disordered phase transition: application to the Al–Ni and Ni–Si systems

Materials Chemistry and Physics 135 (2012) 94e105

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A new approach to establish both stable and metastable phase equilibria for fcc ordered/disordered phase transition: application to the AleNi and NieSi systems Xiaoming Yuan a, Lijun Zhang b, Yong Du a, *, Wei Xiong c, Ying Tang a, Aijun Wang a, Shuhong Liu a a

State Key Laboratory of Powder Metallurgy, Central South University, Changsha, Hunan 410083, PR China Interdisciplinary Centre for Advanced Materials Simulation (ICAMS), RuhreUniversität Bochum, Stiepeler Str. 129, 44801 Bochum, Germany c Department of Materials Science and Engineering, KTH Royal Institute of Technology, SE 100-44 Stockholm, Sweden b

h i g h l i g h t s < We discuss the drawbacks of order/disorder modeling in the NieSi and AleNi systems. < We perform ab initio calculation of thermodynamic properties in the NieSi system. < A CALPHADetype approach is proposed to model the fcc ordered/disordered transition. < The NieSi system was thermodynamically assessed using the new approach.

a r t i c l e i n f o

a b s t r a c t

Article history: Received 7 September 2011 Received in revised form 12 March 2012 Accepted 14 April 2012

Both two-sublattice (2SL) and four-sublattice (4SL) models in the framework of the compound energy formalism can be used to describe the fcc ordered/disordered transitions. When transferring the parameters of 2SL disregarding the metastable ordered states into those of 4SL, inconsistence in either stable or metastable phase diagrams could appear, as detected in both AleNi and NieSi systems. To avoid such a kind of drawback, this behavior was analyzed and investigated in the NieSi and AleNi systems with the aid of firsteprinciple calculations. Furthermore, a new approach considering both the stable and metastable fcc ordered phase equilibria deduced from the firsteprinciples calculations was proposed to perform a reliable thermodynamic modeling for the fcc ordered/disordered transition. The NieSi system was then thermodynamically assessed using the presently proposed approach. The good agreement between the calculation and experiments demonstrates the reliability of the proposed approach. It is expected that the approach is valid for other systems showing complex ordered/disordered transitions. Ó 2012 Elsevier B.V. All rights reserved.

Keywords: Intermetallic compounds Ab initio calculations Computer modeling and simulation Phase equilibria

1. Introduction In the thermodynamic assessments of the fcc ordered phases (L12 and L10), the two-sublattice (2SL) model [1] based on the compound energy formalism (CEF) [2] has been generally accepted in the CALPHAD community due to its simplicity and computational efficiency [3e6]. Recently, the developed four-sublattice (4SL) model is becoming more preferable [7e10] in view of the following aspects. Firstly, the 4SL model, (A,B)0.25(A,B)0.25(A,B)0.25(A,B)0.25, which reflects the crystal structure of fcc lattice, and is more physically sound than the 2SL model. Secondly, 4SL model can describe all the ordered fcc states that only depends on the first nearest neighbors, such as L12 and L10. Thirdly, with the development of computational capability, the high requirement associated * Corresponding author. Tel.: þ86 731 88836213; fax: þ86 731 88710855. E-mail address: [email protected] (Y. Du). 0254-0584/$ e see front matter Ó 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.matchemphys.2012.04.028

with a software to compute the phase equilibria with 4SL model is not problematic now. Therefore, it is appropriate to apply the 4SL model to describe the fcc ordered/disordered transitions. For that purpose, as a first step it is meaningful to transfer the 2SL model into 4SL model because most of the available thermodynamic databases for multiecomponent alloys are modeled by using the 2SL model. Such a transformation was firstly performed by Ansara et al. [1] using the knowledge that the two models are mathematically equivalent. The complete transformation between the 2SL model and 4SL model is deduced in the present work, and presented in Appendix A. The thermodynamic modeling of fcc ordered/disordered transition usually utilizes the 2SL model and ignores the metastable ordered states. Nevertheless, it is welleknown that the ability to reproduce the stable phase diagram is not a sufficient condition to guarantee the reliability of the thermodynamic modeling. The metastable thermodynamic properties and the phase equilibria are

X. Yuan et al. / Materials Chemistry and Physics 135 (2012) 94e105

also important, as they would affect the extrapolation to the higher order systems. The unreliable thermodynamic properties of metastable ordered fcc phases may lead to the incorrect calculation of driving force and the extrapolated phase equilibria. This drawback may be found from the thermodynamic properties associated with the transformed 4SL model. As will be shown later, the metastable NiSieL10 phase turns out to be “stable” when the previous thermodynamic parameters in the form of 2SL model [4,11] are transformed to those of 4SL model. In addition to the inaccurate description of the fcc ordered/ disordered phase equilibria in the NieSi system, the thermodynamic properties of the liquid and the intermetallic compounds in this system are also not well established. The phase diagram calculated by Du et Al. [4] and Tokunaga et al. [11] shows an artificial inverted miscibility gap above 2600  C and 2200  C, respectively. The experimental heat contents [12,13] and heat capacity (CP) [14,15] on the compounds (Ni2Si_L, NiSi, NiSi2) in a wide temperature range were ignored in the previous modeling [4,11,16]. The main purposes of this study are twofold: (1) To demonstrate the shortcomings of modeling the fcc ordered/ disordered transitions disregarding the metastable ordered states via the aid of firsteprinciples calculations. (2) To develop a new approach to establish a thermodynamic description of both stable and metastable phase equilibria for ordered/disordered fcc phases, and to apply it to the thermodynamic assessment of the NieSi system using 4SL. This proposed approach is also applied to the AleNi system to further verify its reliability.

2. Drawbacks of modeling fcc ordered/disordered transition disregarding the metastable ordered states in the NieSi and AleNi systems In the NieSi system, the Ni3SieL12 phase is the only stable fcc ordered phase. Du et al. [4] and Tokunaga et al. [11] assessed this system using 2SL to describe the ordered/disordered transition. Fig. 1 presents the computed NieSi phase diagram using the parameters of 4SL model transformed from those of the 2SL model

Fig. 1. Comparison between the calculated NieSi phase diagrams using the 2SL model and the transformed 4SL model according to Du et al. [4].

95

[4], showing that the L10 phase replaces the intermetallic NiSi phase. Such a phase diagram is in contradiction with the experiments. The stabilization of L10 indicates that the previous modeling [4] is unreliable. This drawback was not found at that time [4], since the 2SL model cannot address the L10 phase. The transformed 4SL model reveals this artifact, as the predicted Gibbs energy (47250 J/ (mol$atoms)) of the L10 by means of 4SL model at 50 at.% Si and 298 K is lower than that (45,289 J/(mol$atoms)) of the stable NiSi phase. This artifact is also observed in the thermodynamic modeling of Tokunaga et al. [11]. The AleNi system depicts another type of drawback. The thermodynamic modeling of the AleNi system was performed by Ansara et al. [1] and the results have been accepted subsequently [3,9]. The parameters were transformed to those of 4SL model in order to assess the AleNiePt system [9] where fcc, L12 and L10 states are stable. The calculated AleNi phase diagram using 2SL and 4SL is identical to each other. However, this does not guarantee that the thermodynamic description of the fcc ordered phase is accurate. In addition to the stable phase diagram, the thermodynamic properties and phase equilibria in the metastable states are also important to check the reliability of thermodynamic modeling. The CALPHADepredicted enthalpies of formation (DHf) for the AlNieL10 and Al3NieL12 are 52.9 and 35.4 kJ/mol$atoms, respectively. Unfortunately, no evidence, either experiment or atomistic simulation, is available to support its correctness. Figs. 2 and 3 depict the calculated metastable phase equilibria including only fccebased phases (fcc, L12 and L10) and liquid phase in the NieSi [4] and AleNi [1] systems, respectively. As shown in Figs. 2 and 3, all the fcc ordered states are “stable” compared with the parent fcc phase, which is doubtful. As will be shown in Section 4, the firsteprinciples calculations demonstrate that this is not the case. According to the above discussions, the modeling of NieSi [4,11] and AleNi [1] systems considering only the stable L12 phase with 2SL model would yield two kinds of drawbacks when the parameters in the form of 2SL model are transformed to those of 4SL model. The inaccurate description of the metastable fcc ordered states even cannot reproduce the stable phase diagram, such as the NieSi system. In the other case, the calculated stable phase diagram is correct, but the predicted thermodynamic properties of the metastable ordered phases are inaccurate, as demonstrated in the

Fig. 2. Calculated NieSi metastable phase diagram including only fccebased phases (fcc, L12 and L10) and liquid phase. The parameters are from Du et al. [4].

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X. Yuan et al. / Materials Chemistry and Physics 135 (2012) 94e105

fccebased SQS-16 [29] is used to mimic the disordered phase. The lattice vectors, atomic positions and correlation function data of the obtained SQS-16 are given in Ref. [25]. As an example, Fig. 4 illustrates one SQS-16 structure with x ¼ 0.5 in its unrelaxed form. It is worth mentioning that the full relaxations are performed in the present firsteprinciples calculations of the NieSi system and the symmetry of the structure is preserved [27].

3.2. Enthalpy of formation at 0 K

Fig. 3. Calculated AleNi metastable phase diagram including only fccebased phases (fcc, L12 and L10) and liquid phase. The parameters are from Ansara et al. [1].

AleNi system. In order to avoid the above drawbacks, it is preferable to model the fcc ordered/disordered transitions with 4SL model and take the information on metastable order phases into consideration. Generally speaking, measuring the metastable phase equilibria is very difficult or even impossible. Therefore, firsteprinciples method could be utilized to provide these metastable thermodynamic properties, which are needed in the CALPHAD modeling.

The Vienna ab initio simulation package (VASP) [30e33] was employed to calculate the DHf of the fcc and its order states at 0 K to check these quantities from the previous assessments [4,11,34]. The calculation was performed within the framework of Density Function Theory (DFT) using the generalized gradient approximation (GGA) of Perdew and Wang [35,36]. The groundestate energy of the compound was calculated with projector augmented wave (PAW) pseudopotentials [30]. The kepoint meshes for Brillouin zone sampling are constructed using the MonkhorstePack scheme [31] and at least 8000 points per reciprocal atom are used. The energy cutoff for plane wave is 400 eV. Total energy calculations are performed by means of the tetrahedron method incorporation with Blöchl corrections [37]. The energy convergence criterion for electronic selfeconsistency is 107 eV per atom. For the fcc alloys, the SQS is fully relaxed with respect to both the volume and shape of the unit cell as well as all the atomic positions. The DHf of fccebased phases in the NieSi and AleNi systems at 0 K is determined by the following equation: F fcc diamond DHfF ¼ ENi  x$EAl  ð1  xÞ$ESi x Si1x

(1a)

F fcc fcc DHfF ¼ EAl  x$EAl  ð1  xÞ$ENi x Ni1x

(1b)

3. Computational methods Firsteprinciples calculation has been demonstrated to be one powerful method to assist CALPHAD modeling [11,17e20]. The important quantity from firsteprinciples calculation which can be incorporated into the CALPHAD modeling is DHf of stoichiometric compounds [17] or the endemembers of intermetallic compounds with certain homogeneity range [21] at 0 K. These calculations reduce the number of parameters to be assessed in the optimization procedure. Thus, the DHf of the fccebased phases in the NieSi and AleNi systems was calculated via firsteprinciples method to check the reliability of previous modeling as well as to assist the present CALPHAD modeling. DHf of fcc disordered solid solution can be computed with the aid of Special Quasirandom Structures (SQS). Subsequently, thermodynamic modeling of the NieSi system based on the compound energy formalism [2] was carried out using the CALPHAD method.

where x (x ¼ 0.25, 0.5 or 0.75) is the composition of the correfcc , Efcc and Ediamond sponding F phase (F ¼ L12, L10 or fcc). EAFx Bð1xÞ , ENi Si Al are the firsteprinciples calculated total energies of the corresponding solid phase and pure elements in their reference states. Moreover, DHf of all the compounds and the endemembers in the NieSi system are calculated with a desire to clarify the experiment and to supply new data to assist modeling using Eq. (1a).

3.3. Thermodynamic models The aseapplied thermodynamic models in this work are the same as those in Du et al. [4] except for those of L12, Ni2Si_L, Ni2Si_H, NiSi and NiSi2 phases.

3.1. Special Quasirandom Structures (SQS) It has been demonstrated that SQS method can be applied to calculate thermodynamic properties of binary solid solutions, such as enthalpy of mixing (DHmix) for fcc structure [22,23]. The SQS proposed by Zunger et al. [24] is a specially designed smalleunitecell with a periodicallyerepeated structure, in which there are only limited atoms of 2e16 in a unit cell in order to closely mimic the most relevant, neareneighbor pair and multisite correlation functions of the random substitutional solid solutions [25]. Due to the one-to-one correspondence between a set of correlation functions and a given structure [26], the SQS can be constructed by comparing the correlation functions of the ordered and the disordered (random) structures by means of ATAT code [27]. More detail about the description of correlation functions and the generation of SQS can be found elsewhere [25,26,28]. In the present work, the

Fig. 4. Crystal structure of the SQS-16 with composition at x ¼ 0.5 in its ideal and unrelaxed form. Large and small spheres represent A and B atoms, respectively.

X. Yuan et al. / Materials Chemistry and Physics 135 (2012) 94e105

To describe both stable and metastable fcc order phases, the L12 phase is described using the ordered/disordered model [1] with 4SL model. Thermodynamic properties, heat contents or heat capacity (CP), for Ni2Si_L, NiSi and NiSi2 are known in a wide temperature range. Thus it is preferable to describe the Gibbs energies of these phases relative to the stable element reference (SER) at 298.15 K and 1 bar. The Gibbs energy of NixSi(1ex) per moleeformula can then be expressed as: Ni Sið1xÞ

Gm x

SER SER  x$HNi  ð1  xÞ$HSi

(2)

¼ a þ b$T þ c$T$ln ðTÞ þ d$T 2 þ e$T 3 þ f $T 1

Ni2Si_H is the high temperature form of Ni2Si_L and shows an observable homogeneity range. Regarding to its crystal structure [38], Ni2Si_H is described as Ni1(Ni,Va)1Si1, where Va represents vacancy. Its Gibbs energy is defined with the following expression: 0 Ni2 Si H Gm


  2 Si H 2 Si H ¼ y00Ni $0 GNi þ y00Va $0 GNi þ R$T$ y00Ni $ln y00Ni Ni:Ni:Si Ni:Va:Si   þ y00Si $ln y00Si þ y00Ni $y00Si $0 LNi:Ni;Va:Si (3a)

Ni2 Si H Ni2 Si H where 0 GNi:Ni:Si and 0 GNi:Va:Si are the Gibbs energies of the corresponding endemembers. The CP of the endemember, (Ni)1(Ni)1(Si)1, of Ni2Si_H is assumed to be the same as that of Ni2 Si H Ni2 Si H and 0 GNi:Va:Si are described as: Ni2Si_L. Therefore, 0 GNi:Ni:Si

0 Ni2 Si H GNi:Ni:Si

2 Si H ¼ A þ B$T þ 0 GNi Ni:Ni:Si

(3b)

0 Ni2 Si H GNi:Va:Si

0 SER ¼ A0 þ B0 $T þ 0 GSER Ni þ GSi

(3c)

where A, B, A0, B0, and a,b,c$$$ in Eqs. (2)e(3) are parameters to be evaluated according to the experimental phase equilibria and thermodynamic properties. 4. Results and discussion 4.1. Firsteprinciples calculations Tables 1 and 2 show the firsteprinciples calculated DHf for the fccebased phases in the NieSi and AleNi systems along with the data in the literature [1,4,18,39e41]. The presently calculated DHf for all the fcc ordered states agree with the previous firsteprinciples calculations and experiments [18,39e41], indicating that the present firsteprinciples calculations are reliable. There is a noticeable contradiction between the firsteprinciples calculation and CALPHAD modeling for the metastable ordered fcc phases in both NieSi and AleNi systems. This difference is as large as 10 kJ/mol$atoms, indicating that the previous modeling of fcc ordered/disordered transition in both NieSi and the AleNi systems needs to be modified. Figs. 2 and 3 present the calculated NieSi and AleNi metastable phase diagrams including only fccebased phases (fcc, L12 and L10) and liquid phase based on the previous thermodynamic parameters [1,4]. However, the comparison of the presently firsteprinciples calculated thermodynamic properties between the fcc order states and its corresponding fcc disordered solid phase indicates that Figs. 2 and 3 should be wrong. The calculated DHf of NiSi-L10 (28,028 J/mol$atoms) and NiSi3eL12 (22,881 J/mol$atoms) at 0 K is much more positive than their corresponding fcc disordered structure, indicating that the fcc lattice is more stable in regard to its order states. As temperature increases, the fcc lattice will be still more stable than the ordered states, since high temperature favors the disorder structure due to the fast increasing entropy. The firsteprinciples calculations indicate that the NiSieL10 and

97

Table 1 Computed enthalpy of formation for the fccebased phases (fcc, L12 and L10) in the NieSi system compared with the literature data. Structure

Composition

DHf, J/(mol$atoms)

Methoda

Reference

(Ni)

Ni0.75Si0.25 b Ni0.5Si0.5 b Ni0.25Si0.75 b Ni0.75Si0.25 Ni0.75Si0.25 Ni0.75Si0.25 Ni0.765Si0.235 Ni0.25Si0.75 Ni0.25Si0.75 Ni0.25Si0.75 Ni0.5Si0.5 Ni0.5Si0.5 Ni0.5Si0.5

27,841 32,385 5940 44,671 44,672 40,582 37,700 22,881 22,867 8610 28028 27,981 40,169

FP FP FP FP FP CALPHAD Experiment FP FP CALPHAD FP FP CALPHAD

This This This This [39] [4] [40] This [39] [4] This [39] [4]

Ni3SieL12

NiSi3eL12

NiSieL10

work work work work

work

work

a

FP ¼ Firsteprinciples calculation. The most negative enthalpy of formation calculated by Firsteprinciples calculation among (Ni), L12 and L10. b

NiSi3eL12 in Fig. 2 should be replaced by disordered fcc phase or other phase, and the Al3NieL12 in Fig. 3 is also metastable compared with the disordered fcc structure. As demonstrated above, the present firsteprinciples calculations detect that the thermodynamic modeling of the ordered/ disordered transition in the NieSi and AleNi systems needs refinement. This kind of drawback is not only limited to the above two systems, but also in other systems with fcc ordered/disordered phase transition. Consequently, a new approach is proposed in the present work to overcome this drawback. The strategy is as follows. First, the DHf of all the fccebased phases are computed using firsteprinciples method. Second, a metastable phase diagram is established via comparing the phase stability between the order phases and their parent disordered phase. Thirdly, the phase stability of fcc ordered and disordered phases is incorporated into the CALPHAD modeling by taking both stable and metastable phase equilibria into consideration. 4.2. Thermodynamic modeling of the NieSi system considering both stable and metastable phase equilibria To validate the presently proposed approach, it is applied to the thermodynamic modeling of the NieSi system. The thermodynamic properties and phase equilibria data for the NieSi system have been critically reviewed by Nash and Nash [42] and Acker and Bohmhammel [43], and this system has been thermodynamically assessed by several groups of authors [4,11,16,34,44]. Since their publications, several research groups [14,15,45e48] have reported Table 2 Computed enthalpy of formation for the fccebased phases (fcc, L12 and L10) in the AleNi system compared with the literature data. Structure

Composition

DHf, J/(mol$atoms)

Methoda

Reference

fcc

b

23,660 45,565 30,607 21,342 35,371 42,209 41,100 40,882 41,300 64,430 52,891

FP FP FP FP CALPHAD FP FP CALPHAD Experiment FP CALPHAD

This This This This [1] This [18] [1] [41] This [1]

Al3NieL12 AlNi3eL12

AlNieL10 a

Al0.75Ni0.25 Al0.5Ni0.5 Al0.25Ni0.75 Al0.75Ni0.25 Al0.75Ni0.25 b Al0.25Ni0.75 Al0.25Ni0.75 Al0.25Ni0.75 Al0.25Ni0.75 b Al0.5Ni0.5 Al0.5Ni0.5

work work work work work

work

FP ¼ Firsteprinciples calculation. The most negative enthalpy of formation calculated by Firsteprinciples calculation among (Ni), L12 and L10. b

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X. Yuan et al. / Materials Chemistry and Physics 135 (2012) 94e105

new experimental data about this system. Acker et al. [14] and Perring et al. [15] measured the CP of NiSi in a wide temperature range. Jung and Yoo [48] determined DHf (37.9 kJ/mol$atoms) of NiSi using Kleppa calorimeter. Witusiewicz et al. [45] and Sudavtsova et al. [46] measured the DHmix in liquid at 1302  C and 1517  C, respectively. In addition to the thermodynamic properties, Oikawa et al. [47] measured the phase equilibria in the Nierich portion and their results agree with the previous measurements [49,50]. 4.2.1. Optimization procedure The optimization was carried out by using the PARROT module in the ThermoCalc software [51], which works by minimizing the square sum of the differences between the measured and calculated values. During the first stage of the optimization, only the thermodynamic data (DHmix of the liquid and fcc phase, DHf and CP of the intermetallics) were considered. DHf was related to A, A0 and a in Eqs. (2)e(3); while CP was only associated with parameters c, d and e in Eq. (2). Next, the other parameters in Eqs. (2)e(3) were optimized using all the invariant reaction data with a desire to describe the general feature of the phase diagram. Thirdly, the solubility, liquidus and solidus data were utilized to refine the preliminary optimization. Fourthly, the metastable and stable information related to L12 phase was considered. Parameters, 4SL G4SL Ni2Si2 and GNiSi3 , were estimated to fit the computed DHf of metastable NiSi-L10 and NiSi3-L12 endemembers and reproduce the deduced metastable ordered/disordered phase equilibria. The relationship between the DHf of the endemembers (Ni3Si-L12, Ni2Si2-L10 and NiSi3-L12) can be described using the following equations:

DHf ðNi3 Si  L12 Þ ¼ 0:75$0:25$ð0 L þ ð0:75  0:5Þ$1 L þ ,,,Þ 4SL þ 0:578$G4SL Ni3Si  0:2109$GNi2Si2 0 4SL  0:0469$G4SL NiSi3  0:75$ LNi;Si:*:*:* 2 4SL  0:375$1 L4SL Ni;Si:*:*:*  0:1875$ LNi;Si:*:*:*

 0:2109$* L4SL Ni;Si:*:*:*

(4)

4SL DHf ðNi2Si2L10 Þ¼0:5$0:5$0 L0:25$G4SL Ni3Si þ0:625$GNi2Si2 0 4SL 0:25$G4SL NiSi3  LNi;Si:*:*:*

0:375$* L4SL Ni;Si:Ni;Si:*:*

DHf ðNiSi3  L12 Þ ¼ 0:75$0:25$ð0 L þ ð0:25  0:75Þ$1 L þ ,,,Þ 4SL þ 0:0469$G4SL Ni3Si  0:2109$GNi2Si2 0 4SL þ 0:578$G4SL NiSi3  0:75$ LNi;Si:*:*:* 2 4SL þ 0:375$1 L4SL Ni;Si:*:*:*  0:1875$ LNi;Si:*:*:*

 0:2109$* L4SL Ni;Si:Ni;Si:*:*

4.2.2. Stable phase equilibria and thermodynamic properties The calculated stable NieSi phase diagram along with the selected experimental data [4,47,49,50,52,53] is presented in Fig. 5, and the computed invariant temperatures in comparison with the literature data [4,49,50,53] are summarized in Table 4. In general, the agreement between the calculated and experimental value is good. The drawbacks in the previous modeling [4,11] are avoided in the present work. Firstly, the artificial inverted miscibility gap in the previous assessments [4,11] is removed. Moreover the phase equilibria in the Nierich side, including the ordered/disordered transformation and the homogeneity range of (Ni), are satisfactorily described for the first time. In comparison, the phase equilibria in the Nierich side were not well reproduced in the previous work [4,11,34]. Finally the modeling of ordered/disordered transition is significantly improved. The stabilization of L10 phase (in Fig. 1) is avoided.

Phase

Model

Parameters

Liquid

(Ni,Si)1

0 Lliq Ni;Si 2 Lliq Ni;Si

(Ni,Si)1

L12

4SLa

0 Lfcc Ni;Si 1 Lfcc Ni;Si 2 GL1 Ni3Si

¼ 215591:5 þ 28:19$T

Ni3/4Si1/4

Ni3Si_H

Ni3/4Si1/4

Ni5Si2

Ni5/7Si2/7 Ni2/3Si1/3 Ni1/3(Ni,Va)1/3Si1/3

Ni3Si2

Ni3/5Si2/5

NiSi

Ni1/2Si1/2

0 GNiSi m

NiSi2

Ni1/3Si2/3

0 GNiSi2 m

4SL model: (Ni,Si)0.25(Ni,Si)0.25(Ni,Si)0.25(Ni,Si)0.25. In J/mole$atoms.

¼ 39423:8

¼ 198751:1 þ 12:25$T ¼ 48824:7

0 Tcfcc Ni;Si

¼ 35712 þ 11:07$T

¼ 3872 2 GL1 ¼ 7649:4 Ni2Si2

¼ 8886:4 þ 3:6585$T

* LL12 Ni;Si:Ni;Si:*:*

2 GL1 ¼ 37790:1 NiSi3

¼ 25458:8

3 1 0 Dia  $0 Gfcc Ni  4 GSi ¼ 39893:7  1:689$T 4 0 GNi3 Si H  3$0 Gfcc  1 0 GDia ¼ 39803:65  1:75325$T m Ni 4 4 Si 0 GNi5 Si2  5$0 Gfcc  2$0 GDia ¼ 46235:03 m Ni Si 7 7 0 GNi2 Si L ¼ 52022:33 þ 109:77$T  20:54$T$ln ðTÞ  0:004525$T 2 þ 42250=T m

Ni2Si_L

b

¼ 88746:1 þ 2:617$T 3 Lliq Ni;Si

0 GNi3 Si L m

Ni2Si_H

a

1 Lliq Ni;Si

¼ 72267:9  39:6$T

1 LL12 Ni;Si:*:*:*

Ni3Si_L

(6)

The invariant reaction and solubility about Ni3Si-L12 were used to adjust the other parameters in the L12 phase. In this work, we * 4SL found that 1 L4SL Ni;Si:*:*:* and LNi;Si:Ni;Si:*:* were necessary to fit the information well. Finally, all the parameters were optimized by using all the experimental data and firsteprinciples derived metastable phase equilibria. Table 3 lists the thermodynamic parameters finally obtained in the present work.

Table 3 Summary of the thermodynamic parameters for the NieSi system.b

(Ni)

(5)

0 GNi2 Si H Ni:Ni:Si

0 GNi3 Si2 m

¼ 50211:53 þ 108:65$T  20:54$T$lnðTÞ  0:004525$T 2 þ 42250=T 0 GNi2 Si H  1 0 Gfcc  1 0 GDia ¼ 36167:7 Ni:Va:Si 2 Ni 2 Si 3 2 0 Dia $  $0 Gfcc  G ¼ 46915:82 Ni Si 5 5

¼ 51655:1 þ 133:315$T  23:3635$T$lnðTÞ  0:0019857$T 2 þ 106697:5=T ¼ 40428:53 þ 148:867$T  25:026$T$lnðTÞ  0:001843$T 2 þ 180550=T

X. Yuan et al. / Materials Chemistry and Physics 135 (2012) 94e105

99

Table 4 Measured and calculated temperatures ( C) of the invariant reactions in the NieSi system. Reaction

Experimental data [53]

Liquid 4 (Ni) þ Ni3Si_H Ni3Si_H 4 Ni3Si_L (Ni) þ Ni3Si_L 4 L12 Liquid þ Ni5Si2 4 Ni3Si_H Ni3Si_L 4 L12 þ Ni5Si2 Liquid 4 Ni5Si2 Liquid 4 Ni5Si2þNi2Si_L Liquid þ Ni2Si_H 4 Ni2Si_L Liquid 4 Ni2Si_H Liquid 4 Ni2Si_H þ NiSi Ni2Si_H þ NiSi 4 Ni3Si2 Ni2Si_H 4 Ni3Si2 þ Ni2Si_L Liquid 4 NiSi Liquid 4 NiSi þ NiSi2 Liquid þ Diamond 4 NiSi2

1151 1119 1040 1163 e e 1242 1242 1285 964 845 806 992 966 993

[49] 1142 e z1035 1170 990 1242 1215 e e e e e e e e

Calculated

[50]

[4]

This work

1158 1125 z1010 1189 1001 1256 1231 e e e e e e e e

1157 1127 z1047 1191 1006 1254 1232 1251 1283 958 860 821 979 953 968

1152 1127 1041.3 1191.8 1005.2 1246 1243.3 1251.3 1280 972.2 860.0 822.3 985.3 946.4 998.3

Fig. 7. Calculated activity of Si in liquid phase at 1580  C along with experimental data. The reference is liquid Si.

Fig. 5. Calculated NieSi phase diagram in this work, compared with the experimental data.

Fig. 6. Calculated enthalpy of mixing in liquid at 1600  C along with the experimental data. The reference states are liquid Ni and liquid Si.

The measured DHmix in liquid [45,46,54e57] are scattering and do not present a clear trend with temperature. As a result, DHmix in liquid is treated as temperature independent in this work. Fig. 6 compares the calculated and measured DHmix in liquid at 1600  C. The fit to the experimental data [54e57] is fairly good. Fig. 7 shows that the predicted activity of Si in liquid is in a reasonable agreement with the reported values [58,59], indicating the rationality of the present modeling. In addition, there is a noticeable improvement for the thermodynamic properties of the compounds. NeumanneKopp rule was applied to all the compounds in the previous modeling [4,11,16,34], while Ni2Si_L, NiSi and NiSi2 are referred to the SER state in this work. The calculated heat contents of Ni2Si_L and NiSi2 are shown in Fig. 8. The present modeling depicts an excellent fit to the experimental data [12,13]. The calculated CP of NiSi2 at 27  C is 22.1 J/  C$mol$atoms, which agrees well with the experimental value [60]. The assessed thermodynamic properties of Ni2Si_L and NiSi2 satisfy

Fig. 8. Calculated heat contents of the Ni2Si_L amd NiSi2 according to the present model and NeumanneKopp rule, compared with the experimental data.

100

X. Yuan et al. / Materials Chemistry and Physics 135 (2012) 94e105

Fig. 9. Calculated heat capacity of NiSi according to the present model and NeumanneKopp rule, compared with the experimental data.

the NeumanneKopp rule, as shown in Fig. 8. Fig. 9 compares the calculated CP of NiSi with experimental values. As shown in Fig. 9, the present modeling fits well with the experimental data measured by Perring et al. [15]. However, it is worth mentioning that the measured Cp by the two groups shows discrepancy at temperature around 100  C. The value measured using adiabatic calorimetry [14] is a little higher than those determined by differential scanning calorimetry [15] with the largest discrepancy about 1 J/ C$mol$atoms. This calculation mainly focuses on the thermodynamic properties above room temperature. Therefore, only the data values measured by Perring et al. [15] were used in the present modeling and the data obtained by Acker et al. [14] were used to check the reliability of present modeling at low temperatures. The discrepancy between the present modeling and measured value [14,15] is within 1 J/ C$mol$atoms and shows agreement at low temperature. In all, the present modeling improves the description

Fig. 11. Metastable NieSi phase diagram computed in the present work, including only fccebased phases (fcc, L12 and L10) and the liquid phase.

of NiSi compared with the calculation using NeumanneKopp rule. The calculated DHf for all the compounds along with the literature data [39,40,48,56,61,62] is presented in Fig. 10. The present CALPHAD modeling agrees well with the experiments. The firsteprinciples calculation reproduces the previous calculation [39] and the calculated values are about several kJ/mol$atoms lower than those of the experiments. 4.2.3. Metastable phase equilibria The CALPHAD calculated DHf of metastable fcc phase at the composition of 25 at.% Si and 75 at.% Si are 27.7 kJ/mol$atoms and 6.0 kJ/mol$atoms, respectively, agreeing well with the firsteprinciples calculations. The calculated metastable ordered/ disordered phase diagram considering only the fccebased phases and liquid is presented in Fig. 11. According to the present modeling, only the Ni3SieL12 is stable relative to the fcc disordered phase, which is in agreement with the firsteprinciples calculations. These calculations fit the phase stability deduced from the firsteprinciples calculations. The reasonable ordered/disordered diagram demonstrates the successful application of the newly proposed approach. It is expected that this approach can be applied to model the ordered/disordered transformations in other systems in order to develop reliable thermodynamic description of multiecomponent alloys with various fcc ordered/disordered transitions. 5. Conclusions

Fig. 10. Calculated enthalpy of formation of the compounds at 298 K in comparison with the literature data.

 The enthalpy of formation for all the fccebased phases (fcc, L12 and L10) in the NieSi and AleNi systems at 0 K have been calculated by firsteprinciples calculations. The present calculations demonstrate that metastable phase equilibria are indispensable in establishing reliable thermodynamic database. By incorporating the metastable fcc order phase diagram deduced from the firsteprinciples calculations, one CALPHADetype approach is proposed to model the fcc ordered/disordered transition.  A four-sublattice model has been utilized to describe the stable and metastable phase equilibria in the NieSi system. In comparison with the previous assessments, noticeable improvements have been made. The artificial inverted miscibility gap is avoided. The overall fit to the experimental data is

X. Yuan et al. / Materials Chemistry and Physics 135 (2012) 94e105

fairly good. The computed phase equilibria associated with ordered and disordered fcc phases are reasonable and inconsistence with the firsteprinciples calculations. The new approach is valid for other systems showing ordered/disorder transitions. Acknowledgments The financial support from the National Basic Research Program of China (Grant No. 2011CB610401) and the National Natural Science Foundation of China (Grant Nos. 50831007, 51021063 and 50971135) are greatly acknowledged. Lijun Zhang would like to thank the Alexander von Humboldt Foundation of Germany for

101

m where DGord i:j , u(v), yi , L and R are the Gibbs energy of corresponding endemembers, lattice fraction of the first (second) sublattice, the site fraction of the constituent i in sublattice m, interaction parameters and gas constant, respectively. Eq. (A2) cannot handle both the L12 and L10 states simultaneously, since the L12 crystal structure corresponds to u ¼ 3/4, v ¼ 1/4 while L10 phase requires u ¼ v ¼ 1/2. This drawback can be solved by introducing the 4SL model (A,B)0.25(A,B)0.25(A,B)0.25(A,B)0.25. In view of the crystallographic equivalence of the four sublattices, the thermodynamic properties of every lattice are equal. Further assuming that the interaction parameters within one sublattice are indes pendent of elements in other sublattices, the Gord m ðyi Þ in Eq. (A1) is represented as follows:

B X B X B X B   X ð1Þ ð2Þ ð3Þ ð4Þ ð1Þ ð2Þ ð3Þ ð4Þ yA ; yA ; yA ; yA ¼ Gord yi yj yk yl DGord i:j:k:l m i¼A j¼A k¼A l¼A

þ

4 X

("

s¼1

) # B 4 X 4 h X      2 i RT X ysi ln ysi þ ysA ysB 0 LsA;B þ ysA  ysB 1 L þ ysA  ysB 2 L þ ysA ysB yvA yvB * L $ 4 s ¼ 1 v>s

(A3a)

i¼A

supporting and sponsoring the research work at ICAMS, RuhreUniversität Bochum, Germany. The ThermoeCalc Software AB in Sweden is gratefully acknowledged for the provision of the ThermoeCalc software. Appendix A. Ordered/disordered model and the transformation of the parameters between the 2SL model and 4SL model. The ordered and disordered states of fcc phase are usually described using a single formula:

where *L is the reciprocal parameter. DGord i:j:k:l has the following constraints due to the crystal symmetry:

DGA:A:A:B ¼ DGA:A:B:A ¼ DGA:B:A:A ¼ DGB:A:A:A ¼ DGA3B DGA:A:B:B ¼ DGA:B:A:B ¼ DGA:B:B:A ¼ DGB:A:B:A ¼ DGB:A:A:B ¼ DGB:B:A:A ¼ DGA2B2 DGB:B:B:A ¼ DGB:B:A:B ¼ DGB:A:B:B ¼ DGA:B:B:B ¼ DGAB3 (A3b)

Gord m

¼

Gdis m ðxi Þ

þ

Gord m

 s  s  yi  Gord m yi ¼ xi

(A1)

Gdis m

where represents the Gibbs energy of the disorder fcc phase. s ord s Gord m ðyi Þ  Gm ðyi ¼ xi Þ denotes the contribution of the order phase to the Gibbs energy. This difference is identical to zero when the disorder state is stable. Thus, Eq. (A1) can represent both the ordered and disordered states of fcc phase. Using 2SL model (A,B)u(A,B)v and assuming that the interaction between the elements within one sublattice is independent of the element in s the other sublattice, Gord m ðyi Þ in a fictitious AeB binary system is described as follows:

ð1Þ

When yi

ð1Þ

ð2Þ

¼ yi

ð2Þ

ð3Þ

ð4Þ

¼ yi syi , 4SL describes the L12 structure. ð3Þ

ð4Þ

When yi ¼ yi syi ¼ yi , it represents the L10 structure. 2SL and 4SL models are mathematically equal in describing the L12 or the L10 phase. Eq. (A2) minus Eq. (A3) must be equal to zero. By this condition, the relationships between the parameters of 2SL model and those of 4SL model can be deduced. For the L12 phase, ð1Þ

considering that ya ¼ y’A ¼ yA y’B

ð1Þ yB

yb ¼ ¼ simplified as:

¼

ð2Þ yB

¼

ð3Þ yB ,

ð2Þ

¼ yA

yB ¼

y’’B

¼

ð3Þ

ð4Þ

¼ yA , yA ¼ y’’A ¼ yA , ð4Þ yB ,

Eq. (A3) can then be

" # B X B B B B i h X X X  0  00   0  0  0 00  X   ord 0 00 0 00 0 1 ord 0 2 2 ord D Gord ; y y $y $ G þ RT u y ln y y ln y y0A $y0B $y00i $ 0 Lord LA;B:i y ¼ þ v þ i:j m A A i j i i i i A;B:i þ yA  yB LA;B:i þ yA  yB i¼A i¼A



B X i¼A

i¼A

i¼A

i¼A

i h  00  00   00 1 ord 00 2 2 ord y0i $y00A $y00B $ 0 Lord Li:A;B þ y0A $y0B $y00A $y00B $0 Lord A;B:A:B i:A;B þ yA  yB Li:A;B þ yA  yB

     3  3 2 y y DG 2 y y þ y y2 y 3 y þ 3y2 y y D DGAB3 þ RTðya lnðya Þ þ yb lnðyb ÞÞ G4L ¼ y y þ 3y þ 3 y G þ y a a B B B A A3B A A2B2 A b b a a a L12 b b b 4 h h i i 1 2 2 0 1 þ RTðyA lnðyA Þ þ yB lnðyB ÞÞ þ 3ya yb yA L þ ðya  yb Þ$ L þ ðya  yb Þ $ L þ 3ya yb yB 0 L þ ðya  yb Þ$1 L þ ðya  yb Þ2 $2 L 4 h h  i i  þya yA yB 0 L þ ðyA  yB Þ$1 L þ ðyA  yB Þ2 $2 L þ yb yA yB 0 L þ ðyA  yB Þ$1 L þ ðyA  yB Þ2 $2 L þ 3 y2a y2b þ ya yb yA yB $* L

ðA2Þ

(A4)

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X. Yuan et al. / Materials Chemistry and Physics 135 (2012) 94e105

Eq. (A2) minus Eq. (A4) and separating DGA3B , DGA2B2 , DGAB3 and * L into various parts yield the following equation:

h i h i h i 4L 2L 4L 2L 4L 0 2SL 0 4L 4L 4L * G2L L12  GL12 ¼ ya yB ΔGA:B  ΔGA3B þ yb yA ΔGA:B  ΔGA3B þ ya yb yA LA;B:A  3 L  λ1 ΔGA3B  λ2 ΔGA2B2  λ3 ΔGAB3  λ4 L h i 1 4L 4L 4L * þya yb yA ðya  yb Þ$ 1 L2SL A;B:A  3 L  x1 ΔGA3B  x2 ΔGA2B2  x3 ΔGAB3  x4 L h i 2 4L 4L 4L * þya yb yA ðya  yb Þ2 $ 2 L2SL A;B:A  3 L  g1 ΔGA3B  g2 ΔGA2B2  g3 ΔGAB3  g4 L h i 0 4L 4L 4L * þya yb yB 0 L2SL A;B:B  3 L  y1 ΔGA3B  y2 ΔGA2B2  y3 ΔGAB3  y4 L h i 1 4L 4L 4L * þya yb yB ðya  yb Þ$ 1 L2SL A;B:B  3 L  z1 ΔGA3B  z2 ΔGA2B2  z3 ΔGAB3  z4 L h i 2 4L 4L 4L * þya yb yB ðya  yb Þ2 $ 2 L2SL A;B:B  3 L  t1 ΔGA3B  t2 ΔGA2B2  t3 ΔGAB3  t4 L h i 0 4L 4L 4L * þya yA yB 0 L2SL A:A;B  L  h1 ΔGA3B  h2 ΔGA2B2  h3 ΔGAB3  h4 L h i 1 4L 4L 4L * þya yA yB ðyA  yB Þ$ 1 L2SL A:A;B  L  f1 ΔGA3B  f2 ΔGA2B2  f3 ΔGAB3  f4 L h i 2 4L 4L 4L * þya yA yB ðyA  yB Þ2 $ 2 L2SL A:A;B  L  d1 ΔGA3B  d2 ΔGA2B2  d3 ΔGAB3  d4 L h i 0 4L 4L 4L * þyb yA yB 0 L2SL B:A;B  L  e1 ΔGA3B  e2 ΔGA2B2  e3 ΔGAB3  e4 L h i 1 4L 4L 4L * þyb yA yB ðyA  yB Þ$ 1 L2SL B:A;B  L  u1 ΔGA3B  u2 ΔGA2B2  u3 ΔGAB3  u4 L h i 2 4L 4L 4L * þyb yA yB ðyA  yB Þ2 $ 2 L2SL B:A;B  L  j1 ΔGA3B  j2 ΔGA2B2  j3 ΔGAB3  j4 L h i 4L 4L 4L * (A5) þya yb yA yB 0 L2SL A;B:A;B  k1 ΔGA3B  k2 ΔGA2B2  k3 ΔGAB3  k4 L

In which, l, x, g, y, z, t, h, f, d, e, u, j and k are coefficients to be confirmed. To satisfy the condition that Eq. (A5) is identical to zero, every part in the center brackets should be zero. To determine the coefficients in Eq. (A5), rewrite the Eq. (A5) as following:

Compared with Eq. (A4), all the above coefficients in Eqs. (A4) 4SL 4SL * and (A6) before DG4SL A3B , DGA2B2 , DGAB3 and L should be identical to each other. Thus the followings equations should be satisfied, respectively:

h i h i h i 2 2 2SL 4L 2L 2L 0 2SL 0 1 2SL 1 2 G2L L12  GL12 ¼ ya yB ΔGA:B þ yb yA ΔGA:B þ ya yb yA LA;B:A  3 L þ ya yb yA ðya  yb Þ$ LA;B:A  3 L þ ya yb yA ðya  yb Þ $ LA;B:A  3 L h h i h i h i i 2 2 2SL 0 1 2SL 1 2 0 2SL 0 þ ya yb yB 0 L2SL A;B:B  3 L þ ya yb yB ðya  yb Þ$ LA;B:B  3 L þ ya yb yB ðya  yb Þ $ LA;B:B  3 L þ ya yA yB LA:A;B  L h h i h i i 2 2 2SL 1 2 0 2SL 0 þ ya yA yB ðyA  yB Þ$ 1 L2SL A:A;B  L þ ya yA yB ðyA  yB Þ $ LA;A:B  L þ yb yA yB LB:A;B  L h i h h i h i 2 2 2SL 1 2 0 2SL þ yb yA yB ðyA  yB Þ$ 1 L2SL B:A;B  L þ yb yA yB ðyA  yB Þ $ LB:A;B  L þ ya yb yA yB LA;B:A;B  ya yB þ λ1 ya yb yA þ x1 ya yb yA ðya  yb Þ þ g1 ya yb yA ðya  yb Þ2 þy1 ya yb yB þ z1 ya yb yB ðya  yb Þ þ t1 ya yb yB ðya  yb Þ2 þh1 ya yA yB

i þ f1 ya yA yB ðyA  yB Þ þ d1 ya yA yB ðyA  yB Þ2 þe1 yb yA yB þ u1 yb yA yB ðyA  yB Þ þ j1 yb yA yB ðyA  yB Þ2 þk1 ya yb yA yB $ΔG4SL A3B h 2 2  λ2 ya yb yA þ x2 ya yb yA ðya  yb Þ þ g2 ya yb yA ðya  yb Þ þy2 ya yb yB þ z2 ya yb yB ðya  yb Þ þ t2 ya yb yB ðya  yb Þ þh2 ya yA yB i þ f2 ya yA yB ðyA  yB Þ þ d2 ya yA yB ðyA  yB Þ2 þe2 yb yA yB þ u2 yb yA yB ðyA  yB Þ þ j2 yb yA yB ðyA  yB Þ2 þk2 ya yb yA yB $ΔG4SL A2B2 h 2 2  yb yA þ λ3 ya yb yA þ x3 ya yb yA ðya  yb Þ þ g3 ya yb yA ðya  yb Þ þy3 ya yb yB þ z3 ya yb yB ðya  yb Þ þ t3 ya yb yB ðya  yb Þ þ h3 ya yA yB þ f3 ya yA yB ðyA  yB Þ þ d3 ya yA yB ðyA  yB Þ2 þe3 yb yA yB þ u3 yb yA yB ðyA  yB Þ þ j3 yb yA yB ðyA  yB Þ2 i h 2 þ k3 ya yb yA yB $ΔG4SL AB3  λ4 ya yb yA þ x4 ya yb yA ðya  yb Þ þ g4 ya yb yA ðya  yb Þ þy4 ya yb yB þ z4 ya yb yB ðya  yb Þ þ t4 ya yb yB ðya  yb Þ2 þh4 ya yA yB þ f4 ya yA yB ðyA  yB Þ þ d4 ya yA yB ðyA  yB Þ2 þe4 yb yA yB þ u4 yb yA yB ðyA  yB Þ i þ j4 yb yA yB ðyA yB Þ2 þk4 ya yb yA yB $* L (A6)

X. Yuan et al. / Materials Chemistry and Physics 135 (2012) 94e105

h

h

103

ya yB þ l1 ya yb yA þ x1 ya yb yA ðya  yb Þ þ g1 ya yb yA ðya  yb Þ2 þy1 ya yb yB þ z1 ya yb yB ðya  yb Þ þ t1 ya yb yB ðya  yb Þ2 þh1 ya yA yB i þ f1 ya yA yB ðyA  yB Þ þ d1 ya yA yB ðyA  yB Þ2 þe1 yb yA yB þ u1 yb yA yB ðyA  yB Þ þ j1 yb yA yB ðyA  yB Þ2 þk1 ya yb yA yB   ¼ y3a yB þ 3y2a yA yb

l2 ya yb yA þ x2 ya yb yA ðya  yb Þ þ g2 ya yb yA ðya  yb Þ2 þy2 ya yb yB þ z2 ya yb yB ðya  yb Þ þ t2 ya yb yB ðya  yb Þ2 þh2 ya yA yB þ f2 ya yA yB ðyA  yB Þ þ d2 ya yA yB ðyA  yB Þ2 þe2 yb yA yB þ u2 yb yA yB ðyA  yB Þ þ j2 yb yA yB ðyA  yB Þ2 þk2 ya yb yA yB   ¼ 3 y2a yb yB þ ya y2b yA

h

h

(A7)

i

(A8)

yb yA þ l3 ya yb yA þ x3 ya yb yA ðya  yb Þ þ g3 ya yb yA ðya  yb Þ2 þy3 ya yb yB þ z3 ya yb yB ðya  yb Þ þ t3 ya yb yB ðya  yb Þ2 þh3 ya yA yB i þ f3 ya yA yB ðyA  yB Þ þ d3 ya yA yB ðyA  yB Þ2 þe3 yb yA yB þ u3 yb yA yB ðyA  yB Þ þ j3 yb yA yB ðyA  yB Þ2 þk3 ya yb yA yB   ¼ y3b yA þ 3y2b ya yB

(A9)

l4 ya yb yA þ x4 ya yb yA ðya  yb Þ þ g4 ya yb yA ðya  yb Þ2 þy4 ya yb yB þ z4 ya yb yB ðya  yb Þ þ t4 ya yb yB ðya  yb Þ2 þh4 ya yA yB þf4 ya yA yB ðyA  yB Þ þ d4 ya yA yB ðyA  yB Þ2 þe4 yb yA yB þ u4 yb yA yB ðyA  yB Þ þ j4 yb yA yB ðyA  yB Þ2 þk4 ya yb yA yB   ¼ 3 y2a y2b þ ya yb yA yB

i

(A10)

Using the constraints: yb ¼ 1  ya and yB ¼ 1  yA to replace yb and yB in the Eqs. (A7)e(10) yields:

y4A ð4j1 Þ þ y3A ð  2u1 þ 8j1 Þ þ y2A ð  e1 þ 3u1  5j1 Þ þ yA ðe1  u1 þ j1 Þ þ ya ð1 þ y1  z1 þ t1 Þ þ ya yA ð  1 þ l1  x1 þ g1  y1 þ z1  t1 þ h1  f1 þ d1  e1 þ u1  j1 þ k1 Þ þ ya y2A ð  h1 þ 3f1  5d1 þ e1  3u1 þ 5j1  k1 Þ þ ya y3A ð  2f1 þ 8d1 þ 2u1  8j1 Þ þ ya y4A ð  4d1 þ 4j1 Þ þ y2a ð  y1 þ 3z1  5t1 Þ þ y2a yA ð  l1 þ 3x1  5g1 þ y1  3z1 þ 5t1  k1 Þy2a y2A ðk1 Þ þ y3a ð  2z1 þ 8t1 Þ þ y3a yA ð  2x1 þ 8g1 þ 2z1  8t1 Þ þ y4a ð4t1 Þ þ y4a yA ð  4g1 þ 4t1 Þ ¼ 3y2a yA þ y3a  4y3a yA (A11)

y4A ð4j2 Þ þ y3A ð  2u2 þ 8j2 Þ þ y2A ð  e2 þ 3u2  5j2 Þ þ yA ðe2  u2 þ j2 Þ þ ya ðy2  z2 þ t2 Þ þ ya yA ðl2  x2 þ g2  y2 þ z2  t2 þ h2  f2 þ d2  e2 þ u2  j2 þ k2 Þ þ ya y2A ð  h2 þ 3f2  5d2 þ e2  3u2 þ 5j2  k2 Þ þ ya y3A ð  2f2 þ 8d2 þ 2u2  8j2 Þ þ ya y4A ð  4d2 þ 4j2 Þ þ y2a ð  y2 þ 3z2  5t2 Þ þ y2a yA ð  l2 þ 3x2  5g2 þ y2  3z2 þ 5t2  k2 Þy2a y2A ðk2 Þ þ y3a ð  2z2 þ 8t2 Þ þ y3a yA ð  2x2 þ 8g2 þ 2z2  8t2 Þ þ y4a ð4t2 Þ þ y4a yA ð  4g2 þ 4t2 Þ ¼ 3ya yA þ 3y2a  3y3a  9y2a yA þ 6y3a yA (A12)

y4A ð4j3 Þ þ y3A ð  2u3 þ 8j3 Þ þ y2A ð  e3 þ 3u3  5j3 Þ þ yA ðe3  u3 þ j3 þ 1Þ þ ya ðy3  z3 þ t3 Þ þ ya yA ð  1 þ l3  x3 þ g3  y3 þ z3  t3 þ h3  f3 þ d3  e3 þ u3  j3 þ k3 Þ þ ya y2A ð  h3 þ 3f3  5d3 þ e3  3u3 þ 5j3  k3 Þ þ ya y3A ð  2f3 þ 8d3 þ 2u3  8j3 Þ þ ya y4A ð  4d3 þ 4j3 Þ þ y2a ð  y3 þ 3z3  5t3 Þ þ y2a yA ð  l3 þ 3x3  5g3 þ y3  3z3 þ 5t3  k3 Þy2a y2A ðk3 Þ þ y3a ð  2z3 þ 8t3 Þ þ y3a yA ð  2x3 þ 8g3 þ 2z3  8t3 Þ þ y4a ð4t3 Þ þ y4a yA ð  4g3 þ 4t3 Þ ¼ yA þ 3ya  6ya yA  6y2a þ 9y2a yA þ 3y3a  4y3a yA (A13)

104

X. Yuan et al. / Materials Chemistry and Physics 135 (2012) 94e105

y4A ð4j4 Þ þ y3A ð  2u4 þ 8j4 Þ þ y2A ð  e4 þ 3u4  5j4 Þ þ yA ðe4  u4 þ j4 Þ þ ya ðy4  z4 þ t4 Þ þ ya yA ðl4  x4 þ g4  y4 þ z4  t4 þ h4  f4 þ d4  e4 þ u4  j4 þ k4 Þ þ ya y2A ð  h4 þ 3f4  5d4 þ e4  3u4 þ 5j4  k4 Þ þ ya y3A ð  2f4 þ 8d4 þ 2u4  8j4 Þ þ ya y4A ð  4d4 þ 4j4 Þ þ y2a ð  y4 þ 3z4  5t4 Þ þ y2a yA ð  l4 þ 3x4  5g4 þ y4  3z4 þ 5t4  k4 Þy2a y2A ðk4 Þ þ y3a ð  2z4 þ 8t4 Þ þ y3a yA ð  2x4 þ 8g4 þ 2z4  8t4 Þ þ y4a ð4t4 Þ þ y4a yA ð  4g4 þ 4t4 Þ ¼ 3ya yA  3ya y2A þ 3y2a  3y2a yA þ 3y2a y2A  6y3a þ 3y4a (A14) Since every part of the polynomial yia yjA between the left and the right sides of the Eqs. (A11)e(14) is equal, the coefficients can then be confirmed. The finally obtained results are presented in the following equation:

0

DGL12 A:B B D L12 B GB:A

1

C C 0 B 0 L12 C 1 B L C B A;B:A C B 0 B 1 L12 C B B LA;B:A C B 3=2 B C B B 2 LL12 C B 3=2 B A;B:A C B B L1 C B 0 B 0L 2 C B B A;B:B C B B 1 L12 C ¼ B 3=2 B L C B 1=2 B A;B:B C B B 2 L12 C B 0 B LA;B:B C B B C B 0 B 0 LL12 C B B i:A;B C B 0 B L1 C B B 1L 2 C @ 0 B i:A;B C B 2 L12 C 0 B L C @ i:A;B A 0 LL12 A;B:A;B

0 0 3=2 3=2 0 3=2 3=2 0 0 0 0 0

0 1 3=2 1=2 0 3=2 3=2 0 0 0 0 0

0 0 3 0 0 3 0 0 1 0 0 0

0 0 0 3 0 0 3 0 0 1 0 0

0 0 0 0 3 0 0 3 0 0 1 0

1 0 0 C C 0 1 3=4 C DG4SL C A3 B 0 C B 4SL C B DGA B C C 2 2 3=4 C C C B C B DG4SL AB3 C 3=4 C B C,B 0 C C L 0 C B C C B 1L 3=4 C C B C @2 A C L 0 C *L 0 C C 0 A 3

For the L10 phase, the relationship between parameters in the two models can be deduced using the above method. The finally derived equations are as follows:

0

0 DGL1 A:B

1

B 0 L10 C 0 0 1 0 B LA;B:A C B C B B 1 LL10 C B 2 1 0 B A;B:A C B 0 0 B 2 L10 C B 0 B L C 0 0 0 B A;B:A C B B B 0 L10 C¼ 0 1 2 B LA;B:B C B B C B 0 0 B 1 L10 C B 0 B LA;B:B C B 0 0 B C @ 0 B 2 LL10 C 4 6 4 @ A;B:B A 0 LL10 A;B:A;B

0 2 0 0 2 0 0 0

0 0 2 0 0 2 0 0

1 0 1 0 0 DG4SL A3 B 0 1=4 C C B C B DG4SL A2 B2 C 0 0 C B C B DG4SL C C AB3 C 2 1=4 C C,B C B 0L C 0 3=4 C B C C B 1L 0 0 C C C B A @ 2L A 2 1=4 *L 0 4 (A16)

[6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

The other parameters can be determined through the symmetry 0 0 ¼ v LL1 , where v represents the order of of the L10 structure: v LL1 i:j j:i the parameter, i and j denote single elements as well as their interaction. Using Eq. (A15)e(16), the parameters in 2SL model can be transformed into those of 4SL model.

[23]

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