- Email: [email protected]
Atomic mobilities and diffusion characteristics for fcc Cu–Ag–Au alloys
CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 314–322
Contents lists available at SciVerse ScienceDirect
CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry journal homepage: www.elsevier.com/locate/calphad
Atomic mobilities and diffusion characteristics for fcc Cu–Ag–Au alloys Yajun Liu a,∗ , Jiang Wang b , Yong Du c , Guang Sheng d , Lijun Zhang c,e , Dong Liang f a
Western Transportation Institute, Montana State University, Bozeman, MT, 59715, USA
b
EMPA, Swiss Federal Laboratories for Materials Testing and Research, Laboratory for Joining and Interface Technology, überlandstrasse 129, CH-8600 Dübendorf, Switzerland
c
State Key Laboratory of Powder Metallurgy, Central South University, Changsha, Hunan, 410083, PR China
d
Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA, 16802, USA
e
Interdisciplinary Centre for Advanced Materials Simulation (ICAMS), Ruhr-Universität Bochum, Bochum, 44801, Germany
f
DNV Columbus, Dublin, OH, 43017, USA
article
info
Article history: Received 4 November 2010 Received in revised form 11 March 2011 Accepted 11 March 2011 Available online 7 April 2011 Keywords: Diffusion Mobility CALPHAD Dictra Fcc Cu–Ag–Au
abstract CALPHAD kinetics has evolved to be a well-established discipline, which allows complex non-equilibrium processes to be fully explored. Such a success relies on the use of Redlich–Kister polynomials to describe atomic mobilities, with the effect of temperature, composition, magnetic ordering and chemical ordering sufficiently taken into consideration. There is thus an increasing demand to construct high-quality atomic mobility databases for alloys of interest. Based on the CALPHAD framework, the atomic mobilities in fcc Cu–Ag–Au alloys are reported in this work, the results of which can provide fruitful information on alloy design. Published by Elsevier Ltd
1. Introduction In recent years, international competition has spurred the development of novel alloys with superior properties. Diffusion plays an important role in a great majority of metallurgical processes, such as heat treatment and welding, which is of great importance for practical and scientific purposes [1,2]. Traditionally, microstructure evolution in metallic materials is studied with experiments, which are laborious and costly. Aimed at higher efficiency and reduced cost [3–6], CALPHAD (CALculation of PHAse Diagram) kinetics has been adopted to address complex diffusioncontrolled problems, thereby serving as an essential roadmap to select candidate alloys. The quality of computational simulation based on CALPHAD kinetics heavily depends on atomic mobility databases, which enable high-order systems to be explored by extrapolation when experimental information is not available [7]. For a binary system, there are two kinds of atomic mobilities and one kind of interdiffusion coefficients. When one more element is added and the binary solution is augmented into a ternary one, the system is characterized by three kinds of atomic mobilities and
∗
Corresponding author. Tel.: +1 404 513-1544. E-mail addresses: [email protected], [email protected] (Y. Liu).
0364-5916/$ – see front matter. Published by Elsevier Ltd doi:10.1016/j.calphad.2011.03.001
four kinds of interdiffusion coefficients defined with respect to a reference element. The construction of atomic mobility databases for solution phases thus presents great challenges. Despite the technological importance of Cu–Ag–Au ternary alloys for dental applications [8,9], little has been done to extract fundamental kinetic information to facilitate alloy design for industry. In this work, the atomic mobilities of fcc Cu–Ag–Au alloys are systematically framed, the results of which shed light on design and application of Cu-based, Ag-based and Au-based alloys. 2. Model description For a substitutional solution containing n elements, the interdiffusion flux of element k, referred to the number-fixed frame of reference, can be given by [10]:
⃗JkN = −
n −1 −
˜ nki ∇ Ci D
(1)
i=1
˜ nki is the interdiffusion coefficient, with element n being the where D dependent one; Ci is the concentration of element i in mole per volume; ⃗JkN denotes the interdiffusion flux of element k; the superscript N stands for the number-fixed frame of reference. In the CALPHAD framework, the molar volume of one phase under consideration is treated as constant.
Y. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 314–322
315
Table 1 Thermodynamic parameters for fcc Cu–Ag–Au alloys (all in SI units). Phase
fcc
Model
Parameters
(Cu, Ag, Au)1 (Va)1
0 fcc LAg,Cu:Va 1 fcc LAg,Cu:Va
= 34 817 − 8.876T [16] = −3270 − 0.570T [16]
0 fcc LAu,Cu:Va
= −28 000 + 78.8T − 10 · T · ln(T ) [31]
1 fcc LAu,Cu:Va
= 6000 [31]
0 fcc LAg,Au:Va
= −15 599 [32]
0 fcc LAg,Au,Cu:Va
= −8998.65 + 2.02T
1 fcc LAg,Au,Cu:Va
= −11 034.57 − 3.13T
2 fcc LAg,Au,Cu:Va
= −10 973.25 − 7.01T
Table 2 Mobility parameters for fcc Cu–Ag–Au alloys (all in SI units). Phase Model
Mobility
Parameters Ag:Va
ΦCu = −179 012 − 87.49T [2] Cu:Va ΦCu = −205 872 − 82.53T [2] 0 Ag,Cu:Va ΦCu = 179 987.65 Cu
fcc
(Cu, Ag, Au)1 (Va)1
Fig. 1. Calculated Cu–Ag phase diagram according to the thermodynamic description of Witusiewicz et al. [16].
Au:Va ΦCu = −167 949.99 − 96.29T [40] 0
Au,Cu:Va ΦCu = 74 434.63 − 19.23T [40]
0
ΦCu
0
ΦCu
1
ΦCu
Ag,Au:Va
= −59 732.54
Ag,Au,Cu:Va
= −398 745.21
Ag,Au,Cu:Va
= 203 150.47
2 Ag,Au,Cu:Va ΦCu
= 198 435.24
Ag:Va
ΦAg = −175 892 − 93.50T [2] Cu:Va ΦAg = −191 533 − 82.93T [2] Ag
0
Ag,Cu:Va
ΦAg
= 120 859.516
Au:Va ΦAg = −169 000 − 97.68T [41] Ag,Au:Va
= −34 686.15 + 22.10T [41]
Ag,Au:Va
= 6316.20 [41]
0
ΦAg
1
ΦAg
0
ΦAg
Ag,Au,Cu:Va
= 119 872.64
Fig. 2. Calculated and experimentally measured Ag tracer diffusion coefficients in fcc Cu–Ag alloys at various temperatures.
Au:Va ΦAu = −176 600 − 95.7T [40] Cu:Va ΦAu = −210 000 − 79.45T [40]
Au
0
Au,Cu:Va ΦAu = 91387.46 − 42.60T [40] Ag:Va
ΦAu
= −202 078.51 − 77.88T [41]
Ag,Au:Va
0
ΦAu
1
Ag,Au:Va ΦAu
0
ΦAu
Ag,Cu:Va
= −21 198.91 + 4.02T [41]
− ∂µj ∂µj (δkj − xk )xj Mj − ∂ xi ∂ xn j
Mk0
Φk
= 59 632.74
where Qk is the activation energy; Mk0 is the frequency factor; R is the gas constant; T is temperature; Φk is defined by −Qk + RT ln(Mk0 ), which is dependent on composition and temperature. Φk can be expanded by the Redlich–Kister polynomial as follows [13]:
(2)
Φk =
− i
xi Φki
+
RT
=
1
= −5093.59 [41]
RT
exp −
Qk
Mk =
The interdiffusion coefficients can be described by the following general expression [11]:
˜ nki = D
where δkj is the Kronecker delta (δkj = 1 if k = j, otherwise δkj = 0). xj , µj , and Mj are the mole fraction, chemical potential and atomic mobility of element j, respectively. According to Andersson and Ågren [12], the atomic mobility of element k can be expressed as
−− i
j >i
RT
xi xj
exp
− r
(3)
RT
r
i ,j Φk
(xi − xj )
r
316
Y. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 314–322
Fig. 3. Calculated and experimentally measured interdiffusion coefficients in fcc Cu–Ag alloys at various temperatures.
Fig. 5. Calculated and experimentally measured fcc miscibility gap in Cu–Ag–Au ternary alloys.
Fig. 6. Calculated and experimentally measured fcc miscibility gap in Cu–Ag–Au ternary alloys.
Fig. 4. Calculated and experimentally measured concentration profiles in Cu/Ag diffusion couples annealed at 1023 K.
+
−−− i
j where Φk
j >i
xi xj xl
l >j
−
vijls s Φki,j,l
(s = i, j, l)
stands for the mobility end-member for element k to difi,j,l
(5)
Assuming a mono-vacancy mechanism for diffusion, the tracer diffusion coefficient of element k(D∗k ) can be correlated with its atomic mobility by the Einstein relation [10]: D∗k = RTMi .
(7)
3. Thermodynamic and kinetic evaluation
fuse in element j; r Φk and s Φk are the binary and ternary inters action parameters, respectively; vijl is defined by
vijls = xs + (1 − xi − xj − xl )/3.
∂ Ck + ∇ · (J˜kN ) = 0. ∂t
(4)
s
i,j
mass conservation form as follows:
(6)
With the interdiffusion fluxes defined above, the temporal and spatial evolution of element k can be given by the Fick law in the
The thermodynamic descriptions for the Cu–Ag binary systems have been reported by Hayes et al. [14], Lim et al. [15], Witusiewicz et al. [16] and He et al. [17]. Based on an unpublished thermodynamic description for the Cu–Ag binary system [18], Kusoffsky [19] assessed the thermodynamic properties of the Cu–Ag–Au ternary system, and provided the ternary interaction parameters for fcc phase. As Witusiewicz et al. [16] has conducted an extensive literature review and provided an updated thermodynamic description for the Cu–Ag binary system, there is an increasing need to reassess the ternary thermodynamic parameters. The phase boundary of fcc miscibility gap has been measured by Yamauchi et al. [20], Sistare [21], Kogachi and Nakahigashi [22], Uzuka et al. [23], Kogachi and Nakahigashi [24], Masing et al. [25], McMullin and
Y. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 314–322
317
Fig. 7. Calculated and experimentally measured fcc miscibility gap in Cu–Ag–Au ternary alloys.
Fig. 9. Calculated and experimentally measured fcc miscibility gap in Cu–Ag–Au ternary alloys.
Fig. 8. Calculated and experimentally measured fcc miscibility gap in Cu–Ag–Au ternary alloys.
Fig. 10. Calculated and experimentally measured fcc miscibility gap in Cu–Ag–Au ternary alloys.
Norton [26], Ziebold and Ogilvie [27], Murakami et al. [28], and Ntukogu and Cadoff [29], the results of which were systematically reviewed by Prince [30]. The thermodynamic descriptions for fcc Cu–Au and Ag–Au alloys based on the CALPHAD framework have been reported by Sundman et al. [31] and Hassam et al. [32], respectively, which are adopted in this work. The assessment of ternary thermodynamic parameters is performed with the Parrot module in Thermo-Calc software, the results of which and those from the literature are presented in Table 1. The mobility end-members for Ag in fcc Ag, Cu in fcc Cu, Ag in fcc Cu and Cu in fcc Ag have been reported by Ghosh [2]. Since these end-members can represent a great majority of the reported self-diffusion and impurity diffusion coefficients, they are adopted in this work. The interaction parameters for Ag and Cu atomic mobilities in fcc Cu–Ag binary alloys have not been assessed by Ghosh [2]. In order to establish a kinetic description for fcc Cu–Ag–Au alloys, such interaction parameters are inversely parameterized in this work. The experimental characterization on diffusion in fcc Cu–Ag alloys has been extensively reviewed by Butrymowicz et al. [33], where the tracer diffusion coefficients of
110 Ag in fcc Cu–Ag alloys from [34–36] and Patil and Sharma [37] were found to be more reliable. Such experimental values are employed in this work to evaluate the interaction parameter for mobility of Ag in fcc Cu–Ag alloys. Despite the abundant investigations on interdiffusion in fcc Cu–Ag binary system, Butrymowicz et al. [33] concluded that a great majority of the reported experimental values are of low quality, and are incorrect in nature. Interested readers can refer to Ref. [33] for more details. In this work, the interdiffusion coefficients from [38] are adopted because the measured values have good extrapolation ability with respect to the impurity diffusion coefficients of Ag in fcc Cu. The concentration curves for Cu/Ag diffusion couples from [39] were utilized to estimate the interaction parameters for the mobility of Cu in fcc Cu–Ag. Ziebold and Ogilvie [27] studied the diffusion characteristics of fcc Cu–Ag–Au alloys with diffusion couples at 998 K, where the interdiffusion coefficients were extracted from concentration curves with the extended Boltzmann–Matano analysis. Such results are taken into consideration during the evaluation of ternary interaction parameters for the Ag, Au and Cu atomic mobilities. In this work, the inverse parameterization
318
Y. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 314–322
Fig. 11. Calculated and experimentally measured fcc miscibility gap in Cu–Ag–Au ternary alloys.
Fig. 14. Calculated and experimentally measured fcc miscibility gap in Cu–Ag–Au ternary alloys.
Fig. 12. Calculated and experimentally measured fcc miscibility gap in Cu–Ag–Au ternary alloys.
Fig. 15. Calculated and experimentally measured fcc miscibility gap in Cu–Ag–Au ternary alloys.
Fig. 13. Calculated and experimentally measured fcc miscibility gap in Cu–Ag–Au ternary alloys.
Fig. 16. Calculated and experimentally measured fcc miscibility gap in Cu–Ag–Au ternary alloys.
Y. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 314–322
Fig. 17. Calculated and experimentally measured fcc miscibility gap in Cu–Ag–Au ternary alloys.
Fig. 18. Calculated and experimentally measured fcc miscibility gap in Cu–Ag–Au ternary alloys.
of the atomic mobilities in fcc Cu–Ag–Au alloys is conducted with the Parrot module in Dictra, which can handle self-diffusion coefficients, impurity diffusion coefficients, intrinsic diffusion coefficients and interdiffusion coefficients concurrently. All the atomic mobilities obtained in this work are given in Table 2, along with those adopted from fcc Cu–Ag [2], Cu–Au [40] and Ag–Au [41] binary systems. 4. Results and discussion 4.1. The Cu–Ag binary fcc alloys The calculated Cu–Ag phase diagram with the thermodynamic description of Witusiewicz et al. [16] is presented in Fig. 1, where the fcc phase forms a miscibility gap across a wide temperature range. As the fcc phase regions are narrow on both Cu and Ag sides, the accurate measurements of diffusion coefficients present great challenge. The calculated and measured tracer diffusion coefficients of Ag in fcc Ag–Cu alloys are shown in Fig. 2. When the mole fraction of Cu approaches zero, the tracer diffusion
319
˜ Au Fig. 19. Calculated and measured ternary interdiffusion coefficients of D CuCu at 998 K; dashed blue isolines (this work) and solid dots [27].
˜ Au Fig. 20. Calculated and measured ternary interdiffusion coefficients of D AgAg at 998 K; dashed blue isolines (this work) and solid dots [27].
coefficients of Ag in fcc Ag-rich alloys are equivalent to the selfdiffusion coefficients of Ag. As the self-diffusion coefficients of Ag in fcc Ag were assessed by Ghosh [2], the tracer diffusion coefficients of Ag in fcc Cu–Ag alloys can be judged from their extrapolation ability on the Ag-rich side. As can be seen in Fig. 2, the experimental data from [34] for 1023 and 1073 K and Sato [35] for 973 K can be reproduced well due to their excellent extrapolation ability. Fig. 3 compares the calculated interdiffusion coefficients in the Cu-rich part at four temperatures with the measured data from [38]. The agreement is excellent at 1273, 1174 and 1076 K. The deviation between the calculated and measured data at 974 K is due to the poor extrapolation ability of the experimental data in the Cu-rich part, where the limiting value is equivalent to the impurity diffusion coefficient of Ag in fcc Cu at 974 K. The further test of the atomic mobilities can be obtained through the simulations of concentration profiles in diffusion couples. The calculated and measured concentration profiles for two Cu/Ag diffusion couples annealed at 1023 K for 1800 and 3600 s are given in Fig. 4, where the agreement is excellent. The sharp and
320
Y. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 314–322
Fig. 21. Calculated and measured ternary interdiffusion coefficient ratio of ˜ Au ˜ Au D CuAg /DCuCu at 998 K; dashed blue isolines (this work) and solid dots [27].
Fig. 22. Calculated and measured ternary interdiffusion coefficient ratio of ˜ Au ˜ Au D AgCu /DAgAg at 998 K; dashed blue isolines (this work) and solid dots [27].
moving interfaces correspond to the tie-line of the two-phase region, which is invariant with time in this binary system. 4.2. The Cu–Ag–Au ternary fcc alloys Ternary fcc Cu–Ag–Au alloys feature a large solubility range, and a miscibility gap is formed around the binary Cu–Ag edge. The calculated isothermal phase boundaries for fcc phase at 1023, 973, 873, 773 and 673 K are presented in Figs. 5–9, and the calculated vertical phase boundaries sectioned at different composition sets are given in Figs. 10–18. For comparison, the experimental data from [20–29] are also superimposed. As can be seen, the agreement is excellent at various temperatures and concentrations. For a ternary system, interdiffusion coefficients are frequently evaluated at intersections of diffusion paths under isothermal conditions. As diffusion paths are one-dimensional geometric elements in two-dimensional composition spaces, intersection points for various diffusion paths are not rarely encountered. As long as the initial alloy compositions are well specified, sufficient numbers of intersection points can be located on the Gibbs triangle,
Fig. 23. Calculated and measured composition profiles for a Cu/Au diffusion couple annealed at 998 K for 48 h.
Fig. 24. Calculated and measured composition profiles for a Cu/Ag0.0865 Au0.9135 diffusion couple annealed at 998 K for 60.8 h.
on which interdiffusion coefficients can be evaluated. A large number of diffusion couples can be set up to generate compositiondependent interdiffusion coefficients over a wide composition range. ˜ Au ˜ Au The calculated dashed blue isolines for D CuCu and DAgAg are presented in Figs. 19 and 20, respectively, together with the experimental values from [27]. As is evident, the weak concentration dependence for such two kinds of interdiffusion coefficients is observed. Both kinds of diagonal interdiffusion coefficients form peaks within the Gibbs triangle, and the calculated isolines generally follow the tendency obtained experimentally. The ratios of ˜ Au ˜ Au off-diagonal to diagonal interdiffusion coefficients, i.e. D CuAg /DCuCu
˜ Au ˜ Au and D AgCu /DAgAg , are also studied with isolines in Figs. 21 and 22, together with the reported values from [27]. Such two ratios have well-defined geometric properties with respect to binary edges. It ˜ Au is known that on the Ag–Au binary edge, D CuAg will be zero, while ˜ Au D CuCu is equivalent to the impurity diffusion coefficients of Cu in ˜ Au ˜ Au fcc Ag–Au alloys. As such, D CuAg /DCuCu is zero on the Au–Ag binary
˜ Au ˜ Au side. Similarly, D AgCu /DAgAg is zero on the Au–Cu binary side, since
Y. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 314–322
Fig. 25. Calculated and measured composition profiles for a Cu/Ag0.309 Au0.691 diffusion couple annealed at 998 K for 60.8 h.
Fig. 26. Calculated and measured composition profiles for a Cu0.674 Au0.326 /Ag0.554 Au0.446 diffusion couple annealed at 998 K for 48 h.
˜ Au ˜ Au D AgCu = 0 and DAgAg denotes the impurity diffusion coefficients of Ag in fcc Au–Cu binary alloys. Such geometric relations can be evidenced not only from the calculated isolines but also from the discrete experimental values from [27]. With the known atomic mobilities and thermodynamic description, it is convenient to calculate the concentration distributions for diffusion couples, as presented in Figs. 23–30. From comparisons in Figs. 23–30, excellent agreements can be obtained, which serve as direct evidence to verify the mobility quality assessed in this work. In Fig. 26, there is one point on the Au concentration profile, on which the concentration gradient of Au is almost zero. At such a point, the concentration gradients of the remaining two elements, i.e. Cu and Ag, are almost equal to each other in magnitude but with opposite signs. Diffusion paths are a kind of plots for concentration profiles in diffusion couples, in which the kinetic information is almost lost, while the concentration relation can be concisely presented on phase diagrams. The diffusion paths calculated in this work are given in Fig. 31 for four diffusion couples, along with the experimental data from [27]. The agreement between the calculated and measured diffusion paths is satisfactory. In Fig. 31, the diffusion path D passes through the critical point
321
Fig. 27. Calculated and measured composition profiles for a Au0.56 Cu0.44 /Ag0.309 Au0.691 diffusion couple annealed at 998 K for 48 h.
Fig. 28. Calculated and measured composition profiles for a Au0.077 Cu0.923 /Au0.446 Ag0.554 diffusion couple annealed at 998 K for 48 h.
Fig. 29. Calculated and measured composition profiles for a Au0.56 Cu0.44 /Au0.446 Ag0.554 diffusion couple annealed at 998 K for 48 h.
322
Y. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 314–322
CALPHAD framework. Comparisons between the calculated and measured data, including phase boundary data, tracer diffusion coefficients and interdiffusion coefficients, are made to verify such thermodynamic and kinetic parameters. In addition, the calculated concentration profiles in diffusion couples and diffusion paths superimposed on one isothermal phase diagram are extensively explored. This work contributes to the fundamental knowledge on microstructure evolution and phase stabilities for fcc Cu–Ag–Au alloys, the results of which can be utilized to solve problems of interest, not only for industrial issues but also for scientific purposes. Appendix. Supplementary data Supplementary material related to this article can be found online at doi:10.1016/j.calphad.2011.03.001. References Fig. 30. Calculated and measured composition profiles for a Au0.56 Cu0.44 /Au0.9135 Ag0.0865 diffusion couple annealed at 998 K for 48 h.
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
[21]
Fig. 31. Calculated and experimentally measured diffusion paths for four ternary diffusion couples annealed at 998 K. (A: Ag0.0865 Au0.9135 /Cu0.44 Au0.56 , 48 h; B: Cu/Ag0.0865 Au0.9135 , 60.8 h; C: Cu0.674 Au0.326 /Ag0.554 Au0.446 , 48 h; D: Cu0.923 Au0.077 /Ag0.554 Au0.446 , 48 h.)
[22] [23] [24] [25] [26] [27] [28]
of the ternary miscibility gap and is tangent to that critical point. Under this case, the composition profiles for diffusion couples around the critical point are expected to be extremely steep and still continuous, because this point denotes a transition between a solution behavior in which the concentration gradients of the three elements are finite and the discontinuous phase interface where two phases are present.
[29] [30] [31] [32] [33] [34] [35] [36]
5. Conclusions The thermodynamic description and atomic mobilities for fcc Cu–Ag–Au alloys are presented in this work, based on the
[37] [38] [39] [40] [41]
Q. Chen, N. Ma, K. Wu, Y. Wang, Scr. Mater. 50 (2004) 471. G. Ghosh, Acta Mater. 49 (2001) 2609. M. Palumbo, M. Baricco, Acta Mater. 53 (2005) 2231. G. Ghosh, G.B. Olson, Acta Mater. 50 (2002) 2099. T. Helander, J. Ågren, Acta Mater. 47 (1999) 1141. T. Helander, J. Ågren, Acta Mater. 47 (1999) 3291. C.E. Campbell, W.J. Boettinger, U.R. Kattner, Acta Mater. 50 (2002) 775. K. Yasuda, H. Metahi, Y. Kanzawa, J. Less-Common Met. 60 (1978) 65. D.C. Wright, R.F. Gallant, L. Spangberg, J. Biomed. Mater. Res. 16 (1982) 509. T. Gomez-Acebo, B. Navarcorena, F. Castro, J. Phase Equilib. Diff. 25 (2004) 237. B. Jönsson, ISIJ Int. 35 (1995) 1415. J.O. Andersson, J. Ågren, J. Appl. Phys. 72 (1992) 1350. G. Lindwall, K. Frisk, J. Phase Equilib. Diff. 30 (2009) 323. F.H. Hayes, H.L. Lukas, G. Effenberg, G. Petzow, Z. Metllkd. 77 (1986) 749. M.S. Lim, J.E. Tibball, P.L. Rossiter, Z. Metallkd. 88 (1997) 236. V.T. Witusiewicz, U. Hecht, S.G. Fries, S. Rex, J. Alloys Compd. 385 (2004) 133. X.C. He, H. Wang, H.S. Liu, Z.P. Liu, CALPHAD 30 (2006) 367. H.L. Lukas, Max-Planck-Institut für Metallforschung, Stuttgart, Germany, 1998, Private communcation. A. Kusoffsky, Acta Mater. 50 (2002) 5139. H. Yamauchi, H.A. Yoshimatsu, A.R. Forouhi, D. de Fontaine, Proc. 4th Precious Metals Conf., Toronto, 1980, Pergamon Press Canada, Ontario, 1981, pp. 241–249. G.H. Sistare, Metals Handbook, 8th ed., vol. 8, American Society for Metals, Metals Park, OH, 1973, pp. 377–378. M. Kogachi, K. Nakahigashi, Japan J. Appl. Phys. 19 (1980) 1443. T. Uzuka, Y. Kanzawa, K. Yasuda, J. Dent. Res. 60 (1981) 883. M. Kogachi, K. Nakahigashi, Japan J. Appl. Phys. 24 (1985) 121. G. Masing, K. Kloiber, Z. Metllkd. 32 (1940) 125. J.G. McMullin, J.T. Norton, Trans. AIME 185 (1949) 46. T.O. Ziebold, R.E. Ogilvie, Trans. AIME 239 (1967) 942. M. Murakami, D. de Fontaine, J.M. Sanchez, J. Fodor, Thin Solid Films 25 (1975) 465. T.O. Ntukogu, I.B. Cadoff, Mater. Sci. Technol. 2 (1986) 528. A. Prince, Int. Mater. Rev. 33 (1988) 314. B. Sundman, S.G. Fries, W.A. Oates, CALPHAD 22 (1998) 335. S. Hassam, J. Agren, M. Gaune-Escard, J.P. Bros, Metall. Trans. A 21 (1990) 1877. B. Butrymowicz, J.R. Manning, M.E. Read, J. Phys. Chem. Ref. Data 3 (1974) 527. R.E. Hoffman, D. Turnbull, E.W. Hart, Acta Metall. 3 (1955) 417. K. Sato, Sci. Rep. Fac. Lit. Sci. 11 (1964) 9. M.E. Yanitskaya, A.A. Zhukhovitskii, S.Z. Bokshtein, Dokl. Akad. Nauk SSSR 112 (1957) 720. R.V. Patil, B.D. Sharma, Trans. Indian Inst. Met. 25 (1972) 1. H. Oikawa, H. Takei, S. Karashima, Metall. Trans. 4 (1973) 653. J. Unnam, C.R. Houska, J. Appl. Phys. 47 (1976) 4336. Y. Liu, L. Zhang, D. Yu, J. Phase Equilib. Diff. 30 (2009) 136. Y. Liu, L. Zhang, D. Yu, Y. Ge, J. Phase Equilb. Diff. 29 (2008) 405.
Contents lists available at SciVerse ScienceDirect
CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry journal homepage: www.elsevier.com/locate/calphad
Atomic mobilities and diffusion characteristics for fcc Cu–Ag–Au alloys Yajun Liu a,∗ , Jiang Wang b , Yong Du c , Guang Sheng d , Lijun Zhang c,e , Dong Liang f a
Western Transportation Institute, Montana State University, Bozeman, MT, 59715, USA
b
EMPA, Swiss Federal Laboratories for Materials Testing and Research, Laboratory for Joining and Interface Technology, überlandstrasse 129, CH-8600 Dübendorf, Switzerland
c
State Key Laboratory of Powder Metallurgy, Central South University, Changsha, Hunan, 410083, PR China
d
Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA, 16802, USA
e
Interdisciplinary Centre for Advanced Materials Simulation (ICAMS), Ruhr-Universität Bochum, Bochum, 44801, Germany
f
DNV Columbus, Dublin, OH, 43017, USA
article
info
Article history: Received 4 November 2010 Received in revised form 11 March 2011 Accepted 11 March 2011 Available online 7 April 2011 Keywords: Diffusion Mobility CALPHAD Dictra Fcc Cu–Ag–Au
abstract CALPHAD kinetics has evolved to be a well-established discipline, which allows complex non-equilibrium processes to be fully explored. Such a success relies on the use of Redlich–Kister polynomials to describe atomic mobilities, with the effect of temperature, composition, magnetic ordering and chemical ordering sufficiently taken into consideration. There is thus an increasing demand to construct high-quality atomic mobility databases for alloys of interest. Based on the CALPHAD framework, the atomic mobilities in fcc Cu–Ag–Au alloys are reported in this work, the results of which can provide fruitful information on alloy design. Published by Elsevier Ltd
1. Introduction In recent years, international competition has spurred the development of novel alloys with superior properties. Diffusion plays an important role in a great majority of metallurgical processes, such as heat treatment and welding, which is of great importance for practical and scientific purposes [1,2]. Traditionally, microstructure evolution in metallic materials is studied with experiments, which are laborious and costly. Aimed at higher efficiency and reduced cost [3–6], CALPHAD (CALculation of PHAse Diagram) kinetics has been adopted to address complex diffusioncontrolled problems, thereby serving as an essential roadmap to select candidate alloys. The quality of computational simulation based on CALPHAD kinetics heavily depends on atomic mobility databases, which enable high-order systems to be explored by extrapolation when experimental information is not available [7]. For a binary system, there are two kinds of atomic mobilities and one kind of interdiffusion coefficients. When one more element is added and the binary solution is augmented into a ternary one, the system is characterized by three kinds of atomic mobilities and
∗
Corresponding author. Tel.: +1 404 513-1544. E-mail addresses: [email protected], [email protected] (Y. Liu).
0364-5916/$ – see front matter. Published by Elsevier Ltd doi:10.1016/j.calphad.2011.03.001
four kinds of interdiffusion coefficients defined with respect to a reference element. The construction of atomic mobility databases for solution phases thus presents great challenges. Despite the technological importance of Cu–Ag–Au ternary alloys for dental applications [8,9], little has been done to extract fundamental kinetic information to facilitate alloy design for industry. In this work, the atomic mobilities of fcc Cu–Ag–Au alloys are systematically framed, the results of which shed light on design and application of Cu-based, Ag-based and Au-based alloys. 2. Model description For a substitutional solution containing n elements, the interdiffusion flux of element k, referred to the number-fixed frame of reference, can be given by [10]:
⃗JkN = −
n −1 −
˜ nki ∇ Ci D
(1)
i=1
˜ nki is the interdiffusion coefficient, with element n being the where D dependent one; Ci is the concentration of element i in mole per volume; ⃗JkN denotes the interdiffusion flux of element k; the superscript N stands for the number-fixed frame of reference. In the CALPHAD framework, the molar volume of one phase under consideration is treated as constant.
Y. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 314–322
315
Table 1 Thermodynamic parameters for fcc Cu–Ag–Au alloys (all in SI units). Phase
fcc
Model
Parameters
(Cu, Ag, Au)1 (Va)1
0 fcc LAg,Cu:Va 1 fcc LAg,Cu:Va
= 34 817 − 8.876T [16] = −3270 − 0.570T [16]
0 fcc LAu,Cu:Va
= −28 000 + 78.8T − 10 · T · ln(T ) [31]
1 fcc LAu,Cu:Va
= 6000 [31]
0 fcc LAg,Au:Va
= −15 599 [32]
0 fcc LAg,Au,Cu:Va
= −8998.65 + 2.02T
1 fcc LAg,Au,Cu:Va
= −11 034.57 − 3.13T
2 fcc LAg,Au,Cu:Va
= −10 973.25 − 7.01T
Table 2 Mobility parameters for fcc Cu–Ag–Au alloys (all in SI units). Phase Model
Mobility
Parameters Ag:Va
ΦCu = −179 012 − 87.49T [2] Cu:Va ΦCu = −205 872 − 82.53T [2] 0 Ag,Cu:Va ΦCu = 179 987.65 Cu
fcc
(Cu, Ag, Au)1 (Va)1
Fig. 1. Calculated Cu–Ag phase diagram according to the thermodynamic description of Witusiewicz et al. [16].
Au:Va ΦCu = −167 949.99 − 96.29T [40] 0
Au,Cu:Va ΦCu = 74 434.63 − 19.23T [40]
0
ΦCu
0
ΦCu
1
ΦCu
Ag,Au:Va
= −59 732.54
Ag,Au,Cu:Va
= −398 745.21
Ag,Au,Cu:Va
= 203 150.47
2 Ag,Au,Cu:Va ΦCu
= 198 435.24
Ag:Va
ΦAg = −175 892 − 93.50T [2] Cu:Va ΦAg = −191 533 − 82.93T [2] Ag
0
Ag,Cu:Va
ΦAg
= 120 859.516
Au:Va ΦAg = −169 000 − 97.68T [41] Ag,Au:Va
= −34 686.15 + 22.10T [41]
Ag,Au:Va
= 6316.20 [41]
0
ΦAg
1
ΦAg
0
ΦAg
Ag,Au,Cu:Va
= 119 872.64
Fig. 2. Calculated and experimentally measured Ag tracer diffusion coefficients in fcc Cu–Ag alloys at various temperatures.
Au:Va ΦAu = −176 600 − 95.7T [40] Cu:Va ΦAu = −210 000 − 79.45T [40]
Au
0
Au,Cu:Va ΦAu = 91387.46 − 42.60T [40] Ag:Va
ΦAu
= −202 078.51 − 77.88T [41]
Ag,Au:Va
0
ΦAu
1
Ag,Au:Va ΦAu
0
ΦAu
Ag,Cu:Va
= −21 198.91 + 4.02T [41]
− ∂µj ∂µj (δkj − xk )xj Mj − ∂ xi ∂ xn j
Mk0
Φk
= 59 632.74
where Qk is the activation energy; Mk0 is the frequency factor; R is the gas constant; T is temperature; Φk is defined by −Qk + RT ln(Mk0 ), which is dependent on composition and temperature. Φk can be expanded by the Redlich–Kister polynomial as follows [13]:
(2)
Φk =
− i
xi Φki
+
RT
=
1
= −5093.59 [41]
RT
exp −
Qk
Mk =
The interdiffusion coefficients can be described by the following general expression [11]:
˜ nki = D
where δkj is the Kronecker delta (δkj = 1 if k = j, otherwise δkj = 0). xj , µj , and Mj are the mole fraction, chemical potential and atomic mobility of element j, respectively. According to Andersson and Ågren [12], the atomic mobility of element k can be expressed as
−− i
j >i
RT
xi xj
exp
− r
(3)
RT
r
i ,j Φk
(xi − xj )
r
316
Y. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 314–322
Fig. 3. Calculated and experimentally measured interdiffusion coefficients in fcc Cu–Ag alloys at various temperatures.
Fig. 5. Calculated and experimentally measured fcc miscibility gap in Cu–Ag–Au ternary alloys.
Fig. 6. Calculated and experimentally measured fcc miscibility gap in Cu–Ag–Au ternary alloys.
Fig. 4. Calculated and experimentally measured concentration profiles in Cu/Ag diffusion couples annealed at 1023 K.
+
−−− i
j where Φk
j >i
xi xj xl
l >j
−
vijls s Φki,j,l
(s = i, j, l)
stands for the mobility end-member for element k to difi,j,l
(5)
Assuming a mono-vacancy mechanism for diffusion, the tracer diffusion coefficient of element k(D∗k ) can be correlated with its atomic mobility by the Einstein relation [10]: D∗k = RTMi .
(7)
3. Thermodynamic and kinetic evaluation
fuse in element j; r Φk and s Φk are the binary and ternary inters action parameters, respectively; vijl is defined by
vijls = xs + (1 − xi − xj − xl )/3.
∂ Ck + ∇ · (J˜kN ) = 0. ∂t
(4)
s
i,j
mass conservation form as follows:
(6)
With the interdiffusion fluxes defined above, the temporal and spatial evolution of element k can be given by the Fick law in the
The thermodynamic descriptions for the Cu–Ag binary systems have been reported by Hayes et al. [14], Lim et al. [15], Witusiewicz et al. [16] and He et al. [17]. Based on an unpublished thermodynamic description for the Cu–Ag binary system [18], Kusoffsky [19] assessed the thermodynamic properties of the Cu–Ag–Au ternary system, and provided the ternary interaction parameters for fcc phase. As Witusiewicz et al. [16] has conducted an extensive literature review and provided an updated thermodynamic description for the Cu–Ag binary system, there is an increasing need to reassess the ternary thermodynamic parameters. The phase boundary of fcc miscibility gap has been measured by Yamauchi et al. [20], Sistare [21], Kogachi and Nakahigashi [22], Uzuka et al. [23], Kogachi and Nakahigashi [24], Masing et al. [25], McMullin and
Y. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 314–322
317
Fig. 7. Calculated and experimentally measured fcc miscibility gap in Cu–Ag–Au ternary alloys.
Fig. 9. Calculated and experimentally measured fcc miscibility gap in Cu–Ag–Au ternary alloys.
Fig. 8. Calculated and experimentally measured fcc miscibility gap in Cu–Ag–Au ternary alloys.
Fig. 10. Calculated and experimentally measured fcc miscibility gap in Cu–Ag–Au ternary alloys.
Norton [26], Ziebold and Ogilvie [27], Murakami et al. [28], and Ntukogu and Cadoff [29], the results of which were systematically reviewed by Prince [30]. The thermodynamic descriptions for fcc Cu–Au and Ag–Au alloys based on the CALPHAD framework have been reported by Sundman et al. [31] and Hassam et al. [32], respectively, which are adopted in this work. The assessment of ternary thermodynamic parameters is performed with the Parrot module in Thermo-Calc software, the results of which and those from the literature are presented in Table 1. The mobility end-members for Ag in fcc Ag, Cu in fcc Cu, Ag in fcc Cu and Cu in fcc Ag have been reported by Ghosh [2]. Since these end-members can represent a great majority of the reported self-diffusion and impurity diffusion coefficients, they are adopted in this work. The interaction parameters for Ag and Cu atomic mobilities in fcc Cu–Ag binary alloys have not been assessed by Ghosh [2]. In order to establish a kinetic description for fcc Cu–Ag–Au alloys, such interaction parameters are inversely parameterized in this work. The experimental characterization on diffusion in fcc Cu–Ag alloys has been extensively reviewed by Butrymowicz et al. [33], where the tracer diffusion coefficients of
110 Ag in fcc Cu–Ag alloys from [34–36] and Patil and Sharma [37] were found to be more reliable. Such experimental values are employed in this work to evaluate the interaction parameter for mobility of Ag in fcc Cu–Ag alloys. Despite the abundant investigations on interdiffusion in fcc Cu–Ag binary system, Butrymowicz et al. [33] concluded that a great majority of the reported experimental values are of low quality, and are incorrect in nature. Interested readers can refer to Ref. [33] for more details. In this work, the interdiffusion coefficients from [38] are adopted because the measured values have good extrapolation ability with respect to the impurity diffusion coefficients of Ag in fcc Cu. The concentration curves for Cu/Ag diffusion couples from [39] were utilized to estimate the interaction parameters for the mobility of Cu in fcc Cu–Ag. Ziebold and Ogilvie [27] studied the diffusion characteristics of fcc Cu–Ag–Au alloys with diffusion couples at 998 K, where the interdiffusion coefficients were extracted from concentration curves with the extended Boltzmann–Matano analysis. Such results are taken into consideration during the evaluation of ternary interaction parameters for the Ag, Au and Cu atomic mobilities. In this work, the inverse parameterization
318
Y. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 314–322
Fig. 11. Calculated and experimentally measured fcc miscibility gap in Cu–Ag–Au ternary alloys.
Fig. 14. Calculated and experimentally measured fcc miscibility gap in Cu–Ag–Au ternary alloys.
Fig. 12. Calculated and experimentally measured fcc miscibility gap in Cu–Ag–Au ternary alloys.
Fig. 15. Calculated and experimentally measured fcc miscibility gap in Cu–Ag–Au ternary alloys.
Fig. 13. Calculated and experimentally measured fcc miscibility gap in Cu–Ag–Au ternary alloys.
Fig. 16. Calculated and experimentally measured fcc miscibility gap in Cu–Ag–Au ternary alloys.
Y. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 314–322
Fig. 17. Calculated and experimentally measured fcc miscibility gap in Cu–Ag–Au ternary alloys.
Fig. 18. Calculated and experimentally measured fcc miscibility gap in Cu–Ag–Au ternary alloys.
of the atomic mobilities in fcc Cu–Ag–Au alloys is conducted with the Parrot module in Dictra, which can handle self-diffusion coefficients, impurity diffusion coefficients, intrinsic diffusion coefficients and interdiffusion coefficients concurrently. All the atomic mobilities obtained in this work are given in Table 2, along with those adopted from fcc Cu–Ag [2], Cu–Au [40] and Ag–Au [41] binary systems. 4. Results and discussion 4.1. The Cu–Ag binary fcc alloys The calculated Cu–Ag phase diagram with the thermodynamic description of Witusiewicz et al. [16] is presented in Fig. 1, where the fcc phase forms a miscibility gap across a wide temperature range. As the fcc phase regions are narrow on both Cu and Ag sides, the accurate measurements of diffusion coefficients present great challenge. The calculated and measured tracer diffusion coefficients of Ag in fcc Ag–Cu alloys are shown in Fig. 2. When the mole fraction of Cu approaches zero, the tracer diffusion
319
˜ Au Fig. 19. Calculated and measured ternary interdiffusion coefficients of D CuCu at 998 K; dashed blue isolines (this work) and solid dots [27].
˜ Au Fig. 20. Calculated and measured ternary interdiffusion coefficients of D AgAg at 998 K; dashed blue isolines (this work) and solid dots [27].
coefficients of Ag in fcc Ag-rich alloys are equivalent to the selfdiffusion coefficients of Ag. As the self-diffusion coefficients of Ag in fcc Ag were assessed by Ghosh [2], the tracer diffusion coefficients of Ag in fcc Cu–Ag alloys can be judged from their extrapolation ability on the Ag-rich side. As can be seen in Fig. 2, the experimental data from [34] for 1023 and 1073 K and Sato [35] for 973 K can be reproduced well due to their excellent extrapolation ability. Fig. 3 compares the calculated interdiffusion coefficients in the Cu-rich part at four temperatures with the measured data from [38]. The agreement is excellent at 1273, 1174 and 1076 K. The deviation between the calculated and measured data at 974 K is due to the poor extrapolation ability of the experimental data in the Cu-rich part, where the limiting value is equivalent to the impurity diffusion coefficient of Ag in fcc Cu at 974 K. The further test of the atomic mobilities can be obtained through the simulations of concentration profiles in diffusion couples. The calculated and measured concentration profiles for two Cu/Ag diffusion couples annealed at 1023 K for 1800 and 3600 s are given in Fig. 4, where the agreement is excellent. The sharp and
320
Y. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 314–322
Fig. 21. Calculated and measured ternary interdiffusion coefficient ratio of ˜ Au ˜ Au D CuAg /DCuCu at 998 K; dashed blue isolines (this work) and solid dots [27].
Fig. 22. Calculated and measured ternary interdiffusion coefficient ratio of ˜ Au ˜ Au D AgCu /DAgAg at 998 K; dashed blue isolines (this work) and solid dots [27].
moving interfaces correspond to the tie-line of the two-phase region, which is invariant with time in this binary system. 4.2. The Cu–Ag–Au ternary fcc alloys Ternary fcc Cu–Ag–Au alloys feature a large solubility range, and a miscibility gap is formed around the binary Cu–Ag edge. The calculated isothermal phase boundaries for fcc phase at 1023, 973, 873, 773 and 673 K are presented in Figs. 5–9, and the calculated vertical phase boundaries sectioned at different composition sets are given in Figs. 10–18. For comparison, the experimental data from [20–29] are also superimposed. As can be seen, the agreement is excellent at various temperatures and concentrations. For a ternary system, interdiffusion coefficients are frequently evaluated at intersections of diffusion paths under isothermal conditions. As diffusion paths are one-dimensional geometric elements in two-dimensional composition spaces, intersection points for various diffusion paths are not rarely encountered. As long as the initial alloy compositions are well specified, sufficient numbers of intersection points can be located on the Gibbs triangle,
Fig. 23. Calculated and measured composition profiles for a Cu/Au diffusion couple annealed at 998 K for 48 h.
Fig. 24. Calculated and measured composition profiles for a Cu/Ag0.0865 Au0.9135 diffusion couple annealed at 998 K for 60.8 h.
on which interdiffusion coefficients can be evaluated. A large number of diffusion couples can be set up to generate compositiondependent interdiffusion coefficients over a wide composition range. ˜ Au ˜ Au The calculated dashed blue isolines for D CuCu and DAgAg are presented in Figs. 19 and 20, respectively, together with the experimental values from [27]. As is evident, the weak concentration dependence for such two kinds of interdiffusion coefficients is observed. Both kinds of diagonal interdiffusion coefficients form peaks within the Gibbs triangle, and the calculated isolines generally follow the tendency obtained experimentally. The ratios of ˜ Au ˜ Au off-diagonal to diagonal interdiffusion coefficients, i.e. D CuAg /DCuCu
˜ Au ˜ Au and D AgCu /DAgAg , are also studied with isolines in Figs. 21 and 22, together with the reported values from [27]. Such two ratios have well-defined geometric properties with respect to binary edges. It ˜ Au is known that on the Ag–Au binary edge, D CuAg will be zero, while ˜ Au D CuCu is equivalent to the impurity diffusion coefficients of Cu in ˜ Au ˜ Au fcc Ag–Au alloys. As such, D CuAg /DCuCu is zero on the Au–Ag binary
˜ Au ˜ Au side. Similarly, D AgCu /DAgAg is zero on the Au–Cu binary side, since
Y. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 314–322
Fig. 25. Calculated and measured composition profiles for a Cu/Ag0.309 Au0.691 diffusion couple annealed at 998 K for 60.8 h.
Fig. 26. Calculated and measured composition profiles for a Cu0.674 Au0.326 /Ag0.554 Au0.446 diffusion couple annealed at 998 K for 48 h.
˜ Au ˜ Au D AgCu = 0 and DAgAg denotes the impurity diffusion coefficients of Ag in fcc Au–Cu binary alloys. Such geometric relations can be evidenced not only from the calculated isolines but also from the discrete experimental values from [27]. With the known atomic mobilities and thermodynamic description, it is convenient to calculate the concentration distributions for diffusion couples, as presented in Figs. 23–30. From comparisons in Figs. 23–30, excellent agreements can be obtained, which serve as direct evidence to verify the mobility quality assessed in this work. In Fig. 26, there is one point on the Au concentration profile, on which the concentration gradient of Au is almost zero. At such a point, the concentration gradients of the remaining two elements, i.e. Cu and Ag, are almost equal to each other in magnitude but with opposite signs. Diffusion paths are a kind of plots for concentration profiles in diffusion couples, in which the kinetic information is almost lost, while the concentration relation can be concisely presented on phase diagrams. The diffusion paths calculated in this work are given in Fig. 31 for four diffusion couples, along with the experimental data from [27]. The agreement between the calculated and measured diffusion paths is satisfactory. In Fig. 31, the diffusion path D passes through the critical point
321
Fig. 27. Calculated and measured composition profiles for a Au0.56 Cu0.44 /Ag0.309 Au0.691 diffusion couple annealed at 998 K for 48 h.
Fig. 28. Calculated and measured composition profiles for a Au0.077 Cu0.923 /Au0.446 Ag0.554 diffusion couple annealed at 998 K for 48 h.
Fig. 29. Calculated and measured composition profiles for a Au0.56 Cu0.44 /Au0.446 Ag0.554 diffusion couple annealed at 998 K for 48 h.
322
Y. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 35 (2011) 314–322
CALPHAD framework. Comparisons between the calculated and measured data, including phase boundary data, tracer diffusion coefficients and interdiffusion coefficients, are made to verify such thermodynamic and kinetic parameters. In addition, the calculated concentration profiles in diffusion couples and diffusion paths superimposed on one isothermal phase diagram are extensively explored. This work contributes to the fundamental knowledge on microstructure evolution and phase stabilities for fcc Cu–Ag–Au alloys, the results of which can be utilized to solve problems of interest, not only for industrial issues but also for scientific purposes. Appendix. Supplementary data Supplementary material related to this article can be found online at doi:10.1016/j.calphad.2011.03.001. References Fig. 30. Calculated and measured composition profiles for a Au0.56 Cu0.44 /Au0.9135 Ag0.0865 diffusion couple annealed at 998 K for 48 h.
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]
[21]
Fig. 31. Calculated and experimentally measured diffusion paths for four ternary diffusion couples annealed at 998 K. (A: Ag0.0865 Au0.9135 /Cu0.44 Au0.56 , 48 h; B: Cu/Ag0.0865 Au0.9135 , 60.8 h; C: Cu0.674 Au0.326 /Ag0.554 Au0.446 , 48 h; D: Cu0.923 Au0.077 /Ag0.554 Au0.446 , 48 h.)
[22] [23] [24] [25] [26] [27] [28]
of the ternary miscibility gap and is tangent to that critical point. Under this case, the composition profiles for diffusion couples around the critical point are expected to be extremely steep and still continuous, because this point denotes a transition between a solution behavior in which the concentration gradients of the three elements are finite and the discontinuous phase interface where two phases are present.
[29] [30] [31] [32] [33] [34] [35] [36]
5. Conclusions The thermodynamic description and atomic mobilities for fcc Cu–Ag–Au alloys are presented in this work, based on the
[37] [38] [39] [40] [41]
Q. Chen, N. Ma, K. Wu, Y. Wang, Scr. Mater. 50 (2004) 471. G. Ghosh, Acta Mater. 49 (2001) 2609. M. Palumbo, M. Baricco, Acta Mater. 53 (2005) 2231. G. Ghosh, G.B. Olson, Acta Mater. 50 (2002) 2099. T. Helander, J. Ågren, Acta Mater. 47 (1999) 1141. T. Helander, J. Ågren, Acta Mater. 47 (1999) 3291. C.E. Campbell, W.J. Boettinger, U.R. Kattner, Acta Mater. 50 (2002) 775. K. Yasuda, H. Metahi, Y. Kanzawa, J. Less-Common Met. 60 (1978) 65. D.C. Wright, R.F. Gallant, L. Spangberg, J. Biomed. Mater. Res. 16 (1982) 509. T. Gomez-Acebo, B. Navarcorena, F. Castro, J. Phase Equilib. Diff. 25 (2004) 237. B. Jönsson, ISIJ Int. 35 (1995) 1415. J.O. Andersson, J. Ågren, J. Appl. Phys. 72 (1992) 1350. G. Lindwall, K. Frisk, J. Phase Equilib. Diff. 30 (2009) 323. F.H. Hayes, H.L. Lukas, G. Effenberg, G. Petzow, Z. Metllkd. 77 (1986) 749. M.S. Lim, J.E. Tibball, P.L. Rossiter, Z. Metallkd. 88 (1997) 236. V.T. Witusiewicz, U. Hecht, S.G. Fries, S. Rex, J. Alloys Compd. 385 (2004) 133. X.C. He, H. Wang, H.S. Liu, Z.P. Liu, CALPHAD 30 (2006) 367. H.L. Lukas, Max-Planck-Institut für Metallforschung, Stuttgart, Germany, 1998, Private communcation. A. Kusoffsky, Acta Mater. 50 (2002) 5139. H. Yamauchi, H.A. Yoshimatsu, A.R. Forouhi, D. de Fontaine, Proc. 4th Precious Metals Conf., Toronto, 1980, Pergamon Press Canada, Ontario, 1981, pp. 241–249. G.H. Sistare, Metals Handbook, 8th ed., vol. 8, American Society for Metals, Metals Park, OH, 1973, pp. 377–378. M. Kogachi, K. Nakahigashi, Japan J. Appl. Phys. 19 (1980) 1443. T. Uzuka, Y. Kanzawa, K. Yasuda, J. Dent. Res. 60 (1981) 883. M. Kogachi, K. Nakahigashi, Japan J. Appl. Phys. 24 (1985) 121. G. Masing, K. Kloiber, Z. Metllkd. 32 (1940) 125. J.G. McMullin, J.T. Norton, Trans. AIME 185 (1949) 46. T.O. Ziebold, R.E. Ogilvie, Trans. AIME 239 (1967) 942. M. Murakami, D. de Fontaine, J.M. Sanchez, J. Fodor, Thin Solid Films 25 (1975) 465. T.O. Ntukogu, I.B. Cadoff, Mater. Sci. Technol. 2 (1986) 528. A. Prince, Int. Mater. Rev. 33 (1988) 314. B. Sundman, S.G. Fries, W.A. Oates, CALPHAD 22 (1998) 335. S. Hassam, J. Agren, M. Gaune-Escard, J.P. Bros, Metall. Trans. A 21 (1990) 1877. B. Butrymowicz, J.R. Manning, M.E. Read, J. Phys. Chem. Ref. Data 3 (1974) 527. R.E. Hoffman, D. Turnbull, E.W. Hart, Acta Metall. 3 (1955) 417. K. Sato, Sci. Rep. Fac. Lit. Sci. 11 (1964) 9. M.E. Yanitskaya, A.A. Zhukhovitskii, S.Z. Bokshtein, Dokl. Akad. Nauk SSSR 112 (1957) 720. R.V. Patil, B.D. Sharma, Trans. Indian Inst. Met. 25 (1972) 1. H. Oikawa, H. Takei, S. Karashima, Metall. Trans. 4 (1973) 653. J. Unnam, C.R. Houska, J. Appl. Phys. 47 (1976) 4336. Y. Liu, L. Zhang, D. Yu, J. Phase Equilib. Diff. 30 (2009) 136. Y. Liu, L. Zhang, D. Yu, Y. Ge, J. Phase Equilb. Diff. 29 (2008) 405.