A NOTE ON MODELS FOR PHASES WITH ORDER/DISORDER TRANSITIONS IN THERMODYNAMIC SOFTWARE AND DATABASES

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1 J. n. etall. Sect. B-etall. 53 (3) B (2017) Journal of nng and etallurgy, Secton B: etallurgy A NOTE ON ODELS FOR PHASES WITH ORDER/DISORDER TRANSITIONS IN THERODYNAIC SOFTWARE AND DATABASES Bo Sundman INSTN, CEA Saclay, France Abstract (Receved 01 August 2017; accepted 20 August 2017) The Calphad method s an mportant tool to assst the development of new alloys. There are several dfferent thermodynamc software and databases avalable for such calculatons. In some of these alloys there are s an mportant er/dser transton, lke n superalloys (N-based) or Al-T alloys. Ths paper descrbes the modelng of such systems and a new software that has mproved the mplementaton of the modelng of ths transton whch can extend the composton range of the applcaton of the model. Keyws: Calphad method; CEF model; Gbbs energy; Open Calphad software; Long range erng. 1. Introducton The use of the Calphad method s descrbed extensvly n many pares and n the book by Saunders and odownk [1]. In the Calphad method each phase s descrbed wth a model for the Gbbs energy as a functon of T, P and the phase consttuton. Wth approprate software and assessed databases the user can calculate phase equlbra, phase dagram and to smulate phase transformatons. The Calphad method has been mplemented n several commercal software and databases [2-5]. ore recently a free software has been developed wth mproved modelng and software technques. Ths wll be brefly descrbed n secton 3. x (2) 1 o where x s the mole fracton of consttuent, G s the Gbbs energy for consttuent relatve to a defned reference state. The second term s the confguratonal 2. The CEF model The Compound Energy Formalsm (CEF) s used n the Calphad method as a framework for several models for dfferent knds of phases. A phase can have several crystallographcally dfferent sublattces and as consttuents n these one can have atoms, ons or molecules and also vacances representng an empty lattce ste. On each sublattce the consttuents are assumed to mx randomly. A substtutonal soluton phase a wth a sngle lattce ste and wthout vacances has a Gbbs energy, G m per mole component descrbed by: o E phys G = (1) m x G RT x ln( x ) Gm G m x 1 * Correspondng author: bo.sundman@gmal.com Fgure 1. Four complex crystallne structures Dedcated to the memory of Professor Dragana Žvkovć DOI: /JB S

2 174 Bo Sundman / J 53 (3) B (2017) entropy where R s the gas constant and T the absolute E temperature. G m s the excess Gbbs energy ncludng phys nteractons between the consttuents and G m can be used to descrbe partcular physcal contrbutons for example from magnetc transtons. Ths and other knds of models are descrbed n detal n the book + by Lukas et al [9]. 2.1 Phases wth long range erng When a crystallne phase has several dfferent knds of lattce stes, for example an ntermetallc lke the, µ or Laves phases ths can be descrbed by specfyng several sublattces wth dfferent sets of consttuents. Several of these sublattces may have vacant stes and rather than model the Gbbs energy per mole components t s convenent to model the Gbbs energy per mole lattce stes.e. per mole of formula unts. Ths s denoted by a subscrpt, G, rather than m as n G m n eq. 1. A generalzaton of eq. 1 usng GCEF s: G ( I y ) o GI RT asy ln( s y ) s s (3) E phys + G G ys 1 (4) where I represent a compound or endmember of the phase specfyng + one consttuent n each sublattce s. I ( y) s the product of the consttuent fractons, ys specfed by I, and ths can be consdered as the probablty of havng ths compound n the I phase. o G I s the Gbbs energy of formaton of endmember I from the reference states of the components. In the confguratonal entropy a s s the number of stes on sublattce s. The mole fracton of a component can be calculated from the consttuent fractons as: s s by j j s x (5) k s s bkj y j js where G b j s the stochometrc factor of component n consttuent j. Ths model has been used successfully to descrbe many dfferent knds of phases wth long range erng (LRO) lke ntersttal soluton [6], ntermetallcs [7] and phases wth er/dser transtons [8]. But frequently the models have smplfed the real crystallography because the number of endmembers ncreases rapdly wth the number of consttuents on the sublattces. For example the phase has 5 crystallographc lattce stes but t s often descrbed wth just 3 to smplfy the calculatons and the database development. Even wth few sublattces there are often not enough data to determne values for all endmembers as the phases are stable only n a lmted composton and temperature range. The use of DFT calculatons to determne endmember energes can smplfy the modelng as shown by Fres and Sundman [10]. Addtonally many endmember energes calculated by DFT for the gven structure have magnary phonon frequences whch mean that they are not mechancally stable. If the unrelaxed value s taken t wll be more postve than could be expected for the endmember and f the structure s relaxed the crystallography wll change and no longer represent the correct structure. However, f the endmember s far fram any stable composton of the phase that may not be very mportant but a more a more severe problems s that the number of endmembers for a phase wth 5 components and where all elements are consdered on all sublattces requre 3125 DFT calculatons. Ths s qute a lot but as wll be explaned n the next secton there are ways to smplfy ths. 2.2 Parttonng of the Gbbs energy of LRO phases any of the complex crystallne lattces can be consdered as a dstorted substtutonal lattce and ths can be used to smplfy the descrpton of phases wth several sublattces. Ths fact suggests that one can partton the Gbbs energy expresson n two parts subst* G G ( x) G ( yjs ) (6) x y sy s s (7) y where y s s the sublattce consttuton, assumng that the consttuents are same as the components, and subst* G ( x ) descrbes a baselne of the Gbbs energy for the phase usng a substtutonal model lke eq. 1, G ( but wthout the confguratonal entropy. G ( yjs ) descrbes the confguratonal entropy and ncludes only those endmember energes that has a G lower Gbbs energy than the baselne,.e. for whch the phase s or could be expected to be stable. All endmembers that have energes above the baselne, n partcular + those representng structures that are mechancally unstable, can be gnored. Ths means there are only a few endmember parameters needed n G part representng compostons close to where the phase s stable or where DFT calculatons predct the phase may be stable. Ths smplfcaton wll reduce the number of endmembers that must be determned by DFT calculatons sgnfcantly. 2.3 Phases wth er/dser transtons A partcular case of LRO occur n some phases wth a smple lattce lke FCC, BCC and HCP. These phases are usually modeled wth a substtutonal

3 Bo Sundman / J 53 (3) B (2017) model lke eq. 1 but n some alloys these phases can have an erng transton where some consttuents prefer a certan set of stes. An example s the Au-Cu system where one can, at low temperature, have several ered structures based on the FCC lattce shown n the phase dagram n Fg.2(a) from an assessment by Sundman et al. [11]. The correspondng superlattces are shown n Fg For physcal reasons the ered forms of FCC should be modeled wth a sngle Gbbs energy functon whch requres at least a 4 sublattce o tetrahedron model n er to have a symmetrcal descrpton. As the transformatons from the A1 to the ered structures are frst er t s n prncple possble to descrbe each ered form as a separate phase but when dealng wth erng of the BCC phase there are frequently second er transtons between the dfferent superlattces, as shown n phase dagram for the Fe-Al system n Fg. 2(b) from an assessment by Sundman et al [13]. In such cases t s mandatory to have a sngle Gbbs energy expresson lke eq. 3 for these yphases. + The ered forms of BCC n Al-Fe are shown n Fg. 4 together wth the Heusler structure whch can appear n ternary system. These can be modeled wth a 4 sublattce model accng to eq. 3 wth a notaton as: (Al,Fe,Va)0.25(Al,Fe,Va)0.25(Al,Fe,Va)0.25(Al,Fe,Va)0.25 where Al and Fe can enter all sublattces. In er to be compatble wth the modelng of the B2 phase n the Al-N system the vacancy, denoted Va s also ncluded on all sublattces because the B2 phase n the Al-N system s stable at hgher concentraton that 50% Al due to the replacement of N atoms by vacances. There are 81 endmembers n the model for the ered BCC n the Al-Fe system but many of them descrbe the same state, they are permutatons and ths wll be further dscussed n secton 2.5. Wrtten explctly for a 4 sublattce model eq. 3 becomes: G y,1 y j, 2 yk,3 yl, 4 ogjkl +RT as ys ln( ys ) E G j k s Fgure 2. Calculated phase dagrams for Au-Cu and Fe-Al l phys G (8) G the expressons for EG and physg please refer For to the book by Lukas et al. [9]. odel compatblty s an mportant feature when ( assessng expermental and theoretcal data to descrbe the Gbbs energy of the phases n bnary and ternary system. The same models must be used for the same phases n dfferent systems n er to combne Fgure 3. The dsered FCC and three ered superlattces

4 176 Bo Sundman / J 53 (3) B (2017) Fgure 4. The dsered BCC and three ered superlattces them n a thermodynamc database. Another essental contrbuton to obtan compatblty s the SGTE unary database [12] whch provdes Gbbs energy functons for the pure elements, o G n many dfferent phases, ncludng lattce stabltes for such phases for whch the pure element s never stable. 2.4 Parttonng of the Gbbs energy for er/dser The FCC, BCC and HCP phases are common n many alloy systems and n er to combne assessments where the these phases have no erng transformaton wth system where the phases are descrbed wth sublattces for erng we can make use of a smlar knd of parttonng as descrbed n secton 2.2. But n ths case the baselne descrbes a real phase and the erng part descrbe an extra contrbuton due to the erng: ds G G ( x) G ( yjs) (9) G (10) G ( yjs) G ( yjs x) d ds where G descrbes the dsered Gbbs energy ( G ( x) wth a substtutonal o model lke eq. 1 and G ( yjs) G ( descrbes the contrbuton due to LRO usng G eq. 3. In ths case the confguratonal entropy s ncluded n all G calculatons. Ths should be zero when the phase s dsered and the smplest way to acheve ths s to calculate the Gbbs energy for the CEF model twce, once wth the orgnal consttuent fractons, y s, and once wth these replaced by the mole fractons usng x sy and take the dfference. s s 2.5 odel parameter permutatons A fnal problem when dealng wth erng n FCC, BCC and HCP s that many endmember parameters represent the same state. For example n Au-Cu system usng a 4 sublattce model for the FCC phase wth all superlattces the L1 2 ered AuCu 3 phase s descrbed by 4 dfferent endmembers that must be equal: o o o o GAu:Cu:Cu:Cu GCu:Au:Cu:Cu GCu:Cu:Au:Cu G Cu:C u:cu:au (11) An ered model for the FCC phase wth 4 consttuents modeled wth 4 sublattces has 256 endmembers but only 35 unque values. Ths s a complcaton for the admnstraton of databases as t s very easy to forget a permutaton whch wll make the phase always ered and gve wrong thermodynamc propertes even n the dsered state. But t s possble to mplement these permutatons n the software and thus the databases need only to provde values for the unque endmembers. 3. The Open Calphad software Snce 2013 there a group of dedcated scentsts has been engaged n the development of a new free thermodynamc software called Open Calphad [14] (OC). The OC software use modern technques to store model parameters together wth the best algorthms [15] to mnmze the Gbbs energy n er to calculate the equlbrum for a flexble set of condtons. There s also an applcaton software nterface whch makes t possble to ntegrate OC n smulaton software [16]. The OC software has mplemented the parttonng of the Gbbs energy n a new way whch reduces the calculaton tmes. Recently the OC software has also mplemented the permutaton of parameters n ered FCC, BCC and HCP phases usng a new technque that reduces the calculaton tmes even more. It s mportant for nterested scentsts to have access to a free software n er to develop new models or mproved versons of exstng models. The OC software s free wth a GNU lcense and can be downloaded from the Opencalphad webste [17] or the development verson from the gthub webste [18]. 4. Summary The method to model the Gbbs energy of phases wth LRO s brefly descrbed tgether wth some detals of handlng ths n the software and databases. In the current databases many smplfcatons n the

5 Bo Sundman / J 53 (3) B (2017) modelng has been made for the database management as well as to speed up the calculatons. For example the phase s often descrbed wth just 3 sublattces and n partcular the erng n FCC, BCC and HCP phases are descrbed wth a 2 sublattce model. In the case of the L1 2 phase descrbed wth 2 sublattces ths leads to a very unsymmetrc model and many ternary and hgher er parameters must be added to compensate for ths. Wth a 4 sublattce model ths s smplfed but a 4 sublattce model leads to a consderable ncrease the tme to calculate an equlbrum. Ths s due to the fact that the calculaton tme s roughly proportonal to the square of the number of consttuents and wth 4 sublattces the number of consttuents are doubled. But wth new software t s possble to fnd ways to speed up the calculatons and make t possble to use models that are closer to realty also n practcal calculatons. References [1] N. Sauders and P. odownk Calphad, Pergamon aterals seres Vol 1, Ed. R.W. Cahn [2] J.-O. Andersson, T. Helander, L. Höglund, P. Sh and B. Sundman, Calphad 26 (2002) [3] C.W. Bale, P. Chartrand, S.A. Degterov, G. Erksson, K. Hack,. Ben ahfound, J. elan con, A.D. Pelton and S. Petersen Calphad 26 (2002) [4] S.L. Chen, S. Danel, F. Zhang, Y.A. Chang, X.Y. Yan, F.Y. Xe, R. Schmd-Fetzer, W.A. Oates, Calphad, 26 (2002), [5] E. Kozeschnk and B. Buchmayr, n athematcal modellng of weld phenomena 5, London Insttute of aterals, (2001) p [6] K. Frsk, L: Dumtrescu,. Ekroth, B. Jansson, O. Kruse and B. Sundman J Phase Equl, 22 (2001) [7] A. Fern`andez Gullermet, Calphad, 6 (1982) [8] N. Dupn, I. Ansara, H. L. Lukas and B. Sundman J. Alloys and Comp, 247 (1997) [9] H.L. Lukas, S.G. Fres and B. Sundman, Computatonal Thermodynamcs, the Calphad method, Cambrdge Unverty Press, 2007 [10] S.G. Fres and B. Sundman, Phys. Rev. B 66 (2002) [11] B. Sundman, S.G. Fres, W.A. Oates, Calphad 22 (1998) [12] A.T. Dnsdale, Calphad 15 (1991) [13] B. Sundman, I. Ohnuma, N. Dupn, U.R. Kattner, S.G. Fres Acta ater. 57 (2009) [14] B. Sundman, U.R. Kattner,. Palumbo and S.G. Fres, Integr. ater. and anufac. Innov., 4:1 (2015) [15] B. Sundman, X.-G. Lu and H. Othan Comp. at. Sc. 101 (2015) [16] B. Sundman, U.R. Kattner, C. Sgl,. Stratmann, R. Le Teller,. Palumbo and S.G. Fres, Comp. at. Sc., 125 (2016) [17] [18] opencalphad repostory at

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