Grain growth behaviour during near-γ solvus thermal exposures in a polycrystalline nickel-base superalloy

Transcription

1 Gran growth behavour durng near-γ solvus thermal exposures n a polycrystallne nckel-base superalloy D.M. Collns, B.D. Condut, H.J. Stone, M.C. Hardy, G.J. Condut & R.J. Mtchell NOTICE: Ths s the author s verson of a work that was accepted for publcaton n Acta Materala. Changes resultng from the publshng process, such as peer revew, edtng, correctons, structural formattng, and other qualty control mechansms, may not be reflected n ths document. Changes may have been made to ths work snce t was submtted for publcaton. A defntve verson was subsequently publshed n Acta Materala 61, 3378 (2013).

2 Gran growth behavour durng near γ solvus thermal exposures n a polycrystallne nckel-base superalloy D.M. Collns a, B.D. Condut b, H.J. Stone c, M.C. Hardy d, G.J. Condut e, R.J. Mtchell f a Department of Materals, Unversty of Oxford, Oxford, OX1 3PH, UK b Strategc Research Centre, Rolls-Royce plc, Snfn-A, PO BOX 31, DE24 8BJ, UK c Rolls-Royce UTC, Department of Materals Scence & Metallurgy, Unversty of Cambrdge, Pembroke Street, Cambrdge CB2 3QZ, UK d Rolls-Royce plc., ELT-10, PO BOX 31, Derby, DE24 8BJ, UK e Theory of Condensed Matter Group, Department of Physcs, Cavendsh Laboratory, 19 J.J. Thomson Avenue, Cambrdge CB3 0HE, UK f Rolls-Royce plc., Raynesway, PO BOX 2000, Derby, DE21 7XX, UK Abstract The gran growth behavour that occurs durng near solvus sothermal and transent heat treatments and the nfluence of prmary γ has been studed n the advanced polycrystallne nckel-base superalloy, RR1000. Expermental observatons showed gran growth can be related to D lm, a theoretcal gran sze lmt that accounts for the tme dependent pnnng contrbuton from the full prmary γ precptate sze dstrbuton. Ths term was also used to descrbe the pnnng contrbuton from MC-carbdes, whch lmts gran sze n the absence of prmary γ. The values of D lm were calculated usng both expermental data and smulated data obtaned from the software, PrecpCalc. The gran growth model proposed by Andersen & Grong [1] was modfed to ncorporate the pnnng effect of prmary γ and MC-carbdes. Ths was used to predct the gran szes for a range of heat treatment tmes, ncorporatng the calculated D lm values. Materal dependent gran growth coeffcents were smultaneously ftted by mnmsng the dfference between the smulated and expermental gran szes. Good correlaton was acheved between the modelled and expermental gran szes, though agreement was superor wth PrecpCalc data. Ths was attrbuted to the removal of expermental scatter, found to be mportant when usng ths method. Key words: gran growth, precptaton, coarsenng, heat treatment 1. Introducton The demandng servce condtons n the hottest stages of gas turbne engnes are almost exclusvely served by components fabrcated from nckel-based superalloys. At elevated temperatures, ths mportant classfcaton of materals have exceptonal mechancal propertes [2]. Ths performance can largely be attrbuted to ts mcrostructure, whch n ts smplest form conssts of a γ, A1 matrx wth a dsperson of coherent γ, L1 2 precptates. In the latest, powder processed turbne dscs, the nckel-base superalloy wll typcally exhbt a trmodal γ precptate dstrbuton. Durng processng, the materal wll be consoldated by hot sostatc pressng (HIP), allowng prmary γ to form. Subsequent subsolvus extruson and forgng affect the volume fracton and precptate sze, whch wll typcally be between 1 µm and 5 µm. The prmary γ has the role of restrctng gran boundary mgraton durng subsolvus soluton heat treatments. Subsequent heat treatments can be used to promote the formaton of two further ntragranular precptate populatons, denoted as secondary γ and tertary γ, wth dameters typcally between 100 & 400 nm and 5 & 50 nm respectvely are also formed [3]. The lattce parameters of the γ and γ phases are talored to gve a low lattce msft, partcularly at servce temperatures, thereby reducng deteroraton of mechancal propertes due to precptate coarsenng or morphologcal changes [4]. The mechancal propertes of nckel-base superalloys are largely governed by the nteracton between the precptates and weakly or strongly coupled dslocatons. These propertes are therefore crtcally controlled by the sze, dstrbuton and morphology of the γ precptates [4, 5]. It s generally consdered desrable to heat treat the alloy to have as many γ precptates as possble wthn the dameter range whch s most benefcal to the mechancal propertes [6, 7]. As the prmary γ precptates are too large to contrbute to the statc strength of the materal, the volume fracton of the prmary γ should therefore be reduced. However, a mnmum volume fracton of γ must reman present to resst gran boundary mgraton and hence preserve a suffcently fne gran sze. Hstorcally, the gran sze n a turbne dsc was selected to be a compromse between the varyng property requrements at dfferent radal locatons across the dsc [6]. At the rm of the dsc, hgh temperatures and moderate stresses demand good creep performance and resstance to dwell crack growth, both of whch mprove wth an ncreased gran sze. Whlst n the bore of the dsc, lower temperatures and hgher stresses demand a fne gran mcrostructure to delver good tensle and low cycle fatgue propertes [8]. The contemporary process of dual mcrostructure heat treatments (DMHT) enables an approprate gran sze to be produced for each radal locaton, thereby offerng mproved performance over the turbne dsc manufacturng Preprnt submtted to Acta Materala February 13, 2013

3 processes hstorcally used. A dual mcrostructure s acheved by subjectng the dsc forgng to locaton specfc heat treatments. Of the large gas turbne manufacturers, each has developed a unque process to provde these heat treatments, controllng the manner n whch heat s delvered and extracted from the component [9, 10, 11]. Common to all, however, s a subsolvus heat treatment close to the dsc bore, retanng the trmodal γ dstrbuton, whereas the rm s subjected to a supersolvus heat treatment, dssolvng the prmary γ and thus permttng gran coarsenng. The advanced polycrystallne nckel-base superalloy, RR1000 [12], s a canddate materal for DMHT [11, 3], and n the present study, the gran growth response s examned for the near-γ solvus heat treatments requred for the optmsaton of such a process. Ths study ncludes both sothermal and transent heat treatments, representatve of the range of condtons used durng component manufacture. Ths s subdvded nto (1) expermental observatons of gran sze, prmary γ and carbde precptate sze dstrbutons (PSDs), and (2) the smulaton of gran sze usng a gran growth model. Comparsons are also made between the precptate pnnng effects from expermentally measured PSDs, and PSDs predcted usng the commercal modellng software PrecpCalc. 2. Background For a sngle phase alloy of mean gran dameter, D, gran growth wll be dependent on the moblty of a gran boundary and the drvng force, whch s proportonal to 2σ/ D, where σ s the surface energy per unt area. Emprcally, the gran growth may be descrbed by the followng relaton [13]: D = k 1 t n (1) where k 1 s a constant wth respect to the system temperature, t s tme and n s a gran growth exponent. The value of n has been dscussed by Fullman et al. [14], who showed that a value of 0.5 wll only be reached n very pure metals and at temperatures close to the meltng pont. In Equaton 1, the startng mean gran dameter at t = 0, s assumed to be equal to zero, and hence ths relaton s only sutable for long annealng tmes where the startng gran dameter may be neglected. For materals operatng n a regme of normal gran growth under sothermal condtons, Hu and Rath [14] descrbed the gran sze by ( ) D 1 n D 1 Qapp n 0 = k 2 exp t (2) RT where k 2 s a constant dependent on the physcal knetcs, D 0 s an ntal mean gran sze, Q app s the apparent gran boundary actvaton energy, R s the molar gas constant, t s the annealng tme and T s temperature. In ths example, n s expected to be between 0.1 and 0.5, dependng on the resstve force to gran boundary mgraton from solute atoms [15]. The drvng force for growth can be attrbuted to an average thermodynamc drvng pressure of a curved gran boundary, P g, wth an assocated gran boundary nterfacal energy, σ gb. Ths relatonshp s gven as P g = k 3 ( σgb D ) (3) 2 where k 3 s a constant dependent on the system characterstcs. Gran growth can be hndered by a dstrbuton of second phase partcles, as frst reported by Zener [16]. For the applcaton to nckel-base superalloys, Song et al. [17] observed the Zener pnnng effect on the growth of γ from both γ and other nert partcles, such as yttrum oxdes or MC-carbdes (M denotes a metallc element). The partcle/precptate szes and volume fractons of the phases were addtonally found to be hghly mportant. The powder process used durng turbne dsc manufacture has been observed to nfluence the gran growth behavour. In the process of soldfcaton, followng gas atomsaton, certan elements segregate to the surface of the powder partcles. These tend to be carbde formers, such as T, Ta, Nb and Hf. Surfaces of the powder partcles also tend to pck up oxygen and ntrogen from atomsaton, screenng and blendng despte the use of a hgh vacuum or an nert atmosphere. Durng hot sostatc pressng, carbon dffuses to the surface of the partcles and reacts wth surface speces to form oxy-carbdes. These decorate the pror partcle boundares (PPBs) and heavly nfluence the gran sze and morphology followng the HIP process. In addton, alloy RR1000 contans very small HfO 2 partcles, whch also provde a contrbuton to gran boundary pnnng [18]. Opposng the drvng pressure for gran growth s the Zener drag pressure, denoted as P z, whch s a functon of the precptate volume fracton, φ, and the partcle radus, r, gvng ( φ ) P z = k 4 σ gb (4) r The constant, k 4, s a system dependent coeffcent. The relatonshp between the thermodynamc drvng pressure and the Zener drag has been used by Hu and Rath [14], to descrbe the velocty of the mgratng boundary, v, v = M ( P g P z ) 1 n 1 (5) where M s the moblty of the gran boundary. The drvng force per unt volume of boundary s smply the dfference between the thermodynamc drvng pressure, P g, and the resstve pressure due to Zener drag, P z. When n = 0.5, the gran boundary velocty vares lnearly wth the drvng pressure. When n < 0.5, the gran boundary velocty vares wth a power of the drvng pressure [14]. Modellng of gran growth n nckel-base superalloys wth the ncluson of pnnng effects usng phase feld methods can also be performed, e.g. [19]. Whlst such models have the potental to nclude all of the governng physcs, they arguably reman unattractve, partcularly for commercal applcatons due to the requrement of sgnfcant computatonal power, the number of assumptons made and the need for consderable thermophyscal data. One alternatve may be to use a mean-feld statstcal method for gran growth, whch agan may ncorporate a statc or non-statc pnnng effect [20]. An analytcal model whch descrbes gran growth behavour n the presence of precptates that are allowed to grow or dssolve as the grans are smultaneously growng has been developed by Andersen & Grong[1]. Ths model permts expermental lnear ntercept measurements of mean gran sze, denoted as

4 D, to be converted to equvalent 3D measurements, and uses these to predct the 3D gran growth behavour for an arbtrary thermal cycle. Therefore, from expermentally determned 2D mean gran dameters, a converson of D = 3 D 2 s used, assumng a log-normal gran sze dstrbuton [14, 1]. Ths model has been successfully used to descrbe normal [1] or abnormal gran growth [21], and has been appled to a number of dfferent metallurgcal systems.e. [22, 23, 24]. Followng the model for normal gran growth, gven n Equatons 3 & 4, the velocty, v, and moblty, M, are descrbed by the followng expressons; v = 1 2 (d D/dt) and M = M 0 exp( Q app /RT) [14] and substtuted nto Equaton 5 to gve a generc gran growth model whch accounts for the presence of mpurtes and pnnng second phase partcles. M 0 s a knetc constant n the moblty. In ths way, the rate of change of gran sze s gven by d D dt = 2M 0 (k 3 σ gb ) 1/n 1 exp ( Qapp RT ) [ 1 D k 4φ k 3 r ] 1 n 1 (6) Ths dfferental equaton can be ntegrated to evaluate the gran sze as a functon of tme. It s therefore possble to nclude a dynamc process such as precptaton, ncorporatng the temporal evoluton of volume fracton and partcle radus. Wth ths approach, gran growth can be evaluated n systems of stable, growng or dssolvng precptates. Many of the constants can be collapsed nto smpler forms, specfcally, M0 = 2M ( ) 1 n 0 k3 σ 1 gb (7) k = k 3 k 4 (8) where M0 and k are both constants based on physcal parameters. The value of k was gven as 4/3 by Zener, though ths value has been dsputed and recalculated numerous tmes wth lmted agreement, as summared by ref. [25], showng a dependence on the assumptons used to model the drag force [15]. By agglomeratng the constants n ths way, Equaton 6 can be smplfed to gve ( ) [ d D = M0 dt exp Qapp RT 1 D 1 ] 1 n φ 1 (9) k r The lmtng gran sze, D lm, s therefore related the radus and volume fracton of the pnnng precptates, as well as the Zener coeffcent, by D lm = k ( r φ ). (10) Practcally, grans wll be allowed to grow as the volume fracton of pnnng precptates decreases or ther radus ncreases. Equaton 9 can be rearranged and solved as defnte ntegrals for D and t as follows, D t D 0 d D ( 1 D ) D 1 1 lm t2 = n 1 t 1 M 0 exp ( Qapp RT ) dt. (11) For the condton of n = 0.5, Equaton 11 can be solved for D analytcally. For n 0.5, the ntegral has to be solved usng a numercal method Expermental detals In ths nvestgaton, the relatonshp between gran sze, and the sze dstrbuton of the prmary γ and carbdes of the polycrystallne nckel-base superalloy, RR1000, were nvestgated as a functon of heat treatment condtons. The startng mcrostructure of the materal was dctated by the pror thermomechancal deformaton, havng been HIPed and extruded n a subsolvus condton. Ths bllet materal was sectoned nto cubes measurng mm and subjected to the matrx of heat treatments lsted below. 1. Isothermal heat treatments Samples were heat treated for 2, 4, 6, 8, 10, 20, 30, 40, 60, 90 and 120 mnutes at temperatures between 1120 C and 1200 C n 10 C ntervals. 2. Transent heat treatments Samples were subjected to varyng thermal cycles wth a monotonc heatng rate of 222 C h 1. Ths rate was selected to closely represent a typcal thermal cycle experenced by a turbne dsc durng a processng heat treatment. Samples were heated from 3 startng temperatures: 1038 C, 1052 C and 1066 C. The heat treatments were nterrupted when selected temperatures had been reached, rangng from 1090 C to 1200 C n 10 C ntervals. For all of the samples, after they were nserted nto the furnace, the temperature ntally dropped. As a result, the heat treatment tme was started once the furnace temperature had returned to the target temperature. Ths was consstently observed to be less than one mnute. Each heat treated sample was mounted, ground and polshed to a 1 µm fnsh. The samples were then electrolytcally etched at 4 V usng an aqueous soluton of 10 vol.% phosphorc acd, leavng the γ precptates n relef from the γ surface enablng mcrostructural magng usng optcal and electron mcroscopy. A number of mcrostructural features present n RR1000 that are consdered n ths study are shown n Fgure 1. The optcal mcrograph, (a), shows a supersolvus mcrostructure where only γ gran boundares can be observed at ths magnfcaton. A backscattered SEM mcrograph, (b), shows the fne gran structure of a sample havng receved a subsolvus heat treatment, wth prmary γ present at the gran boundares. A hgher magnfcaton TEM mage, (c) prepared usng a carbon replca method [26], shows the ntragranular secondary and tertary γ. A hgh voltage backscattered electron mage s shown n (d), and reveals the dstrbuton of carbdes present n the materal. Whlst observaton of prmary γ was possble usng optcal mcroscopy, large dscrepances were found wth precptate szes obtaned usng SEM. The etchng process leaves the γ n relef and presents a problem when usng optcal mcroscopy as the rased features glare when observed, resultng n ther volume fractons and precptate szes beng overestmated. The relable charactersaton of the sze, dstrbuton, morphology and volume fracton of γ, can therefore only be acheved wth electron mcroscopy. Optcal mcroscopy was therefore used for gran sze assessment only, as these measurements were not compromsed by ths phenomenon.

5 Mean 2D gran szes were obtaned usng the lnear ntercept method [27] for each materal condton. Backscattered electron mcrographs were captured usng a JEOL 6340F FEGSEM at the same magnfcaton. Images were taken at a seres of random locatons throughout each sample, for at least 200 precptates to be measured. The prmary γ0 sze dstrbutons were obtaned from the raw mages usng the mage analyss software, ImageJ [28, 29]. Durng mage processng, the prmary γ0 precptates were manually solated by selectng regons of the mcrograph that exceeded a greyscale threshold. These selected solated regons were next converted to bnary mages and ndvdually measured. The measured area of each precptate was then converted to an equvalent radus, assumng the precptate was sphercal. The total area fracton occuped by the precptates was used to provde an equvalent measurement of the volume fracton [30]. a 20 µm b gran boundares prmary 4. Expermental Results 4.1. Materal Startng Condton The materal charactersed n ths study had an ntal gran sze of 3.0 µm ± 0.2 µm, where the error s gven as one standard devaton. The partcle szes of the prmary γ0 are presented n Fgure 2 (a) along wth a log-normal sze dstrbuton ftted to the data. The precptates dentfed as prmary γ0 were observed to be present on the gran boundares and at trple ponts, showng qute rregular, globular morphology. The volume fracton of prmary γ0 n ths startng condtons was found to be 0.17 ± 0.01, wth a mean precptate radus of 0.39 µm ± 0.01 µm. 1 µm c tertary 4.2. Pnnng from Carbdes Mcrostructural analyses were also conducted to characterse the carbdes present. Backscattered electron magng was performed usng a CamScan MX2600 SEM at an appled voltage of 30 kev. An example mcrograph s shown n Fgure 1 (d). Measurement of these carbdes gave a volume fracton of ± and a mean partcle dameter of 140 ± 34 nm. An example sze dstrbuton from these carbdes s shown n Fgure 2 (b). Equlbrum thermodynamc calculatons at the heat treatment temperature usng JMatPro [31] predcted MC carbdes are the only stable carbdes at heat treatment temperatures n the vcnty of the γ0 solvus temperature. The MC carbdes are predcted to be present wth a volume fracton of at 1100 C; and, that ths volume fracton remans approxmately constant between 960 C and 1260 C. Ths s n good agreement wth the expermental measurements made n ths study, and wth other results reported n pror nvestgatons on RR1000 for a range of heat treatments [32]. Addtonally, the measured carbde partcle sze dstrbutons dd not notceably change wthn the matrx of heat treatment condtons studed. secondary 100 nm d gran boundary annealng twns carbdes twn boundary 1 µm Fgure 1: Example optcal mcrographs of RR1000 heat treated n the condtons (a) supersolvus, 1170 C for 40 mnutes, and (b) subsolvus, 1140 C for 40 mnutes. Smaller ntragranular secondary and tertary γ dstrbutons are observed at hgher magnfcaton usng TEM. Carbon replcas were used to produce the mcrograph shown n (c). Identfcaton of carbdes and other mcrostructural features n RR1000 from backscattered SEM mage at 30 kev shown n (d) Isothermal Heat Treatments The measured gran szes obtaned followng the sothermal heat treatments are presented n Fgure 3 (a). These results wll be dscussed wth reference to the correspondng prmary γ0 volume fractons, shown n Fgure 6 (a). Wth a startng 2D 4

6 (a) Prmary radus / m Carbde radus / m (a) Fgure 2: Partcle sze dstrbutons for (a) prmary γ precptates from the startng, bllet condton, and (b), carbdes partcles, measured from materal subjected to an sothermal heat treatment of 2 mnutes at 1200 C. These dstrbutons have been ftted wth a log-normal functon. mean gran dameter of 3.0 µm at 1120 C, the growth s restrcted by pnnng prmary γ and does not exceed 3.8 µm after 120 mnutes at ths temperature. Durng ths perod, the γ volume fracton s ntally 0.17 and drops to After ths tme at 1120 C, t s expected that the volume fracton wll have approached equlbrum. At the hgher temperature of 1130 C, the γ dssoluton s ncreased, wth the volume fracton reducng to 0.09 n just 6 mnutes. After 120 mnutes at 1130 C, the volume fracton drops to approxmately 0.04, allowng steady, but lmted, gran growth up to an observed mean gran dameter of 5 µm. Rapd gran growth occurs at temperatures of 1140 C and above. At 1140 C, the γ volume fracton drops to 0.04 wthn 8 mnutes, and falls further to approxmately 0.02 after 120 mnutes. Though prmary γ remans present, t appears that the pnnng effect of the precptates has been reduced enough to no longer effectvely resst the mgratng gran boundares. After all of the prmary γ has dssolved, the rate of gran growth shows comparatvely small ncreases wth ncreasng temperature. Ths suggests that the gran sze s now lmted by the pnnng from a dfferent partcle dstrbuton once a certan gran sze s reached. It s proposed that the presence of carbdes of suffcent sze and volume fracton may account for the observed retardaton of gran growth at these temperatures. Evdence of nert partcles ncludng MC carbde pnnng has also been reported n other N-base superalloys [33, 17]. However, the lower volume fracton and fner sze of the carbdes, means they do not affect the gran growth behavour when the grans are fne, nstead the pnnng effect s domnated by prmary γ and wll therefore be controlled by the r/φ rato of these phases, as expressed n Equaton 10. At temperatures of 1150 C and above, complete γ dssoluton s expected, as shown by a dffer- Fgure 3: Gran sze as a functon of tme at varous sothermal heat treatment temperatures. Expermental observatons are shown n (a). In (b), modelled gran growth results are shown usng D lm /k values from measurements, and (c), D lm /k values are calculated from PrecpCalc smulatons. Both models shown had all data ftted smultaneously. (b) (c) 5

7 perfect ft (a) Fgure 4: The modelled gran growth szes usng (a) D lm /k values determned from expermental measurements of prmary γ, and (b) wth D lm /k values determned from Precpcalc predctons of prmary γ, are compared to expermentally measured gran szes. ental scannng calormetry (DSC) measurement, predctng the onset of prmary γ dssoluton at 1145 C at a constant heatng rate [7]). Thermal exposures at 1150 C, 1160 C and 1170 C all showed rapd gran growth as the γ volume fracton reduces to zero. These sothermal exposures once agan appeared to reach a lmtng gran sze of approxmately 25 µm. As expected, at 1180 C, 1190 C and 1200 C, the gran growth rate s further ncreased, whch can be attrbuted to both the fast dssoluton of prmary γ, and the temperature dependent thermal actvaton of gran boundary mgraton. Once agan, these thermal exposures appeared lmted by carbdes, wth grans not growng beyond 25 µm. It therefore appears from ths evdence that the r/φ value assocated wth the carbdes remans approxmately constant for the test duratons and temperature ranges examned durng ths nvestgaton, consstent wth the nvarablty of the MC-carbdes observed from the mcrostructural observatons and confrmed by the thermodynamc equlbrum calculatons Transent Heat Treatments Followng the forgng process of a turbne dsc, the tme taken for the component to heat up to the desred heat treatment temperature may be sgnfcant, and therefore understandng the gran growth response to a transent thermal cycle s valuable, partcularly when conductng supersolvus heat treatments. The mean gran sze and volume fracton measurements obtaned from the transent heat treatments are shown n Fgure 5 (a) and Fgure 7 (a) respectvely. As temperature ncreases, the volume fracton of prmary γ can be seen to drop, regardless of the startng temperature. Between 1090 C and 1130 C, the mean gran sze appears unaffected by the ncreased temperature and assocated drop n volume fracton. Gran growth begns only once the temperature reaches 1140 C, for materal wth a startng temperature of 1038 C. From ths startng temperature, the mean gran sze ncreases rapdly to 17 µm once 1150 C s reached. An ntal acceleraton of gran growth s quckly followed by a slower growth rate as the pnnng effect from carbdes becomes ncreasngly nfluental. Unlke the sothermal heat treatments, where the gran growth rate was observed to accelerate between 1150 C and 1200 C, a slowly ncreasng rate of gran growth s observed. Ths may smply be attrbuted to the ncreased thermal energy avalable to (b) 6 the mgratng gran boundares. For the heat treatments startng at 1052 C and 1066 C, a smlar trend s observed, though the rapd burst of gran growth lags behnd the observed mean gran szes n the samples started at 1038 C. The lower startng temperature s clearly a contrbutor to gran growth, where the tme at elevated temperatures must be crtcal. It would be expected that the samples havng started at a lower temperature, and thus havng a longer heat treatment, wll result n lower volume fractons of prmary γ at the correspondng nterrupted temperatures. However, t s not possble to unequvocally conclude that any dfference n gran growth between the startng temperatures s solely dependent on the dfferences n the γ volume fracton that as a result of an extended tme at elevated temperature beyond expermental error. 5. Modellng To successfully smulate the gran gran behavour n nckelbase superalloys, one must account for the temporal γ precptate behavour and ts tme dependant pnnng effect on the mgratng γ gran boundares. The followng secton descrbes frstly the predcton of γ PSDs, and secondly, a gran growth model whch ncorporates the modelled γ PSD predctons Smulaton of γ Precptaton The modellng software PrecpCalc s capable of smultaneous predcton of precptate nucleaton, growth and coarsenng. It s based on the Kampmann-Wagner (KW) [34] soluton to the well-known Langer and Schwartz (LS) [35] algorthm, descrbng nucleaton and growth of precptates n a near-crtcal, metastable lqud and has been calbrated for dfferent alloy systems ncludng RR1000 [36]. For a detaled dscusson of the modellng strategy and knetc equatons used, the reader s referred to [37], and for lmtatons and applcatons to RR1000 [7]. To assess the extent to whch ths software could predct the mcrostructural changes assocated wth the heat treatments nvestgated n ths study, smulatons were run wth PrecpCalc usng data on the ntal mcrostructure determned expermentally. In Fgure 6 (c) & (d), the prmary γ volume fractons and D lm /k values calculated wth PrecpCalc for the sothermal heat treatments are shown. The predctons made at 1120 C and 1130 C show a steady ncrease n D lm /k, as the lmtng gran sze ncreases, as functon of heat treatment tme, wth a concomtant decrease n the prmary γ volume fracton. However, the D lm /k values become extremely large wth longer tme at hgher temperatures as the dssoluton rate has been overestmated by the model. The predctons for 1140 C and above show complete dssoluton, whch s expected as the γ solvus temperature s approxmately 1149 C [3]. In general, the predcted trends n D lm /k are n good agreement wth the expermental data,.e. suggestng slow and steady growth at low temperatures, and rapd growth at hgh temperatures, although there s some dsparty between the absolute values. A number of serratons can be observed n the 1150 C sothermal thermal cycle. Close to the γ solvus temperature, the PrecpCalc model predcts small fluctuatons n dssoluton and reprecptaton of γ. These serratons are beleved to be artefacts of the

8 perfect t (a) smulaton and are not observed n the expermental observatons wthn expermental error. The prmary γ volume fractons and D lm /k values predcted usng PrecpCalc for the transent thermal exposures are shown n Fgure 7 (c) & (d). These data appear to be n good agreement wth the values obtaned from the transent heat treatments n Fgure 5 (a). Wth the sothermal heat treatments, dssoluton of γ above the solvus temperature s rapd, and hence descrbng ths transformaton s dffcult at nterrupted tme ntervals. The slower dssoluton rate of prmary γ observed wth the transent thermal exposures enables D lm /k values to be obtaned wth greater confdence from expermental measurements, and are found to be well reproduced by the PrecpCalc smulatons. The precptaton smulatons show an ntal ncrease n volume fracton, wth an assocated drop n the D lm /k value at 10 mnutes, or before (dependng upon the startng temperature). However, the tme resoluton of the expermental measurements was not small enough to ascertan whether ths subtle effect s seen n realty Smulaton of Gran Growth Behavour (b) For the applcaton of the Andersen gran growth model to a mcrostructurally complex system, such as a polycrystallne nckel-base superalloy, t s necessary for the model to be approprately modfed. The Andersen model consders only the pnnng effect of a sngle dstrbuton of partcles, where only the mean radus of the partcle dstrbuton s consdered. Improvements can therefore be made by consderng the effect of the whole sze dstrbuton, rather than a mean sze alone. Dfferences between the r/φ estmatons can be attrbuted to the hgh polydspersty of γ dstrbutons present n nckel-base superalloys, and how the standard devaton of the dstrbuton changes wth tme. Calculatons usng a mean dstrbuton radus may therefore produce a based Zener pnnng constant, k, when estmatng D lm. In the followng modfcaton, D lm s calculated by consderng multple hstogram bns. Frstly, followng Equaton 10 for the prmary γ dstrbuton; D lm = k r γ φ γ (12) where r γ s the mean dstrbuton radus for the prmary γ precptates and φ γ s the volume fracton of the prmary γ dstrbuton. Each hstogram bn has a measured frequency for a radus range, between a lower radus, r 1, and an upper radus, r 2. For each class,, the average radus, r s gven smply as Fgure 5: Gran sze as a functon of tme for transent thermal cycles from dfferent startng temperatures. (a) Expermental observatons of transent heat treatments. Modelled gran growth results are ftted smultaneously usng (b) D lm /k values from measurements, and (c) D lm /k values calculated from PrecpCalc smulatons. The modelled 2D gran szes versus the expermental gran szes are also shown wth each modelled example n the nset graphs. (c) r = 1 2 (r 1 + r 2 ) (13) Assumng the precptates are spheres, for each hstogram bn the precptate volume s, V = 4 3 π r3. A scalng parameter, α, s defned to normalse each hstogram bn aganst the dstrbuton volume fracton, φ, such that α = 1 φ N V f (14) =1 7

9 where N s the maxmum hstogram bn number and f s the measured frequency for the th bn. Each hstogram bn can now be defned n terms of ts contrbuton to the volume fracton, denoted as φ φ = V f α (15) As N =1 φ = φ, the scalng factor can be elmnated by combnng Equatons 14 and 15, gvng φ = V f φ N =1 V f (16) Each bn n the partcle sze dstrbuton contrbutes a proporton of the overall pnnng. The value of D lm (appled to any dstrbuton) can now be better descrbed by; 1 N φ D lm = k r (17) =1 A smlar treatment of a sze dstrbuton of precptates on the drag pressure has been nvestgated by Evan et al. [38]. In that study, the sgnfcance of a varyng dstrbuton polydspersty to the consequent pnnng effect was noted. For ths reason, the whole dstrbuton was consdered n ths study. As a crude estmate, the D lm /k values should provde a good ndcator of gran growth behavour. The D lm /k values calculated from the sothermal heat treatments measurements are shown n Fgure 6 (b). Smlarly, n Fgure 7 (b), the equvalent calculaton for the transent heat treatment measurements are presented. Assumng the pnnng effect can be adequately descrbed by the D lm /k values, these can be compared to the gran szes obtaned expermentally. As a general observaton, the D lm /k values accountng for the whole γ dstrbuton predct hgher values than the calculatons ncorporatng mean precptate szes only. Typcally, consderng the whole dstrbuton a D lm /k value 30-40% larger s obtaned than consderng only the mean precptate sze. Ths dfference demonstrates precptates wth szes devatng from the mean may contrbute sgnfcantly to the pnnng effect and should therefore not be neglected. The Andersen & Grong [1] gran growth model enables a mean gran sze to be predcted for any thermal cycle, as a functon of the heat treatment tme. From the results obtaned n the precedng secton, the values obtaned for D lm appear hghly dependent on the D lm /k quantty. The Andersen & Grong model can therefore be modfed to account for the polydspersvty of the γ dstrbuton through the use of the modfed D lm descrbed by Equaton 17. In addton, above the γ solvus temperature, the γ grans appear to be pnned by carbdes, and as such, the expresson used to descrbe gran boundary mgraton should also account for the contrbutons from both the prmary γ dstrbuton and the carbde dstrbuton. Lmtng gran szes for both prmary γ pnnng and carbde pnnng can be descrbed as D lm1 = k 1 N =1 φ γ r γ 1 (18) 8 and D lm2 = k 2 N =1 φ carbde r carbde 1 (19) where D lm1 and D lm2 are the gran sze lmts wth pnnng by prmary γ and carbdes respectvely. The Zener pnnng coeffcents are assumed to be ndependent for γ & carbdes and are denoted as k 1, and k 2 respectvely. The volume fracton and precptate rad of the th bn wthn the dstrbuton are denoted as φ γ & r γ for γ and φ carbde & r carbde for the carbde dstrbuton. From mcroscopy observatons, the carbdes are not observed to decorate the gran boundares when the prmary γ remans present, and therefore do not contrbute to the pnnng effect untl all of the γ s dssolved. Instead of smply addng the terms 1/ D lm1 and 1/ D lm2 n seres, each s rased by the power, m, followed by the m th root taken of the sum of the two pnnng contrbutons. For the calculatons conducted n ths study, m was arbtrarly taken to be 5, whch s a suffcently large number to enable γ to domnate when present, and the carbdes to domnate once the γ has dssolved. The fts were seen to be nsenstve to the value of m selected for m 5. Pnnng by secondary and tertary γ may also be dsmssed. DSC analyss has prevously revealed that complete dssoluton of secondary and tertary γ occurs once temperatures of 1115 C and 1000 C respectvely had been reached [7]. Therefore, at the soluton heat treatment temperatures consdered n ths study, these precptate dstrbutons would have ether entrely dssolved, or wll be present only n very small volume fractons. It s hence assumed that the pnnng contrbuton of ntragranular γ s neglgble. Accountng for the approprate RR1000 specfc pnnng contrbutons, the Andersen & Grong model now takes the followng form; D t D 0 d D [( ) m ( 1 D 1 D lm1 + = t2 t 1 ) m ] 1 m 1 D lm2 M 0 exp = I (20) 1 n 1 ( Qapp RT ) dt Equaton 20 may be solved by varous methods. Frstly, the dervatve of the gran growth functon (n the format of Equaton 9) could be ftted to the expermental data. Ths s possble when the dervatve of the expermental data can be relably obtaned. In practce, ths s very smple to obtan, although ts accuracy s largely dependent on the number of expermental data ponts and the degree of scatter. Ths may be aded by parametersng the data, though ths approach once agan reles upon a large number of data ponts for confdence. To help the fttng of the data, a smoothng algorthm could also be used. Followng smoothng, the expermental data may then be vsually nspected to ensure ths process had not lost nformaton to descrbe the trends n the gran growth response. In ths study, the ntegral n Equaton 20 was evaluated numercally, wth the expermental tmes used to calculate the

10 (a) (b) Volume fracton Volume fracton (c) D lm /k D lm /k (d) Fgure 6: Volume fracton and calculated D lm /k values for sothermal exposures; (a) & (b) from measurements, and (c) & (d) from PrecpCalc smulatons, respectvely. These data can be used to approxmate the lmtng gran sze for each thermal cycle, at any correspondng tme. mean gran sze, D t. The ntegral was solved at 500 nterpolated ncrements between each value of t. Where t = 0, D 0 = 4.5 µm (from a 2D expermental measurement of 3.0 µm) was used. The left hand sde ntegral of Equaton 20 was evaluated usng Smpsons rule. After an ntal guess for D t, convergence of ths value was found usng the bsecton method for each tme step, t, re-evaluatng the LHS ntegral wth each teraton untl ths was approxmately equal to the RHS ntegral. The functon could then be ftted drectly to the expermentally measured values of D by varyng, Q app, k 1, k 2, n and M0. For each tme step solved, the correspondng D lm1 value was used. These were ether expermentally determned or calculated from PrecpCalc predctons. Mcroscopy observatons show the carbdes dd not appear to change as a functon of tme or temperature, gvng a fxed value of D lm /k throughout the smulatons. Usng the measured carbde dstrbuton, the D lm2 = N =1 r /φ value was fxed at 49.2 µm, where k represents the greatest 3D mean gran sze (n mcrons) for ths materal. In summary, the model nput conssts of gran sze measurements, D, the assocated D lm1, D lm2, and temperature. For each tme step solved, a fttng ndex,, was assgned along wth a correspondng fttng score, δ. A lnear weghtng was used, based upon the followng expresson: δ = Icalculated I target (21) hence, when δ = 1, a perfect ft s obtaned. The values of I target were calculated by solvng the RHS ntegral of Equaton 20 and I calculated was calculated from the LHS ntegral of Equaton 20 usng teratve guesses of D t. A fttng ndex for all tme steps was then calculated by δ = N δ (22) =1 The above expresson was calculated durng each teraton of a fttng procedure n whch all the data obtaned followng the varous sothermal or transent heat treatments were ftted smultaneously. Ths enabled the global mnmum of all data to be found at the correspondng mnmum value of δ. The best ft was obtaned by smultaneously changng each varable usng a specally developed searchng algorthm. 9

11 (a) (b) Volume fracton Volume fracton (c) Temperature / C Temperature / C (d) Volume fracton or r/ vs tme 1060 Temperature vs tme Fgure 7: Volume fractons and calculated D lm /k values for the transent heat treatments. (a) and (b) are from measurements, and (c) and (d) are derved from predctons made usng PrecpCalc. For reference, the temperature s also plotted wth respect to tme wth the volume fracton data Gran Growth Model Results and Dscusson The model predctons of the gran growth that occurs durng the sothermal heat treatments are shown n Fgure 3 (b) and (c), where the D lm /k values shown n (b) were obtaned by fttng all of the expermental data smultaneously, whlst the D lm /k values shown n (c) were obtaned by fttng to the results obtaned from PrecpCalc smulatons. These results have been converted from 3D to 2D gran szes for drect comparson aganst the expermental measurements. A summary of the fttng coeffcents obtaned for all plotted smulatons are shown n Table 1. In Fgure 4, the success of fttng the sothermal heat treatments s shown by plottng the modelled 2D gran szes versus the expermental 2D gran szes. Fgure 4 (a) corresponds to data ftted usng D lm /k values from expermental measurements and Fgure 4 (b) corresponds to D lm /k values determned from PrecpCalc smulatons. Errors have been plotted for expermental measurements and for modellng results whch used expermental nput data for D lm /k values. Ths was not possble when usng PrecpCalc data as uncertantes are not generated from precptaton predctons. It can be seen from Fgure 3 (b) and Fgure 4 (a) that fttng all of the sothermal expermental data smultaneously provdes good correlaton to the 10 expermental data, partcularly for temperatures 1140 C and above. However, the modelled gran szes for the heat treatments at 1120 C and 1130 C were underestmated. At these temperatures, the gran growth behavour s heavly restrcted by the presence of pnnng prmary γ, and expermentally, lttle growth was observed. As the measured D lm1 ncreases, the model predcts sgnfcantly more gran growth than s observed expermentally. The sudden growth rate change observed expermentally between 1130 C and 1140 C ndcates the behavour s very senstve to temperature. Clearly, small expermental dscrepances n the temperature or precptate measurements could account for some of ths error, though t s unlkely fully explan the dfference. If a sngle D lm1 value s overestmated, the gran sze for the correspondng tme nterval would be permtted to ncrease. As gran sze s only predcted to ncrease or reman constant, ths error would not be corrected durng the modellng of consequent tme steps. Therefore, expermental scatter may sgnfcantly nfluence the predcted behavour. Ths ssue does not arse wth the D lm /k calculated from the PrecpCalc predctons, due to the absence of scatter from these data. In Fgure 3 (c) and Fgure 4 (b), the gran growth modellng predctons are shown usng the D lm /k values calculated from PrecpCalc. These results were able to better reproduce the

12 sothermal expermental data. For the 1120 C sothermal heat treatment, some gran growth s predcted, though s more restrcted than than usng D lm /k values obtaned from the expermental measurements. Above 1120 C, PrecpCalc predcts the D lm /k reduces sgnfcantly (see Fgure 6 (d)), and consequently gran growth above ths temperature s accelerated. A prevous study by Collns et al. [7] has shown PrecpCalc underestmates the prmary γ solvus temperature of RR1000, whch drectly accounts for the gran growth model dscrepancy observed n ths study. The nfluence of scatter n D lm /k values suggests that modellng the behavour of pnnng precptates s benefcal, though wll be lmted by the accuracy of dssoluton temperatures. Furthermore, as PrecpCalc s only capable of a 1-D predcton of precptaton behavour t does not account for the precptate spacng, and can therefore not be used to asses the nfluence of ths factor on the gran growth behavour. Only by accountng for ths, can the ndvdual fttng coeffcents be further scrutnsed to elmnate other nfluencng factors, such as temperature dependence. The maxmum gran szes, lmted by the carbde D lm2 value, were shown to be n good agreement wth expermental observatons. For the carbde dstrbuton, where D lm2 was fxed at 49.2 µm, assumng k 2 = 0.835, for example, and convertng to 2D, the mean gran sze expected from a lnear ntercept method s therefore 27.4 µm. Ths compares favourably wth the maxmum gran sze observed n the sothermal heat treatments, as shown by Fgure 3 (a). Usng Equaton 10, where k s assumed to be the Zener constant, 4/3, φ = and r = 70 nm, a maxmum 2D gran sze of 28.3 µm s obtaned, whch slghtly overestmates the maxmum gran sze observed for sothermal heat treatments. Furthermore, the value, k 2 s qute dfferent from the Zener constant as the full partcle sze dstrbuton has been accounted for. Therefore, the value of k 2, or ndeed k 1, may be thought of as a sze dstrbuton dependent gran sze lmt constant. It s accepted that although carbdes other than MC are thermodynamcally stable at lower temperatures, ther occurrence s knetcally nhbted. Hence, the volume fracton of any such speces wll be small. Among such phases, the M 23 C 6 s formed by the decomposton of MC (va MC + γ M 23 C 6 + γ [39]), though ths occurs at lower servce-type temperatures ( 750 C [6]). In ths study, ths dstrbuton have been classfed as carbdes, ths may nclude some oxy-carbdes of a smlar sze. Any smaller scale partcles, such as nm-scale HfO 2 partcles, have not been measured n ths study. Importantly, no evdence of gran boundary pnnng by other phases, such as topologcally close packed (TCP) phases or bordes was observed, nor wth annealng twns. Neglectng oxy-carbdes may explan why the value of k 2 has been predcted to be lower than k 1. The modellng results of the transent thermal exposures are shown n Fgure 5 (b) and (c). In (b), the D lm /k values from the expermental measurements have been used, fttng all measurements from both transent and sothermal heat treatments smultaneously. In (c), the D lm /k values were obtaned from the PrecpCalc smulatons, agan fttng all (sothermal and transent) data smultaneously. The fttng coeffcents obtaned for the transent heat treatments are summarsed n Table 1. The 11 Table 1: Fttng coeffcents for the sothermal heat treatments usng D lm /k values determned from (a) expermental measurements and (b) PrecpCalc smulatons. Both sothermal and transent heat treatment data sets are ftted together, agan usng D lm /k values from (c) expermental measurements and (d) PrecpCalc smulatons. Ftted M 0 k 1 k 2 Q app n data set (m 2 s 1 ) (kj mol 1 ) (a) (b) (c) (d) scatter n the expermentally measured D lm /k values gve rse to dfferent ntal growth rates dependent on the startng temperatures, whch s not observed for D lm /k values calculated from the PrecpCalc smulatons. Agan, the rate of dssoluton for the PrecpCalc s too rapd, whch explans the onset of gran growth at an earler tme than s observed experentally. Attempts were made to ft the transent heat treatments smultaneously, wthout ncludng the sothermal heat treatments. Ths resulted n unrelable fttng of the gran growth coeffcents, beng most sgnfcant for the value of k 2. As the transent heat treatments are not expermentally observed to reach a maxmum gran sze, the model becomes unrelable wthout confdently determnng a D lm2 value. Usng both the sothermal and transent heat treatment data together elmnates ths problem. Ths observaton ndcates that when an addtonal coeffcent s ntroduced, the expermental data must descrbe the behavour unque to ths parameter, ensurng relable fttng. As a comparson wth other metallurgcal systems that have used the Andersen & Grong model, the moblty constant, M 0 for mcroalloyed-nb steel, mcroalloyed-t and austentc steel were m 2 s 1 [1], m 2 s 1 [1], and m 2 s 1 [23] respectvely. In these examples, Equaton 11 was evaluated assumng n = 0.5. However, Andersen demonstrated that the gran growth model s hghly senstve to the coeffcent, and wth values typcal of ths study of n 0.45, the gran growth rate s sgnfcantly lower than when n = 0.5. Fxng the hghest permssble value of n wll therefore overestmate the rate of growth, and hence when fttng, wll be compensated by lowerng the value M0. A value of n 0.5 s true only for a pure materal, and hence, t s unsutable to assume ths value for a hghly alloyed materal. Furthermore, the treatment of D lm n ths study to account for a full partcle sze dstrbuton wll requre a hgher value of M0 than f a mean partcle sze alone was used to calculate D lm. The apparent gran boundary actvaton energy values of 220 kj mol 1 for mcroalloyed steel [1] or 291 kj mol 1 used by Seetharaman et al. [22] for a gamma-ttanum alloy ndcates the value of 340 kj mol 1 found n ths study s reasonable for an engneerng alloy. Referrng to the gran growth coeffcents obtaned from the fttng of each data set n Table 1, the values appear smlar rrespectve of the type of data beng ftted. Notably n, Q app and k 2 show very hgh agreement. The largest dsagreement of M0 and k 1 occurs when fttng data wth D lm /k data from measure-

13 ments. Ths may be expected as data fttng was found to be very senstve to scatter n the D lm /k values. Sgnfcant dsagreement s not observed wth nput data used from Precp- Calc smulatons. In ths case, the value of k 1 of 1.5 s closer to the classcal value of 4/3 predcted by Zener. As stated before, ths value s expected to dffer n the present study due to the model accountng for the whole precptate sze dstrbuton, rather than a mean sze alone. Temperature Fgure 8: Predcted tme-temperature-transformaton graph of gran szes usng prmary γ data predcted by PrecpCalc, between 0 & 20 mnutes and 20 & 120 mnutes n (a) and (b), respectvely. The ultmate applcaton of the gran growth measurements and predctons obtaned durng ths study s to ad the selecton of sutable heat treatments durng a DMHT manufacturng process. The fttng strategy employed has enabled gran growth coeffcents to be obtaned, thus descrbng the behavour of RR1000 durng near-solvus heat treatments. Usng ths calbrated data, Equaton 20 can be re-solved for gran sze, D, for any desred tme, t. Smulated D lm /k data was then obtaned from PrecpCalc at temperatures between 1120 C and 1200 C n 1 C ncrements, for tmes up to 2 hours n 5 second steps. Ths enabled a gran sze, D, to be obtaned for each correspondng D lm /k and t. In the example shown here, values of the necessary coeffcents from the sothermal thermal exposures wth PrecpCalc measurements (Table 1) were used, enablng the rght hand sde of Equaton 20 to be recalculated. The modelled gran szes wth respect to tme and temperature are presented as a contour plot n Fgure 8, whch may also be consdered a tme-temperature-transformaton (TTT) representaton. Gran szes have been converted to 2D lnear ntercept measurements. Any tme and temperature can subsequently be selected, provdng a mean 2D gran sze predcton. As PrecpCalc allows the nput of any arbtrary thermal cycle, as does the Andersen & Grong [1] model, smlar gran sze predctons could be obtaned for any complex processng heat treatment. It has therefore been shown that a seres of gran sze measurements alone, ensurng they cover a sutable tme and temperature range, enables hgh resoluton predcton of gran sze; a capablty hghly desrable for components subjected to locaton specfc heat treatments Summary In ths study, the gran growth behavour of the polycrystallne nckel-base superalloy, RR1000, has been nvestgated. Isothermal and transent heat treatments have been carred out n the vcnty of the prmary γ solvus temperature, to understand the relatonshp between the pnnng prmary γ and the gran growth behavour. Mcroscopy revealed the presence of MC carbdes that mpede gran boundary mgraton above the γ solvus temperature, further lmtng gran growth. Volume fracton and precptate szes were measured, provdng predcted values of D lm for each heat treatment assessed. The gran growth behavour was subsequently modelled usng an adapted verson of the model proposed by Andersen & Grong [1], accountng for the presence of carbdes, and the ncluson of whole prmary γ PSD. In ths way the gran behavour could be approxmated by 5 ndependent parameters. All of the parameters can be ftted usng the gran growth data alone, wthout requrng any of these parameters to be obtaned from other experments. The results obtaned wth such models were seen to be senstve to the accuracy wth whch the parameter, D lm /k could be determned. By usng PrecpCalc to smulate the dssoluton of prmary γ, smooth D lm /k values were obtaned, thereby provdng superor fttng results, and thus fttng coeffcents. Usng these fttng coeffcents, a hgh resoluton TTT plot of gran sze wth respect to tme and temperature was produced, wth suffcent detal for the applcaton to component manufacture wth locaton specfc heat treatments, such as DMHT. Acknowledgments The authors would lke to acknowledge and thank the EPSRC and Rolls-Royce plc. for ther fnancal support. HJS & BDC were supported by the EPSRC/Rolls-Royce Strategc Partnershp (EP/H500375/1) and DMC by an EPSRC studentshp. Thanks are extended to J.R. May for assstance durng metallographc preparaton and J.W. Aveson for useful dscussons. References [1] Andersen, I., Grong, Ø.. Acta Metall Mater 1995;43:2673. [2] Sms, C., Stoloff, N., Hagel, W., edtors. Superalloys II- Hgh Temperature Materals for Aerospace and Industral Power. John Wley & Sons, Inc; [3] Connor, L.. The Development of a Dual Mcrostructure Heat Treated N-base Superalloy for Turbne Dsc Applcatons. Ph.D. thess; Unversty of Cambrdge; [4] Jackson, M., Reed, R.. Mater Sc Eng A 1999;259:85. [5] Kozar, R., Suzuk, A., Mllgan, W., Schrra, J., Savage, M., Pollock, T.. Metall Mater Trans A 2009;40:1588. [6] Reed, R.. The Superalloys - Fundamentals and Applcatons. Cambrdge, UK: Cambrdge Unversty Press;