Gibbs energy modeling of Fe–Ta system by Calphad method assisted by experiments and ab initio calculations

CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Gibbs energy modeling of Fe–Ta system by Calphad method assisted by experiments and ab initio calculations V.B. Rajkumar, K.C. Hari Kumar n Department of Metallurgical and Materials Engineering, Indian Institute of Technology Madras, Chennai, Tamil Nadu 600 036, India

art ic l e i nf o

a b s t r a c t

Article history: Received 27 October 2014 Received in revised form 7 December 2014 Accepted 29 December 2014

In this work we report the Gibbs energy model parameters of equilibrium phases of Fe–Ta system obtained by the Calphad approach. In order to assist the Gibbs energy modeling new constitutional and enthalpy increment data were obtained by experiments. Further, the energy of formation of intermediate phases and the energy of mixing of bcc solid solution were calculated by ab initio method. These results were combined with selected experimental data from the literature in order to optimize the model parameters of the Gibbs energy functions. Calculated phase diagram and the thermochemical properties are in good agreement with the experimental results. & 2014 Published by Elsevier Ltd.

Keywords: Fe–Ta Phase diagram Thermodynamic modeling Constitutional studies Ab initio calculations Calphad

1. Introduction Tantalum is an important alloying element in reduced activation ferritic–martensitic steels (RAFM) used in the fabrication of critical components of nuclear power plants. Knowledge of thermochemistry and constitution of multicomponent steels containing tantalum is therefore important in designing new alloys with improved properties. In this context, the Gibbs energy modeling of Fe–Ta system gains significance. The critical review of the constitutional, crystallographic, and thermochemical data of the system was reported by several authors [1–3]. Fig. 1 shows the Fe–Ta phase diagram, reproduced from [2,4]. In this system there are two non-stoichiometric intermediate phases (ε and μ) and five invariant reactions. There is disagreement between various authors regarding the mode of formation of the intermediate phases. Refs. [2,5–7] regarded the ε phase as forming congruently from the liquid (L ⇋ ε ), whereas Ref. [3] considered it being formed by a peritectic reaction (L + μ ⇋ ε ). In the case of the μ phase, Refs. [2,3,5,6] considered it as forming congruently from liquid (L ⇋ μ), whereas Ref. [7] proposed its formation by a peritectic reaction (L + ε ⇋ μ). The arguments by [7] regarding the mode of formation of these phases are accepted in the present work. Many authors have reported the thermodynamic modeling of n

Corresponding author. E-mail address: [email protected] (K.C. Hari Kumar).

the system [2,5–9]. In the reports by [2,8] the phases ε and μ are modeled as stoichiometric compounds. Ref. [5] modeled the ε phase as a non-stoichiometric compound while the μ phase was modeled as a stoichiometric compound. Thermodynamic modeling by [6,7,9] treated both the intermediate phases as non-stoichiometric. The Gibbs energy description of liquid reported by [7] uses too many model parameters. Activity of Ta in (γFe) and ε reported by [10] was not incorporated by [7]. Thermodynamic assessments by [2,5–9] indicate the Gibbs energy of mixing of equiatomic bcc phase at room temperature to be positive, suggesting the existence of a metastable miscibility gap. We believe that this is not true, but the Gibbs energy of mixing is negative at this temperature, similar to the Fe–Nb system [11]. Optimization of Gibbs energy functions for the intermediate phases by [7] relied on the enthalpy of formation data whose credibility is questionable, while ignoring the emf data on the Gibbs energy of formation (see Section 2.2). This has probably resulted in the calculated entropy and enthalpy of formation of ε phase to have opposite signs, which is unusual. In the present work, in addition to the carefully selected constitutional and thermochemical data from the literature as input, we have made use of liquidus temperature measured using an arc melting-pyrometer setup, tie-lines measured using EPMA, enthalpy increment data obtained by inverse drop calorimetry and results from ab initio calculation, to model the Gibbs energy functions.

http://dx.doi.org/10.1016/j.calphad.2014.12.006 0364-5916/& 2014 Published by Elsevier Ltd.

Please cite this article as: V.B. Rajkumar, K.C. Hari Kumar, Gibbs energy modeling of Fe–Ta system by Calphad method assisted by experiments and ab initio calculations, Calphad (2014), http://dx.doi.org/10.1016/j.calphad.2014.12.006i

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Fig. 1. Fe–Ta phase diagram as reported in [4,2].

2. Literature review

2.2. Thermochemical data

Tables 1 and 2 summarize the published constitutional data. For a detailed discussion concerning the constitutional data up to 1995, the reader may refer to the critical reviews by [1–3]. A brief discussion of literature data beyond this period is given below.

Table 3 gives a summary of the published thermochemical data. Calorimetric data on integral enthalpy of mixing of liquid [24,25] show negative deviation from ideal behavior. Activity of Fe at 1873 K in liquid, ε, μ and (Ta) phases was determined by KnudsenCell mass spectrometry by [22]. In addition to this, Ref. [10] has measured the activity of Ta in (γFe) and ε phases. Enthalpy of formation of the ε phase was determined calorimetrically by [7,26]. Their values are in reasonable agreement with each other. The Gibbs energy of formation of ε phase was determined by EMF method by [27,28]. Their results also agree with each other. However, if we consider these two data together, one would end up with a positive entropy of formation for ε. It is to be noted that according to [26] their data on enthalpy of

2.1. Constitutional data Melting point of ε phase was measured using a pyrometer in a levitation melting setup by [12]. Their result is in good agreement with the recent DTA results of [7]. It was also established by [7] that the μ phase is formed by a peritectic reaction (L + ε ⇋ μ ) at 2056 K, rather than by a congruent reaction as thought earlier. Table 1 Summary of constitutional data.

Metallography, XRD, Magnetic measurements Thermal analysis, Metallography Thermal analysis, Metallography XRD, Incipient melting Metallography, XRD, Hardness XRD XRD, Metallography Dilatometry, Metallography, Thermal analysis Metallography, Magnetic measurements EPMA, XRD, Metallography, Ion intensity thermal analysis Thermal analysis EPMA, DTA, XRD, Metallography XRD, Thermal analysis, Dilatometry SEM/EDX, XRD, DTA Cooling curve analysis, EPMA

Phase region investigated/Composition range (at% Ta)/Temperature range (K)

Reference

((αFe)þ ε), (μþ(Ta)), 5.2–48, 1473 (γFe)/((γFe)þ(δFe)), ((γFe)þ (δFe))/(δFe), (αFe)/((αFe)þ (γFe)), ((αFe)þ(γFe))/(γFe), (γFe)/((γFe)þε), (αFe)/((αFe)þ ε), 0–2.4, 1182–1661 L/(Lþ (αFe)), L/(Lþ ε), 0.13–7.2, 1715–1791 L ⇋ ε , 32.6, 2048 Intermetallic phases, 0.4–60, 1473–1723 (αFe)/((αFe)þ ε), 0.17–0.28, 873–1173 ((αFe)þ ε), (εþ μ), (μþ (Ta)), ε, μ, 20–90, 1573 L/(Lþ (αFe)), L /(L þε) (L þ (δFe))/(δFe), (γFe)/((γFe)þ(δFe)) (L þ (δFe)), ((δFe)þ ε), (δFe), (γFe), 0–15, 1183–1824 ((γFe)þ(δFe)), ((γFe)þ (αFe)) ((γFe)þε), (γFe), 0.18–1.6, 1198–1648 (L þ ε), ε–μ, μ–(Ta), L, ε, μ, 11.1–64, 1873 L/(Lþ (Ta)), 68–77, 2084–2556 (αFe)–ε, ε–μ, μ–(Ta), ε, μ, 20–70, 1673 L ⇋ ε , 33.4, 2152

[13]

L/(Lþ ε), L/(L þμ), (Lþ ε)/ε, L ⇋ ε , (L þ μ)/μ, (μþ (Ta)), ε, 15.6–64.2, 1043–2137 L/(Lþ (Ta)), μ–(Ta), L ⇌ μ+ (Ta), 70–100, 1623–3160

[14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [3] [12] [7] Present work

Please cite this article as: V.B. Rajkumar, K.C. Hari Kumar, Gibbs energy modeling of Fe–Ta system by Calphad method assisted by experiments and ab initio calculations, Calphad (2014), http://dx.doi.org/10.1016/j.calphad.2014.12.006i

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3. Experimental details

Table 2 Coordinates of invariant equilibria. Equilibrium

Composition (at% Ta)

T (K)

Reference

L⇋ε

32.6 33.4 34.8 35.98 –, 42.4, 49.0 50.67, 42.80, 49.69

2048 2152 2136 2137 2056 2065 2023

[16] [12] [7] Calculated [7] Calculated [3]

66.0, 91.5, 61.0

2035 ∼2000 2015 1714

[7] Measured Calculated [22]

7.9, 2.8, 33.3 7.2, –, 27.7 6.29, 2.28, 29.83 1.1, 0.5, 33.3

1713 1716 1721 1512

[20] [7] Calculated [20]

1.5, 1.0, 33.3 1.4, 0.9, 33.3 1.56, 0.917, 31.01 0.3, 33.3, 0.6

1493 1488 1516 1504 1245

[14] [21] [7] Calculated [20]

0.4, 33.3, 0.7 0.2, 33.3, 0.3 –, –, 0.6 0.392, 32.18, 0.836

1238 1247 1246 1245

[21] [14] [7] Calculated

L+ε⇋μ L ⇋ μ + (Ta)

65.03, 92.23, 61.53

L ⇋ (δ Fe) þ ε

(δ Fe) ⇋ (γ Fe) þ ε

(γ Fe) + ε ⇋ (αFe)

Table 3 Summary of thermochemical data. Technique

Quantity measured

Composition range (at% Ta)/ Temperature range (K)

Reference

Calorimetry

Δmix H L

2–12, 1930

[25]

2.5–15, 1866 Δmix H L Mass 4.8–82.8, 1873 aFe in L, spectrometry ε, μ, (Ta) EMF studies a Ta along (γFe)/((γFe)þ ε) 0.29–0.36, 1273– 1373 32.2–64.1, 298 Calorimetry Δf H ε, μ Calorimetry 33, 298 Δf H ε

[10]

EMF studies

Δf

Gε

33, 1070–1300

[27]

EMF studies

Δf G ε

33, 1260–1340

[28]

Estimated

Δf H ε

33, 298

[29]

6.48–33, 298–1448

Present work

Calorimetry

Calorimetry

HT −

(αFe), ((αFe) + ε) H298

[24] [22]

[7] [26]

formation of ε phase should only be taken as indicative because of the presence of a second phase in the alloys investigated. Therefore, we have not used the data by [7,26] in the optimization. Table 4 lists crystallographic information [12,16,30–35] and stability ranges for equilibrium phases of the system.

Table 4 Crystallographic data. Phase

Homogeneity range (at% Ta) [2]

3

Pearson symbol

Struktur– Bericht

Prototype T (K)

3.1. Liquidus measurement of Ta-rich alloys Alloys were prepared by arc melting of high purity iron (99.99%) filings and tantalum (99.95%) wire (from Alfa Aesar) in an argon atmosphere. The description of the setup used and the principle of liquidus temperature estimation is detailed in a recent publication [36]. Compositions of alloys investigated were Fe-70.8, 87.2 and 99.9 at% Ta. Each alloy was subjected to several cycles of melting and cooling. The temperature profile of alloys during the thermal cycles as a function of time for Fe-70.8 at% Ta, Fe-87.2 at% Ta and Fe-99.9 at% Ta is shown in Fig. 2a, b and c respectively. The liquidus temperatures obtained from the temperature profiles are given in Table 6. In addition to the spike corresponding to the liquidus, Fe70.8 at% Ta and Fe-87.2 at% Ta alloys show another spike (Fig. 2a, b) in their cooling curves around ∼2000 K, which corresponds to the invariant L ⇌ μ + (Ta). 3.2. Tie-line determination EPMA (JEOL Ltd., Model JXA-8530F) was used for the determination of tie-lines. Same alloys used for liquidus measurement were placed in separate thick quartz capsules, evacuated and sealed under vacuum and subsequently heat treated at different temperatures as detailed in Table 5 for 36,000 s. Samples were quenched in cold water. The compositions of the observed phases analyzed using EPMA are also given in Table 5. The electron micrograph and the XRD pattern of the alloys Fe70.8 at% Ta (Fig. 3) and Fe-87.2 at% Ta (Fig. 4) show that their microstructural constituents are (Ta) and μ phases, whereas the Fe99.9 at% Ta alloy (Fig. 5) shows the presence of only (Ta) phase. The compositions of the observed phases in the two-phase alloys were analyzed using EPMA and the results are given in Table 5. 3.3. Enthalpy increment Results of enthalpy increment (HT − H298) measurement of the Fe-6.48, 6.98, 29.8 at% Ta alloys as a function of temperature using inverse-drop calorimetry (Multi HTC-96, Setaram Instrumentation) are shown in Fig. 6. XRD analysis of the alloys annealed at 1473 K indicated that the Fe-6.48 and 6.98 at% Ta alloys consisted of two phases ((αFe)þ ε), whereas Fe-29.8 at% Ta alloy had only one phase (ε).

4. Ab initio calculations As mentioned in Section 2.2, discrepancy exists between the measured values of thermochemical properties of the ε phase. In order to ensure the trustworthiness of thermochemical properties used as input for the optimization, an attempt was made to compute the energy of formation of μ and ε phases by ab initio methods. Further, energies of formation of the metastable endmembers of the sublattice formulations for these phases were also calculated. Lastly, energy of mixing of bcc phase was calculated by the cluster expansion method. 4.1. Energy of formation

BCC_A2 ((αFe), 0–0.7 (δFe), 0–2.5 (Ta)) 93–100

cI2

A2

W

<1515 1515–1811 <3293

FCC_A1 ((γFe)) 0–0.9 LAVES_C14 (ε) 28–36 MU_PHASE (μ) 49–54

cF4 hP12 hR13

A1 C14 D85

Cu MgZn2 Fe7W6

1667–1185 <2136 <2056

The details of the software package and the input parameters used for the calculation are given in Table 6. The predicted values of formation energy of ε and μ phases are given in Table 7. The table also lists lattice parameters of optimized structures obtained in the present work. The ab initio values agree quite well with the

Please cite this article as: V.B. Rajkumar, K.C. Hari Kumar, Gibbs energy modeling of Fe–Ta system by Calphad method assisted by experiments and ab initio calculations, Calphad (2014), http://dx.doi.org/10.1016/j.calphad.2014.12.006i

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Fig. 2. Temperature profiles of (a) Fe-70.8 at% Ta, (b) Fe-87.2 at% Ta, and (c) Fe-99.9 at% Ta alloys. Table 5 Results of tie-line, liquidus and invariant temperature measurement of selected Fe– Ta alloys. Composition, at% Ta T (K)

Phase(s)

Phase composition, at% Ta

Liquidus, Invariant , T (K)

70.8

1623 7 2 (μþ (Ta)) μ, 55.3 70.5(Ta), 93.8 7 0.5

>2343, 1953

87.2

1648 7 2 (μþ (Ta)) μ, 56 70.5(Ta), 94.7 7 0.5

>2754 , 2057

99.95

16737 2 (Ta)

(Ta), 99.99 7 0.0

>3160 , –

estimated values from [29,37] as well as the values obtained from the EMF measurements [27,28], whereas it deviates considerably from the calorimetric values [7,26]. 4.2. Energy of mixing Energy of mixing of bcc phase as a function of composition at 0 K was calculated by cluster expansion method, employing ATAT suite of computer programs [39]. Results (Fig. 7) indicate that energy of mixing is negative at 0 K, which is in qualitative agreement with predictions using embedded-atom model [40]. On the other hand, it contradicts with the values calculated using the

model parameters of bcc reported in previous thermodynamic assessments [2,5–9]. They all predicted the energy of mixing values to be positive, which indicates the presence of a metastable miscibility gap in the bcc phase. We believe that bcc phase of Fe– Ta system has negative energy of mixing similar to Fe–Nb system [11]. Fig. 7 shows the presence of ground states at 0, 50 and 100 at% Ta. The energy of ground state at 50 at% Ta is  15,449 J mol  1. This value is the residue corrected energy. The residue corresponds to the error that arises due to the cluster expansion fit. In the present work this value was used as input for the thermodynamic optimization.

5. Thermodynamic models The LIQUID (L), BCC_A2 ((αFe), (δFe), (Ta)) and FCC_A1 ((γFe)) phases are modeled as random solutions. Magnetic contribution to the Gibbs energy of BCC_A2 and FCC_A1 phases are described using the Hillert–Jarl model [41]. All intermediate phases (LAVES_C14 and MU_PHASE) are modeled using appropriate sublattice formulations [42,43], compatible with their crystallography and homogeneity range (Table 8). The Gibbs energy parameters for the end-members of LAVES_C14 were constrained according to the relation suggested by [44], which is now a standard practice for modeling phases with narrow homogeneity range within the

Fig. 3. Fe-70.8 at% Ta: (a) electron micrograph and (b) XRD pattern.

Please cite this article as: V.B. Rajkumar, K.C. Hari Kumar, Gibbs energy modeling of Fe–Ta system by Calphad method assisted by experiments and ab initio calculations, Calphad (2014), http://dx.doi.org/10.1016/j.calphad.2014.12.006i

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Fig. 4. Fe-87.2 at% Ta: (a) electron micrograph and (b) XRD pattern.

framework of Wagner–Schottky defect model [45]. This kind of a constraint was not used by [7] in their modeling. For the MU_PHASE the sublattice formulation suggested by [46,47] is used. Note that in this model there are two end-members, viz. Fe7Ta6 and Fe6Ta7, whose compositions are very close to each other and lie within the reported homogeneity range of the phase. Both these end-members should be playing a major role in the stability of the phase as its homogeneity range widens at higher temperatures.

500

((αFe)+ε)

400 HT-H298, J g-1

ε 300 200 100

6. Optimization

Fe-29.8 at.% Ta Fe-6.98 at.% Ta Fe-6.48 at.% Ta Calphad

0 300

600

900

1200

1500

Temperature, K Fig. 6. Results of enthalpy increment measurements.

Optimization proceeded by introducing one phase at a time. The sequence in which phases were introduced was mainly decided by relative abundance of data. Every time a new phase was

(011) b

Optimization of the Gibbs energy parameters was carried out using the PARROT module [48] of the Thermo-Calc [49] databank system using a subset of carefully selected data as input. It should be mentioned that in addition to the ab initio calculated energy of formation of intermediate phases (Table 7), we have made use of experimental thermochemical data from [10,22,24,25] in the optimization. Thermochemical data was particularly useful to obtain meaningful start values for some of the optimizing variables. We have also made use of the so-called “alternate mode” of PARROT to generate start values wherever it was necessary.

b - (Ta)

(112) b

30

40

50

(022) b

(002) b

(Ta)

Intensity, arb. units

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

5

60

70

80

90

2θ, Degrees Fig. 5. Fe-99.99 at% Ta: (a) electron micrograph and (b) XRD pattern.

Please cite this article as: V.B. Rajkumar, K.C. Hari Kumar, Gibbs energy modeling of Fe–Ta system by Calphad method assisted by experiments and ab initio calculations, Calphad (2014), http://dx.doi.org/10.1016/j.calphad.2014.12.006i

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6

2000

Table 6 VASP input parameters.

predicted energy vasp calculated energy ground state

0

introduced only its parameters were optimized until the sum of squares of error did not change significantly, but all optimizing variables were simultaneously optimized in the subsequent step. Based on this logic the sequence was as follows: Liquid, BCC_A2, Liquid þBCC_A2, μ, Liquid þ BCC_A2 þ μ, ε, and Liquid þBCC_A2 þ μ þ ε. The model parameters of FCC_A1 were constrained by assuming that T0 boundary is located midway in the ((γFe) þ(αFe)) phase field [50]. In the final run of the optimization all input data were used simultaneously. At any stage of the optimization if the relative standard deviation of a parameter happened to be large, it was taken off from the list of optimizing variables. Finally, all parameters were rounded-off using the relative standard deviation criterion suggested by [50]. Table 9 gives the energies of metastable end-members of the μ phase. During the optimization, the enthalpy parameters of all metastable endmembers of the μ phase were kept fixed at the ab initio calculated values. Final optimization was done including all phases and data simultaneously, until the sum of squares of error did not change significantly.

7. Results and discussion

-2000

-1

Version 4.6 Potential PAW Exchange-correlation functional GGA-PBE Energy cut-off 500 eV k-point spacing ≤0.2/ Å Geometric convergence factor 0.02 eV/Å SCF convergence 10  7 eV Integration scheme Methfessel–Paxton Smearing width 0.2 eV

Energy, J.mol

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 Q4 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

-4000 -6000 -8000 -10000 -12000 -14000 -16000

Fe

0.2

0.6

0.8

Ta

Fig. 7. Energy of mixing of bcc calculated using cluster expansion technique. Table 8 Sublattice formulations for ε and μ. Phase

Sublattice formulation

LAVES_C14 MU_PHASE

(Fe , Ta)2 (Fe, Ta )1 (Fe , Ta)6(Ta)4(Fe, Ta )2(Fe , Ta)1

Note: Underline indicates major constituent in the sublattice.

Table 9 Formation energies (J mol  1(of formula units)) of metastable end-members of sublattice formulation for μ phase calculated using VASP. End-members

Final set of Gibbs energy parameters that gave the lowest reduced sum of squares of error is listed in Table 10. Fig. 8 shows the calculated phase diagram. Calculated coordinates of the invariant reactions are listed in Table 2 along with data from the literature. Fig. 9 depicts the comparison of calculated phase diagram with the experimental data. The quality of fit of the phase diagram data is nearly the same as in [7]. Fig. 6 compares experimental incremental enthalpy data with the corresponding calculated ones. A T ln T term in the Gibbs energy description of the ε phase accounts for its small deviation from the Kopp–Neumann rule. Fig. 10 is the comparison of calculated enthalpy of mixing of liquid phase with the experimental data from [24]. In Fig. 11 activity of Fe calculated at 1873 K is

0.4

Mole fraction, xTa

Stoichiometry

MU_PHASE (Fe ,Ta)6 (Fe,Ta )2 (Ta)4 (Fe ,Ta)1 Fe:Fe:Ta:Fe Fe9Ta4 Fe:Fe:Ta:Ta Fe8Ta5 Ta:Fe:Ta:Fe Fe3Ta10 Ta:Fe:Ta:Ta Fe2Ta11 Ta:Ta:Ta:Fe Fe1Ta12 Ta:Ta:Ta:Ta Ta13

Energy

 86,940  113,490 þ414,750 þ368,270 þ 80,020 þ139,230

Underline indicates major constituent in the sublattice.

compared with the experimental data from [22]. The calculated enthalpy of formation of the intermediate phases matches very well with most data from the literature and ab initio calculations (Table 7). As expected, agreement with the data from [7,26] is

Table 7 Formation energies (J mol  1) and lattice parameters (Å) of intermediate phases. Laves phase (ε)

x Ta 0.322 0.362 0.333 0.333 0.333 0.333 0.333 0.333

Mu phase (μ)

Δf H °  5700 [7]a  5400 [7]a  6300 [26]a  18,610 [38]c  19,250 [29]b  19990 [37]b  19,767c  20,118d

Lattice parameters

a = b = 4.759, c = 7.866 c a = b = 4.817, c = 7.822 [2]

x Ta 0.523 0.641 0.462 0.462 0.539 0.462 0.539

Δf H °  7100 [7]a  5800 [7]a  22,586 [37]b  19,868c  17,684c  20,529d  17,224d

Lattice parameters

a = b = 4.86, c = 26.81c a = b = 4.94, c = 27.09c a = b = 4.93, c = 27.1 [2]

a

Measured. Estimated. c Ab initio. d Calphad. b

Please cite this article as: V.B. Rajkumar, K.C. Hari Kumar, Gibbs energy modeling of Fe–Ta system by Calphad method assisted by experiments and ab initio calculations, Calphad (2014), http://dx.doi.org/10.1016/j.calphad.2014.12.006i

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3500

Table 10 Thermodynamic description of the Fe–Ta (in SI units).

G(LIQUID,FE;0)-H298(BCC_A2,FE;0) ¼ þGFELIQ G(LIQUID,TA;0)-H298(BCC_A2,TA;0) ¼ þGTALIQ L(LIQUID,FE,TA;0) ¼  63,613þ 12.922nT L(LIQUID,FE,TA;1) ¼ þ11,687

3000 Temperature, K

Phase name: Liquid (LIQUID) Constitution: (Fe,Ta)1

Phase name: α (BCC_A2) Constitution: (Fe,Ta)1(Va)3 Note: This phase is ferromagnetic

Phase name: μ (MU_PHASE) Constitution: (Fe,Ta)6(Fe,Ta)2(Ta)4(Fe,Ta)1 G(MU_PHASE,TA:TA:TA:TA;0)-13nH298(BCC_A2,TA;0) ¼ þ 13nGHSERTA þ139230 G(MU_PHASE,FE:TA:TA:FE;0)-7nH298(BCC_A2,FE;0)-6nH298(BCC_A2, TA;0) ¼ þ7nGHSERFE þ 6nGHSERTA-330,960 þ66.951nT G(MU_PHASE,FE:TA:TA:TA;0)-6nH298(BCC_A2,FE;0)-7nH298(BCC_A2, TA;0) ¼ þ6nGHSERFE þ 7nGHSERTA-281,029 þ70.779nT G(MU_PHASE,FE:FE:TA:TA;0)-8nH298(BCC_A2,FE;0)-5nH298(BCC_A2, TA;0) ¼ þ8nGHSERFE þ 5nGHSERTA-113,490 G(MU_PHASE,FE:FE:TA:FE;0)-9nH298(BCC_A2,FE;0)-4nH298(BCC_A2, TA;0) ¼ þ9nGHSERFE þ 4nGHSERTA-86,940 G(MU_PHASE,TA:TA:TA:FE;0)-H298(BCC_A2,FE;0)-12nH298(BCC_A2, TA;0) ¼ þGHSERFE þ 12nGHSERTA þ 80,020 G(MU_PHASE,TA:FE:TA:TA;0)-2nH298(BCC_A2,FE;0)-11nH298(BCC_A2, TA;0) ¼ þ2nGHSERFE þ 11nGHSERTA þ 368,270 G(MU_PHASE,TA:FE:TA:FE;0)-3nH298(BCC_A2,FE;0)-10nH298(BCC_A2, TA;0) ¼ þ3nGHSERFE þ 10nGHSERTA þ 414,750 L(MU_PHASE,FE:FE,TA:TA:FE;0) ¼  79,399

(δFe)

1500

(γFe)

1504

ε

μ

(Ta)

1245

(αFe)

0.2

0.4 0.6 0.8 Mole fraction, xTa

Ta

Fig. 8. Calculated phase diagram.

3500

Present work [14] [3] [9] [16] [18] [10] [19] [12] [20] [13]

3000

Phase name: ε (LAVES_C14) Constitution: (Fe,Ta)2(Fe,Ta)1 G(LAVES_C14,FE:FE;0)-3nH298(BCC_A2,FE;0) ¼ þ 15,000þ3nGHSERFE G(LAVES_C14,TA:TA;0)-3nH298(BCC_A2,TA;0) ¼ þ 15,000þ3nGHSERTA G(LAVES_C14,FE:TA;0)-2nH298(BCC_A2,FE;0)-H298(BCC_A2,TA;0) ¼  79,607 þ39.984nT-3.183nTnLN(T) þ2nGHSERFE þ GHSERTA G(LAVES_C14,TA:FE;0)-H298(BCC_A2,FE;0)-2nH298(BCC_A2,TA;0) ¼ þ79,607  39.984nT þ 3.183nTnLN(T) þGHSERFE þ 2nGHSERTA þ30,000 L(LAVES_C14,FE:FE,TA;0) ¼ þ 19,926 L(LAVES_C14,FE,TA:TA;0) ¼ þ 26,806

2015

2065 1721

Fe

Temperature, K

G(FCC_A1,FE:VA;0)-H298(BCC_A2,FE;0) ¼ þ GFEFCC G(FCC_A1,TA:VA;0)-H298(BCC_A2,TA;0) ¼ þ GTAFCC TC(FCC_A1,FE:VA;0) ¼  201 BMAGN(FCC_A1,FE:VA;0) ¼  2.1 L(FCC_A1,FE,TA:VA;0) ¼  36,165 þ14.191nT

2137

2000

500

G(BCC_A2,FE:VA;0)-H298(BCC_A2,FE;0) ¼ þ GHSERFE G(BCC_A2,TA:VA;0)-H298(BCC_A2,TA;0) ¼ þ GHSERTA TC(BCC_A2,FE:VA;0) ¼ þ1043 BMAGN(BCC_A2,FE:VA;0) ¼ þ2.22 TC(BCC_A2,FE,TA:VA;0) ¼ þ600 L(BCC_A2,FE,TA:VA;0) ¼  57,995 þ26.801nT L(BCC_A2,FE,TA:VA;1) ¼  6642 L(BCC_A2,FE,TA:VA;2) ¼ þ 23,837 Phase name: γ (FCC_A1) Constitution: (Fe,Ta)1(Va)1 Note: This phase is antiferromagnetic

L

2500

1000

2500 2000 1500

( δ Fe) ( γ Fe)

1000

( α Fe)

[21]

L ε μ

( Ta)

500 Fe

0.2

0.4 0.6 0.8 Mole fraction, xTa

Ta

2000 L Temperature, K

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

7

1800 (δFe)

1600 1400

(γFe)

1200 (αFe) 1000 Fe

0.02

0.04 0.06 0.08 Mole fraction, xTa

0.10

Fig. 9. Calculated phase diagram and its comparison with experimental data. (a) Tie-lines and phase boundaries and (b) Fe-rich corner (expanded).

equiatomic BCC_A2 phase at 300 K is  13,622 J mol  1. This is in good agreement with the cluster expansion results. It is to be noted that in most other thermodynamic optimizations, including the one by [7], this value is positive.

8. Conclusions

poor. In Table 11 the calculated Gibbs energy of formation of ε phase is compared with the data from [27,28]. It is clear that the enthalpy of formation data used in the optimization is consistent with the Gibbs energy of formation data, since agreement with the Calphad values are quite good in both the cases. Moreover, calculated entropy and enthalpy of formation of intermediate phases have same sign (i.e. both negative). The enthalpy of mixing of

Constitutional and thermochemical data pertaining to the Fe– Ta system were carefully selected and optimized using the Calphad method. The present description uses latest recommendations of Gibbs energy models for ε and μ phases. We believe that the use of ab initio calculations and the judicious use of thermochemical data have greatly improved reliability of the generated Gibbs energy functions, and thermochemistry of the system is well represented. This is the most significant difference between the present work

Please cite this article as: V.B. Rajkumar, K.C. Hari Kumar, Gibbs energy modeling of Fe–Ta system by Calphad method assisted by experiments and ab initio calculations, Calphad (2014), http://dx.doi.org/10.1016/j.calphad.2014.12.006i

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8

0 -2 ΔmixHL,kJ mol-1

-4 -6 -8 -10 -12 -14 -16

[25]

-18 Fe

0.2

0.4 0.6 0.8 Mole fraction, xTa

Ta

Ashutosh S. Gandhi, Indian Institute of Technology Madras (IITM), Chennai, India is gratefully acknowledged. Professor G. Phanikumar, of IITM is gratefully acknowledged for his suggestions for liquidus measurements. Sincere thanks to Dr. L. Chandrasekaran, (Retired) SMA Technology Specialist, Principal Scientist QuinetiQ Ltd, Farnborough, UK for his valuable input pertaining to the experimental techniques. Support extended by Dr. K. Nagarajan, Dr. K. Ananthasivan and S. Raju of Indira Gandhi Center for Atomic Research (IGCAR), Kalpakkam, India for their support in carrying out liquidus measurements and calorimetric measurements. Support of Professor Aloke Paul, Indian Institute of Science (IISc), Bangalore, India for EPMA measurements is gratefully acknowledged. Thanks to S. Balakrishnan, K. Ambika, Dr. B. Jeya Ganesh, S. Ghosh of IGCAR, Chadrasekhar Tiwary of IISc and Ajit Srivastav, Mohan Babu of IITM for help in doing experiments. Also thanks to J. Deepak and B.S. Srinivas Prasad of MIT, Boston, USA for providing certain key references.

Fig. 10. Enthalpy of mixing of liquid at 1930 K.

Appendix A. Supplementary material

1.0

L

Supplementary data associated with this paper can be found in the online version at http://dx.doi.org/10.1016/j.calphad.2014.12. 006.

[22]

0.8 Activity, aFe

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

ε

0.6

References

0.4 μ

0.2

(Ta)

0 Fe

0.2

0.4 0.6 0.8 Mole fraction, xTa

Ta

Fig. 11. Activity of Fe at 1873 K.

Table 11 The Gibbs energy of formation (J mol  1) of ε phase at 33.33 at% Ta and the temperature indicated. Laves phase (ε) T (K)

1070 1185 1300 1260 1300 1340

Δf G° Experimental

Calphad

 19,628 [27]  19,458 [27]  19,287 [27]  22,381 [28]  22,299 [28]  22,217 [28]

 19,793  19,384  18,891  19,060  18,891  18,725

and the ones reported earlier. Availability of more experimental thermochemical data, especially for the liquid and the μ phase, would definitely allow further improvement in the thermodynamic description of the system.

Acknowledgments Support for doing heat treatment extended by Professor

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