Development of an atomic mobility database for

D. Liu et al.: Development of an atomic mobility database for disordered and ordered fcc phases in Al alloys

Dandan Liua,b , Lijun Zhangc , Yong Dub , Senlin Cuib , Wanqi Jied , Zhanpeng Jina a School

of Materials Science and Engineering, Central South University, Changsha, Hunan, P. R. China Key Laboratory of Powder Metallurgy, Central South University, Changsha, Hunan, P. R. China c Interdisciplinary Centre for Advanced Materials Simulation, Ruhr-Universität Bochum, Bochum, Germany d State Key Laboratory of Solidification Processing, Northwestern Polytechnical University, Xi’an, Shanxi, P. R. China

2013 Carl Hanser Verlag, Munich, Germany

www.ijmr.de

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

b State

Development of an atomic mobility database for disordered and ordered fcc phases in multicomponent Al alloys: focusing on binary systems An atomic mobility database for binary disordered and ordered fcc phases in multicomponent Al – Cu – Fe – Mg – Mn – Ni – Si – Zn alloys was established based on a critical review of diffusion data in various constituent binary systems via the DICTRA (DIffusion Controlled TRAnsformation) software package. The mobility parameters for selfdiffusion in the metastable fcc structure were determined through a semi-empirical method. An effective strategy, which takes the homogeneity range and defect concentration into account, was used to optimize the atomic mobilities of L12 phase in the Fe – Ni system. Comprehensive comparisons between various calculated and measured diffusivities show that most of the experimental data can be well reproduced by the presently obtained atomic mobilities. The general agreement between the model-predicted concentration profiles and the experimental ones in the Al – Ni – Si, Al – Mg – Zn and Cu – Mn – Ni – Zn diffusion couples validates the potential application of the present atomic mobility database to predict the concentration profiles in higher order systems. An 8-elemental diffusion couple was also simulated with the present database. Keywords: Diffusion; Atomic mobility; Al alloys; DICTRA; Order/disorder transition

1. Introduction Diffusion is a fundamental property for an understanding of many important phenomena, such as homogenization of alloys, grain boundary migration, recrystallization, creep, solidification and precipitation [1, 2]. In order to simulate the diffusion-controlled transformations in a multicomponent system, the DICTRA (Diffusion Controlled TRAsformation) software has been developed [1] in the frame work of the CALPHAD (CALculation of PHAse Diagram) method. Based on a sharp interface and the local equilibrium hypothesis, DICTRA has been successfully used to simulate various diffusion processes [3 – 5]. The quality of such Int. J. Mater. Res. (formerly Z. Metallkd.) 104 (2013) 2

DICTRA-type simulations is critically dependent on the accuracy of both thermodynamic and atomic mobility databases. Nowadays, reliable thermodynamic databases for various multicomponent alloy systems are available [6 – 8]. In contrast, the atomic mobility database only exists for limited multicomponent alloys, e. g. steel [9] and fcc Nibased superalloys [10]. Thus, there is an urgent need to remedy this situation. Al alloys are widely used in aerospace, electrical, packaging, transport, building and architecture industries. In the production of commercial Al alloys, knowledge of diffusivity is essential for a quantitative description of microstructural evolution and controlling processing conditions for optimal engineering properties. The Al – Cu – Fe – Mg – Mn – Ni – Si – Zn system covers all the major elements in most commercial Al alloys, and the important disordered and ordered fcc phases in this multicomponent system were thus chosen as the target in the present work. Most recently, a reliable thermodynamic database of this multicomponent system has been established in our research group [8], which can provide reliable thermodynamic factors for the present atomic mobility assessment. In order to establish a high-quality atomic mobility database for fcc and L12 phases in such an 8-elemental system, critical assessment of the atomic mobilities for fcc and L12 phases in 28 binary systems is a prerequisite. For disordered fcc phase, atomic mobilities in 14 out of a total of 28 binary systems, i. e. Al – Cu [11, 12], Al – Mg [13], Al – Ni [14], Al – Zn [15], Cu – Fe [16], Cu – Mg [13], Cu – Mn [17], Cu – Ni [17], Cu – Zn [12, 18, 19], Fe – Mn [20], Fe – Ni [21], Fe – Si [22], Mn – Ni [17], Ni – Zn [15], have been assessed in our research group and other groups. While for ordered L12 phase, only the atomic mobility in the Al – Ni system has been assessed by Zhang et al. [14]. Thus, atomic mobilities in the remaining 14 binary fcc phases and 27 binary L12 phases need to be evaluated in the present work. In addition, the mobilities for self-diffusion in pure elements are the \building blocks" for the development of an atomic mobility database for multicomponent alloys, and thus need to be unified based on the results from different sources. For 135


www.ijmr.de



example, the mobility for self-diffusion in pure Ni based on the Fe – Ni system evaluated by Jönsson [21] has been recently updated by Zhang et al. [14] in the Al – Ni system by considering all the experimental diffusivities for pure Ni. Consequently, the main purposes of the present work are 1. to propose a set of mobility parameters for self-diffusion in pure Al, Cu, Fe, Mg, Mn, Ni, Si and Zn; 2. to establish the atomic mobilities of binary fcc and L12 phases in the Al – Cu – Fe – Mg – Mn – Ni – Si – Zn system by means of DICTRA software, and 3. to validate the present preliminary atomic mobility database in several binary, ternary and quaternary diffusion couples and to apply the established mobility database to simulate the concentration profiles in a fictitious diffusion couple involving all the 8 elements.

2. Models

As mentioned above, a reliable thermodynamic database for the Al – Cu – Fe – Mg – Mn – Ni – Si – Zn system has been established in our research group [8] via an integrated approach of key experiments, first-principles calculations and the CALPHAD approach, providing the reliable thermodynamic factors for the present evaluation of atomic mobilities. According to Du et al. [8], (A, B)1 and (A, B)0.75(A, B)0.25 are utilized to describe the disordered fcc and ordered L12 phases, respectively. In order to represent the Gibbs energies of the ordered-disordered transition using a single function, the equation derived by Ansara et al. [23], which allows the thermodynamic properties of the disordered phase to be evaluated independently, is employed in the present work. This is done by resolving the Gibbs energy into three terms. The molar Gibbs energy of fcc and L12 phases is given by the following equation, 0 00 L12 2 ð1Þ yi ; yi GL1 Gm ¼ Gfcc m ð xi Þ m ð xi Þ þ G m

where Gfcc m ðxi Þ is the molar Gibbs energy of the fcc phase, 00 0 2 ; y y GL1 i i the molar Gibbs energy of L12 phase as dem 2 scribed by the sub-lattice model, and GL1 m ðxi Þ the term which represents the energy contribution of the disordered state to the ordered phase. xi is the mole fraction of element 00 0 i (i = A or B). yi and yi are the site fractions of i in the different sub-lattices. The last two terms cancel each other when the site fractions are equal, thus corresponding to a disordered phase. The molar Gibbs energy for the disordered fcc phase (model (A, B)1) is expressed as: X X xi 0 Gfcc ðxi ln xi Þ þ ex Gfcc ð2Þ Gfcc m ¼ i þ RT ex

Gfcc ¼

i

XX i

j>i

xi xj

X k

k fcc Lij

xi x j

k

ð3Þ

where 0 Gfcc i is the reference energy of element i, R the gas constant (J mol–1 K–1), T the temperature in K, ex Gfcc the excess energy, and k Lfcc ij the interaction parameters. For the ordered L12 phase with the model (A, B)0.75(A, B)0.25, the mo136

i

j

þ RT 0:75

0

0

X

yi ln yi þ 0:25

0

2 yk LL1 i;j:k þ

0

þ

XX

þ

XXXX

i

i

j6¼i

j6¼i

X

00

yi ln yi

XX i

k

k

00

00

i

i

yi yj

X

2 yi yj yk yl LL1 i;j:k;l 0

0

00

00

j6¼i

00

00

yi yj

!

X

2 yk LL1 k:i;j 0

k

ð4Þ

l6¼k

2 where i, j, k, l = A or B, 0 GL1 i:j is the Gibbs energy of the hyL12 L12 2 pothetical compound i0.75j0.25, and LL1 i;j:k , Lk:i;j and Li;j:k;l the interaction parameters of the ordered L12 phase.

2.2. Diffusion models

2.1. Thermodynamic models

i

2 lar Gibbs energy GL1 m is given by the classical sublattice formalism via the following equation, X X 0 00 L1 2 yi yj 0 Gi:j 2 GL1 m ¼

2.2.1. Disordered fcc phase From the absolute-reaction rate theory arguments, the atomic mobility for an element i, Mi, may be divided into a frequency factor Mi0 and an activation enthalpy Qi [1, 21]. According to the suggestion by Andersson and Ågren [2] and Jönsson [21], Mi can be expressed as: RT ln Mi0 Qi 1 mg Mi ¼ exp exp X ð5Þ RT RT RT where R is the gas constant and T the absolute temperature. mg X 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 [24]. Thus for the disordered fcc phase, RT ln Mi0 and Qi can be merged into one parameter, i ¼ Qi þ RT ln Mi0 , which can be represented by the Redlich–Kister expansion as below: X A;B r B i ð xA xB Þ r ð6Þ i ¼ xA A i þ xB i þ x A x B r

are the model parameters to be where A , B and r A;B i evaluated from the experimental data. xA and xB are the molar fractions of A and B, respectively. Assuming a mono-vacancy atomic exchange as the main diffusion mechanism, the tracer diffusivity Di can be related to the atomic mobility Mi by the Einstein relation: Di ¼ RTMi

ð7Þ

In the volume-fixed frame of reference or number-fixed frame of reference, the interdiffusion coefficient Dnkj , which relates the flux of element k with the gradient of component j and reference component n, is given by Andersson and Ågren [2]: X qi qi n Dkj ¼ ðik xk Þ xi Mi ð8Þ qxj qxn i

where li is the chemical potential of element i and dik is the Kronecker delta, i. e., dik = 1 if i = k and dik = 0 otherwise.

Int. J. Mater. Res. (formerly Z. Metallkd.) 104 (2013) 2



www.ijmr.de


2.2.2. Ordered L12 phase For an ordered phase, the composition dependence of the atomic mobility should include the effect of chemical ordering. Based on Girifalco’s conclusion [25] that the increase in the activation energy due to chemical ordering depends quadratically on the long-range order parameter, Helander and Ågren [26] suggested incorporating the effect of chemical ordering by dividing the activation energy into two terms. The first term represents the contribution from the disordered state, Qdis k , which can be expressed by Eq. (6), and the other term represents the contribution from the ordered state, Qord k , ord Qk ¼ Qdis k þ Qk

where Qord k is defined as h i XX a b Qord Qord kij yi yj xi xj k ¼ i

ð9Þ

ð10Þ

i6¼j

where Qord kij is a parameter describing the contribution of the component k as a result of the chemical ordering of the i – j atoms on the two sublattices and yai is the site fraction of component i on the a sublattice, yai ¼

Nia a Ntotal

ð11Þ

in which Nia is the number of sites on the a sublattice that are a occupied by an i atom and Ntotal is the total number of sites on the a sublattice. Though this phenomenological model developed by Helander and Ågren [26] was for an AB (B2) alloy where diffusion occurs via jumps between two metal sublattices, Tôkei et al. [27] verified that it is also valid for A3B (L12) structure where diffusion occurs via a network of nearest neighbor jumps. Moreover, this model has been successfully applied to assess the atomic mobililities in ordered L12 phases of the Al – Ni [14] and Al – Cr – Ni [28] systems. An effective strategy, which takes the homogeneity range and defect concentration into account and greatly reduces the number of parameters to be optimized, was used in the present work and will be described later. 2.2.3. Metastable disordered fcc phase When the fcc state of some pure elements is metastable, it is impossible to assess the corresponding mobility parameters without experimental data. In this case, semi-empirical relations [29 – 31] or atomistic simulations [32, 33] are needed to evaluate the mobility parameters for self-diffusion in metastable fcc phase. Among those methods, the semi-empirical relations reviewed by Askill [34] have been applied successfully to calculate the self-diffusion coefficients of fcc-Zn [35] and fcc-Mg [36]. They were thus employed in the present work to calculate the self-diffusivity of fcc-Si. According to Askill [34], the activation energy Q and the frequency factor D0 for self-diffusion are expressed as, Q ¼ ðK0 þ 1:5 VÞRTm

ð12Þ

D0 ¼ 1:04 103 Qa2

ð13Þ


where K0 is a crystal structure factor: 15.5 for fcc structure; V is valence of the metal: 1.5 for Group IVB (Ti, Zr, Hf), 3.0 for Group VB (V, Nb, Ta), 2.8 for Group VIB (Cr, Mo, W), 2.6 for Group VIIB (Mn, Re) and 2.5 for other transition elements; Tm is melting point in Kelvin and a is lattice parameter in Å.

3. Critical evaluation of literature data on diffusivities Among the remaining 14 binary fcc phases to be evaluated in the present work, the experimental information is only found for the Al – Fe, Fe – Zn, Al – Si, Cu – Si and Ni – Si systems. For the remaining 9 systems, i. e. the Al – Mn, Fe – Mg, Mg – Mn, Mg – Ni, Mg – Si, Mg – Zn, Mn – Si, Mn – Zn and Si – Zn systems, either a very narrow homogeneity range for fcc phase region or metastable fcc phase exists. As for the ordered L12 phase, experimental information only exists in the Fe – Ni system among the remaining 27 systems. Therefore, the experimental data about these 6 systems, i. e. fcc Al – Fe, fcc Fe – Zn, fcc Al – Si, fcc Cu – Si, fcc Ni – Si and L12 Fe – Ni, are critically evaluated as follows. 3.1. The Al – Fe system Up to now, no systematic investigation has been performed for fcc phase in the Al – Fe system except for one assessment from Campbell et al. [37] in 2004. However, they took Fe Al Al and Fe directly from Engström and Ågren [38] and Jönsson [21], respectively. The mobility parameter of Al Fe was optimized on the basis of the impurity diffusion coefficients of Fe in fcc-Al from Hood [39], Tiwari and Sharma [40], and Beke et al. [41]. As for Fe Al , Campbell et al. [37] just employed Ni from Engström and Ågren [38] as an apAl proximation considering that there is not enough data available in the literature. In addition, Campbell et al. [37] also introduced an interaction term 0 Al;Fe Fe , the use of which is in question because there are no tracer or chemical diffusion coefficients for fcc phase in the Al – Fe system. Moreover, Zhang et al. [14] have updated Al Al very recently by taking into account almost all the experimental Al self-diffusivities. Du et al. [42] conducted a critical assessment of the impurity diffusivities of Fe in fcc-Al recently by means of the least-squares method and semi-empirical correlaAl Fe tions. Thus, Al Al , Fe and Fe are taken directly from Zhang et al. [14], Jönsson [21] and Du et al. [42], respectively, while Fe Al is assessed in the present work. For the impurity diffusion of Al in fcc-Fe, there exists limited experimental information due to lack of suitable radioactive tracer for Al. The two early contributions are from Grobner et al. [43] in 1955 and Akimova et al. [44] in 1983. The experimental data from both groups agree reasonably with each other. However, the data due to Akimova et al. [44] indicate that the influence of magnetic ordering on the diffusion of Al is negligible, which is not consistent with the current opinion. Thus, the reliability of these two pieces of experimental data is in question. By using the sandwich diffusion couples, Fe-1.5 at.% Al/Fe/Fe-5.1 at.% Al and Fe-1.5 at.% Al/Fe/Fe-5.1 at.% Al, coupled with electron probe micro-analysis (EPMA), Bergner and Khaddour [45] determined the temperature-dependent impurity diffusion coefficients of Al in fcc-Fe in 1993. Both Boltz137


www.ijmr.de



mann–Matano and Hall methods were utilized by them. By comparing with the results in paramagnetic and ferromagnetic bcc Fe, it can be concluded that the influence of magnetic ordering on the diffusion of Al cannot be negligible. The data from Bergner and Khaddour [45] show large discrepancy with those earlier data from Grobner et al. [43] and Akimova et al. [44]. Using lathe sectioning and instrumental analyses, Taguchi et al. [46] studied the impurity diffusion of Al in fcc-Fe from 1 273 to 1 396 K by means of absorptionmetry, and their results agree reasonably with those of Bergner and Khaddour [45]. Later on, Taguchi et al. [47] also measured the impurity diffusion coefficients of Al in fcc-Fe by using glow discharge spectroscopy (GDS) in the temperature range between 1 280 and 1 380 K. However, their obtained impurity diffusivities [47] are one order smaller than those from their previous work [46]. One of the probable reasons is that the oxygen exists near the surface according to Taguchi et al. [47], and the oxygen may cause the trap effect. Considering the fact that the data from Bergner and Khaddour [45] and Taguchi et al. [46] agree with each other, and show large differences with those questionable data of Grobner et al. [43], Akimova et al. [44], and Taguchi et al. [47], the data from Refs. [39, 40] are employed in the present assessment, while the others are not. 3.2. The Fe – Zn system The self-diffusivities of fcc-Fe and fcc-Zn are taken from Jönsson [21] and Cui et al. [35], respectively. There is no report about the impurity diffusion coefficient of Fe in fccZn or that of Zn in fcc-Fe. The only experimental information for fcc Fe – Zn system is from Budurov et al. [48] who determined the chemical diffusivity within the temperature range of 1 273 – 1 425 K by using vapor solid diffusion couples and EPMA. The only piece of experimental information was used to evaluate the atomic mobilities in the Fe – Zn system. 3.3. The Al – Si system The self-diffusivity of fcc-Al and the impurity diffusivity of Si in fcc-Al are directly taken from Zhang et al. [14] and Du et al. [42], respectively. The fcc-Si is a metastable phase in the Al – Si system. Considering lack of experimental data, Franke and Inden [22] assumed the self-diffusivity of fccSi to be equal to the impurity diffusion of Si in fcc-Fe, during the assessment of atomic mobility in the fcc Fe – Si system. The recent work on the Co – Si system by Zhang et al. [49] assumed self-diffusivity in fcc-Si to be 10 times of the self-diffusivity of Fe. Though the good agreement between the calculation and the experiment data in the above Fe – Si [22] and Co – Si [49] systems has been achieved, these kinds of simplification cannot be applied in a general atomic mobility database. Therefore, in order to obtain the mobility parameter for self-diffusion in metastable fcc-Si and keep consistent with the calculation of self-diffusivity in fcc-Zn [35] and fcc-Mg [36], the self-diffusivity of fccSi is also determined from the semi-empirical relations of Askill [34]. The calculation of various diffusivities in fcc Fe – Si system with the newly determined mobility for selfdiffusion in fcc-Si makes no difference with the result of Franke and Inden [22]. Thus, the mobility for self-diffusion 138

in fcc-Si is used to establish the atomic mobility database in the present work. There are several contributions to the measurements of interdiffusion coefficients in fcc Al – Si alloys. Freche [50] investigated the average rate of diffusion of silicon from the alloy core into the high-purity aluminum coating at 783 K from the quantitative spectrographic analysis. Using the bulk diffusion couple, Beerwald [51] determined the interdiffusion coefficients in Al-rich alloys up to 1.8 at.% Si within the temperature range of 738 – 873 K. Mehl et al. [52] measured the interdiffusion coefficients in Al-rich side of Al – Si alloys in the temperature range of 673 – 774 K using the diffusion couple method. By means of the diffusion couple method and microhardness tests, Bückle [53] measured the interdiffusion coefficients of Al-rich fcc alloys from 743 to 853 K. Bergner and Freiberg [54] investigated the interdiffusion coefficients in Al-0.5 at.% Si alloys from 618 to 904 K by using the bulk diffusion couple method. Fujikawa et al. [55] determined the interdiffusion coefficients in Al – Si alloys through the Matano method in the temperature range from 753 to 893 K with the couple consisting of pure aluminum and an Al – Si alloy. They also calculated the intrinsic diffusion coefficients in Al – Si alloy/Al couples at the marker position. Since the experimental diffusivities reported in the literature [50 – 55] agree well with each other, they are all utilized in the present assessment. 3.4. The Cu – Si system The self-diffusivity of fcc-Cu has been critically assessed by Ghosh [56], while that of metastable fcc-Si can be taken from the above Al – Si system. Since there is no a stable Sirich fcc phase in the Cu – Si system, only the impurity diffusion coefficient of Si in fcc-Cu, the tracer diffusion coefficient of Cu in fcc Cu – Si alloys and the interdiffusion coefficients were reported in the literature. The only existing experimental data for the impurity diffusion coefficient of Si in Cu is from Fogel’son et al. [57], who investigated the diffusion of Si in Cu by means of Xrays and obtained the average diffusion coefficients between 973 and 1 323 K. In addition, the impurity diffusion coefficients of Si in Cu could be estimated by extrapolating ~ to infinite dilution [58 – the interdiffusion coefficient (D) 60]. Since those extrapolated data are not original experimental ones, they are not directly utilized in the present mobility assessment, but employed to check the reliability of the presently obtained atomic mobilities. Rhines and Mehl [58] employed several bulk diffusion couples to determine the interdiffusion coefficients in the Cu-rich side at 0 – 5.1 wt.% Si and 973 – 1 075 K. By means of semi-infinite diffusion couples with the Kirkendall markers, Iijima et al. [59] measured the interdiffusion and intrinsic diffusion coefficients in Cu-rich Cu – Si alloys containing up to 8 at.% Si in the temperature range between 900 and 1 150 K. Furthermore, the tracer diffusion coefficient of 67 Cu in pure Cu and the alloys containing up to 1.8 at.% Si at 1 130 K was determined via the serial sectional method. Minamino et al. [60] investigated the interdiffusion in the Cu – Si solid solution of Cu – Si alloys in the temperature range of 998 – 1 173 K by using the bulk diffusion couple method. Aaronson et al. [61] used the moving interphase boundary method to investigate the interdiffusion coefficients in a-Cu – Si alloys from 928 to 1 048 K. All the exInt. J. Mater. Res. (formerly Z. Metallkd.) 104 (2013) 2


perimental data from Refs. [57 – 61] are used in the present work.


www.ijmr.de


3.5. The Ni – Si system Du and Schuster [62] have performed an assessment of atomic mobilities of Ni and Si in fcc Ni – Si alloys during the study of fcc Cr – Ni – Si alloys. However, some experimental diffusivities available in the literature [63 – 65] were not considered by them [62]. In addition, the self-diffusivity of Ni has been critically assessed and updated by Zhang et al. [14]. Hence, the atomic mobilities in fcc Ni – Si system need to be updated in the present work. Considering the fact that there is no stable Si-rich fcc phase, only the impurity diffusion coefficient of Si in fcc-Ni, the tracer diffusion coefficient of Ni in fcc Ni – Si alloys and the interdiffusion coefficients are considered in the present work. Swalin et al. [66] measured the diffusion rates of silicon in binary Ni – Si alloy containing less than 1 at.% Si as a function of temperature by using spectrophotometric methods. Allison et al. [67] investigated the volume and grain boundary diffusion rates of silicon, which is alloyed as a single additive in nickel between 1 073 and 1 243 K. In addition, the impurity diffusion coefficients of Si in Ni could be estimated by extrapolating the interdiffusion coefficient ~ to infinite dilution [63, 68, 69]. Since those extrapolated (D) data are not original experimental ones, they are not utilized in the present mobility assessment, but employed to check the reliability of presently obtained mobilities. Johnston [68] studied the diffusion of silicon in Ni-rich Ni – Si solid solutions in the temperature range of 1 323 – 1 523 K using binary diffusion couple experiments. Gülpen et al. [69] employed the bulk diffusion couple method to measure the interdiffusion coefficients in the Ni(Si) solid solultion between 1 093 and 1 423 K. Muralidharan et al. [63] investigated the interdiffusion coefficients in Ni – Si alloys at 1 373 K from 0 to 10 at.% Si. Rastogi and Ardell [70] used the bulk diffusion couple method to measure the interdiffusivities in Ni-rich fcc Ni – Si alloys from 873 to 1 048 K. Assassa and Guiraldenq [64] studied the activation energy and frequency energy for volume self-diffusion of 63Ni in 0 – 4 wt.% Si. Faupel et al. [65] measured the non-linear enhancement of solvent diffusion in Ni by alloying Si in the temperature range of 1 330 – 1 470 K, employing 59Co as a substitute tracer for a nickel isotope. The data from Assassa and Guiraldenq [64] show that the Arrhenius relationship remains the same regardless of the silicon content, which is inconsistent with other results. Moreover, the result from Faupel et al. [65] was not directly obtained from a nickel isotope. As a consequence, the data from Assassa and Guiraldenq [64] and Faupel et al. [65] are not used in the present optimization, while all the other data [63, 66 – 70] are utilized in the present work. 3.6. The Fe – Ni system The atomic mobilities of both Fe and Ni in fcc Fe – Ni alloys have been critically assessed by Jönsson [21]. However, as mentioned in Section 1, the self-diffusivity of pure Ni has been updated by Zhang et al. [14], and this change in atomic mobility of pure Ni does not affect other calculated results of Jönsson [21]. Hence, the self-diffusivity of pure Ni determined by Zhang et al. [14] and all the other atomic mobiliInt. J. Mater. Res. (formerly Z. Metallkd.) 104 (2013) 2

ties of fcc Fe – Ni system obtained by Jönsson [21] are used in the present work. For L12 phase in Fe – Ni system, there exist limited experimental diffusion coefficients. The longrange order (LRO) parameter evolution for L12-type Ni3Fe was used by Radchenko et al. [71, 72] to calculate the diffusivities for Permalloy. The LRO was described by means of the static concentration waves’ method and self-consistentfield approximation. This is the only experimental diffusion coefficient and thus employed in the present optimization.

4. Results and discussion The assessment of the atomic mobilities in the fcc and L12 phases was performed by means of the PARROT module of the DICTRA software package [1, 73]. The strategy used in the present work is as follows: For the fcc phases with some experimental tracer or interdiffusivities (e. g. the Fe – Zn, Cu – Si and Ni – Si systems), the interaction parameters are assessed. While for the system with a narrow homogeneity range for fcc phase region (i. e. the Al – Fe, Al – Si, Al – Mn, Fe – Mg, Mg – Mn, Mg – Ni, Mn – Si and Mn – Zn systems) or metastable fcc phase (i. e. Mg – Si, Mg – Zn and Si – Zn systems), the impurity diffusivity of A in B is assumed to be equal to the self-diffusivity of B in B. Here, A and B are the two elements of the binary A – B system. For L12 Fe – Ni system, the ordered parameters are evaluated on the basis of the experimental data. As for the systems with stable L12 phase but without experimental information, like Mn – Ni and Ni – Si, their ordered parameters are set to be equal to those of Fe – Ni and Al – Ni systems, respectively, considering the similarity of phase diagrams. While for the systems with no stable L12 phase, the ordered parameters are all set to be 0 for simplification. The finally obtained atomic mobilities for Al, Cu, Fe, Mg, Mn, Ni, Si and Zn in binary fcc and L12 phases of this 8-elemental system have been compiled into the commercial diffusivity database MOBAL2 for multi-component Al alloys [74]. 4.1. The self-diffusion in pure elements Among the eight elements in this Al – Cu – Fe – Mg – Mn – Ni – Si – Zn system, Al, Cu, Fe, Mn and Ni have stable fcc phases, while Mg, Si and Zn metastable ones. The self-diffusivities of fcc-Al, fcc-Cu, fcc-Fe, fcc-Mn and fcc-Ni have been assessed by Zhang et al. [14], Ghosh [54], Jönsson [21], Liu et al. [20] and Zhang et al. [14], respectively, and are thus directly used in the present work. The self-diffusivities of metastable fcc-Mg and fcc-Zn are calculated by Yao et al. [36] and Cui et al. [35], respectively, based on the semi-empirical relations reviewed by Askill [34]. The same method is also employed in the present work to calculate the self-diffusivity of fcc-Si. According to Eq. (12) and Eq. (13), the calculated activation enthalpy (Q) for self-diffusion in metastable fcc-Si is 155 635 J mol–1, and the frequency factor D0 = 5.855 10–5 m2 s–1. Based on the calculated values of Q and D0, we could estimate the selfdiffusivity of fcc-Si phase. 4.2. The Al – Fe system Considering the narrow homogeneity range and no tracer or chemical diffusion coefficient for fcc phase in the Al – Fe 139

system, no interaction terms were introduced in the present work. The comparison between the experimental impurity diffusion coefficients [39 – 41, 75 – 79] and the simulated ones from Du et al. [42] is presented in Fig. 1a. The assessed result from Campbell et al. [37] is also appended for comparison. 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. As can be clearly seen, the result from Du et al. [42] can describe nearly all the experimental data over a wide temperature range, while that from Campbell et al. [37] can not. Figure 1b shows the comparison between the presently calculated impurity diffusivities of Al in fcc-Fe and the experimental data [43 – 47]. The work from Campbell et al. [37] is


www.ijmr.de



also superimposed for a direct comparison. It can be seen from Fig. 1b that the present result can give an excellent fit to two reasonable sets of experimental data, while that from Campbell et al. [37] cannot describe any of them. 4.3. The Fe – Zn system The self-diffusivity of Zn obtained by Cui et al. [35] using the semi-empirical relation and the self-diffusivity of fccFe assessed by Jönsson [21] were directly used here. Other parameters were assessed in the present work. Figure 2 shows the calculated interdiffusivity compared with the experimental ones [48]. A good agreement between the calculated and the measured data is obtained.

Fig. 2. Comparison between the presently calculated interdiffusion coefficients of fcc Fe – Zn alloys and the experimental data [48]. A constant, M, is added in order to separate the data for different compositions in the figure.

Fig. 1. (a) Comparison between the experimental impurity diffusion coefficients of Fe in Al [39 – 41, 75 – 79] and the simulated ones of Du et al. [42]. (b) Comparison between the presently calculated impurity diffusivities of Al in Fe and the experimental data [43 – 47]. The result from Campbell et al. [37] is also appended for a comparison. Dashed lines refer to the diffusion coefficients with a factor of 2 or 0.5 from the model-predicted diffusion coefficients in this work.

140

Fig. 3. Comparison between the calcualted impurity diffusion coefficients of Si in pure Al from the CALPHAD method [42] and first-principles approach [80].





www.ijmr.de

Fig. 4. Comparison between the presently calculated interdiffusion coefficients (a) at x(Si) = 0.005 with the experimental data [54]. (b) at 673, 721, 810 and 853 K with the experimental data [51 – 53]. (c) at 740, 773, 783, 823 and 873 K with the experimental data [51 – 53, 55]. A constant, M, is added in order to separate the data for different temperatures in the figure. Dashed lines refer to the diffusion coefficients with a factor of 2 or 0.5 from the model-predicted diffusion coefficients in this work.


141



www.ijmr.de


4.4. The Al – Si system Since fcc-Si is a metastable state, the impurity diffusion of Al in fcc-Si cannot be obtained directly through experimental methods. When x(Si) is close to 1, it is assumed Si that DSi Al is identical to DSi . Thus the atomic mobility for Al in fcc-Si is assumed to be identical to the self-diffusion of fcc-Si. Figure 3 shows the comparison between the impurity diffusion coefficients of Si in pure Al through the CALPHAD method [42] and first-principles approach [80]. As can be seen from the figure, the results from the two methods are in good accordance. The comparisons between the calculated interdiffusion coefficients and the measured ones [51 – 55] are illustrated in Fig. 4, where an excellent fit can be found. Figure 5 presents the calculated and the experimental intrinsic diffusion coefficients of Al and Si in Al – Si alloy/Al couples at the marker position [55]. As the figure shows, the calculated results agree well with the experimental ones, and the intrinsic diffusion of Al is more dependent on concentration than that of Si.

present work can describe the extrapolated data [63, 68, 69] better and are a little smaller than the experimental data from Refs. [66, 67]. As mentioned in Section 3.5, the impurity diffusion of Si in Ni was performed in Ni – Si alloys containing up to 1 at.% Si by Swalin et al. [66], while the results from Allison and Samelson [67] contained a contribution of grain boundary diffusion, both of which may add to the measured impurity diffusivities of Si. Hence, it is considered to be reasonable that the present results are consistent with the extrapolated data [63, 68, 69]. Figure 11 provides the comparison between the calculated and the experimental interdiffusion coefficients [63, 68 – 70]. The results from Du and Schuster [62] are also

4.5. The Cu – Si system For the Cu – Si system, the atomic mobility for Cu in fcc-Si is assumed to be identical to that for self-diffusion in fcc-Si. The remaining parameters are assessed from the experimental data reviewed in Section 3.4. Figure 6 shows the comparison between the calculated impurity diffusion coefficients of Si in pure Cu and the experimental values [57 – 60]. The calculated interdiffusion coefficients are compared with the experimental ones at different temperatures [58 – 60] in Fig. 7. Obviously, the calculated values are in good accordance with the experimental ones. Figure 8 shows the comparison between the calculated and the experimental tracer diffusion coefficients of Cu in different fcc Cu – Si alloys at 1 130 K [59]. It can be concluded from the figure that the agreement between the calculated results and the experimental data is reasonable. The calculated diffusivities agree well with the measured data when the concentration of Si is very small. As the concentration of Si increases, the deviation between the calculation and the experimental data increases, but still falls in the maximal errors. The comparison between the model-predicted concentration profiles and the experimental ones [58] in three Cu – Si diffusion couples are given in Fig. 9. As can be seen from the figure, the model-predicted results and the experimental data are in good agreement with each other.

Fig. 5. Comparison between the calculated and experimental intrinsic diffusion coefficients of Al and Si at different fcc Al – Si alloys [55]. A constant, M, is added in order to separate the data for different elements in the figure. Dashed lines refer to the diffusion coefficients with a factor of 2 or 0.5 from the model-predicted diffusion coefficients in this work.

4.6. The Ni – Si system For the Ni – Si system, the atomic mobility for Ni in fcc-Si is assumed to be identical to the self-diffusion of fcc-Si. The remaining parameters are assessed from the experimental data reviewed in Section 3.5. Figure 10 shows the comparison between the impurity diffusion coefficients of Si in pure Ni and the experimental values [63, 66 – 69]. The results from Du and Schuster [62] are also included for a comparison. As can be seen, the 142

Fig. 6. Comparison between the calculated and experimental data of impurity diffusion coefficient of Si in Cu [57 – 60]. Dashed lines refer to the diffusion coefficients with a factor of 2 or 0.5 from the modelpredicted diffusion coefficients in this work.





www.ijmr.de

Fig. 7. Comparison between the presently calculated interdiffusion coefficients of fcc Cu – Si alloys (a) at 900, 973, 1 025, 1 074, 1 126 and 1 173 K with the experimental data [58 – 60]. (b) at 950, 1 000, 1 050, 1 100 and 1 150 K with the experimental data [59, 60]. A constant, M, is added in order to separate the data from different temperatures in the figure. Dashed lines refer to the diffusion coefficients with a factor of 2 or 0.5 from the model-predicted diffusion coefficients in this work.

superimposed in the figure. As the figure shows, the present work yields a slightly better fit to the experimental data in comparison with the previous assessment [62], for instance, close to pure Ni side at higher temperatures. In Fig. 12, the model-predicted concentration profiles in the Ni/Ni-9 at.% Si diffusion couple annealed at 1 523 K for 619 ks and the Ni/Ni-12.5 at.% Si diffusion couple annealed at 1 373 K for 1 296 ks are compared with the corresponding experimental information from Johnson [68] and Muralidharan et al. [63]. The results from Du and Schuster [62] are also included for comparison. As can be seen in Fig. 12a, the model-predicted concentration profiles in the Ni/Ni9 at.% Si diffusion couple due to the present mobilities show a slightly better agreement with the experimental data than the previous assessment [62]. As for the Ni/Ni-12.5 at.% Si diffusion couple, the present work and Du and Schuster [62] can give the same fit to the experimental data [63]. 4.7. The Fe – Ni system As mentioned in Section 3.6, the atomic mobility of pure Ni is taken from Zhang et al. [14], and all the other kinetic paInt. J. Mater. Res. (formerly Z. Metallkd.) 104 (2013) 2

rameters of fcc Fe – Ni system are taken from Jönsson [21]. According to Eqs. (9 – 11), four endmembers, Qord FeFeNi , ord ord , can be used to describe the and Q , Q Qord NiNiFe NiFeNi FeNiFe order contributions to the activation energy in the L12 phase. Since the diffusion in L12 phase occurs via jumps between two different sublattices, these contributions are not symmetric, as pointed out by Campbell [28]. Therefore, no ord similar simplification in B2 phase, like Qord AAB ¼ QABA , can be made for end members in L12 phase. However, an effective strategy, which takes the homogeneity range and defect concentration into account, was proposed by Zhang et al. [14] to choose the minimum but reasonable number of parameters during the assessment of atomic mobilities of L12 phase. Thus, this strategy is also utilized in the present Fe – Ni system. From a thermodynamic point of view, it is easy to know that yaNi ybFe yaFe ybNi for Ni3Fe since antisite defect concentrations are very small in the ordered state, as shown in Fig. 13 where the Fe – Ni phase diagram and the site fractions of Fe and Ni in the two sublattices of L12 phase at 673 K are presented. This means that the contributions for the Fe : Ni configuration could be negligible proord vided the corresponding parameters, Qord FeFeNi and QNiFeNi , 143

are not exceptionally large. Consequently, only Qord FeNiFe and were chosen to be optimized in the present assessQord NiNiFe ment. Figure 14 shows the comparison between the calculated and the experimental interdiffusion coefficients of L12 and fcc phases [71, 72, 81 – 83]. As shown in the figure, a good fit to the experimental data is obtained for both L12 and fcc phases.

5. Validation of the atomic mobility database In order to evaluate the reliability of the present mobility database, its application to predict the concentration profiles in several ternary and quaternary diffusion couples by means of the atomic mobility parameters in the binary systems is performed in the present work. In view of the exis-

Fig. 8. Comparison between the calculated and experimental tracer diffusion coefficients of Cu in different fcc Cu – Si alloys at 1 130 K [59]. Dashed lines refer to the diffusion coefficients with a factor of 2 or 0.5 from the model-predicted diffusion coefficients in this work.


www.ijmr.de



Fig. 9. Model-predicted concentration profiles of the Cu-10.7 at.% Si/ Cu, Cu-10.7 at.% Si/Cu, Cu-10.8 at.% Si/Cu diffusion couples annealed at 973, 1 023 and 1 075 K for 1 548, 590 and 2 036 ks, respectively, compared with the experimental data [58]. A constant, M, is added in order to separate the data for different diffusion temperatures.

144

tence of experimental data, the fcc Al – Ni – Si, Al – Mg – Zn and Cu – Mn – Ni – Zn systems were chosen. All the thermodynamic parameters were taken from Du et al. [8]. Figure 15a predicts the comparison between the model-predicted and the measured concentration profiles of the Ni12.5 at.% Al/Ni-12.5 at.% Si diffusion couple annealed at 1 373 K for 1 296 ks [63]. As can be seen from the figure, the model-predicted concentrations profiles based on the presently obtained binary atomic mobilities agree reasonably with the experimental data [63], except for a slight deviation for the concentration of Si and Ni, which might be improved by inclusion of ternary interactive parameters. In Fig. 15b, the experimental ternary diffusion paths of several fcc Al – Mg – Zn diffusion couples annealed at 755 K for 56.9 ks by Takahashi et al. [84] are compared with the presently predicted ones. It is obvious that the agreement is pretty good even without ternary interaction parameters.

Fig. 10. Comparison between the calculated and the measured impurity diffusion coefficient of Si in pure Ni [63, 66 – 69]. The result from Du and Schuster [62] is also included for a comparison. 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. 11. Comparison between the calculated and the experimental interdiffusion coefficients of fcc Ni – Si alloys [63, 68 – 70]. The result from Du and Schuster [62] is also included for comparison.


The comparison between the model-predicted and the measured concentration profiles of the Cu-10.6 at.% Ni8.7 at.% Zn-9.6 at.% Mn/Cu-22.3 at.% Ni-11.6 at.% Zn and Cu-21.3 at.% Ni-17.6 at.% Zn/Cu-30.4 at.% Ni9.9 at.% Mn quaternary diffusion couples annealed at 1 048 K for 172.8 ks [85] are shown in Fig. 15c and d, respectively. As can be seen in the figures, it is amazing that the calculated profiles due to the present atomic mobility database with only binary interaction parameters agree very well with the measured ones in quaternary diffusion couples. The generally good agreement between the calculations and the experimental data demonstrates the reliability of the present atomic mobilities and their good extrapolation to higher-order systems in the Al – Cu – Fe – Mg – Mn – Ni – Si – Zn system. One fictitious diffusion couple involving all the 8 components was designed for diffusion simulation based on

the thermodynamic database developed by Du et al. [8]. The nominal alloy compositions for the two sub-systems \Alloy1" and \Alloy2" are given in Table 1. Both alloys were set to be in the single fcc phase for simulation though Alloy2 would contain 98 % fcc and 2 % Al62Mn12Ni4 (mole-percent) under equilibrium condition at 793 K. This simulation illustrates some of the interesting diffusion behaviors that may occur in multicomponent systems. Figure 16 shows the predicted concentration profiles after annealing at 793 K for 36 ks. As the simulation illustrates, Zn is the fastest diffuser in the system, which has a diffusion distance of 0.4 mm after 36 ks at 793 K. Si, Mg, Ni and Cu are slower diffusers, while Fe and Mn the slowest diffusers. The diffusion distance of Mn is 0.014 mm at the same condition.

6. Conclusions A set of atomic mobilities for self-diffusion in pure Al, Cu, Fe, Mg, Mn, Ni, Si and Zn are proposed in the present work


www.ijmr.de



Fig. 12. (a) Model-predicted concentration profile of the Ni/Ni-9 at.% Si diffusion couple annealed at 1 523 K for 619 ks, compared with the experimental data [68]. (b) Model-predicted concentration profile of the Ni/Ni-12.5 at.% Si diffusion couple annealed at 1 373 K for 1 296 ks, compared with the experimental data [63]. The result from Du and Schuster [62] is also included for comparison.


Fig. 13. (a) The phase diagram of the Fe – Ni system. (b) The site fractions of Fe and Ni in both sublattices of L12 phase at 673 K. L12 phase is modeled as (Fe, Ni)0.75(Fe, Ni)0.25, and the underlined element represents the main element in the sublattice.

145

Fig. 14. Comparison between the calculated and experimental interdiffusion coefficients of L12 and fcc phases of the Fe – Ni system [71, 72, 81 – 83]. Dashed lines refer to the diffusion coefficients with a factor of 2 or 0.5 from the model-predicted diffusion coefficients in this work.

based on the critical assessment of different researchers. A reliable atomic mobility database for the binary disordered and ordered fcc phases in multicomponent Al – Cu – Fe – Mg – Mn – Ni – Si – Zn system is established by utilizing a phenomenological model and the DICTRA software. The comparison between various calculated and experimental diffusion coefficients shows that the present database can satisfactorily reproduce most of the experimental diffusion data in the literature. The good agreement between the model-predicted and the measured concentration profiles for a variety of diffusion couples, especially for the ternary and quaternary diffusion couples, further validates the present atomic mobility database. In addition, the concentration evolution in an eight-elemental diffusion couple between two sub-systems was predicted based on the present diffusion database and the simulated diffusion behavior demonstrated the importance of establishing a reliable diffusion database. The present work indicates that by only using the binary atomic mobilities, the diffusion-related phenomena in multicomponent Al alloys, such as the concentration profiles, can be reasonably described. It is expected that the presently proposed approach to establish the atomic mobility


www.ijmr.de



Fig. 15. (a) Model-predicted concentration profiles of the Ni-12.5Al/Ni-12.5Si (at.%) diffusion couple annealed at 1 373 K for 1 296 ks, compared with the experimental data [63]. (b) Model-predicted diffusion paths of several fcc Al – Mg – Zn diffusion couples annealed at 755 K for 56.9 ks, compared with the experimental data [84]. (c) Model-predicted concentration profiles of the Cu - 10.6 at.% Ni-8.7 at.% Zn-9.6 at.% Mn/Cu22.3 at.% Ni-11.6 at.% Zn diffusion couples annealed at 1 048 K for 172.8 ks, compared with the experimental data [85]. (d) Model-predicted concentration profiles of the Cu-21.3 at.% Ni-17.6 at.% Zn/Cu-30.4 at.% Ni-9.9 at.% Mn diffusion couples annealed at 1 048 K for 172.8 ks, compared with the experimental data [85].

146



Table 1. Composition of each element (mole fraction) in two sub-systems \Alloy1" and \Alloy2".

Al Cu Fe Mg Mn Ni Si Zn

Alloy1

Alloy2

0.9695 0.015 – 0.012 – – 0.0035 –

0.9493 – 0.0007 – 0.004 0.001 – 0.045


www.ijmr.de


database in Al alloys can be equally applied to other multicomponent alloys. Thus our proposed approach can reduce the huge amount of work associated with the establishment of diffusivity databases for other multicomponent alloys.

Fig. 16. Simulated concentration profiles resulting from the annealing at 793 K for 36 ks between diffusion couple Alloy1 and Alloy2. (a) Cu, Mg and Zn profiles. (b) Fe, Mn, Ni and Si profiles. Dashed lines represent the original concentration profiles at time 0 s.


The financial support from the Creative Research Group of National Science Foundation of China (Grant No. 51021063), the National Basic Research Program of China (Grant No. 2011CB610401), the Key Program of the National Natural Science Foundation of China (Grant No. 50831007) and Thermo-Calc Software AB under the Aluminum Alloy Database Project is acknowledged. Lijun Zhang would like to acknowledge the Alexander von Humboldt Foundation of Germany for supporting and sponsoring the research work at ICAMS, Ruhr-Universität Bochum, Germany. References [1] A. Borgenstam, A. Engström, L. Höglund, J. Ågren: J. Phase Equilib. 21 (2000) 269. DOI:10.1361/105497100770340057 [2] J.O. Andersson, J. Ågren: J. Appl. Phys. 72 (1992) 1350. DOI:10.1063/1.351745 [3] L.J. Zhang, Y. Du, I. Steinbach, Q. Chen, B.Y. Huang: Acta Mater. 58 (2010) 3664. DOI:10.1016/j.actamat.2010.03.002 [4] L. Höglund, J. Ågren: Acta Mater. 49 (2001) 1311. DOI:10.1016/S1359-6454(01)00054-4 [5] H. Strandlund, H. Larsson: Acta Mater. 52 (2004) 4695. DOI:10.1016/j.actamat.2004.06.039 [6] N. Dupin, B. Sundman: Scand. J. Metall. 30 (2001) 184. DOI:10.1034/j.1600-0692.2001.300309.x [7] S.L. Shang, H. Zhang, S. Ganeshan, Z.-K. Liu: Jom. 60 (2008) 45. DOI:10.1007/s11837-008-0165-1 [8] Y. Du, S.H. Liu, L.J. Zhang, H.H. Xu, D.D. Zhao, A.J. Wang, L.C. Zhou: CALPHAD 35 (2011) 427. DOI:10.1016/j.calphad.2011.06.007 [9] J.O. Andersson, T. Helander, L. Höglund, P. Shi, B. Sundman: CALPHAD 26 (2002) 273. DOI:10.1016/S0364-5916(02)00037-8 [10] C.E. Campbell, W.J. Boettinger, U.R. Kattner: Acta Mater. 50 (2002) 775. DOI:10.1016/S1359-6454(01)00383-4 [11] D.D. Liu, L.J. Zhang, Y. Du, H.H. Xu, S.H. Liu, L.B. Liu: CALPHAD 33 (2009) 761. DOI:10.1016/j.calphad.2009.10.004 [12] H. Chang, L. Huang, J.J. Yao, Y.W. Cui, J.S. Li, L. Zhou: CALPHAD 34 (2010) 68. DOI:10.1016/j.calphad.2009.12.002 [13] W.B. Zhang, Y. Du, D.D. Zhao, L.J. Zhang, H.H. Xu, S.H. Liu, Y.W. Li, S.Q. Liang: CALPHAD 34 (2010) 286. DOI:10.1016/j.calphad.2010.05.003 [14] L.J. Zhang, Y. Du, Q. Chen, I. Steinbach, B.Y. Huang: Int. J. Mater. Res. 101 (2010) 1461. DOI:10.3139/146.110428 [15] S.L. Cui, Y. Du, L.J. Zhang, Y.J. Liu, H.H. Xu: CALPHAD 34 (2010) 446. DOI:10.1016/j.calphad.2010.08.002 [16] Y.J. Liu, J. Wang, Y. Du, L.J. Zhang, D. Liang: CALPHAD 34 (2010) 253. DOI:10.1016/j.calphad.2010.04.002 [17] W.B. Zhang, Y. Du, L.J. Zhang, H.H. Xu, S.H. Liu, L. Chen: CALPHAD 35 (2011) 367. DOI:10.1016/j.calphad.2011.04.009 [18] S.L. Cui, L.J. Zhang, Y. Du, D.D. Zhao, H.H. Xu, W.B. Zhang, S.H. Liu: CALPHAD 35 (2011) 231. DOI:10.1016/j.calphad.2010.10.002 [19] E. Kozeschnik: Z. Metallkd. 91 (2000) 57. [20] Y.J. Liu, L.J. Zhang, Y. Du, D. Yu, D. Liang: CALPHAD 33 (2009) 614. DOI:10.1016/j.calphad.2009.07.002 [21] B. Jönsson: Scand. J. Metall. 23 (1994) 201. [22] P. Franke, G. Inden: Z. Metallkd. 88 (1997) 795. [23] I. Ansara, B. Sundman, P. Willemin: Acta Metall. 36 (1988) 977. [24] B. Jönsson. Z. Metallkd. 85 (1994) 498 – 501. [25] L.A. Girifalco: Phys. Chem. Solids 25 (1964) 323. DOI:10.1016/0022-3697(64)90111-8 [26] T. Helander, J. Ågren: Acta Mater. 47 (1999) 1141. DOI:10.1016/S1359-6454(99)00010-5 [27] Z. Tôkei, J. Bernardini, P. Gas, D.L. Beke: Acta Mater. 45 (1997) 541. DOI:10.1016/S1359-6454(96)00196-6 [28] C.E. Campbell: Acta Mater. 56 (2008) 4277. DOI:10.1016/j.actamat.2008.04.061 [29] S. Dushman, I. Langmuir: Proc. Am. Phys. Soc. (1922) 113. [30] C. Zener: J. Appl. Phys. 22 (1951) 372. DOI:10.1063/1.1699967 [31] R.A. Swalin: J. Appl. Phys. 27 (1956) 554. DOI:10.1063/1.1722421 [32] M. Mantina, Y. Wang, R. Arroyave, L.Q. Chen, Z.K. Liu, C. Wolverton: Phys. Rev. Lett. 100 (2008) 215901/215901. DOI:10.1103/PhysRevLett.100.215901

147


www.ijmr.de



[33] D.D. Zhao, Y. Kong, A.J. Wang, L.C. Zhou, S.L. Cui, X.M. Yuan, L.J. Zhang, Y. Du: J. Phase Equilib. Diffus. 32 (2011) 128. DOI:10.1007/s11669-011-9854-5 [34] J. Askill: Tracer Diffusion Data for Metals, Alloys, and Simple Oxides, New York, IFI, Plenum (1970). [35] Y.W. Cui, K. Oikawa, R. Kainuma, K. Ishida: J. Phase Equilib. Diffus. 27 (2006) 333. DOI:10.1361/154770306X116261 [36] J.J. Yao, Y.W. Cui, H.S. Liu, H.C. Kou, J.S. Li, L. Zhou: CALPHAD 32 (2008) 602. DOI:10.1016/j.calphad.2008.04.002 [37] C.E. Campbell, J.C. Zhao, M.F. Henry: J. Phase Equilib. Diffus. 25 (2004) 6. DOI:10.1361/10549710417966 [38] A. Engström, J. Ågren: Z. Metallkd. 87 (1996) 92. [39] G.M. Hood: Phil. Mag. 21 (1970) 305. DOI:10.1080/14786437008238419 [40] G.P. Tiwari, B.D. Sharma: Phil. Mag. 24 (1971) 739. DOI:10.1080/14786437108217047 [41] D.L. Beke, I. Godeny, I.A. Szabo, G. Erdelyi, F.J. Kedves: Philos. Mag. A 55 (1987) 425. DOI:10.1080/01418618708209907 [42] Y. Du, Y.A. Chang, B.Y. Huang, W.P. Gong, Z.P. Jin, H.H. Xu, Z.H. Yuan, Y. Liu, Y.H. He, F.Y. Xie: Mater. Sci. Eng. A 363 (2003) 140. DOI:10.1016/S0921-5093(03)00624-5 [43] P. Grobner: Hutn. Listy 10 (1955) 200. [44] I.A. Akimova, V.M. Mironov, A.V. Pokoev: Fiz. Met. Metalloved. 56 (1983) 1225. [45] D. Bergner, Y. Khaddour: Diffus. Defect Data, Pt. A 95 – 98 (1993) 709. DOI:10.4028/www.scientific.net/DDF.95-98.709 [46] O. Taguchi, M. Hagiwara, Y. Yamazaki, Y. Iijima: Diffus. Defect Data, Pt. A 194 – 199 (2001) 91. DOI:10.4028/www.scientific.net/DDF.194-199.91 [47] O. Taguchi, Y. Iijima, S. Suzuki, T. Nakamura, Y. Hirano, H. Kono: Diffus. Defect Data, Pt. A 237 – 240 (2005) 474. DOI:10.4028/www.scientific.net/DDF.237-240.474 [48] S. Budurov, P. Kovatchev, Z. Kamenova: Z. Metallk. 64 (1973) 652. [49] L.J. Zhang, Y. Du, Y. Ouyang, H.H. Xu, X.G. Lu, Y.J. Liu, Y. Kong, J. Wang: Acta Mater. 56 (2008) 3940. DOI:10.1016/j.actamat.2008.04.017 [50] H.R. Freche: Tech. Pub. 714 (1936) 325. [51] A. Beerwald, Z. Elektrochem: Angew. Phys. Chem. 45 (1939) 789. [52] R.F. Mehl, F.N. Rhines, K.A. von den Steinen: Met. Alloys 13 (1941) 41. [53] H. Bückle, Z. Elektrochem: Angew. Phys. Chem. 49 (1943) 238. [54] D. Bergner, E. Cyrener: Neue Huette 18 (1973) 356. [55] S. Fujikawa, K. Hirano, Y. Fukushima: Metall. Trans. A 9 (1978) 1811. DOI:10.1007/BF02663412 [56] G. Ghosh: Acta Mater. 49 (2001) 2609. DOI:10.1016/S1359-6454(01)00187-2 [57] R.L. Fogel’son, Y.A. Ugai, A.V. Pokoev, I.A. Akimova, V.D. Kretinin: Fiz. Metal. Metalloved. 35 (1973) 1307. [58] F.N. Rhines, R.F. Mehl: Trans. AIME 128 (1938) 185. [59] Y. Iijima, Y. Wakabayashi, T. Itoga, K. Hirano: Mater. Trans., JIM 32 (1991) 457. [60] Y. Minamino, T. Yamane, T. Kimura, T. Takahashi: J. Mater. Sci. Lett. 7 (1988) 365. DOI:10.1007/BF01730745 [61] H.I. Aaronson, H.A. Domian, A.D. Brailsford: Trans. Am. Inst. Min., Metall. Pet. Eng. 242 (1968) 738. [62] Y. Du, J.C. Schuster: Z. Metallkd. 92 (2001) 28. [63] G. Muralidharan, M.C. Petri, J.E. Epperson, H. Chen: Scr. Mater. 36 (1996) 219. DOI:10.1016/S1359-6462(96)00362-4 [64] W. Assassa, P. Guiraldenq: C.R. Acad. Sci., Ser. C 279 (1974) 59. [65] F. Faupel, C. Kostler, K. Bierbaum, T. Hehenkamp: J. Phys. F 18 (1988) 205. DOI:10.1088/0305-4608/18/2/005 [66] R.A. Swalin, A. Martin, R. Olson: J. Metals 9 (1957) 936. [67] H.W. Allison, H. Samelson: J. Appl. Phys. 30 (1959) 1419. DOI:10.1063/1.1735346 [68] G.R. Johnston: High Temp. – High Pressures 14 (1982) 695.

148

[69] J.H. Gülpen, A.A. Kodentsov, F.J.J. van Loo: Z. Metallkd. 86 (1995) 530. [70] P.K. Rastogi, A.J. Ardell: Acta Met. 19 (1971) 321. DOI:10.1016/0001-6160(71)90099-X [71] T.M. Radchenko, V.A. Tatarenko, S.M. Bokoch: Metallofiz. Noveishie Tekhnol. 28 (2006) 1699. [72] T.M. Radchenko, V.A. Tatarenko: Diffus. Defect Data, Pt. A 273 – 276 (2008) 525. DOI:10.4028/www.scientific.net/DDF.273-276.525 [73] B. Sundman, B. Jönsson, J.O. Andersson: CALPHAD 9 (1985) 153. DOI:10.1016/0364-5916(85)90021-5 [74] http://www.thermocalc.com/MOBDATA.htm, 2012. [75] W.B. Alexander, L. Slifkin: Phys. Rev. B 1 (1970) 3274. DOI:10.1103/PhysRevB.1.3274 [76] S. Mantl, W. Petry, K. Schroeder, G. Vogl: Phys. Rev. B 27 (1983) 5313. DOI:10.1103/PhysRevB.27.5313 [77] G. Rummel, T. Zumkley, M. Eggersmann, K. Freitag, H. Mehrer: Z. Metallkd. 86 (1995) 122. [78] K. Hirano, R.P. Agarwala, M. Cohen: Acta Metall. 10 (1962) 837. DOI:10.1016/0001-6160(62)90100-1 [79] K. Soerensen, G. Trumpy: Phys. Rev. B 7 (1973) 1791. DOI:10.1103/PhysRevB.7.1791 [80] M. Mantina, Y. Wang, L.Q. Chen, Z.K. Liu, C. Wolverton: Acta Mater. 57 (2009) 4102. DOI:10.1016/j.actamat.2009.05.006 [81] J.I. Goldstein, R.E. Hanneman, R.E. Ogilvie: Trans. Am. Inst. Min., Metall. Pet. Eng. 233 (1965) 812. [82] Y. Nakagawa, Y. Tanji, H. Morita, H. Hiroyoshi, H. Fujimori: J. Magn. Magn. Mater. 10 (1979) 145. DOI:10.1016/0304-8853(79)90166-5 [83] E.A. Balakir, Y.P. Zotov, V.B. Kosachev, A.A. Mukhametova, Y.S. Stark, A.S. Chavchanidze: Poverkhnost (1988) 112. [84] T. Takahashi, Y. Minamino, K. Hirao, T. Yamane: Mater. Trans., JIM 40 (1999) 997. [85] K.E. Kansky, M.A. Dayananda: Metall. Trans. A 16 (1985) 1123. DOI:10.1007/BF02811681

(Received February 10, 2012; accepted July 21, 2012; online since September 14, 2012) Bibliography DOI 10.3139/146.110846 Int. J. Mater. Res. (formerly Z. Metallkd.) 104 (2013) 2; page 135 – 148 # Carl Hanser Verlag GmbH & Co. KG ISSN 1862-5282 Correspondence address Professor Dr. Yong Du State Key Laboratory of Powder Metallurgy Central South University Changsha Hunan 410083 P.R. China Tel.: +86 731 88836 213 Fax: +86 731 88710 855 E-mail: [email protected]

You will find the article and additional material by entering the document number MK110846 on our website at www.ijmr.de


Development of an atomic mobility database for

Development of an atomic mobility database for

Suggest Documents

Development of an atomic mobility database for liquid

Development of an atomic mobility database for liquid

SOURCES OF DEVELOPMENT AND MOBILITY: AN EMPIRICAL

SOURCES OF DEVELOPMENT AND MOBILITY: AN

Development of an env gp41-Based Heteroduplex Mobility Assay for ...

ATOMIC PROCESSES IN PLASMA AND DATABASE OF ATOMIC ...

Development of Atomic Force Microscope for ...

The development of an image/threshold database for designing and ...

development of an online database and web application for ... - ESCMID

The development of an image/threshold database for designing and ...

Development of a Prosodic Database for an Argentine Spanish ... - DCC

The development of an online database for ... - Wiley Online Library

Development of an Improved Pesticide Properties Database for Risk ...

Development of Gesture Database for an Adaptive ... - CiteSeerX

Development of an object-orientated database for ...

mobility in cities database - UITP

mobility in cities database - UITP

An Application for Image Database Development for Mobile Face ...

Synthesis of Concurrent Systems for an Atomic Read/Atomic Write ...

Database Development

Development of the Atomic Theory

QUANTUM NUMBERS OF AN ATOMIC MODEL An atomic orbital is ...

An Efficient Database Design for IndoWordNet Development Using ...

An Efficient Database Design for IndoWordNet Development Using ...