Cluster dynamics modeling of materials: advantages and limitations. A. Barbu and E. Clouet

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1 Preprt of the artcle publshed Sold State Pheo. 9 (007), pp Cluster dyacs odelg of aterals: adatages ad ltatos. A. arbu ad E. Clouet Serce de Recherche de Métallurge Physque, CEA/Saclay 99 Gf-sur Yette, Frace ala.barbu@cea.fr Keywords: Cluster dyacs, rate theory, precptato, pot defect clusters, rradato. Abstract. The a of ths paper s to ge a short reew o cluster dyacs odelg the feld of atos ad pot defects clusterg aterals. It s show that ths ethod, due to ts low coputer cost, ca hadle log ter eoluto that caot, ay cases, be obtaed by Lattce Ketc Mote Carlo ethods. Ideed, such a possblty s acheed thaks to a portat drawback that s the loss of space correlatos of the eleets of the crostructures. Soe exaples, the feld of precptato ad rradato of etallc aterals are ge. The ltatos ad dffcultes of ths ethod are also dscussed. Usurprsgly, t s show that t goes a ery satsfactory way whe the objects are dstrbuted hoogeeously. Coersely, the source ter descrbg the prary daage uder rradato, by ature heterogeeous space ad te, s trcky to troduce especally whe dsplaceet cascades are produced. Itroducto. I ateral scece, cluster dyacs (CD) s based o ketc equatos descrbg the forato ad eoluto of clusters of solute atos or pot defects such as acaces or self tersttal atos (SIA). It s a ery effcet ethod ter of coputatoal cost. Ths effcecy s due to a drawback cog fro the basc hypothess of ufor dstrbutos of clusters: the real syste s replaced by a effecte edu whch all processes occur cotuously te ad space. The spatal correlatos betwee clusters s cosequetly ot cosdered explctly. Lattce Ketc Mote Carlo ethods do ot suffer of such a ltato. Howeer, they are ery te cosug copared wth the CD eso-scale odelg ad, are ee uable to address log te agg ay cases. Cosequetly, CD reas at ths te a rreplaceable ethod for predctg log ter eoluto of aterals. I a frst part, the CD odelg s preseted. I the secod oe, exaples of applcatos to precptato ad clusterg of pot defects uder rradato are ge. Cluster dyacs odels I these odels, the syste s see as a gas of clusters ade of ooers that ca be solute atos, acaces, self tersttal atos. A cosequece s that the total olue fracto occuped by the clusters ust be sall. The eoluto of the uber desty of clusters of each sze ad of a certa kd s treated wth the fraework of the checal rate theory. I the ost geeral case,.e. whe soe speces are o-coserate, t s descrbed by a set of dfferetal equatos of the for: dc = J J q + G KC dt () where q C s the cluster desty of sze ad class. It s ge by: J = () w C J the cluster flux fro the class of sze to the

2 wth w, the trasto rate per ut cocetrato fro the class of sze to the class of sze. G, the producto rate of clusters of sze ad K, the loss of clusters of sze at soe fx sks such as dslocatos, gra boudares or surfaces, ust oly be take to accout the case of o coserate speces as, acaces or SIAs. Puttg () () we hae: dc = G + w C w qc KC dt q (3) The cluster dstrbutos ca oly be calculated uercally. Keepg d that the terest of CD s ts short coputer te, the axu sze of the cluster that ca be cosdered s lted by the uber of dfferetal equatos. A detaled descrpto, as the oe ge by eq.3, s portat for sall cluster szes. O the cotrary, a approxated descrpto ca be used for large szes. For ths purpose, two approaches hae bee cosdered: The frst oe, troduced by Ghoe et al [], cossts deelopg eq.3 to the secod order about. Ottg G ad K, the syste eoluto s descrbed by the Fokker Plack type equato : C( x) = [ f ( x) C( x) ] + [ d( x) C( x) ] dt x x (4) where C (x) s the sze dstrbuto fucto. f (x) ad d(x) are fuctos of the trasto rates. The sze of the cluster, x, s ow a cotuous arable. Ths cotuous equato ca be uercally soled dscretzg the cotuous arable x, wth Δx creasg wth x. Ths ethod s sple but has the dsadatage of beg ustable f the uercal schee s ot adequate ad does ot strctly cosere the atter. The stabltes ca be aoded by usg a uercal schee surg the posteess of the cocetratos. Cocerg the atter losses, the dscretzato ust be chose to ze t. I ay case, the losses ust be checked afterward to kow f they are acceptable. The secod oe s the groupg ethod cosstg replacg a group of aster equatos by oly oe equato represetg the class []. It has bee show that the result ca be ery bad f the groupg s ot carred out properly. Furtherore, as the preous oe, t does ot cosere strctly the atter ee wth a optzed groupg. Recetly, Golubo [3] proposed a ew groupg ethod that ca cosere at the sae te the atter ad the cluster uber. For ths purpose, oly the frst ad the secod oet of the dstrbuto are cosdered. Two equatos for each class.e. twce ore are to be cosdered. The equatos o the frst oet esure the coserato of the uber of precptates ad those o the secod oet, the coserato of atter. Applcato to precptato. Let us cosder a sold soluto of ato a atrx A. Assug that, () oly sgle atos are oble, () oly sgle atos ca be etted fro a precptate, () precptates hae a well defed cocetrato ad shape such a way that they are oly characterzed by oe uber, the uber of atos they cota, the flux of clusters s ge by: J = + C C α + C + (5)

3 Where C s the uber of clusters per ut olue ad ade of atos, s the codesato rate ad α, the esso rate of atos. If the codesato s cotrolled by dffuso, the expresso of, geerally used s: = (6) where 4πr D r s the capture radus of a cluster cotag, atos, C ad D the cocetrato ad dffuso coeffcet of atos. Eq.6 s prcpal oly ald for a solated precptate a fte eda. ralford ad al [4] hae ge a self-cosstet expresso showg a uber desty ad radus precptate depedece (ul-sk correcto). Howeer, eq.6 s usually ald as far as the olue fracto of precptate s ot ery hgh. To calculate the esso rate, t s classcally assued that t s a trsc property of the cluster ad cosequetly that t does ot deped o the cocetrato the sold soluto. So, t s the sae as the oe calculated for a sold soluto at equlbru for whch J + s equal to zero. The eaporato rate s the ge by: D α = 4πr G G G exp ( ) Ω (7) Where [ kt ] / G s the free eergy of a cluster cotag atos ad, Ω, the atoc olue. I the fraework of the capllary odel, curretly used the classcal theory of ucleato, ad assug that the atoc olue of solute atos s the sae as the solet ato oe, (o sze effect cotrbuto), G s ge by: G 0 0 uc 0 36π / 3 [ μ( C) μ A( C) ] G ( C ) Ω = + Δ + σ c p c p (8) 0 Δ G uc C s the ucleato free eergy of a soluto of cocetrato C 0 ad Where ( ) terface free eergy of a cluster cotag, atos. Ths yeld: α (9) D 36π Ω exp Ω cp / 3 3 = 4πr { / σ ( ) σ σ }/ Δ does ot appear the expresso of the rate of esso CD odelg. The reaso s that the gas of cluster s by tself a therodyac odel of the alloy [5]. A typcal eoluto of the sze hstogras of Al 3 Sc precptates calculated wth paraeters represetate of the AlSc0.8at% sold soluto s ge o fgure [6]. / 3 kt 0 Cotrary to the classcal theory of ucleato, the ucleato free eergy G uc ( C ) σ the

4 Fgure : Sze hstogra eoluto of Al 3 Sc precptates the AlSc0.8at% sold soluto therally treated at 350 C. A coparso betwee CD ad KMC results, calculated wth the sae paraeters, shows that CD ges a good extrapolato of the ea radus eoluto obta wth KMC at te that caot be reached by KMC. Further, t s clear that oly CD sulatos allow coparsos wth experets that, ths case, are ery good. Fgure : Mea radus ad uber desty of precptates as a fucto of the agg te for the Al Sc 0.8 at%sc sold soluto aged at 350 C as ge by cluster dyacs ad by Ketc Mote Carlo. Experetal results are also reported [7-8]. It s worth otcg that the surface eergy of the precptates σ, used the CD calculatos to hae a good agreeet wth KMC ust deped o. It s ge by a law 3 of the for: ( / 3 σ = + + / σ c d ) where σ ad the c ad d coeffcets are deduced fro the Ketc Mote Carlo results [6]. Furtherore, the dscrete alues ust be used for <0. Applcato to crostructural eoluto uder rradato. The a dffculty wth the clusterg of pot defects uder rradato s that all the objects we hae to deal wth are ot coserate: tersttal ad acacy type objects are produced durg the collsos of rradatg partcles wth the atos of the ateral. They ca also recobe or dsappear o fx sks (dslocatos, surfaces or gra boudares). It s geerally cosdered that, at least ery pure etals, the tersttal ad acacy clusters are oble ( alloys, the pot defect clusters are lkely less oble). The aster equato for tersttal ad acacy cluster eolutos are cosequetly ore coplex. Assug for splcty that oly sgle acacy ad SIAs are oble, t s, for tersttal clusters such as >, ge by:

5 + ( ( ) C ) C( ) + ( ( + ) C + α ( + ) ) ( α + C + C ) C dc = G C( + ) dt (0) where: C s the cocetrato of SIA clusters cotag SIAs. G s the producto rate of SIA clusters. C s the aggloerato rate of a SIA wth a SIA cluster of SIAs, producg a tersttal cluster of + SIAs. C s the reacto rate of a acacy wth a SIA cluster of SIAs, leadg to a tersttal cluster of SIAs. α the esso rate of a SIA fro a tersttal cluster of SIAs. α the esso rate of a sgle acacy fro a tersttal cluster of SIAs. A syetrcal equato wth syetrcal deftos yelds for acacy clusters. The equato for s sgfcatly dfferet for the oble speces, here the sgle acaces ad SIAs. For SIAs (syetrcal equato for acaces), t s ge by: dc = G + C C C C C C C CC dt = = () + α C + α C + α C = = Ideed, the oble speces ca react wth all clusters whateer ther type (thrd ad ffth ter). Further, clusters of two ooers ad ore ca prcpal et a sgle defect of the sae or of the opposte type (seeth ad th ters). The probablty for a acacy cluster to et a SIA s realty so low that ths ter ca be erased. The reerse s ot true: the probablty for a tersttal dslocato loop to et a acacy caot be gored f the sze of the loop ad the teperature are large eough. The reaso of the fourth ter s that whe a ooer of oe type ecouter a ooer of the sae type, two ooers dsappear ad for the eghth ter, that whe a der dssocates, two ooers appears. Note that these equatos, the classcal recobato ter betwee a acacy ad a SIA s spltted two ters: the frst ter of the secod su ad the sxth ter. The aggloerato coeffcet s ge by a relato of the sae type as eq.6 [9-]. Howeer, the effecte capture radus depeds the shape of the cluster (dslocato loops for tersttal clusters ad dslocato loops, ods or stakg fault tetrahedras for acacy clusters). The teracto betwee pot defects ad the pot defect clusters ust be cosdered. It s partcularly true for SIA ad dslocato loops whateer ther ature. The esso coeffcets are ge by a relato of the sae for as eq.7 [9-]. I the case of esso of a SIA fro a SIA cluster, t ca be rewrtte: [ E kt ] D α = 4πr( ) exp / () Ω Where r s the sphercal capture radus of a SIA by a SIA cluster cotag SIAs ad E the bdg eergy of a SIA wth a SIA cluster cotag SIAs. The sae kd of relato holds for the other ettg coeffcets. As for precptato, the E s ust be calculated at the atoc leel for sall clusters. I order to lt the uber of paraeters E, extrapolato ca be used for large clusters. The extrapolato laws hae usually the sae for as the capllary law used for precptato of solute atos []. The reacto rate for the elato of defects o surfaces or gra boudares has bee ge by ullough et al [3]. I ths case, cotrary to pot defect clusters, the ult-sk effect ca eer be otted as, t appears the frst order ter.

6 The large uber of paraeters eterg ths kd of odelg s ofte a object of crtcss. They are o our opo o releat. Ideed, all paraeters correspod to a physcal realty. Ether they are assued ot to be portat ad, ths case, CD odelg s a good ethod to check t, or they are portat ad, ths case, the odels gorg the for splfcato, does ot correspod to the physcal realty. Fgure 3: Eoluto of the sze hstogras of (left) tersttal loops ad (rght) acacy loops. Cotuous sotheral rradato. A exaple of the calculate eoluto of the dstrbuto of pot defect clusters the FeCu 0. at%, durg cotuous rradato wth MeV electros, at a daage rate of dpa/s s ge o fgure 3 []. I agreeet wth trassso electro croscopy experetal obseratos, tersttal loops ucleate ad grow quckly up to large szes whe acacy clusters rea uder the obserato lts. Fgure 4: Eoluto of the statoary uber desty of tersttal loops of radus larger tha oral ro. The paraeters eeded such sulatos are uerous. All are ot are ot kow. Adjustets caot be aoded. The paraeters used to ft the experetal results, as for stace the statoary total uber of precptates, are, as far as possble, closed to those calculated by atostc ethods (Fg.4). Howeer, the effecte acacy grato eergy used s approxately twce the calculated alue ultra pure ro to take to accout the effect of the 00atpp carbo presets the alloys [4]. Furtherore, t ust be assued that SIA clusters but sgle SIAs are oble, cotradcto wth the data obtaed by atostc calculatos ultra pure ro. Ths apparet cotradcto could coe to the trappg of SIA clusters by carbo atos [4] or ery sall carbo clusters. O the effect of spatal correlatos. I a partcular case, a ult scale odelg has bee perfored usg alost oly paraeters obtaed fro frst prcples. It s the sulato of the recoery of, at 4K, electro rradated ultra pure ro, durg sochroal aealg. The forato ad grato eerges of defect were obtaed by usg a ab to ethods ad the ketcs, the Eet Ketc Mote Carlo (EKMC) code called JERK [5]. The arous stages correspodg to the trggerg of the oblty of oe or ore defects were correctly reproduced [6]. A coparso betwee CD ad EKMC were carred

7 out to check f the spatal correlatos that appear the defect creato process, hae a portat pact o the results [7]. Ideed, ee wth electro rradato that does ot duce dsplaceet cascades, but solated FP, acaces ad SIAs are ot created at rado: the dstace betwee acacy ad SIA of the sae FP, typcally of four lattce paraeters, s short coparso wth de ea dstace betwee the ceters of graty of FPs. Fgure 5: Coparso of sochroal recoery of Fe electro rradated at 4K obtaed wth CD ad EKMC. As show the fgure 5, CD ges good results f the tal codtos are ot the total uber of FPs after rradato but what s ge by KMC by the ed of the stage I, correspodg to the recobato of PFs (aly correlated). Ideed, f the tal pot defect cocetrato s drectly troduced at the begg of the CD sulato, () the I D ad I E are erged together as the frst oe s due to correlated FPs ad, () the testy of the frst peak ge by CD s saller wth the cosequece that ore pot defects beg preset by the ed of stage I, the testy of hgher teperature peaks are te tes too hgh. Ths effect s less proouced for hgher tal cocetratos of FP as the rato of the dstace betwee FPs to the dstace betwee the acacy ad the SIA, of the sae FP, decreases. The dfferece betwee full KMC ad full CD sulatos would be ore proouced for rradato wth partcles ducg dsplaceet cascades such as eutros or os. As for electro rradatos, the source ter used CD should ot be the oe ge by the ed of olecular dyacs (MD) sulatos but the oe obtaed after a short agg of MD cascades by KMC. The deterato of the correct agg te to erase, as uch as possble, the spatal correlatos s ot obous. It s a portat ssue of the ult scale odelg usg CD for ketcs. Aother key pot s the oblty of SIA clusters actual steels that are certaly less oble tha what s ge by atostc sulatos ultra-pure ateral. The reaso s that they are trapped by purtes ad alloyg eleets. It s worth otcg that the proble of correlatos s less portat for hoogeeous precptato as far as the precptate olue fracto s sall. Ideed the solute dstrbuto at the begg of the agg s ery ofte rado. It s also the case of the aealg of crostructures produced by cotuous rradatos up to hgh flueces, whch are curretly rado. It has bee show recetly that CD ca reproduced correctly the crostructure aealg of rradated zrcou alloys [8]. Cocluso The basc aspect of the Cluster Dyac odelg (CD) ad soe applcatos relatoshp wth experets are preseted ths short paper. CD odelg s, ay cases, the oly way to sulate log ter eolutos of the crostructures of ateral subtted to theral treatets or rradato. The a adatage of CD s ts low coputatoal cost allowg short te calculato. Such a possblty s obtaed thaks to a portat drawback that s the loss of space correlatos of the eleets of the crostructures. Ths aspect ust be cosdered partcularly for rradato uder cascade codtos. The oly way to oercoe ths proble s to age, durg a short te, by

8 a KMC ethod, the dsplaceet cascades ge Molecular Dyacs, order to erase as far as possble the spatal correlatos. Refereces [] Ghoe N.M. ad Sharafat S., J. Nucl. Mat., 9 (980). [] Krta M., J. Phys. Soc. Japa, 37 (974) 53. [3] Golubo S.I., Ochareko A.M., arashe A.V., ad Sgh.N., Phl. Mag., A8,, (00) 643. [4] ralford A.D., ullough R., hays M.R., J. Nucl. Mat. 60 (976) 46. [5] Mart G., Sold-State Phase Trasforato Iorgac aterals, Vol., TMS 005, (005) 9. [6] Clouet E., arbu A., Laé L. ad art G., Acta Mater. 53 (005) 33 [7] Marqus E.A., Seda D.N., Acta. Mater. 49 (00) 909. [8] Nooty G.M., Ardell A.J., Mater. Sc. Eg. A38 (00) 44. [9] Masur L.K. Nucl. Techol. 40 (978) 5. [0] ralford A.D., ullough R., Phlos. Tras. Roy. Soc. Lodo 30 (98) 87. [] Hardou Duparc A., Mogeo C., Setasky-de grade N., arbu A., J. Nucl. Mat., 30, (00) 43. [] Dubko V.I., Abyzo A.S., Turk A.A., J. Nucl. Mat., 336, (00). [3] ullough R., Hays M.R., Wood M.H., J. Nucl. Mat. 90 (980) 44. [4] Fu C.C. to be publshed. [5] Dalla Torre J., ocquet J.L.,, Doa N.V., Ada E. ad arbu A., Phl. Mag., 85, 4-7 (005) 549. [6] Fu C.C., Dalla Torre J., Wllae F., ocquet J.L. ad arbu A., Nature aterals, 4 (005) 68. [7] Dalla Torre J., Chu-Chu Fu, Wllae F., arbu A. ad ocquet J.L., J. Nucl. Mat., 35, -3, (006) 49. [8] arbu A., to be publshed.