Assessment of atomic mobilities of Al and Cu in fcc Al–Cu alloys

CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 33 (2009) 761–768

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Assessment of atomic mobilities of Al and Cu in fcc Al–Cu alloys Dandan Liu a , Lijun Zhang a , Yong Du a,∗ , Honghui Xu a , Shuhong Liu a , Libin Liu b a

State Key Laboratory of Powder Metallurgy, Central South University, Changsha, Hunan 410083, PR China

b

School of Materials Science and Engineering, Central South University, Changsha, Hunan 410083, PR China

article

info

Article history: Received 17 June 2009 Received in revised form 10 October 2009 Accepted 11 October 2009 Available online 22 October 2009 Keywords: Diffusivity Atomic mobility Fcc Al–Cu alloys Diffusion couple

abstract Based on various kinds of experimental diffusivities and thermodynamic parameters available in the literature, the atomic mobilities of Al and Cu in face-centered cubic (fcc) Al–Cu alloys have been assessed as a function of temperature and composition by means of DIffusion Controlled TRAnsformation (DICTRA) software package. Comprehensive comparisons between the calculated and measured diffusivities show that most of the experimental data can be well reproduced by the presently obtained atomic mobilities. In addition, the atomic mobilities obtained in the present work can also reasonably predict the concentration profiles for a variety of diffusion couples in fcc Al–Cu alloys including the one prepared in the present work. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction Diffusion plays a very important role in designing and understanding many important phenomena, such as precipitation, homogenization of alloys, recrystallization, grain boundary migration, creep, solidification and protective coatings [1,2]. It is generally accepted that estimating the rate of a phase transformation from the rate of volume diffusion of different components is a wellestablished procedure. Recently, computer simulation has become an important and effective tool to gain insight into complex material processes [3–5]. In order to simulate the diffusion controlled transformations in a multicomponent system, the software DIffusion Controlled TRAnsformation (DICTRA) has been developed [6]. This software is based on a sharp interface and the local equilibrium hypothesis. If the thermodynamic properties and the volume diffusivities of a system are known, it is then possible to estimate the migration rate of a phase interface using this software. Generally speaking, reliable thermodynamic databases for alloy systems are available nowadays. However, this is not the case for the diffusion kinetic database because experimentally determined diffusion coefficients are not generally assessed, and the data from different sources are often not mutually consistent. Thus, there is an urgent need to remedy this situation. Due to technological importance and fundamental scientific interest for various diffusion phenomena, the atomic mobility for fcc Al–Cu alloys is evaluated in the present work. The preliminary atomic mobilities for the fcc phase in the Al–Cu system are available in the MOB2 database [7]. Such atomic mobilities

∗

Corresponding author. Tel.: +86 731 8836213; fax: +86 731 8710855. E-mail address: [email protected] (Y. Du).

0364-5916/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.calphad.2009.10.004

can be refined by using all the diffusion coefficients available in the literature, as demonstrated in the present work. Most recently, the atomic mobilities of Al and Cu corresponding to selfdiffusivities have been updated by Zhang and Du [8] and Ghosh [9], respectively. Consequently, the main objectives of the present work are: (I) to evaluate all the diffusion coefficients associated with fcc Al–Cu alloys available in the literature; (II) to obtain a self-consistent set of atomic mobility parameters for both Al and Cu in the fcc Al–Cu alloys using DICTRA software, and (III) to validate the presently obtained atomic mobilities via comparisons of calculated diffusivities and measured ones as well as modelpredicted concentration profiles and measured ones in various binary diffusion couples. The presently measured concentration profile is also compared with the model-predicted one. 2. Evaluation of experimental data in the literature Since the self-diffusivities of Al and Cu used in the present work are taken from Zhang and Du [8] and Ghosh [9], respectively, only three kinds of experimental diffusivities in the fcc Al–Cu alloys are considered in the present work. They are (a) impurity diffusion coefficient of Al in Cu and that of Cu in Al, (b) interdiffusion coefficients in the fcc Al–Cu alloys, and (c) tracer diffusion coefficients of Cu in the fcc Al–Cu alloys. All the experimental data are summarized in Table 1 and categorized in the following. 2.1. Impurity diffusivity There only exists limited experimental information about impurity diffusion of Al in Cu due to lack of appropriate isotope. The only existing experimental data are from Hirvonen [10], who used

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D. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 33 (2009) 761–768

Table 1 Summary of various diffusion coefficients in the fcc Al–Cu alloys. Temperature range (K)

Method

Ref.

Codea

648–823 1043–1203 977–1277 985–1270 973–1176 1023–1223

27

Al+ : (p, γ )-resonance broadening technique Extrapolated valuesb Extrapolated valuesb Extrapolated valuesb Extrapolated valuesb Extrapolated valuesb

[10] [11] [12] [13] [14] [15]

     

594–928 858, 930

[16] [17]

 

648–892 706–924.8 667–930 623–903 648–892 762, 881 473–923 793

Radioactive 67 Cu: bremsstrahlung irradiation, SSMb , RAMb Radioisotopes 67 Cu–64 Cu mixture:electroplate and evaporation Radioactive 64 Cu: RAMb Radioactive 64 Cu: TSTb Radioisotope 67 Cu: SSMb Radioactive 64 Cu: RAMb Radioactive 64 Cu: RAMb Radioactive 64 Cu: TSTb EPMAb , RMb ICP-AESb

[18] [19] [20] [21] [22] [23] [24] [25]

      

Al-rich side

730–838 682–893 698–808 778–908 773–823 775–811 673–777

BDCb , EPMAb BDCb , EPMAb , MMb BDCb , EPMAb , MMb BDCb ,SMb , RAb BDCb , MTb BDCb , PBMb BDCb , EPMAb

[26] [27] [28] [29] [30] [31] [32]

      

Cu-rich side

1043–1203 977–1277 985–1270 973–1176 1023–1223 1162–1250 778–1138

BDCb , EPMAb ,MMb BDCb , EPMAb ,BMMb BDCb , EPMAb ,BMMb BDCb , CAb , GJMb BDCb , EPMAb ,BMMb BDCb , CAb ,MMb XRDb

[11] [12] [13] [14] [15] [33] [34]

   M M  

D∗Cu

1073–1313 762, 881

Isotope 64 Cu: RAMb Radioactive 64 Cu: TSTb

[36] [23]

 

Type of data

DCu Al (Al in Cu)

Impurity diffusion coefficients

DAl Cu (Cu in Al)

Interdiffusion coefficient

Tracer diffusion coefficient

DCu AlAl

+

a Indicates whether the data are used or not used in the atomic mobility assessment: , used; , not used but considered as reliable data for checking the parameters; M, partly used; +, not used. b Extrapolated values = Extrapolated impurity diffusion coefficients of Al in Cu from the interdiffusion coefficients; SSM = serial sectioning method; RAM = residual activity method; TST = tracer section technique; EPMA = electron probe microscopy analysis; RM = resistance method; MM = Matano method; SM = spectrophotometric method; RA = radioactivation analysis; BDC = bulk diffusion couple; MT = microhardness test; PBM = phase boundary motion; BMM = Boltzmann–Matano method; CA = chemical analysis; GM = Grube method; XRD = X-ray diffraction.

the resonance broadening technique to investigate the diffusion of 27 Al+ in ion-implanted Al–Cu solid solutions. The activation energy of Al in Cu reported by Hirvonen [10] is judged to be reasonable since this energy is of the same magnitude order in comparison with the reported activity energies of several elements with fcc structure in Cu [35]. Consequently, the impurity diffusion coefficients of Al in Cu from Hirvonen [10] are used in the present optimization. In addition, the impurity diffusion coefficients of Al in Cu could be estimated by extrapolating interdiffusion coefficient ˜ to infinite dilution [11–15]. Since those extrapolated data (D) [11–15] are not original experimental ones, they are not utilized in the present mobility assessment, but employed to check the reliability of presently obtained mobilities. The impurity diffusion coefficients of Cu in pure Al have been measured by many groups of authors [16–25] using various techniques. By means of the serial sectioning method and the residual activity method, Fujikawa and Hirano [16] measured the impurity diffusion of 67 Cu in pure Al in the temperature range of 594–928 K. Peterson and Rothman [17] evaporated the mixture radioisotopes of 67 Cu − 64 Cu in vacuum onto freshly etched single Al crystals to study diffusion of Cu in pure Al at 858 and 930 K. Using the residual activity method, Hirano and Fujikawa [18], Anand et al. [21] and Fujikawa and Hirano [22] determined diffusivities of radioactive 64 Cu in pure Al in the temperature range of 648–892 K, 623–903 K and 648–892 K, respectively. Utilizing the tracer sectioning technique, Peterson and Rothman [19] and Alexander and Slifkin [23] measured the

diffusion of 64 Cu in Al single crystals from 706 to 924.8 K and from 762 to 881 K, respectively. Ushino et al. [20] reported the impurity diffusion of 67 Cu in pure Al in the temperature range of 667–930 K using the serial sectioning method. Bergner [24] studied the impurity diffusion of Cu in Al in the temperature range of 473–923 K by means of the resistance method. With the application of inductively coupled plasma-atomic emission spectrometry (ICPAES), Combe and Cabane [25] determined the impurity diffusion coefficients of Cu in Al at 793 K by depositing thin copper film on the surface of Al. Since these data [16–24] are in good agreement with one another, they are employed in the present optimization. Only one experimental point from Ref. [25] is not considered in the present work due to its inconsistency with the other data [16–24].

2.2. Interdiffusion coefficient The measurements of the interdiffusion coefficients in fcc Al–Cu alloys have been conducted by a number of groups [11–15,26–34]. The authors of Refs. [26–32] focused on the Al-rich side, while those of [11–15,33,34] on the Cu-rich side. Using the bulk diffusion couple, Beerwald [26] reported the interdiffusion coefficients in Al-rich alloys up to 0.395 at.% Cu within the temperature range of 730–838 K. Minamino and Yamane [27] investigated the interdiffusion of the Al-rich solid solution in the composition range of 0 to 1.52 at.% Cu from 682 to 893 K by using singlephase diffusion couples. Funamizu and Watanabe [28] measured

D. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 33 (2009) 761–768

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investigated the solid state diffusion characteristics in the Cu-rich solid solution in the temperature range of 1023 to 1223 K. By measuring the movement of the marker in the Cu/Cu-7 wt% Al diffusion couples, Silva and Mehl [33] obtained the interdiffusion for alloys up to 16 at.% Al within the temperature range of 1162 to 1250 K. To investigate the effect that an internal strain brings to the diffusion of alloy elements, Hasiguti [34] reported the interdiffusion coefficients of Cu-15.7 at.% Al alloy with and without internal strain. Due to the damage during the sample preparation, those measured interdiffusion coefficients at 1023 K [14,15] are higher than those reported in Refs. [12,13] at the same temperature. As a result, except for the experimental data reported in Refs. [14,15], all the other experimental data [11–15,33,34] are utilized in the present assessment. 2.3. Tracer diffusivity Fig. 1. Calculated temperature dependence of self-diffusion coefficients of Cu and Al, according to the atomic mobilities from Zhang and Du [8] and Ghosh[9].

the interdiffusion coefficients of fcc Al–Cu alloys in Al-rich region in the temperature range of 698–808 K by utilizing Al/Cu diffusion couples. Murphy [29] studied the interdiffusion in dilute Al–Cu solid solution by roll bonding super-purity Al cladding to fcc Al–Cu solid solution core within 0–0.5 wt% Cu and 778–908 K. Using diffusion couple method and microhardness tests, Bückle [30] measured the interdiffusion coefficients of Al-rich fcc alloys from 773 to 923 K. Cahoon [31] studied the interdiffusion in the Alrich corner of the Al–Cu system within the temperature range of 775–811 K and reported the average interdiffusion coefficients at 775, 783, 791, 802 and 811 K. Mehl et al. [32] measured the interdiffusion coefficients in Al-rich side of Al–Cu alloys in the temperature range of 673–777 K using diffusion couple method. All these data [26–32] are used in the present optimization in view of their mutual consistency. Takahashi et al. [11] determined the interdiffusion coefficients in Cu-8 at.% Al alloy within the temperature range of 1043-1203 K by utilizing diffusion couple method. Using Cu/Cu-12.2 at.% Al diffusion couple, Matsuno and Oikawa [12] measured the interdiffusion coefficients of Cu-rich fcc alloys in the temperature range of 977 to 1277 K. Oikawa et al. [13] used the polycrystalline Cu-rich alloys to determine the interdiffusion coefficients in the temperature range of 985–1270 K. Rhines and Mehl [14] used several bulk diffusion couples to determine the interdiffusion coefficients in Cu-rich side in the ranges of 0–8.65 wt % Al and 973–1176 K. Using a single-phase bulk diffusion couple between Cu and Cu-10 at.% Al, Laik et al. [15]

a

The tracer diffusion coefficients of Cu in various fcc Al–Cu alloys have been reported by two groups of authors [23,36]. By means of the tracer sectioning technique, Alexander and Slifkin [23] studied the tracer diffusion of 64 Cu in Al-1 at.% Cu alloy at 762 and 881 K. Using residual activity method, Kucera and Million [36] measured the tracer diffusion coefficients of 64 Cu in Cu–Al solid solutions in the composition range of 0 to 19 at.% Al from 1073 to 1313 K. Since the experimental data from Alexander and Slifkin [23] and Kucera and Million [36] agree well with each other, these data are used in the present mobility assessment. 3. Experimental procedure Al pieces (purity 99.999 wt%) and Cu pieces (purity 99.99 wt%) were used as the starting materials. First, one Cu-12 at.% Al alloy was prepared in an arc melting furnace (WKDHL-1, Optoelectronics Co. Ltd., Beijing, China) under a high purity argon atmosphere using a non-consumable W electrode. The button was re-melted three times to improve its homogeneity. Then, one Cu piece and one Cu-12 at.% Al alloy were cut into bars with approximate dimension of 2 × 4 × 14 mm3 . After being grounded and polished, they are bound together with Mo wires to make the Cu/Cu-12 at.% Al diffusion couple. The diffusion assemblies were then sealed in a quartz tube under vacuum (1 Pa). The diffusion couple was subjected to annealing at 1173 K for 130 ks, followed by water quenching. The determination of phase composition was performed via electron probe microanalysis (EPMA) (JXA-8100, JEOL, Japan) to obtain the Al concentration–penetration profile. The obtained concentration–penetration profile is compared with the one computed via atomic mobilities of Al and Cu in fcc phase in order to further verify the reliability of the atomic mobilities.

b

Fig. 2. (a). Comparison between the presently calculated temperature dependence of impurity diffusion coefficients of Al in fcc Cu and the experimental ones [10–15]. (b). Comparison between the presently calculated temperature dependence of impurity diffusion coefficients of Cu in fcc Al and the experimental ones [16–24].

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D. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 33 (2009) 761–768

a

b

Fig. 3. Comparison between the presently calculated temperature dependence of interdiffusion coefficients: (a) at x(Cu) = 0.002 and x(Cu) = 0.01 with the experimental values [27–29,31]. (b) at x(Al) = 0.08 and x(Al) = 0.157 with the experimental values [11–15,33,34]. A constant, M, is added in order to separate the data for different compositions in the figure. The light solid lines denote the calculation using only the atomic mobilities corresponding to the four end-members, while the dashed lines denote the maximal errors associated with the assessment.

a

b

c

d

Fig. 4. Comparison between the presently calculated interdiffusion coefficients on Al-rich side (a) at 678, 725, 770 and 784 K with the experimental values [26–28,31,32]. (b) at 710, 745, 775 and 788 K with the experimental values [26–28,30,32]. (c) at 803, 808, 817, 823 and 838 K with the experimental values [26–28,30]. (d) at 833, 847, 860 and 893 K with the experimental values [27]. A constant, M, is added in order to separate the data for different temperatures in the figure. The dashed lines denote the maximal errors associated with the assessment.

4. Diffusion modeling

where Ck is the concentration in moles per volume, and ∇ denotes

In multicomponent systems, the diffusion equation describing the variation of the concentration of species k as a function of time and space is of the following form:

∂ Ck = −∇ · Jk (k = 1, . . . , n) ∂t

(1)

the divergence operator. The flux Jk of species k depends on the gradients of the concentrations of all n species:

Jk = −

n X j =1

Dkj ∇ Cj .

(2)

D. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 33 (2009) 761–768

a

b

c

d

e

f

765

Fig. 5. Comparison between the presently calculated interdiffusion coefficients on Cu-rich side (a) at 975 and 985 K with the experimental values [12–14]. (b) at 1007, 1025, 1050 and 1075 K with the experimental values [12–15]. (c) at 1123, 1173 and 1199 K with the experimental values [13,15]. (d) at 1120, 1167 and 1177 K with the experimental values [12–14,33]. (e) at 1222, 1230 and 1250 K with experimental values[12,15,33]. (f) at 1270 and 1275 K with the experimental values [12,13]. A constant, M, is added in order to separate the data for different temperatures in the figure. The light solid lines denote the calculation using only the atomic mobilities corresponding to the four end-members, while the dashed lines denote the maximal errors associated with the assessment. Table 2 Summary of the atomic mobility parameters for Al and Cu in fcc phase of the Al–Cu system assessed in the present work, only considering the four end-members.* Mobility

Parameter

Reference

Mobility of Al

∆GAl Al = −123 111.6 − 97.34 × T ∆GCu Al = −188 717.9 − 92.59 × T

[8] This work

Mobility of Cu *

∆GCu Cu = −205 872 − 82.52 × T ∆GAl Cu = −134 393.4 − 81.97 × T

[9] This work

In J/mol-atoms. Temperature (T ) in Kelvin.

Since there is a relation among the n concentration gradients in Eq. (2) and for practical calculations one usually chooses to

eliminate one of them. The reduced diffusivity in a volume-fixed frame of reference, where it is assumed that only the substitutional species contribute to the volume (i.e. the reference employed within DICTRA frame), is expressed as, Dnkj = Dkj − Dkn Dnkj

(when j is substitutional)

= Dkj (when j is interstitial)

(3) (4)

where n is taken as the dependent species. Using these diffusivities, Eq. (2) then becomes, Jk = −

n −1 X j =1

Dnkj ∇ Cj .

(5)

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D. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 33 (2009) 761–768 Table 3 Summary of the atomic mobility parameters of Al and Cu in fcc phase of Al–Cu system assessed in the present work, considering the interaction terms.* Mobility

Parameter

Mobility of Al

∆GAl Al ∆GCu Al 0 Al,Cu ∆ GAl

[8] This work

∆GCu Cu = −205 872 − 82.52 × T ∆GAl Cu = −133 184.4 − 83.65 × T ,Cu ∆0 GAl = −31 461.4 + 78.91 × T Cu

[9] This work

Mobility of Cu *

= −123 111.6 − 97.34 × T = −181 583.4 − 99.80 × T = −183 094.3 + 159.01 × T

Reference

This work

This work

In J/mol-atoms. Temperature (T ) in Kelvin.

5. Results and discussion

Fig. 6. Comparison between the presently calculated tracer diffusion coefficients of Cu in different fcc Al–Cu alloys and the experimental ones [36]. A constant, M, is added in order to separate the data for different compositions in the figure. The dashed lines denote the maximal errors associated with the assessment.

Eq. (5) contains the so-called interdiffusion coefficients Dnkj . Dnkj in a substitutional solution phase can be given by the following expression [1,37], Dnkj =

  X ∂µi ∂µi (δik − xk ) · xi · Mi · − ∂ xj ∂ xn i

(6)

where xi and µi are the mole fraction and chemical potential of element i, respectively. δik is the Kronecker delta, i.e., δik = 1 if i = k and δik = 0 otherwise, n, k and j in Eq. (6) are the dependent, diffusing and gradient elements, respectively. Mi is the composition dependent atomic mobility. From the absolute reaction rate theory arguments, Mi may be divided into a frequency factor Mi0 and an activation enthalpy Qi [1,37]. According to the suggestion by Jönsson [37], the Mi is expressed by an equation of the form:

 Mi = exp

RT ln Mi0 RT



 exp

−Qi RT



1

mg

RT

Ω

(7) mg

where R is the gas constant and T the absolute temperature. Ω is a factor taking into account a ferromagnetic contribution to the diffusion coefficient. For fcc alloys it was concluded that the ferromagnetic effect on diffusion is negligible [38]. Thus for fcc alloys, RT ln Mi0 and Qi can be merged into one parameter, ∆Gi = −Qi + RT ln Mi0 Consequently, ∆Gi for the fcc Al–Cu phase can be described by the following expression: Al ∆Gi = xCu ∆GCu i + xAl ∆Gi

+ xCu xAl

n X

,Al ∆(j) GCu (xCu − xAl )j i

(8)

j =0 Cu,Al

where ∆(j) Gi (i = Cu or Al) is the interaction term for diffusion between Cu and Al. Assuming a mono-vacancy atomic exchange as the main diffusion mechanism, the tracer diffusivity D∗i can be related to the atomic mobility Mi by the Einstein relation: D∗i = RTMi .

(9) ∗

In a binary system, the tracer diffusivity Di can also be used to cal-

˜ by Darken’s equation [39]: culate the interdiffusion coefficient D ˜ = (xAl D∗Cu + xCu D∗Al )φ D

(10)

where φ is the thermodynamic factor, and can be expressed as:

∂ ln γi xi dµi = (i = Cu or Al) (11) ∂ ln xi RT dxi where xi , γi and µi (i = Cu or Al) are the mole fraction, the activity φ =1+

coefficient and the chemical potential of Cu or Al, respectively.

The evaluation of the model parameters in Eq. (8) is attained by means of the PARROT module of the DICTRA software package [40,41], based on all the critically reviewed experimental data from the literature as described in Section 2. The thermodynamic data are taken from [42]. Tables 2 and 3 list the atomic mobility parameters of Al and Cu in fcc phase obtained in the present work, as well as those of pure Al and Cu taken from Zhang and Du [8] and Ghosh [9]. The assessment of atomic mobilities for the fcc phase in the Al–Cu system is conducted in two steps. In the first step, only the atomic mobilities corresponding to the four end-members (∆GAl Al , Cu Al ∆GCu , ∆ G , ∆ G Table 2. However, ) are considered, as shown in Al Cu Cu it turns out to be that a better fit to the experimental diffusivities can be obtained when the interaction term is introduced, as shown later. Thus in the second step, the atomic mobilities corresponding to the effect of the interaction between Al and Cu on the diffusion flux are assessed by considering the measured interdiffusion coefficients, and thus optimized atomic mobilities are shown in Table 3. During the optimization, it was found that the interaction parameters mainly affect the interdiffusion coefficients in Cu-rich side. This is understandable since the solubility of Al in Cu is large in comparison with the negligible solubility of Cu in Al. Fig. 1 shows the calculated temperature dependence of selfdiffusion coefficients of Al and Cu according to Refs. [8] and [9], respectively. Fig. 2(a) illustrates the calculated temperature dependence of Al impurity diffusion coefficients in pure Cu compared with the experimental data [10], where an excellent fit can be found. The extrapolated impurity diffusion data [11– 15] at higher temperatures are also appended for comparison. Although these data [11–15] were not used in the assessment, the presently obtained parameters mainly based on Ref. [10] can roughly reproduce them. Fig. 2(b) shows the calculated impurity diffusion coefficients of Cu in pure Al as a function of inverse temperature, compared with the corresponding experimental data [16–24]. It can be seen that the fit to all the experimental data [16–24] is quite good. The comparisons between the presently calculated interdiffusion coefficients and the measured ones [11–15,27–29,31,33,34] are shown in Fig. 3. Since many experimental data are included in the diagrams, the error bars associated with the experimental data are not presented in order to facilitate the reading. However, the maximal error of the assessment is given in the form of dashed lines. As can be seen from the figure, the calculated interdiffusion coefficients in the present work are in good agreement with the experimental ones [11–15,27–29,31,33,34]. Fig. 4 presents the calculated interdiffusion coefficients in Alrich side of fcc Al–Cu alloys at different temperatures together with the experimental data [26–28,30–32]. Fig. 5 compares the calculated interdiffusion coefficients in Cu-rich side with the experimental data [12–15,33,34]. Considering the large number of the experimental data from different sources, one may conclude

D. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 33 (2009) 761–768

a

767

b

c

d

e

Fig. 7. (a) Model-predicted concentration profiles of the Cu-18.2 at.% Al/Cu, Cu-17.8 at.% Al/Cu and Cu-17.6 at.% Al/Cu diffusion couples annealed at 973, 1023 and 1176 K for 1937, 1210 and 93 ks, respectively, compared with the experimental data [14]. (b) Model-predicted concentration profiles of the Cu/Cu-16 at.% Al diffusion couple annealed at 1162, 1221 and 1250 K for 1098, 629 and 670 ks, respectively, compared with the experimental data [33]. (c) Model-predicted concentration profiles of the Al-0.0174 at.% Cu/Al diffusion couple annealed at 773, 788 and 823 K, respectively, for 79 ks, compared with the experimental data[30]. (d) Model-predicted concentration profiles of the Al/Al-0.2 at.% Cu diffusion couple annealed at 883 K for 92 ks, compared with the experimental data [29]. (e) Model-predicted concentration profile of the Cu/Cu-12 at.% Al diffusion couple annealed at 1173 K for 130 ks, compared with the presently measured experimental data. A constant, M, is added in order to separate the data for different diffusion temperatures.

from Figs. 4 and 5 that the presently obtained mobility parameters can reasonably reproduce most of the experimental interdiffusion coefficients available in the literature. Furthermore, it can be seen from the figures that the interdiffusion coefficients decrease slightly with the increase of Cu concentration in Al-rich side of the Al–Cu alloys, while obviously increase with the increase of Al concentration in Cu-rich side of the Al–Cu alloys. In Fig. 3(b) and Fig. 5, the light solid lines denote the calculated diffusivities using only the atomic mobilities corresponding to the

end-members. And in these figures, the heavy solid lines represent the calculated diffusivities using the interaction terms. As can be seen from the diagrams, an improved fit to the experimental data is obtained when the interaction term is employed. Since the solubility of Cu in Al is very small, the calculated interdiffusion coefficients in Al-rich side of the fcc phase are not sensitive to the employed interaction term. The calculated tracer diffusion coefficients of Cu in fcc Al–Cu alloys are compared with the experimental data [36], as shown in

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D. Liu et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 33 (2009) 761–768

Fig. 6. The fit to the experimental values of Cu from Ref. [36] is good except for a slight deviation at xAl = 0.182. The main reason for this deviation is that this experimental value above 1073 K is located in two-phase region of fcc and bcc (body-centered cubic), but not in single fcc phase. Fig. 7 provides the most severe check on the validity of the presently obtained atomic mobilities, where the model-predicted concentration profiles in a variety of binary Al–Cu diffusion couples are compared with the corresponding experimental data [14,29,30,33]. Fig. 7(a) shows the model-predicted concentration profiles of the Cu-18.2 at.% Al/Cu, Cu-17.8 at.% Al/Cu and Cu-17.6 at.% Al/Cu diffusion couples annealed at 973, 1023 and 1176 K for 1937, 1210 and 93 ks, respectively, compared with the experimental data by Rhines and Mehl [14]. As can be seen in the figure, the agreement between the presently calculated composition profiles and the experimental data [14] is good. Fig. 7(b) illustrates the predicted concentration profiles of the Cu/Cu-16 at.% Al diffusion couple annealed at 1162, 1221 and 1250 K for 1098, 629 and 670 ks, respectively, along with the experimental data of da Silva and Mehl [33]. As shown in Fig. 7(b), the modelpredicted composition profiles of Cu/Cu-16 at.% Al annealed at 1162 and 1250 K for 1098 and 670 ks, respectively, can reasonably reproduce the corresponding experimental data from [33]. The scattering experimental data [33] corresponding to the diffusion couple annealed at 1221 K for 629 ks can be also reasonably described by the calculation. Fig. 7(c) presents the modelpredicted concentration profiles of the Al-0.0174 at.% Cu/Al diffusion couple annealed at 773, 788 and 823 K for 79 ks, respectively, compared with the experimental data of Bückle [30]. As shown in the figure, the agreement between the calculations and the experimental values is reasonable. Fig. 7(d) shows the modelpredicted concentration profiles of the Al/Al-0.2 at.% Cu diffusion couple annealed at 883 K for 92 ks, compared with the experimental data of Murphy [29]. As can be seen from the figure, the calculated result agrees well with the experimental data from [29] except for some deviation close to the Al-0.2 at.% Cu point. The slight decrease of the concentration–distance curve close to the Al–0.2 at.% Cu point may be due to the fact that diffusion had occurred before and/or during rolling the couples, as pointed out by Murphy [29]. Finally, Fig. 7(e) shows the model-predicted concentration profile of the Cu/Cu-12 at.% Al diffusion couple annealed at 1173 K for 130 ks, compared with presently measured experimental data. The fit to the experimental data is reasonable. The slight increase of the measured data close to Cu side of the diffusion couple may be due to the same reason, as mentioned by Murphy [29]. 6. Conclusions Based on the critically reviewed diffusion coefficients and the thermodynamic parameters available in the literature, the atomic mobilities of Al and Cu in fcc Al–Cu alloys have been assessed as a function of temperature and composition, using the DICTRA software package. The presently obtained atomic mobilities can satisfactorily reproduce most of the experimental diffusion data in fcc Al–Cu alloys from different sources. The concentration profiles for various single-phase diffusion couples in fcc Al–Cu alloys including the presently measured one can be also well reproduced by the established atomic mobilities.

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