CVM CALCULATION OF THE PHASE DIAGRAM OF ... - Science Direct

Acta metall, mater. Vol. 41, No. 4, pp. 1109-1118, 1993

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CVM CALCULATION OF THE PHASE DIAGRAM OF b.c.c. Fe-Co-AI C. C O L I N E T l, G. I N D E N 2 and R. K I K U C H I 3 tLaboratoire de Thermodynamique et de Physico-Chimie Metallurgiques, E.N.S.E.E.G., St Martin d'Heres, France, 2Max-Planck-lnstitut fiir Eisenforschung GmbH, Dfisseldorf, Germany and 3Department of Materials Science and Engineering, UCLA, Los Angeles, Calif., U.S.A. (Received 28 August 1992)

Almtraet--The Cluster Variation Method (CVM) in the irregular tetrahedron approximation is used to calculate the phase diagram of the b.c.c, ternary system Fe-Co-AI. At first paramagnetic states are treated and chemical pair interactions between first and second nearest neighbours are taken into account in the calculation. The numerical values for these interactions are taken from separate studies. Complete isothermal sections at T = 1400, 1000, 800, 700, 600, 300 K and several vertical sections are presented. A ternary miscibility gap is found between ordered and disordered phases, (B2 + A2) as well as (DO3 + A2), and between the two ordered phases B2 and DO3. This miscibility gap closes at multicritical fines defined by the intersection of the second order critical temperature surfaces with the miscibility gap. In the binary F¢-Co system the Curie temperature is much higher than the critical temperature of chemical ordering. Therefore magnetic interactions cannot be neglected in this system nor in the Fe- and Co-rich ternary alloys. Magnetic interactions have been considered between nearest neighbours in a spin 1/2 treatment. Each element was treated as two components, AT and A~, and the CVM treatment thus dealt with a six component system. For T = 600 K a complete isothermal section is presented which includes the magnetic effects. R$mm&--Le mtthode variationeUe des amas dans l'approximation du tttratdre irrtgulier a 6t6 utiliste pour calculer le diagramme de phases du systtme ternaire cubic centr6 Fe-Co-AI. D'abord les 6tats paramagn&iques ont 6t6 considerts en tenant eompte des interactions entre premiers et seconds voisins dont les valeures numtriques ont 6t6 prises d'ttudes anttrieures. Les sections isothermes de T = 1400, 1000, 800, 700, 600, 300 K et plusieures sections verticales sont prtsenttes. Les calculs donnent une large lacune de miscib/lit6 entre phases d6sordonntes et ordonntes (types A2 + B2, DO3 + A2, B2 + DO3) qui se ferme par une ligne multicritique. La temptrature de Curie 6rant suptrieure aux transitions ordre-dtsordre atomiques dans des alliages riches en F e e t Co, les calculs ont 6t6 performts en tenant compte d'une interaction magnttique entre premiers voisins dans rapproximation de spin 1/2. Ce cas a 6t6 trait6 en prenant deux esptces, AT et A J,, pour chaque 616ment, ce qui conduit d un traitement CVM d six constituants. La section isotherme/t 600 K est prtstntte pour le cas ferromagnetique. Zmammenfamuug--Die Phasengleichgewichte des kubisch raumzentrierten Systems Fe-Co-AI wurden mit der Cluster Variationsmethode berechnet. Als Hauptcluster wurde das irregul~re Tetraeder gewihlt, das n/ichste und zweitn/ichste Paaarbindungen enthglt. Zun/ichst wurden die paramagnetischen Gleichgewichtszust~nde unter Berficksichtigung n~chster and zweitn/ichster Nachbarwechselwirkungen berechnet, deren Zahlenwerte aus frfiheren Arbeiten iibernommen wurden. Es werden isotherme Schnitte fiir die Temperaturen T = 1400, 1000, 800, 700, 600, 300 K und verschiedene vertikale Schnitte vorgestellt. Die Rechnungen ergeben tern/ire Mischungsl/icken vom Typ (B2 + A2), (1303+ A2) und (B2 + DO3). Diese Mischungslficken schlie~n sich in multikritischen Linien, die sich aus dem Schnitt zwischen den Flgchen der kritischen Fernordnungstemperaturen mit den Mischungsliicken ergeben. Im bin/iren Fe-Co System ist die Curie Temperatur sehr viel h6her als die kritische Fernordnungstemperatur. Daher k6nnen magnetische Effekte in den Fe- und Co-reichen Legierungen nicht vernachl~ssigt werden. Zur Beriiksichtigung dieser Effekete wurde den Atomen der Spin 1/2 zugeordnet und eine magnetische Paarwechselwirkung zwischen n/ichsten Nachbarn eingef/ihrt. Das fiihrt in der CVM zur Behandlung eines 6-Stoffsystems, for das ein voilst/indiger isothermer Schnitt bei 600 K berechnet wurde.

1. INTRODUCTION The ordering reactions and associated equilibria in binary body centered cubic (b.c.c.) alloys have been analyzed recently by Ackermann et al. [1] in a systematic study using the Cluster Variation Method (CVM) with an irregular tetrahedron as the basic cluster and using the M o n t e Carlo (MC) simulation. F o r a representative set of different numerical values

of first and second neighbour interchange energies it could be shown that both methods yield almost identical phase diagrams. This encourages us to apply the same type of C V M calculation to a ternary system taking the values of the interchange energies from the associated binary subsystems. The F c - C o - A I system is a very candidate since the binary systems F ¢ - C o and F e - A I are well known and exhibit wide ranges of composition where the b.c.c, structure is stable. It can

1109

1110

COLINET et al.: CVM CALCULATION OF b.c.c. Fe-Co-AI

thus be expected that this structure is also stable in a large composition range in the ternary system. The CVM has been written in [1] in the general form for any number of components. To the authors knowledge there exists no calculation in the literature of a phase diagram for a ternary b.c.c, system using the CVM with an irregular tetrahedron as a basic cluster. In this first attempt we consider a system with strong ordering tendencies in all three subsystems. We will consider only chemical interactions and therefore neglect the magnetic interactions although we know that these magnetic interactions cannot be neglected in the Fe-Co-A1 system. However, the magnetic effects on the phase diagram are confined to temperatures close to and below the Curie temperature. In the Fe-Co-A1 system the Curie temperature decreases strongly from the binary F e - C o system when A1 is added. Therefore these effects are sensible only close to the binary Fe-42o system and at low temperatures. Most of the equilibria reported here remain unaffected if magnetic interactions are included. The diagrams we will calculate describe "incoherent" phase equilibria, despite the fact that the considered phases are coherent. The word coherent is used here to denote phases which differ from each other, except for composition, merely by the arrangement of the atomic species on a common network of lattice sites. In the Fe-Co-A1 system we distinguish the disordered or short range ordered solid solution A2 and the ordered superstructures B2 and D03t. In the calculations all these phases, ordered or not, are assumed fully coherent with the parent b.c.c, lattice. In reality, however, the lattice parameters vary slightly with composition or degree of order. Therefore, in reality, the "coherent" phase equilibrium is subject to the network constraint which imposes a continuity in displacement along the interface and thus involves elastic energy terms in the equilibrium [2-4]. Since the present calculations do not take such elastic effects into account the present results should be compared with experimental diagrams obtained from "incoherent" equilibria. These equilibria are observed if the stresses can be partially or fully relaxed with the help of defects such as dislocations or grain boundaries. In the case of Fe-A1 both types of equilibria have been observed experimentally, one "coherent" [5] and one "incoherent" [6]. For a detailed discussion see Allen and Cahn [7, 8]. The experimental information about the ternary system F e - C o - A I is scarce, see the review by Raynor [9]. Ivanov [10] and Abson [11] reported the

~'Strukturbericht symbols characterizing binary superstructures. The same symbols will be used here for ternary superstructures if they are built according to the same scheme of sublatticc occupation, see equation (1). Some ternary superstructures have got individual symbols (L20 and L2t for ternary B2 and D03, respectively) but they are not commonly used.

existence of a miscibility gap in the ternary system. The term miscibility gap is usually used to characterize a separation into two phases of the same structure, thus differing only in their composition. This term is also used here to characterize a phase separation into an ordered and a disordered or into two ordered phases of the same crystal structure, here b.c.c. This miscibility gap was recently analyzed experimentally by Miyazaki et al. [12], by Ackermann and Inden [13] and by Argent [14]. The existence of a miscibility gap has been confirmed by the three groups, but Miyazaki arrived at a direction of the tie-lines vertical to the direction obtained by [13, 14]. This will be discussed later on, on the basis of the present calculations. The b.c.c./f.c.c, phase boundaries were analyzed by [13, 14]. The outline of this paper is as follows. In Section 2 we review some basic concepts of the CVM for multicomponent systems using the irregular tetrahedron as the basic cluster, In Section 3 we explain how the energy parameters, i.e. the first and second nearest neighbour interchange energies and the nearest neighbour magnetic interactions, have been obtained for the limiting binary subsystems. We present the calculated phase diagrams of the three binary subsystems. In Section 4 we present the results of our ternary and six component calculations as isothermal and vertical sections. Finally Section 5 summarizes our conclusions. 2. T H E O R Y

2.1. Tetrahedron approximation o f C V M The simplest cluster including first and second nearest neighbour interactions is an irregular tetrahedron with edges made of two different lengths as shown in Fig. 1. The sites called ct, fl, ~, t~ define four interpenetrating f.c.c, sublattices with a total number of N/4 lattice points each. The edges ~-fl and y-6 are second neighbour bonds and the rest are first neighbour bonds. The species Fe, Co, AI occupying the lattice points will be indicated by roman indices i , j , . . , which can take the values 1, 2 and 3 in the case of this ternary system since we are not considering spins of the atoms. We define the mean occupation of

L

a

Fig. 1. Irregular tetrahedron cluster used in the present calculations. This cluster includes first (~-~,, ~-~, fl-~, fl-J) and second (~-fl, ~ ) nearest neighbour bonds.

COLINET et al.: CVM CALCULATION OF b.c.c. Fe-Co-AI a sublattice site a by the species i as X7 = N~/(N/4), where N~ is the number of species i on the sublattice a. The various superstructures can then be defined by the equivalence of sublattice occupations A2:

x~ = x~ = x ; = x¢

B2:

x~ = x,~ # x~ = x , ~

D0~:

x , ~ = x~ # x~ # x¢

1111

2.3. Entropy expression

In the case of the b.c.c, lattice Kikuchi and Van Baal [15] were the first authors to give the expression of the configurational entropy for a general ordered state using an irregular tetrahedron as basic cluster L (Z,j~,)

S = -ksN(6~ (

ijkl

F213m: B32:

x~ = x~ ~ x~ = x~.

(1)

We note that any tetrahedron in the b.c.c, lattice always shares the vertices with each of the four sublattices. We thus have to consider only one type of tetrahedron. The basic variables of the treatment then are the probabilities Zist that a tetrahedron takes a configuration ijkl on the sublattice points a, fl, 7 and 6. The variables for subclusters, e.g. U~ 6 for the triangle, V~a and V~ for the pairs of second neigh° bours, Y,~ for the pairs of first neighbours and X~ for the point probabilities are derived as linear combinations of the Z ' s b) the reduction relations. This has been outlined explicitly in Ref. [1] using the same nomenclature. One additional quantity is useful in characterizing the system, the mole fraction x~ of each species. It is obtained from the relation x, = ~(x~ - x~ + x; + x~). 2.2. Energy expression

U = 6N ~

eijkIZijkl.

(2)

~jk~

In the present instance we assume that only pairwise interactions ~A0 U between first and second neighbours i and j (k = 1,2) are dominant. Then the tetrahedron energy eUkt can be expressed in terms of these pairwise interactions according to 7- e, + ~jk

~,~u + e~,~))• (3)

The fractions take care of the fact that first and second neighbour bonds are shared with 6 and 4 tetrahedra, respectively. It is common use to take the energy of the pure components in the same structure as the alloy as reference. This reference energy is U o = g ~ ( 4 ~ ) + 3£~2i})xi.

(4)

i

With this reference state the internal energy can be rewritten in terms of the so-called interchange energies WIk) = - z~-(k)u+ e I~) + ~z'(k)which take positive values for an ordering tendency (unlike bonds more strongly attractive than the average between like bonds) I t'l',Iz(l)-L

I£r(l)

1 I" blZ(2) 4-~,--,~

w~2).

euk~= - ~ , , -

qk

+EtL(r,7)

+ L(Y,k~' ) + L(Y{~) + L(Y~)]

ik

- 4,1~ [L(x~) + L ( x { ) + L ( x D + L(x~)]}

~ ~ ,,, + -

bl, r(I) rr jk

+ W~)))

(5)

(O)

where k B is Boltzmann's constant, N is the total number of lattice points and L ( x ) = x In(x). 2.4. Grand potential

In our phase diagram calculation we fix the chemical potentials rather than the compositions in the calculation. The thermodynamic function to be minimized in this instance is the grand potential D defined as f~(V, T, Iti) = U - T S - ~ It, N,

The internal energy is usually written in terms of cluster interactions of the basic cluster, here tetrahedron interactions. Since in the b.c.c, structure there are six tetrahedra per lattice point we may write

eUkl = 6 ~

- 3 y~ [L(U,~) + L ( V ~ ~) + L(U~*) + L (V~?)]

(7)

i where Iti are the chemical potentials of the species i. In our treatment the size of the system is determined by the total number of atoms, N, rather than by volume which is regarded as given by N. Furthermore, since we are not treating vacancies, the chemical potentials are not all independent and we may introduce a new set of chemical potentials It* which fulfill the relation Eit* =o. i

It is easy to see that the new set is given by It* = Iti - ½(itA+ Its + Itc)- The equilibrium state at constant temperature and constant N is then obtained by minimizing the potential fl* - (T, N, ItA, Its ) = U - T S - (NAIt* + Nsit* + Ncit~ ).

(8)

The potentials It* vary in the interval [ - o o , + oo]. The choice of the two independent components [i = A, B in (8)] is, of course, arbitrary. 2.5. Minimization o f grand potential by Natural Iteration Method ( N I M )

Combining equations (2), (6) and (8) we obtain the CVM grand potential which must be minimized in order to yield the equilibrium state. In binary systems two different minimization algorithms have

1112

COLINET et al.: CVM CALCULATION OF b.c.c. Fe-Co-AI

been used with success, the Newton Raphson method and the natural iteration method (NIM) developed by Kikuchi [16]. In the NIM the grand potential is minimized with respect to the tetrahedron probabilities using a Lagrange multiplier to take the normalization equation of the tetrahedron probabilities into account. The noteworthy feature of NIM is its stability: it always converges whatever initial values chosen in the iteration procedure provided that they are within the interval of existing states. The Newton Raphson method works with a set of independant variables. Sanchez and de Fontaine [17] and independently Aggarwal [18] proposed to use cluster correlation functions. The grand potential is then minimized with respect to these correlation functions using the Newton Raphson method. The method of correlation functions has recently been generalized to systems with any number of components by Inden and Pitsch [19]. Compared to the NIM the Newton Raphson method is a quadratic iteration scheme and thus converges faster than the NIM. In this work we have chosen the NIM to solve the minimization equations which can be written (see [1]) Zijkl : U

1/2 V -

l/4~i" l/24

Y-1/6 e x p ( - flEijkl)

(~(~t + ~?24+ ~t + ~t)~ (~ -)exp~-~-)

x exp-

(9)

where fl = l / ( k T ) and 2 is a Lagrange parameter introduced for the normalization of Z's. The U product takes account of the four triangles in the tetrahedron U = U~Pkr U~'fl~ U ~ U ~ , Y and V account for the pairs of nearest and next nearest neighbours Y = Y~ Y~ Y~ Y~6, V = V~a V~, and X is for the four points X = xTx~x~x~. The powers 1/2, 1/24, etc. originate in the entropy expression. The natural iteration algorithm for fixed temperature and chemical potentials is the following: (i) choice of a starting distribution Zijkl, (ii) calculation of the subcluster probabilities using the reduction relations, (iii) calculation of the new distribution ZUk~ from equation (9) with the value of 2 determined from the normalization equation, (iv) take these output Z ' s as new input values for the next iteration. After the iteration has converged, the Lagrange multiplier 2 is identified as

2.6. Phase diagram calculation If we denote by ~p and ~, two different ordered phases and take It*, /~* as independent potential variables the phase transition tp/~ is obtained from the condition (temperature and volume fixed) t~.*~ = f~m**,

~F = ~*,

M ~ = M*.

A first order phase transition is thus obtained by determining the intersection of the grand potentials of the two phases. Figure 2 shows an example of this kind of transition. The iterative calculation proceeds as follows: (i) choice of starting values for #* and /z*, (ii) calculation of the grand potentials f~** and f~*¢', (iii) modification of the values of the chemical potentials until the absolute value of the difference [ f l * * - t~**[ is less than the desired precision. A second order transition can be considered as a limiting situation of the preceding case: the branches f~** and f~** do not really intersect, they contact at the transition point with the same slope, see Fig. 3. The algorithm for finding the point of equal grand potentials for both phases can still be used in this case. The ordered phase has no metastable continuation into the range of stable lro states. The branch of the sro phase has continuation into the range of stable Iro states corresponding to an unstable state, i.e. a maximum of the grand potential. A second order transition point can be determined by calculating the second Hessian determinant, or we may plot [~**-f~**[ against the chemical potential and see the plot vanish linearly at the transition point.

-4O ~

"~

C

T=600K 0 *=

o

t(" +

stableB2 1

stableA2

L I

-42

.

.

.

.

39

.

. . 40

.

. 41

2 = f~*. laFe As mentioned before the NIM is not very sensitive to the choice of the numerical values of the starting distribution ZUkt. However, this method conserves symmetries. If the starting values represent some symmetry (e.g. B2) it is possible to converge to solutions which share the same symmetry elements (e.g. A2) but is not possible to converge to a less symmetrical solution (e.g. D0~ or B32).

[ k J/mol ]

Fig. 2. Variation of the grand potential of the ordered phase B2, ~,a2, and of the short range ordered phase A2, f~,A2, as a function of the chemical potential #~e= t + P-co+/~AI) at constant temperature and /~o #Fc-- ~(#F~ (the index * shall indicate that the equiatomie alloy has been taken as referencestateand that the total number of atoms has been fixed). At #fc = 39.7 kJ/mol a firstordertransition B2/A2 occurs given as an intersection of the corresponding grand potentials.

COLINET et al.: CVM CALCULATION OF b.c.c. Fe-Co-A1

1113

1600 -43.5

Tc ~

T = 1200 K

",,,,~ __ O

B2

I.tCo= 0

.--,

1200

#

"'q~" •

"44.0

800

.

.

.

-

te

@

, , • A2 E

4t

,

"g

4O0 stable B2 -44.5

.

.

.

.

38

.

, stable A2 .

.

.

I~Fe

0

40

39

0.0

[ k J/tool ]

i

i

i

i

0.2

0.4

0.6

0.8

Fe

Fig. 3. Variation of the grand potential of the ordered phase B2 and of the short range ordered phase A2 as a function of the chemical potential of Fe at constant temperature and #co.* At/~F~*----39.35 kJ/mol a second order transition B2/A2 occurs given by the endpoint of the grand potential f~,B2 which osculates f~,A2. 3. FIRST AND SECOND NEIGHBOUR PAIR INTERACTIONS

Since we are treating the CVM in the irregular tetrahedron approximation we can account only for first and second nearest neighbour pair interactions. Their numerical values can be derived from experimental data like critical temperatures, enthalpies of ordering, antiphase boundary energies. The numerical values are to be taken as "effective" interaction energies since it cannot be excluded that higher order interactions are different from zero and should be taken into account explicitly.

3.1. Fe-Co The b.c.c. F e - C o alloys are ferromagnetic and the atomic order/disorder transition B2/A2 takes place well below the Curie temperature. For most alloys T~c'c is well above the f.c.c./b.c.c, transition [22], see Fig. 4. Therefore a magnetic interaction has to be included which will be done in the way outlined in [1]: _:!9=,j _,jV!!)-Jl!)aiaj, where V(J) is the nonmagnetic interaction energy and -./!9 is the magnetic -ij interaction energy (proportional to the exchange integral). The parameter ai takes the values _+ 1 in a spin 1/2 treatment which will be used here. The numerical values of OF~F, ~(') and OCo~ r(])^ are fixed by the Curie temperatures of pure Fe and pure Co and J(IL F e t . o is obtained from the variation of T~ ¢c" in the binary system. This yields the numerical values given in Table 1.

Co

Fig. 4. Phase diagram of the Fe42o system. The solid lines indicate the f.c.c./b.c.c, transition taken from [27]. The magnetic (T~c'c') and atomic (T B2~A2) order/disorder transition temperatures have been calculated with the interchange energies in Table 1. The second order nature of these transitions is indicated by a hachure. Experimental data: (A2/B2 transition) • and • Ref. [23], ~ Ref. [28], O Ref. [29], [] Ref. [30], A Ref. [31], Curie temperature: V, • Ref. [22]. The chemical interchange energies w(k),, F~Co(k = 1, 2) have been determined previously [20] from the critical temperature T~02.~)A2 at the equiatomic composition and from the enthalpy of mixing. This led to the values given in Table 1. Figure 4 shows the transition temperature T a2 ~ A2 calculated for ferromagnetic alloys with the CVM using the values of the energy parameters in Table 1. The agreement with the experiments is excellent. For paramagnetic alloys the transition temperature is lower than for ferromagnetic alloys.

3.2. F e - A I In the Fe-AI system two order/disorder transitions are observed, D03/B2 and B2/A2. The critical temperatures of these transitions yield two independent pieces of information which can be used to determine wt~At (k = 1, 2). This has been done in [23] using CVM and Monte Carlo simulations. The resulting values are given in Table 1. The phase diagram calculated with these values is shown in Fig. 5 together with the experimental diagram. The overall agreement is quite satisfactory although there are still qualitative and quantitative discrepancies: (i) The experimentally observed two-phase field A2 + B2 is not obtained from the CVM calculations. They yield only a second order transition for B2/A2. This has recently been discussed [24] and it seems that a

Table 1. Chemicaland magnetic interchangeenergiesin meV

AM 41/4--I

1.0

i-j

WI~)

"" vW~9

j~l)

jo)

Fe-Co Fe-A1 Co-Al

43 145 310

0 64 130

-- 14 - 14 - 18.8

-- 18.8 0 0

- ij10)

- 2.1 - 3.3 - 3.3

Ref.

[20] [19, 21] [I9]

1114

COLINET et al.: CVM CALCULATION OF b.c.c. Fe42o-Al 1900

3.3. Co-AI liquid

AI

~"

% A2

14oo

0

|

I-

9o0

~

400

.

.

.

B2

,

i

0

.

.

.

.

•

25

§0

Fe

FeAI

Fig. 5. Phase diagram of the Fe-AI system: thin lines represent the experimental diagram [26, 27], thick lines represent the present calculations using the interchange energies in Table 1. Experimental data: I-1, V, • Ref. [25]; C) Ref. [36]. separation tendency between fourth neighbours is required in order to reproduce this two-phase field. (ii) A difference exists between the calculated and observed critical temperatures T m ~ ~ in the composition range 0.25 ~