Modelling simultaneous precipitationreactions in austenitic stainless steels

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1 C L P H A A D Computer Couplng of Phase Dagrams and Thermochemstry 27 (23) Modellng smultaneous precptatonreactons n austentc stanless steels T. Sourmal,H.K.D.H. Bhadesha Department of Materals Scence and Metallurgy, Unversty of Cambrdge, Pembroke Street, Cambrdge CB2 3QZ, UK Receved 3 June 23 Abstract A physcal model has been developed for smultaneous precptaton reactons n austentc stanless steels, takng nto account nteractons between phases. Comparsons aganst expermental results show satsfyng agreement, wth the model successfully predctng the exstence of transent phases that are often observed, but not predcted n equlbrum calculatons. An analyss of σ -phase knetcs, especally the effect of gran sze, supports the hypothess that the formaton of ths phase depends more on the avalablty of hgh energy nucleaton stes than on the drvng force for ts formaton. The study underlnes a lack of thermodynamc data for mportant phases n creep-resstant austentc stanless steels. 23 Elsever Ltd. All rghts reserved. 1. Introducton Precptaton phenomena n creep-resstant austentc stanless steels are numerous and complex, wth a large number of possble phases formng durng long term exposure to hgh temperature [1]. The occurrence of these dfferent precptates are nterdependent and senstve to small composton changes [1, 2]. Any attempt to model the decomposton of austente nto several phases should therefore take nto account the competton for solute and nucleaton stes and space. Recent works by Robson and Bhadesha [3], or Fujta and Bhadesha [4] have demonstrated the possblty to model nteracton between varous precptaton reactons n steels. Ths work s concerned wth the development and mprovement of a smlar model for austentc stanless steels. 2. Model 2.1. Growth rate n bnary systems Precptates have been approxmated to be of sphercal shape throughout. The dffuson-controlled growth rate of Correspondng author. E-mal address: ts228@cus.cam.ac.uk (T. Sourmal). asphercal partcle n a bnary system s a well-studed problem [5], forwhch the soluton can be wrtten: ψ = S D 2 (1) t where ψ s the velocty of the nterface, D the dffusvty of solute n the matrx, and S s the soluton of: S 3 = 2 Ω exp( S2 4 ) Φ(S) wth Ω = (c c γθ )/(c θγ c γθ ) where c θγ refers to the concentraton of the phase θ n equlbrum wth γ,andc s the bulk composton of the bnary system (Fg. 1). Φ(S) s gven by: ( ) Φ(S) = u 2 exp S2 du. (3) S 4 For small supersaturatons (Ω ) ths can be approxmated by S 2Ω whle for large supersaturatons (Ω 1), S 6/(1 Ω); further detals can be found n [6] Growth rate n a multcomponent alloy In a multcomponent alloy, assumng local equlbrum at the nterface, the nterface composton s gven by the telne satsfyng the set of equatons [7]: (2) /$ - see front matter 23 Elsever Ltd. All rghts reserved. do:1.116/j.calphad

2 17 T. Sourmal, H.K.D.H. Bhadesha / Computer Couplng of Phase Dagrams and Thermochemstry 27 (23) (a) c θγ c 2 θ r c P c θ γ 2 c 2 M c γ θ 2 (b) T c γθ γ c 1 r γ θ c γ θ 1 c θ γ 1 A c γθ c c θγ Fg. 1. (a) The concentraton profle of the element as a functon of the dstance from the nterface matrx/precptate. (b) Relatonshp wth the phase dagram. r s the radal coordnate, the value of whch s r at the nterface. B r c 1 Fg. 2. The flux-balance te-lne gong through M can be sgnfcantly dfferent from the mass-balance one, that passes through P the bulk composton. J = ψ(c θγ c γθ ) = 1,...,n 1 (4) where n s the number of elements n the system, the ndex refers to the element, wth c γθ the concentraton n γ n equlbrum wth θ. When two solutes have sgnfcantly dfferent dffusvtes, the partcular te-lne whch allows the set of Eq. (4) to be satsfed,.e. the flux-balance te-lne, wll n general be dfferent from that passng through c,asllustrated n Fg. 2 for a ternary system Nucleaton rate Classcal theory for nucleaton suggests that the nucleaton rate I s gven by: ( ) ( ) I θ = N exp G θ ν exp G t (5) RT RT where G θ s the actvaton energy for the nucleaton of θ and G t the actvaton energy for thetransfer ofatomsacross the γ/θ nterface; N s the number densty of nucleaton stes and ν s an attempt frequency taken as beng kt/h. Fora sphercal precptate θ, theactvaton energy for nucleaton s gven by [5]: G θ = 16π 3 σ 3 γθ V θ 2 m G 2 m,θ where V θ m s the molar volume of θ, σ γθ the energyperunt area of the nterface γ/θ, G m,θ the drvng force for the precptaton of θ, per mole of components n θ. (6) 2.4. Overall transformaton knetcs Calculatons were carred out n dscrete tme steps to evaluate the volume fractons of the dfferent precptates expected to form. At the end of each tme step, the average matrx concentraton was updated accordng to: dc = (c c θγ )dv f,θ c (7) V θ f,γ where dc s the change of matrx composton, dv f,θ the change of volume fracton for the precptate θ. The computer program for the model was nterfaced wth MT- DATA [8], n orderto re-evaluatethe drvngforcesand local equlbra after each modfcaton of the matrx composton. Nucleaton growth rates are then re-calculated and the process s repeated n order to follow the evoluton of the mcrostructure. In the classcal Avram theory, or n ts adaptaton to smultaneous reactons [3], the ncrease n volume fracton s corrected to account for the presence of transformed regons. Ths correcton was neglected throughout, a reasonable approxmaton gven the small volume fractons nvolved. Table 1 The dffuson coeffcent of Cr n AISI 316 at 75 Cascalculated and found n dfferent studes Materal Calculated (m 2 s 1 ) Lterature (m 2 s 1 ) AISI 316 [12] Cr 14N [13]

3 T. Sourmal, H.K.D.H. Bhadesha / Computer Couplng of Phase Dagrams and Thermochemstry 27 (23) Dffuson coeffcents and other parameters A number of recent studes have focussed on provdng dffuson data n a framework smlar to what CALPHAD s to phase calculatons. Data publshed by Jönsson [9], Ågren [1] andandersson and Ågren [11] havebeen used to calculate the dffuson coeffcentof carbon and chromum as a functon of composton. Ths calculaton s performed wthn the overall model usng the ntal composton of the matrx. Table 1 provdes examples of calculated dffuson coeffcents together wth values found n the lterature. Jönsson [9] made extended comparson wth measurements of the carbon dffusvty and t was verfed that hs results were satsfynglyreproduced. Data were less readly avalable for Nb, T and Mo for these steels, and these coeffcents, when requred n calculatons were set to be equal to that of chromum. Smlarly, n all cases, the actvaton energy for transfer across the nucleus nterface G t was taken to be about the actvaton energy for chromum dffuson, 25 kj mol 1. Fe n / wt% Sze / nm Cr control Cr and C control soactvty of carbon (I) Fe Cr Cr n / wt% 3. Implementaton and results 3.1. Identfyng the flux-balance te-lne Of all precptate phases lkely to form n austentc stanless steels (MX,, σ,etc.,see[1]), the dfference between flux and mass-balance te-lnes was found to be sgnfcant only for.thesmallpredcted dfference for MX type precptates (NbC, TC, NbN, TN) s because these phases are modelled n the thermodynamc databases as pure substances, whose compostons are fxed by stochometry. However, expermental results [14, 15] haveshownthatsuch precptates start growng wth a composton dfferent from equlbrum, the dfference beng attrbuted to the shft n the flux-balance te-lne as precptaton progresses [1, 14]. Because of the assumed stochometry, t s not possble n the present work to properly model MX precptates. For ntermetallcs (σ, Laves, etc.), the dffusvtes of the dfferent elements nvolved are smlar therefore leadng to only a small dfference between the mass and flux-balance te-lne. For the case of,thefollowng approxmatons were made when evaluatng the local equlbrum at the transformaton front. Because the dffuson coeffcent of carbon s larger than that of the substtutonal elements, t was assumed that carbon acheves a unform actvty nstantaneously. Once the te-lne consstent wth carbon soactvty was dentfed, the nterface velocty was calculated on the bass of the correspondng chromum flux: J Cr = D Cr c Cr D CrC c C D Cr c Cr (8) where the cross-dffuson D CrC term can be reasonably neglected Fg. 3. Comparson between measured and calculated sze for n AISI 316 aged at 65 C, usng dfferent models, and example of evoluton of the composton of wth tme. As llustrated n Fg. 3, the calculated growth rate of compares well wth data obtaned by Záhumenský et al. [12] foratype 316 steel at 65 C. Alternatve models have used the Cr gradent as defned by the mass-balance te-lne [3, 16], labelled Cr-control n Fg. 3, or the zero carbon-gradent te-lne [4], labelled Cr C control n Fg. 3. These methods have been shown to under or over estmate the actual growth rate respectvely. Table 2 The Cr/Fe rato n n dfferent cases, for the AISI 34 steel studed by Boeuf et al. The last method clearly gves satsfyng agreement wth expermental results Cr/Fe rato Equlbrum 12 Boeuf et al. at t = 2.1 Carbon zero-gradent.72 Carbon soactvty 2.2 Further valdaton for ths method s obtaned when comparng measured and predcted Cr/Fe ratos n :t has frequently been reported [15, 17, 18]thatthe rato Cr/Fe s sgnfcantly lower at the early stages of precptaton than t s at equlbrum, and gradually ncreases wth ageng tme. Such behavour s correctly predcted by the model (Fg. 3).

4 172 T. Sourmal, H.K.D.H. Bhadesha / Computer Couplng of Phase Dagrams and Thermochemstry 27 (23) Volume fracton.4.2 TC Temperature o C σ Fg. 4. The calculated volume fracton of and TC as a functon of tme n an AISI 321 steel, durng ageng at 75 C. s expected as atransent phase only. The volume fracton of TC does not nclude the amount left undssolved after soluton treatment. Table 2 compares the predcted ntal composton of n AISI 34 to the measured values gven by Boeuf et al. [17] Competton for solute As mentoned earler, the nteractons between precptaton reactons are essental n the understandng of the sequences of reactons occurrng n austentc stanless steels. A typcal example s the case of stablsed austentc stanless steels, where addton of strong carbo-ntrde formers such as Nb or T are made to suppress the formaton of.thelatter s however reported to exstasatransent phase (Fg. 4). The AISI 321 steel (17.1Cr, 12.6N, 1.5Mn,.5S,.49T,.11C wt%) as studed by Thorvaldsson [19] s a good example of the mportance of nteractons n precptaton sequences. It should be noted that the behavour and fnal amount of phases are dctated by the underlyng thermodynamc models. In ths case, TC s predcted to be the stable phase. Dscrepances have been reported [1]onthebehavour of T stablsed steels. In general, there s lttle quanttatve nformaton regardng precptaton n austentc stanless steels. Furthermore, results obtaned by dfferent methods dffer sgnfcantly. For example, the work utlsed n the prevous secton, from Thorvaldsson and Dunlop [19], ndcates that the maxmum volume fracton of MC type carbdes (manly TC) s reached after about 1 h, on the bass of TEM nvestgatons. On the other hand, Thorvaldsson et al. [15], n a dfferent publcaton on a smlar steel, for dentcal condtons, report the maxmum volume fracton of MC type carbdes (mxed (T, Nb)C) to be reached after 3 8 h, on the bass of resstance measurements. There are no comments from the authors as to the largely dfferent knetcs of precptaton. Ths makes t dffcult to estmate the adjustable parameters N and σ γθ for these phases. However, the tme scale for the formatonof such carbdesand ntrdesremans Fg. 5. TTP dagram for the formaton of σ n an AISI 34 steel of composton: 18.7Cr, 9.N, 1.73Mn,.6S,.5C wt%. Predctons (bottom), wth ndcaton of the temperature n C, are compared to the TTP dagram obtaned by Mnam et al. [2] (top). The nose of the TTP curve s n both cases obtaned at 75 C. small n comparson wth the typcal lfetme n creep, and therefore the error ntroduced n the precptaton sequence s of lttle consequence The formaton of σ -phase n the AISI 3 seres Identfyng the nucleaton parameters for AISI 34 The case of σ -phase, whch ncdentally s beleved to be more relevant to long term creep propertes than the varous carbdes, s better defned. Mnam et al. [2] provde data on the mcrostructural evoluton of the man AISI 3 seres steels, whch have been used to refne the parameters N σ and σ γσ (respectvely the number densty of nucleaton stes for σ -phase and the nterfacal energy between σ -phase and austente). Wth N σ = m 3 and σ γσ =.252 J m 2, satsfyng agreement was obtaned between the predctons and both the tme temperature precptaton (TTP) dagram and the quanttatve measurements provded by these authors (Fgs. 5 and 6). The dsagreement at hgh temperatures s essentally due to thermodynamc data on whch the calculatons rely: the NPL PLUS database (based on SGTE SSOL) predcts an equlbrum amount of σ -phase (1.7%) whch s not consstent wth the observatons made by Mnam et al. [2].

5 T. Sourmal, H.K.D.H. Bhadesha / Computer Couplng of Phase Dagrams and Thermochemstry 27 (23) (a) (b) (c) Fg. 6. The rate of formaton of σ -phase at 7 Cn(a) AISI 34, (b) AISI 316 and (c) AISI 347. For 316 and 347 (1) the calculated lne s obtaned wth the prefactor ftted wth the data for 34 [2], corrected for the gran sze dfference. Furthermore, ths fttng can only be sem-quanttatve, as there remans the problem of the detecton lmt, whch may not scale wth the volume fracton: at hgh temperature, few precptates of large sze are expected, whle at low temperature, one expects smaller but more numerous precptates. Ths means that, unless an exact determnaton of the volume fracton s carred out, σ -phase mght be detected at lower volume fracton at hgh temperatures Observed and predcted trends n the AISI 3 seres Fg. 6 shows the evoluton of the volume fracton of σ -phase as a functon of ageng tme at 7 C, n dfferent steels of the AISI 3 seres. Intally, the nucleaton parameters obtaned for AISI 34 were used to predct the formaton of σ -phase n 316 (34 wth Mo) and 347 (Nb-stablsed). The results were opposte to the observed trends, wth the formaton of σ predcted to be slowest n 347, where t s observed to be the fastest. Ths problem s drectly related to the predcted drvng forces n the dfferent steels, shown n Table 3. However, examnaton of the dfferent mcrographes publshed n [2] revealed consderable dfference n gran szes between 34 and 347. The mean lnear ntercept was estmated to be 3.7 tmes smaller n the latter steel. The nucleaton ste densty was therefore corrected so as to account for the smaller gran sze: the model essentally consders gran boundary nucleaton, mplyng that the number densty should be greater n 347 than n 34. Once ths correcton s made, better agreement s obtaned as 347 s correctly predcted to occur faster n 347 than n 34, despte the lower drvng force. Calculatons wth the corrected nucleaton parameters are shown n Fg. 6. The predcted ncrease s generally slghtly steeper than that measured. It was verfed however, that saturaton of the nucleaton stes was reached n the very early stages of the precptaton, whch corresponds to the least steep curve for agvengrowthrate. The explanaton for the dscrepancy was therefore expected to be found n the smple growth model adopted. It must be emphassed at ths pont that very few of the parameters n ths model are user-selected; for example, dffuson coeffcents, nterface compostons and drvng forces are ndrectly provded by thermodynamc databases, there s therefore lttle latttude to modfy the calculated growth rate for a gven phase usng the natural parameters such as dffuson coeffcent, supersaturaton etc. By artfcally multplyng the nterface velocty for σ -phase by a factor varyng between.2 and 1.5, the effect on the steepness of the growth curves could be nvestgated. It was found that, once the nucleaton ste densty modfed to obtan curves located at the same tmes, the gradent of the volume fracton vs. tme curves was not sgnfcantly changed. Furthermore, the fnal partcle szes obtaned for σ -phase were n best agreement wth typcal szes observed n [2] when the growth rate was left unchanged, wth predcted szes of 3.8 µmafter5 h at 7 C, compared to an observed sze of 4.2 µmntheaisi 347 studed n [2]. It s therefore concluded that the dfference s not the result of the smple growth model adopted, but s most lkely the result of the mean-feld approxmaton, as only an enhanced soft-mpngement could further decrease the steepness of the volume fracton vs. tme curves. Further work s requred n order to mprove on ths approxmaton. Barck [21] proposed that σ phase formaton s most strongly affected by gran sze, that s, by nucleaton ste densty. Mnam et al. [2]have opposed ths vewpont, and proposed that the drvng force s a predomnant factor n the rate of σ -phase formaton. However, the present study, on the bass of ther own data, ndcates that, although the

6 174 T. Sourmal, H.K.D.H. Bhadesha / Computer Couplng of Phase Dagrams and Thermochemstry 27 (23) Table 3 The composton of dfferent steels studed by Mnam et al. [2], after precptaton of all carbdes, calculated wth MT-DATA [8], and the drvng force for the formaton of σ -phase from the austente of ths composton, expressed n joules per mole of components. All calculatons are for 7 C. All alloys have an ntal carbon content of.5 wt%, the ntal substtutonal content of 34 and 316 s hardly changed, whle n 347 the ntal Nb content s of.59 wt%, reduced by precptaton of NbC Element wt% Cr N Mn S T Nb C G m (Jmol 1 ) AISI AISI AISI 316 (2.3 Mo) Volume fracton Laves (Fe 2 Mo) + Fe 2 Mo + Fe 2 Mo + σ phase phase σ Fg. 7. The evoluton of the volume fracton of dfferent phases n an AISI 316 steel at 7 C, and on the upper part, the phases observed expermentally by Mnam et al. [2]. expermental calculated the work of Mnam et al. [2]. However, at 75 C, χ phase s found expermentally, but s not predcted to form by the model. Ths s because, accordng to MT-DATA, the drvng force for ts formaton s zero throughout the precptaton sequence. It s mportant to note that a smple equlbrum calculaton wll not predct the presence of Laves phase (Fe 2 Mo), whch only exsts as a transent phase, and dssolves as σ -phase precptaton progresses. Although there s no drvng force for the formaton of Laves phase once σ -phase s formed, ths s not the case f the supersaturated matrx composton s consdered. In a smlar way, there s a drvng force for precptaton of from the ntal supersaturated matrx, n the example llustrated n Fg. 4,however TC s ultmately the only stable carbde. 4. Conclusons drvng force mght have an effect, t s expected to be of small nfluence compared to the gran sze, or any other parameter whch could result n an ncrease n nucleaton ste densty. In fact, use of the drvng force alone leads to predctons of trends opposte to the observed ones. Calculatons ndcate, n general, that σ -phase s consderably more dffcult to nucleate than most carbde phases, and therefore confned to relatvely hgh-energy defects. Ths suggests that the method recently used to control senstsaton [22], by whch the average gran boundary energy s reduced by a carefully desgned thermomechancal treatment, may also lead to suppresson, or sgnfcant delay, of σ -phase formaton. The valdty of these conclusons s obvously dependent on that of the predcted drvng forces, that s, ultmately, on the thermodynamc database used n ths study (NPL plus, based on SGTE SSOL), and on the hypothess that the nterfacal energy σ γσ does not vary sgnfcantly between the three grades. Ths seems a reasonable assumpton gven ther smlar compostons Example of full precptaton sequence Fg. 7 shows the complete precptaton sequence for an AISI 316 steel, at 7 C, whch sngood agreement wth A physcally-based model to predct precptaton sequences has been presented that accounts for nteractons between the dfferent knds of phases formng n austentc stanless steels. The underlyng growth model was shown to gve satsfyng quanttatve agreement when requred thermodynamc data were avalable. The overall knetcs are n satsfyng agreement wth the publshed observatons, partcularly n regard of the relatve smplcty of the model. However, ths success s necessarly lmted to phases for whchthermodynamc data are avalable. Ths s unfortunately not the case for a number of phases observed n austentc stanless steels: n Nb rch steels (AISI 347) n whch Fe 2 Nb and Fe 3 Nb 3 Careexpected to form when Nb s n excess [1]. Although the formaton of the former s correctly predcted, ts dssoluton for the latter cannot be accounted for, snce there are no thermodynamc data for Fe 3 Nb 3 C. Phases such as Cr 3 N 2 SX or Z-phase are not represented n the SGTE database, therefore makng dffcult any meanngful predcton on ntrogen bearng steels. It was shown that the composton change of can be correctly predcted whch suggests t could be used as an ndcator of the progress of the precptaton. The present model also supports the suggeston that σ -phase formaton depends above all on the number of hgh-energy nucleaton stes, the drvng force beng of lesser mportance.

7 T. Sourmal, H.K.D.H. Bhadesha / Computer Couplng of Phase Dagrams and Thermochemstry 27 (23) The code for the correspondng software s freely avalable on the data used for the predctons shown n the present publcaton. Acknowledgements The authors are grateful to Innogy Plc and EPSRC for fundng the project of whch ths work s part, and to Pr. D. Fray for provson of laboratory facltes. References [1] T. Sourmal, Mater. Sc. Technol. 17 (21) [2] A.F. Padlha, P.R. Ros, I.S.I.J. Internatonal 42 (22) [3] J.D. Robson, H.K.D.H. Bhadesha, Mater. Sc. Technol. 13 (1997) [4] N. Fujta, H.K.D.H. Bhadesha, Mater. Sc. Technol. 15 (1999) [5] J.W. Chrstan, Theory of Transformaton n Metals and Alloys, Pergamon Press, Oxford, [6] T. Sourmal, Ph.D. Thess, Department of Materals Scence and Metallurgy, Unversty of Cambrdge, 22. Avalable from [7] D.E. Coates, Metall. Trans. 3 (1972) [8] MT-DATA, Natonal Physcal Laboratory, Teddngton, UK (1989). [9] B. Jönsson, Z. Metallkd. 85 (1994) [1] J. Ågren, I.S.I.J. Internatonal 32 (1992) [11] J.-O. Andersson, J. Ågren, J. Appl. Phys. 72 (4) (1992) [12] P. Záhumenský, P. Ševc, J. Janovec, Kovo. Mater. 37 (1999) [13] W. Assassa, P. Guraldenq, Met. Corros. Ind. 621 (1977) [14] H.O. Andrén, A. Henjered, H. Norden, J. Mater. Sc. 15 (198) [15] T. Thorvaldsson, H. Rubnszten-Dunlop, H.-O. Andrén, G.L. Dunlop, Quanttatve Mcroanalyss wth Hgh Spatal Resoluton, The Metals Socety, 1981, pp [16] R.G. Faulkner, H. Jang, Mater. Sc. Technol. 9 (1993) [17] A. Boeuf, R. Coppola, F. Zambon, J.P. Morlevat, F. Rustchell, D. Wenger, J. Mater. Sc. 16 (1981) [18] J. Barck, Mater. Sc. Technol. 4 (1988) [19] T. Thorvaldsson, G.L. Dunlop, Metal Sc. 16 (1981) [2] Y. Mnam, H. Kmura, Y. Ihara, Mater. Sc. Technol. 2 (1986) [21] J. Barck, Metall. Trans. A 14 (1983) [22] M. Shmada, H. Kokawa, Z.J. Wang, Y.S. Sato, I. Karbe, Acta Mat. 5 (22)