Ordering and phase separation in the b.c.c. phase of the Fe–Al–Ti system

PII:

Acta mater. Vol. 46, No. 6, pp. 2083±2094, 1998 # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain 1359-6454/98 $19.00 + 0.00 S1359-6454(97)00376-5

ORDERING AND PHASE SEPARATION IN THE b.c.c. PHASE OF THE Fe±Al±Ti SYSTEM I. OHNUMA1, C. G. SCHOÈN2, R. KAINUMA1, G. INDEN2 and K. ISHIDA1 1

Department of Materials Science, Faculty of Engineering, Tohoku University, Sendai 980-77, Japan and 2Max-Planck-Institut fuÈr Eisenforschung, D-4000 DuÈsseldorf, Germany (Received 15 July 1997; accepted 15 October 1997)

AbstractÐOrdering and phase separation in a-Fe alloys of the Fe±Al±Ti system have been investigated by di€erential scanning calorimetry (DSC), transmission electron microscopy (TEM) and electron-probe microanalysis (EPMA). The results indicate that the partial replacement of Fe with Ti in the Fe-rich Fe±Al binary alloys results in drastically stabilizing the B2 (FeAl) and D03 (Fe3Al) phases and increasing the A2± B2±D03 successive order±disorder transition temperatures. It is con®rmed that there are two kinds of phase separations of the BCC phase, (A2 + D03) and (B2 + D03) in the composition range below 25 at.%Al. The width of the two-phase region drastically increases with decreasing Al composition. The two-phase regions close at ternary tricritical points. Calculations of BCC phase equilibria have been performed with the cluster variation method (CVM) using the irregular tetrahedron approximation. First and second nearest pair interactions as well as tetrahedron interactions are taken into account. Starting from binary and ternary phase equilibria the numerical values of these interactions have been determined and an excellent agreement between calculation and experiments was obtained. # 1998 Acta Metallurgica Inc. ReÂsumeÂÐLes reÂactions ordre±deÂsordre et de deÂmixtion dans la phase cubique centreÂe des alliages Fe±Al± Ti ont eÂte eÂtudieÂes par calorimeÂtrie di€eÂrentielle aÁ balayage (DSC), par microscopie eÂlectronique aÁ transmission (MET) et par microsonde. Les reÂsultats montrent que le remplacement de Fe par Ti stabilise les phases ordonneÂes B2 (FeAl) et D03 (Fe3Al) en augmentant les tempeÂratures critiques correspondantes. Deux reÂgions de deÂmixtion ont eÂte observeÂes dans le domaine de concentrations infeÂrieur aÁ 25 %at Al, (A2 + B2) et (B2 + D03). La largeur des domaines biphaseÂs deÂcroõà t avec addition d'Al et ils se ferment aÁ un point tricritique ternaire. Les eÂquilibres de phases ont eÂte calculeÂs par la meÂthode variationelle d'amas (CVM) dans l'approximation du teÂtraeÁdre irreÂgulier. Des interactions de paires entre premiers et seconds voisins ainsi que des interactions teÂtraeÂdriques ont eÂte deÂtermineÂes aÁ partir des systeÁmes binaires limitrophes. En prenant ces valeurs binaires sans introduire des termes ternaires un bon accord entre calculs et expeÂriences a eÂte obtenu. # 1998 Acta Metallurgica Inc. ZusammenfassungÐOrdnungs- und Entmischungsreaktionen im a-Mischkristallbereich des Systems Fe±Al± Ti wurden mit Hilfe der Di€erential-Scanning-Kalorimetrie (DSC), der Transmissionselektronenmikropskopie (TEM) und der Mikrosonde (EPMA) bestimmt. Die Ergebnisse zeigen, daû der Austausch von Fe durch Ti die Ordnungsphasen B2 (FeAl) und D03 (Fe3Al) stabilisiert und die entsprechenden Fernordnungstemperaturen erhoÈht. Im Konzentrationsbereich unter 25 at.% Al wurden die Zweiphasengebiete (A2 + D03) und (B2 + D03) beobachtet. Diese Zweiphasengebiete verengen sich mit zunehmendem AlGehalt und schlieûen sich in einem ternaÈren trikritischen Punkt. Es wurden Rechnungen mit der Cluster Variationsmethode (CVM) in der TetraedernaÈherung durchgefuÈhrt unter BeruÈcksichtigung von Paarwechselwirkungen zwischen ersten und zweitnaÈchsten Nachbarn sowie Tetraederwechselwirkungen. Diese Wechselwirkungsparameter wurden in den binaÈren Randsystemen zahlenmaÈûig bestimmt und bei den Gleichgewichtsrechnungen im ternaÈren System verwendet. Es konnte eine gute UÈbereinstimmung zwischen Rechnung und Experiment erzielt werden. # 1998 Acta Metallurgica Inc.

1. INTRODUCTION

{Strukturbericht designation of binary b.c.c. ordered structures. This notation will also be used for ternary variants of B2 and D03 rather than the historic notation L20 and L21. As shown in Table 3 the D03 phase must be written A2AB rather than A3B in order to make the di€erence between A-atoms with di€erent neighborhood. It is only for a particular ratio between ®rst and second neighbor interactions that these positions become equivalent. It is unlikely to encounter this special case in nature. Consequently, the D03 phase is the binary limitrophe Heusler phase A2BC.

Iron aluminides such as Fe3Al and FeAl, which exhibit the D03 and B2 structures{ as indicated in Fig. 1, have received considerable attention as candidates for high temperature structural materials due to their low costs, high strength and good oxidation resistance [1]. The disordered a-Fe (A2: disordered b.c.c.) alloys with less than 20 at.% Al also have attracted attention as corrosion resistant materials [2]. However, for practical applications the workability and the transition temperature

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OHNUMA et al.: ORDERING AND PHASE SEPARATION

Fig. 1. Phase diagrams of the Fe±Al and Fe±Al±Ti systems.

from ductile to brittle fracture need to be improved. Figure 1(a) shows the phase diagram of the Fe± Al binary system [3]. There are two kinds of 2ndorder order±disorder transitions in the a-Fe phase region, A2/B2 and B2/D03, and two kinds of phase separations, (A2 + B2) and (A2 + D03). It is known that the mechanical properties are a function of the degree of order of the materials. Therefore, it is important to know the e€ect of ternary additions on the ordering behavior of these materials for improving the mechanical properties by alloying elements. Previous investigations utilizing transmission electron microscopy (TEM) and thermal analysis of the Fe±Al±Ti alloys have shown that the e€ects of Ti addition are to increase the order±disorder transition temperatures [4±7] and to expand the (A2 + D03) phase ®eld [4]. Very recently, Palm et al. have determined the isothermal sections of the Fe±Al± Ti system at 8008 and 10008C and clari®ed that the a-Fe single-phase region exists in the composition triangle spanned by the composition lines FeAl±Fe3Al, Fe3Al±Fe2AlTi and Fe2AlTi±FeAl, as shown in the phase diagram of Fig. 1(b) [8]. It is important to know the regions of existence of homogeneous ordered phases and of two-phase regions in the BCC a-Fe phase ®eld. In the present study the e€ect of Ti addition on the long range ordering reactions and on decompositions in Fe±Al±Ti alloys are investigated.

Table 1. Nominal alloy composition and critical temperature of order±disorder transitions Alloy composition (at.%) Al

Ti

20 20 22 (22.1) 22.5 25 25 24.9 25 (24.0) 25 25 (25.1) 25 (25.9) 25 (24.2) 25 25.5 27.5 27.5 30 30 30 30 30 (30.8) 35 35 35 35 (36.0) 40 (39.5)

5 10 5 (5.2) 2.5 0 2.5 5 5 (6.8) 5 10 (10.3) 15 (15.5) 20 (20.5) 25 5 15 20 2.5 5 10 15 20 (19.8) 0 5 10 15 (14.2) 10 (9.9)

Transformation temp. (8C) T(A2/B2) C Ð Ð Ð 783 777 939 Ð 1099 Ð 1202 Ð Ð Ð Ð Ð Ð 1102 1174 1256 Ð Ð Ð 1264 Ð Ð Ð

The composition determined by chemical-analysis are indicated in the parentheses.

TC…B2=D03 † Ð Ð Ð 643 547 705 Ð 842 Ð 1018 1155 1193 1212 Ð 1164 1213 657 799 1034 1172 1210 Ð 791 Ð 1094 850

OHNUMA et al.: ORDERING AND PHASE SEPARATION

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2. EXPERIMENTAL PROCEDURES

The alloys were fabricated by an electromagnetic levitation furnace from pure Fe(99.9 wt%), pure Al(99.99 wt%) and pure Ti(99.7 wt%) under an argon atmosphere. The nominal compositions are given in Table 1. Some of them were chemically analyzed for checking the di€erence between nominal and actual compositions. The results of chemical analysis are shown in parentheses in Table 1. It is seen that the composition di€erence between nominal and analyzed data is within 1 at.% in most cases. Each alloy was cast into a Cu mould with 12 mm inner diameter. The order±disorder transition temperatures were determined by di€erential scanning calorimetry (DSC) in the temperature range below 10008C and by di€erential thermal analysis (DTA) in the range above 10008C. Cubic specimens with 3 mm edge were cut from the ingots after a solution treatment at 12008C for 24 h followed by ordering treatments at 6008C or 7008C for 168 h to obtain a high degree of D03 order. The DSC and DTA measurements were performed by heating or cooling at a rate of 38C/min. Transmission electron microscopic (TEM) observations were performed to identify the crystal structure. The specimens were equilibrated at 10008C or 11008C for 24 h up to 168 h in sealed quartz glass tubes with Ti getter and then quenched into 10% NaCl solution. Disk specimens with 3 mm in diameter and 1 mm thickness were cut by spark erosion out of the heat-treated specimens. Thin foils were prepared from the disk specimens by mechanical grinding to 0.2 mm thickness and dimpling and then electropolishing by twin jet method. The electrolyte consisted of one part of perchloric acid to six parts of butylglycol to ten parts of methanol. If this procedure was not successful, ion-milling was performed instead. The TEM observations were carried out using a JEM2000EX microscope. Di€usion couples and multiphase alloys were prepared for determining the phase equilibria. The multiphase specimens were solution treated at 12008C for 24 h and equilibrium treated at 8008C or 9008C for 672 h. Transparent quartz tubes were used to seal the specimens which were wrapped in Mo thin foil to avoid any contact with the tubes. After the equilibrium treatment, they were quenched into 10% NaCl solution. The preparation of di€usion couples was carried out in almost the same way as described in the previous paper [9]. Equilibration of the di€usion couples was conducted at 9008C for 336 h under the same condition as for the multiphase specimens. Equilibrium compositions were determined by electron-probe microanalysis (EPMA).

Fig. 2. DSC traces of Fe±25 at.% Al±5 at.% Ti alloy. 3. EXPERIMENTAL RESULTS

3.1. Order±disorder transition temperatures Figure 2 shows typical DSC and DTA curves with the order±disorder transition reactions. The ordering temperatures were determined by exothermic peaks during cooling. The results obtained from the DSC and DTA analysis are shown in Figs 3 and 4 and in Table 1. In order to con®rm that the observed signals are due to the ordering reactions, TEM observations were performed at room temperature. It is very dicult to suppress a second order transition by quenching. However, it is possible to conclude from TEM observations whether a disordered or an ordered state existed at high temperature by looking at the size of antiphase domain boundaries (APBs) which can be observed in dark-®eld images obtained with superlattice re¯ections. In the present instance the superlattice re¯ections (200) and (222) appear in both B2 and D03 superstructures, while the (111) re¯ection only appears in the D03 superstructure (indices according to the enlarged BCC unit cell a = 2ao). During quenching from a disordered state, the ordering reactions start simultaneously at many places without long range correlation of the choice of sublat-

Fig. 3. Order±disorder transition temperatures in the vertical sections (isopleths) Fe3Al±Fe2AlTi and Fe2AlTi±FeAl.

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OHNUMA et al.: ORDERING AND PHASE SEPARATION

Fig. 4. Order±disorder transition temperatures: symbols indicate experimental data, broken lines represent calculated second order iso-critical temperature contours using CVM and the energy parameters in Table 5. (a) A2/B2 transition. (b) B2/D03 transition. The second order lines become ®rst order at a multicritical point. Starting from this point the solid lines represent the D03 boundary of the (B2 + D03) two-phase ®eld.

tices. Thus, a large number of small antiphase domains is formed. On the contrary, if the ordering reactions take place at the annealing temperature there is time for the antiphase domains to grow much larger, usually larger than a selected

area of TEM observation. Consequently, one can identify the state of ordering at elevated temperature by an analysis of APB size appearing in the {200} and {111} dark-®eld images according to the following scheme:

OHNUMA et al.: ORDERING AND PHASE SEPARATION

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Fig. 6. SEM micrograph taken from the Fe±22.5 at.% Al± 5 at.% Ti alloy showing the A2(matrix) + D03 (precipitates) two-phase structure.

obtained from the TEM observation are plotted in Fig. 3. These data are in agreement with those from DSC measurements, and it is thus con®rmed that the exothermic reactions in the DSC and DTA curves shown in Fig. 2 correspond to TA2/B2 and C 3 T B2=D0 , respectively. C It is seen from Figs 3 and 4 that the addition of Ti into the Fe±Al binary a-Fe phase produces a strong stabilization of the B2 and D03 phases since 3 the critical temperatures TA2/B2 and T B2=D0 C C increase with increasing Ti content as already reported in previous investigations [4±7]. Figure 3 4(b) shows that a maximum of T B2=D0 is observed C near the composition Fe2TiAl (Fe±28 at.% Al± 20 at.% Ti). A previous investigation using MoÈssbauer spectroscopy has shown that Ti substitutes selectively on the Fe sublattice with Fe as nearest neighbors, according to the formula Fe2(Fe, Ti)Al [10]. These facts con®rm that the D03 phase is in fact the Fe2AlTi Heusler phase and not (Fe, Ti)3Al. 3.2. Phase separation Fig. 5. TEM observations with an incident beam direction [110]D03, alloy Fe±25 at.% Al±5 at.% Ti. (a) superlattice re¯ections (200) and (111). (b) dark-®eld image taken with (200)D03. (c) dark-®eld image taken with (111)D03.

(case I) there are the APBs in both the dark-®eld images: the alloy has been quenched from an A2 disordered structure, (case II) only the {111} dark-®eld image shows APBs: the alloy has been quenched from a B2 structure, (case III) there are no APBs in both images: the alloy had a fully ordered structure D03 before quenching [4]. An example is shown in Fig. 5. Since the Fe± 25 at.% Al±5 at.% Ti alloy quenched from 11008C is to be classi®ed into case (I) it can be concluded that this alloy was A2 phase at 11008C. All results

Two-phase microstructures were observed in the a-Fe region at 8008 and 9008C. A typical microstructure is shown in Fig. 6. The results of EPMA examinations are shown in Fig. 7 and Table 2. In the Fe±Al±Ti ternary system, a (B2 + D03) twophase ®eld appears in addition to an (A2 + D03) two-phase ®eld. It is seen in Fig. 7 that there is a strong relationship between the second order transition temperatures (marked by the hatching) and the boundaries of the two-phase ®elds opening up at multicritical points: the phase boundary located towards the less ordered phases (B2 relative to D03, A2 relative to B2 or D03) asymptotes the second order transition line, while the opposite boundary starts with a ®nite angle from the critical point. This behavior is analogous to binary alloys for which this has been discussed in detail in [11]. In the case of 8008C, the phase constitution changes

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OHNUMA et al.: ORDERING AND PHASE SEPARATION

Fig. 7. Isothermal sections showing the two-phase ®elds (A2 + B2) and (B2 + D03) in the BCC existence range. Hatched lines represent second order transitions. (a) 9008C. (b) 8008C.

from the (A2 + D03) to the (B2 + D03) two-phase structures with increasing Al composition due to the presence of the A2±B2 second-order transition. It is interesting to see that this two-phase ®eld opens up towards the Fe±Ti binary system, see Figs 7 and 8. Figure 9 shows the Fe±5 at.% Ti vertical section. The present results are in agreement with those reported by Mendiratta et al. [4]. It can be concluded that the addition of Ti in the Fe±Al alloys results in widening up the (A2 + D03) region.

4. CVM CALCULATIONS

4.1. Energy parameters The ordering reactions of the b.c.c. Fe±Al±Ti system will now be modelled using the cluster variation method in the irregular tetrahedron (IT) approximation. Details of the formulation of the CVM for multicomponent systems can be found in [12]. Here only those aspects shall be presented which are pertinent to the Fe±Ti±Al system. It will be shown that the ternary system can be treated purely on the

Table 2. Tie-line compositions of bcc phases in equilibrium Sample (at.%)

Temp. (8C)

Time (h)

Fe±20Al±5Ti

800

672

Fe±22.5Al±5Ti

800

672

Fe±24.9Al±5Ti

800

672

Fe±25Al±5Ti

800

672

Fe±20Al±10Ti (three-phase)

900

336

Di€usion couplea (Fe±20Al±10Ti/Fe±25Al)

900

336

Di€usion coupleb (Fe±20Al±10Ti/Fe±25Al)

900

336

a

Taken from two-phase structure formed in the di€usion zone of the couple. Taken by extrapolation to the phase boundary in concentration pro®le.

b

Phase

A2 D03 A2 D03 B2 D03 B2 D03 B2 D03 B2 D03 B2 D03

Equilibrium composition (at.%) Fe

Al

Ti

77.6 64.4 76.1 66.7 71.6 69.3 70.1 69.2 68.3 66.2 70.9 65.2 75.4 64.2

19.2 23.6 20.7 24.0 24.2 24.9 25.1 25.7 24.8 25.4 22.2 23.9 18.0 22.7

3.3 12.0 3.2 9.3 4.3 5.8 4.2 5.7 6.9 8.4 6.9 10.9 6.6 13.1

OHNUMA et al.: ORDERING AND PHASE SEPARATION

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Fig. 9. Vertical section at constant 5 at.% Ti.

of formation de®ne the corresponding tetrahedron energies. There are thus four energy parameters entering the description. Usually, the tetrahedron energies eabgd ijkl are expressed in terms of pair interactions between ®rst and second nearest neighbors, as, e.g. in [12]. These pair interactions may be considered as the leading terms of internal energy. Higher order cluster interactions may then be introduced as correction terms: Fig. 8. Vertical sections at constant Al contents. Hatched lines represent second order transitions. (a) section with 25 at.% Al. (b) section with 23 at.% Al.

basis of the binary subsystems, i.e. without introducing ternary interactions. However, the assessment of the binary subsystems Fe±Al and Fe±Ti has shown that in order to get a reasonable description of both the order±disorder transition temperatures and of the thermodynamic properties higher order cluster interactions than pairs have to be used. The CVM free energy functional O(r) is de®ned as NX …m * ‡ mj * ‡ mk * ‡ ml *†rabgd O…r†  U ÿ TS ÿ ijkl 4 i,j,k,l i

…1† …1† …1† …2† 1 …1† 1 …2† eabgd ijkl ˆ 6 …eik ‡ eil ‡ ejk ‡ ejl † ‡ 4 …eij ‡ ekl †

~ abd ~ agd ~ bgd ~ ijkl ‡ 12 …~eabg ijl ‡ e ijk ‡ e ikl ‡ e jkl † ‡ e

Setting the reference state as the mechanical mixture of the pure components the following transformations can be made: 1 X e~ mmmm , o ijkl ˆ ÿ~eijkl ‡ 4 mˆi,j,k,l o ijk ˆ ÿ~eijk ‡

i,j,k,l

In a binary system four stoichiometric compounds can be de®ned with a tetrahedron. Their enthalpies

1 X e~ nnn , 3 nˆi,j,k

1 …k† 1 …k† …k† o …k† ij ˆ ÿeij ‡ eii ‡ ejj 2 2

…1† where N is the total number of lattice points, {m*} = {mFe*, mAl*, mTi*} is a special choice of chemical potentials satisfying the relation aimi* = 0 [12], and rabgd ijkl is the probability to ®nd the species i, j, k, l on the sublattices a, b, g, d (Fig. 10 and Table 3), respectively. The sum over i, j, k, l runs over all species Fe, Ti and Al. The expression of the entropy in the IT approximation of the BCC lattice can be found in Ref. [12]. The internal energy is given by: X abgd abgd eijkl rijkl …2† U ˆ 6N

…3†

…4†

The energy of formation is then given as X X abgd abgd DF U ˆ U ÿ x i
0 Ui ˆ ÿ6N o ijkl
rijkl iˆA,B,...

i,j,k,l

…5† 0

where Ui are the energies of the pure components i. In a binary system the four stoichiometric compounds de®ned with the irregular tetrahedron are: A2AB (D03-I), AABB (B2), ABAB (B32), ABB2 (D03-II). The four corresponding tetrahedron energy parameters can be obtained from equation (5) using experimental enthalpies of formation of these stoichiometric compounds. Any other choice of four parameters out of those in

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OHNUMA et al.: ORDERING AND PHASE SEPARATION Table 3. Strukturbericht designation of b.c.c. ordered structures Space group

Sublattice

Positions

Atoms binary

Fm3m

a b g d

0, 0, 0 1 1 1 2,2,2 1 1 1 4,4,4 3 3 3 4,4,4

ternary

D03

B2

L21

L20

8A

8A

8A

8A

4A 4B

8B

4B 4C

8B/C

a

Strukturbericht.

Table 4. Relation between energy parameters and energies of formation of stoichiometric compounds. The superlattice designation is according to the Strukturbericht notation a

Con®guration

Superlattice

1 2 3 4 5 6

AAAA BBBB AAAB AABB ABAB ABBB

A2 A2 D03 (I) B2 B32 D03 (II)

System A B A±B

Energy of formation of stoichiometric compound 0 0

3 (2) ÿ2o(1) ABÿ2oAB ÿ4o(1) AB (2) ÿ2o(1) ABÿ3oABÿ6oABAB (1) 3 (2) ÿ2oABÿ2oABÿ6oABBB

Table 5. Numerical values of the parameters (1 k-unit = 1 kBK = 8.6  10ÿ5 eV) System A±B Fe±Al Ti±Fe Al±Ti

o(1) AB/k (meV)

o(2) AB/k (meV)

oabgd ABAB/k. . . (meV)

oabgd ABBB/k

840 (72) 790 (68) 1210 (104)

375 (32) ÿ525 (ÿ45) 600 (52)

ÿ35 (ÿ3) 450 (38.7) 0

0 0 0

equation (4) can be made, provided that one tetrahedron term is within the set [13]. Using less than four parameters implies that the enthalpies of formation are not all independent. For instance taking only pair interactions of ®rst and second neighbors introduces the constraints DF H D03 ÿI ˆ DF H D03 ÿII and DF H B32 ˆ 2
DF H D03 ÿI ÿ 12 DF H B2 . The ®rst constraint lends a symmetry to the system with respect to the equiatomic composition, the second disposes of the value of the enthalpy of the B32 phase. In the present instance pair interactions between (2) ®rst and second neighbors, o(1) AB and oAB, and two tetrahedron energies oABAB and oABBB have been selected. Table 4 shows the relation between the enthalpies of formation and the energy parameters. At ®rst sight this choice seems to introduce some bias into the treatment since tetrahedron terms only appear in the B-rich con®gurations. This is only apparent. Any other choice would lead to the same values in the expressions (equations (1) and (2)), although with di€erent values for the interaction parameters. In the three binary subsystems Fe±Al, Fe±Ti and Ti±Al the b.c.c. structure is stable only in limited composition and temperature ranges. The available {In Refs [12 and 15] the pair interaction parameters (k) were de®ned as W(k) FeAl=2oFeAl.

experimental data do not allow to conclude about any asymmetry in the enthalpy of formation. Consequently, oABBB=0 has been used as for a symmetric system. The numerical values for the binary subsystems will now be derived. 4.2. Fe±Al The values for the binary Fe±Al system were obtained in the following way: the pair interactions o(k) FeAl for ®rst and second neighbors, Table 5, have been taken from [12]{. These values were derived 3 =B2 from the critical temperature T D0 at 30 at.% Al C and from an extrapolated maximum temperature at 50 at.% Al, TB2/A2 11600 K. The calculated phase C

Fig. 10. Irregular tetrahedron and sublattice positions.

OHNUMA et al.: ORDERING AND PHASE SEPARATION

Fig. 11. Calculated b.c.c. phase diagrams of the Fe±Al system: 0: CVM calculation using only ®rst and second neighbor pair interactions o(k) FeAl=72,32; k = 1, 2 (meV). 1: CVM calculation using pair and tetrahedron interabgd actions o(k) FeAl=72,32; k = 1, 2 and oFeAlFeAl= ÿ 3 (meV). MC: Monte Carlo simulation using pair interactions up to 5th neighbors (o(k) FeAl/meV = 70.4, ÿ16.4, ÿ8.0; k = 1, 5) [20].

diagram is shown in Fig. 11 (labelled ``0'' for no tetrahedron interaction). While the D03/B2 transition temperature is rather well reproduced, the calculated B2/A2 transition temperature is not. According to this CVM calculation the extrapolated 11600 K is too low. In fact, summit value TB2/A2 C alloys with Al contents around 40±50 at.% Al are

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Fig. 13. Superposition of the experimental and calculated (b.c.c.) phase diagram of Ti±Al. Solid lines: Experimental diagram [19], dotted lines: CVM calculation using the pair interactions in Table 5.

known to exhibit a very high thermal vacancy concentration (up to several at.%) at temperatures close to the melting point [14]. This e€ect is not included in the CVM. Therefore, in this work only TB2/A2 values in the composition range up to C 40 at.% Al have been used to determine the tetrahedron interaction oFeAlFeAl given in Table 4. The corresponding calculated phase diagram is labelled ``1'' in Fig. 11. With this tetrahedron interaction the agreement with the experiments is excellent except in the range close to 50%Al. 4.3. Fe±Ti The values for the binary Ti±Fe system have been obtained by taking the ®rst neighbor pair interaction term from the enthalpy of formation as already done in [15]. The second neighbor pair and the tetrahedron interaction terms were obtained trying to ®t the phase boundaries of the two-phase ®eld (A2 + B2) at the titanium rich side of the phase diagram where the b.c.c. phase is stable. The numerical values obtained are given in Table 5 and the calculated phase diagram is shown in Fig. 12 together with experimental data for the (A2 + B2) two-phase equilibrium [16, 17] and the published phase diagram [18]. It was not possible to obtain better agreement. 4.4. Ti±Al

Fig. 12. Superposition of the experimental and calculated (b.c.c.) phase diagram of Fe±Ti. Dotted lines: stable phase diagram [18], solid lines: CVM calculation using the pair and tetrahedron interactions in Table 5.

The Ti±Al parameters could not be obtained by the same procedure since the BCC phase (b-Ti) is stable only in Ti-rich alloys at high temperatures. No information about atomic ordering in this existence domain has been reported. With this lack of information only pair parameters have been considered which had to be derived from experimental data in ternary alloys. The two pair energies in

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OHNUMA et al.: ORDERING AND PHASE SEPARATION

Fig. 14. Calculated isothermal section at 1173 K of the b.c.c. phase equilibria. Symbols: experiments (this work). CVM calculations using the binary interactions in Table 5.

Table 5 have been obtained from the position in composition and temperature of the maximum of the 2nd order transition D03/B2 shown in Fig. 4. In Fig. 13 the calculated b.c.c. phase diagram is superimposed to the experimental equilibrium diagram. In conformity with the experimental data the calculated b.c.c. phase equilibria are metastable, except at high Al contents where the A2/B2 transition temperature penetrates into the b-phase ®eld. So far there exists no evidence for such ordering in the binary system. It has to be mentioned, however, that a bending of the b/a phase boundary has been observed recently which has been interpreted as an indication of such ordering [19]. Evidence for this B2 ordering would require high temperature measurements, since the transition b(BCC) 4 a(hex.) cannot be suppressed during quenching. 4.5. Fe±Ti±Al The results of the ternary CVM calculations using the parameters in Table 5 are shown in Figs 4, 14 and 15. The slope of the A2/B2 isocritical temperature lines in Fig. 4(a) is in good agreement with the experiments. The same holds for the absolute values, except for the very high temperatures. This di€erence is inherited from the Fe±Al system where this di€erence has been discussed and justi®ed. {In Ref. [17] the pair interaction parameters were (k) de®ned as W(k) FeAl=2oFeAl.

Figure 4(b) shows the B2/D03 isocritical temperature contours. The agreement between experiment and calculation is acceptable. Position and temperature of the maximum has been used to derive the interactions of Ti±Al. Figures 14 and 15 represent calculated isothermal sections of the ternary system, together with the measured tie-lines. A surprisingly good agreement of both the phase boundaries and of the tie lines with experiments is obtained. 5. DISCUSSION

The good agreement between experiment and calculation in the ternary system is encouraging. It shows that a good thermodynamic description of ternary systems can be obtained already without invoking ternary interaction parameters. In the binary Fe±Al system there is still one qualitative feature in the phase diagram which is not reproduced with the set of interaction parameters in Table 5, that is the tricritical point at the A2/B2 transition at low temperatures. This e€ect cannot be obtained in the tetrahedron approximation. By means of Monte Carlo calculations it has been shown [20] that pair interactions between 4th neighbors and their sign control the tricritical point. The numerical values of the energy parameters used in [20] (o(k) FeAl/meV = 70.4, ÿ16.4, ÿ8.0; k = 1.5){ were derived from di€use scattering experiments [21]. The calculated phase diagram is shown in Fig. 11 (labelled MC). Although the

OHNUMA et al.: ORDERING AND PHASE SEPARATION

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Fig. 15. Calculated isothermal section at 1073 K of the b.c.c. phase equilibria. Symbols: experiments (this work). CVM calculations using the binary interactions in Table 5.

qualitative result of a tricritical point is obtained, the calculated critical temperatures are much too low and the quantitative description remains poor. Furthermore, the enthalpy of formation of the ordered phases turns out too small by a factor of about 1.5 compared to those obtained with the parameters in Table 5. It is interesting to notice that the shape of the calculated phase diagram seems to be all right if temperature or, equivalently, all the parameters are scaled up by the same factor of about 1.5. This has been done in [22] and the result with respect to the phase diagram is a rather good ®t of the experimental data. After this scaling up the parameters are, of course, no longer in accordance with the di€use scattering data. The value o(1) FeAl=70 meV of the ®rst neighbor pair interactions in [20] compares very well with o(1) FeAl=71 meV of the present work. A big di€erence appears in the value of o(2) FeAl. While in the present work the ordering in the second shell (D03type) is controlled by the interaction o(2) FeAl=32 meV, the corresponding value used in [20] is very small (4 meV). The D03 ordering must be induced by o(3) FeAl. In the BCC structure 3rd neighbors are located on the same sublattice. Such contributions cannot contribute to equation (5). In the Monte Carlo calculations a larger cluster than a tetrahedron must carry the e€ect of this interaction. Fourth neighbor interactions are between positions belonging to sublattices which are connected by nearest neighbor relationships, e.g. a±g, a±d etc.

Such contributions add to equation (5) even though the 4th neighbor distance is not within the tetrahedron cluster. A negative value of o(4) FeAl produces a weakening of the ordering tendency between 1st neighbors.

6. CONCLUSIONS

The A2/B2 and B2/D03 order±disorder transition temperatures in the BCC phase of the Fe±Al±Ti system were determined. Ti addition into Fe-rich Fe±Al binary alloys stabilizes both the B2 and the D03 phases resulting in an increase of the critical temperatures. The phase separations, (A2 + D03) and (B2 + D03), were observed and the equilibrium compositions of the phases A2, B2 and D03 (L21) were determined. These two-phase regions exist in the composition range below 25 at.% Al and the width of the two-phase region drastically increases with decreasing Al contents. Using only interaction parameters of the binary subsystems the BCC phase equilibria of the ternary system could be obtained in excellent agreement with experiments using the irregular tetrahedron CVM. AcknowledgementsÐThis work was supported by the Grant-in-Aids for Scienti®c Research from the Ministry of Education, Science, Sports and Culture, Japan. The authors acknowledge the support from the TEPCO Research Foundation. C. G. S. gratefully acknowledges

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the support of the German Academic Exchange Oce (DAAD) under grant no. A/93/17173. REFERENCES 1. Mckamey, C. G., DeVan, J. H., Tortorelli, P. F. and Sikka, V. K., J. Mater. Res., 1991, 6, 1779. 2. Davies, R. G., J. Phys. Chem. Solids, 1963, 24, 985. 3. Massalski, T. B., et al. (ed.), Binary Alloy Phase diagrams, 2nd Ed. ASM International, Metals Park, OH, 1990, p. 147. 4. Mendiratta, M. G., Ehlers, S. K. and Lipsitt, H. A., Metall. Trans., 1987, 18A, 509. 5. Fortnum, R. T. and Mikkola, D. E., Mater. Sci. Eng., 1987, 91, 223. 6. Sellers, C. H., Hyde, T. A. and O'Brien, T. K., J. Phys. Chem. Solids, 1994, 55, 505. 7. Anthony, L. and Fultz, B., Acta metall. mater., 1995, 43, 3885. 8. Palm, M., Thomas, N. and Inden, G., J. Phase Equil., 1995, 16, 209. 9. Kainuma, R., Palm, M. and Inden, G., Intermetallics, 1994, 2, 3221.

10. Athanassiadis, G., le Caer, G., Foct, J. and Rimlinger, L., Phys. Status Solidi (a), 1977, 40, 425. 11. Allen, S. M. and Cahn, J. W., Bull. Alloy Phase Diagrams, 1982, 3, 287. 12. Colinet, C., Inden, G. and Kikuchi, R., Acta metall. mater., 1993, 41, 1109. 13. SchoÈn, C. G. and Inden, G., J. Chim. Phys. , 1997, 94, 1143. 14. Krachler, R., Ipser, H., Sepiol, B. and Vogl, G., Intermetallics, 1994, 3, 83. 15. Inden, G., Z. Metallkde., 1975, 66, 648. 16. van Thyne, R. J., Kessler, H. D. and Hansen, M., Trans. ASM, 1952, 44, 974. 17. Dew-Hughes, D., Metall. Trans. A, 1980, 11, 1219. 18. Murray, J. L., in Phase Diagrams of Binary Titanium Alloys, ed. J. L. Murray. ASM International, Metals Park OH, 1987, pp. 99±111. 19. Kainuma, R., Palm, M. and Inden, G., Intermetallics, 1994, 2, 321. 20. Bichara, C. and Inden, G., Scr. Met., 1991, 25, 2607. 21. Schweika, W., Mater. Res. Soc. Symp. Proc., 1990, 166, 249. 22. Bichara, C. and Inden, G., Prog. Theor. Phys. , 1994, 115, 171Supplement.