AN INTRODUCTION TO THE CALPHAD METHOD AND THE COMPOUND ENERGY FORMALISM (CEF)

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1 a Revew Artcle AN INTRODUCTION TO THE CALPHAD METHOD AND THE COMPOUND ENERGY FORMALISM (CEF) Gabrele Caccaman 1 Abstract The CALPHAD (Calculaton of phase dagrams) method and the Compound Energy Formalsm (CEF) are brefly ntroduced and a few selected examples of ther applcaton are reported. Keywords: CALPHAD; Thermodynamcs; Compound energy formalsm; Alloys; Intermetallcs. UMA INTRODUÇÃO AO MÉTODO CALPHAD E AO FORMALISMO DA ENERGIA DOS COMPOSTOS (CEF) Resumo O método CALPHAD (Calculaton of Phase Dagrams) e o Formalsmo da Energa dos Compostos (CEF) são apresentados e alguns exemplos de suas aplcações são dscutdos. Palavras-chave: CALPHAD; Termodnâmca; Formalsmo da energa dos compostos; Lgas metálcas; Compostos ntermetálcos. 1 INTRODUCTION Desgn and preparaton, use, and eventually, recyclng and dsposal of materals requre a detaled knowledge of ther consttutonal propertes among whch, n partcular, phase stabltes and equlbra as a functon of the condtons the materals are subjected. Such knd of nformaton s provded, n a complete and concse way, by phase dagrams, whch are wdely used, especally n the case of non-stochometrc materals such as alloys, ceramcs, salt mxtures, etc. Tradtonally, phase dagrams have been determned by means of expermental nvestgatons whch become more and more dffcult and tme consumng wth the ncreasng number of components. Moreover, phase equlbra and thermodynamc propertes, unless strctly correlated, are generally determned by ndependent experments and may result nconsstent wth each other. The CALPHAD approach helps to overcome these dffcultes because t reduces the number of experments needed to nvestgate a system, assures the self-consstency between phase equlbra and thermodynamcs, makes easy to calculate any knd of sectons n mult-component phase dagrams, and provdes the thermodynamc bass for the subsequent modelng of knetc processes whch determne behavor and propertes of real materals. The CALPHAD approach was ntroduced n the early 70s of the past century by L. Kaufman and H. Bernsten [1] and later developed by several authors: see for nstance P.J. Spencer [2] for a bref hstory of CALPHAD. Moreover, the CALPHAD method and ts theoretcal as well as practcal aspects have been comprehensvely treated n a number of books: we may menton N. Saunders and A.P. Modownk [3], M. Hllert [4], H.L. Lukas et al. [5]. Nowadays CALPHAD s a very complex process where not only experments but also quantum mechancal and statstcal thermodynamc calculatons contrbute to the determnaton of the parametrsed expressons of the thermodynamc functons of the phases. Then, any calculaton or further modellng based on these parametrsed functons (e.g. phase feld modellng of crystal growth, smulaton of dffuson-controlled transformatons, etc.) s consdered part (or an extenson) of the CALPHAD approach. Nevertheless, the core of the method s stll represented by the block dagram shown n Fgure 1. It conssts n the determnaton of the Gbbs energy functon of all the phases nvolved n the system under 1 Unverstà degl Stud d Genova UNIGE, Genova, Lgura, Italy. E-mal: caccaman@chmca.unge.t. Alloy Phase Dagram Internatonal Commsson APDIC, Orlando, Florda, Unted States of Amerca / 2016 Assocação Braslera de Metalurga, Materas e Mneração. Publshed by ABM. Ths s an open access paper, publshed under the Creatve Commons CC BY-NC-ND lcense (Attrbuton-NonCommercal-NoDervs) - by-nc-nd/4.0/. 16 Tecnol. Metal. Mater. Mner., São Paulo, v. 13, n. 1, p , jan./mar. 2016

2 An ntroducton to the calphad method and the compound energy formalsm (CEF) 1.2 Selecton and Evaluaton of Input Data All the data whch may be related to the Gbbs energy of the phases must be used to evaluate the emprcal parameters a k. To ths end, several nput data may be consdered: - Expermental results (e.g. from DTA, DSC, calormetry, EMF, vapor pressure, LOM, SEM, X-ray dffracton, etc.); - Calculaton results (e.g. from ab nto and frst prncples calculatons, from sem-emprcal models, etc.); - Estmated data (e.g. by extrapolaton / nterpolaton procedures, or by applyng chemcal crtera based on the smlarty between elements, etc.); Fgure 1. Flowchart representng the core of the CALPHAD method. consderaton. They are parametrzed functons n that they depend on the state varables (pressure, temperature, composton, etc.) and on a seres of emprcal parameters characterstcs of a specfc phase n a specfc system. Once G (P, T, x) functons have been determned for each phase, they are collected n a thermodynamc database whch can be used to calculate phase equlbra, phase dagram and thermodynamc propertes of complex mult component systems. More n detal, the steps of the optmzaton process nclude: 1.1 Selecton of Models for the Gbbs Energy Functons The Gbbs energy of each phase competng for equlbrum n a gven system s expressed as a functon of the state varables (x ) and a number of emprcal parameters (a k ) (Equaton 1): G = G x 1,, x, a 1,..., a (1) k The G functons assume dfferent forms dependng on the nature of the phase : ths can be a stochometrc compound, a metallc lqud, a dsordered sold soluton, an ordered metallc soluton, an onc lqud, etc. The composton dependence of G s usually expressed accordng to the Compound Energy Formalsm (CEF), a general formalsm applcable to dfferent knd of phases. Notce that G must be defned and evaluated n a wde range of condtons, generally much wder than the stablty range of n the system under consderaton. A prelmnary crtcal evaluaton of the avalable lterature s mandatory to select the most relable data to be used. 1.3 Optmzaton of Model Parameters Emprcal parameters a k are evaluated by an error mnmzaton procedure such as non-lnear least squares (dfferent software packages use dfferent algorthms). An approprate weght factor must be assocated to each nput datum. Correlaton between nput data and model parameters must be checked. When necessary, constrants and approxmatons may be ntroduced to reduce the number of ndependent parameters to be evaluated. 1.4 Calculatons and Comparsons The Gbbs energy functons contanng the newly optmzed a k parameters are used to calculate phase and property dagrams whch may be compared to known equlbra or data. Metastable equlbra and thermodynamc functons n metastable condtons must also be checked. Extrapolaton to hgher order systems and comparson wth known data s also recommended. Unsatsfactory comparsons may lead to the adopton of more approprate thermodynamc models for selected phases or to the selecton of new or dfferent nput data. 1.5 Applcatons The optmzed nteracton parameters are generally used for the mplementaton of new databases or the mprovement of exstng ones. Then, databases for specfc materals may be used to predct phase equlbra and thermodynamc propertes n hgher order systems, or, n combnaton wth other data, to predct varous materals propertes such as mcrostructure and knetc-related Tecnol. Metal. Mater. Mner., São Paulo, v. 13, n. 1, p , jan./mar

3 Caccaman propertes, mechancal propertes, corroson resstance, etc. or to smulate materals behavor durng processng. One of the most attractve aspects of the CALPHAD approach s ts ncreasng predctve ablty. For nstance, as the number of system components ncreases, the number of new emprcal parameters needed to obtan a satsfactory descrpton of the Gbbs energy of the phases tend to decrease. In other words, f the optmzaton of bnary systems may requre the evaluaton of a relatvely hgh number of parameters (and, consequently, the need of a large number of expermental data), the optmzaton of a ternary system may requre the ntroducton of a lmted number of new parameters (n addton to those already evaluated for the bnary sub-systems), and consequently a relatvely lmted number of new expermental data. In practce, once the bnary and ternary nteracton parameters have been evaluated, quaternary and hgher order systems may be relably calculated wthout need of new emprcal parameters. Thanks to ths behavor t can be sad that the Calphad approach s essentally an optmzaton method for 2-3 component systems, whle, for mult-component systems t assumes, more and more, the characterstcs of a predcton method. Ths makes Calphad partcularly useful for the development of technologcally relevant materals whch are typcally mult-component. On the contrary, the man lmtaton of the Calphad approach s the mpossblty to predct the stablty of a mult-component phase not already present n lower-order systems. In the more recent years, ths lmtaton has partally been overcome by couplng thermodynamc modellng and ab nto calculatons. In fact, the stablty of phases whch may be expected to compete for equlbrum n a specfc systems can be calculated (typcally by DFT method) and ncluded n the Calphad equlbrum calculatons. 2 COMPOUND ENERGY FORMALISM The thermodynamc modelng of the phases,.e. the way we use to express ther thermodynamc propertes as a functon of the state varables, s certanly one of the crucal ponts n the CALPHAD approach. The mportance of the modelng was well expressed by Mats Hllert (p. 400)[4]: The purpose of modelng s two-fold. From a scentfc pont of vew one lkes to learn how nature functons. One way to gan knowledge s to defne some hypothess resultng n a model and test t by comparng the predctons from the model wth expermental nformaton. [ ] From a more technologcal pont of vew one lkes to predct the propertes of a partcular system n order to put t to effcent use n some practcal constructon or operaton. Then t s often most convenent to have a model whch yelds an analytcal expresson. In the smplest case modelng s just the selecton of a mathematcal form whch has proved useful, whether t s based on some physcal model or not. However experence shows that a model s usually more powerful f t s based upon physcally sound prncples. Wth such a model one can hope to make predctons outsde the tested range wth some confdence. If the Gbbs energy dependence on temperature and pressure, though complcated, may be represented by a general expresson, applcable to all knd of phases, the dependence on composton s strongly affected by nature and strength of the atomc nteractons. In ths case t s not possble to adopt a sngle model for all phases. The best soluton proposed untl now s the Compound Energy Formalsm (CEF) [6], a general formalsm whch allows to select the most approprate model for each phase n the framework of the same formalsm. Accordng to the CEF, the dfferent consttuents of a phase (neutral atoms, ons, vacances, etc.) are dstrbuted n one or more thermodynamc sublattces where they mx accordng to the classcal soluton theory (adoptng deal, regular, subregular, etc. behavor) wth dfferent nteracton parameters n dfferent sublattces. The Gbbs energy of the phase wll then depend on the ste fractons (.e. the mole fractons relatve to each sublattce) nstead of the overall composton. Thermodynamc sublattces can have dfferent physcal meanngs n dfferent phases: n onc compounds, for example, they can be used to separate anons and catons; n alloys they can be dentfed wth crystallographc sublattces. In ths case, for nstance, t s clear that atoms n the same crystallographc ste have the same nteracton wth ther surroundngs, whle atoms n dfferent stes generally have dfferent coordnaton and, then, dfferent nteracton wth the rest of the structure. Formally, the Gbbs energy of a phase can be expressed as the sum of three man terms as shown n Equaton 2: the reference term, one deal mxng term and an excess term. They can be followed by further terms n case further nteractons (such as magnetc, electrcal, etc.) occur: ref d ex mag G = G + G + G + G (2) In case of a pure element, second and thrd terms are mssng and any dependence on composton of course dsappears. Equaton 2 becomes Equaton 3: ref mag G = G + G (3) Where the terms n Equaton 3 are gven by Equatons 4 and 5: 18 Tecnol. Metal. Mater. Mner., São Paulo, v. 13, n. 1, p , jan./mar ln... ref 2 G T = A +B T+C T T+D T + (4) ln( 1) τ β mag G =RT f x + (5)

4 where A, B, C, D, are the emprcal parameters whch must be evaluated for any specfc element, on the bass of the expermental nformaton. In the magnetc term (Equaton 5), based on the model ntroduced by Inden [7] and subsequently adapted by Hllert and Jarl [8], β s the average magnetc moment per mole of atoms n Bohr magnetons, τ s the rato T/T C (T C = crtcal temperature for magnetc orderng), and f(τ) s a polynomal expresson obtaned by expandng Inden s descrpton of the magnetc heat capacty nto a power seres of τ. Temperature and magnetc parameters for pure elements are avalable n the PURE database by SGTE [9]. If two or more components are present, mxng can occur and all terms of Equaton 2 are sgnfcant. Dependng on the mxng (or non mxng) mechansm a few typcal stuatons may be dstngushed: 2.1 Substtutonal Random Solutons In the smplest case of random mxng, as n the lqud or n the termnal sold solutons, only one sublattce s needed and the three man terms of Equaton 2 are gven n Equaton 6: ref G = xg T ln d G =RT x x G = G + G +... ex ex 2 ex 3 where: x s, at the same tme, the ste fracton n the unque sublattce and the overall mole fracton of component. G ( T ) s the Gbbs energy of the pure component n the 2 phase (Equaton 3), ex 3 G and ex G are the bnary and ternary excess terms, respectvely. They are expressed accordng to Equatons 7 and 8: ex 3 n 1 n ex 2 ν ν j,j j = 1 j=+ 1 ν ( ) Tecnol. Metal. Mater. Mner., São Paulo, v. 13, n. 1, p , jan./mar An ntroducton to the calphad method and the compound energy formalsm (CEF) (6) G = xx L T x x (7) ( ) n 2 n 1 n j k j j k k = 1 j=+ 1k=j+ 1 G = x x x ul T +ul T +u L T (8) respectvely, wth the coeffcents gven by Equaton 9: 1 x xj xk 1 x xj xk u =x +, u j =x j+, x xj xk u k =x k+ 3 L temperature dependence s smlar to that of G T n Equatons 7 and 8 are emprcal parameters whose T gven n Equaton 4, as presented n Equaton 10: (9) 2 L T = A +B T+C Tln T+D T +... (10) For a bnary phase wth magnetc orderng the composton dependence of mag G (Equaton 5) results from the composton dependence of T C and β, whch are expressed by a Redlch Kster seres expanson (Equaton 11): ( ) T x =x T A +x T B +x x T x x C A C B C A B C A B = 0.. n ( x ) =xa ( A ) +xb ( B ) +xaxb ( xa xb) β β β β = 0.. n (11) where T C and β n Equaton 11 are expanson parameters to be evaluated on the bass of the expermental nformaton avalable. 2.2 Stochometrc Phases (or Compounds) Stochometrc phases or compounds are modelled by an n-sublattce model wth each sublattce occuped by only one element. As n the case of pure elements, mxng terms of the Gbbs energy are null. In ths case, Equaton 6 reduces to Equaton 12: G = xg T + G T (12) ref SER form SER where G n ts standard element reference state (SER) and form G T s the Gbbs energy of the pure component T s the Gbbs energy of formaton of the compound expressed, as a functon of temperature, accordng to Equaton Ordered Solutons Ordered solutons are ntermetallcs wth a reference stochometry, but charactersed by more or less extended solublty ranges. The solublty mechansm s generally due to the recprocal substtuton between components n ther crystallographc stes. In ths case each component can occupy two or more (possbly all) sublattces. A typcal example may be an ordered bnary soluton represented by a two-sublattce model where each sublattce s manly occuped by one component but ts substtuton by the other element s allowed. Then both elements can occupy both sublattces. When a new component s added, t wll preferentally occupy one of the sublattces, but t wll also be allowed to occupy the other one. Whenever more components are allowed to mx n two or more sublattces the Gbbs energy s expressed (s) as a functon of the ste fractons y, the mole fractons of each component n the sublattce s. Ste fractons obey the condtons gven by Equaton 13: (s) y = 1 s (s) n s (s) (s) n y ( 1 y ) VA =x (13) 19

5 Caccaman (s) where n are the stochometrc coeffcents relatng the sublattces. Notce that Equaton 13 has been formulated assumng that vacances (VA) can also be ncluded n one or more sublattces. Ths accounts for a second solublty mechansm, namely vacancy formaton n one or more sublattces. Equaton 6 becomes Equaton 14: ref ( 1) ( 2) ( s) G =... y yj... yk Gj : :...: k j k d ( s) ( s) ( s) G = RT n y ln ( s) y s ex ex 2 ex 3 s 1 n G = G + G +... = ( s) ( s) ( r) ex 3 y yj yk L,j:...: k T G+... s j r s k (14) where G j : :...: k n ref G are the Gbbs energes of all the so-called end members, the stochometrc compounds (ether stable, metastable or unstable) formed when only one consttuent s present n each sublattce. L,j:...:k ( T ) n ex 2 G are bnary nteracton parameters correspondng to the mxng of components and j on the sublattce s whle the other sublattces are sngly occuped. More terms can 2 be added to ex G, correspondng to smultaneous mxng on two sublattces whle the remanng sublattces are sngly 3 occuped. ex G accounts for the possble nteractons of three elements n a gven sublattce. A smple case s represented by the Laves phase MgCu 2 type (Pearson symbol cf24, strukturbercht desgnaton C15). It s descrbed n the space Group Fd-3m (num. 227). Mg occupes the 8a Wckoff poston, whle Cu s n 16d, but both elements are allowed to occupy the other poston, at least partally. Then the phase may be modelled by a 2-sublattce model: (8a) 1 (16d) 2 (Mg, Cu) 1 (Mg, Cu) 2 where each sublattce corresponds to a crystallographc ste and the 1:2 stochometry correspond to the rato 8:16 between the ste multplctes. Each sublattce s typcally occuped by the element reported n bold, but all the elements are allowed to enter all sublattces to reflect the substtuton mechansm responsble for the homogenety range of the phase. It s mportant to notce that the model adopted generates four so called end-members,.e. four stochometrc compounds whch correspond to all the possble combnatons wth each sublattce occuped by a sngle element: (Mg) 1 (Mg) 2 = 3 Mg (Mg) 1 (Cu) 2 = MgCu 2 (Cu) 1 (Mg) 2 = Mg 2 Cu (Cu) 1 (Cu) 2 = 3 Cu Among them, only the second one corresponds to a stable phase and can be evaluated on the bass of expermental nformaton. However the Gbbs energes of formaton of all of them, G Mg: Mg, G Mg: Cu, G Cu: Mg, G Cu: Cu, appear n Equaton 14. The three values whch correspond to metastable or even unstable compounds cannot be derved from expermental data, but, n many cases, can be drectly calculated by ab nto methods, or ndrectly evaluated on the bass of expermental phase equlbra. 2.4 Ordered Phases Related to Random Solutons A specal case, relatvely frequent n ntermetallc systems, are the ordered phases whch may be related to dsordered ones, as n the case of the AuCu 3 type phase (Pearson symbol tp2, strukturbercht desgnaton L1 2 ) related to the dsordered fcc structure (Cu type, Pearson symbol cf4, strukturbercht desgnaton A1), or the CsCl type phase (Pearson symbol cp2, strukturbercht desgnaton B2) related to the dsordered bcc structure (W type, Pearson symbol ci2, strukturbercht desgnaton A2). In some cases ordered and dsordered structures are stable n the same system and can transform to each other by a frst order as well as a second order transformaton. A typcal example of ths behavour can be found n the Al-Fe system. In ths system, when the transformaton s second order, A2 and B2 are n equlbrun at the same composton (no two-phase feld between them can exst) and ths mples that there s no dscontnuty between the Gbbs energy of the two phases (or between ther composton dervatves) at the equlbrum composton. Such a behavour may be modelled by ntroducng one sngle Gbbs energy expresson for both ordered and dsordered phases. The orderng state wll depend on the occupaton of the dfferent sublattces. Ths means that, n the case of A2/B2 orderng n Al-Fe, we can assume a two-sublattce model, (Al,Fe) 0.5 (Al,Fe) 0.5, whch become equvalent to (Al,Fe) 1 when the two sublattces have dentcal occupaton and, consequently, become ndstngushable. Ths s realsed by ntroducng the expresson for the Gbbs energy gven by Equaton 15: A 2 where G G =G x +G y,y G x,x (15) A 2/B2 A2 B2 B2 1 2 x s the Gbbs energy of the dsordered phase calculated at the concentraton x accordng to the one sublattce B2 B2 model (Equaton 6), whle G ( y 1,y 2) and G ( x, x ) are the Gbbs energes of the ordered phase (Equaton 14) calculated frst at the ste fractons y 1,y 2 (ordered state) and then at the ste fractons x, x (dsordered state). So, the dfference B2 B2 G ( y 1, y 2) G ( x, x ) s the Gbbs energy of orderng. If a 4 sublattce model s ntroduced, a larger number of ordered phases related to the same bcc lattce can be descrbed. Ths s llustrated n Fgure 2. The correspondence between ste occupances and structures s the followng: 20 Tecnol. Metal. Mater. Mner., São Paulo, v. 13, n. 1, p , jan./mar. 2016

6 An ntroducton to the calphad method and the compound energy formalsm (CEF) Fgure 2. Ordered structures based on the bcc lattce whch may be represented by a four sublattce model. Sublattce occupaton Correspondng structure (A)1/4(A)1/4(A)1/4(A)1/4 A2 ci2-w (A)1/4(A)1/4(B)1/4(B)1/4 B2 cp2-cscl (A)1/4(B)1/4(A)1/4(B)1/4 B32 cf16-natl (A)1/4(B)1/4(C)1/4(C)1/4 cf16-alscag2 (A)1/4(B)1/4(B)1/4(B)1/4 D03 cf16-bf3 (A)1/4(B)1/4(C)1/4(D)1/4 cf16-allmgsn (A)1/4(B)1/4(C)1/4(A)1/4 L21 cf16-alcu2mn All of them are specal cases of the general model: (A,B,C,D)1/4(A,B,C,D)1/4(A,B,C,D)1/4(A,B,C,D)1/4 Whch s used to express thermodynamc functons of ths group of phases. Smlar 4-sublattce models may be used to descrbe the ordered structures related to the fcc or hcp lattces. To summarse, a selecton of examples of CEF modellng of dfferent knd of phases s reported n the Table 1 below: 3 SELECTED EXAMPLES In the followng a few examples of CALPHAD applcatons are reported, taken from recent results obtaned n the author s lab. These examples show how CALPHAD Tecnol. Metal. Mater. Mner., São Paulo, v. 13, n. 1, p , jan./mar can be used to understand and/or predct the behavour of real systems and how t can be used to solve applcaton problems. 3.1 CALPHAD assessment supported by quantum DFT calculatons: the Fe-N system Fe-N s a key system for a large number of technologcal alloys. From a scentfc pont of vew t shows an nterestng combnaton of structural and magnetc orderng n conjuncton wth very expermental dffcultes due to the very slow knetcs at temperatures lower than about 500 C. For all these reasons t s a very nterestng playground for CALPHAD modellng. Recently [10] we assessed the system by usng four sublattce models for the fcc- and bcc-based ordered phases and used DFT atomstc calculaton to evaluate the enthalpy of formaton of all ordered end-members. As a result we were able: 1) to confrm the presence of the trcrtcal pont ( Nshzawa horn ) related to the magnetc orderng of the fcc sold soluton (Fgure 3a); 2) to confrm the low temperature stablty of the L10 fcc ordered phase (expermentally supported by the analyss of meteorc samples), as well as the metastable, low temperature mscblty gap n the fcc sold soluton, 3) to underlne the role of the magnetc nteractons n the establshment of the Fe-N phase equlbra; 4) to calculate the thermodynamc stablty of the stable and/or metastable 21

7 Caccaman Table 1. Examples of CEF modellng Phase type Typcal sublattce model Example phase Example sublattce model Substtutonal dsordered solutons (A,B) lqud-(cu,n) fcc-(co,n) (Cu,N) (Co,N) Stochometrc compounds (A) u (B) v Al 3 N (Al) 3 (N) 1 Substtutonal solutons wth ntersttals (A) u (B,Va) v bcc-(fe-n) (Fe) 1 (N,Va) 3 Substtutonal ordered solutons (A,B) u (A,B) v C15-Cu 2 Mg (Cu,Mg) 2 (Cu,Mg) 1 Ionc solds (Catons) u (Anons) v NaCl+CaCl 2 spnel (Ca 2+,Na + ) 1 (Cl,Va) 2 (Mg 2+,Al 3+ ) 1 (Mg 2+,Al 3+ ) 2 (O 2 ) 4 Semconductors wth pont defects (A,B,B + ) (A,A,B) (Va,e - ) (Va,h + ) GaAs (As,Ga,Ga + ) (As,As,Ga) (Va,e ) (Va,h + ) Order/dsorder relatons (A,B) and (A,B) u (A,B) v fcc-(au,cu) and (AuCu 3 ) (Au,Cu) and (Au,Cu) 1/4 (Au,Cu) 3/4 Fgure 3. (a) the calculated Fe-N phase dagram; (b) the calculated enthalpy of mxng of the ordered and dsordered fcc and bcc Fe-N phases compared wth results of ab-nto calculatons. ordered and dsordered phases (Fgure 3b). Ths s especally mportant for the modellng of hgher order systems. 3.2 CALPHAD smulaton of a non-equlbrum process: damond synthess by mcro-plasma CVD It s well known that the equlbrum synthess of damond crystals requres very hgh pressures and temperatures. By a non-equlbrum process, however, small damond crystals can be obtaned at low pressure and relatvely low temperature. When a hgh temperature plasma jet formed by a mxture of hydrogen and methane mpnges on a metal substrate at a temperature of about C varous carbon structures (graphte, damond, graphene, nanotubes...) may grow on the metal surface, dependng on the process parameters, namely total pressure, metal substrate temperature, plasma temperature, gas composton. A possble explanaton of the damond formaton s that the hghly energetc atomc hydrogen present n the plasma s adsorbed on the carbon surface and promotes the transton of carbon from the sp2 to the sp3 hybrdsaton states. Once the hydrogen s released as H 2 molecules, the sp3 hybrdsaton s mantaned and a new layer of damond structure has grown. Ths non-equlbrun non-dsspatve process may be modelled accordng to the CALPHAD approach by ntroducng a hypothetcal new phase, called actvated graphte (G*), whch represents graphte excted by the nteracton wth atomc hydrogen. Its thermodynamc propertes then depend on a combnaton of the propertes of regular graphte at the temperature of the substrate and the propertes of atomc hydrogen at the temperature of the plasma. Accordng to ths model, frst ntroduced by Wang et al. [11], damond can grow when t s more stable than the actvated graphte. In a recent paper [12] we calculated, n a pressure vs. temperature dagram, the damond growth regon and verfed the good agreement of the model calculatons wth the expermental results, as shown n Fgure 4. Notce that exotc carbon forms are obtaned n the regon where actvated graphte s 22 Tecnol. Metal. Mater. Mner., São Paulo, v. 13, n. 1, p , jan./mar. 2016

8 An ntroducton to the calphad method and the compound energy formalsm (CEF) more stable than damond, whle, on the opposte sde of the damond regon, regular graphte s obtaned, as expected. 3.3 CALPHAD smulaton of wettng experments: N alloys n contact wth UHTC UHTCs (Ultra Hgh Temperature Ceramcs) are materals charactersed by meltng temperatures n excess of 3000 C. Among them dbordes of the group 4 metals (T, Zr, Hf) are canddate materals for applcaton n extreme condtons such as leadng edges of vehcles for hypersonc flght, atmospherc re-entry vehcles, rocket propellers, refractory lnngs, electrodes, cuttng tools, etc. Wettng and jonng propertes of these materals wth metals and alloys are scarcely known. So they have been recently nvestgated, especally by means of wettng experments carred out usng the sessle drop technque and asssted by CALPHAD calculatons [13-17]. In one of the mentoned works [16], for example, phase dagram calculatons easly explaned the observed sequence of unusual sothermal transformatons occurrng durng a wettng experment of a N-B alloy n contact wth a ZrB 2 substrate. As shown n Fgure 5, the N-17at%B alloy n contact wth ZrB 2 at 1130 C melts and reacts wth the ceramc substrate. Durng the sothermal reacton the composton of the lqud drop moves along the lne shown n the Fgure 5a Fgure 4. The pressure vs. temperature phase dagram of carbon where the stablty areas of damond and actvated graphte are supermposed and compared wth expermental results. The coloured area represent the range where damond can be obtaned by mcro-plasma CVD process. Fgure 5. Evaluaton of the reacton between a N-B alloy drop and a ZrB 2 substrate. a) and b) Equlbrum durng wettng. c) Equlbrum durng coolng. Tecnol. Metal. Mater. Mner., São Paulo, v. 13, n. 1, p , jan./mar

9 Caccaman untl t reaches the saturaton composton, n equlbrum wth ZrB 2. Dong so t crosses the two-phase feld where the ternary compound Zr 3 N 20 B 6 becomes stable. Accordngly, durng the sothermal wettng experment precptaton and subsequent re-dssoluton of a sold phase s observed, as evdenced n Fgure 5b. Once the equlbrum between ZrB 2 and the drop s reached, the system s cooled down. In Fgure 5c the sold phases formed durng coolng are shown. They correspond to the phases actually dentfed n the soldfed drop. The same procedure has been successfully used to predct and/or nterpret the results of many more wettng experments. 4 CONCLUSION CALPHAD s a very useful and powerful method n materals scence and processng. Thank to ts ablty to predct phase equlbra and thermodynamc functons n mult-phase and mult-component systems t has been successfully appled to the descrpton and smulaton of very complex technologcal materals and processes. Thanks to the ntegraton of CALPHAD modellng wth DFT and/or molecular dynamcs calculatons, on one sde, and knetc smulatons, on the other sde, an all scales modellng of materals may be acheved. REFERENCES 1 Kaufman L, Bernsten H. Computer calculaton of phase dagrams. New York: Academc Press; Spencer PJ. A bref hstory of CALPHAD. Calphad. 2008;32: Saunders N, Modownk AP. CALPHAD (calculaton of phase dagrams): a comprehensve gude. Oxford: Pergamon; Hllert M. Phase equlbra, phase dagrams and phase transformatons: ther thermodynamc bass. 2nd ed. Cambrdge: Cambrdge Unversty Press; Lukas HL, Fres SG, Sundman B. Computatonal thermodynamcs: the CALPHAD method. Cambrdge: Cambrdge Unversty Press; Hllert M. The compound energy formalsm. Journal of Alloys and Compounds. 2001;320(2): Inden G. The role of magnetsm n the calculaton of phase dagrams. Physca B+C. 1981;103(1): Hllert M, Jarl M. A model for alloyng n ferromagnetc metals. Calphad. 1978;2: Dnsdale AT. SGTE data for pure elements. Calphad. 1991;15: Caccaman G, Dnsdale A, Palumbo M, Pasturel A. The Fe N system: thermodynamc modellng asssted by atomstc calculatons. Intermetallcs. 2010;18: Wang J-T, We Zhang D, Dng S-J, Wang P-F. A new feld of phase dagrams of statonary nonequlbrum states. Calphad. 2000;24: Ghezz F, Caccaman G, Canello R, Toncu DC, Causa F, Dellasega D, et al. Carbon structures grown by drect current mcroplasma: damonds, sngle-wall nanotubes, and graphene. The Journal of Physcal Chemstry C. 2014;118: Kaufman L, Caccaman G, Muolo ML, Valenza F, Passerone A. Wettablty of HfB2 by molten N(B) alloys nterpreted by CALPHAD methods, Part 1: defnton of the B Hf N system. Calphad. 2010;34: Caccaman G, Ran P, Valenza F. Equlbrum between MB2 (M=T,Zr,Hf) UHTC and N: a thermodynamc database for the B Hf N T Zr system. Calphad. 2011;35: Valenza F, Muolo ML, Passerone A, Caccaman G, Artn C. Control of nterfacal reactvty between ZrB2 and N-based brazng alloys. Journal of Materals Engneerng and Performance. 2012;21: Artn C, Muolo ML, Passerone A, Caccaman G, Valenza F. Isothermal sold lqud transtons n the (N,B)/ZrB2 system as revealed by sessle drop experments. Journal of Materals Scence. 2013;48: Artn C, Muolo ML, Passerone A, Valenza F, Manfrnett P, Caccaman G. Expermental nvestgatons and thermodynamc modelng n the ZrB2N secton of the BNZr system. Journal of Alloys and Compounds. 2014;592: Receved: 27 Nov Accepted: 18 Dez Tecnol. Metal. Mater. Mner., São Paulo, v. 13, n. 1, p , jan./mar. 2016