. Liu, Appl. Phys. Lett. 86 (2005) 216104. [12] J.A. Somoza, L.J. Gallego, C. Rey, H.M. Fernandez, Philos. Mag. B 65 (1992) 5. [13] L.J. Gallego, J.A. Somoza, J.A. Alonso, J. Phys.: Condes. Mat. 2 (1990) 6245. [14] A. Takeuchi, A. Inoue, Mater. Trans. 42 (2001) 7. [15] H.N. Su, Thermodynamic model of Al and Ni based ternary alloy, Dissertation of PhD, Illinois Institute of Technology, 2004. [16] A.P. Goncalves, M. Almeida, Physica B 228 (1996) 289. [17] B.W. Zhang, W.A. Jesser, Physica B 315 (2002) 123. [18] H. Wang, R. Luck, B. Predel, J. Phase. Equilib. 14 (1993) 184. [19] V. Ganesan, F. Schuller, H. Feufel, F. Sommer, Z. Metallkd. 88 (1997) 701. [20] U.K. Stolz, I. Arpshofen, F. Sommer, Z. Metallkd. 84 (1993) 552. [21] K. Fitzner, O.J. Kleppa, Metall. Trans. A 24 (1993) 1827. [22] I. Arshofen, M.J. Pool, F. Sommer, U. Gerling, Z. Metallkd. 74 (1983) 25.
A thermodynamic model proposed for calculating the standard formation enthalpies of ternary alloy systems W.C. Wang, J.H. Li, H.F. Yan and B.X. Liu* Advanced Materials Laboratory, Department of Materials Science and Engineering, Tsinghua University, Beijing 100084, China Received 21 December 2006; revised 30 January 2007; accepted 31 January 2007 Available online 6 March 2007
Based on Miedema’s method for binary alloys, a thermodynamic model is proposed for calculating the standard formation enthalpies of ternary alloy systems. In the proposed model, both the atomic size difference of the constituent metals and the interaction between the third metal and the other two are taken into account. Compared with the Miedema’s original model, the newly proposed model can considerably improve the precision of calculation. Ó 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Thermodynamics model; Alloys; Formation enthalpy
It is of vital importance to develop relevant theories to provide thermodynamic and kinetic insight into basic issues in the field of physical metallurgy, especially issues related to the formation and transformation of equilibrium and nonequilibrium phases. Since the 1950s, great efforts have been made to build theoretical models, among which Miedema’s theory, which combines concepts from the thermodynamics of solids and energy band theory, is one of the most successful [1–4]. Miedema’s theory has been widely used to calculate the formation enthalpies of liquids, solid solutions and compounds, as well as to construct Gibbs free energy diagrams of the alloy systems which could provide thermodynamic insight into the relative stability of various alloy phases in the respective systems [5,6]. Meanwhile, some research groups have performed extensive experimental studies to obtain the thermodynamic properties of binary as well as ternary alloy systems. For instance, Kleppa et al. measured some 260 standard formation enthalpies of binary intermetallic compounds [7,8]. Some researchers have found that there is a systemic deviation between the measured experimental values and the results calculated by Miedema’s theory for binary alloy systems. In addition, results calculated by Miedema’s theory are even worse for ternary alloys. Consequently, the proposal of new theories to solve these problems is of importance. In the present work,
* Corresponding author. Tel.: +86 10 6278 1255; e-mail: dmslbx@ tsinghua.edu.cn
we propose a model which can calculate standard formation enthalpy for ternary as well as binary alloys. In Miedema’s theory, the enthalpy effect upon alloying is due to the electron redistribution generated at the contact surfaces between dissimilar atomic cells. In order to calculate the standard formation enthalpies of the binary alloy systems, Miedema’s theory [4] presented a method by calculating microscope interfacial energy, which can be described as DH cA
in B
¼ xA fBA DH int A in B ;
ð1Þ fBA
is a funcwhere xA is the mole fraction of metal A. tion which accounts for the degree to which atoms of type A are surrounded by atoms of type B, and can be given by 2
fBA ¼ xSB ½1 þ cðxSA xSB Þ ; xSA ¼ xSB ¼
2=3 A
xA V ; 2=3 2=3 xA V A þ xB V B xB V B2=3 : 2=3 2=3 xA V A þ xB V B
ð2Þ ð3Þ ð4Þ
In Eq. (2), c is an empirical parameter which is used to describe the short-range-order difference of the liquid and the ordered compound, for which it is usually taken to be 0 and 8, respectively. xSA and xSB are the fraction of surface area of metal A and B, respectively. DH int A in B is the amplitude concerning the magnitude of the electron redistribution interaction and is a constant for a specific binary alloy system, i.e. the interfacial
1359-6462/$ - see front matter Ó 2007 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.scriptamat.2007.01.044
976
W. C. Wang et al. / Scripta Materialia 56 (2007) 975–978
energy for 1 mol metal A ideally surrounded by metal B. DH int A in B can be obtained by DH int A in
B
¼
V 2=:3 A
2
1=3
ðnWS Þav
1=3 2
½P ðD/ Þ þ QðDnWS Þ R ; ð5Þ
where VA is the volume of one mole metal A; P, Q and 1=3 R* are three empirical constants; ðnWS Þav is the mean value of the electron density at the boundary of the Wigner–Seitz cell as derived for the pure elements in 1=3 1=3 the metallic state ðnWS ÞA and ðnWS ÞB ; D/* describes the electronegativity difference between D/A and D/B ; 1=3 and DnWS describes the electron density difference between the two constituent metals. It has been mentioned above that there is a systemic deviation between the formation enthalpies calculated by Miedema’s theory and experimental values of binary systems. Based on Miedema’s theory, the contact between two dissimilar Wigner–Seitz unit cells is ideally matched. However, this is not the case because of the atomic size difference between the constituent metals. Therefore, a pre-factor S(x), which is proposed to take into account the significant effect of the atomic size difference, is introduced into Miedema’s theory. Two forms of S(x) have been proposed, by Zhang et al. and Chen et al. [9–11], respectively. However, the formula of formation enthalpy is unsymmetrical by using Zhang’s S(x), while it is symmetric in Miedema’s theory. Though the formula is symmetric in form when introducing Chen’s complex S(x), the physical meaning of Miedema’s theory is lost. We revise the method of calculating the interfacial energy DH int A in B by introducing a new factor S(x), which is more reasonable than that published before, and is described as 2=:3
DH int A in
B
¼
SðxÞ V A 1=3
ðnWS Þav
1=3
½P ðD/ Þ2 þ QðDnWS Þ2 R ; ð6Þ
SðxÞ ¼ 1
C xA xB jV A V B j ; xA xA V A þ xB xB V B
ð7Þ
where C is an empirical parameter that describes the effect of the atomic size difference in a semi-quantitative manner and is taken as 0.5 and 2.0 for the liquid alloy and the ordered compound, respectively. By introducing the new S(x), it turns out that the modified Miedema’s model accords well with the experimental values for binary alloy systems, as shown in Figure 1b. As a comparison, one can find calculated results by Miedema’s original model in Figure 1a. For calculating formation enthalpies of ternary systems, Miedema’s theory presents a simple method based on the method used to calculate binary systems [4,12– 14]. A predigestion is employed which turns a ternary problem into three binary problems and can be described as DH ABC ¼ DH cA in B þ DH cB in C þ DH cA in C ;
ð8Þ
where DHABC is the standard formation enthalpy of the ternary intermetallic compound or ternary liquid alloy.
Figure 1. Comparison of the experimental values with the calculated results of standard formation enthalpies by (a) the original Miedema’s model [9] and (b) the modified Miedema’s model with the new prefactor S(x), for binary compounds. The linear line y = x represents 100% agreement between the calculated and experimental values, and two dashed and dotted lines set the defined data zone, with an error bar to be ±23 kJ mol1. AE: average relative error, AE ¼ P set jy i xi j 1 N jxi j ; R: correlation factor. N
DH cA in B is the heat of solution of metal A in metal B, regardless of the effect of metal C, i.e. the method of calculating DH cA in B , DH cB in C and DH cA in C for ternary systems is similar to that for binary systems, respectively. Using the original Miedema model (OMM), formation enthalpies of about 90 ternary intermetallic compounds are calculated and compared with corresponding experimental values. Formation enthalpies for corresponding systems are also calculated by the revised Miedema model (RMM), in which the pre-factor S(x) is introduced in calculating DH cA in B ; DH cB in C and DH cA in C . Figure 2a and b illustrates the calculated results compared with measured experimental values [15,16]. From the figure, one can see that the results in Figure 2b are a little better than those in Figure 2a, which shows that the pre-factor S(x) has an effect in calculating ternary alloy systems. However, even the RMM cannot produce a precise result. There is one point that should be paid attention: by using Eq. (8), the formation enthalpy of ternary alloys can be split into three items corresponding to three binary alloys. The predigestion itself assumes that the third metal has no effect on the other two metals in calculating the interfacial energy. In reality, introducing a third 1=3 metal can affect D/* and DnWS in Eq. (5). Therefore, the predigestion may be responsible for Miedema’s model, even when modified by S(x), losing precision in predicting the thermodynamic properties of ternary alloy systems. Considering the interaction of the elements, we propose a two-step calculation method (TSCM) for calculating DHABC, which could solve the fatal problem. Under the TSCM hypothesis, the alloying process is split into two steps. The first step is the alloying of metals
W. C. Wang et al. / Scripta Materialia 56 (2007) 975–978
977
As the above three processes are about equal, taking an average from the above three equations, i.e. Eqs. (11)– (13), will achieve a best approximation as follows: 1 c DH A in B þ DH cC in AB DH ABC ¼ 3 þ DH cB in C þ DH cA in BC þ DH cA in C þ DH cB in AC : ð14Þ
Figure 2. Comparison of the experimental values with the calculated results of standard formation enthalpies by (a) the OMM; (b) the RMM; and (c) the PM, respectively, for 90 Al-based and Mg-based ternary intermetallic compounds.
A and B according to the alloy composition, and the second step is the introducing the third metal, metal C, into the mixture of metals A and B. The heat of solution of the two steps can be described as DH cA in B and DH cC in AB , respectively, and can be calculated by ð9Þ DH cA in B ¼ xA fBA DH int A in B ; C DH cC in AB ¼ xC fAB DH int C in AB :
ð10Þ
Employing the proposed model (PM) of Eq. (14), the standard formation enthalpies of the ternary intermetallic compounds and the mixing enthalpies of the ternary liquid alloys can be calculated. Accordingly, 90 Al-based and Mg-based ternary intermetallic compounds have been calculated by the PM and the results compared with those obtained in experiments reported in the literature [15,16]. Figure 2c displays the calculated results obtained by the PM, and can be compared with Figure 2a and b, which represents the results calculated by the OMM and RMM, respectively. One can see that the PM gives results with better precision than the OMM statistically, with over 85% of the calculated results improved. For instance, Table 1 lists the data for the Al–Ni–Y system, clearly showing that the PM can improve the precision compared with the OMM. Also, from the average relative error and the correlation factors in Figure 2, one can conclude that the PM is better than both the OMM and RMM. Some 420 ternary liquid alloys have also been calculated by the PM and OMM, respectively, and then compared with the experimental values obtained from the literature [17–22]. Figure 3 shows the results calculated by the PM and OMM. It can be seen that, as the heat of formation becomes more negative, the improvement by the PM becomes more obvious. The figure can thus be divided into two regions, I and II. In region I, the calculated results by the PM or OMM are almost the same as the experimental values, and the degree of improvement by the PM is not obvious. For the ternary alloy systems in this region, we find that the difference of the property (V, /, nws) among the three elements is Table 1. The standard formation enthalpies (kJ mol1) of Al–Ni–Y ternary intermetallic compounds measured in experiments and calculated by the PM and OMM, respectively
DH cC in AB ,
In calculating the properties of the mixture of metals A and B comply with the following reasonable assumptions: xAB = xA + xB; VAB = (xAVA + xBVB)/ xAB; /AB = (xA/A + xB/B)/xAB; nwsAB = (xA nwsA + xB nwsB)/xAB. By considering the entire process of alloying, the formation enthalpy can be calculated by ð11Þ DH ABC ¼ DH cA in B þ DH cC in AB : There are another two alternative processes that could achieve the same ternary alloy, i.e. by first alloying metals B and C and metals A and C as the first step, respectively, and then introducing metals A and B, respectively. Accordingly, the following two equations are for calculating DHABC: DH ABC ¼ DH cB in C þ DH cA in BC ; DH ABC ¼
DH cA in C
þ
DH cB in AC :
ð12Þ a
ð13Þ
Compounds
Experimental PM values (EXP)
OMM
Improvement by the PM (%)a
Al0.08Ni0.67Y0.25 Al0.15Ni0.68Y0.17 Al0.18Ni0.55Y0.27 Al0.25Ni0.58Y0.17 Al0.33Ni0.33Y0.33 Al0.50Ni0.25Y0.25 Al0.50Ni0.33Y0.17 Al0.58Ni0.25Y0.17 Al0.60Ni0.07Y0.33 Al0.60Ni0.20Y0.20 Al0.67Ni0.17Y0.16 Al0.69Ni0.23Y0.08 Al0.70Ni0.18Y0.12
37.90 47.33 48.50 48.30 54.10 62.80 62.80 61.30 56.70 59.80 54.00 47.00 50.60
54.42 62.01 83.58 81.05 107.41 97.73 90.60 81.82 74.57 80.68 66.70 53.09 58.51
42.96 20.57 71.48 60.26 93.94 46.75 35.86 20.32 21.85 22.00 9.13 4.39 0.00
Improvement ð%Þ ¼
38.14 42.38 48.08 51.94 56.59 57.23 57.52 53.24 51.21 52.07 46.23 42.98 42.69
EXP jDH PM ABC DH ABC j 100%. OMM jDH ABC DH EXP ABC j
978
W. C. Wang et al. / Scripta Materialia 56 (2007) 975–978
The authors are grateful for the financial support from the National Natural Science Foundation of China (50531040), The Ministry of Science and Technology of China (2006CB605201) and the Administration of Tsinghua University.
Figure 3. Comparison of the experimental values with the calculated results of mixing enthalpies by the OMM and the PM, respectively, for 420 ternary liquid alloys.
not distinct. Therefore, the influence of atomic size and of the interaction among elements on the results calculated by the OMM can be neglected. However, for the ternary alloy systems in region II, the difference of the property (V, /, nws) among the three elements is distinct and cannot be neglected. Therefore the calculated error of the OMM is larger. This is also the reason why there is more scatter in this region, and why the improvement by the PM in this region is more obvious. In Figure 3, 100% of the circles fall in the data zone in region II, whereas only about 85% of squares fall in the same data zone, indicating that the PM could achieve more precise results than the OMM in calculating the mixing enthalpies of ternary liquid alloys. The improvement of the PM for ternary systems originates from two important factors. Firstly, the atomic size difference of the constituent metals is taken into consideration by introducing a new pre-factor, S(x). Secondly, the interactions between the third metal and the other two are taken into account by employing a TSCM method. The ternary model is verified by calculating formation enthalpies of about 90 intermetallic compounds and by calculating mixing enthalpies of about 420 liquid alloys. The continuity of the ternary model is also analyzed, and it is confirmed that the model can be applied to binary alloys. In addition, the PM is equal to the RMM for binary alloy systems.
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