Development of an atomic mobility database for liquid

S. Wang et al.: Development of an atomic mobility database for liquid phase in multicomponent Al alloys

Shaoqing Wanga , Dandan Liua,b , Yong Dua , Lijun Zhanga , Qing Chenc , Anders Engströmc a State

Key Laboratory of Powder Metallurgy, Central South University, Changsha, Hunan, PR China of Materials Science and Engineering, Central South University, Changsha, Hunan, PR China c Thermo-Calc Software AB, Stockholm, Sweden

2013 Carl Hanser Verlag, Munich, Germany

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Not for use in internet or intranet sites. Not for electronic distribution.

b School

Development of an atomic mobility database for liquid phase in multicomponent Al alloys: focusing on binary systems An atomic mobility database for binary liquid phase in multicomponent Al – Cu – Fe – Mg – Mn – Ni – Si – Zn alloys was established based on critically reviewed experimental and theoretical diffusion data by using DICTRA (Diffusion Controlled TRAnsformation) software. The impurity diffusivities of the elements with limited experimental data are obtained by means of the least-squares method and semiempirical correlations. Comprehensive comparisons between the calculated and measured diffusivities indicate that most of the reported diffusivities can be well reproduced by the currently obtained atomic mobilities. The reliability of this diffusivity database is further validated by comparing the simulated concentration profiles with the measured ones, as well as the measured main inter-diffusion coefficients of liquid Al – Cu – Zn alloys with the extrapolated ones from the present binary atomic mobility database. The approach is of general validity and applicable to establish mobility databases of other liquid alloys. Keywords: Liquid Al alloy; Atomic mobility; Diffusivity; DICTRA

1. Introduction Diffusion plays a significant role in material design and understanding many important phenomena, such as precipitation, homogenization of alloys, recrystallization, grain boundary migration, creep, solidification and protective coatings [1, 2]. Much diffusion behavior can be simulated using DICTRA (Diffusion Controlled TRAsformation) software [2], which is based on thermodynamic and kinetic databases. Accurate thermodynamic and kinetic databases are a prerequisite for describing various kinds of diffusioncontrolled transformation processes in multi-component alloys. Nowadays, extensive efforts have been made to construct reliable thermodynamic databases for multicomponent Al alloys [3]. Nevertheless, much fewer investigations have been done for diffusivities in the multicomponent Al alloys, especially in the liquid systems, because of the rigorous experimental conditions and hence the lack of reliable diffusivity data. Consequently, the establishment of the diffusivity database for multi-components liquid alloys by utilizing both the experimental and theoretical methods is of vital and fundamental importance. Int. J. Mater. Res. (formerly Z. Metallkd.) 104 (2013) 8

Du et al. [4] systematically investigated the diffusion of several transition elements (Ti, V, Cr, Mn, Fe, Co, Ni, Cu, and Zn) and a few non-transition elements (Mg, Si, Ga, and Ge) in liquid Al, based on critically reviewed and assessed impurity diffusion coefficients by means of the least-squares method and semi-empirical correlations. Most recently, on the basis of experimental and theoretical data about diffusivities, Zhang et al. [5] established a complete diffusivity database for ternary Al – Fe – Ni melts through kinetic modeling via DICTRA software, and then applied the database successfully to the simulation of microstructural evolution during solidification. Due to technological importance and scientific interest of various diffusion phenomena, the atomic mobilities for liquid Al alloys are to be evaluated in the present work. The main objectives of the present work are: (i) to evaluate all the diffusion coefficients associated with binary liquid systems of the Al – Cu – Fe – Mg – Mn – Ni – Si – Zn system available in the literature; (ii) to obtain a self-consistent set of atomic mobilities for the elements in the liquid binary systems using DICTRA software; and (iii) to verify the atomic mobilities obtained in the present work via the simulation of concentration profiles from several diffusion couples and the prediction of main inter-diffusion coefficients of liquid Al – Cu – Zn alloys.

2. Literature review Currently one obstacle that hampers the establishment of an atomic mobility database in liquid Al alloys is that the reported diffusivity data are usually very limited due to experimental difficulties. Two categories could be found to determine diffusivities in liquid phase. On one hand, diffusivities obtained from experimental measurements such as the long-capillary (LC) method and the quasi-elastic neutron scattering (QNS) method are adopted as reliable data. On the other hand, theoretical methods such as Molecular Dynamics (MD) simulations and first-principles calculations have demonstrated their promising potential to predict diffusivities. Consequently, diffusion data obtained from these theoretical methods can also be used to establish the diffusivity database for liquid phase. Table 1 shows the experimental and theoretical self-diffusion information in pure liquid Cu, Mg, Mn, Si and Zn. Table 2 presents the experimental and theoretical impurity 721



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Table 1. Summary of self-diffusion coefficients in the pure liquid metals. Type of data

Temperature range (K)

Methodb

Ref.

Codea

DCu Cu

1 370 – 1 620 1 410 – 1 520 1 370 – 1 440 1 413 – 2 231 1 398 – 3 198 1 356 1 356 1 358 1 357, 1 423 1 423 1 356

QNS LC MD MD-EAM MD-EAM MD-HSF MD-EAM MSEF MD MD MD

[7] [6] [10] [16] [13] [14] [11] [15] [12] [8] [9]

& + & & & & & & & & &

DMg Mg

450 – 1 130 923 – 1 300 923 – 1 223 970 – 2 000 953

MD MD MD MD MD

[19] [17] [14] [20] [18]

+ & & + &

DMn Mn

1 553 – 1 993

Calculated from viscosity data

[21]

&

DSi Si

1 687 1 687 – 1 853 1 800 1 800 1 740 1 780 1 780 1 700 1 683 1 691

pulsed laser melting SEF ab initio MD ab initio MD MD-TB MD-TB MD-TB MD-SW MD-SW MD-SW

[22] [31] [23] [25] [24] [26] [27] [28] [29] [30]

+ & & & & & & + + +

DZn Zn

718 – 873 693 – 873 600 – 800 693

open capillary capillary-dip method SEF SEF

[32] [33] [34] [35]

& & & &

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; +, not used.

b

QNS = quasi-elastic neutron scattering; LC = long capillary method; MD = molecular dynamics; EAM = embedded atom method; HSF = hard sphere fluid; MSEF = modified Stokes – Einstein formula; SEF = Stokes–Einstein formula; TB = Tight-binding; SW = Stillinger–Weber.

diffusivities in liquid Cu and Zn. Table 3 summarizes the experimental and theoretical tracer diffusivities and interdiffusivities in the Al – Cu, Al – Si, Al – Zn, Fe – Mn, Fe – Si, Mg – Zn and Ni – Si binary systems. As for the ternary systems, only limited main inter-diffusion information was found for the Al – Cu – Zn system, which is also shown in Table 3. The diffusivities for the above systems are critically evaluated as follows. 2.1. Self-diffusivity in liquid Al, Cu, Fe, Mg, Mn, Ni, Si and Zn Recently, Zhang et al. [5] investigated the self-diffusion in liquid Al, Fe and Ni, based on critically reviewed self-diffu722

sion coefficients. Since the atomic mobilities obtained by Zhang et al. [5] can describe most of the diffusivities well, their parameters are directly utilized in the present work. Table 1 summarizes the self-diffusion coefficients in liquid Cu, Mg, Mn, Si and Zn obtained from both the experimental and theoretical methods. Using the LC radiotracer method, Henderson and Yang [6] determined the Cu self-diffusion coefficient in liquid Cu. Recently, Meyer [7] measured the self-diffusion coefficient of liquid Cu using the QNS method over the temperature range of 1 370 – 1 620 K. Values from Henderson and Yang [6] are significantly larger than those reported by Meyer [7] and exhibit a relatively larger scatter because gravity-driven convective flow enhances the diffusion proInt. J. Mater. Res. (formerly Z. Metallkd.) 104 (2013) 8


Type of data

Temperature range (K)

Methodb

Ref.

Codea

DCu Fe

1 373 – 1 573

capillary reservoir method

[36]

&

DCu Ni

1 373

capillary method

[38]

&

DCu Zn

1 353, 1 473

revolving disk method

[37]

&

DZn Al

723 – 793

Sutherland–Einstein Enskog Expression Semiemipirical Formulas MD

[41]

&

DZn Fe

723 – 973

DC

[39]

&

DZn Ni

723 – 823

dissolution method

[40]

&

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.

b

MD = molecular dynamics; DC = diffusion couple method.

Table 3. Summary of tracer and chemical diffusion coefficients in the binary and ternary melts. Temperature range (K)

Methodb

Ref.

Codea

1 000 – 1 800

QNS

[43]

&

1 000 – 1 800

QNS

[44]

&

973, 1 173, 1 373

QNS

[42]

+

983 – 1 173

XRR

[44]

&

963 – 1 053

LC

[45]

&

DSi

923 – 1 473

rotating disk

[46]

+

DSi

858 – 1 123

capillary-reservoir

[47]

&

DAl DSi

930 – 2 173

ab initio MD

[48]

+

DAl DSi

943

HSF

[49]

+

Al – Zn

DZn AlAl

723 – 793

LC

[41]

&

Fe – Mn

DMn FeFe

1 823 – 1 973

DC

[50]

&

1 850 – 1 960

DC

[51]

+

1 823 – 2 000

DC

[50]

&

1 850 – 1 960

DC

[51]

&

DMg

926

MSEF

[52]

&

DZn

926

DNi

1 263 – 1 700

QNS

[31]

&

DSi

1 417 – 1 994

SEF

[53]

DAl CuCu DCu ZnZn

973 – 1 173

DC

[54]

Type of data Al – Cu

DCu

DCu AlAl

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Table 2. Summary of impurity diffusion coefficients in the liquid metals.


Al – Si

Fe – Si

Mg – Zn

Ni – Si

Al – Cu – Zn

a b

DSi FeFe

&

973 – 1 273

Indicates whether the data are used or not used in the atomic mobility assessment: &, used; +, not used. QNS = quasi-elastic neutron scattering; DC = diffusion couple method; LC = long capillary method; XRR = X-ray radiography; MD = molecular dynamics; HSF = hard sphere fluid; MSEF = modified Stokes–Einstein formula; SEF = Stokes–Einstein formula.

Int. J. Mater. Res. (formerly Z. Metallkd.) 104 (2013) 8

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cess. So the data from Henderson and Yang [6] are not used in the present work. Several authors have investigated the self-diffusion coefficient in liquid Cu at different temperatures by means of MD method, hard sphere model and modified Stokes–Einstein formula. Their data are scattered around the experimental values obtained by Meyer [7]. Therefore the data from Meyer [7] are utilized to evaluate the atomic mobility for pure liquid Cu in the present work, while the theoretically predicted ones [8 – 16] are used only for a comparison with the obtained atomic mobilities. As shown in Table 1, only theoretically predicted selfdiffusion coefficients via MD method exist for liquid Mg. The reported self-diffusion coefficients from Yang et al. [17], Protopapas et al. [14] and Alemany et al. [18] agree well with each other, while the values reported by Li et al. [19] and Wax et al. [20] are larger and smaller than the data from Refs. [14, 17, 18], respectively. Therefore, those data from Refs. [14, 17, 18] are used to assess the atomic mobility for pure Mg in the present work. Levin et al. [21] determined the viscosity by using the logarithmic decrement of torsioned oscillations and rotating magnetic field technique. They then calculated the self-diffusion in liquid Mn over the temperature range of 1 553– 1 993 K based on their measured viscosity data. This is the only information for self-diffusion of liquid Mn available in the literature. Those data are thus utilized to evaluate the atomic mobility for pure liquid Mn. There is only one paper reporting the experimental measurement of self-diffusion in liquid Si. Sanders and Aziz [22] determined the self-diffusivity of liquid Si at 1 687 K by pulsed laser melting of 30+Si ion implanted silicon-on-insulator thin films. But their values are much larger than the ones obtained by other authors [23 – 30] using molecular dynamics simulation method. Recently, Pommrich et al. [31] calculated the self-diffusion coefficients in liquid Si from viscosity data of pure liquid Si from 1 687 to 1 853 K. The calculated data are in accordance with the tracer diffusion of Ni in liquid Si-rich Ni – Si alloys [31]. Moreover those calculated data could fit most of the theoretically predicted ones acceptably. Therefore, the calculated self-diffusion data from Pommrich et al. [31] are utilized to evaluate the atomic mobility for pure liquid Si, while the data of Sanders and Aziz [22] are not used in the present assessment. Self-diffusion in liquid Zn has been measured by Nachtrieb et al. [32] using the open capillary technique over the temperature range of 723 – 873 K. By means of the capillary-dip method and Zn65, Lange el al. [33] determined the self-diffusion coefficients in liquid Zn over the temperature range of 693 – 873 K. Vadovic and Colver [34] calculated the self-diffusion of liquid Zn through Stokes–Einstein equation using the Goldschmidt diameter over the temperature range of 600 – 800 K. Recently Iida et al. [35] also investigated the self-diffusivity of liquid Zn by means of a modified Stokes–Einstein formula. All the reported selfdiffusion coefficients in liquid Zn agree well with each other, and they are thus employed to determine the atomic mobility for liquid Zn. 2.2. Impurity diffusivity in liquid Al, Cu and Zn The investigation for the impurity diffusion of Cu, Fe, Mg, Mn, Ni, Si and Zn in liquid Al was performed by Du et al. [4], who critically reviewed and assessed the impurity diffu724

sion coefficients by means of the least-squares method and semi empirical correlations. Since the activation energies and pre-exponential factors obtained by Du et al. [4] can describe most of the diffusivities well, their atomic mobilities are directly utilized in the present assessment. For other elements in the present Al – Cu – Fe – Mg – Mn – Ni – Si – Zn system, only impurity diffusion coefficients in liquid Cu and Zn could be found and are summarized in Table 2. Ejima and Kameda [36] measured the impurity diffusion coefficients of Fe in liquid Cu by means of the modified capillary reservoir method over the temperature range of 1 373 – 1 573 K. These data are utilized in the present assessment. Lukashenko et al. [37] measured the impurity diffusion coefficients of Zn in liquid Cu using the capillary method with 65Zn. Shurypin and Shantarin [38] determined the impurity diffusivity of Ni in liquid Cu at 1 473 K from the loss in weight of a revolving disk. Those limited experimental data from Refs. [37, 38] are used only for a comparison with the results predicted by the presently obtained atomic mobilities. Kato and Minowa [39] measured the impurity diffusion coefficients of Fe in liquid Zn in the temperature range of 723 – 973 K. Langberg and Nilmani [40] calculated the impurity diffusion coefficients of Ni in liquid Zn in the temperature range of 723 – 823 K based on the measured dissolution rate of Ni plates immersed in Zn melt. Yang et al. [41] calculated the impurity diffusion coefficients of Al in liquid Zn by using several theoretical methods, including the Sutherland–Einstein equation, the Enskog expression, the \hole" theory, the fluid theory and the MD method. Those impurity diffusivities in liquid Zn from Refs. [39 – 41] are utilized in the present assessment. 2.3. Tracer diffusivity and interdiffusivity in binary systems Among the 28 binary systems in the Al – Cu – Fe – Mg – Mn – Ni – Si – Zn alloy, the Al – Fe, Al – Ni and Fe – Ni systems are assessed by Zhang et al. [5]. For the remaining 25 systems, the tracer diffusion and inter-diffusion information is only found for Al – Cu, Al – Si, Al – Zn, Fe – Mn, Fe – Si, Mg – Zn and Ni – Si binary systems. Table 3 summarizes the tracer diffusion coefficients and the inter-diffusion coefficients in liquid binary alloys from the literature. 2.3.1. Al – Cu system Dahlborg et al. [42] determined the tracer diffusion coefficient of Cu in Al1–xCux alloys (x = 0.10, 0.17 and 0.25) by means of quasielastic cold neutron scattering experiments at 973, 1 173 and 1 373 K. Brillo et al. [43] measured the shear viscosity and the tracer diffusion coefficients of Cu in liquid Al80Cu20 alloy over the temperature range of 1 000 – 1 800 K by means of the QNS method. They also calculated the tracer diffusion coefficient of Cu from the experimental viscosity data using the Stokes–Einstein equation. Using time-resolved X-ray radiography technique and the QNS method, Zhang et al. [44] investigated the inter-diffusion coefficient in liquid Al81.3Cu18.7 melt and the tracer diffusion coefficients of Cu in Al80Cu20 melt respectively. Lee and Cahoon [45] determined the inter-diffusion coefficients in the Al – Cu melts by means of the long capillary technique from 963 to 1 053 K. A linear extrapolation of the tracer diffusion coefficients of Cu to xCu = 0 shows Int. J. Mater. Res. (formerly Z. Metallkd.) 104 (2013) 8


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that the impurity diffusion coefficients from Dahlborg et al. [42] are about a factor of 4 larger than the ones obtained by Du et al. [4]. Thus their data are not used in the present assessment. The tracer diffusion data measured by Zhang et al. [44] and Brillo et al. [43] agree well with each other. However, the tracer diffusion coefficients calculated from viscosity data deviate from the measured ones at low temperature. Therefore, only the experimental tracer diffusion and inter-diffusion data are utilized to evaluate the atomic mobility in Al – Cu liquid alloys, while the calculated tracer diffusion coefficients are used for comparison with the values predicted with the presently obtained atomic mobilities.

By means of the diffusion couple method and EPMA, Calderon et al. [50] determined the inter-diffusion in molten Fe – Si system in the temperature range of 1 823 – 2 000 K at various concentrations. Using diffusion couple method and chemical analysis method, Saito and Maruya [51] measured the inter-diffusion coefficients in liquid Fe – Si alloys. The inter-diffusion data measured by Calderon et al. [50] and Saito and Maruya [51] agree well with each other. Therefore, their experimental inter-diffusion data are utilized to evaluate the atomic mobility in Fe – Si liquid alloys

2.3.2. Al – Si system

2.3.6. Mg – Zn system

Lozovskii et al. [46] detected the tracer diffusion coefficients of Si in Al – Si melt at 923 – 1 473 K by using the rotating disk method. Petrescu [47] investigated the tracer diffusion of Si in liquid Al-12.5 wt.% Si alloy over the temperature range of 858 – 1 123 K by means of the capillary-reservoir method. Ji and Gong [48] calculated the tracer diffusion coefficients in liquid Al-16.1 wt.% Si alloy at 930 – 2 173 K using the ab-initio MD method. Mishra and Venkatesh [49] theoretically calculated the tracer diffusion coefficients in Al – Si melt at 943 K based on hard sphere method. The calculated results of Al-16.1 wt.% Si alloy from Ji and Gong [48] are smaller than the experimental ones of Al-12.5 wt.% Si alloy from Petrescu [47]. While the diffusion data from Refs. [46, 49] show a tendency that the tracer diffusion of Si increases with the Si concentration. Thus these data from Ref. [48] are not used in the present assessment. Besides, an extrapolation of the tracer diffusion of Si to xSi = 0 indicates that the impurity diffusion coefficients from Lozovskii et al. [46] and Mishra and Venkatesh [49] are much larger than the ones obtained by Du et al. [4]. Consequently, those data from Refs. [46, 49] are not used in the present assessment either, and only the tracer diffusion coefficients of Si measured by Petrescu [47] are utilized in the present work.

There is limited diffusion information for the liquid Mg – Zn system in the literature. Rao and Bandyopadhyay [52] studied structural and transport properties of liquid Mg – Zn alloy by means of the modified Stokes–Einstein formula, and obtained the tracer diffusion coefficients of both Mg and Zn in liquid Mg – Zn alloy.

2.3.3. Al – Zn system Yang et al. [41] experimentally investigated the diffusion coefficient of Al in molten Zn using the long capillary technique. However, their measured diffusion coefficients are treated as impurity ones. Considering the fact that the concentration of Al in the sample is 2 at.%, the diffusion data should be considered as inter-diffusion coefficients in Al – Zn melts. These data are used in the present assessment.

2.3.5. Fe – Si system

2.3.7. Ni – Si system Pommrich et al. [31] measured the tracer diffusion coefficients of Ni in liquid Si95Ni5, Si90Ni10, Si80Ni20 alloys by means of the QNS method over the temperature range of 1 263 – 1 700 K. Wang et al. [53] measured the surface tension of liquid Ni-5 wt.% Si alloy by means of an electromagnetic oscillating drop method over a wide temperature range of 1 417 – 1 994 K. Then they calculated the solute diffusion coefficient of this alloy based on the measured data of surface tension, which can be treated as tracer diffusion coefficients of Si in Ni-5 wt.% Si melts. 2.4. Diffusion in Al – Cu – Zn system Only one piece of information about the inter-diffusion coefficients in the Al – Cu – Zn melt is available in the literature. Uemura [54] measured the main inter-diffusion coefficients in Al-rich region of Al – Cu – Zn ternary melts from 973 to 1 273 K using the diffusion couple method. In order to check the reliability of the presently obtained atomic mobility, the inter–diffusion coefficients from Uemura [54] are not used in the assessment procedure, but are used for comparison with the inter-diffusion coefficients directly extrapolated from the atomic mobilities of the sub-binary systems.

2.3.4. Fe – Mn system

3. Modeling

By means of diffusion couple method and electron probe microanalysis (EPMA), Calderon et al. [50] determined the inter-diffusion in molten Fe – Mn system in the temperature range of 1 823 – 1 973 K at various concentrations of Mn. Using the diffusion couple method and the chemical analysis method, Saito and Maruya [51] measured the inter-diffusion coefficients in liquid Fe – Mn alloys. But the alloys are carbon-saturated in the experiments of Saito and Maruya [51]. Therefore, only the experimental inter-diffusion data from Calderon et al. [50] are utilized to evaluate the atomic mobility in Fe – Mn liquid alloys.

3.1. Diffusion modeling


The Arrhenius equation is usually employed to describe the self-diffusion and impurity diffusion in liquids though the identity of the activation energy is uncertain [4]. In addition, the Darken equation is also widely used to estimate the inter-diffusion coefficients in simple liquids [55]. Such treatments suggest that the concept of atomic mobility and the model for multicomponent diffusion in simple phases introduced by Anderson and Ågren [1] can also be successfully applied to the liquid binary Al alloys. 725


According to the absolute-reaction rate theory arguments, the atomic mobility for an element B in disordered liquid phase, MB, can be expressed as: RT


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MB ¼ exp

ln MB0 RT

exp

QB 1 RT RT

ð1Þ

where QB is the activation enthalpy; MB0 is the frequency factor; R is the gas constant; T is the temperature. In the spirit of the CALPHAD approach [56], RT ln MB0 and QB can be described by the following expression: " # m X XX X i;j r r ð2Þ UB ¼ xi UiB þ xi xj UB ðxi xj Þ i

i

j>i

r¼0

where UB ¼ QB þ RT ln MB0 ; xi and xj are the mole fractions of elements i and j respectively; UiB is the value of UB for B in pure i; r UiB is the interaction term. In a volume-fixed frame of reference, the inter-diffusion coefficient Dnkj , which relates the flux of element k with the gradient of component j and reference component n, can be given by the following expression: Dnkj ¼

X i

ðdik

xk Þ xi Mi

q i qxj

q i qxn

ð3Þ

where dik is the Kronecker delta (dik = 1 if i = k, otherwise dik = 0). xi and li are the mole fraction and the chemical potential of element i, respectively. Mi is the composition-dependent atomic mobility for element i. Assuming a mono-vacancy atomic exchange mechanism for diffusion and neglecting correlation factors, the tracer diffusivity Di is directly related to the atomic mobility Mi via the Einstein relation: Di ¼ RTMi

ð4Þ

In a binary system, the tracer diffusivity Di correlates to the inter-diffusion coefficient D~ through Darken’s equation [57]: D~ ¼ ðxA DB þ xB DA Þ where as: ¼1þ

ð5Þ

is the thermodynamic factor, and can be expressed q ln ci xi d i ¼ q ln xi RT dxi

ð6Þ

where xi , ci and i are the mole fraction, the activity coefficient and the chemical potential of i, respectively. 3.2. Semi-empirical correlations to estimate impurity diffusivity in liquid phase In the case of impurity diffusion in liquid phase, as described in Section 2, the measurement is very limited due to the experimental difficulties and there exist even no diffusion data for some elements. Such a situation makes it difficult to assess the corresponding atomic mobilities in liquid systems. Thus semi-empirical correlations are employed to obtain the impurity diffusion coefficients in liquid phase. Based on an electrostatic model using a Thomas–Fermi approximation [58] for the electric potential around a solute 726

atom, Swalin [59] derived the following semi-empirical correlation: qlog10 D0 ½ð1 þ qrÞ þ 0:75ð q3 r 3 þ 6q2 r2 þ 5qr þ 5Þ ¼ 2:3R½1 0:25ðq2 r 2 5qr 5Þ qQ ð7Þ where is the thermal expansion coefficient of the solvent, R is the gas constant, r is the interatomic distance of the solvent, q is the screening constant, Q is the activation energy and D0 is the pre-exponential factor. Based on both experimental investigations and theoretical analysis, Ejima et al. [60] found that there is a correlation between the activation energy and the position of a solute in the periodic table, and the valence relative to the solvent is an appropriate parameter to describe the position of the solute. Both of the above semi-empirical relations were successfully employed to describe the diffusion of some solutes in fcc and liquid Al [4]. Consequently, they are utilized to estimate the impurity diffusivity of Ni and Zn in liquid Cu in the present work. As for the systems with no impurity diffusion information, i. e. Al – Cu, Al – Si, Fe – Mn, Fe – Zn, Mg – Zn and Ni – Si systems, the two end-members, UBA and UAB , were set to be equal to UBB and UAA , respectively, for the sake of simplification.

4. Results and discussion The evaluation of the model parameters in Eq. (2) is attained by means of the PARROT module of the DICTRA software package [2, 61] applied to all the critically reviewed experimental data in Section 2. In order to calculate the thermodynamic factor ( ) in Eq. (6) during the assessment of mobility parameters, the thermodynamic parameters for the liquid systems are necessary, which were taken directly from Du et al. [3]. For each binary system, the assessment procedure was performed in the following steps. Firstly, the values of the end-members were evaluated by using the DICTRA software and the semi-empirical relation [59, 60] or taken directly from the literature , as in the Al – Fe, Al – Ni and Fe – Ni systems [4, 5]. Next, the tracer diffusion and inter-diffusion coeffcients in binary systems were calculated by using the above-obtained values of the endmembers, and the computed diffusivities were compared with the corresponding experimental data. From such comparsion, we can judge if the interaction parameters should be employed in the assessment. The finally obtained atomic mobilities for Al, Cu, Fe, Mg, Mn, Ni, Si and Zn in binary liquid phases of this 8-elemental system have been compiled into the commercial diffusivity database MOBAL2 for multi-component Al alloys [62]. 4.1. Self-diffusivity in liquid Al, Cu, Fe, Mg, Mn, Ni, Si and Zn Among these 8 elements, the self-diffusivities in the liquid Al, Fe and Ni are taken from Zhang et al. [5], while the self-diffusivities in the remaining 5 liquid elements are optimized based on the reliable diffusion coefficients, as mentioned in Section 2.1. Int. J. Mater. Res. (formerly Z. Metallkd.) 104 (2013) 8

Figure 1a and b shows the self-diffusivites in the liquid elements, compared with the diffusion coefficients in the literature [7 – 15]. As shown in the figure, the calculated Cu self-diffuison coefficients are in excellent agreement with the experimental values measured by Meyer [7], lower than the theoretically predicted ones [10 – 15] and higher than the valuses of Refs. [8, 9]. The calculated liquid Mg self-diffusion coefficients locate around the center of all the theoretically predicted data, and give a good fit to the

data from Yang et al. [17], Protopapas et al. [14] and Alemany et al. [18]. The calculated self-diffusion coefficients in liquid Mn are in excellent agreement with the experimental values from Levin et al. [21]. The calculated self-diffusion coefficients of liquid Si are in excellent agreement with the values from Pommrich et al. [31] and fit the values calculated via MD methods reasonably. The calculated values of self-diffusivity in liquid Zn are in good accordance with the measured self-diffusion data [32 – 34].


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Fig. 1. Calculated self-diffusion coefficients in liquid (a) Cu, Fe, Mn, Ni and Si and (b) Al, Mg and Zn. The mobility parameters for Al, Fe and Ni are directly taken from Zhang [5]. A constant, M, is added in order to separate the data from different elements in the figure.


727


The impurity diffusion parameters of Cu, Fe, Mg, Mn, Ni, Si and Zn in liquid Al are reasonably established by Du et al. [4]. The activation energy of Cu, Fe and Ni versus valence relative to Al and the logarithmic pre-exponential factors against activation energies for these elements are used as a basis to predict the diffusion coefficients of Mn, Si and Zn in liquid Al [4]. The successful application of this approach to estimation of the impurity diffusion coefficients for Mn, Si and Zn in liquid Al indicates that the approach is equally valid for the elements in the same row of the periodic table.

Figure 2 presents the calculated impurity diffusivity of Fe in liquid Cu, which fits well with the experimental data from Ejima et al. [36]. Based on the well estimated diffusivity of Fe and Co in liquid Cu and the self-diffusivity of Cu, we could plot the activation energy versus valence relative to Cu, and the logarithmic pre-exponential factors against activation energies as shown in Fig. 3 and Fig. 4, respectively. Then we can obtain the estimated activation energy for Ni and Zn from the least-squares fitting curve in Fig. 3. Based on the estimated activation energy, the pre-exponential factors are calculated by using the correlation line in Fig. 4. In Fig. 5, the predicted impurity diffusion coefficients of Ni and Zn in liquid Cu are compared with the lim-

Fig. 2. Calculated impurity diffusion coefficients of Fe in liquid Cu, compared with the experimental data [36].

Fig. 4. Logarithmic pre-exponential factors against activation energies for impurity diffusion of different solutes in liquid Cu.

Fig. 3. Activation energies for impurity diffusion of different solutes in liquid Cu.

Fig. 5. Model-predicted and measured impurity diffusion coefficients of Zn and Ni in liquid Cu [37, 38]. Dashed lines refer to the diffusion coefficients with a factor of 2 or 0.5 from the model-predicted diffusion coefficients in this work.


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4.2. Impurity diffusiviy in liquid Al, Cu and Zn

728


ited experimental data [37, 38]. Here the dashed lines refer to the diffusion coefficients with a factor of 2 or 0.5 from the model-predicted diffusion coefficients in this work. Such a factor is a generally accepted experimental error for measurement of diffusivities. A reasonable agreement can be seen from Fig. 5. Figure 6 shows the calculated and measured impurity coefficients of Fe and Ni in liquid Zn. The calculations can well reproduce the experimental values. Figure 7 shows the presently calculated impurity coefficients of Al in liquid Zn, which fit the theoretically predicted ones acceptably.

Fig. 6. Calculated impurity diffusion coefficients of Fe and Ni in liquid Zn, compared with the experimental data [39, 40].

4.3. Diffusivities in binary and ternary melts 4.3.1. Al – Cu system Figure 8 shows the calculated tracer diffusion coefficients of Cu in Al80Cu20 melts compared with both experimental data and those calculated from the experimental viscosity data using Stokes–Einstein equation [42 – 44]. It is found that the presently calculated tracer diffusion coefficients agree well with the experimental values, and the values calculated using the Stokes–Einstein equation could fit the experimental data at high temperatures. In Fig. 9a, the calculated concentration dependence of inter-diffusion coefficients in liquid Al – Cu alloys at 983 K are compared with the data from experimental measurement [44]. It can be seen that the inter-diffusion coefficient increase slightly with the concentration of Cu in Cu-rich side. The calculated inter-diffusion coefficients in Al-18.7 at.% Cu are compared with the experimental data from Zhang [44], as shown in Fig. 9b. The presently predicted results are found to agree satisfactorily with the experimental values. Figure 9c presents the calculated and measured inter-diffusion coefficients at different temperatures between 963 and 1 073 K. The calculated ones can well reproduce the experimental values from Lee and Cahoon [45]. Figure 10a presents the model-predicted concentration profile of the Al/Al-4.5 at.% Cu diffusion couple annealed at 983 K for 175 s and 1 045 s. Figure 10b shows the model-predicted concentration profile of the Al-21.3 at.% Cu /Al diffusion couple annealed at 1 053 K for 10 800 s. The profiles predicted by the present mobility database are in excellent accordance with the experimental data [44, 45], which verifies the validity of the currently diffusivity database.


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Fig. 7. Calculated impurity diffusion coefficients of Al in liquid Zn, compared with the theoretical predictions [41]. Dashed lines refer to the diffusion coefficients with a factor of 2 or 0.5 from the model-predicted diffusion coefficients in this work.


Fig. 8. Calculated tracer diffusion coefficients in liquid Al – Cu alloys, compared with the experimental data [43, 44].

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Fig. 10. Model-predicted concentration profile of (a) the Al/Al4.5 at.% Cu diffusion couple annealed at 983 K for 175 s and 1 045 s; (b) the Al-21.3 at.% Cu /Al diffusion couple annealed at 1 053 K for 10 800 s.

4.3.2. Al – Si system Figure 11 shows the calculated and measured tracer diffusion coefficients of Si in Al-12.5 wt.% Si melt. It can be seen from the figure that the present parameters can well reproduce the experimental data from Ref. [47]. 4.3.3. Al – Zn system Fig. 9. (a) Calculated concentration dependence of inter-diffusion coefficients in liquid Al – Cu alloys at 983 K, and (b) Calculated inter-diffusion coefficients in Al-18.7 at.% Cu, compared with the experimental data [44]; (c) Calculated and measured inter-diffusion coefficients in Al – Cu melts at different temperatures between 963 and 1 073 K [45].

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Figure 12 presents the calculated inter-diffusion coefficients in Zn-2 at.% Al melts at 723, 753, 773 and 793 K, compared with the experimental data from Yang et al. [41]. Considering the fact that each measured inter-diffusion coefficient is the average value of one diffusion couInt. J. Mater. Res. (formerly Z. Metallkd.) 104 (2013) 8

Fig. 11. Calculated tracer diffusion coefficients of Si in Al-15 wt.% Si melts, compared with the experimental data from Petrescu [47].


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Fig. 12. Calculated inter-diffusion coefficients in Al2Zn98 melts, compared with the experimental data from Yang et al. [41].

ple, the calculated values can reproduce the measured ones within the experimental error range. The presently model-predicted concentration profiles of Zn/Zn-2 at.% Al diffusion couples annealed for 1 800 s at 723 K and 753 K are compared with the corresponding experimental data [41], as shown in Fig. 13a and b. There are some deviations between the calculated curves and the experimental ones, as indicated in Fig. 13a and b. Considering the fact that this binary system is assessed based on limited experimental data, the presently calculated results are still acceptable within the experimental error. Int. J. Mater. Res. (formerly Z. Metallkd.) 104 (2013) 8

Fig. 13. Calculated concentration profiles of Zn-2 at.% Al/Zn diffusion couple, annealed at (a) 723 and (b) 753 K for 1 800 s.

4.3.4. Fe – Mn system The calculated inter-diffusion coefficients in Fe-7.5 wt.% Mn are shown in Fig. 14a, along with the experimental data [50]. The presently predicted results are found to agree satisfactorily with the experimental values. In Fig. 14b, the calculated concentration dependence of inter-diffusion coefficients in liquid Fe – Mn alloys at 1 873 K are compared with the experimental data [50]. It can be seen that the model-predicted inter-diffusion coefficients are in good accordance with the measured ones. 731


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Fig. 14. Calculated (a) temperature-dependence and (b) concentration-dependence of inter-diffusion coefficients in Fe – Mn melts, compared with the experimental data [50].

Figure 15a and b shows the predicted concentration curves of Fe-15 wt.% Mn/Fe diffusion couple annealed at 1 873 K for 480 s and 1 800 s and the Fe-30 wt.% Mn/Fe diffusion couple annealed at 1 873 K for 3 208 s, compared with the experimental measured data [50]. The satisfactory agreement between the calculated results and the experimental data verify the validity of the currently diffusivity database. 4.3.5. Fe – Si system The calculated inter-diffusion coefficients in Fe-2 wt.% Si, Fe-2.2 wt.% Si, Fe-12.5 wt.% Si, Fe-20 wt.% Si alloys are compared with the corresponding experimental ones [50], as shown in Fig. 16a. The presently predicted results are 732

Fig. 15. Model-predicted concentration profiles of (a) Fe-15 wt.% Mn/Fe diffusion couple annealed at 1 873 K for 480 s and 1 800 s and (b) the Fe-30 wt.% Mn/Fe diffusion couple annealed for 3 208 s, compared with the experimentally measured data [50].

found to agree satisfactorily with the experimental values. In Fig. 16b, the calculated concentration dependence of inter-diffusion coefficients in liquid Fe – Si alloys at 1 873 K are compared with the experimental data [50, 51]. As can be seen in the figure, the model-predicted inter-diffusion coefficients fit well with the measured ones. Figure 17a presents the concentration-distance profile of the Fe/Fe-34.8 at.% Si diffusion couple predicted by the atomic mobilities obtained in this work. This couple diffuses at 1 873 K for 7 200 s. Figure 17b shows the calculated concentration profile of the Fe-4.4 wt.% Si/Fe diffusion couple annealed at 1 873 K for 480 and 7 200 s. It can be seen from the figure that the predicted profiles fit perfectly Int. J. Mater. Res. (formerly Z. Metallkd.) 104 (2013) 8


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Fig. 16. Calculated (a) temperature-dependence and (b) concentration-dependence of inter-diffusion coefficients in liquid Fe – Si system, compared with the experimental data [50, 51].

with the experimental ones [50, 51]. It can be concluded that the presently obtained atomic mobilities of Fe and Si in the Fe – Si melts are reliable. 4.3.6. Mg – Zn and Ni – Si systems Figure 18 illustrates the presently calculated inter-diffusion coefficients in Mg – Zn alloys at 926 K. Symbols in the figure present inter-diffusion coefficients calculated by Eq. (4) using the tracer diffusion data obtained by Rao and Bandyopadhyay [52]. As shown in the figure, the agreement between the calculated inter-diffusivities and the experimental ones is satisfactory. These atomic mobilities of the endmembers can well describe the inter-diffusion coefficients, so there is no need to introduce the interaction parameters. Int. J. Mater. Res. (formerly Z. Metallkd.) 104 (2013) 8

Fig. 17. Model-predicted concentration-distance profiles of (a) the Fe/ Fe-34.8 at.% Si diffusion couple annealed at 1 873 K for 7 200 s and (b) the Fe-4.4 wt.% Si /Fe diffusion couple annealed at 1 873 K for 480 and 7 200 s.

Figure 19 presents the calculated tracer diffusion coefficients of Ni and Si in the Ni – Si melt, compared with the experimental and theoretical data from Pommrich et al. [31] and Wang et al. [53], respectively. The calculated diffusivities could well reproduce the experimental ones from the literature. 4.4. Validation of the present atomic mobility database In order to evaluate the reliability of the present mobility database, its application to predict the ternary inter-diffusion coefficients of Al – Cu – Zn system was performed. For the Al – Cu – Zn system, only Uemura [54] has measured a number of main inter-diffusion coefficients. The atomic mobilities of the three sub-binary systems can reasonably describe the inter-diffusion coefficients in the Al – 733

Fig. 18. Model-predicted inter-diffusion in liquid Mg – Zn system, compared with the theoretical results from Rao and Bandyopadhyay [52].


Fig. 20. Model-predicted inter-diffusion coefficients in liquid Al – Cu – Zn system, compared with the experimental data [54]. Dashed lines refer to the diffusion coefficients with a factor of 2 or 0.5 from the model-predicted diffusion coefficients in this work.

were critically reviewed. Semi-empirical relations were employed to evaluate the impurity diffusivities in liquid Cu. The atomic mobilities of binary melts in the Al – Cu – Fe – Mg – Mn – Ni – Si – Zn system were then evaluated using the DICTRA software. Comprehensive comparisons show that most of the diffusivities from different sources can be well reproduced by the presently obtained atomic mobilities. The good agreement between the model-predicted concentration profiles and the experimental data in 7 binary systems indicates that the atomic mobilities obtained in the present work are reliable. The well-reproduced main inter-diffusion coefficients of liquid ternary Al – Cu – Zn system based on the direct extrapolation from the subsystems can further validate the present diffusivity database.

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Fig. 19. Calculated tracer diffusion coefficients of Ni and Si in liquid Ni – Si system, compared with the theoretical results [31, 53].

The financial support from the National Natural Science Foundation of China (Grant No. 51071179), the Sino-German Center for Promotion of Science (Grant No. GC755), the Creative Research Group of National Science Foundation of China (Grant No. 51021063), the National Basic Research Program of China (Grant No. 2011CB610401) and Thermo-Calc Software AB under the Aluminum Alloy Database Project is acknowledged. References

Various experimentally measured and theoretically predicted diffusivities available in the literature for binary melts in the Al – Cu – Fe – Mg – Mn – Ni – Si – Zn system

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Cu – Zn melt, as shown in Fig. 20. The experimental data are reasonably consistent with the presently calculated ones. It can be seen from the figure that though no ternary interaction parameters were introduced, the presently calculated results agree reasonably well with the experimental values within the acceptable error.

5. Conclusion


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(Received October 20, 2012; accepted January 4, 2013; online since February 18, 2013) Bibliography DOI 10.3139/146.110923 Int. J. Mater. Res. (formerly Z. Metallkd.) 104 (2013) 8; page 721 – 735 # Carl Hanser Verlag GmbH & Co. KG ISSN 1862-5282 Correspondence address Professor Dr. Yong Du State Key Laboratory of Powder Metallurgy Central South University Hunan, 410083 China Tel.: +86 731 88836213 Fax: +86 731 88710855 E-mail: [email protected]

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