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Published quarte* as the organ of the twndry Cornmlssron or tne PM&n Acaaemy of Saences

Screening effect during the growth of spheroidal grains vs deviations from Kolmogorov- Johnson-Mehl-Avrami

equation A.A. ~urbe lko~*

Tactdty of Foundry Engineering, AGE1 University of Science and Technology. Rcyrnonta 23,30-059 Kmkow. Poland *E-mail address: abur@agh.edu.pl

Rcccived 1 1.03.2008; accepted in rcvised form 17.03.2008

Abstract

To forccast rhc kinc~ics o f phasc transformations which consist in nucleation and growth of rhc grains o f a ncw phnsc, rhc wcll-known Kolmogorov-lohson-Mchl-Avrami (KJM A) equation has becn used. I t is gencr;llly known that in lhc case of pnrsllcl gr;cin prowlh in diffcrcnt phascs proceeding at difrcrcnl vclocirics. tbc rcsulis OF thc ralcularions arc burdcncd with an crror. 111 this study. applying rhc assumpfions o f n stnist ical thcory or thc scrccncd grain growlh. an attcrnpt has bccn madc to cstimatc this crror ror thc casc or prowl11 of thc sphcroidnl grains of two rypcs. Thc ohiaincd rcsulls indicate that thc vatuc of an crror in typical equations incrcsscs with iacrcasing dilrfercncc in the growth vclocity of thc particlcs of bath typcs.

Kcywnrds: Mc~nllognphy, Kinctics o r transCorrnations. Pamllcl growth

1. Introduction tsnnsformntinns taking plncc in rnclallic marcriats arc dcscrihnl in publications writlcn by Johnson and Mchl 141 and Avrami 15. 6 i~nd 71.

3hc mcchanisrn that drivcs numcmus pmccsscs changing rhc This ar~iclc discusses thc snsc disrcgardd hy thc classical milt'cr is Ihc nuclcntion growth 'lcrncntary tflcory [3]+ This is lhc cast or ;r parallel ymlah

ohjccrs or a ncrv suhs~nncc within thc substance subjcct to particlcs, whcn cach of them has a tliffcrcnr vclociry o f prowih. tr;uisforrn;~tions. 'Thc plrnomcna o f this type takc place in - . rnctnlIic tnatcrinls &ring t hc sol idificnt ion a n i phasc transformatinns in solid starc. They arc also obserwcd in multi- phirsc chemical rcxtions during polymcrisation of plastics. and 2, Limitations of the Statistical KmA cvcn ill somc hiolopical systcms [I. 21. No mattcr how much thcsc phcnorncna may dirfcr, changcs in thc volumc o f thc

Theory of Solidification tnnsformcd Tractions arc dcscribcd h i thc samc slalistical ~hcory Thc gcncnl cquarion of thc aa!istical thmry of .sol id ificaricln 13 1 , dcvclopcd in ~ h c laic thirlics o f past century. Thc thcorcticaI cdlod Kalmopmv equation, mahlm us pmlict rhc rmsfomcd fundamcn~als of thc rnathcmatical rormulac uscd nowadays wcrc fncrion yotumc fmm zhc, so cdld. cx,mdcd spccilic ,,olumc (il) dcvclopcd hy Koimogortw [3 1. Comprchc~~sivc casc studies of zhc

A R C H I V E S o f F O U N D R Y ENGINEERING Volume 8 , S p e c i a l Issue 112008. 35-40 35

lcngth of thc r c s p t i v c crystaI faccts hS(iiF.t) and assumcs thc valuc:

w h c t-the tirnc. 7hc vduc of R is cdmlatcd from somc gmmchicd rulcs, allowing

For thc shnpc, si7c md quantity of p-iicles in a unit volumc but disrcgding ccimin limitations resulting from their inter;iction.

For constant numhcr OF grains and stable mc of l i n w growth. quaiion ( I) wsurncs thc wcll-known form of the A v m i quation:

where: a-thc constant depmding on the grains s h , ~ numbcr d velocity of p w r h . One of the conditions indiqmsablc to satisfy both of thc ahvc

rnentioncd equations is to have oquJ vclocitics or growth of dl the p i n s in a g i m dirmion arxl ;II a given time instant. In d i t y , this condition is not always misfid. d actual kinctics o f thc process diffm lrom lhat ddcrmind by cquaions ( 1 ) and (2). Typical cxnmplcs whcn thew mnditions m nm ~ i s f i c d arc dcxribcd in [R]. This study dim= rhccxamplc of a simultaneous growth of thc sphcmidal p i n s of rwo rypcs whcn mch olrhcm has a diffccrcnt gmwh vclocily.

2. Extended Range of Application of the Statistical KJMA Theory

Numcrous attcmpts arc known that aim at an impmvcment o f the stnristical thcory of phasc transformations to describe in a mrrcct way thc kinctics OF these transformations in situations when thc abovc mcntioncd conditions arc not satisfied. Yct. as reported in [9]. nll thcsc improvcmcnts arc not of a general character but refer to somc specific czcs only.

In [ 101 it has been provcd that thc most frcquent cause of dc- vialions is the eflect OF screening shown in Figurc 1 . Thc cx- tended valumc i s cqual to a sum o l all volumcs o l all thc panicles, and the superposing areas (rhc hatchcd arcas in Figurc t ) are counted several tlmcs. Equation ( 1 ) allows for ~ h c cftcct o f pains supepsing. However, in situations whcn thc assumptions dc- scrihcd in [3] are not satisfied. thc cxrcndcd valumc may inctudc, besides the areas supepsing (the hatched arcas in Pigurc 1) and non-suvrposing (the whitc areas in Figurc I), also the mixed ("overgrown") layers (grcy areas PA and PB in Figurc I ) , in technical litcraturc called phantoms. I n placcs wherc thc grains conracn carh orhcr, thcy mutually impcdc thcir growth. nnd zoncs o f thc PA and Pa typc arc not formed.

Thc solu[ion prcsentcd in further part of this study i s bas& on a statis~ical rheory of the screencd growth discussod in [ t I]. In this study it has hccn assumcd that wc know thc function S(tr,#), dctcrmining thc ficld o f an cxternat boundary of thc cxtcndcd grains, lthc growth of which at a given lime instant r takcs phcc at a velocity nnt grcatcr than lr (this function does not allow lor thc sc~cncd surfaces, i.e. for thc houndarics of areas PA and Ps in Figurc 1). For thc facctcd and sphcroidal grains this function is running in inrcrvals, while for orhcr non-facctcd grains i t is of a continuous character. In its continuous scctions S(u.t), ~ h c

function S'(u,t) i s cqual to its partial dcrivativc dS (rr,t)/au , and at the points of discontinuity, whcsc S(u,t), i t s step is cqual to thc

whcrc: t r ~ - rhc w i v c p w r h vclocity on thc sudacc.

Fig. I. Screening cCbcca Cor thc case of non-convex grain growth: A, I3 - concave grains. PA. Ps -phantom zoncs

Undcr such assumptions. thc vclocity of the cxtcndcd grain surface growth can hc cxprcsscd wiih Slicttjcs integer:

whcrc the second lcrm allows for thc f a c a d or sphcroidal growth velocity, while the first tcrm allows for thc non-Sacctcd grntuh vclocity.

Thc cxtcndcd volumc i s dctcrmincd hy intqration of cqtrntion (4) aftcr the tirnc:

whcrc: u,. - the maximum mi@ion vclocity or thc bor~nrlny. The screening vclwity in the casc of thrcc-rlimcnsional

growth [ l I ] is:

w h c stl- thc vclbcity of growth OF thc screened sttrfxc, 1r l - thc integration v d 8 l c . If. within thc intcgrntion ranp, thcrc arc points OF

discontinuity of rhc fitnction S(lr), St iclljcs integral applies:

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3. Analysis of thc Scrcencd Growth for Two Types of thc Spheroidal Grains

k t us considcr thc growth of thc spheroidal patlicles of two types, i.e. A and R. pmcccding at diffcrcnl vcloci[ics. I t is assumed that thc nuclci ob thesc particles arc Forrncd at thc samc rime instant t=Q in a numhcr n~ and I X B , Let pariiclcs grow at consrant vcIocities ~ I A md us. and Ict I I A > ug, In thc C ~ S C under considcntion. for the slow growh I l lg ) screening docs not occur, and the velocity of scrccning Ihc surfacc of grains A is:

or according lo definition (31:

whcrc: Sri(1) - thc rotnl ficld of thc suriaccs of all thc cxtcndcd !3 zypc grains.

Thc scrccning cffcct is probably rcsponsiblc Tor the dccrcasing run of both functions S and S', while the grain gmwh makcs rhcm incrcasc. A more rapid growth of grains A may bc scrccncd on thc cxtcndcd boundary of grains B. with the sizc of zhc boi~ndary assuming a value cgual to:

The houndary S, (I) = ~ ( r i , , t ) of the errended grains A is growing with ~ h c growing radius o f thcsc grains. and from yeomcrricaE rclat ions for thc panially scrccncd sphcmidal grains follows:

Sincc in the cxamincd case thc velocities arc t imc- indcpcndcnr, cquation (5) is rcduccd to thc following form:

Difrcrcntinring fi~nclinn S' for grains o f type A along their radius, we obtain:

Thc mtc of changcs in S'(rtA,l) dcpcnds on thc two compctitivc proccsxs, i.c. an incrcxc of thc grain dimcnsions and scrccning of thc surhce:

On substituting ro this cquation thc dcrivarivcs (9) and (1 3) we obtain:

whcretrom allowing for (2) and (10):

An integer of this cquation cnablcs calculation of thc non- scrccncd boundary of grains A:

I rr: 2 3 l n ~ , ( t ) = lnt--rt,n--(rr, -u,) t +C (E7) 3 Ll,

Sincc at thc instant of nucleation. duc to a small sizc of zhc grain bnundarics, scrccning can bc ncglcctcd, wc havc:

and it i s possiblc to dcrcrininc ~ h c integration constan! C:

Final ty, the non-scrccncd boundary or grains A will hc calculated rrom thc foltowing cqunrion:

The total cxtcndcd volume of ~ h c products olgrowtb, calcu ta!cd according to ( I2), is:

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wherc t hc first tcrm in the sum refers to grains of type A (aA), md the sccond to grains of typc R (no):

4 a, (t) = , I I , K ( L ~ , t y

Differentiating equation ( I ) in the casc of the growth of two phases. we obtain:

The numerical integration of equation (24) using (22) and (23) enables us to analyse thc kinetics of transrormations in thc casc under consideration.

Analysis was m d c for thc growth of two types of the spheroidal partictcs with ~ h c ratio of growth velocities differing from t : t to 1: 16. In all calculations it has bccn assumed that the growth velocity and [he nurnbcr of grains of type I3 (blocking) is constant. Thc gmwlh velocity and the number of grains of type A (screened) was in cach individual case calculated from thc reIa1ion:

Using this rcIation without allowing for the scmning cffcct resutts in this that, independent of thc velocity ralio, during transforrna!ion. thc sizc of thc extendcd volume of the grains of bolh typcs is equal. Equal in this casc arc also the fractions of grains of each Iypc (Vn= VB). Pig. 2 shows changes during lransformalion in excess volume fraction of thc grains of type B, characlerised by the growth vclocity lower than that of the grains of typc A. Thc rcsults OF mdclling thc growth whcn thc boundaries arc migrating at thc same vclocity arc consisrent with the rcsults of the cIassical thcory. With increasing diffcrcnce in the velocities is increasing the prcvalence of the volume fraction of grains characteriscd by a lower vclocity of the boundary migration.

It is worth noting that the difference in volume fractions for the velocities ratio of 1:16 i s maximum at a level of 6 ve1.8. The order o f magnitude of this value approaches the accuracy of some methods used to measure volume fractions.

4. Conclusions

When diffcrenccs exist in thc vcIocity of g r o ~ h on thc surface of thc grains growing during phasc transformations. the equation of

the statistical thcory ef solidification is burdcncd by w crror resulting from thc fact hat the screening cffcct is ncglcctcd.

In thc prcscnt study a method has bccn proposed which cnabIcs thc scmning cffcct to bc takcn into considcntion in thc cql~ations of statistical thcory. Thc mcthod also allows for the diffcrcnt grain growth velocities.

Thc efrect that lhc dific'crcncc in thc vcIocitics of thc sphcroidnl pmiclcs grow~h may havc on a relationship of the tnnsformcd fraction volumcs has k e n illustratted with an cxarnple

Written under own research project AGFI No. IO, 10.170.297.

Rcfercnces

[ I ] Ramos R.A., Rikvold P.A.. Novotny M.A.: Test of ?/LC

Koltnogorov-Johnson-MEhl-Avrntni picrttre of tncrasrable decay in a model with tnicroscopic djvlmnics. Physical Rcview B, 59, 1999.9053-9069..

[21 Andrienko Y.A., Brilliantov N.V, Krapivsky P.L. Nucleation and grolvflz C sysferns 1vit11 tnnr~y sfoble plroses. Physica! Review A, V. 45, No 4, 1992,2263-2269.

[3] Kolmegorov A.N. K staris~iczeskoj tearii kristalizncfi ~nefallov. Izvcstija Akadc~nii Nauk SSSR. Scrija matcmaticzcskaja, or 3. 1937,355-359 (in Russian).

[4] Johnson W.A., Mchl R.P.: Reaction KEnetics irr Pracerses ol N~tcleafion and Grolvth. Transaction Mctnllurgical Socicty AIME, V. 135.. 1939,4 14-442.

[ S ] Avrami M.: Kinetics of Phase Change. I. GenernI T ! I P o ~ . Journal of Chem. Phys, Vol. 7., 1939, 1 103-1 1 12.

[6] Avrarni M.: Kinetics of Plzase Clzange. 11. TransJ3ntiario1r- Titne Relations for Randorn Dislribrifio~~. Journal oS Chcm. Phys. Vol. 7., 1940.2 12-224.

[7] Avrami M.: Ki~terics of Pltase Change. 111. Grorr~rlarinn, Plrase Change, and Micrasrruc~rtr~. Journal of Chcin. Phys. Vol. 9., 1941, 177-t84.

[8] Burbelko A. SzybkoSt ekranowanin tvzrosfrr c ~ q s ~ c k liniowycll IV pmstrzerti jedno\vy~niaro~vej. XXX Konfcrcnqia Naukowa z okazji Swiga Odlewnikn, WO AGI I, Krakdw, 2006,99-103 (in Polish).

[9] Kooi B. J, E~rensiorl of the Jolollnson-M~hl-A vrarni- Kol~nogorov flteoty incorpora ring a~tisorrol~ic gro~vtj~ studied by Monfe Carlo sirn~tlations. Physical Rcvicw U. V. 73,2006,054 1 03-( 1 - 13)

[ 101 Bclen'kij W.Z.: Gee~nefrike-iverojaft~osl~~yjc morlcli krisfallizacii. Fenotnenolug iczeskij podchod, Mosk wa, Nauka, I980 (in Russian).

[ E 1 ] Burbelko A.: Probabilistyc~a reoria ekrarlowania \vzrosrrr czqstek. Inromatyka w technotogii mnterialdw. Nr 4. t. 3,.

2002, 106-120.

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Time

Fig. 2. The cvrrlution of the volume fraction diffcrcnccs bctween thc spheroidal particles of two typcs (I3 and A) for the diffcrcnt grain growth velocity ratios

Ekranawanie pmy wzroicie ziaren kulistych a adchyIcnia od rownania Kolmagorava-Johnsona-Mehla-Avramiego

Dla prognozozvania kinctyki prxcmian razowych polcgajqcyci~ nn znrodkowaniu i \\*rrobcic ziaren nowcj fnzy utyrva sic znanc r6wnanie Kolmogorova-Johsona-Mehla-Avramiego. Wiadomo, 2c w przypadku rbwnolcgtcgo wzrostu ziarcn r6tnych faz z rbZnq pr~dkoiciq wyniki obliczcri obarczonc bt(;dcm. W ninicjszcj pracy z wykorzystrtniern zalozcli stnlystyczncj tcorii rvzrostu ckranorvnnego pod,iqto prdbe oszacowania tcgo blqdu dla przypadku wzrostu ziarcn kulistych dwu typbw. Otrzymanc wyniki tvskazujq, ic wiclkoSt btcdu klasyczncgo rhwnania wzrasta ze zwiekszenicm rhtnicy prqdkoSci wzrostu czqstek obu typbw.

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