Macrosegregation During Solidification(1)

8/7/2019 Macrosegregation During Solidification(1)

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M a c r o s e g r e g a t i o n D u r in g S o l id i f i c a t io nR e s u l t in g F r o m D e n s i t y D i f f e r e n c e si n t h e L i q u i d

N. STREAT AND F. WEINBERGMacro segre gat ion has been observe d in dire ct io nal l y sol idified Pb-20 pct Sn al loys, over arange of f reezing ra tes and tempe ratur e gradients . The macrosegre gat ion was shown to re-sul t from the upward flow of less dense, t in rich, inte rden dri t ic l iquid during sol idificat ion,us ing rad ioact ive t race r t echniques . For compar i son , i t was shown that macrose gregat io nocc urr ed in the opposi te direct i on in a Sn-4 pct Pb al loy, where the int erde ndri t ic l iquidwas lead rich, and conseq uently more dense. Shrinkage trai ls and pipes were observe d insome of the Pb-20 pct Sn ingots , s imi la r to " f re ckl es" observed in d i rect ional ly cas tsuperal loys .A mathemat ica l model for macros egrega t ion in ver t i ca l ly so l id i f ied ingots i s presented ,the dr iv ing force being the dens i ty d i f ferences in the in terdendr i t i c l iquid dur ing so l id i f i -cat ion. Liquid flow through the dendri t ic ar ra y is est im ated by cons ider ing the part ial lysol idified al loy as a porous med ium of var iabl e porosi ty. For s imp lici ty, the model ne-glects backflow due to volume shrinkag e (invers e segregat ion). The expe rime ntal resu l tsare compared to the model predic t ions .

MACROSEGREG ATION caused by in terdendr i t i c f lu idflow has been trea ted analy t ical l y in a numb er of rec entpapers . Most of these models are for inverse segr ega-t ion, where an analyt ic al solut ion can be obtained bycons ider ing backflow through a volume eleme nt as thesol id and l iquid contract during sol idificat ion. Chil lface segregat ion under these condi t ions was f i rs t p r e-dicted by Scheil ~ and his model was lat er extended byKirkaldy and Youdelis to predict the solute distr ibut io nalong the whole casti ng. 2'3

A more general so lu t ion for macr osegreg at ion con-s ider ing f lu id f low in three d imens ions for a b inary a l -loy with a constant part i t ion rat io was fi rs t publishedby Flemi ngs and Nereo 4 and used to predict solute dis -t ribu t ions for inv erse se gregat ion in A1-Cu al loys. Themodel was ba sed on the lafann equation , which givesthe composi t io n of sol id at the sol id- l iquid interfa ce( C s ) when there is complete mixing in the adjoiningliquid:

C s = kCo(1 - fs)/~lwhere

k = equi l ibr ium par t i t ion ra t iof s = weight fract ion sol id ( i . e . , compos i t ions are

in wt pet)C O = ini t i al al loy composi t io nThe model has subse quentl y been re fined and ex-

tended s '6 and has be en applied by Mehr abian, Keaneand F lem ings 7 to predic t macros egreg at ion caused bya combinat ion of sol idificat ion contract io n and solute

N. STREAT, formerly Research Associate, Department of Metal-lurgy, University of British Columbia, is now Metallurgical ResearchEngineer, British Columbia Hydro and Power Authority, Vancouver,Canada. F. WEINBERG is Professor, Department of Metallurgy, Uni-versity of British Columbia, Vancouver, Canada.Manuscript submitted October 22, 1973.

convect ion. They consider the fluid dynamics througha volume element where the forces act ing are so l idcontra ct ion, l iquid contra ct ion and gravi ty. The l iquidis of var iabl e densi ty, and the sol i d-l iq uid region ist reated as a porous medium of var iab le poros i ty . Equa-t ions are der ived re la t ing in terdendr i t i c f luid press ure ,interdendri t ic flow veloci ty, fract ion l iquid and l iquidcomposi t ion which can, in theory, be solved to givethese va ria ble s as a funct ion of posi t ion. In pract i ce,solut ions for the genera l cas e are difficul t to obtain,since this would involve the solut ion of s imulta neouspar t i a l d i f ferent ia l equat ions . They have thereforemade the fol lowing simplifyi ng assu mptio ns:1) For the pur pose of calculat i ng flow veloci t ies , i tis ass umed that the fract ion l iquid var ies with posi t ionin the mushy zone only as a funct ion of tem per atu re,and i s calcu la ted by assuming s teady-s ta te , un id i rec-t ional heat and fluid flow. A new distribu t ion of frac -t ion l iquid in the mush y zone is su bseque ntly predictedwhen these flow veloci t ies are included in the macro-segregat ion equat ion der ived by Nereo and F lemings . 4

2) Pla nar isot her ms are a ssume d, so that for a con-stant l iquidus slope, the l iquid composi t io n var ies l in-early with posi t ion in one direct ion.

3) The densi ty of l iquid var ies l ine arl y with compo-si t ion.4) The de nsit y of solid is consta nt.The model has been applied to the special case ofhor izontal , un id i rect ional heat f low and s t eady-s ta teso l id i ficat ion, where a value for the para mete r char -acter iz ing the s t ruc ture and thermal condi t ions hasbeen assumed.In addit ion, Mehrabian e t a l have d i scussed themechani sm whereby, under cr i t i ca l condi t ions , up-ward flowing l iquid beco mes superh eated and can dis-solve dendri te branches in i ts path to form visibletra i ls or "pi pe s" of high solute content . These have

METALLURGICAL TRANSACTIONS VOLUME 5, DECEMBER 1974-2539


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b e e n o b s e r v e d i n e x p e r i m e n t s o n A 1 -C u a l l o y s , e a n dt h e y p r o p o s e t h a t t h e c r i t i c a l c o n d i t i o n f o r t h e f o r m a -t i o n of c h a n n e l - t y p e d e f e c t s i s w h e n th e d i r e c t i o n o f t h ei n t e r d e n d r i t i c f l u i d f l ow v e c t o r g o e s f r o m t h e c o l d e rt o t h e h o t t e r r e g i o n s o f t h e c a s t i n g . U p w a r d f l o w i n gc h a n n e l s h a v e b e e n o b s e r v e d b y o t h e r a u t h o r s i n t r a n s -p a r e n t a m m o n i u m c h l o r i d e - w a t e r m o d e l s a n d t h ism e c h a n i s m i s t h o ug h t to be r e s p o n s i b l e f o r t h e f o r -m a t i o n o f f r e c k l e s i n n i c k e l - b a s e s u p e r a l l o y s , 9' lo a n dA s e g r e g a t e s i n s t e e l , n

T h e p u r p o s e o f t h e p r e s e n t i n v e s t i g a t i o n i s to e x -p e r i m e n t a l l y d e t e r m i n e t h e ex t e n t o f m a c r o s e g r e g a -t i o n a n d f r e c k l e f o r m a t i o n i n v e r t i c a l l y s o l i d i f i e d l e a d -t i n a l l o y s a s a f un c t i on o f th e s o l i d i f i c a t i o n v a r i a b l e s .S i n c e i t w o u l d b e d i f f i c u l t t o a p p l y t h e m o d e l d e v e l o p e db y M e h r a b ia n e t a l t o t h e t y p e o f i n g o t a n d a l l o y s y s -t e m u s e d in t h i s i n v e s t ig a t i o n , a s i m p l e m a t h e m a t i c a lm o d e l w a s d e v e l o p e d w h i c h w o u l d t a k e i n to a c c o u n tt h e e ff e c t o f g r o w t h r a t e , t e m p e r a t u r e g r a d i e n t a n ds t r u c t u r e . S o m e o f t h e a s s u m p t i o n s f o r t h is m o d e ld i f f e r fr o m t h o s e u s e d b y M e h r a b i a n e t a l , t h e m a j o rd i f f e r e n c e s b e i n g :

1 ) T h e p a r t i t i o n r a t i o v a r i e s a s a f un c t io n o f t e m -p e r a t u r e .

2 ) T h e d e n s i t y o f t h e l i q u i d i s a f u n c t i o n o f t e m p e r -a t u r e a n d c o m p o s i t i o n , o b ta i n e d f r o m e x p e r i m e n t a ld a t a .

3 ) T h e s t r u c t u r e o f t h e m u s h y z o n e is c h a r a c t e r i z e db y a p a r a m e t e r w h i c h i s a l s o o b ta i n e d f r o m e x p e r i m e n -t a l d a t a .

4 ) S i n c e t h e m o d e l i s a p p l i e d t o t h e s o l i d i f i c a t i o n o fP b - S n a l l o y s , b a c k f l o w d ue to v o l u m e s h r i n k a g e i s n e -g l e c t e d .

E X P E R I M E N T A L P R O C E D U R ECylindrical ingots 14 c m long and 1.27 c m in diana

w e r e s o li d i fi e d i n g r a ph i t e m o l d s b y l o w e r i n g t h em o l d s t h r o u g h a t w o z o n e f u rn a c e. B y a d j us t i ng t h et e m p e r a t u r e o f t h e f u r n a c e z o n e s , t h e g ra d i en t a n dg r o w t h r a te c o u l d b e v a r i e d i n d ep e n de n t ly . D u r i n gsolidification, the alloy wa s que nch ed wh en desir edb y s u r r o u n d i n g t he m o l d b y w a t er . Q u e n c h i n g o c c u r r e din less than 20 sec onds.

Fo r eac h set of solidification conditions, four ingotsw e r e m a d e . T h e t e m p e r a t u r e g r ad ie nt a n d f r e e zi n gr a te w e r e e s t ab l i sh e d i n o n e i n g o t u s i n g t h r e e t h e r m o -couples positioned along the axis. Th e cast structurew a s d e t e r m i n e d f r o m l on gi t ud in a l a n d t r a n s v e r s e s e c -t i o ns o f t w o o f t h e i n g o t s w h i c h w e r e p o l i s h e d a n d e t c h e da f te r s e c ti o ni n g . T h e f o ur t h i n g o t w a s u s e d f o r m e a -s u r e m e n t s o f m a c r o s e g r e g a t i o n . R a d i oa c t i ve t ra c e rwa s well m ix ed into the liquid prior to solidification,and after solidification the ingot wa s place d in a lathea n d c u t t i ng s w e r e t a k e n i n a p l a n e p e r p e n d i c u l a r t o th eingot axis. Starting at one end, the cuttings we re col-lected a t 0 .3 7 c m i n t e r v a l s a n d t h e a c t i v i t y o f e a c hg r o u p o f c u t t i n g s w a s m e a s u r e d i n a s c i n t i l l a t i o nc o u n t e r .

T h e i s o t o p e s u s e d w e r e e i t h e r S n n 3 (h a l f l i f e 1 12d a y s , p r i m a r i l y a l o w e n e r g y ~ e m i t t e r ) o r T 1 a~ ( h a l fl i f e 4 .1 y e a r s , p r i m a r i l y a # e m i t t e r , b u t a l s o s o m el o w e n e r g y ~ ) . T h e S n u 3 w a s u s e d i n t h e l e a d r i c h a l -l o y s , a n d T 1 a~ i n t h e t i n r i c h a l l o y s . F r o m t h e m e a -s u r e d a c t i v i t y a n d t h e w e i g ht o f e a c h s a m p l e , t h e s o l -

u t e c o n t e n t w a s d e t e r m i n e d a s a f u n c t i o n of d i s t a n c ef r o m t h e b o t t o m o f th e i n g o t .

C o m p o s i t i o n p r o f i l e s f o r i n g o ts q u e n c he d d i r e c t l yf r o m t h e l i q u i d w e r e c o m p a r e d t o t h e d i r e c t i o n a l l ys o l i d i f i e d in g o t s , to c o n f i r m t h a t t h e e f fe c t s o b s e r v e df o r t h e l a t t e r w e r e d u e to t h e s o l i d i f i c a t i o n c o n d i ti o n s ,a n d n ot d u e to p o o r m i x i n g o f th e o r i g i n a l a l l o y c h a r g e ,o r o t h e r e x t r a n e o u s f a c t o r s .

T o o b s e r v e c o n v e c t i o n t h r o u g h t h e b u l k l i q u i d a n ds o l i d - l i q u i d r e g i o n s , 0 .1 g p e l l e t s o f th e c a s t i n g a l l o y ,c o n t a i n i n g S n n s , w e r e p l a c e d i n th e l i q u i d a t t h e t o p o ft h e m o l d d u r i n g s o l i d i f i c a t i o n . S o l i d i f ic a t i o n w a s c o n -t i n u e d f o r o n e h o u r ; f o l l o w i n g w h i c h t h e i n g o t s w e r eq u e n c h e d i n th e f u r n a c e , t h e n s e c t i o n e d a n d p o l i s h e d .A u t o r a d i o g r a p h s o f t r a n s v e r s e a n d l o n g i t u d i n a l s e c -t i o n s s h o w e d t h e e x te n t of t r a c e r m o v e m e n t . A s i m i -l a r e x p e r i m e n t w a s p e r f o r m e d o n a n i n g o t o f t h e s a m ec o m p o s i t i o n h e l d c o m p l e t e l y l i q u id u n d e r t he s a m et e m p e r a t u r e g r a d i e n t f o r o n e h o u r . I n t h i s c a s e ,s p r e a d i n g o f t h e t r a c e r c o u l d b e a t t r i b u t e d t o t h e c u -m u l a t i v e e f f e c t s o f d i s t u r b a n c e s a s s o c i a t e d w i t h a d d -i n g t h e p e l l e t a n d q u e n c h i n g . T h e f l u i d f l o w r e s u l t i n gf r o m t h e s o l i d i f i c a t i o n p r o c e s s c o u l d b e e v a l u a t e d b yc o m p a r i n g t h e r e s u l t s o f t h e s e t e s t s .A s m a l l s a m p l e ( 15 0 g) o f t h e m o l t e n a l l o y w a sp l a c e d i n a g r a p h i t e c r u c i b l e a n d o b s e r v e d d u r i n gs o l i d i f i c a t i o n u n d e r v a c u u m . N o b u b b l e s w e r e s e e n .T h i s e v i d e n c e , t o g e t h e r w i t h i n f o r m a t i o n i n t h e l i t e r -a t u r e t h a t o n l y o x y g e n i s v e r y s l i g h t l y s o l u b le i n m o l -t e n F o - S n a l l o y s , l a c o n f i r m e d t h a t g a s e v o l u t io n c o u l db e i g n o r e d a s a p o s s i b l e f r e c k l i n g m e c h a n i s m i n t h i ss y s t e m .

R E S U L T Si . C o m p o s i t i o n P r of i l es

T h e c o m p o s i t i o n p r o fi l e s o f t h e i n go t s , d e t e r m i n e db y t h e a c t i v i t y m e a s u r e m e n t a n a l y s i s , a r e s h ow n inF i g s . 1 a n d 2 . T h e s o l i d l i n e s a r e t h e o r e t i c a l c u r v e sc a l c u l a t e d f r o m t h e m a t h e m a t i c a l m o d e l . T h e s o l i d i -f i c a t i o n v a r i a b l e s a n d t h e m e a s u r e d v a l u e s o f m a c r o -s e g r e g a t i o n , t o g e t h e r w i t h th e r e s u l t s f o r q u e n c he di n g o t s , a r e g i v e n in T a b l e I . C o o l i n g c u r v e s s h o w e dt h a t e s s e n t i a l l y c o n s t a n t g ro w t h r a t e s a n d t e m p e r a -t u r e g r a d i e n t s w e r e m a i n t a i n e d u n d e r t h e s l o w f r e e z -i n g c o n d i t i o n s i m p o s e d i n t h e s e e x p e r i m e n t s .

M a c r o s e g r e g a t i o n ( A C) , i n t h i s c a s e , h a s b e e n d e -f i n e d a s t h e d i f f e r e n c e i n a v e r a g e c o m p o s i t i o n b e -t w e e n t h e u p p e r a n d l o w e r h a l v e s o f t h e i n g o t . I t i sc o n s i d e r e d p o s i t i v e w he n th e s o l u t e c o n c e n t r a t i o n i n -c r e a s e s i n t h e d i r e c t i o n o f s o l i d i f ic a t i o n , a n d n e g a t i v ef o r t he r e v e r s e .C o m p a r i n g t h e m a c r o s e g r e g a t i o n f o r d i r e c t io n a l l ys o l i d i f i e d an d q u e n c h e d c a s t i n g s , l i s t e d i n T a b l e I , i tc a n b e s e e n t h a t t h e s o l u t e d i s t r i b u t i o n i s a f u n c ti o no f t h e s o l i d i f i c a t i o n c o n d i t i o n s . I n g e n e r a l , t h e a m o u n to f m a c r o s e g r e g a t i o n f o r i ~ - 2 0 p c t Sn a l l o y s d e c r e a s e da s t h e p r i m a r y d e n d r i t e s p a c i n g d e c r e a s e d . F i g s . l ( a )a n d (b ) s h o w t h a t m a c r o s e g r e g a t i o n i n c r e a s e s f o r t h es l o w e r g r o w t h r a t e a t t h e s a m e t e m p e r a t u r e g r a d ie n t ,a n d F i g s . l ( b ) a n d ( c ) s h o w t h e s a m e e f f e c t f o r t h es h a l l o w e r g r a d i e n t w h e n t h e g r o w t h r a t e s a r e a l m o s tt h e s a m e . F o r t h e S n - 4 p e t P b a l l o y ( F i g . 2 ), t h e r ew a s m o r e s o l u t e a t th e b o t t o m o f t h e c a s t i n g a n d l e s sa t t h e t o p , r e s u l t i n g i n s o m e n e g a t i v e m a c r o s e g r e g a -t i o n .

2540-VOLUME 5, DECEMBER 1974 METALLUR GICAL TRANSACTIONS


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AVERAGE TEMPERATURE GRADIENT 1.5*C/cmAVERAGE GROWTH RATE 0-0 047 cr n /sec

--0 0 2 : O 4 ~ 0 670 ' S~O ' IO:ODISTANCE FROM BOTTOM OF CASTING (ca t ) ' 12~0 ' 14"0(a)

AVERAGE TEMPERATURE GRADIENT 1'5*C/eraAVERAGE GROWTH RATE O-OIScm/sec

-OO 2:0 ' 4?0 " 6?0 ' e~o ' ~o'oD~STANCE FROM BOTTOM OF CASTING (cm) 12'0 14'0

~ ] AVERAGETEMPERATURE GRADIENT 2.3"C/cm. AVERAGE GROWTH RATE OOII cm/sec

. /

--OO 2:O 4:O 6:O 8~O ' IO~C) ' 12'O ' 14'ODISTANCE FROM BOTTOM OF CASTING (cml(c )

Fig. l--Solute profiles for direetionally solidified Pb-20 petSn alloys under the conditions shown. The solid lines aretheoretical curves calculated using the data in Table ITI.

The reproducibility of the analysis technique wastested by counting a single sample of turnings severaltimes, emptying and refilling the container in which itwas held, to vary the geometry of packing. The errorbars shown in Figs. 1 and 2 are 2s limits based onthese tests, and for castings with a me an compositionof Fo-20 pct Sn, they represent a scatter of ~0.38 pctSn (percentage error = 1.9 pct). This was consideredacceptable, since the composition difference betweenthe ends of the castings was, in general, significantlylarger than the scatter.For the Sn-4 pct Pb castings, the error bars repre-sent a scatter of +0.22 pct Pb, which is a larger per-centage error (4.3 pct), probably due to greater ab-sorption of the low energy emission f rom Tl 2~

2 . Co nv ect io n in the LiquidThe resu l t s dem o nstra t ing co nv ect io n in the l iqu idduring so l id i f i ca t io n a re g iv en in F ig . 3 (a ) . The a u to -ra d io g ra phs sho w n a re of sec t io ns pa ra l l e l to the

freez ing d i rect io n o f a 1 ~ -2 0 pet Sn a l l o y under thecondit ions l i sted in Table II . In Fig . 3 (a) , the reg ionswhich a re un i fo rm ly da rk ( sec t io ns i - i v ) ind ica te tha tthe l tqutdus i so therm pa ssed thro ug h th i s reg io n a f t ertra ce r ha d bec o m e m ix ed thro ug h the bu lk l iqu id . Insect io n (v ), o n ly the in terdendri t i e reg io n s a re da rk ,Ind ica ting t ra ce r penetra t io n in to the so l id - l iqu id zo ne .Fig . 3 (b) shows a utoradtog raphs for an ingot quenchedfro m the liqu id . Tra c er ha s m o v ed do wn l es s tha n 3 cm ,a s co m pa red to 6.5 cm fo r F ig . 3 (a ) . The d i f f erence in

AVENGE TEMPERATURE GRAOIENT 1.9*C/cmAVERAGE GROWTH RATE 0OO33cm/sec

2"0 4YO 670 SYO I0'0 12'0 E4'ODISTANCE FROM BOTTOM OF CASTING (cm)F i g . 2 - - S o l u t e p ro f i l e for d i rec t i on a l l y so l i d i f i ed S n -4 p c tPb . N o raaerose grega t i o n i s p red i c ted b y the mod e l for th i sa l l o y , shown by the sol id l i n e .

Table I. Solidification Variablesand Ma aro ~ on

AverageAlloy TemperatureFi g. Com posit ion Gradient

No. (wt pet) (~

CalculatedAverage Dist ance BetweenGrowth Liquidus andRate Solidus Isotherms(cm/sec) (cm)

Pr imaryDendriteSpacing(microns) Structure

Macro-segregation

(txC pet)l(a ) Pb-20 Sn 1.5 0.05l(b ) Pb-20 Sn 1.5 0.2l(c) Pb-20 Sn 2.3 0.3- Pb-20 Sn -

2 Sn-4 Pb 1.9Sn-4 Pb

0.0047 62.0 2060.013 62.0 1720.011 40.4 119quenched - -0.0033 24.7 -

quenched

columnar 1.07half columnar 0.73hal f equiaxedequiaxed 0.13equ iaxed -0 .04half columnar -0.3 5half equiaxedequiaxed 0.11

M E T A L L U R G I C A L T R A N S A C T I O N S V O L U M E 5 , D E C E M B E R 1 9 7 4 - 2 5 4 1


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Fig. 3-Autoradiographsshowing the extent oftracer movement one hourafter tracer was added (a)directionally solidified (b)quenched from liquid.Solidification conditionsgiven in Table II. Magnifi-cation 2 times.

(a )penet ra t ion i s a t t r ibu ted to so lu te convect ion associ -ated with the sol idificat i on proc ess. Ther e is no indi-cat ion, however, of the flow pat tern which caused mix-ing.

3 . F recklesEvidence of s t ruc tures rese mbl in g f reckles was seen

in ingots sol idified at the slowest growth rate s . In onecase (Fig. 4(a)), the outer surface showed a shrinkagetra i l appr oximat ely 7 cm long near the top. A shr ink -age defect of this type indicates that a long channel ofeutect ic l iqu id was prese nt jus t before the cas t ing be-came complete ly so l id . This bears a close rese m-blance to previ ousl y published photographs of freck lesin n ickel -ba se superal loys . 9

Fig. 4(b) shows longitudinal sect ions through a notheringot which containe d an inter nal t rai l that could beclassed as a f reckle . An enlarged view of a t ransv ers esection (Fig. 5(a)) shows that the trail had a fine den-2542-VOLUME 5, DECEMBER 1974

(b )dri t ic s t ru ctur e. Fig. 5(b) is an enlar ged view of thelowest port ion of the t rai l , showing that i t originate din the inte rior of the ingot as an inter dendr i t ic chan-nel , which subse quentl y widened and moved towardsthe mold wall , in the same d irec t ion as the pr ima rydendri te s ta lks. The sol idificat ion condit ions for theingots in Figs. 4 and 5 are given in Table II.

D I S C U S S I O N O F R E S U L T ST h e c u r v e s i n F i g. 1 s h o w i n g p os i ti ve m a c r o s e g r e -

g a t io n r e s e m b l e c u r v e s f o r n o r m a l s e g r e g a t i o n w i thdiffusion controlled mi xin g a hea d of a planar solid-liquid interface. The re is, how eve r, consid erable ex-p e r i m e n t a l evid ence in the li te ra tu re 13'x4 to show thatonly a negligible amount of solute is re ject ed aheadof dendri te t ips when growth is not plana r. Normalsegre gat io n takes place over a distance of the orderof micr ons in the l iquid between dendri te branch es,leading to micros egre gat i on. However, one would not

METALLURGICAL TRANSACTIONS


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Fig.No.

Table II . Cooling ConditionsCalculatedAverage Avera ge Distan ceBetweenAl lo y Tem pera tu re Growth Liqu idus andComposition Gradient Rale Solidus Isotherms(wt pct) (~ (cm/sec) (cm)

3(a) Pb-20 Sn 1.9 0.0033 48.93(b) Pb-20 Sn - quenched -4(a) Pb-20 Sn 1.5 0.0047 62.04(b),5 Pb-20 Sn 1.9 0.0033 48.9

(a)

(a) (b)Fig . 4 - - ( a ) Sh r inkage t r a i l , app rox ima te ly 7 cm long , a longthe ou t s ide o f an ingo t so l id i f i ed unde r cond i t ions g iven inTable I I. M agnif ica t ion 2 t im es. (b) Longitudinal sect ionsshowing a f r eck le t r a i l on r igh t s ide . Hor izon ta l wh i t e a reasa r e s p a c e s b e t w e e n s e c t i on s c a u s e d b y t r a n s v e r s e s a w c u ts .Magn i f i ca t ion 5 t imes .e x p e c t a n e t m o v e m e n t o f s o l u t e i n t h e d i r e c t i o n o fg r o w t h u n l e s s t h e r e w a s l i q u i d m i x i n g o n a m a c r o -s c o p i c s c a l e . T h i s m i x i n g c o u l d t a k e p l a c e e i t h e rw i t h i n th e s o l i d - l i q u i d z o n e , o r b e t w e e n t h i s z o n e a n dt h e b u l k l i q u i d a h e a d o f t h e d e n d r i t e t i p s .

A n e s t i m a t e o f t h e s i g n i f i c a n c e o f t h e m a c r o s e g r e -g a t i o n v a l u e s i n T a b l e T c a n b e o b t a i n e d b y u s i n g e l e -m e n t a r y s t a t i s t i c s . T h e c o m p o s i t i o n m e a s u r em e n t sf o r e a c h s a m p l e a r e s u b j e c t to m i c r o s e g r e g a t i o n a n dg e o m e t r i c a l s c a t t e r , h o w e v e r , i t i s r e a s o n a b l e t o a s -s u m e t h a t th e m e a n o f s a m p l e s o v e r h a l f th e i n g o tMETALLURGICAL TRANSACTIONS

~ '~ -" % * :~ ~ ~' 7-- . . -- --~r" ~ , i ~i.... i ..~

? ?- : ..(/

If,rI~--;.~7:7~- "r i,

(b )Fig . 5 - - ( a ) Tran sve r se sec t ion o f f r eck le t r a i l i n Fig . 4(b)showing the f ine dendr i t i c s t ruc t u re wi th in the t r a i l . (b) Lon-g i tud ina l s ec t ion showing tha t the t r a i l o r ig ina te s f rom widen -ing in te rden dr i t i c channe l s in the in te r io r o f the ingo t . Mag-n i f i ca t ion 25 t imes .

l e n g t h w i ll o n ly b e s u b j e c t t o g e o m e t r i c a l s c a t t e r ,s i n c e m i c r o s e g r e g a t i o n o n l y e x t e n d s o v e r r e l a t i v e l ys h o r t d i s t a n c e s . K n o w i n g t h e m e a n s a n d s t a n d a r d d e -v i a t i o n s f o r t h e t o p a n d b o t t o m h a l v e s o f t h e i n g o t , t h e" t - t e s t " m a y be u s ed to c h e c k w h e t h e r t h e c o m p o s i -t i o n d i f f e r e n c e s a r e s i g n i f i c a n t .

F o r a s i g n i f i c a n c e l e v e l o f 0 . 01 , o n e m a y c o n c l u d et h a t t h e r e i s n o s i g n i f i c a n t m a c r o s e g r e g a t i o n w h e nA C < 0 . 1 9 p c t Sn f o r Pb - 2 0 p c t Sn i n g o t s , o r A C < 0 . 1 1p c t la b f o r S n - 4 p c t P b i n g o t s . T h u s t h e m e a s u r e d v a l -u e s o f A C in T a b l e I f o r b o t h q u e n c h e d i n g o t s s h o w n o

VOLUME 5, DECEMBER 1974-2543


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s i g n i f i c a n t m a c r o s e g r e g a t i o n , n o r c a n t h e v a l u e o f 0 .1 3p e t S n f o r t h e i n g o t i n F i g . l ( c ) b e r e g a r d e d a s a s i g -n i f i c a n t d i f f e r e n c e . T h e r e m a i n i n g i n g o t s , h o w e v e r ,s h o w a s i g n i f i c a n t a m o u n t o f m a c r o s e g r e g a t i o n c o m -p a r e d t o t h e q u e n c h e d i n g o t s , a n d s i g n i f i c a n t d i f f e r -e n c e s w h e n c o m p a r e d to o n e a n o t h e r .

T h e e x p e r i m e n t w h e r e t r a c e r w a s a d d e d t o t h e l i q u ida t th e t o p of t h e c a s t i n g ( F i g . 3 ) c o n f i r m s t h a t s o l u t ec o n v e c t i o n t oo k p l a c e , w h i c h i s a t t r i b u t e d t o th e f o r m a -t i o n o f l o w e r d e n s i t y l i q u i d i n th e s o l i d - l i q u i d r e g i o n .I n t h e c a s e o f P o - 2 0 p c t S n , t h e i n t e r d e n d r i t i c l i q u i db e c o m e s e n r i c h e d i n t i n , u p t o t h e e u t e c t i c c o m p o s i -t i o n ( 6 2 p c t S n ) . T h e d e n s i t y o f t h e b u l k l i q u i d a t th ei n t e r f a c e w o u l d b e 9 . 7 g / c m 3, a n d t h e e u t e c t i c w o u l db e 8 . 2 g / c m 3 , TM t h u s t h e r e w o u l d b e a d e n s i t y i n v e r s i o nt h r o u g h t h e s o l i d - l i q u i d r e g i o n w h i c h c a u s e s t h e l e s sd e n s e l i q u i d t o r i s e .

O n e c an t h e r e f o r e c o n c l u d e t h a t th e s o l u t e p r o f i l e si n F i g . 1 a r e a f u n c t i o n o f t h e g r o w t h r a t e , t e m p e r a -t u r e g r a d i e n t a n d d e n d r i t e s p a c i n g . H o w e v e r , i t i s no tp o s s i b l e to d r a w f i r m c o n c l u s i o n s r e g a r d i n g t h e e f f ec to f v a r y i n g a n y o n e o f t h e s e p a r a m e t e r s a l o n e , b e c a u s eo n l y o n e o f t h e t h r e e v a r i a b l e s c o u l d b e h e l d c o n s t a n tf o r a n y t w o i n g o t s . F o r t h i s r e a s o n , a s i m p l e m a t h e -m a t i c a l m o d e l w a s d e r i v e d , b a s e d o n th e c o n c e p t ofm a s s t r a n s f e r t h r o u g h t h e s o l i d - l i q u i d r e g i o n c a u s e db y d e n s i t y d i f f e r e n c e s i n t h e l i q u id .

T h e S n - 4 p c t l ~b a l l o y w a s c h o s e n a s a n e x a m p l e o fa c o m p o s i t i o n w h e r e t h e i n t e r d e n d r i t i c l i q ui d w o ul d b em o r e d e n s e t h a n t h e b u l k li q u i d a b o v e . T h e c o m p a r a b l ed e n s i t i e s w o u l d b e 7 .1 g / c m 3 i n th e b u l k l i q u i d , a n d 8 .2g / c m 3 i n t h e e u t e c t i c . ~s T h i s d e n s i t y c o n f i g u r a t i o nw o u l d b e s t a b l e , a n d o n e w o u l d n o t e x p e c t a n y c o n v e c -t i o n . T h e r e s u l t i n g c o m p o s i t i o n p r o f i l e ( F i g . 2) s h o w sa s m a l l i n c r e a s e i n l e a d c o n te n t c l o s e t o t he b o t t o mo f t h e i n g o t , w h i c h i s p o s s i b l y d u e t o t h e e f f e c t o f i n -v e r s e s e g r e g a t i o n .

C o n s e q u e n t l y , t h e f r e c k l e t r a i l s o b s e r v e d i n i n g o tss o l i d i f i e d a t t h e s l o w e s t f r e e z i n g r a t e s a r e a t t r i b u t e dt o t h e e f f e c t of s o l u t e c o n v e c t i o n . A s t h e l e s s d e n s el i q u id t o w a r d s t h e b o t t o m o f t h e m u s h y z o n e b e g i n s t or i s e , i t b e c o m e s s u p e r h e a t e d a n d c a n d i s s o l v e d e n d r i t eb r a n c h e s i n i t s p a t h . I f s u f f i c i en t i n t e r d e n d r i t i c c h a n -n e l s w i d e n in t h is f a s h i o n , t h e y c a n c o n v e r g e a n d r e -s u l t in t h e f o r m a t i o n o f a l a r g e v e r t i c a l p i p e . D i r e c te v i d e n c e o f t h i s m e c h a n i s m i s s ho w n i n F i g . 5 (b ) . T h ee x p e r i m e n t a l e v i d e n c e s u g g e s t s t h a t f r e c k l e s d o n o ta l w a y s f o r m w h e n s o l u t e c o n v e c t io n t a k e s p l a c e , b u tt h e y a p p e a r w h e n t h e v e l o c i t y o f t h e r i s i n g i n t e r d e n -d r i t i c l i q u i d r e a c h e s a c r i t i c a l v a l u e .T o e x a m i n e t h e m a g n i t u d e o f i n v e r s e s e g r e g a t i o n i nt h e l a b - S n s y s t e m , t h e S c h e i l m o d e l ~ w a s u s e d t o c a l -c u l a t e th e m a x i m u m c h i l l f a c e s e g r e g a t i o n f o r a l a r g en u m b e r o f a l l o y c o m p o s i t i o n s . B o t h t h e p a r t i t i o n r a t i oa n d t h e s o l i d i f i c a t i o n s h r i n k a g e w e r e c o n s i d e r e d a sf u n c t i o n s o f t e m p e r a t u r e a n d c o m p o s i t i o n , r e s p e c t i v e l y .T h e r e s u l t s o f t h i s c a l c u l a t i o n a r e g i v e n in F i g . 6 . F o rS n - 4 p c t P b , i n v e r s e s e g r e g a t i o n w o u l d c a u s e t h e c o m -p o s i t i o n a t t h e c h i l l f a c e t o r i s e b y 0 . 2 p c t l ~ b, w h i c hc o r r e s p o n d s t o t h e e x p e r i m e n t a l r e s u l t i n F ig . 2. F o rP b - 2 0 p c t S n, i n v e r s e s e g r e g a t i o n w o u l d c a u s e a n i n -c r e a s e o f 0 . 5 5 p c t S n a t t h e c h i l l f a c e . T h i s e f f e c tw o u ld o p p o se t h e p o s i t iv e m a c r o s e g r e g a t i o n c a u s e db y s o l u t e c o n v e c t i o n i n t h e l i q u i d . I n v i e w o f t h e o b -s e r v e d c h i l l f a c e s e g r e g a t i o n i n F i g . 1 ( w h ic h r a n g e s

o 71# O6

0.5

~ 0. 4g,~ 03$

0. 1

~0 20 30 40 50 60 70 80 90 I00We~ t Percent Tm

F i g . 6 - - M a x i m u m c h i l l f a c e s e g r e g a t i o n f o r P b - S n a l l o y s ,c a l c u l a t e d u s i n g t h e S c h e i l m o d e l , tfr om - 1.8 to - 1.0 pct Sn), the effect of inver se seg-regation is conside red to be sma ll for this alloy sys-t e m .

M O D E L O F T H E S O L I D I F I C A T I O NP R O C E S S

T h e m a t h e m a t i c a l m o d e l a s s u m e s t h a t s ol id if ic at io nis des cri bed using the Pfan n equatio n in exactly thes a m e m a n n e r a s p r e v i o u sl y p r e s e n t e d b y K i r k a l d y a n dYoudelis. 2 Th e partition ratio (k) can be cons ider ed avariable duri ng solidification if one as su me s that it re-. ~ ma l ns c o n st a n t o v e r a s m a l l t e m p e r a t u r e i n te r va l .L i q ui d c o m p o s i t i o n , a v e r a g e s o l i d c o m p o s i t i o n a n d t heweigh t fractions of solid and liquid ca n be defined ata n y t e m p e r a t u r e b e t w e e n t h e l i q ui d u s a n d s o l id u s b yu s i n g d a t a f r o m t h e p h a s e d i a g r a m .

In addition, if density data are available as a func-t i o n o f t e m p e r a t u r e a n d c o m p o s i t i o n , v o l u m e f r a ct i o nsa n d d e n si t i es a r e a l s o d e f in e d a t a n y t e m p e r a t u r e .

I N T E R D E N D R I T IC F L U I D F L O W M O D E LT h e m o d e l o f t h e s o l i d i f i c a t i o n p r o c e s s d e s c r i b e sh o w s o l u t e i s r e d i s t r i b u t e d p e r p e n d i c u l a r t o t h e d e n -d r i t e s t a l k s . I f n o i n t e r d e n d r i t i c f l u i d f l ow o c c u r s , t h ei n g o t w o u l d s h o w m i c r o s e g r e g a t i o n o n a s c a l e e q u i v a -l e n t to t h e d e n d r i t e s p a c i n g , b u t t h e r e w o u l d b e n o n e tm o v e m e n t o f s o l u t e o v e r g r e a t e r d i s t a n c e s . T h e a s -s u m p t i o n i s no w m a d e t h a t s o l u t e c a n b e m o v e d o v e rm u c h g r e a t e r d i s t a n c e s b y i n t e r d e n d r t t i c f l u i d f l owc a u s e d b y d e n s i t y d i f f e r e n c e s i n t h e l i q u id . T h e r e i s

e x p e r i m e n t a l e v i d e nc e t h a t D a r c y ' s L a w ca n b e a p -p l i e d t o i n t e r d e n d r i t i c f l u i d f l o w , *a'*7 t h e r e f o r e , t h ef lo w v e l o c i t y ( v) i s p r o p o r t i o n a l t o t h e p r e s s u r e d r o pa c r o s s t h e m u s h y z o n e ( & P ) :

- Kv = - ~ & Pw h e r e

K = p e r m e a b i l i t y ( in un i t s o f a r e a )/1 = v i s c o s i t y o f t h e l i q u i dh = l e n g t h o f t h e m u s h y z o n eF u r t h e r m o r e , t h e p e r m e a b i l i t y o f t h e d e n d r i t i c n e t -

w o r k c a n b e e x p r e s s e d i n t e r m s o f f lo w th r o u g h a b u n -d l e o f c a p i l l a r y t u b e s : ~ ' ~ '

K = - -247rm-s

2544-VOLUME 5 , DECEMBER 1974 METALLURGICAL TRANSACTIONS


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where g L is the volume fract ion l iquid, n is the num-ber of channel s , and ~ is a tortuosi t y factor, whichtakes into account the fact that the inter dend ri t i c chan-nel s are nei ther s t ra ight nor symmetr ical .

P revious exper imental work TMhas shown that K ispropor t ional to g~ , when g L is les s than 0.3. In theabsence of a bet ter model , i t is assumed that the aboveequation holds for al l values of g L " Thus , the in terden-dri t ic fluid flow model is esse nt ia l ly the same as thatused by Mehrabian e t a l ~ with the fol lowing difference s:

1) the nu mber of channels (n) is cal culated from thepr ima ry dendr i te spacing (;~), and is taken to be equalto 1/X 2.

2) the tortu osi ty factor is taken from e xper imen talwork done in this lab ora tor y, ~v and is equal to 4.6.

UNIDIRECTIONAL SOLIDIFICATION OFA VERTICAL CASTING

As an example of the applic at ion of the model , c on-sider the vert i cal sol idific at ion of an ingot of constantcro ss sect ion, of the type examined in Fig. 1. To sim-plify the calcula t ions, c onstant growth rate (R) andtempe ratur e gra dient (G) are assumed. In addit ion ,since the model wil l be applied to Pb- Sn al loys, wherethe volume change on free zing is smal l (of the order of2 pct) , backflow due to volume shrinkage is assumed tobe negligible. This assu mption is just i fied in view ofthe max imu m chil l face com posi t ion s shown in Fig. 6.

The ingot is divided into a number of horizontal lay-ers , and sol idificat ion is consid ered to begin when thetem per atu re of the bot tom layer is equal to the l iquidust em p era t u re T L . F reez i n g p ro g res s es i n d i s c re t e t i m esteps (At) and within each step, sol idifica t ion and fluidf low are t reated separate ly . Conservat ion of solu tewithin each layer applies during each sol idificat ionstep, and con serva t ion of solute through the whole in-got applie s in each fluid flow step. Fluid flow throughany layer s tops when i ts temperature fal ls below theeutect ic t emperature T E , and the fina l composi t io n ofthe ingot is obtained when the top layer reaches T E .

Fig. 7 gives a schemat ic re pre sen tat ion of the ingotat an in termediate t ime, and shows the temperatur e ,composi t ion and densi t y profi le s in the l iquid, obtainedusing the sol idificat ion model ( i . e . , the Pfann equation)for an al loy where incr eas ed amounts of solute causethe l iquid to beco me le ss dense.The driving force for flow through the mushy zoneis given by the densi ty differ ence between T L and T E ,and is equal to A p L g h , where A p L is the densi ty dif-fer enc e in the liquid, and g is gravi ty. Since the liquid

f. . . . o - - - -T L -- _ ~

r - - i - d ---1

T T E M P E R A T U R E

CEL I Q U I D L I Q U ~ OC O M P O S I T I O N D E N S I T Y

Fig. 7--Unidire ctional ingot divided into layer s. Temp era -ture, composition and density profiles given by the solidifi-cation model.

o

01~o

~i 2814

L: 37L= I

length of cost lnq ( H ) : 1 4 c mf e m pe ra lu re g ra d ie n t ( G ) = 1 5 ~grow J h ra t e ( R ) = 0 005 c m / s e cn u m b e r o f c h a n n e l s ( N C ) = 3 5 x tC r~viscosity of the l iquid = 0 0 3 p o i s e

%'0 ' Z'O ' 4'0 ' 6:0 ' 8'0 ' I0:0 ' /2~0DISTANCE FROM BOTTOM OF CASTING (cm)

Fig. 8--Solute distribution as a function of the number oflayers.is a continuum, the drivin g force is cons ider ed to beconstan t through the sol id- l iquid region, durin g thet ime in terval At .

The redi stri but i on of solute is then calculated by as-suming a flow pat te rn within the sol id l iquid region.Taking into account the driving force acr oss each layer,and the resis tance to flow of the dendri t ic network, sol-ute mass balances can be wri t ten for each layer of theingot . This is desc ribe d in more detai l in Appendix A.

The working of the model can be su mmar ized br ief lyas follows: when the Pfann equation is applied to eachlaye r of the ingot as i t sol idifies , the av erage compo-si t ion of both sol id and l iquid in the layer remainsequal to the mean al loy composi t io n C O. The additionaleffect of interdendri t ic fluid flow is to cause l iquid witha high tin content, close to the bottom of the mushyzone, to flow upwards, and be rep lace d by l iquid of alower t in content . Consequently, the average co mpos i-t ion of layers towards the top of the ingot wil l r iseabove Co, and those towards the bottom will fall belowC o .

RESULTS OF CALCULATIONS FORTHE SOLIDIFICATION OF A Pb-Sn ALLOY

The model was used to calculate the final solute dis-tri but ion in a Pb- 20 pct Sn alloy as a function of thesol idifica t ion condit ions. Data for the part i t i on rat io kand the eq uil ib rium l iquidus l ine were obtained fromthe phase diagr am, and the densi t ie s of l iquid Pb-Snalloys as a funct ion of tem per atu re and composi t io nwere avai lable in the for m of a table. 15Fig. 8 shows the solute distrib ut ion for diffe rentnumb ers of layers , when the length of the ingot H, G,R, ~, and the "effe ct ive numb er of cha nne ls , " N C ,have the values shown. The curves show that , exceptfor the ends of the ingot , as the number of layers in-cre ases the so lu te d i s t r ibu t ion converges to a s ingleso lu t ion . As the number of l ayers increases ( i . e . , th esize of laye rs de crea ses) , the as sumpt ion that the flowrate between laye rs (q) is small (see Appendix A) com-pared to the amount of l iquid in each layer, no longerholds. This breakdow n would appear f i rs t at the ex-treme ends of the ingot , therefore the end composi-tions shown in Figs. 8 through 12 are not con sid ere dmeaningful when they differ by m ore than about 20 pctof the mean value. However, within this l i mitat ion, theshape of the curves and the integrated amount of solute

METALLURGICALTRANSACTIONS VOLUME 5, DECEMBER 1974-2545


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0

a t

o

D O

NC = 1 '7 x 10 eNC = 3 " 5 x I 0 ~

~ ~ ~ N C = 6 " 0 X I 0 "

length of cost ing (H) = 14crnt e m p e r a t u r e g r a d i e n t (G ) : I 5 = C/ cmg r ow t h r o t e ( R } = 0 0 0 5 c m / s e cvisr of the l i q u i d = O 0 3 poisen u m b e r o f l a y e r s ( L ) = 2 8

' 2r0 ' 470 ' 6tO ~ 8TO IOTO 1270 14'0D I S T A N C E F R O M B O T T O M O F C A S T I N G ( c m )Fig. 9--Solute distribution as a function of the st ructure (ef -fective number of channels).

~ M = ~ _ _ = ; H : [ 4 r ~H= S cm~ . H = l O c m

~ " _

E pn u m b e r o f c h a n n e l s ( N C ) = O - 6x IO s/o j v i s c o s i t y o f t h e l i q u i d 9 O-O~peisen u m b e r O f l a y e r s ( L ) - 2 B

9| o:, dz o:3 o~, o:5 o~ dr o'.a o:9 ,.'oDISTANCE F~ B O T T O M O F CASTING( X / H )Fig. 10--Solute distribut ion as a function of ingot height.

w h i c h h a s m o v e d f r o m t h e b o t t o m o f t h e i n g o t t o th e t o pa r e a m e a s u r e o f t h e r e l a t i v e a m o u n t o f m a c r o s e g r e g a -l i o n .

T h e s t r u c t u r e o f t h e s o l i d li q u i d r e g i o n i s e x p r e s s e di n t e r m s o f t h e " e f f e c t i v e n u m b e r o f c h a n n e l s " ( N C )w h e r e N C = n T 3 A ( A = c r o s s s e c t i o n a l a r e a o f t h e i n g o t )a n d th e s o l u t e p r o f i l e s f o r d i f f e r e n t v a l u e s o f N C a r es h o w n i n F i g . 9 . I t c a n b e s e e n t h a t t h e a m o u n t o f m a c -r o s e g r e g a t i o n , c o n s i d e r e d i n t e r m s o f t h e am o u n t o fs o l u t e w h i c h m o v e s f r o m t h e b o tt o m h a l f o f t h e c a s t i n gt o t h e t o p , i n c r e a s e s a s N C d e c r e a s e s . S i n c e N C i s r e -l a t e d to th e d e n d r i t e s p a c i n g , t h i s m e a n s t h a t f o r l a r g e rs p a c i n g s t h e r e s i s t a n c e t o f lo w th r o u g h t h e s o l i d - l i q u i dr e g i o n d e c r e a s e s . T h e r e f o r e , f o r t he s a m e p r e s s u r ed r o p t h e r e i s m o r e f l o w .

F i g . 1 0 s h o w s t h e s o l u t e d i s t r i b u t i o n a s a f u n c t i o n o fi n g o t h e i g h t . A s th e h e i g h t i n c r e a s e s , s o t he f lu i d h e a dw i l l i n c r e a s e , c a u s i n g m o r e f l o w t h r o u g h th e m u s h yz o n e . H o w e v e r , t h i s o n l y a p p l i e s w h e n t h e l e n g t h o ft h e m u s h y z o n e c a n h e e q u a l t o th e i n g o t h e i g h t . T h et h e o r e t i c a l l e n g th o f t h e m u s h y z o n e i s (T L - TE)/ G ,w h i c h f o r t h e c o n d i t i o n s u s e d i n F i g . 1 0 i s 6 2 c m .

F i g . 1 1 s h o w s t h a t t h e a m o u n t o f m a c r o s e g r e g a t i o ni n c r e a s e s a s th e g r o w t h r a t e d e c r e a s e s . T h i s w o u l db e e x p e c t e d , s i n c e t h e a m o u n t o f t i m e a v a i l a b l e f o r t h ef lo w i n c r e a s e s , a s R d e c r e a s e s . F i g . 1 2 s h o w s t h a t t h ea m o u n t of m a c r o s e g r e g a t i o n i n c r e a s e s a s th e t e m p e r -a t u r e g r a d i e n t i n c r e a s e s , c o n t r a r y t o th e s e m i - q u a n t i -t a t i v e t h e o r y p r o p o s e d b y C o p le y , G i a m e i , e t a l . z~ T h i si s d ue to s t e e p e r c o m p o s i t i o n g r a d i e n t s t h r o u g h t hem u s h y z o n e a t h i g h e r t e m p e r a t u r e g r a d i e n t s , w h i c hw o u l d le a d to a h i g h e r d e n s i t y d i f f e r e n c e a n d c o n s e -

~ R 0 2 5 r n / , e o/ i R = O I O "

/ / ~ R = O 0 5 "

o J . . , e . g , , o f c o s t , . Q o H ) : , 4 c mtemperature qrod ient (G) = I 5~cm/sec number a t channels (NC)= 0 6 x I C5, ~ t v i sco s i t y ~ t h e l i q u id = 0 " 0 3 p o i se

n u m b e r o f l a y e r s ( L ) = 2 8O J - w~ 0 O ~ ] 0 4 ~ 0 6 F O ' 8 : 0 ' t O l O 1 ~ 2 ' 0 ' t 4 1 CO I S TA N C E F R O M B O T T O M O F C A S T I N G ( c m )

Fig. ll--Solute distr ibution as a function of growth rate.

n u m b e r o f l a ye rs (L ) = 2 8

~DO 0 2 : 0 4 '0 6 : 0 8 '0 I 0 : 0 1 2 '0 1 4 'OD I S T A N C E F R O M B O TT O M O F C A S T I N G ( c m }Fig. 12--S01ute distribution as a function of temperaturegradient.

T ab l e I I I . S o l i d i f i ca t i o n V ar i ab l es U sed f o r T h eo r e t i ca l P l o t sFig. No. 1( a ) 1(b) 1(c)

T e m p e ra t u re Gra d i e n t (G) ~ 1 .5 1 .5 2 .3G r o w t h R a t e ( R ) cm/sec 0 .0047 0 .013 0 .011L e ng t h o f Ca s t i ng ( / / ) c m 14 14 14Num be r o f Cha nne l s (NC) = n~-3A 2 .96 X l 0 s 4 .24 l 0 s 8 .86 X l OsVi sc os i t y o f t he L i qu i d (p ) po i se 0 .03 0 .03 0 .03Num be r o f l a ye r s 28 28 28AC ( t he o re t i c a l ) 1 .95 0 .39 0 .31~ C ( e xpe r i m e n t a l ) 1 .07 0 .73 O. 13

q u e n t ly m o r e f l o w , w he n a l l o t h e r v a r i a b l e s a r e h e l dc o n s t a n t . T h e r e a s o n w h y t h i s a p p e a r s t o c o n t r a d i c te x p e r i e n c e i s t h a t h i gh t e m p e r a t u r e g r a d i e n t s a r e u s u -a l l y a s s o c i a t e d w i t h h ig h g r o w t h r a t e s , a n d i t i s n o tn o r m a l l y f e a s i b l e to v a r y t h e se t w o p a r a m e t e r s i n d e -p e n d e n t l y .

C O M P A R I S O N W I T H E X P E R I M E N TT h e d a t a u se d to g e n e r a t e t h e t h e o r e t i c a l c u r v e s i n

F i g . 1 , t o g e t h e r w i t h t h e o r e t i c a l a n d e x p e r i m e n t a l v a l -u e s o f m a c r o s e g r e g a t i o n a c c o r d i n g to t he p r e s e n t d e f t -n i t i o n , a r e g i v e n in T a b l e H I . I n g e n e r a l , t h e m o d e lp r e d i c t s p r o f i l e s o f t h e s a m e s h a p e a s th e e x p e r i m e n -t a l p l o t s , b u t t h e c o m p o s i t i o n s a t t he e n d s o f t h e i n g o td o n o t a l w a y s a g r e e w e l l w i t h t h o s e p r e d i c t e d . T h i s i sd u e t o t h e a s s u m p t i o n s u s e d i n d e r i v i n g t h e m o d e l . I naddition to those already discussed, the Pfann equationdoes not take into account that the liquid composition

2 5 4 6 - V O L U M E 5 , D E C E M B E R 1 9 74 M E T A L L U R G I C A L T R A N S A C T I O N S


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I r - ' - ~ ( - ~ l TL 6 ,R4,q 44 R3 ,q 3

(a) (b)F i g . 1 3 - - ( a ) A s s u m e d f l o w p a t t e r n s h o w i n g t w o m a i n f l o wc e l l s . ( b) R e s i s t a n c e s R I _~ , a n d f l o w r a t e s q t _ ~ f o r f l o w b e -t w e e n s i x l a y e r s .c a n n o t r i s e a b o v e t h e e u t e c t ic c o m p o s i t i o n .

F o r F i g s . l ( a ) a n d (b ) w h i c h w e r e s o l i d i f i e d u n d e rt h e s a m e t e m p e r a t u r e g r a d i e n t , b u t w it h d i f f e r e n tg r o w t h r a t e s a n d d e n d r i t e s p a c i n g s , b o th th e t h e o r e t i -c a l a n d e x p e r i m e n t a l r e s u l t s s h o w m o r e m a c r o s e g r e -g a t i o n a t t h e l o w e r g r o w t h r a t e . F o r F i g s . l ( b ) a n d ( c)w h i c h w e r e s o l i d i f i e d a t a p p r o x i m a t e l y t h e s a m e g r o w t hr a t e , b u t w i t h d i f f e r e n t t e m p e r a t u r e g r a d i e n t s a n d d e n -d r i t e s p a c i n g s , t h e t h e o r y p r e d i c t s s l i g h t l y m o r e m a c -r o s e g r e g a t i o n a t t h e l o w e r t e m p e r a t u r e g r a d i en t . T h i si s q u a l i t a t i v e l y i n a g r e e m e n t w i t h e x p e r i m e n t . I t s h o u l db e n o t e d t h a t a lt h o ug h F i g . 12 p r e d i c t s l e s s m a c r o s e g -r e g a t i o n a t l o w e r t e m p e r a t u r e g r a d i e n t s ( w he n a l l o t h e rv a r i a b l e s a r e h e l d c o n s t a n t ) , th i s e f f e c t h a s b e e n o u t -w e i g h e d b y t he d i f f e r e n c e i n d e n d r i t e s p a c i n g . T h i sa l s o c o r r e s p o n d s w i t h a p r e v i o u s i n v e s t i g a t i o n o ff r e c k l i n g , ~9 w h e r e i t w a s r e p o r t e d t h a t a r e d u c t i o n i nd e n d r i t e s p a c i n g e l i m i n a t e d f r e c k l e s i n c o n s u m a b l ea r c m e l t e d i n g o ts .

U s i n g th e m o d e l p r e d i c t i o n s , i t i s p o s s i b l e t o r e c -o m m e n d a n u m b e r o f c h a n g e s i n c a s t i n g p r a c t i c e t h a tw o u l d r e d u c e g r a v i t y s e g r e g a t i o n e f f e c t s i n v e r t i c a ld i r e c t i o n a l c a s t i n g s :

1) R e f i n e m e n t o f t h e d e n d r i t i c s t r u c t u r e w i l l i n -c r e a s e t h e r e s i s t a n c e t o f lo w th r o u g h th e m u s h y z o n e.2 ) R e d u c t i o n o f i n g o t h e i g h t f o r a l l o y s w i t h a w i d ef r e e z i n g r a n g e w i l l r e d u c e t h e d r i v i n g f o r c e f o r f lo w .3 ) I n c r e a s i n g t h e g r o w t h r a t e w i l l r e d u c e t h e t i m ea v a i l a b l e f o r f l ow .

4 ) D e c r e a s i n g t h e t e m p e r a t u r e g r a d i e n t w il l r e d u c et h e d r i v i n g f o r c e f o r f l ow . H o w e v e r , t h i s r e c o m m e n -d a t i o n s h o u l d b e a p p l i e d w i t h c a r e . A l t h o u g h t h e m o d e lp r e d i c t s a d e c r e a s e i n m a c r o s e g r e g a t i o n , t h i s w ou ldo n l y b e t h e c a s e i f a l l o t h e r s o l i d i f i c a t i o n v a r i a b l e sr e m a i n c o n s ta n t .

C O L IC L U S I O N SM a c r o s e g r e g a t i o n c a n o c c u r d u r i n g u n i d i r e c t i o n a l

s o l i d i f i c a t i o n o f l e a d - t i n a l l o y s d u e t o c o n v e c t i v e f l owc a u s e d b y t h e f o r m a t i o n o f a l e s s d e n s e l i q u id i n th ei n t e r d e n d r i t i c r e g i o n s . U n d e r t h e s e c o n d i t i o n s , t h e r ei s a n e t f lo w o f s o l u t e a l o n g t h e c a s t i n g w h i c h c a n p r o -d u c e a c o m p o s i t i o n p r o f i l e s i m i l a r t o n o r m a l s e g r e -g a t i on f o r a p l a n a r s o l i d - l i q u i d in t e r f a c e . E x p e r i m e n t ss h o w e d t h a t t h i s p r o f i l e i s a f u n c t i o n o f t h e g r o w t h r a t e ,t e m p e r a t u r e g r a d i e n t , d e n d r i t e s p a c i n g , a n d a l l o y c o m -p o s i t i o n .A n u m e r i c a l m o d e l h a s b e e n p r o p o s e d w h i ch p r e -d i c t s t h e c o m p o s i t i o n p r o f i l e s a s a f u n c t io n o f t h e s o l -i d i f i c a t io n v a r i a b l e s . T h e d r i v i n g f o r c e f o r m a c r o s e g -r e g a t i o n i s th e d e n s i t y d i f f e r e n c e i n th e i n t e r d e n d r i t i cMETALLURGICAL TRANSACTIONS

l i q u id , a n d t h e d e n d r i t i c s t r u c t u r e i s c o n s i d e r e d t o b ea p o r o u s m e d i u m o f v a r i a b l e p o r o s i t y . S h r i n k a g e t r a i l sa n d p i p e s w h i ch o r i g i n a t e f r o m t h e w i d e n i n g o f i n t e r -d e n d r i t i c c h a n n e l s h a v e b e e n p r o d u c e d i n t h e P b - S ns y s t e m w h e n a d e n s i t y i n v e r s i o n e x i s t s d u r i n g s o l i d i -f i c a t io n a n d t h e g r o w th r a t e s a r e v e r y l o w . T h e s e f i n d -i n g s s u p p o r t t h e m e c h a n i s m f o r f r e c k l e f o r m a t i o n p r e -v i o u s l y p r o p o s e d .

A P P E N D I X AR E D I S T R I B U T I O N O F S O L U T E B Y

F N T E R D E N D R I T I C F L U I D F L O WT h e a s s u m e d f l o w p a t t e r n t h r o u g h t h e s o l i d - l i q u i d

r e g i o n c o n s i d e r s t h a t f lo w c a n t a ke p l a c e v e r t i c a l l yf r o m o n e l a y e r t o t h e n e x t, a n d h o r i z o n t a l l y t h r o u g ht h e la y e r . F o r v e r t i c a l f lo w , h a l f t h e c r o s s s e c t i o n a la r e a c o n t r i b u t e s t o d o w n w a r d f lo w , a n d h a l f t o u p w a r df l o w . F i g . 1 3 (a ) s h o w s t h e a s s u m e d f l o w p a t t e r n w i t ht w o m a i n f l ow c e l l s . H o w e v e r , p r o v i d e d d o w n w a r d a n du p w a r d f l ow e a c h o c c u p y h al f th e c r o s s s e c t i o n a l a r e a ,t h e a c t u a l n u m b e r o f f lo w c e l l s i s u n i m p o r t a n t . T h em o d e l d i f f e r s f r o m t h e p h y s i c a l s i t u a t i o n o b s e r v e d i nF i g . 3 i n t h a t i t d o e s n o t c o n s i d e r f l u i d f l o w b e t w e e nt h e m u s h y z o n e a n d t h e l i q u i d z o n e .T h e r e s i s t a n c e o f t h e d e n d r i t i c n e t w o r k t o f l ui d fl o wi s r e p r e s e n t e d s c h e m a t i c a l l y in Fi g . 13 (b ). T h e r e s i s -t a n c e s y m b o l s r e p r e s e n t p o r o u s m e d i a o f a r e a e q u alt o h a l f t h e c r o s s s e c t i o n a l a r e a o f t h e c a s t i n g , a n dl e n g t h e q u a l t o t h e l e n g t h o f t h e l a y e r s . W e a s s u m et h e r e i s no r e s i s t a n c e t o h o r i z o n t a l f l o w t h r o u g h t h el a y e r s , s i n c e t h e d i s t a n c e s w i l l b e s h o r t , e s p e c i a l l yf o r a l a r g e n u m b e r o f f l ow c e l l s .P o r o u s l a y e r s s t a c k e d i n t h i s m a n n e r o b e y t h e l a w so f s e r i e s r e s i s t a n c e s , t h e r e f o r e , s i n c e th e m a g n i t u d eo f t h e r e s i s t a n c e c a n b e c a l c u l a t e d i n t e r m s o f t h el i q u id f r a c t i o n a n d s t r u c t u r e , a n d t h e p r e s s u r e d r o pi s k no w n , t h e v e l o c i t y o f t h e i n t e r d e n d r i t i c l i q u i d( v / g L ) c a n b e c a l c u l a t e d u s i n g D a r c y ' s L a w . T h e fl o wr a t e o f i n t e r d e n d r i t i c l i q u i d ( q ) i s t h e n e q u a l t o 2 v / A g L ,w h e r e A is t h e c r o s s s e c t i o n a l a r e a o f t h e in g o t . F o rb r i e f t i m e i n t e r v a l s A t , t h e q u a n t i t y o f l i q u i d w h i c hf l o w s b e t w e e n l a y e r s w i l l b e s m a l l , t h e r e f o r e t h e d r i v -i n g f o r c e f o r f l ow i s a s s u m e d t o r e m a i n c o n s t a n t .I n F i g . 1 3 ( b) , s i x l a y e r s a r e s h o w n , t h e r e s i s t a n c e sb e t w e e n t h e l a y e r s a r e n u m b e r e d R I _ ~, a n d t h e fl o wr a t e s a r e q ~ -s . T h e l i q u id c o m p o s i t i o n s o f e a c h l a y e r ,e x p r e s s e d a s w e i g h t p e r u n i t v o l u m e , a r e e q u a l to( p L C L ) I _ 6 . T h e v o l u m e s o f t h e l a y e r s , V I_6, r e m a i nc o n s t a n t .E a c h l a y e r e x c h a n g e s l i q u i d w i th t h e a d j o i n in g l a y -e r s , a n d f o l l o w i n g th i s n o t a t i o n , t h e v o l u m e f l o w r a t ea c r o s s t h e to p a nd b o t t o m s u r f a c e s o f t h e i - t h l a y e ra r e q i a n d q i - 1 , r e s p e c t i v e l y . A s o l u te m a s s b a l a n c ec a n , t h e r e f o r e , b e w r i t t e n f o r th e i - t h l a y e r :

dV i ~ ( p L C L ) i = q i . l ( P L C L ) i _ l + q i ( P L C L ) i . l- - ( q i + q i - 1 ) ( P L C L ) i

T h i s m a s s b a l a n c e c a n b e w r i t t e n fo r e a c h l a y e r ,g i v in g a s e r i e s o f s i m u l t a n e o u s o r d i n a r y d i f f e r e n t i a le q u a t i o n s w h i c h c a n b e s o l v e d f o r th e c o m p o s i t i o n o fe a c h l a y e r a f t e r a t i m e i n t e r v a l Z ~t u s i n g s t a n d a r d n u -m e r i c a l m e t h o d s.

VOLUME 5 , DECEMBER 1974-25 47


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A C K N O W L E D G M E N TF i n a n c i a l s u p p o r t p r o v i d e d b y t h e N a t i o n a l R e s e a r c h

C o u n c i l a n d t h e K i l l a m F o u n d a t io n i s g r a t e f u l l y a c k -n o w l e d g e d .R E F E R E N C E S

1. E . Schei l : Metallforsch., 1942 , vo l . 20 , p . 69 .2 . J . S . Ki rka l dy a nd W . V. Youde l i s : Trans. TMS-AIME, 1958 , vo l . 58 , p . 212 .3 . W. V. Youde l i s : The Solidification of Metals, I . S. I . Publ i ca t ion 1 IO, Decem-

be r 1967 , p . 112 .4 . M. C . F l e m i ngs a nd G. E . Ne re o : Trans. TMS-AIME, 1967 , vo l . 239 , pp .

1449 -61 .5 . M. C . F l e m i ngs , R . Me hra b i a n , a nd G. E . N e re o : Trans. TMS-AIME, 1968, vol .

2 4 2 , p p . 4 1 - 4 9 .6 . M. C . F l e m i ngs a nd G. E . N e re o : Trans. TMS-AIME, 1968 , voL 242 , pp . 50 -55 .7 . R . Me hra b i a n , M. Ke a ne , a nd M. C . F l e m i ngs : Met. Trans. , 1970, vol . 1 , pp.1209 -20 .

8 . R . M e hra b i a n , M. Ke a ne , a nd M. C . F l e m i ngs : Met. Trans. , 1970, vol . 1 , pp.3 2 3 8 - 4 1 .9 . A . F . G i a m e i a nd B . H . Ke a r : Met. Trans., 1970 , vo l . I , pp . 2185 -91 .

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2 5 4 8 - V O L U M E 5 , D E C E M B E R 1 9 74 M E T A L L U R G I C A L T R A N S A C T I O N S

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Macrosegregation During Solidification(1)

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