First-principles calculations and thermodynamic modeling of the Ni Mo system

Transcription

1 Materals Scence and Engneerng A 397 (2005) Frst-prncples calculatons and thermodynamc modelng of the N Mo system S.H. Zhou, Y. Wang, C. Jang, J.Z. Zhu, L.-Q. Chen, Z.-K. Lu Department of Materals Scence and Engneerng, The Pennsylvana State Unversty, Unversty Park, PA 16802, USA Receved 17 December 2004; receved n revsed form 9 February 2005; accepted 9 February 2005 Abstract The phase equlbra and thermodynamc propertes of the N Mo system were analyzed by combnng a frst-prncples approach and calculaton of phase dagram (CALPHAD) technque. The frst-prncples calculaton results ndcate that N 2 Mo and N 8 Mo are stable n addton to N 3 Mo and N 4 Mo, and -NMo s not stable at 0 K, both n contradcton to the exstng phase relatonshps n the N Mo system. The enthalpes of the mxng of the bcc and fcc sold soluton phases were also predcted by frst-prncples calculatons usng the specal quasrandom structures. In the present work, the non-stochometrc -NMo and N 3 Mo phases were modeled usng three- and two-sublattce models, respectvely. The N 2 Mo, N 4 Mo and N 8 Mo phases were treated as stochometrc compounds. Based on the frst-prncples data from the present work and expermental data n the lterature, the Gbbs energy functons of ndvdual phases were evaluated, and a new N Mo phase dagram was presented Elsever B.V. All rghts reserved. Keywords: Mo N; phase dagram; Thermodynamc modelng; Frst-prncples calculaton 1. Introducton The performance of N-base superalloys depends on ts ablty to wthstand hgh temperatures wthout falure. A tremendous amount of research has been devoted to mprovng ther performance at hgher temperatures and for longer perods of tme [1 7]. Understandng of alloyng effects on phase stablty s crtcal n provdng quanttatve gudelnes for alloy development and processng desgn. The present work focuses on the thermodynamc study of the N Mo system through frst-prncples calculatons and CALPHAD modelng. Thermodynamc modelng of the N Mo system was frst carred out by Frsk [8] n whch sx phases were consdered. The lqud, fcc and bcc phases were treated wth the one-sublattce model, the -NMo phase wth three-sublattce model as (N) 24 (Mo, N) 20 (Mo) 12, and the N 3 Mo and N 4 Mo Correspondng author. Present address: Metals and Ceramcs Scences Program, Ames Laboratory, Department of Energy, 116 Wlhelm, Ames, IA 50011, USA. Tel.: ; fax: E-mal address: szhou@ameslab.gov (S.H. Zhou). phases as two-sublattce stochometrc compounds. Based on Frsk s work [8], Cu et al. [9] modfed the thermodynamc descrpton of the N Mo system by modelng the N 3 Mo phase wth a two-sublattce model, (Mo, N) 3 (Mo, N) 1, to account for ts homogenety range. The calculated phase dagram by Cu et al. [9] revealed unrealstc behavor of the -NMo phase at low temperature as shown n Fg. 1. The N Mo system was also studed by Morshta et al. [10] n the N Mo B ternary system to desgn a new NMo 2 B 2 -dspersed N-based alloy. In ther work, the Gbbs energy functons of lqud, fcc and -NMo (as a stochometrc compound) n the N Mo system were evaluated, whle other compounds were not consdered n ther work. In addton to the three ntermetallc phases consdered by Frsk [8] and Cu et al. [9], lterature data ndcated the possble exstence of N 2 Mo [11] and N 8 Mo [12].Tonvestgate the stablty of the compounds n the N Mo system, Wang et al. [13] calculated ther enthalpes of formaton usng VASP [14]. Furthermore, the recent development n the frstprncples calculatons enables us to calculate the enthalpy of mxng n fcc and bcc usng the specal quasrandom structures (SQS) [15,16], whch mmc the random mxng n fcc /$ see front matter 2005 Elsever B.V. All rghts reserved. do: /j.msea

2 S.H. Zhou et al. / Materals Scence and Engneerng A 397 (2005) Table 1 Intermetallc compounds n the N Mo system Phases Prototype Structure-bercht Space group Ref. -NMo NMo P [18,33,34] N 2 Mo Pt 2 Mo Immm [38] N 3 Mo Cu 3 T D0 a pmmn [18] N 4 Mo N 4 Mo D1 a I4/m [18] N 8 Mo Pt 8 T 4/m2/m2/m [12] Fg. 1. The phase dagram calculated usng the parameters of N 3 Mo (threesublattce model) wth the expermental data ( ) three-phase equlbrum [12]; ( ) three-phase equlbrum [17]; ( ) sngle phase feld, ( ) lqud phase feld, ( ) two-phase feld wth lqud phase, ( ) two-phase feld [18]; ( ) phase boundary [19] and ( ) bcc phase boundary [20]; The dotted-lnes calculated phase dagram usng the parameters by Cu et al. [9]. and bcc usng relatve small supercells. These data are used to gude the evaluaton of thermodynamc parameters. Based on the newly obtaned enthalpy of formaton of compounds by frst-prncples calculatons [13], the enthalpy of mxng n fcc and bcc by SQS n the present work, and expermental data n the lterature, the thermodynamc modelng of the N Mo system was carred out, and a new N Mo phase dagram was predcted. In ths paper, the expermental nformaton n the lterature and frst-prncples calculatons wll be revewed frst, followed by thermodynamc modelng and dscussons. 2. Expermental nformaton and frst-prncples calculatons In thermodynamc modelng, the Gbbs energy functons of ndvdual phases are modeled wth respect to temperature, pressure and composton. Models are based on crystal structures of phases, and model parameters are evaluated from the phase equlbrum and thermochemcal data. The crystal structure nformaton of the N Mo system s shown n Table 1, and phase equlbrum and thermochemcal data are revewed n ths secton Expermental nformaton The expermental data n the lterature, revewed n detal by Frsk [8] and Cu et al. [9], are brefly presented here. The three-phase equlbra n N Mo were measured by Grube and Schlecht [17] usng the thermal analyss method. They also studed the lqud + fcc two-phase equlbrum n the composton range of at.% N. The lqudus boundary was determned by Casselton and Hume-Rothery [18] usng the thermal analyss, metallurgraphc, and X-ray dffracton methods. They also nvestgated the equlbra among bcc, fcc, -NMo, N 3 Mo and N 4 Mo. The phase equlbra at temperatures below 1600 K were nvestgated by Hejwegen and Reck [19] by means of the dffuson couple technque and electron probe mcroanalyss (EPMA). The soldus boundary of bcc was nvestgated by Kang et al. [20] through the powder snterng technque and the electrolytc phase separaton method and usng EPMA. The compound N 2 Mo was found by Pearson et al. [11] n a 76N 13Al 9Mo 2Ta alloy aged at 1033 K usng an electron mcroscope, whle Hejwegen and Reck [19] dd not fnd t n bnary dffuson couple samples treated between 1073 and 1550 K. Thus the decomposton temperature of N 2 Mo seems between 1033 and 1073 K. The stablty of the compound N 8 Mo was nvestgated by Mayer and Urban [12] usng a hgh-voltage electron mcroscope. The decomposton temperature was found to be 555 ± 10 K. The enthalpy of formaton of -NMo was measured by Spencer and Putland [21] and Kubaschewsk and Hoster [22] usng a hgh-temperature adabatc calormeter. The actvty data n fcc and Gbbs energy of mxng of fcc were measured by varous authors [23 27] usng electromotve force (EMF) method. The actvtes of Mo n fcc measured by Meshkov et al. [24] at 1273 K are compatble wth those by Tsa [25] at 1250 K, and slghtly dfferent from those by Katayama et al. [23] at 1223 K and Pejryd [26] at 1273 K. The data by Koyama et al. [27] show a very strong temperature-dependency of Gbbs energy of mxng n fcc whch results n a postve Gbbs energy of mxng n N Mo fcc when extrapolated to room temperature (see Fg. 2) Frst-prncples calculatons The total energy of fcc-n and bcc-mo and all bnary compounds were calculated by means of VASP [14] wth Vanderblt ultrasoft pseudopotental [28] and the generalzed gradent approxmaton (GGA) [14] usng the hgh precson choce. The 3d4s and 4d5s shells were treated as valence states wth core rad of 2.43 and 2.75 a.u. for N and Mo, respectvely. Monkhost k ponts were used for the pure elements N and Mo, k ponts for N 2 Mo, N 3 Mo, N 4 Mo, and N 8 Mo and k ponts for the followng end-member of -NMo: N 24 (Mo 20 )Mo 12,N 24 (N 4,

3 290 S.H. Zhou et al. / Materals Scence and Engneerng A 397 (2005) Fg. 2. The Gbbs energy of mxng of fcc extrapolated from the data ( ) [27] at composton N Mo wth the correspondng frst-prncples data ( ). Mo 16 )Mo 12, N 24 (N 16, Mo 4 )Mo 12, and N 24 (N 20 )Mo 12 [13]. To ensure that the unt cell of unstable end-member stll corresponds to stable structure, we only relaxed the cell volume of the unstable end-member wthout local lattce relaxatons due to the structure nstablty [29]. The enthalpy of formaton of a compound Φ, Hf Φ, s calculated by subtractng the total energy of fcc-n and bcc-mo, ETOT fcc-n and Ebcc-Mo TOT, from the total energy of the compound Φ, ETOT Φ H Φ f = E Φ TOT xφ N Efcc-N TOT xφ Mo Ebcc-Mo TOT (1) where x Φ s the mole fracton of component n Φ. The calculated enthalpes of formaton are plotted n Fg. 3 and lsted n Table 2. Due to the lack of experments, the sostructure enthalpes of mxng of the bcc and fcc sold soluton phases were calculated n the present study usng 16-atom SQS s [15,16] for the compostons x N = 0.25, 0.5 and 0.75, respectvely. The dervatons of the SQS s used n the present calculatons were dscussed n detals n the lterature [16]. The energetcs of the SQS s as well as bcc-n and fcc-mo were calculated usng the GGA pseudopotentals wthn VASP. Spn-polarzed calculatons were performed to account for the ferromagnetc nature of N. The frst-prncples data shows that energy dfference, G bcc fcc ( G bcc fcc = G fcc G bcc ), of element Mo s kj/mol, whle G bcc fcc calculated by the CAL- PHAD approach usng pure element database by Dnsdale [30] s kj/mol. Ths dscrepancy s due to the nstablty of fcc-mo as dscussed n the lterature [29]. Usng the same arguments as that of the unstable end-members of the compounds, we only relaxed the cell volume of the SQS s Fg. 3. The calculated enthalpy of formaton ( ) of the bnary N Mo system and enthalpy of mxng (- - -) of the compound -NMo at 0 K wth the frst-prncples calculaton results [13] of ( ) -NMo, ( )N 2 Mo, ( ) N 3 Mo, ( ) N 4 Mo, and ( ) N 8 Mo and the dotted-lne ( ) calculated usng the parameters by Cu et al. [9]. Reference state: fcc-n and bcc-mo. wthout local lattce relaxatons due to the structure nstablty of fcc-mo and bcc-n [29]. The calculated total energes of the SQS s n both fcc and bcc structures at compostons N 0.75 Mo 0.25,N 0.5 Mo 0.5 and N 0.25 Mo 0.75 are gven n Table 2 together wth the derved sostructure enthalpes of mxng of bcc and fcc plotted n Fg. 4. The enthalpes of mxng n Table 2 were used n the thermodynamc modelng. In Fg. 3, the frst-prncples calculated enthalpy of formaton of the -NMo phase s above the sold-lne between the pure bcc-mo and N 2 Mo phases. Ths ndcates that -NMo s not stable at 0 K. The dotted-lne n Fg. 3 llustrates the enthalpy of formaton calculated usng the parameters by Cu et al. [9] at 298 K, whch shows sgnfcant devatons from the present frst-prncples data. 3. Thermodynamc models Thermodynamc models n the present work are based on the sublattce models developed by Sundman and Agren [31]. The number of sublattce and the number of stes per sublattce are constructed based on crystal structures of phases though combnaton of smlar sublattces s often needed due to lmted data avalable Lqud, fcc and bcc phases Lqud, fcc and bcc were modeled as random solutons wth the one-sublattce model (Mo, N). The Gbbs energy s

4 S.H. Zhou et al. / Materals Scence and Engneerng A 397 (2005) Table 2 Frst-prncples calculaton results Phase Prototype Formula ev/atom H (kj/mol) Ref. N fcc N [13] N bcc N Ths work Mo bcc Mo [13] Mo fcc Mo Ths work N 0.25 Mo 0.75 bcc-sqs16 N 0.25 Mo Ths work N 0.50 Mo 0.50 N 0.50 Mo Ths work N 0.75 Mo 0.25 N 0.75 Mo Ths work N 0.25 Mo 0.75 fcc-sqs16 N 0.25 Mo Ths work N 0.50 Mo 0.50 N 0.50 Mo Ths work N 0.75 Mo 0.25 N 0.75 Mo Ths work -NMo NMo N 24 (Mo 20 )Mo Ths work N 24 (N 4 Mo 16 )Mo [13] N 24 (N 8 Mo 12 )Mo Ths work N 24 (N 12 Mo 8 )Mo Ths work N 24 (N 16 Mo 4 )Mo Ths work N 24 (N 20 )Mo Ths work N 2 Mo Pt 2 Mo N 2 Mo [13] N 3 Mo Cu 3 T Mo 3 Mo Ths work N 3 Mo [13] Mo 3 N Ths work N 3 N Ths work N 4 Mo N 4 Mo N 4 Mo [13] N 8 Mo Pt 8 T N 8 Mo [13] expressed as G Φ m = x GΦ + RT x ln x + xs G Φ m (2) =Mo,N =Mo,N where G Φ ( = Mo, N) s the Gbbs energy of the pure element wth the structure Φ from the database by Dnsdale [30]. xs G Φ m s the excess Gbbs energy, expressed n the Redlch Kster polynomal [32], n xs G Φ m = x Mox j N L Φ Mo,N (x Mo x N ) j (3) j=0 where j L Φ Mo,N s the jth nteracton parameter between component Mo and N, expressed as j a Φ + j b Φ T, wth j a Φ and j b Φ beng model parameters to be evaluated δ-nmo phase Fg. 4. The calculated sostructure enthalpy of mxng of bcc ( ) referred to bcc-n and bcc-mo and fcc (- - -) referred to fcc-n and fcc-mo wth the frst-prncples calculaton data ( ) for bcc and ( ) for fcc. -NMo has the pseudo-tetragonal structure wth space group P [13,33,34]. The unt cell of - NMo has 56 atoms dstrbuted n four sublattces,.e. (N) (Mo, N)14 20 (Mo)15 8 (Mo)16 4, where the superscrpt s the coordnaton number and the subscrpt s the number of the stes n the sublattce. Accordng to the analyss by Andersson et al. [35], the sublattce wth the coordnaton number of 12 s preferred by fcc elements, whle sublattces wth the coordnaton numbers of 14 and 15 are favorable for bcc elements. In ths work, the reference state fcc was selected for the frst sublattce and bcc for other sublattces. To smplfy the descrpton, the thrd and forth sublattces occuped by Mo only were combned, and the three-sublattce model, (N) 24 (Mo, N) 20 (Mo) 12 was used, the same as n Refs. [8,9],

5 292 S.H. Zhou et al. / Materals Scence and Engneerng A 397 (2005) N 3 Mo phase N 3 Mo has the D0a structure and the space group pmmn [18]. The unt cell of N 3 Mo has eght atoms dstrbuted n three-sublattces,.e. N 4 N 2 Mo 2. The expermental data by Hejwegen and Reck [19] revealed a small composton homogenety of N 3 Mo. The non-stochometrc N 3 Mo phase s thus descrbed usng the three-sublattce model, (Mo, N) 4 (N) 2 (Mo, N) 2, wth the Gbbs energy gven as: G N 3Mo m = y I =Mo,N j=mo,n + RT =Mo,N yj III G N 3Mo :N (4y I ln yi + 2yIII ln y III )+ xs G N 3Mo m (7) Fg. 5. The calculated enthalpy of formaton of -NMo (x N = 0.5) n comparson wth the frst-prncples calculaton data ( ) and the expermental data ( ) [21] and ( ) [22]. Reference state: fcc-n and bcc-mo. and ts Gbbs energy s gven as: G m = =Mo,N y II G N::Mo + 20RT =Mo,N ln y II + xs G m (4) where G N::Mo represents the Gbbs energy of the endmember (N) 24 () 20 (Mo) 12 wth the colon separatng components n dfferent sublattces and s gven as: G N::Mo = 24 G fcc N + 20 G bcc + 12 G bcc Mo + G N::Mo n Eq. (4) s the ste fracton of component n the second sublattce. G fcc N, G bcc Mo, and G bcc ( = N or Mo) are the Gbbs energes of the pure fcc-n, bcc-mo and bcc-n, respectvely, from the database by Dnsdale [30]. Accordng to the frst-prncples data and expermental data [21,22] n Fg. 5, the enthalpy of formaton of -NMo s temperaturedependent. The Gbbs energy of formaton, G N::Mo, y II s thus expressed as an::mo + b N::Mo T + c N::MoT ln(t ). xs G m n Eq. (4) s the excess Gbbs energy, expressed n Redlch Kster polynomal [32] as ( ) xs G m = yii N yii Mo k L N:N,Mo:Mo (yii Mo yii N )k (6) k=0 where k L s the kth nteracton parameter, expressed as k a + k b T. a N::Mo, b N::Mo, c N::Mo, k a, and k b are model parameters to be evaluated. y II (5) Due to the frst and thrd sublattces manly occuped by N and Mo, respectvely, the reference state bcc was selected for the thrd sublattce and fcc for the other sublattces. The Gbbs energy of the end-member :N n Eq. (7), G N 3Mo :N, s gven as follows: G N 3Mo :N = 4 G fcc + 2 G fcc N + 2 G bcc j + G N 3Mo :N (8) where y I and y III are the ste fractons n the frst and thrd sublattces, respectvely, G fcc and G bcc j (, j = Mo and N) the Gbbs energy of the elements Mo and N from the database by Dnsdale [30]. As mentoned n Secton 3.2, the enthalpy of formaton of -NMo s temperature-dependent. The same behavor s assumed for N 3 Mo. The Gbbs energy of formaton of the end-member :N, G N 3Mo :N, s thus expressed as a N 3Mo :N + b N 3Mo :N T + cn 3Mo :N T ln(t ). xs G N 3Mo m n Eq. (8) s the excess Gbbs energy, expressed n the Redlch Kster polynomal [32] as xs G N 3Mo m = y I Mo yi N + y III Mo yiii =Mo,N y III k=0 N y I =Mo,N k=0 k L N 3Mo Mo,N:N: (yi Mo yi N )k k L N 3Mo :N:Mo,N (yiii Mo yiii N )k where k L N3Mo s the kth nteracton parameter, expressed as k a N3Mo + k b N3Mo T. a N 3Mo :N, bn 3Mo :N, cn 3Mo :N, k a N3Mo, and k b N3Mo are model parameters to be evaluated. The two-sublattce model was also used n the lterature to model the D0a structure [9]. To be compatble wth those descrptons, we also develop another set of model parameters wth N 3 Mo beng descrbed by the two-sublattce model,.e. (Mo, N) 3 (Mo, N) 1, denoted by N 3 MoII G N 3MoII m = y I =Mo,N j=mo,n yj II GN 3MoII + RT =Mo,N (9) (3y I ln yi + yii ln y II ) + xs G N 3MoII m (10)

6 S.H. Zhou et al. / Materals Scence and Engneerng A 397 (2005) The Gbbs energy of the end-member n Eq. (10), G N 3MoII, s gven n Eq. (11): G N 3MoII = 3 G ref + G ref j + G N 3MoII (11) where y I and y II are the ste fractons n the frst and second sublattces, respectvely, G ref and G ref j (, j = Mo and N) the Gbbs energy of the elements Mo and N n ther stable structure at 298 K [30]. The Gbbs energy of formaton of the end-member, G N 3MoII, s expressed as a N 3MoII + b N 3MoII T + c N 3MoII T ln(t ). xs G N 3MoII m n Eq. (10) s the excess Gbbs energy, expressed n the Redlch Kster polynomal [32] as xs G N 3MoII m = ymo I yi N y II k L N 3MoII Mo,N: (yi Mo yi N )k + y II Mo yii N =Mo,N k=0 y I =Mo,N k=0 k L N 3MoII :Mo,N (yii Mo yii N )k (12) where k L N3MoII s the kth nteracton parameter, expressed as k a N3MoII + k b N3MoII T. a N 3MoII, b N 3MoII, c N 3MoII, k a N3MoII, and k b N3MoII are model parameters to be evaluated N 2 Mo, N 4 Mo, and N 8 Mo phases N 2 Mo, N 4 Mo and N 8 Mo were all treated as stochometrc compounds. Ther Gbbs energy functons are expressed by: G N Mo m = a NMo + b NMo T + c NMo T ln(t ) + G fcc N + G bcc Mo (13) where a NMo, b NMo, and c NMo are the model parameters to be evaluated. 4. Model parameter evaluaton and result dscusson The model parameters dscussed n Secton 3 were evaluated usng the Parrot module [36] n Thermo-Calc [37], based on the expermental data selected n Secton 2. The Parrot module s able to take varous types of expermental data smultaneously and mnmzes the error of sum wth each of the selected data values gven a certan weght. The weght s chosen and adjusted based upon the data uncertantes gven n the orgnal publcatons, the ntrnsc uncertantes of varous expermental technques, and the modeler s judgment by examnng all data concurrently. The parameter evaluaton process started wth lqud, followed by bcc and fcc solutons phases and then n the order of -NMo, N 3 Mo, N 4 Mo, N 2 Mo and N 8 Mo. The model parameters for the lqud, bcc and fcc phases n Eq. (3) were evaluated wth the phase dagram data [17,20], Table 3 Thermodynamc propertes of the N Mo system per mole of the formula unt Phase (model) Parameters Value Lqud(Mo, N) 0 L lq Mo,N T 1 L lq Mo,N T 2 L lq Mo,N T bcc(mo, N) 0 L bcc Mo,N L bcc Mo,N fcc(mo, N) 0 L fcc Mo,N T 1 L fcc Mo,N T 2 L fcc Mo,N T -NMo (N) 24 G N:Mo:Mo T (N, Mo) 20 (Mo) T ln(t) G N:N:Mo T T ln(t) 0 L N:Mo,N:Mo T 1 L N:Mo,N:Mo T N 2 Mo (N 2 Mo) N 3 Mo (Mo, N) 4 (N) 2 (Mo, N) N 3 MoII(Mo, N) 3 (Mo, N) 1 N 4 Mo(N 4 Mo) N 8 Mo(N 8 Mo) G N 2Mo N:Mo T T ln(t) G N 3Mo Mo:N:Mo G N 3Mo N:N:Mo T T ln(t) G N 3Mo N:N:N L N 3Mo Mo,N:N:Mo G N3MoII Mo:Mo G N 3MoII N:Mo G N 3Mo Mo:N:N /2 G N 3MoII Mo:N G N 3Mo Mo:N:Mo /2 G N 3MoII N:N G N 3Mo N:N:N /2 0 L N 3MoII Mo,N:Mo L N 3MoII N:Mo,N T G N 4Mo N:Mo T T ln(t) G N 8Mo N:Mo T T ln(t) the actvty of Mo n fcc [23 26], the Gbbs energy of mxng of fcc [27], and the enthalpy of mxng of bcc and fcc n Table 2 by frst-prncples calculatons. For fcc and bcc, the phase equlbrum data [17 19] and thermochemcal data [23 26] were consdered as key data and were set to hgher weghts than the frst-prncples calculated enthalpes of mxng due to the structure nstablty and Gbbs energy of mxng of fcc n Ref. [27] due to ts too strong temperature-dependent as dscussed n Secton 2. The model parameters for -NMo are an::mo, b N::Mo, cn::mo, k a, and k b n Eqs. (4) (6) wth representng ether Mo or N. The parameters an::mo were fxed by the frstprncples calculaton data [13]. The parameters of the endmember (N) 24 (Mo) 20 (Mo) 12, bn:mo:mo and c N:Mo:Mo, were determned usng the bcc + -NMo + lqud three-phase equlbrum data [17], the bcc + -NMo two-phase equlbrum data [19], and the enthalpy of formaton [21,22]. The parameters of the end-member (N) 24 (N) 20 (Mo) 12, bn:n:mo and

7 294 S.H. Zhou et al. / Materals Scence and Engneerng A 397 (2005) Fg. 6. The calculated actvty of Mo n fcc, N 4 Mo, N 3 Mo, -NMo and bcc at 1150 K ( ), 1273K (- - -) and 1373K ( ) wth the expermental data at 1373 K (+) and 1273 K ( ) [23]; 1273 K ( ) and 1123 K ( ) [24]; 1150 K ( ), 1250 K (*), and 1550 K ( ) [25]; 1150 K ( ) [26]. Reference state: fcc-n and bcc-mo. cn:n:mo, were determned through the fcc + -NMo + lqud three-phase equlbrum data [17], the fcc + -NMo twophase equlbrum data [19], the actvty of Mo n fcc and -NMo [23 25], and the enthalpy of formaton [21,22]. The nteracton parameters, 0 a, 0 b, 1 a, and 1 b were determned usng the frst-prncples data at the confguratons of N 24 (Mo 20 )Mo 12,N 24 (N 4,Mo 16 )Mo 12,N 24 (N 16, Mo 4 )Mo 12 and N 24 (N 20 )Mo 12 n Table 2, the phase equlbrum data [17,18], and the expermental enthalpy of formaton [21,22]. For N 3 Mo, the parameters a N 3Mo :N n the three-sublattce model and a N 3MoII n the two-sublattce model were fxed by the frst-prncples data. The parameters b N 3Mo N:N:Mo and c N 3Mo N:N:Mo, bn 3MoII N:Mo and c N 3MoII N:Mo and nteracton parameters 0 L N3Mo were evaluated usng the fcc + -NMo + N 3 Mo three-phase equlbrum data [17], the fcc + N 3 Mo two-phase boundary data [19]. The parameters a NMo for N 4 Mo, N 2 Mo, and N 8 Mo were fxed by the frst-prncples data. The parameters b N 4Mo and c N4Mo were evaluated wth the fcc + N 3 Mo+N 4 Mo three-phase equlbrum data [18] and the fcc phase boundary data [18,19]. For N 2 Mo, the -NMo + N 2 Mo+N 3 Mo three-phase equlbrum temperature was assumed to be K accordng to the expermental nvestgaton by Pearson et al. [11] and Hejwegen and Reck [19]. Due to the lmt expermental data, the parameter, c N2Mo, was neglected frst. Then the parameter, b N2Mo, could be evaluated wth the -NMo + N 2 Mo+N 3 Mo three-phase equlbrum temperature. However, ths makes N 3 Mo and N 4 Mo unstable between temperatures 200 and 1000 K because the en- Fg. 7. The calculated Gbbs energy of mxng of fcc at dfferent temperatures wth the expermental data ( ) at 1183; ( ) at 1233 K; ( ) at 1273 K; ( ) at 1323; ( ) at 1373 K; ( ) at 1423 K [27]. Reference state: fcc-n and bcc-mo. thalpes of formaton of N 3 Mo and N 4 Mo are consdered as temperature-dependent. Thus both parameters, b N 2Mo and c N 2Mo, have to be consdered for N 2 Mo and were evaluated through the N 2 Mo+N 3 Mo + -NMo three-phase equlbrum temperature and the Gbbs energy functons of N 3 Mo and -NMo. Wth the obtaned parameters and three- and two-sublattce models of N 3 Mo, the calculated -NMo, Fg. 8. The calculated phase dagram of the bnary N Mo system usng the parameters of N 3 Mo and N 3 MoII for the sold-lnes and dashed-lnes, respectvely, wth the expermental data ( ) three-phase equlbrum, ( ) two-phase feld wth the lqud phase [17]; ( ) sngle phase feld, ( ) fcc phase boundary, (+, ) two phase feld wth lqud phase, ( ) two-phase feld [18]; ( ) N 3 Mo phase boundary, ( ) phase boundary [19].

8 S.H. Zhou et al. / Materals Scence and Engneerng A 397 (2005) Table 4 Invarant reactons of the N Mo system Reacton Three-sublattce model s calculated results Two-sublattce model s calculated results Expermental results Ref. Lqud + bcc -NMo T (K) [17] x (lqud,n) x (bcc,n) x (,N) Lqud -NMo + fcc T (K) [17] x (lqud,n) x (,N) x (fcc,n) fcc + -NMo N 3 Mo T (K) [17] x (fcc,n) x (,N) x (N3 Mo,N) fcc+n 3 Mo N 4 Mo T (K) [17] x (fcc,n) x (N3 Mo,N) x (N4 Mo,N) NMo + N 3 Mo N 2 Mo T (K) [11,19] x (,N) x (N3 Mo,N) x (N2 Mo,N) fcc+n 4 Mo N 8 Mo T (K) [12] x (fcc,n) x (N4 Mo,N) x (N8 Mo,N) NMo bcc+n 2 Mo T (K) x (,N) x (bcc,n) 0 0 x (N2 Mo,N) N 2 Mo and N 3 Mo three-phase equlbrum temperatures are 1035 and 1040 K, respectvely. Smlar to N 2 Mo, the parameters, b N8Mo and c N8Mo, were evaluated wth ts decomposton temperature beng 555 K [12] and the Gbbs energy functons of N 4 Mo and fcc. The parameters, b N2Mo and c N 2Mo of N 2 Mo and b N8Mo and c N8Mo,ofN 8 Mo are desrable to be evaluated wth more expermental data n future. Iteratons were needed to evaluate the model parameters when more and more phases were ntroduced. At the end, all model parameters were allowed to change smultaneously wth all selected data ncluded. The fnal model parameters after truncatons are shown n Table 3, and the calculated nvarant reactons are lsted n Table 4. Wth the obtaned parameters, the calculated enthalpy of formaton at 0 K as a functon of composton s shown n Fg. 3 wth the frst-prncples calculaton data supermposed [13]. The standard devaton, defned as (A B ) 2 /N to represent the degree of agreement, s 0.68 kj/mol. The calculated enthalpes of mxng of bcc and fcc are plotted n Fg. 4 together wth the frst-prncples data wth the standard devatons of 3.63 and 3.58 kj/mol for bcc and fcc, respectvely. The enthalpy of formaton of -NMo wth x N = 0.5 calculated as a functon of temperature s shown n Fg. 5 wth the standard devaton of 0.51 kj/mol. The actvty data of Mo n fcc, N 4 Mo, N 3 Mo, -NMo and bcc [23 26] are shown n Fg. 6 wth the standard devatons of Fg. 7 shows the calculated Gbbs energy of mxng of fcc wth the standard devaton 0.2 kj/mol from the expermental data [27]. A comparson of the calculated phase dagrams usng the three- and two-sublattce models of N 3 Mo wth the expermental data show n the sold lnes and dashed-lnes n Fg. 8. The complete phase dagram of the N Mo system calculated usng the parameters n Table 3 s plotted n Fg. 1. The three-phase equlbrum data [12,17] were compared wth the calculated three-phase equlbrum temperature, and the standard devaton s 2.7 K. The rest expermental phase equlbrum data [17 20] were compared wth the calculated equlbrum composton wth the standard devaton beng 0.02.

9 296 S.H. Zhou et al. / Materals Scence and Engneerng A 397 (2005) Summary A new N Mo bnary phase dagram was proposed based on frst-prncples calculatons and expermental data. The compounds N 2 Mo and N 8 Mo were consdered as stable phases, whch were not ncluded prevously. The -NMo phase stable at hgh temperature was found not stable at 0 K by the frst-prncples calculaton. The enthalpes of mxng of bcc and fcc were calculated usng the SQS s and VASP and used n evaluatng model parameters. A self-consstent thermodynamc descrpton of the N Mo bnary system s obtaned by ntegratng the frst-prncples and CALPHAD approaches and serves as part of the thermodynamc database for N-base superalloys. Acknowledgements Ths work s supported fnancally by The Ultra Effcent Engne Technology Program at NASA Glenn Research Center under grant NCC3-920 and the Natonal Scence Foundaton through the CAREER Award DMR and the Informaton Technology Research Grant DMR The Thermo-Calc program s lcensed from The Foundaton for Computatonal Thermodynamcs and used for all calculatons. References [1] S. Chakrovorty, D.R.F. West, Mater. Sc. Technol. 1 (1985) [2] E. Ageltnger, M. Kersker, Met. Forum 4 (1981) [3] B. Grushko, S. M, J.G. Hghfeld, J. Alloys Compd. 334 (2002) [4] R. Jayaram, J.J. Hren, M.K. Mller, Surf. Sc. 246 (1991) [5] R. Jayaram, M.K. Mller, Surf. Sc. 266 (1992) [6] P. Nash, D.T.F. West, Met. Sc. 13 (1979) [7] P. Wllemn, M. Durand-Charre, I. Ansara, Lqud Sold Equlbra n the Systems N3Al N3Ta and N3Al N3T, Hgh Temperature Alloys for Gas Turbnes and Other Applcatons, D Redel Pub Co., Belgum, [8] K. Frsk, CALPHAD 14 (1990) [9] Y. Cu, Z. Jn, X. Lu, Met. Mater. Trans. A 30A (1999) [10] M. Morshta, K. Koyama, S. Yag, G. Zhang, J. Alloys Compd. 314 (2001) [11] D.D. Pearson, B.H. Kear, F.D. Lemkey, n: B. Wlshre, D.R.J. Owen (Eds.), Creep and Fracture of Engneerng Materals and Structures, Pnerdge, Swansea, [12] J. Mayer, K. Urban, Phys. Stat. Sol. (a) 90 (1985) [13] Y. Wang, C. Woodward, S.H. Zhou, Z.K. Lu, L.Q. Chen, Scrpta Mater. 52 (2005) [14] J.P. Perdew, J.A. Chevary, S.H. Vosko, K.A. Jackson, M.R. Pederson, D.J. Sngh, C. Folhas, Phys. Rev. B 46 (1992) [15] A. Zunger, S.H. We, L.G. Ferrera, J.E. Bernard, Phys. Rev. Lett. 65 (1990) [16] C. Jang, C. Wolverton, J. Sofo, L.-Q. Chen, Z.K. Lu, Phys. Rev. B 69 (2004) [17] G. Grube, H. Schlecht, Z. Elektrochem. 44 (1938) [18] R.E.W. Casselton, W. Hume-Rothery, J. Less-Common Met. 7 (1964) [19] C.P. Hejwegen, G.D. Reck, Z. Metallkd. 64 (1973) [20] S.J. Kang, Y.D. Song, W.A. Kaysser, H. Hofmann, Z. Metallkd. 75 (1984) [21] P.J. Spencer, F.H. Putland, J. Chem. Thermodyn. 7 (1975) [22] O. Kubaschewsk, T. Hoster, Z. Metallkd. 74 (1983) [23] I. Katayama, H. Shmatan, Z. Kozuka, J. Jpn. Inst. Met. 37 (1973) [24] L.L. Meshkov, L.S. Guze, E.M. Sokolovskaya, Zh. Fz. Khm. 49 (1975) [25] H.L. Tsa, Ph.D. Thess, Unversty of Tennessee, [26] L. Pejryd, Ph.D. Thess, Unversty of Umea, [27] K. Koyama, Y. Hashmoto, K. Suzuk, S. Kameyama, J. Jpn. Inst. Met. 53 (1989) [28] D. Vanderblt, Phys. Rev. B 41 (1990) [29] Y. Wang, S. Curtarolo, C. Jang, R. Arroyave, T. Wang, G. Ceder, L.-Q. Chen, Z.-K. Lu, CALPHAD 28 (2004) [30] A.T. Dnsdale, CALPHAD 15 (1991) [31] B. Sundman, J. Agren, J. Phys. Chem. Solds 42 (1981) [32] O. Redlch, A.T. Kster, Ind. Eng. Chem. 40 (1948) [33] C.B. Shoemaker, D.P. Shoemaker, Acta Cryst. 16 (1963) [34] A.V. Vrkar, A. Raman, Z. Metallkd. 60 (1969) [35] J.O. Andersson, A.F. Gullermet, M. Hllert, B. Jansson, B. Sundman, Acta Metall. 34 (1986) [36] B. Jansson, Evaluaton of Parameters n Thermochemcal Models Usng Dfferent Types of Expermental Data Smultaneously, TRITA-MAC-0234, Royal Insttute of Technology, Stockholm, Sweden, [37] J.O. Andersson, T. Helander, L.H. Hoglund, P.F. Sh, B. Sundman, CALPHAD 26 (2002) [38] S.K. Das, G. Thomas, Phys. Stat. Sol. 21 (1974)