Co-Cr–Al phase diagram

L

P

H

A

A

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Computer Coupling of Phase Diagrams and Thermochemistry 27 (2003) 335–342

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CVM calculation of the b.c.c. Co–Cr–Al phase diagram Luiz Tadeu Fernandes Eleno1, Cl´audio Geraldo Sch¨on∗ Computational Materials Science Lab., Department of Metallurgical and Materials Engineering, Escola Polit´ecnica da Universidade de S˜ao Paulo, Av. Prof. Mello Moraes, 2463 - CEP 05508-900 S˜ao Paulo-SP, Brazil Received 9 October 2003; received in revised form 27 November 2003; accepted 30 November 2003 This work is dedicated in memoriam to Prof. Dr. Ryoichi Kikuchi

Abstract The cluster variation method (CVM) has been used as a tool for modelling the thermodynamics of the b.c.c. Co–Cr–Al system within the irregular tetrahedron approximation. The interaction parameters (nearest and next-nearest neighbour pairs, as well as tetrahedron interactions) for the three binary sub-systems have been derived following the so-called phenomenological approach, i.e. the interaction parameters have been fitted to experimental phase diagram and/or thermochemical data. As a result, the three binary phase diagrams of the system and four isothermal sections of the ternary phase diagram have been obtained. The results show that the CVM is thermodynamically self-consistent. © 2004 Elsevier Ltd. All rights reserved. Keywords: Ternary alloy systems; Order/disorder transformations; Thermodynamic and thermochemical properties; Phase diagram prediction; Ordering energies

1. Introduction Ordered intermetallics based on B2 structures (NiAl or FeAl type) have received considerable attention due to their potential for high temperature applications as structural materials. These materials show good mechanical properties at high temperatures (∼700 ◦C), low density (∼6 g/cm3 ), and improved corrosion resistance [1]. The Co–Al system shows a large single-phase field of B2 phase at nearly equiatomic compositions. This phase melts congruently around 1640 ◦C (1913 K) and x Al = 0.50 [2]. Materials based on this compound are suitable candidates, therefore, for structural applications at high temperatures. Addition of chromium has great potential for alloy development based on B2-CoAl compounds since both Al–Cr and Co–Cr binary systems show large stability fields of the bodycentred cubic (b.c.c.) solid solution [3]. Equilibria involving the disordered phase and the B2-CoAl compound are expected to dominate over large concentration ranges of ∗ Corresponding author. Tel.: +55-11-3091-5726; fax: +55-11-30915243. E-mail address: [email protected] (C.G. Sch¨on). 1 Present address: Max-Planck-Institut f¨ur Eisenforschung, Postfach 140444, D-40074 D¨usseldorf, Germany.

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the phase diagram, allowing the production of two-phase microstructures containing a strong intermetallic and a ductile second phase (A2). Chromium may also be used to improve the corrosion resistance of these alloys. In a recent work Ishikawa et al. [4] experimentally determined four isothermal sections of the ternary Co–Al–Cr system. The experimental isothermal sections are dominated by a large two-phase field between the ordered B2-CoAl phase and the disordered chromium-rich A2 phase. The equilibrium is of second-order at low-chromium concentrations, becoming first-order at a multicritical point. The composition of this point is temperature-dependent. The extent of the two-phase field (A2 + B2) decreases as the temperature increases, but its total disappearance has not been observed in the investigation due either to temperature limitations or to competing equilibria with other phases, not based on the b.c.c. lattice [4]. In the present work the phase relationships between the B2 and A2 phases for the ternary Co–Cr–Al system using the cluster variation method (CVM) will be investigated. This kind of thermodynamic modelling allows for extrapolations to metastable equilibria outside the composition and temperature ranges where experimental data can be obtained. The CVM is based on the solution of a (factorizable) partition function [5, 6]. Its sound statistical

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2.2. Formalism

β γ

δ

α

Defining the basic cluster we may write the corresponding thermodynamic functions. First, the internal energy U is given by
αβγ δ αβγ δ εi j kl ρi j kl , (1) U = 6N i j kl

Fig. 1. Irregular tetrahedron in the b.c.c. lattice.

mechanical foundation allows for better confidence in those extrapolations [7], provided the used model shows what Kaufman defines as “thermodynamic self-consistency” (i.e. modelling the phase equilibria implies the modelling of other thermochemical properties and vice versa) [8]. Previous works on the Fe–Co–Al [9] and in the Fe–Al–Ti [10] systems show that CVM-based thermodynamic models present evidence of thermodynamic self-consistency, but this aspect was not the main concern of these works. The aim of the present work is to test the thermodynamic self-consistency of the CVM by using phenomenological calculations of the extrapolated ternary Co–Cr–Al system, based only on parameters obtained by fitting thermochemical data for the three binary sub-systems. The test will be performed by comparison of the phase diagram topology obtained in the calculations with the experimental data of Ishikawa et al. [4].

2. CVM—cluster variation method 2.1. Definitions The CVM was initially proposed as a general formalism to approximate the configurational entropy of crystalline systems [11]. This is achieved defining a basic cluster which includes all relevant atomic correlations. The cluster is defined as a geometric figure (usually in three dimensions, but not necessarily) containing a fixed number of atoms and geometry. Fig. 1 shows, as an illustration, the irregular tetrahedron cluster used in the present work. The irregular tetrahedron is the simplest three-dimensional cluster to take into account first (α–γ , α–δ, β–γ , and β–δ) and second (α–β and γ –δ) nearest-neighbour interactions in the b.c.c. lattice [9]. Tetrahedron correction terms must be included in order to obtain a proper modelling of the system [10, 12]. The thermodynamic species are the alloy components—Co, Cr, and Al—, which may occupy any of the positions of the tetrahedron. The specific order of occupation defines a certain configuration. A configuration is denoted by {i j kl} where i , j , k, and l indicate which species occupies the positions α, β, γ , and δ of the tetrahedron, respectively. The probability of finding a given cluster with configuration {i j kl} in the system is denoted αβγ δ by ρi j kl .

where 6N is the total number of tetrahedra in a b.c.c crystal containing N lattice points (equivalent to the total number of atoms, since no vacancies are taken into account here) αβγ δ and εi j kl is the interaction eigen-energy2 of configuration {i j kl}. Instead of the interaction eigen-energies defined above, it is a common procedure in the literature to write the internal energy in terms of pair interactions, because the latter are considered the leading terms of the energy, with additional tetrahedron correction terms to complete the description of αβγ δ the εi j kl . Hence, the interaction eigen-energies of Eq. (1) are written as αβγ δ

(1) (1) εi j kl = 16 (wik + wil(1) + w(1) jk + w jl ) (2) + 14 (wi(2) i j kl . j + wkl ) + w

(2)

In Eq. (2), the w(1) and w(2) terms represent respectively the nearest- and next-nearest-neighbour pair interactions. The factors 1/4 and 1/6 correct for the right number of pair clusters in the set of 6N tetrahedra. The term w i j kl is αβγ δ an “excess” term which includes all contributions to εi j kl which cannot be ascribed to pair interactions. The reference state is set to the mechanical mixture of the pure elements αβγ δ (i.e. εiiii = 0 for any i ) [12]. As pointed out by Sch¨on and Inden [12, 13], the parameters in Eq. (2) are not all independent. In a binary system A–B, for example, the energy model is fully determined by four parameters. The remaining parameters are obtained by application of the symmetry operations of = w(n) the lattice (e.g. wi(n) j j i ) or set to zero otherwise. For a ternary A–B–C system six additional terms must be introduced [14]. For practical purposes, however, the experimental data used in setting values to these parameters are available only in limited composition and temperature ranges and some of these parameters cannot be determined. The usual procedure in these cases is to set a value of zero for these parameters. The derivation of the CVM entropy formula has been thoroughly outlined in several reviews [13, 15–17] and shall not be discussed here. For the irregular tetrahedron approximation in the b.c.c. lattice, the configurational

2 The formalism used in the present work is equivalent to the Ising model of statistical mechanics. In this sense the eigen-energies defined above are to be understood as the eigen-values of the Hamilton operator of the Ising model for the given configuration (see [13]).

L.T.F. Eleno, C.G. Sch¨on / Computer Coupling of Phase Diagrams and Thermochemistry 27 (2003) 335–342 0

entropy is written as 
αβγ δ αβγ δ S = −NkB 6 ρi j kl ln ρi j kl

+




βγ δ ln ρi j k )

+

αβ ln ρi j

γδ + ρi j

αγ

αγ

βδ

1 4

βγ

CVM calculation

−2000 0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

x Cr Fig. 2. CVM adjust to the experimental data of Havrankova et al. [18] at T = 1673 K. The fitted CVM parameters are shown on Table 1.



γ

−1200

−1800

βγ

+ ρi ln ρi + ρiδ ln ρiδ )

−800 −1000

−1600


β β (ρiα ln ρiα + ρi ln ρi i

Experimental (Vrestal et al.)

−400

−1400

γδ ln ρi j )

αδ (ρi j ln ρi j + ρiαδ j ln ρi j + ρi j ln ρi j

+ ρi j ln ρi j ) − γ

αβ (ρi j

ij

ij βδ

3 2




µ*Co (kB.K)

i j kl

ijk

−200 −600


αβγ αβγ αβδ αβδ αγ δ αγ δ −3 (ρi j k ln ρi j k + ρi j k ln ρi j k + ρi j k ln ρi j k βγ δ + ρi j k

337

(3)

where kB is the Boltzmann constant and the point, pair and triangle probabilities that appear in Eq. (3) are obtained from the tetrahedron probabilities using the so-called reduction αβδ relations [11]. For example, ρi j l , the probability of finding the configuration {i jl} in the triangle αβδ, is given by
αβγ δ αβδ ρi j kl . (4) ρi j l =

The minimization of Eq. (5) is performed by a numerical algorithm called NIM—natural iteration method [17]— in which the function F is minimized through a selfconsistent algorithm which searches for fixed points in the tetrahedron probability space. These fixed points are either the global (possibly degenerate) or local minima of the function F(T, {µ∗i }). Stable and metastable configurations may be mapped as functions of T and {µ∗i }, resulting in the phase diagram of the system.

k

The relations for the other cluster probabilities may be obtained similarly. An equilibrium state is defined as the set of cluster probaαβγ δ bilities {ρi, j,k,l } which minimizes the function F defined by N
αβγ δ ∗ F = U −TS − ρi j kl (µi + µ∗j + µ∗k + µl∗ ), 4

(5)

i, j,k,l

where S is the entropy, T is the temperature and the µ∗i is the generalized chemical potential of element i [15], defined as n 1
µi , (6) µ∗i = µi − n

3. Results and discussion 3.1. Co–Cr Havrankova et al. [18] experimentally determined the activities of Cr and Co in the Co–Cr system by Knudsen cell mass spectrometry at temperatures where the disordered b.c.c. phase is observed. These data may be used for setting the CVM parameters for the Co–Cr system. The fit of the CVM interaction parameters to the experimental data is shown in Fig. 2. The corresponding parameters are shown in Table 1.

j =1

where n is the total number of species of the system and µi is the absolute chemical potential of element i . The µ∗i variables are referred to in this work as the “baricentric” chemical potential of the system. The reason for this is the property
µ∗i = 0, (7)

Table 1 CVM interaction parameters used in the calculations. Units are in kB K (1 kB K = 8.314 51 J/mol) System (A–B)

wAB

(1)

wAB

(2)

wABAB

wABBB

Co–Cr Cr–Al Co–Al

+295 −660 −1800

−390 −290 −600

0 −60 0

+50 0 0

i=Co,Cr,Al

which can be easily proved using Eq. (6). The adoption of the baricentric chemical potential allows for the reduction by one unit of the degrees of freedom of the system [15], which is necessary due to the mass balance condition
Ni = N (8) i=Co,Cr,Al

(Ni is the number of atoms of species i in the system).

It is important to stress that the raw experimental data for T = 1673 K, i.e. the difference of chemical potentials for Cr and Co as functions of the alloy composition, as published by Havrankova et al. [18] have been used for this fit. These data are directly related to the measured effusion rates. The fitting of the interaction parameters based on these experimental data corresponds to a simple trial-and-error

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3.2. Co–Al

G

A2 (A)

σ

A2 (B) B32

A

x BA2/σ x BA2/B32

xB

B

Fig. 3. Schematic representation of free energy G vs. composition xB curves for an A–B binary alloy. The σ/A2 is the stable equilibrium, while the B32/A2 is a metastable one.

procedure. In a first step the system is assumed to behave like (1) a regular solution and a suitable guess for wCo,Cr is deduced based on the data. A first CVM calculation is compared with the experimental data and proper changes in the interaction parameters are made. The procedure is repeated until a close agreement is obtained. The scatter of the experimental data is usually wider than the sensitivity of the calculation to the interaction parameters, so that in all cases investigated in the present work the use of this simplified procedure is justified. The calculated b.c.c. Co–Cr phase diagram is shown in Fig. 4(a). The only stable superlattice is the B32 phase, which shows a somewhat narrow stability range at equiatomic composition, bounded by large first-order equilibria with A2(Co) and A2(Cr). The equilibria is of firstorder at all temperatures and the congruent disordering point is located at 942 K and x Cr = 0.47. These phase relationships for the metastable equilibria are compatible with the experimental phase diagram, since at these compositions and temperature ranges, equilibria involving the σ -phase are observed [19]. The logic of this reasoning is the following: since the σ phase cannot be included in the present calculation no direct inference can be obtained about the competition of the calculated A2-B32 and the experimental σ -A2 equilibria, but an indirect inference can be obtained. This is schematically shown in Fig. 3. The basic assumption is that the A2-B32 equilibrium is less stable than the A2-σ one (that is, the A2/σ former would be metastable). As a consequence, xB < A2/B32 , that is, the two-phase A2 + B32 field must be xB fully contained within the interval where either the twophase σ + A2 or the single phase σ field are experimentally observed in the phase diagram. This condition is satisfied by the present calculation since, according to the published Co–Cr phase diagram [19], the congruent decomposition of the sigma phase occurs around x Cr = 0.59 and T = 1283 ◦C (=1556 K).

For the Co–Al phase diagram, we used as a starting point the parameters determined by Colinet et al. [9]. These parameters were based on the experimental phase diagram according to Hansen and Anderko [20] and were limited to the pair approximation, which we kept in the present work. As discussed by Colinet et al. [9] the parameters used in their work needed to be reassessed, since they resulted in a D03 field at about 30 at.% Cr up to T = 1600 K, which is located inside the B2 field in the experimental phase diagram. There is no experimental evidence for D03 ordering in the system. (2) Following the suggestion of the original work, the wCoAl parameter was reduced from −754 kB K to −600 kB K in order to correct this behaviour. No further changes in the parameters have been attempted since this could affect the agreement obtained by Colinet et al. [9] in the Fe–Co–Al system. Fig. 5 shows the comparison of the results obtained with the present set of parameters and aluminium activity data at T = 1273 K published by Ettenberg et al. [21]. The results show that the activity data are correctly reproduced, especially considering the spread dispersion of the experimental data. Fig. 4(b) shows the calculated b.c.c. Co-Al phase diagram. The CVM parameters are found in Table 1. 3.3. Cr–Al The b.c.c. Cr–Al phase diagram was modelled using experimental aluminium activity data listed by Johnson et al. [22] at T = 1273 K. The parameters were derived by one of the authors in a previous work using the procedure described in the present work [13]. The information was available only at the chromiumrich side, so that the calculated phase diagram was kept symmetrical relative to the equiatomic composition (corresponding to wCrAlAlAl = 0). The parameters of the assessment are shown in Table 1. The calculated phase diagram is shown in Fig. 4(c). The maximum of the critical temperature for the B2/A2 second-order transition is located at T = 999 K and x Al = 0.5. The maxima of the critical temperature for the D03 /B2 second-order transition are located at T = 612 K and x Al ∼ = 0.34 and x Al ∼ = 0.66. These limits are compatible with the experimental phase diagram [23], since they are contained inside stable two-phase fields involving other intermetallics and are, thus, metastable (as previously discussed for the case of Co–Cr). It is interesting to compare the predicted critical temperature at x Al ≈ 0.39 (T ≈ 855 K) with the observation by Helander and Tolochko [24] of a small B2 field just above the A2 → Cr2 Al + Al8 Cr5 eutectoid (at x Al = 0.39 and T = 1143 K). A full description and analysis of the b.c.c. Cr–Al system will be the object of a separate publication.

L.T.F. Eleno, C.G. Sch¨on / Computer Coupling of Phase Diagrams and Thermochemistry 27 (2003) 335–342

339

4500

1100

(a)

(b)

1000

4000

900

3500

800

3000

A2

700

T (K)

T (K)

B32 B2

2500

A2

600

2000

500

1500

400

1000

300

500

D03

D03 0 Co

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

x Cr

1

0

Cr

Co

0.1

0.2

0.3

0.4

0.5

0.6

0.7

xAl

0.8

0.9

1 Al

1100

(c) 1000 900

T (K)

800

B2

700

A2

600 500

D03

D03

400 300

0

0.1

0.2

0.3

Co

0.4

0.5

x Al

0.6

0.7

0.8

0.9

1 Al

Fig. 4. CVM calculated phase diagrams with the parameters on Table 1. (a) Co–Cr; (b) Co–Al; (c) Cr–Al.

3.4. Co–Cr–Al As already mentioned, ternary calculations were performed without the inclusion of the ternary interaction parameters. Hence, the CVM parameters used for the ternary calculation are the same as Table 1. Four isotherms of the CVM-calculated b.c.c. Co–Cr–Al phase diagram are shown in Figs. 6–9, at 1273 K (Fig. 6), 1473 K (Fig. 7), 1573 K (Fig. 8) and 1623 K (Fig. 9). These temperatures correspond to the published data of Ishikawa et al. [4]. The figures also show the relevant experimental tie-lines and second-order critical compositions for A2/B2 phase equilibria as determined by those authors. An important characteristic of these calculations is the remarkable agreement between the slopes of the experimental and calculated tie-lines for the B2 + A2

two-phase equilibria. This result is quite relevant for the aim of the present work, since tie-lines join, by definition, different phases which are stable at the same value of chemical potential (or, equivalently, of thermochemical activity). In other words, tie-lines indicate the position of isoactivity lines in a region of the phase diagram. The agreement between calculation and experiment, thus, shows that the functional relationships between species activities and composition in the ternary phase diagram are being, at least qualitatively, reproduced by the extrapolation procedure using the data fitted to the binary limit systems. The topology of the calculated phase diagram is also in agreement with the experimental results. All four isotherms show a two-phase field for the A2 + B2 equilibria, which is stable at chromium-rich compositions and closes at two multicritical points as the chromium content of the alloy

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0.0

1.0

3.5 0.1

−log (aAl )

3

0.9

0.2

2.5 2

0.8

0.3

0.7

1.5 1 0.5

Experimental (Ettenberg et al.)

0.42

0.44

0.46

0.48

0.5

0.52

0.54

0.56

0.58

0.7

Fig. 5. Decadic logarithm of the aluminium activity in the Co–Al system at 1273 K as a function of the molar aluminium fraction. The solid line indicates the CVM with the parameters in Table 1, while the points are experimental data obtained by Ettenberg et al. [21]. The reference state is the liquid phase at T = 1273 K.

1.0 0.9

0.2

0.7

0.4

0.6

0.5

x Cr 0.5

A2

0.4

0.7

0.3 B2

0.8

0.2

0.9

Co

0.1 A2

D03

0.1

0.2

0.3

D03

0.4

0.3 0.2

B2

0.9 1.0 0.0

A2 0.1

0.2

0.3

0.4

0.5

x Al

0.6

0.7

0.8

0.9

0.1 0.0 1.0 Al

0.8

0.3

0.6

A2

Fig. 7. B.c.c. Co–Cr–Al isotherm at 1473 K (1200 ◦ C) calculated with the CVM using the parameters in Table 1. Circled-ended lines: experimental B2/A2 tie-lines obtained by Ishikawa et al. [4]; squares: experimental 2nd order B2/A2 transition [4]; solid lines: calculated phase diagram; dashed lines: calculated tie-lines.

Cr

0.1

0.4

0.8

Co

0.0

x Cr 0.5

0.6

0.6

xAl

x Co

0.6

0.5

CVM calculation

0 0.4

1.0 0.0

0.4

x Co

0.5

x Al

0.6

0.7

0.8

0.9

0.0 1.0 Al

Fig. 6. B.c.c. Co–Cr–Al isotherm at 1273 K (1000 ◦ C) calculated with the CVM using the parameters in Table 1. Circled-ended lines: experimental B2/A2 tie-lines obtained by Ishikawa et al. [4]; squares: experimental 2nd order B2/A2 transition [4]; solid lines: calculated phase diagram; dashed lines: calculated tie-lines.

is reduced (both in the directions towards cobalt-rich and aluminium-rich alloys). In the experimental results only the cobalt-rich multicritical point can be observed, and only above T = 1473 K, probably due to the existence of competing, more stable equilibria with other phases at aluminium-rich compositions. Finally, the quantitative comparison between the calculated phase diagrams and the experimental data shows that the extent of the two-phase field between the B2 and A2 phases is underestimated in the calculation. The solubility limit of the A2 phase, however, is correctly reproduced at

the 1273 and 1473 K isotherms. At higher temperatures the calculated solubility limits of the A2 and of the B2 phases are both overestimated in the calculation. Several explanations may be proposed to justify the quantitative disagreement observed between the calculated and experimental two-phase fields in the ternary system which will be described in detail in what follows. Recent calculations by the present authors based on the Fe–Rh–Ti ternary system indicate that ternary interaction parameters are essential for the correct modelling of ternary systems if a quantitative agreement between calculation and experiments is targeted [14]. Indications for the importance of ternary interaction parameters in ternary b.c.c. alloys are also found at the stability region of the Heusler phase X2 TiAl (with X = Fe, Co, Ni and Cu) by Ishikawa et al. [25]. These authors attribute the variations of the relative stabilities of the Heusler and of the B2 phase, observed in these systems, to an “electron-concentration dependent” Ti–Al next-nearest neighbour pair interaction, which are equivalent in the formalism of configuration dependent cluster interactions used in the present work to ternary cluster interactions for configurations with the X2 AlTi composition. As discussed before, the aim of the present work is to verify the quality of the extrapolation obtained based on the fitting of binary thermochemical data, so no attempt was made to fit ternary interaction parameters for the system. The neglected spin degrees-of-freedom of cobalt and chromium are another possible source of error for the disagreement. It is a common assumption in the literature (see e.g. [9]) that the effect of the magnetic degrees-offreedom is negligible for equilibria above the Curie or N´eel

L.T.F. Eleno, C.G. Sch¨on / Computer Coupling of Phase Diagrams and Thermochemistry 27 (2003) 335–342 Cr 0.0

Cr 1.0

0.1

0.0 0.9

0.2

0.8 0.7

0.4

x Cr

0.4 A2

0.6 0.3

Co

0.4

0.4

A2

0.3

0.8

0.2 B2

B2

0.3

x Cr 0.5

0.7 0.2

0.2

0.6

0.5

0.8

0.1

0.7

0.4

x Co

0.5

0.6

1.0 0.0

0.8

0.3 0.6

0.9

0.9

0.2

0.5

0.7

1.0

0.1

0.3

x Co

341

A2 0.5

x Al

0.6

0.7

0.8

0.9

0.1

0.9

0.0 1.0 Al

Fig. 8. B.c.c. Co–Cr–Al isotherm at 1573 K (1300 ◦ C) calculated with the CVM using the parameters in Table 1. Circled-ended lines: experimental B2/A2 tie-lines obtained by Ishikawa et al. [4]; squares: experimental 2nd order B2/A2 transition [4]; solid lines: calculated phase diagram; dashed lines: calculated tie-lines.

temperatures. Recent results obtained by one of the present authors [13], however, show that short-range magnetic order may significantly influence the thermochemical properties, as is the case of the activities of components, and introduce asymmetries in otherwise symmetrical calculations. Some evidence for this, in the present calculation, would be the unusual asymmetry that has to be introduced in the interaction energies for fitting the thermochemical data on Co–Cr [18]. Another possible reason for the quantitative disagreement could be the fact that CVM calculations provide information about the so-called coherent phase equilibria (since the molar volume is not included as a variable in the present formalism), while experiments usually refer to the incoherent phase equilibria. The influence of large concentrations of thermal vacancies has also been postulated to explain deviations of observed and calculated secondorder B2/A2 phase equilibria in the Fe–Al system at high temperatures [12]. Finally, the disagreement could indicate simply that some non-configurational contribution (e.g. a temperature dependence of the interaction energies) is necessary for the proper modelling of the system. In fact, the agreement improves for temperatures closer to those used for fitting the binary data (1273 K). In spite of this disagreement, however, two results of the present work assure the thermodynamical self-consistency of CVM: • the qualitative features of the ternary phase diagram (i.e. the phase diagram topology) are correctly reproduced by the extrapolation from the binary data and

1.0 0.0

A2 0.1

0.2

0.3

0.4

0.5

x Al

Co

0.6

0.7

0.8

0.9

0.1 0.0 1.0 Al

Fig. 9. B.c.c. Co–Cr–Al isotherm at 1623 K (1350 ◦ C) calculated with the CVM using the parameters on Table 1. Circled-ended lines: experimental B2/A2 tie-lines obtained by Ishikawa et al. [4]; squares: experimental 2nd order B2/A2 transition [4]; solid lines: calculated phase diagram; dashed lines: calculated tie-lines.

• the experimental and calculated slopes of the tie-lines for the first-order equilibria agree remarkably well, showing that the qualitative features of the concentration dependence of the activity of the species are correctly reproduced in the calculation. The last result, in particular, is important since the concentration dependence of thermal and thermochemical properties is predicted to be strongly nonlinear (see [26, 27]). The use of the CVM in the CALPHAD approach has the potential, thus, to allow the modelling of these complex, nonlinear dependencies with a relatively simple energy model, as the one used in the present work. 4. Conclusions Four isothermal sections of the ternary Co–Cr–Al system have been obtained by CVM modelling in the present work. The energy parameters used in the calculation have been derived by fitting to thermochemical data for the binary subsystems and checked for consistency based on experimental phase diagram data, both on the binaries and on the ternary systems. The results show a remarkable agreement between calculated and experimentally observed slopes of firstorder tie-lines in the ternary phase diagram, which indicate that the qualitative features of the composition dependence of activities in this system can be well reproduced by extrapolation from binary data using the CVM. The correct prediction of the phase diagram topology is also worth mentioning and shows that CVM based extrapolations have

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