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1 Open Archve Toulouse Archve Ouverte (OATAO) OATAO s an open access repostory that collects the work of Toulouse researchers and makes t freely avalable over the web where possble. Ths s an author-deposted verson publshed n: Eprnts ID : 2306 To lnk to ths artcle : URL : To cte ths verson : onnetable, Damen and Lacaze, Jacques and Maugs, Phlppe and Sundman, B. ( 2008) A alphad assessment of Al Fe system wth the carbde modelled as an ordered form of the fcc phase. alphad, vol. 32 (n 2). pp ISSN Any correspondence concernng ths servce should be sent to the repostory admnstrator: staff-oatao@np-toulouse.fr

2 A alphad assessment of Al Fe system wth the κ carbde modelled as an ordered form of the fcc phase Damen onnetable a, Jacques Lacaze a, Phlppe Maugs b, Bo Sundman a, a IRIMAT, UPS/INPT/NRS ENSIAET, Toulouse, France b Arcelor Research SA, BP 30320, Mazères-lès-Metz, France Abstract As part of an extensve study devoted to the development of new hgh-al steels, a ALPHAD-type assessment of the Al Fe system has been carred out. In place of the usual cementte, these steels show precptaton of the so-called κ carbde that s an ordered form of austente. Inconsstences between the scarce expermental nformaton n the Al Fe system and extrapolatons from the avalable Al Fe and Al descrptons made t necessary to revse them. Ths has been done n part by usng new ab nto calculatons as well as phase dagram data. The κ carbde has been descrbed as an ordered form of the fcc phase, wth ts Gbbs energy calculated as the sum of a dsordered and an ordered contrbuton. The addtonal parameters needed to express the ordered part were evaluated usng phase equlbra nformaton and new ab nto data related to the κ carbde. Wth ths approach, addton of new alloyng elements, lke Mn or N, for extrapolaton to hgher-order systems should be straghtforward. Keywords: omputatonal thermodynamcs; Ab nto; Alloys; Phase dagram; Al Fe 1. Introducton The ternary system Al Fe has nterestng propertes for a new class of hgh-al steels. Addton of carbon to Al Fe expands the fcc regon (also called austente or γ and wth Strukturbercht desgnaton A1) n the ternary system and there s a cubc carbde called κ, wth the Strukturbercht E2 1, whch can form a eutectod structure wth bcc (also called ferrte and wth Strukturberct A2) n the same way as cementte when austente s cooled. The E2 1 structure, also known as perovskte, s an ordered L1 2 structure based on the fcc lattce wth Al at the cube corners and Fe on the cube faces and a carbon atom n the central octahedral ste. The Al Fe system has prevously been assessed wth the alphad method and usng ab nto data by Ohtan et al. [28]. They took nto account the smlarty of the L1 2 and the κ carbde but the extended austente regon n the ternary system and the solublty range of the κ carbde were not well orrespondng author. E-mal address: bosse@mse.kth.se (B. Sundman). descrbed. Furthermore, ther results from ab nto calculatons were found to dffer consderably from smlar data reported n the lterature and thus new calculatons were performed for the present study. Usng avalable assessments for the three bnary systems t was found that the extrapolatons gave too small solublty of Al n the austente. As explaned below ths could not be descrbed by any ternary parameter wthout a modfcaton of the thermodynamc model parameters for the fcc phase n Al Fe and Al. Ab nto calculatons of the metastable ordered and dsordered fcc phase n Al Fe and for a metastable cubc Al carbde were made to mprove the relablty of these extrapolatons. The use of ab nto data for alphad assessments s further dscussed n [37]. A alphad descrpton of the whole system ncludng the κ carbde has been assessed whch s compatble wth the exstng multcomponent databases [35]. 2. Lower-order systems The Al system has been assessed by Gröbner et al. [21], the Al Fe by Seersten [19] and revsed by Ohnuma [25] and do: /j.calphad

3 (a) Al. (b) Al Fe. (c) Fe. Fg. 1. Bnary phase dagrams. For the Al Fe dagram the second-order transton between A2 (bcc) and B2 (ordered bcc) and between paramagnetc (pm) and ferromagnetc (fm) regons are dashed. See the text for more detals. the Fe system by Gustafson [11]. The calculated phase dagrams are shown n Fg. 1 and n the Al Fe dagram the dashed lnes represent the second-order ferromagnetc orderng and the chemcal orderng between A2 and B2. The more complex bcc orderng wth Strukturbercht D0 3 has not been ncluded n the model and the dagram s thus not correct below 800 K on the Fe-rch sde. As explaned below the bnary systems Al and Al Fe were partally re-nvestgated for several reasons. Such modfcatons should not change the stable bnary systems unless there are new data and one must also check that there are no unexpected effects n ternary or hgher-order systems that have been assessed usng the orgnal bnary. The models used for the dfferent phases are dscussed n Secton 3 as they were selected n functon of the extrapolaton nto the ternary system. See ths secton for an explanaton of the notaton for the dfferent parameters used also n ths secton. For a comprehensve dscusson of thermodynamc modellng see Lukas et al. [34] Revson of the Al system In the Al assessment by Gröbner et al. [21] the sold fcc-al phase was never nvestgated and the parameters for ths phase found n most databases have been nvented wthout checkng the expermental data. The same s true for the parameters of the bcc phase, whch s not stable n ths bnary system. The solublty of n Al has been reported to be around 0.03 at.% by Shunk et al. [8]. The Gbbs energy of formaton of a metastable cubc Al carbde wth B1 structure was calculated by ab nto to be J/mol atom, as descrbed n Appendx. A regular soluton parameter was then ftted to the expermental solublty. The new parameters for the fcc phase are: G fcc Al: Gfcc Al G graphte = (1) L fcc Al:,Va = T. (2) Addtonally the parameters for the metastable bcc phase wth all ntersttal stes flled wth carbon were changed from zero to be of the same order of magntude as that for bcc ron and the nteracton parameter was ftted to the maxmum solublty of n bcc at 1473 K n Al Fe from Palm and Inden [20]. G bcc Al: Gfcc Al 3G graphte = T (3) L bcc Al:,Va = T. (4) These changes do not nfluence the phase dagram n Fg. 1(a) but the temperature for 3-phase equlbrum lqud +fcc + Al 4 3 changes from K to K. Ohtan et al. [28] had also revsed the Al 4 3 phase n ths system but the heat capacty for ther descrpton dffers consderably from that assessed by Gröbner et al. [21] above 1500 K. In ths work t was judged that the expermental nformaton from the ternary Al Fe dd not support any modfcaton of the descrpton of the Al 4 3 phase Revson of the Al Fe system The assessment by Seersten [19] has been used for more than a decade wth good results for many calculatons of Albased alloys. The B2-ordered phase n the Fe-rch sde was not of prmary mportance n the orgnal assessment and a revson of parameters for the B2 and D0 3 orderng, ncludng the magnetc parameters, has been made by Ohnuma [25]. In order to be compatble wth the B2 orderng n the Al N system substtutonal vacances were added by Dupn [26] but ths has no nfluence on the stable phase dagram. In the ternary Al Fe system some te-lnes for fcc + bcc and fcc + graphte were measured by Palm and Inden [20] and n Fg. 2(a) the extrapolaton from the bnares at 1473 K s shown together wth three expermental te-lnes. The drectons of the calculated te-lnes are good but the carbon content of the fcc phase n equlbrum wth bcc s almost double that of the expermental value. A ternary parameter n fcc can ncrease ts stablty and the solublty of Al n fcc n equlbrum wth bcc and thus decrease the carbon content. But t wll also ncrease the solublty of n fcc n equlbrum wth graphte and that solublty should decrease accordng to the thrd expermental te-lne between fcc and graphte. To ncrease the solublty of Al n fcc along the Al Fe sde the most mportant parameters are the bnary nteractons for fcc n Al Fe. These have been ftted to descrbe the small solublty of Al n fcc-fe and the

4 (a) The extrapolaton usng orgnal Al and Al Fe bnares. (b) The extrapolaton usng revsed Al and Al Fe bnares. Fg. 2. Ternary extrapolatons at 1473 K usng the old (left) and new (rght) bnares for Al and Al Fe. Two expermental ternary te-lnes between fcc and bcc from [20] and one between fcc and graphte are ncluded. The fcc, bcc and graphte phases only were consdered. even smaller solublty of Fe n fcc-al only. Thus there are reasons to beleve that the Gbbs energy for the metastable fcc phase relatve to the stable bcc phase at hgher Al contents s not correctly descrbed n the assessment by Seersten [19]. It s possble to change the descrpton of the fcc phase n the bnary Al Fe wth almost no changes to the stable bnary dagram. For such a change t s useful to have ab nto calculatons of the metastable fcc and also compare wth other systems based on the Al Fe bnary wth extended solublty of Al n fcc, for example the Al Fe N and Al Fe Mn systems. hanges must be made wth great care as they lead to modfcatons n several hgher-order systems. Ab nto calculatons of the Al Fe system should be consdered wth some crtcsm because they show that the ordered L1 2 phase, based on the fcc lattce, should be more stable than the ordered D0 3 phase although n realty D0 3 s the stable phase at low temperatures around 25 at.% Al. However, modellng the D0 3 s outsde the scope of ths work. New ab nto calculatons have been made n ths study n order to descrbe better the stablty of the ordered and dsordered fcc phase. The values are shown n Table 1 for the ordered forms of fcc and the B2 phases and n Table 2 for the dsordered fcc and bcc phases. The methods of the new calculatons are descrbed n the Appendx. As can be seen there s a wde range of calculated values for the same structure by dfferent authors due to dfferent approxmatons made n the calculatons, for example the values by Watson [23] do not nclude the magnetc propertes of Fe. When usng ab nto data n an assessment they should thus be treated as expermental data wth reasonable error estmates. Usng these data together wth the expermental data for the stable fcc phase n the bnary,.e. the γ -loop close to pure Fe and the solublty of Fe n sold Al, a new set of nteracton parameters of the dsordered fcc phase was assessed: 0 L fcc Al,Fe:Va = T 1 L fcc Al,Fe:Va = (5) 2 L fcc Al,Fe:Va = T. Table 1 Ab nto calculated enthalpes of formaton of ordered Al Fe alloys and alphad type assessed values for the L1 2, L1 0 and B2 structures Structure Mole fracton Al Enthalpy (kj/mol atom) Reference L LDA [31] L GGA [31] L LAG [31] L [33] L [23] L [30] L [36] L [28] L Assessed, ths work L [23] L [30] L Assessed, ths work B [36] B [30] B Assessed [19] L [23] L [30] L [36] L [28] A Assessed, ths work The values for B2 are ncluded for comparson. These regular soluton parameters descrbe an fcc phase wthout long-range order but nclude a short-range order (sro) contrbuton. It s possble to calculate the mxng enthalpy of an fcc phase wthout sro usng ab nto technques as descrbed by Zunger et al. [14]. These can be compared wth the excess Gbbs energy n the dsordered fcc phase wth the sro contrbuton removed accordng to an approxmaton by Abe and Sundman [29]: 0 L no sro Al,Fe:Va = T + 1.5u AlFe 1 L no sro Al,Fe:Va = (6) 2 L no sro Al,Fe:Va = T 1.5u AlFe u AlFe = T, (7)

5 Table 2 Ab nto calculated values by onnetable et al. [36] for dsordered bcc and fcc compared wth values from the assessed Al Fe system Mole fractons Structure E f E f Assessed value Al Fe (mev/atom) (kj/mol atom) (kj/mol atom) 1 0 A A A A A A A A A A A A A A (a) The γ -loop n Al Fe. The loop assessed by Seersten s short dashed. Expermental data from [1,2,6]. (b) The solublty of Fe n Al, the curve s dentcal to the prevous assessments. Expermental data from [3,4,13]. (c) The metastable fcc-ordered phase dagram for Al Fe. There s no ordered L1 2 phase on the Al-rch sde. Fg. 3. The stable and metastable fcc regons n Al Fe. where u AlFe represents the average bond energy between Al and Fe n the fcc tetrahedron descrbng the long-range order. Ths s used for the Gbbs energes of the ordered end members: o G fcc Al:Al:Al:Fe = 3u AlFe o G fcc Al:Al:Fe:Fe = 4u AlFe (8) o G fcc Al:Fe:Fe:Fe = 3u AlFe The values of the excess parameters n Eq. (5), the bond energy u AlFe and the ordered end member parameters were ftted to the ab nto data n Table 1, the ab nto calculated energes for the dsordered fcc phase n Table 2 and the telnes between fcc and bcc at 1473 K, see Fg. 2(b). In the same assessment the nformaton from the stable Al Fe phase dagram.e. the γ -loop on the Fe-rch sde and the expermental data on the solublty of Fe n fcc-al were also ftted. The changes of the γ -loop and solublty of Fe n Al are shown n Fg. 3. The new γ -loop s compared wth the prevous assessment of Seersten [19] whch s plotted wth dashed lnes and expermental nformaton. The expermental data on the solublty of Fe n fcc-al s shown n Fg. 3(b) together wth the calculated solublty lne. Ths solublty s very crtcal as Fe s an mportant mpurty n alumnum alloys. The assessed metastable-ordered fcc phase dagram s shown n Fg. 3(c). The stablty at hgher temperatures s lmted by the fact that the fcc phase and ts ordered forms must not become stable relatve to bcc. On the Al-rch sde the L1 2 - ordered phase s metastable even relatve to dsordered fcc. The ab nto calculatons of the dsordered fcc and bcc phases and the assessed values for dfferent compostons are lsted n Table 2. The agreement between the values s reasonable. 3. Thermodynamc models All models used are based on the ompound Energy Formalsm (EF), ths uses the pont approxmaton for the confguratonal entropy of mxng and sublattces to descrbe long-range order. Except for the E2 1 phase these models have been used many tmes n publshed assessments and only a bref summary of the equatons s gven here.

6 3.1. The lqud phase The lqud phase s modelled wth a substtutonal regular soluton model usng Redlch Kster seres for the excess Gbbs energy. G m = x o G +RT x ln(x ) + E G m (9) where x are the mole fracton of components and o G are the Gbbs energes of the components n the lqud state relatve to the SER (Stable Element Reference) state, the stable state at K and 1 bar. Functons for these Gbbs energes for the lqud and other phases are taken from SGTE [15]. The term multpled wth RT, where R s the gas constant and T the absolute temperature, s the confguratonal entropy and E G m s the excess Gbbs energy takng nto account both bnary and ternary nteractons E G m = ( x x j L j + ) x k L jk. (10) j> k> j The bnary nteracton parameters can be composton dependent usng a Redlch Kster seres L j = n (x x j ) ν ν L j. (11) ν= The bcc phase wth B2 orderng The B2 orderng n Al Fe s a second-order transformaton from the dsordered bcc phase (A2) and one must thus have the same Gbbs energy functon for both A2 and B2. To descrbe the B2 orderng the substtutonal sublattce s dvded nto two dentcal sublattces and one must take nto account that there are both ant-ste defects and vacancy defects (structural vacances) n the B2 lattces. The phase wll be called B2 also when there s ntersttal carbon n the phase. As there are 3 tmes as many ntersttal stes as there are substtutonal the model for B2 s: (Al, Fe, Va) 0.5 (Al, Fe, Va) 0.5 (Va, ) 3. (12) The fracton of the consttuents on each sublattce s denoted by the consttuent fracton y (s) where s s the sublattce and s the consttuent. The mole fracton of a component can be calculated by summng the content of ths component over all sublattces excludng the vacances: x = s s a s y (s) a s (1 y (s) (13) Va ). The contrbuton to the Gbbs energy due to the ferromagnetc transton, magn G m s descrbed wth a phenomenologcal model proposed by Inden [10] as a functon of the ure temperature and the Bohr magneton number, both of whch can be composton dependent. The Gbbs energy functon for the A2 and B2 phases s G m = j + RT s k y (1) y (2) a s j y (3) k o G jk y (s) ln(y (s) ) + magn G m + E G m. (14) The o G jk parameters represent the Gbbs energy of formaton of a compound wth a sngle consttuent n each sublattce wthn the soluton phase. The term multpled wth RT s the confguratonal entropy as a sum of the deal mxng n each sublattce separately weghted by the number of stes, a s. The excess Gbbs energy, E G m, ncludes terms representng nteracton between consttuents on each sublattce and also smultaneous nteracton between consttuents on dfferent sublattces. Explanatons of ths and the magn G m terms can be found n n the book by Lukas et al. [34] together wth a detaled dscusson of the orderng model Parttonng of the Gbbs energy The model n Eq. (14) can descrbe both the ordered B2 and the dsordered A2, n the latter case the ste fractons on the substtutonal sublattces are dentcal and equal to the mole fractons: y (1) = y (2) = x. (15) But as there are many systems wth a bcc phase wthout B2 orderng t s convenent, when developng multcomponent databases, that one separates the dsordered part of the Gbbs energy functon that can be descrbed wth the model (Al, Fe, Va) 1 (Va, ) 3 (16) from the ordered part and wrtes the Gbbs energy as a sum of two parts: G m = G ds m (x ) + G ord m (y ) (17) G ord m = Gord m (y ) G ord m (y = x ). (18) The dsordered part descrbes the whole Gbbs energy when the phase s dsordered,.e. when Eq. (15) apples. G ord m must thus be zero when the phase s dsordered and ths s acheved n Eq. (18) by calculatng the same Gbbs energy functon twce, once wth the orgnal ste fractons and once wth the ste fractons set equal to ther dsordered values. The functon G ord m thus contans parameters related to orderng only. Note that the ntersttal sublattce s not affected by the order/dsorder transton and all parameters for the ntersttal consttuents should be gven n the dsordered part Restrctng the thermal vacances The model for B2 has structural vacances n the substtutonal sublattces. In Al Fe system the fracton of such vacances s very small up to equatomc composton and then the B2 phase s no longer stable. However, n systems lke Al N the vacancy fracton n the B2 phase s very mportant and thus they must be ncluded n the model for any B2 phase

7 (and subsequently the A2 phase) whch should be compatble wth the Al N system. In the A2 phase these vacances have the same functon as thermal vacances but those are normally not ncluded n the modellng. In order to keep the fracton of these vacances very low the followng parameter s recommended o G A2 Va:Va = 30T. (19) Ths parameter ensures that there wll never be any sgnfcant fracton of thermal vacances n the A2 phase. In the B2 phase one may assess parameters lke L B2 L B2 Va:,Va:Va,Va:Va:Va = to descrbe the structural vacances as has been done for the Al N system by Dupn [26] The fcc phase and ts ordered forms The fcc phase, Structurbercht A1, wth an octahedral ntersttal sublattce for carbon has a B1 structure wth a hgh amount of vacancy defects n the ntersttal sublattce. There are other systems, for example T, wth a B1 structure wth lower fracton of vacances but for smplcty the fcc phase wth a small fracton of ntersttal carbon can be denoted A1. There are many types of orderng based on the fcc lattce but the most mportant cases are the L1 2 and L1 0 orderng whch can be modelled wth 4 substtutonal sublattces wth equal number of stes. Addng the ntersttal sublattce the model s (Al, Fe) 0.25 (Al, Fe) 0.25 (Al, Fe) 0.25 (Al, Fe) 0.25 (Va, ) 1. (20) As descrbed for the A2 and B2 phases the Gbbs energy for ths phase s usually parttoned nto a dsordered fcc phase wth one substtutonal and one ntersttal sublattce and an ordered part wth four substtutonal sublattces for the metallc components and one ntersttal. The dsordered model s: (Al, Fe) 1 (Va, ) 1. (21) The Gbbs energy expressons are smlar for the fcc-ordered phase as for the B2 gven n Eq. (14) except that there are fve sublattces nstead of three. The fcc phase s dsordered when t has the same fractons on all four substtutonal sublattces. Both the dsordered fcc phase and the ordered phases formed from the fcc lattce wll be descrbed wth the same Gbbs energy functon and t wll be denoted fcc unless the orderng s mportant. In the fcc phase sro s much more mportant than n the A2 phase. An approxmaton of ths sro for EF was derved by Sundman and Mohr [17] The κ phase The κ carbde wth Structurberct E2 1, has orderng both on the substtutonal and the ntersttal sublattces. On the metallc sublattces one has an L1 2 orderng wth Fe on the cube surfaces and Al at the cube corners. On the ntersttal sublattce carbon prefers the octahedral ste n the mddle of the cube where t has only Fe atoms as nearest neghbours,.e. 1/4 of the avalable stes. As the ntersttal sublattce s an fcc lattce ths s n prncple an L1 2 orderng of carbon and vacances. It s possble to model the E2 1 as an ordered fcc phase wth 4 sublattces for the metallc components and 4 for the ntersttal. However, ths would gve 8 sublattces wth 2 8 = 256 end members and although many of them are dentcal such a complex model s outsde the scope of ths paper. The necessary smplfcaton wll be to assume that there s only vacances n the 3 ntersttal sublattces that should be empty n the E2 1 structure but there may be vacances n the remanng one. Ths means that the κ phase wll be descrbed wth the followng model: (Al, Fe) 0.25 (Al, Fe) 0.25 (Al, Fe) 0.25 (Al, Fe) 0.25 (Va, ) (22) Wthout any ntersttal carbon ths model s dentcal to the fcc phase wth orderng descrbed for the bnary Al Fe system n Secton 2.2 and the correspondng parameters for the κ phase are coped from the fcc phase. As carbon enters only n 1/4 of the ntersttal stes that s avalable n the dsordered fcc phase accordng to Eq. (20), the values for the end member parameters o G E2 1 Fe:Fe:Fe:Fe: and o G E2 1 Al:Al:Al:Al: are approxmated by the values of the Gbbs energy of the fcc phase at these compostons accordng to Eq. (25). The maxmum solublty of wth ths model s 20 at.% whch s well outsde the stable range of the phase. The parameters for the ordered E2 1 phase wth the ntersttal sublattce flled wth carbon can be ftted to expermental solubltes and ab nto data. In the model for the E2 1 phase ts Gbbs energy s parttoned nto an ordered and dsordered phase as for fcc and bcc phases, Eq. (17). When extrapolatng to hgher-order systems one can take advantage of the fact that the E2 1 phase has a smlar model to fcc and copy known fcc parameters to the E2 1 model and obtan reasonable ft to expermental data wth no or few addtonal parameters. 4. Expermental data and results of the optmzaton The expermental work by Palm and Inden [20] at three temperatures, 1073, 1273 and 1473 K was the man nformaton used n addton to ther expermental nformaton for the lqudus. The expermental data was used to ft ternary parameters n lqud, fcc and κ phase. The calculated sothermal phase dagrams at 1073 K, 1273 K and 1473 K are shown n Fg. 4(a) (c) and the lqudus lnes n Fg. 4(d). onsderng the scatterng and possble naccuraces n the expermental nformaton, the overall ft obtaned s satsfactory. At 1073 and 1273 K there s no two-phase regon between bcc and graphte on the Al-rch sde of the κ phase as found expermentally. It would be possble to modfy the stablty of the Al 4 3 carbde slghtly to obtan a 3-phase regon bcc + κ + graphte but such a change was consdered outsde the scope of ths assessment. At 1273 K the two-phase regon fcc+bcc s more narrow than the expermental one but to obtan a better ft for ths regon the solublty lne for graphte n fcc should be shfted to hgher carbon content and ths would lead to too hgh solublty at 1473 K. The κ phase regon s smaller than expermentally found at both 1273 and 1473 K and extends a bt too much towards the bnary Al Fe sde at 1073 K.

8 (a) 1073 K. (b) 1273 K. (c) 1473 K. (d) Monovarant lqudus lnes. Fg. 4. alculated sothermal sectons and the monovarant lqudus lnes. For the sothermal sectons expermental data from [20] are shown, te-lnes are delmted by symbols, two-phase regons are denoted by the symbol and three-phase regons by. In (d) the frst sold phase found expermentally s denoted by for fcc, for bcc, for graphte and for the κ phase. At 1073 and 1273 K the calculated solublty of n bcc s much smaller than the expermental data but t would requre a sgnfcant change n the model parameters for the bcc to accommodate such a hgh solublty, n partcular snce the solublty seems to decrease wth ncreasng temperature. It should be nterestng to have further expermental work on ths. The calculated monovarant lnes of the lqud n equlbrum wth two sold phases are shown n Fg. 4(d). For the lqudus surface the dfferent symbols show the frst sold phase at varous compostons from Palm and Inden [20]. The temperatures n degree elsus at the nvarant four-phase equlbra are shown n the dagram and n Table 3 there s a comparson wth other assessments of the temperatures for the nvarants. Palm and Inden [20] assessed also two sopleths for fxed contents of and one for a fxed content of Al. Smlar sectons have been calculated and are shown n Fg. 5(a) (c) together wth expermental nformaton. The general agreement between the assessed and calculated sopleth sectons s very good. One has to take nto account that the lnes n these dagrams can shft sgnfcantly wth a very small amount of the phases present. In the present assessment the Al 4 3 phase s n equlbrum wth the κ phase at low temperatures and thus there are some addtonal calculated lnes compared to the dagram by Palm and Inden [20] Assessed parameters As the data nvolvng the lqud s for a very small temperature range a sngle temperature-ndependent parameter was used and ts fnal value was: L lq Al,,Fe = (23) For the dsordered fcc phase two parameters were used, the regular soluton parameters for Al and Fe wth the ntersttal sublattce flled wth. Ther fnal values were 0 L fcc Al,Fe: = T 1 L fcc Al,Fe: = (24) Addng more parameters dd not sgnfcantly mprove the ft and t became very dffcult to restrct the parameter values wthn reasonable lmts as there were so few expermental ponts.

9 Table 3 alculated temperatures n for the nvarant equlbra wth the lqud compared wth assessed values from other authors Phases Ths work [28] [20] [16] [12] lqud + fcc + bcc + κ lqud + bcc + graphte + κ lqud + fcc + graphte + κ (a) 5 at.%. (b) 10.5 at.%. (c) 22 at.% Al. (d) Magnfcaton of 22 at.% Al. Fg. 5. Isopleth calculatons for two fxed carbon compostons and one fxed Al composton. The symbols are expermental data by [20] and the sngle phase regons are denoted for fcc and for κ, two-phase regons for fcc + bcc, for fcc + graphte, + for fcc + κ, for bcc + graphte, for bcc + κ and the three-phase regons for fcc + bcc + graphte, for fcc + bcc + κ, for fcc + graphte + κ and for bcc + graphte + κ. The magnfed secton for 22 at.% Al shows the phase regons around the nvarant lqud + fcc + bcc + κ reacton at For the κ phase the parameters for the bnary Al Fe are almost dentcal to the fcc phase as descrbed above. The end members Al 0.25 and Fe 0.25 were set equal to the Gbbs energy of the fcc phase at x = 0.2.e. y = 0.25 usng the followng formula where M s ether Al or Fe: o G κ M: o G fcc M 0.25 o G graphte = 0.25( o G fcc M: o G fcc M o G graphte ) ( o G fcc M:Va o G M ) + RT (0.25 ln(0.25) ln(0.75)) L fcc M:,Va. (25) Note that the parameter o G κ M: s for 1.25 moles of atoms. The fnally assessed parameters for the dsordered kappa, those wthout carbon coped from the fcc phase n Al Fe, are o G Al: o G fcc Al 0.25 o G graphte = T (26) o G Fe: o G fcc Fe 0.25 o G graphte = T (27) o G Al:Va o G fcc Al = 100 (28) o G Fe:Va o G fcc Fe = 100 (29) L Fe:,Va = 2000 (30) 0 L Al,Fe:Va = 0 L fcc Al,Fe:Va = T (31) 1 L Al,Fe:Va = 1 L fcc Al,Fe:Va = (32) 2 L Al,Fe:Va = 2 L fcc Al,Fe:Va = T (33)

10 Table 4 The ab nto calculated values for the ordered E2 1 (κ) phase at varous compostons together wth the correspondng values from the alphad assessment Mole fractons E f E f Reference Al Fe (mev/atom) (kj/mol atom) (kj/mol atom) [36] Assessed ths work [36] Assessed ths work [28] [36] Assessed ths work [36] Assessed ths work 0 L Al,Fe: = T (34) 1 L Al,Fe: = T (35) L Al,Fe:,Va = T. (36) The end member parameters for pure Al and Fe n Eqs. (28) and (29) are set 100 J/mol hgher than the correspondng for the fcc phase to emphasze the smlarty but to avod that the κ phase becomes stable n the bnary Al Fe system. The temperature dependence may seem rather hgh but the rato between enthalpy and entropy parts of the parameters s around 1000 or larger whch s qute normal. However, one must be careful that the values obtaned do not lead to unreasonable extrapolatons at low or hgh temperatures. The assessed parameters for the ordered end members wth carbon are u AlFe = T (37) o G Al:Fe:Fe:Fe: = 3u AlFe (38) o G Al:Al:Fe:Fe: = 4u AlFe 5200 (39) o G Al:Al:Al:Fe: = 3u AlFe (40) L Al,Fe:Al,Fe: : : = L Al,Fe: :Al,Fe: : = = u AlFe, (41) where u AlFe s the average bond energy between Al and Fe n the fcc tetrahedra wth a carbon atom n the central octahedral ste. The astersk * n a sublattce means that the parameter s ndependent of the consttuent n that sublattce. The end member value for the Al 3 Fe compound could not be ftted to the ab nto value n Table 4 as that made t dffcult to descrbe the stable range of the κ phase, nstead t was assumed to be the same as that for the stable AlFe 3 end member. The end member values for Fe 4 and Al 4 for the κ phase was set smlar to the dsordered fcc phase at 20 at.% from the already assessed Fe and Al bnares as gven by Eq. (25). Several other parameters were allowed to vary durng the optmzaton but none of them had a sgnfcant nfluence and they were fnally set to zero. There s no nformaton how far below 1073 K the κ phase s stable and wthout ths nformaton the assessment gave that the κ phase became unstable around 800 K. That s not mpossble because graphte has dffcultes both to nucleate and to grow but a fcttous expermental pont was added that the κ should be stable at least at 500 K. Ths dd not change the assessed parameters sgnfcantly but made the κ phase stable down to 0 K. The metastable phase dagrams for the dsordered and dfferent ordered forms of the κ phase at 973 and 1273 K are shown n Fg. 6. In addton to the stable Fe 3 Al phase there are also metastable-ordered phases at the compostons Al 2 Fe 2 and Al 3 Fe respectvely wth varyng carbon content. As the κ phase s almost dentcal to the fcc phase along the Al Fe bnary t s nterestng to note that the κ phase jons wth the metastable-ordered L1 2 on the Al Fe bnary at 973 K. Also, there s a mscblty gap n the fcc phase on the Al-rch sde at 973 K. 5. Summary The Al Fe ternary system has been assessed usng both ab nto data and expermental nformaton on the phase dagram. The extrapolaton of the fcc and bcc phases from the bnares nto the ternary system ndcated that the relatve stablty of the bcc and fcc phases was not correct, see Fg. 2, and could not be corrected by any ternary parameters. Thus a reassessment of Al and Al Fe was made wth mnmal changes of the stable bnary descrptons. Other extrapolatons from the new bnary Al Fe were also checked, for example nto the Al Fe N system. Wth the orgnal Al Fe system the extrapolaton nto the ternary ftted well expermental data wthout any ternary parameters. Wth the modfed Al Fe bnary a small postve ternary parameter n the fcc phase was needed to have the same good ft but ths was consdered acceptable. In the Al Fe Mn system a prevously negatve ternary nteracton n fcc could be set to zero to obtan a reasonable ternary phase dagram. Whenever ab nto data s used to descrbe metastable orderng, lke here for the fcc phase n Al Fe, t s mportant to assess a reasonable descrpton for all possble ordered forms allowed by the model, not just the one currently mportant, because the bnary s used n many dfferent ternary assessments and one cannot revse t too many tmes. But there s stll a need to reassess the stable Al Fe bnary to ft the D0 3 orderng usng 4 sublattces also for the bcc phase for example to extrapolate nto the Al Fe T system. As already mentoned the solublty of carbon n the Al-rch bcc phase needs to be confrmed by new expermental work. The complete set of parameters for ths system can be obtaned as a TDB fle from the webste bosse. Acknowledgements omputer resources for ths job were provded by ALMIP (Toulouse, France) and the computer center GRID 5000 Naton-Wde Grd Expermental platform funded by the French Mnstery of Research through the AI GRID program. INRIA, NRS and RENATER and other contrbutng partners (see GRID5000 France. One of the authors, BS, s grateful for a senor research grant from NRS.

11 (a) 973 K. (b) 1273 K. Fg. 6. Isothermal metastable sectons at two temperatures wth only κ and A1. Appendx. Frst prncple methodology Full detals of the calculatons reported here can be found n onnetable et al. [36]. The ab nto approach s based on the Densty Functon Theory (DFT) developed by Hohenberg and Kohn [7,5]. The Perdew Burke Ernzerhof (PBE) [22] generalzed gradent approxmaton (GGA) has been employed for the exchange and correlaton functonal n ts spnpolarzed verson. The Venna ab nto smulaton package VASP developped by the Hafner s group [18], mplementng the projector augmented wave (PAW) method [24] was used. Brlloun-zone samplng was performed usng the Monkhorst- Pack scheme [9] wth a mesh grd centered on the k-pont. For the phases, the plane-wave energy cutoff s 600 ev. The on relaxatons and cell relaxatons were performed usng the standard conjugate gradent algorthms mplemented n the VASP code. The forces are fully relaxed wth a crteron for stoppng the structural optmzaton of 0.05 ev/å. The chosen energy cutoff, k-ponts and convergence parameters were checked to ensure a convergence n energy of the order of 1 mev per atom. The formaton enthalpes at 0 K are calculated takng as reference the followng pure phases: Fe n the bcc ferromagnetc state, Al n the fcc state, and n the damond state. The conventon that the formaton enthalpy s negatve for a stable phase and postve n the opposte case was used. Enthalpes are gven ether n kj per mole of atoms ether n mev per atoms. To smulate dsordered Fe x Al 1 x bnares the SQS ( Specal Quasrandom Structures ) framework as proposed by A. Zunger [14] was employed. In ths formalsm the fcc and bcc Fe x Al 1 x random bnary compounds for dfferent values of x was studed. The structures proposed by Jang et al. [27] was used for the bcc structure (x = 0.25, and x = 0.75) and the SQS structures gven by Sluter [32] for fcc structures (x = 1/6, 0.25, 1/3 and 0.5). References [1] M. Isawa, T. Murakamm, Knsoku-no-Kenkyu 4 (1927) 467. [2] F. Wever, A. Muller, Zetsch. Anorg. hem. 192 (1930) 340. [3] I.K. Edgar, Trans. AIME 180 (1948) 225. [4]. russard, F. Aubertn, Rev. Metal XLVI (1949) 661. [5] W. Kohn, Phys. Rev. Lett. 2 (1959) 393; W. Kohn, L.J. Sham, Phys. Rev. 140 (1965) A1133. [6] A.G. Lsnk, V.P. Skvorchuk, Dopovd Akad. Nauk. Ukr. RSR (1960). [7] P. Hohenberg, W. Kohn, Phys. Rev. 136 (1964) B864. [8] K. Anderko, F.A. Shunk, onsttuton of Bnary Alloys, Suppl. 2, McGraw-Hll, NY, ISBN: , [9] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (1976) [10] G. Inden, Physca 103B (1981) 82. [11] P. Gustafson, Scan. J. Metall. 14 (1985) 259. [12] E. Schürman, J. Schwenchen, Gessereforschung 38 (1986) 125. [13] A. Oscarsson, W.B. Hutchnson, H.-E. Ekström, D.P.E. Dckson,.J. Smensen, G.M. Raynaud, Z. Metallkde 79 (1988) 600. [14] A. Zunger, S.-H. We, L.G. Ferrera, J.E. Bernard, Phys. Rev. Lett. 65 (1990) 353. [15] A.T. Dnsdale, ALPHAD 15 (1991) 317. [16] H.K.. Kumar, V. Raghavan, J. Phase Equl. 12 (1991) 275. [17] B. Sundman, T. Mohr, Z. Metallkde 81 (1990) 251. [18] G. Kresse, J. Hafner, Phys. Rev. B 47 (1993) 558; 49 (1994) 14251; G. Kresse, J. Furthmüller, Phys. Rev. B 54 (1996) 11169; G. Kresse, J. Furthmüller, omput. Mater. Sc. 6 (1996) 15. [19] M. Seersten, SINTEF report STF-28F93051, Oslo, Norway, [20] M. Palm, G. Inden, Intermetallcs 3 (1995) 443. [21] J. Gröbner, H.L. Lukas, F. Aldnger, ALPHAD 20 (1996) 247. [22] J.P. Perdew, K. Burke, M. Ernzerhof, Phys. Rev. Lett. 77 (1996) (1997) [23] R.E. Watson, M. Wenert, Phys. Rev. B 58B (1998) [24] G. Kresse, D. Joubert, Phys. Rev. B 59 (1999) 1758; P.E. Blöchl, Phys. Rev. B 50 (1994) [25] I. Ohnuma, prvate communcaton [26] N. Dupn, I. Ansara, B. Sundman, ALPHAD 25 (2001) 279. [27]. Jang,. Wolverton, J. Sofo, L.-Q. hen, Z.-K. Lu, Phys. Rev. B 69 (2004) [28] H. Ohtan, M. Yamano, M. Hasebe, ISIJ Int. 44 (2004) [29] T. Abe, B. Sundman, ALPHAD 27 (2003) 403. [30] F. Lecherman, M. Fähnle, J.M. Sanchez, Intermetallcs 13 (2005) [31] H.L. Skrver, [32] M. Sluter, prvate communcatons. [33] M. Wdom, [34] H.L. Lukas, S.G. Fres, B. Sundman, omputatonal Thermodynamcs, ambrdge Unv. Press, ambrdge UK, ISBN: , [35] TFE5, thermodynamc steel database [36] D. onnetable, to be publshed. [37] P.E.A.,Turch, et al., ALPHAD 31 (2007) 4.