Interfacial energy determination of nano-scale precipitates by CALPHAD description of Gibbs Thomson effect

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1 J Mate Sci (2007) 42: DOI /s z Inteacial enegy detemination o nano-scale pecipitates y CALPHAD desciption o Gis Thomson eect Sina Shahandeh Æ Hamed Aami Æ S. K. Sadnezhaad Received: 11 Noveme 2006 / Accepted: 9 May 2007 / Pulished online: 27 July 2007 Ó Spinge Science+Business Media, LLC 2007 Astact A theoy ased on calculation o phase diagams in the inay systems was developed that descies Gis Thomson eect. In this model eect o oth inteacial enegy and inteace coninement (Laplace Young pessue) ae included in enegy shit o alloys and phases. By using the CALPHAD model, inteacial enegy o Cu 4 Ti pecipitates in Cu Ti system was otained which shows ette consistency with expeimental esults o Gis Thomson eect o nm adius pecipitates. Intoduction Usually expeimental values o inteacial enegy o pecipitates ae scatteed in a wide ange. Most o the mentioned data ae otained om coasening expeiments. Size evolution o the paticles can e measued y TEM, SAXS o SANS and y elating thei size distiution to aging time and use o Lishitz Slyozov Wagne (LSW) theoy [1, 2], o its modiied vesions [3], the intephase inteacial enegy can e calculated as ollows: ¼ Kt ð1aþ K ¼ 8D avm Ca B 1 Ca B 2 ð1þ 9RT C B Ca B S. Shahandeh H. Aami S. K. Sadnezhaad (&) Mateials and Enegy Reseach Cente, P.O. Box , Tehan, Ian sadnezh@shai.edu whee is the mean adius o pecipitates in time t and 0 is initial adius o the paticles. a is the inteacial enegy, C B a and C B ae equiliium concentation o B in a (matix) and (pecipitate) phases, espectively, D is the intediusion coeicient o B in the matix and V m is mola volume o pecipitate phase. As it is seen, the diusivity o the atoms in the system is also necessay o detemination o a, which itsel has noticeale scatteed values. Theeoe, esides the unavoidale expeimental eos in estimation o size distiution o the pecipitates, this will also esult in uncetain values o inteacial enegy. On the othe hand, the govening diving oce o coasening is the dieence etween matix concentation in the vicinity o small and lage paticles, which leads to diusion lux o solute atoms om smalle paticles to lage ones. This phenomenon is caused y Gis Thomson eect that changes the phase equiliia etween nanosized pecipitates and matix and also inceases the solute concentation aound small paticles due to inteacial cuvatue. Dieent models wee poposed o desciing this eect. Fist model was poposed y Thomson [4], Gis [5] and Feundlich [6], which is known as Gis Thomson omula: XB a ðþ ¼Xa B ð1þ exp 2 avm RT ð2þ This model diectly elates the equiliium matix concentation to the adius o paticles. By use o analytical electon micoscopy [7] o electical esistivity measuements [8], one can evaluate matix concentation aound the pecipitates and calculate the inteacial enegy o the paticles, consideing Gis Thomson eect. The emaining pat is inding an accuate model o this eect.

2 J Mate Sci (2007) 42: In this study we poposed a theoetical model ased on CALPHAD method o desciing Gis Thomson eect in inay alloys and compaed it with othe omulas. Then this method was used to ind coheent inteacial enegy o Cu 4 Ti in FCC Cu Ti matix. Gis Thomson eect in inay alloys In this section some omulas o dilute and concentated inay alloys ae summaized. The simplest elation is Eq. 2 that assumes some simpliications that ae not eal o most o the alloy systems, such as dilute solution ehavio o the matix phase and X B = 1 o pecipitates. In Hillet [9] omula o dilute alloys the concentation o pecipitate is included: 1 X 1 XB a B ln ðþ 1 XB að1þ þ X B ln XB aðþ XB að1þ ¼ 2 av m RT ð3þ Bette models should emace moe complicated themodynamics o matix solution. Theeoe, concentated solution ehavio o matix is included in Daken acto [10]:! XB a ðþ ¼Xa B ð1þ 1 þ 1 Xa B ð1þ 2 a Vm X B ð1þ Xa B ð1þ RTe a B ðxa B ð1þþ whee e B a is unction o X B a and is given y: ð4þ e a o ln c a B Xa B ¼ 1 þ B o ln XB a ð5þ whee c a B is activity coeicient o B in matix. This model has a simpliication o the small vaiation o matix composition at cuved inteaces. The omula o Qian and Lim [11] uses Daken acto and dose not have this simpliication: 1 X 1 XB a B ln ðþ 1 XB að1þ ¼ þ X B ln XB aðþ XB að1þ 2 a Vm RTe a B ðxa B ð1þþ ð6þ Note that Daken acto is only evaluated at X B a ( ), the equiliium concentation o matix at lat inteace. Moe accuate esult can e otained i one consides themodynamic ehavio o matix phase at highe concentations at cuved inteace. This was included in a omula y Matin and Dohety [12]: XB a ln ðþ! 1 XB a XB að1þ ¼ ðþ 2 a Vm X B ð1þ Xa B ð1þ RT ln c a XB aðþ c ð a XB að1þþ ð7þ whee c a XB aðþ is activity coeicient o B in matix nea small pecipitate. Equations 3, 6 and 7 ae implicit and should e solved vesus X a B () to ind matix concentation in equiliium with a paticle with adius. All o these models wee developed ased on assumption o constant pecipitate concentation [11, 13]. Compaison etween these omulas and some o thei simpliications can also e ound in eeences [11, 13, 14]. Reades ae eeed to the eeence o each model o thei poo and deivation. CALPHAD model o Gis Thomson eect in inay alloys The decease in cuvatue o pecipitates with sizes smalle than 100 nm esults in appeaance o two eects; ist is an incease o the total inteacial aea pe constant volume o pecipitates [15] and second is an incease o intenal pessue that is called Laplace Young pessue [16]. In most o the pevious aticles, the intenal pessue was intoduced as the only govening eason o Gis ee enegy shit o the pecipitates and consequently occuence o Gis Thomson eect. As it will e shown, although this leads to coect answes, om themodynamics points o view, anothe tem should also e included to otain moe accuate esults. Gis ee enegy o pecipitate and matix will e changed with the omation o nano-stuctues; the simplest case is small spheical pecipitates in ininite matix. I we assume that a micostuctue consists o n p paticles with mean adius o p, the total intephase inteace is S p =4p 2 n p. Fo one mole o pecipitate the total volume is V m, theeoe one mole o consists o n p ¼ V m = 4 3 p3 paticles. Thus, y sustitution, the additional enegy o inteace ðdg s Þ is: DG s ¼ S p inteace ¼ 3 inteacev m ð8þ In addition, when the inteace conines a small paticle, the suace tension aound the pecipitate causes an intenal pessue inside the paticle. This pessue which is adius dependent is [17]: DP ¼ 2 inteace ð9þ

3 9442 J Mate Sci (2007) 42: This pessue causes an incease in Gis ee enegy o the phase, which can e expessed in om o dg = VdP SdT. Because tempeatue is constant and mola volume o pecipitates is assumed to e constant with vaiation o P, the Gis enegy shit due to Laplace Young pessue (Eq. 9), can e witten in the om o: DG ¼ 2 inteace V m ð10þ Contiution o these two paametes changes the Gis ee enegy o matix and pecipitate and consequently the state o equiliium etween them. Figue 1 shows a themodynamic loop etween two states o ulk phases and nano-size pecipitates. In state 1 oth phases with ee enegy o G a ( ) and G ( ), and composition o X a B ( ) and X B ( ), ae in equiliium. In state 3 dispese spheical pecipitates ae inside matix phase. One o the aims o this study is to detemine the dieence etween enegy o each phase in the states 1 and 3. Because Gis enegy is independent o path, we can deine an auxiliay state etween them and move towad state 1 to 2 and then to 3. In state 2, phase is divided into the same volume o spheical pecipitate with adius. Also inside a phase, thee is the same nume o spheical holes with the same adius. In this state the additional suace enegy o is DG s ¼ 3 Vm = and o a is DGa s ¼ 3V m = a; whee a and ae inteacial enegy o phases in equiliium with thei vapos. phase also has the additional enegy o Laplace Young pessue which is DG P ¼ 2 Vm =: When we put into a holes the ee suace o paticles and matix is eplaced with intephase inteace. In the same way one can simply ind the enegy vaiations as it is shown in Fig. 1. By adding enegy change o path 1-2-3, enegy change etween states 1 and 3 can e ound: DG a ðþ ¼3 av m DG ðþ ¼3 av m þ 2 av m ð11aþ ð11þ These additional enegies ae not equal and theeoe, they will alte the state o equiliium. Fo inding equiliium point one should use CALPHAD methods. Two methods can e used o inding the equiliium in inay systems. One is tangent constuction [18] and solution o two nonlinea equations and the othe is minimization o total enegy that is moe common in CALP- HAD techniques [19]. Fo desciing the Gis Thomson eect in inay alloys one should add the excess enegy o the inteace and Laplace Young (Eq. 11) to the Gis ee enegy o mixing and minimize the total enegy. A stating point and an alloy point ae necessay o unning o the optimization algoithm. Figue 2 shows mixing enegy o matix and pecipitate phases. Deceasing the size o the pecipitates shits the cuves to highe levels o enegy, which altes the minimum point in Gis enegy-composition space. The assumed alloy point is in the middle o two stating points in the ee enegy cuves o a and phases (Fig. 2, coss make). The total enegy o this alloy can e expessed as: Fig. 1 Vaiation o the Gis ee enegy o matix and pecipitate etween two conditions o ulk phase and nano-scale pecipitates. Total enegy change om path 1 to 3 is equal to state 1 to intemediate state 2 and then to inal state 3 Fig. 2 Gis ee enegy o matix and pecipitate phases. Dashed line is o ulk phase and solid line is o nanomete scale pecipitates. Coss points ae initial equiliium compositions which move to eal equiliia points y optimization o the total enegy. Total enegy deceases om point 1 to 2 y optimization pocess. The consequence o the enegy shit is the change o equiliium composition o the matix and pecipitate which is indicated y pointed aows

4 J Mate Sci (2007) 42: G X 0 ðx a B ; X B Þ¼ a DG a mix; ðxa B Þþ DG mix; ðx B Þ ð12þ whee DG a; mix; is mixing Gis ee enegy o a and plus additional enegy o inteace ðdg a; Þ: a, is mola action o the phases, which can e ound y leve ule. As it is shown in Fig. 2, y moving though equiliium composition in Gis enegy cuves, total enegy o the alloy deceases om point 1 to 2. This pocess is govened y optimization algoithm which minimizes two vaiale unction G X 0 ðxb a; X B Þ: All o the pevious models (Eqs. 2 6) assume that X B is constant, ut as it can e seen, the pecipitate composition should also vay with enegy shit. The amount o mutual composition change is depending on the shape o Gis ee enegy cuves o the matix and pecipitate. Theeoe, in some cases this assumption can lead to consideale inaccuacy. On the othe hand, CALPHAD model has no assumption o themodynamic ehavio o solution o o composition change. Anothe impotant point that was ogotten in pevious models is enegy shit o the matix. Conside a paticula alloy with composition X 0, that tansoms into matix and pecipitate phases with compositions X a B ( ) and X B ( ) and mola actions o a and, espectively. When dispeses into small paticles thee is a shit in enegy o alloy with espect to the ulk condition. This enegy shit o the alloy is shown with aow c in Fig. 2. This shit has two components, one o additional enegy o the pecipitate (aow ) and the othe o additional enegy o the matix (aow a). I like pevious models, only the Laplace Young pessue is consideed o calculation o the enegy shit o the phases, the contiution o the additional intephase inteace to the additional enegy o the alloy will e neglected. In this model, additional inteacial enegy is also added to the total ee enegy o a and phases, which leads to calculation o the actual Gis ee enegy o the alloy. This is impotant especially i anothe phase with dimensions dieent om the paticles is in equiliium with matix o pecipitate. In this case, vaiation o the Gis enegy changes the equiliium composition. Howeve, in two-phase systems, the inteacial enegy does not change the equiliia signiicantly, since oth matix and pecipitate gain the same additional enegy. Details o CALPHAD method will e discussed in inding inteacial enegy o Cu 4 Ti in Cu Ti system. Inteacial enegy o Cu 4 Ti pecipitates Geneally, in the Gis Thomson eect and othe inteace elated phenomena inteace chaacteistics ae unknown. Although it is possile to measue the incease o solute concentation aound the paticles, no one eally knows the easons such as inteace tension and enegy. The measued values o inteacial enegy o Cu 4 Ti phase ae scatteed. The values o 67 mj/m 2 [20] and 31 mj/m 2 [21] wee measued. Also, Bouches et al. [22] calculated value o 47 mj/m 2 y pai inteaction model. Anothe accuate expeimental esult was otained y Miyazaki et al. [23]. They measued concentation o the matix in equiliium with the pecipitates y macoscopic composition gadient method. Paticles adius vaied om 9 nm to 27 nm. They also calculated the inteacial enegy o Cu 4 Ti pecipitates y linea vesion o Gis Thomson omula (Eq. 2). Because o low accuacy o that omula they epoted the value o 100 mj/m 2. Equation 2 undeestimates composition changes; theeoe it esults in highe inteacial enegy to compensate its low vaiation ate. Qian and Lim [24] used a omula simila to Eq. 3 to e-evaluate expeimental esults o Miyazaki et al. In thei wok, a ette model o dilute inay systems led to values o 53 mj/m 2 (at 823 K) and 63 mj/m 2 (at 873 K) that ae the most easonale esults otained o the inteacial enegy o Cu 4 Ti pecipitates. In this model some simpliications wee incopoated such as dilute solution ehavio o matix and pecipitate phases and constant pecipitate composition. In ode to minimize Eq. 12 as a unction o matix and pecipitate compositions, one should know values o DG a mix and DG mix : Advantage o CALPHAD method is that it uses a vesatile solution themodynamics model. In this system themodynamic ehavio o FCC matix phase was descied y a semi-egula model and o complex intemetallic pecipitates sulattice model [19] was used. Assuming a single lattice andom solid solution o matix, the mixing ee enegy o the matix can e witten in om o: DG a mix ¼ X i X a i 0 G i þ RT X i X a i lnðxa i Xn 1 X n Þþ i j¼iþ1 X a i Xa j X i;j ð13þ whee 0 G i is the dieence etween Gis enegy o element i in eeence state and in FCC phase. X i;j is inay inteaction paamete and ased on Ridlich Kiste omalism [19] it can e expanded as: X i;j ¼ P X k k i;j X i X j k X k i;j ¼ a k þ k T ð14þ Kuma et al. [25] modeled Cu 4 Ti pecipitates which consisted o two sulattices one o Cu and one o Ti. The anti-stuctual atoms in the sulattices om wide homogeneity ange o this phase. Using sulattice omalism o (Cu, Ti) 4 (Cu, Ti), the ee enegy o Cu 4 Ti can e omulated:

5 9444 J Mate Sci (2007) 42: DG Cu 4Ti mix ¼ y 1 Cu y2 Ti 0 G Cu:Cu þ y 1 Cu y2 Ti 0 G Cu:Ti þ y 1 Ti y2 Cu 0 G Ti:Cu þ y 1 Ti y2 Ti 0 G Ti:Ti " # þ RT 4 X i y 1 i ln y1 i þ X i y 2 i ln y2 i þ y 1 Cu y1 Ti ðy2 Cu L Cu;Ti:Cu þ y 2 Ti L Cu;Ti:TiÞ þ y 2 Cu y2 Ti ðy1 Cu L Cu;Cu:Ti þ y 2 Ti L Cu;Ti:TiÞ ð15þ whee y i s is actional site occupation and is deined as the atio o nume o i atoms in sulattice s to the total nume o sites in the sulattice. Theeoe, the mola action o i atoms can e expessed as X i ¼ 0:8y 1 i þ 0:2y 2 i : In this model ou ist tems stand o Gis enegy eeence state with these constains: 0 G Cu:Cu ¼ 25; 000 J/mol; 0 G Ti:Ti ¼ 25; 000 J/mol ð16þ 0 G Ti:Cu ¼ 0 G Cu:Cu þ 0 G Ti:Ti 0 G Cu:Ti The paametes indicated as 0 G A:B coespond to the Gis enegy o the stoichiometic compound A a B that is elated to the omation enegies and elements standad eeence states: 0 DG A ab A:B ¼ DHA ab TDS A ab þ a 0 G e A þ 0 G e B ð17þ whee DH A ab and DS A ab ae compound omation enthalpy and entopy, espectively. 0 G e i is the Gis enegy o the pue component (i) in its standad eeence state. Fou L i in Eq. 15 ae inteaction paametes o two atoms etween two sulattices and epesent non-ideal ehavio o Cu 4 Ti phase. In this model only ist linea inteactions ae included. Fo peoming the equiliium calculations, it was assumed that Cu is in standad condition o FCC phase and Ti is in eeence state o HCP. The optimized themodynamics data o this system, which ae listed in Tale 1, wee otained om eeences [24 26]. In sulattice model the Gis enegy o the phase is elated to the site actions y i. Figue 3 shows ee enegy o Cu 4 Ti as a unction o site actions. Site actions vay Tale 1 Themodynamics popeties o solution o matix and pecipitates FCC X 0 Cu;Ti X 1 Cu;Ti Cu 4 Ti 0 G Cu:Ti 4 0 G cc Cu 0 G hcp Ti L Cu;Ti: L *:Cu,Ti 9,882 J/mol 115,777 J/mol 5( T) 17,089 J/mol 15,767 J/mol Fig. 3 (a) Gis ee enegy o Cu 4 Ti as a unction o Cu site action in two sulattices o (Cu,Ti) 4 (Cu,Ti). () Contou view o (a). y 1 and y 2 should e on the dotted lines in ode to peom constained minimization om zeo to one and composition and stuctue o the phases change with them. Fou cone o the coodination space coespond to ou dieent stuctues and concentation o Cu vaies om one cone to the othes as indicated y iso-concentation lines in Fig. 3. In each X Ti, o inding DG mix as a unction o Ti mola action, we should optimize Eq. 15 with constain 0:8y 1 Ti þ 0:2y2 Ti ¼ X Ti : A geometical intepetation o this pocedue is choosing the minimum enegy in each olique line in Fig. 3 and connecting these points in an odinay ee enegy-composition diagam. This cuve is plotted in Fig. 4 o oth matix and pecipitate. Ate minimizing DG mix in 500 composition points, an accuate esult o unknown concentation can e otained y a spline intepolation within these known data. Fo

6 J Mate Sci (2007) 42: Fig. 4 Gis enegy o matix and pecipitates in two states o ulk phase and dispesed spheical pecipitates with p = 2 nm. Coss and cicle makes show equiliium composition o the matix and pecipitates, espectively. a was assumed to e 72 mj/mol DG a mix we can diectly use Eq. 13. Odinay simplex o gadient ase methods [27] can e used to minimize Eq. 12 and inding the equiliia. Now, one can peom the method descied in section CALPHAD model o Gis Thomson eect in inay alloys to ind the eect o inteacial cuvatue on incease o matix concentation. Figue 4 shows Gis enegy shit o matix and pecipitate with p = 2 nm and a = 72 mj/mol. By assuming a value o intephase inteacial enegy o the matix and pecipitates, the enegy shit, DG ; was calculated as a unction o paticle adius (Eq. 11) and was added to mixing enegy o the matix and pecipitates (Eqs. 13 and 15). Total enegy (Eq. 12) o an alloy X 0 = 0.1, was minimized in ode to ind alteed equiliia in each paticle adius. The expeimental data o Miyazaki et al. is plotted in Fig. 5. The unknown paamete o a in this model was changed until accuate ageement etween expeiments and the model was achieved. The value o a = 72 mj/mol can it the CALPHAD model on the expeimental esults. As shown in Fig. 5, with consideation o this inteacial enegy, othe dilute models like Hillet (Eq. 3) and Gis Thomson omula (Eq. 2) ae not consistent with the expeiments. It should e noted that with vaiation o the inteacial enegy, slope o the lines changes. Fo example i one assumes lowe a (53 mj/mol), slope o the line deceases and Hillet model it on the data as epoted y Qian and Lim [24]. Highe values o a its Gis Thomson omula on the data. On the othe hand, the concentated solution models have esults simila to CALPHAD model, especially at Fig. 5 Incease o matix concentation as a unction o inteace adius pedicted y dieent models with value o a = 72 mj/mol. Inteacial enegy was otained ased on est itting o expeimental data o Miyazaki et al. y CALPHAD model lowe inteacial cuvatues, whee the change in concentation is small. This consistency continues in a ange that the concentation vaiation o the matix and pecipitate is so small that the assumption o constant pecipitate composition and constant Daken acto (Eq. 7) ulills. Conclusion In this aticle, CALPHAD methodology was utilized to descie Gis Thomson eect in Cu Ti alloy which decomposes to FCC matix and Cu 4 Ti pecipitates. The theoy o Gis enegy shit o the phases due to Laplace Young pessue and inteacial enegy was coected. This new theoy can tuly descie enegy shit o the phases and alloys which om nano-scale stuctues. Expeimental esults o Miyazaki et al. on matix composition change vesus pecipitates adius wee modeled By CALPHAD method. It was shown that othe concentated solution models o Gis Thomson eect can also pedict the concentation change. The value o 72 mj/mol was otained o inteacial enegy o coheent Cu 4 Ti pecipitates. With this value, CALPHAD and othe concentated solution models give consistent esults with expeiments. Reeences 1. Lishitz IM, Slyozov VV (1961) J Phys Chem Solids 19:35 2. Wagne C (1961) Z Elektochem 65: Wagne R, Kampman R, Voohees PW (2001) In: Kostoz G (ed) Phase tansomations in mateials. WILEY-VCH, Weinheim

7 9446 J Mate Sci (2007) 42: Thomson W (Lod Kelvin) (1871) Phil Mag 42: Gis JW (1878) Am J Sci Se 3: Feundlich H (1923) Colloid and capillay chemisty. Dutton, New Yok 7. Miyazaki T, Koyama T, Koayashi S (1996) Metall Mate Tans A 27: Nole B, Bay SE (1999) Mate Sci Eng A 266:80 9. Hillet M (1975) In: Aaoson HI (ed) Lectues on the theoy o phase tansomations. TMS-AIME, New Yok, NY, pp Pudy GR (1971) Met Sci J 5: Qian M, Lim LC (2000) Metal Mate Tans A 31: Matin JW, Dohety RD (1976) Staility o micostuctue in metallic systems. Camidge Univesity Pess, Camidge 13. Qian M (2002) Metal Mate Tans A 33: Peez M (2005) Scipta Mate 52: Shiinyan A, Wautelet M, Belogoodsky Y (2006) J Phys: Condens Matte 18: Wang CX, Yang GW (2005) Mate Sci Eng R 49: Glicksman ME (2000) Diusion in solids: ield theoy, solid-state pinciples and application. John Wiley & Sons Inc., New Yok 18. Lupis CHP (1983) Chemical themodynamic o mateials. Not- Holand: Elsevie Science, New Yok 19. Saundes N, Miodowink AP (1998) CALPHAD (Calculation o Phase Diagams): a compehensive guide. Elsevie Science Ltd 20. Kampmann R, Wagne R (1984) In: Haasen P, Geold V, Wagne R, Ashy MF (eds) Decomposition o alloys: the ealy stages. Pegamon Pess, New Yok, pp Boches C (1999) Philos Mag A 79: Boches C, Bomann R (2005) Acta Mate 53: Miyazaki T, Koyama T, Koayashi S (1996) Metall Mate Tans A 27: Qian M, Lim LC (1998) Scipta Mate 39: Kuma KCH, Ansaa I, Wollants P, Delaey L (1986) Z Metallkd 87: Aoyave R, Eaga TW, Kauman L (2003) J Alloys Comp 351: Optimization Toolox (2005) Matla Ò sotwae documentations. Mathwoks Inc