Assessment of the atomic mobilities for ternary Al–Cu–Zn fcc alloys

CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 34 (2010) 68–74

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Assessment of the atomic mobilities for ternary Al–Cu–Zn fcc alloys Hui Chang a,∗ , Lei Huang a , Jianjun Yao b , Y.-W. Cui c , Jinshan Li a , Lian Zhou a a

State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xi’an 710072, PR China

b

Department of Materials Science and Engineering, Virginia Tech, Blacksburg, VA 24061, USA

c

Department of Materials Science and Engineering, The Ohio State University, Columbus, OH 43210, USA

article

info

Article history: Received 24 July 2009 Received in revised form 7 December 2009 Accepted 9 December 2009 Available online 22 December 2009

abstract Abundant experimental diffusion data were evaluated to assess the atomic mobilities for the fcc phase of the Al–Cu, Cu–Zn and Al–Cu–Zn systems. Comprehensive comparisons show that good agreements were obtained between calculated and experimental values not only for the diffusion coefficients, but also for diffusion/solidification processes resulting from interdiffusion, such as the Kirkendall shift and solidification curve. © 2009 Elsevier Ltd. All rights reserved.

Keywords: Atomic mobility DICTRA software Al–Cu–Zn ternary alloys Kirkendall effect Solidification curve

1. Introduction High-strength 7000 series alloys are generally heat-treated to achieve superior mechanical properties for aircraft and other high-strength applications [1–4]. Diffusion knowledge of the quaternary Al–Cu–Mg–Zn system, a base system of 7000 series alloys, is helpful in understanding the microstructure formation and further design of heat-treatment schedules. Our group has been conducting a program to develop an atomic mobility database for the Al–Cu–Mg–Zn quaternary system by means of a phenomenological treatment, which is commonly performed in the DICTRA software (Diffusion Controlled Transformation) [5]. The mobility database can be used in conjunction with the CALPHAD-base (Calculation of Phase Diagram) thermodynamic database [6] to offer a full diffusion picture of Al-based quaternary alloys of interest without extra experimental measurements. One of the most important ternary subsystems, Al–Mg–Zn, has been critically assessed [7]; however, another key ternary system, Al–Cu–Zn, remains unaddressed. It is noticed that commercially available mobility databases with the DICTRA software, i.e. MOB2 and MOBAL1 [8], can be used to address the issue of diffusion in Al–Mg–Zn and Al–Cu–Zn ternary alloys. However, care should be taken particularly when using these two databases for alloys containing significant amounts of alloying elements, because none

of the three constituent binaries, nor the Al–Cu–Zn ternary, has been critically assessed. This is also due to the fact that mobility parameters of Al and Zn have been recently updated [9] and a subsequent update for the ternary Al–Cu–Zn system is necessary. Therefore, the objective of this work is to assess atomic mobilities for Al–Cu–Zn ternary fcc alloys, and then to predict some in-depth details resulting from interdiffusion by applying the assessed atomic mobilities. 2. Model and selection of experimental diffusion data 2.1. Atomic mobility and diffusivity Andersson and Agren [10] suggested that the atomic mobility Mi of species i could be expressed as a function of temperature T , Mi = Mi0 exp

0364-5916/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.calphad.2009.12.002

RT



1 RT

,

(1)

" X

p xp Qi

p

Corresponding author. E-mail address: [email protected] (H. Chang).

−QiS

where QiS is the activation energy, Mi0 is the frequency factor and R is the gas constant. Similar to the phenomenological CALPHAD technique, the parameter Qi (either QiS or Mi0 ) is assumed to be composition dependent, and can be expressed by the Redlich–Kister polynomial in composition [11], Qi =

∗



+

+

XX p

XXX p

q>p v>q

q >p

xp xq

p,q rQi

X

# (xp − xq )

r

r =0,1,2,... p,q,v

s s xp xq xv [vpq v Qi

],

(s = p, q, v),

(2)

H. Chang et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 34 (2010) 68–74

69

Table 1 Summary of the experimental data used in the present assessment. Alloy systems

Diffusion data type

Method

Temperature (K)

Composition range

Ref.

˜ ∗Cu D ˜ ∗Cu D ˜ ∗Cu D ˜ ∗Cu D ˜ ∗Cu D ˜ ∗Cu D ˜ Cu D Al

LS RAT LS RAT LS RM DC

667–993 594–928 706–924 623–923 768–781 714–887 996–1127

1.0 ∗ 10−4 at.% Cu 1.0 ∗ 10−4 at.% Cu 1.0 ∗ 10−3 at.% Cu 5.0 ∗ 10−2 at.% Cu 1.0 ∗ 10−3 at.% and 1.0 at.% Cu 5.0 ∗ 10−3 at.% Cu Up to 10 at.%

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

Cu–Zn

˜ ∗Cu D ˜ ∗Cu D ˜ ∗Cu D ˜ ∗Cu D DCu DZn ˜ Zn D Cu

LS LS LS LS VSP DC DC

1068–1175 973–1313 896–1316 993–1219 973–1173 1168 997–1188

Up to 30 at.% Zn Up to 30 at.% Zn Up to 30 at.% Zn Up to 30 at.% Zn ∼30 at.% Zn Only up to 8 at.% Zn Up to 25.3 at.% Zn

[25] [26] [27] [28] [29] [30] [31]

Al–Cu–Zn

˜ ∗Zn D ˜DCu ˜ Cu ˜ Cu ˜ Cu Zn,Zn DAl,Al DAl,Zn DZn,Al

LS DC

714–893 1043–1203

– –

[32] [33]

DC

817

–

[34]

Al–Cu

˜ Al ˜ Al ˜ Al ˜ Al D Cu,Cu DCu,Zn DZn,Zn DZn,Cu

LS denotes the lathe-sectioning technique, DC denotes the diffusion couple, RAT denotes the residual activity technique, VSP denotes the vapor–solid couples, RM denotes the resistance method. p

where xp is the mole fraction of species p, Qi is the value Qi of p,q p,q,v species i in the pure species p, r Qi and s Qi are the binary and s ternary interaction parameters, and the parameter vpq v is given by s vpqv = xs + (1 − xp − xq − xv )/3. The tracer diffusion coefficient D∗i is related to the atomic mobility by the relation D∗i = RTMi . The interdiffusion coefficient can be derived from n

˜ pq = D

X i=1


∂µi δip − xp xi Mi , ∂ xq

(3)

where the Kronecker delta δip = 1 when i = p and 0 otherwise, and µi is the chemical potential of species i. Based on the above p p,q p,q,v relations, the mobility parameters, Qi , r Qi and s Qi , can be numerically assessed by fitting the calculated diffusion coefficients with their experimental values. It should be pointed out that some important assumptions and/or simplifications have been made in the DICTRA-type mobility formalism. These are, for example, (i) the Darken relation is used, i.e. no vacancy wind or any other correction term is considered; (ii) off-diagonal mobility terms are ignored (i.e. correlation effects are assumed to be negligible); (iii) all the species have the same partial molar volumes, and (iv) the composition dependence of the mobility is assumed to follow the Redlich–Kister expansion. These treatments make DICTRA-type modeling practical for creating kinetic databases and performing diffusion modeling for multicomponent systems, particularly because it defines one unique mobility for each component in a multicomponent system, and it works in conjunction with the CALPHAD-base thermodynamic database to obtain the thermodynamic quantities. In the case of a diffusion couple where the partial molar volume is assumed to be constant, the variation of composition with diffusion time, t, in the diffusion zone, can be described by the equation of continuity, 1 ∂ xi Vm ∂ t

=

X

˜ ikn ∇ [µk − µn ] , ∇ ·M

(4)

k=i,j

˜ ikn is the chemical mobility where Vm is the molar volume and M of species i in the composition gradient of species k with n as the dependent species [12]. Eq. (4) can be solved numerically, corresponding to different initial conditions and boundary conditions. If diffusion occurs by a vacancy mechanism, the non-uniform velocity of the inert markers, with respect to the laboratory-fixed frame of reference, is dependent on the difference of the intrinsic

diffusivities of the species and the composition gradient [13–15]. For an A–B binary system, it reads

v = −Vm (JA + JB ) = (DB − DA )

∂ xB , ∂z

(5)

where JA and JB are the fluxes of species A and B relative to a latticefixed frame, DA and DB are the intrinsic diffusivities of species A and B, and z is the distance. In a diffusion-controlled interaction, the Kirkendall plane is the only plane that stays at a constant composition during the whole diffusion annealing, and it moves parabolically in time with a velocity

vk =

dz dt

=

zk − zk0 2t

=

zk 2t

,

(6)

where zk and zk0 (t = 0) are the positions of the Kirkendall plane at times t = t and t = 0, respectively. As recently proposed [15–17], the Kirkendall plane can be located by finding a crossover point in the Kirkendall velocity construction, i.e. a plot of the velocity of the inert markers as a function of the distance. The lattice plane displacement can then be computed from the velocity of the inert markers by integrating

∂y y − 2v t = , ∂ z0 z0

(7)

where z0 is the position of the original location of the markers, and y is the displacement of the markers relative to z0 . 2.2. Selection of experimental data The experimental diffusion data adopted in this work to assess the mobility include the tracer diffusivity, intrinsic diffusivity and interdiffusion coefficient, which are summarized in Table 1. Selection criteria primarily depend on the extent of agreement on diffusion data among different resources. The Cu tracer diffusivity in fcc-Al has been measured by many researchers [18–23]. The data from their works show good consistency and thus were accepted in this work. The interdiffusion data reported by Matsuno et al. [24] were employed to assess the mobility of Al in fcc-Cu and the interaction parameters of mobility for Al–Cu alloys. The assessment for the Cu–Zn system was primarily based on the Cu tracer diffusivity in the Cu–Zn binary alloys measured by DeHoff et al. [25–28] and intrinsic diffusivities of Cu and Zn by

70

H. Chang et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 34 (2010) 68–74

Table 2 Assessed atomic mobilities for the fcc phase of the Al–Cu–Zn system. Mobility

Parameter (J/mol)

Ref.

−126 719 − 92.92 ∗ T −19 433 084.1 ∗ T −83 255 − 92.92 ∗ T

Present work [9]

Mobility of Al Al QAl Cu QAl Zn QAl

Al,Zn

[9] [9]

0

QAl

+30 169 − 111.8 ∗ T

1

QAl

+11 835 + 39 ∗ T

[9]

−47 405

Present work

[36]

Zn QCu

−136 773 − 79.4 ∗ T −205 872 − 82.5 ∗ T −58 891 − 90.3 ∗ T

QCu

251 370

Present work

QCu

−69 263

Present work

−120 033 − 88.34 ∗ T −188 010 − 88 ∗ T −76 569 − 86.21 ∗ T

Present work

Al,Zn

Al,Cu QAl

Mobility of Cu Al QCu Cu QCu Al,Cu Cu,Zn

Present work Present work

Mobility of Zn Al QZn Cu QZn Zn QZn

[9]

0

QZn

Al,Zn

−40 720 + 31.7 ∗ T

[9]

1

QZn

Al,Zn

+147 763 − 133.7 ∗ T

[9]

Cu,Zn

QZn

−33 071

Present work

QZn

139 758

Present work

Cu,Al

Fig. 1. Arrhenius plot of Cu tracer diffusion in fcc-Al.

[9]

Resnick et al. [29,30]. The interdiffusion data from Horne et al. [31] were used to adjust the interaction parameters in order to obtain a better overall fit. Zn tracer diffusivities determined in several dilute Al–Cu and Al–Cu–Zn fcc alloys by Beke et al. [32] served as a baseline to assess the ternary parameters. The ternary interdiffusion coefficients obtained by Takahashi et al. [33,34] were also used to adjust the ternary interaction parameters, but were given low weight in the optimization process in view of the relatively larger uncertainty in determination of interdiffusion coefficients. 3. Results and discussions The thermodynamic description of the Al–Cu–Zn system was taken from the work of Liang et al. [35]. The mobility parameters of the Al–Zn system were taken from the assessment of Cui et al. [9], and those of Cu self-diffusivity were from Ref. [36]. The other parameters were assessed by fitting to the selected experimental data. All mobility parameters we obtained are listed in Table 2. 3.1. The Al–Cu system As shown in Fig. 1, the calculated impurity diffusivity of Cu in fcc-Al compares favorably with the experimental data. Fig. 2 illustrates the excellent agreement between the experimental interdiffusion coefficients and the assessed results in the Al–Cu alloys with the content of Al ranging from 0 to 10 at.%. 3.2. The Cu–Zn system Comparisons for the Cu–Zn binary alloys in Fig. 3(a) and (b) show good agreements for both Cu and Zn tracer diffusivities obtained by Anusavice [26], and in Fig. 3(c) for Cu tracer diffusivities measured by Peterson [28], respectively. Note that both tracer diffusivities increase with the content of Zn. The agreement with the experimental interdiffusion coefficients [31] is not as strong; see Fig. 4. Attempts were made to give a better fit, but these failed as long as we were trying to keep the mobility parameters physical, therefore indicating that the measured

Fig. 2. Interdiffusion coefficients of fcc Al–Cu binary alloys.

tracer diffusivities [25–28] are not essentially compatible with the reported interdiffusion coefficients [31]. As it is generally believed that the latter have a larger uncertainty, the interdiffusion coefficients were given a low weight during the optimization process. Please note that Kozeschnik conducted an intrinsic diffusion simulation for the Cu–Zn system [37], in which he instead chose to trust the intrinsic diffusion data, and from those calculated tracer diffusivity and interdiffusion coefficients in order to obtain the best fitting to measurements of lattice shifts. 3.3. The Al–Cu–Zn systems Good agreement was generally obtained for the Zn tracer diffusivities in Al–Cu–Zn ternary alloys. An example is shown

H. Chang et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 34 (2010) 68–74

71

a

b

Fig. 4. Interdiffusion coefficients of fcc Cu–Zn binary alloys.

c

Fig. 5. Arrhenius plot of Zn diffusion in the fcc Al–Cu–Zn ternary alloys with 2.04 at.% Zn. An arbitrarily chosen scaling factor S is added to clearly represent the data at different temperatures.

4. Diffusion simulations Fig. 3. Arrhenius plots of (a) Cu diffusion and (b) Zn diffusion in fcc Cu–Zn alloys; (c) comparison between the calculated Cu tracer diffusivities and their experimental values at 1167 K and 1219 K.

4.1. Diffusion couple and solidification

in Fig. 5, where the Zn tracer diffusivity is found to have weak composition dependence.

Validation of the assessed diffusion mobility not only includes comparison with the experimental diffusion coefficients, but also

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H. Chang et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 34 (2010) 68–74

a

a

b b

c Fig. 7. Simulated diffusion paths of the diffusion couples at (a) the Cu-rich corner annealed at 1173 K for 10 500 s (4.175 h); and (b) the Al-rich corner annealed at 777 K for 1308 000 s (363.3 h). Symbols are experimental data.

d

Fig. 6. Simulated concentration profiles of (a) the Al/Al–3.32 wt% Cu binary couple and (b) the Al–11.8 wt% Zn–3.66 wt% Cu/Al–12.6 wt% Zn ternary couple at 777 K for 1308 000 s (363.3 h); (c) the Cu/Cu–30.06 at.% Zn binary couple at 1057 K for 2487 600 s (691 h); and (d) the Al–4.0 at.% Zn/Al–0.99 at.% Cu ternary couple at 850 K for 259 200 s (72 h).

comparison between the predicted and observed in-depth diffusion processes essentially resulting from interdiffusion. Diffusion simulation has been set up to model a number of semi-infinite Al/Al–Cu, Cu/Cu–Zn, Al–Zn/Al–Cu and Al–Cu–Zn/Al–Cu diffusioncouple experiments under different initial conditions and heattreatment schedules. Good agreement was obtained for most of the diffusion couples. Four examples are shown in Fig. 6 for the Al–3.2 wt% Cu/Al diffusion couple annealed at 777 K for 363.3 h [38], the Al–11.8 wt% Zn–3.66 wt% Cu/Al–3.32 wt% Cu annealed at 777 K for 363.3 h [38], the Cu/Cu–30.06 at.% Zn diffusion couple annealed at 1057 K for 691 h [39,40], and the Al–4.0 at.% Zn/Al–0.99 at.% Cu ternary couple annealed at 850 K for 72 h [34], respectively. Fig. 7 compares simulated diffusion paths with their experimental data [33,38], in which the composition of the Kirkendall plane for each couple is marked by a star symbol as well. As can been seen, the agreement for the Al-rich couples at 777 K is very satisfactory [38], but it is not as good for some of the Cu-rich couples at 1173 K, i.e. A4 to A6, B4 and B6 [33]. Careful inspection reveals that the experimental diffusion paths of B4 and B6 do not cross the straight lines joining the terminal compositions of the couples (Fig. 7(a)), in contradiction to the mass balance requirement. In addition, the experimental paths of A4 to A6 exhibit straight lines, indicating that the off-diagonal diffusion coefficients are negligible and the two diagonal diffusion coefficients are likely or approximately equal [41]. However, our calculation shows the diagonal coefficients are different and predicts S-shaped diffusion paths. A 3D plot of atomic mobility surfaces in the Al-rich side is given in Fig. 8. The second kind of simulation was used to model the solidification curves of the Al–Cu binary alloys, which is a base

H. Chang et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 34 (2010) 68–74

73

a × 10-16

Atioic mobility

1.5

1

0.5 0 0.02 0 0

0.04 0.02

0.04

0.06 0.06

0.08

Mole fraction of Cu

0.08 0.1

Mole fraction of Zn

Fig. 8. A 3D view of the atomic mobility surfaces for Al, Cu and Zn at the Al-rich side at 777 K.

Temperature (°C)

b

Fraction Solid Fig. 9. Simulated solidification curves of the Al–4.5 wt% Cu alloy.

system of the high-strength 2090 and 6061 series alloys. The Al–4.5 wt% Cu alloy and the Al–10 wt% Cu alloy were selected for study, but only the first is presented as a representative. For each alloy, three different models were used. Two of the most fundamental models are the equilibrium solidification model and the Scheil model. The third model is performed in the DICTRA software, in which diffusion in the solids is treated by using our mobility database. The atomic mobility of the liquid is taken from MOB2 [8]. The simulated solidification curves for the Al–4.5 wt% Cu alloy are shown in Fig. 9 in comparison with the experimental points [42]. With the cooling rate of 2 K/s, the DICTRA modeling agrees well with the coupled DTA measurement due to Chen et al. [42]. As read from the Al–Cu phase diagram, the equilibrium solidification model shows no secondary solidification, while the Scheil model and the DICTRA modeling support a eutectic secondary phase, as found from the coupled DTA measurement. However, there are discrepancies between the measured and predicted curves, particularly a distinctly lower temperature at the completion of solidification even though the undercooling effect was further included. This suggests that the alloying effect of the other elements cannot be neglected for this alloy.

Fig. 10. (a) Kirkendall velocity construction for the Cu/Cu–12.2 at.% Al diffusion couple at 1078 K; (b) Lattice plane displacement of the Cu–12.2 at.% Al/Cu couples annealed at 1078 K for 36, 191, 322.7, 437.2 and 568.4 h.

4.2. Kirkendall velocity construction and marker displacement As stated previously, the velocity of the inert markers can be determined from the knowledge of intrinsic diffusivities and the composition gradient at the marker position. Fig. 10(a) illustrates the Kirkendall velocity construction for the Cu/Cu–12.2 at.% Al diffusion √couple annealed at 1078 K. Note √ that the√reduced velocity curve v t intersects the straight line v t = z /2 t at a point with a positive gradient. As a result, there is an unstable Kirkendall plane for the Cu–12.2 at.% Al/Cu diffusion couple annealed at 1078 K. The shift of the Kirkendall plane is plotted in Fig. 10(b) as a function of the square root of diffusion time for the Cu/Cu–12.2 at.% Al diffusion couple. The displacement of the Kirkendall marker is about 55 µm toward the Al side of the couple, after annealing for 568.4 h. Appreciable discrepancies were observed for the long diffusion times, which might be caused by the experimental errors associated with the displacement measurement [43,44], or the consequence of the assumptions in the DICTRA formalism. 5. Conclusion Existing experimental diffusion data were used to assess atomic mobilities for the fcc phase of the Al–Cu, Cu–Zn and Al–Cu–Zn systems. Good agreement was obtained from comprehensive comparisons between calculated and experimental diffusion coefficients.

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The assessed mobility was further validated by simulations of the diffusion-couple experiments and solidification curves. Acknowledgements This work was supported by the National Basic Research Program of China (973 Program, No. 2007CB607603) and the National Natural Science Foundation of China (No. 50601011, No. 50432020). One of the authors, J. Li, would like to thank the Program for New Century Excellent Talents in University, China, for financial support. Appendix. Supplementary data Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.calphad.2009.12.002. References [1] M. De Sanctis, Mater. Sci. Eng. A 141 (1) (1991) 103. [2] N. Fridlyander, V.G. Sister, O.E. Grushko, V.V. Berstenev, L.M. Sheveleva, L.A. Ivanova, Met. Sci. Heat Treat. 44 (2002) 365. [3] E.A. Starke Jr., J.A. Wert, in: G.J. Hildeman, M.J. Koczak (Eds.), High Strength Powder Metallurgy Aluminum Alloys II, The Metallurgical Society, Warrendale, PA, 1986, p. 3. [4] H.G.F. Wilsdorf, in: Y.-W. Kim, W.M. Griffith (Eds.), Dispersion Strengthened Aluminium Alloys, The Metallurgical Society, Warrendale, PA, 1988, p. 3. [5] C.E. Campbell, W.J. Boettinger, U.R. Kattner, Acta Mater. 50 (2002) 775. [6] C.E. Campbell, Acta Mater. 56 (2008) 4277. [7] J. Yao, Y.-W. Cui, H.S. Liu, H.C. Kou, J.S. Li, L. Zhou, CALPHAD 32 (2008) 602. [8] MOB2, Mobility Database, Thermo-Calc Software AB, Stockholm, 1998. [9] Y.-W. Cui, K. Oikawa, R. Kainuma, K. Ishida, J. Phase Equillib. Diff. 27 (2006) 333. [10] J.O. Andersson, J. Agren, J. Appl. Phys. 72 (4) (1992) 1350.

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