on the growth kinetics of grain boundary ferrite allotriomorphs

On the Growth Kinetics of Grain Boundary Ferrite AIIotriomorphs

C. ATKINSON, H. B. AARON, K. R. KINSMAN, AND H. I. AARONSON

Prev ious work has shown that the thickening k ine t ics of proeutectoid f e r r i t e a l lo t r iomorphs in an Fe-0 .11 pct C alloy a re often more rapid than the k ine t ics ca lcula ted for volume d i f fus ion-con t ro l f rom the D u b e - Z e n e r equation for the migra t ion of a p lanar boundary of infini te extent , a s suming the diffusivi ty of carbon in aus ten i te , D, to be constant at that of the carbon content of the A e 3 . Recalcu la t ing the thickening k ine t ics , us ing a n u m e r i c a l ana lys i s of the infini te p lanar boundary p rob lem prev ious ly developed by Atkinson in which the var ia t ion of D with composi t ion is taken fully into account , was found to i n c r e a s e this d i sc repancy . M e a s u r e m e n t s were then made of the lengthening as wel l as the thickening k ine t ics of g ra in boundary a l lo t r iomorphs in the same alloy. Applicat ion to these data of A tk inson ' s n u m e r i c a l ana lys i s of the growth k ine t ics of an oblate e l l ipsoid , in which the compos i t ion-dependence of D is s i m i l a r l y cons idered , produced an acceptable account ing for nea r ly a l l of the data. It was concluded that the growth of f e r r i t e a l lo t r iomorphs is p r i m a r i l y cont ro l led by the vo lume diffusion of carbon in aus ten i te ; the p r e sence of a sma l l p ropor t ion of d is locat ion facets along one of the broad faces of the a l l o t r i omorphs , however , usua l ly r e s u l t s in growth k ine t ics which a re somewhat s lower . An a l t e rna te t r e a t m e n t of the lengthening and thickening data upon the bas i s of the theory of in te r fac ia l d i f fus ion-a ided growth of a l lo t r iomorphs indicated that, in the t e mpe r a t u r e range i n v e s t - . o . . .

1gated (735 to 810 C) , the dlffumvxties of carbon along V:7 and 7 : a boundar ies r equ i r ed for this mechan i sm to make a s igni f icant cont r ibut ion to growth a re too high to be phys ica l ly p laus ib le .

S E V E R A L yea r s ago, K insman and Aaronson 1 r e - por ted m e a s u r e m e n t s of the thickening kinet ics of g ra in boundary a l lo t r iomorphs of proeutectoid f e r r i t e in an Fe-0 .11 pct C and in two F e - C - X a l loys , obta ined by means of the rmionic e m i s s i o n mic roscopy . The parabol ic ra te cons tant , a , for these kinet ics*

*The parabolic rate constant is the constant of proportionality between growth distance and the square root of the growth time.

was de t e rmined f rom the expe r imen ta l data and also calcula ted f rom the Dub62- Zene r 3 equation for the mig ra t ion k ine t ics of a p l ana r , d i so rde red in te rphase boundary of infini te extent . This equation was der ived under the a s sumpt ion that the diffusivi ty , D, in the ma t r ix phase is i nva r i an t with composi t ion . Since the diffusivi ty of carbon in aus ten i te i n c r e a s e s rapidly with carbon content , 4 a p a r t i c u l a r value of this d i f - fus iv i ty had to be chosen. Fol lowing Wagner , 5 the one se lec ted was that at the carbon content of the A e 3 or 7 / ( a + 7) phase boundary . In the Fe -C alloy and even more pronouncedly in one of the F e - C - X a l loys , how- eve r , many of the expe r imen ta l ly m e a s u r e d va lues of a re f lec ted growth kinet ics fas te r than those ca lcula ted . If the ca lcula ted k ine t ics a re accepted as exact , this r e s u l t indicates that the in te r face composi t ions at the broad faces of the a l lo t r iomorphs depar ted a p p r e - c iably f rom their equ i l i b r ium values dur ing growth.

C. ATKINSON is Lecturer, Department of Mathematics, Imperial College of Science and Technology, London, United Kingdom. H. B. AARON, formerly with Scientific Research Staff, is now Chem-Met Specialist, Automotive Assembly Division, Ford Motor Company, Dearborn, Mich. 48121. K. R. KINSMAN is with the Scientific Re- search Staff, Ford Motor Company. H. I. AARONSON, formerly with Scientific Research Staff, Ford Motor Company, is now Professor, De- partment of Metallurgical Engineering, Michigan Technological Uni- versity, Houghton, Mich. 49931.

Manuscript submitted March 23, 1972.

In view of the sma l l undercool ings be low the A e 3 at which much of these data were obtained and the p r e - dominant ly d i so rde red s t r uc t u r e which n o r m a l l y c h a r - a c t e r i z e s the broad faces of a l l o t r i omorphs , this ex- p lanat ion s e e ms unl ikely . The r e s u l t ob ta ined was, in fact , a lmos t the r e v e r s e of that ant ic ipated. The ex- p e r i m e n t a l o~'s were expected to be less than those calcula ted as a consequence of the p r e sence of a s m a l l p ropor t ion of d is loca t ion or pa r t i a l ly coheren t facets along one of the broad faces of the a l l o t r i omorphs . This expectat ion is supported by the cons ide rab le sca t t e r in the a va lues . The sca t t e r is c l ea r ly in ex- cess of that r e su l t ing f rom expe r imen ta l e r r o r , but is cons i s t en t with the concept that the propor t ion of d i s - locat ion facets va r i e s with the angle between the habit plane of the nucleus and the g ra in boundary plane,6 and is thus usua l ly d i f ferent f rom one a l lo t r iomorph to the next.

In view of the tendency for the m e a s u r e d a ' s to fal l i nc r ea s ing ly above those calcula ted with dec reas ing t e m p e r a t u r e , a poss ib le explanat ion of this unan t i c i - pated r e s u l t is that the growth of f e r r i t e a l l o t r i o - morphs is acce l e ra t ed by in te r fac ia l diffusion. 7 In the p r e s e n t context , the m e c h a n i s m of this acce l e ra t i on would cons i s t of the diffusion of carbon along the p r e - dominant ly d i so rde red aus teni te : f e r r i t e boundar ies of the a l l o t r i omorphs , followed by carbon diffusion along the aus teni te g ra in boundar ies (also expected to be d isordereda) , and f inal ly volume diffusion f rom the gra in boundar ies into the aus teni te ma t r i x . The c i r - cums tance that the absolute value of the average r e - act ion t e mpe r a t u r e employed is l ess than 0.6 the absolute mel t ing t empera tu re* and the finding that the

*In substitutional alloys with an fcc matriX phase, the growth of grain boundary allotriomorphs appears to be controlled by the interfacial diffusion-aided mechanism below ca. 0.9 Trn.

VOLUME 4, MARCH 1973-783 METALLURGICAL TRANSACTIONS

growth of pearlite in Fe-C alloys can be aided by interracial diffusion io-12 lend plausibility to this sug- gestion. However, interstitial solutes are usually considered unlikely candidates for rapid diffusion along internal boundaries and carbon has yet to be proved conclusively to do so by means of the usual macro- scopic type of diffusion experiment.

Another type of explanation for the anomalous result, which must be examined in detail before an interfacial diffusion mechanism can be accepted, is that the result arose from an insufficiently accurate calculation of ~. An obvious source of inaccuracy is the approximate method which had to be used to compensate for the composition dependence of the diffusivity of carbon in austenite. Shortly afterwards, however, Atkinson 13-~4 described a numerical method for taking accurate account of the role which this factor plays in the growth of a variety of precipitate morphologies. Application of this method to recalculate ~ forms the basis of the present investigation. In order to avoid introducing the additional complication of the influence of alloying el- elements upon phase boundary compositions,~s-~7 this study was confined to the Fe-0.11 pet C alloy.

The following approach was adopted. Retaining the simple assumption that the broad faces of ferrite aUo- triomorphs can be regarded as effectively infinite planes, ~ will be recalculated using Atkinson's ~3 analysis for this geometry. If this calculation fails to provide satisfactory agreement with the experimental data, the hypothesis will then be examined that the thickness/length or aspect ratio of ferrite allotriomorphs is too large to make the infinite plane assumption valid. New data on lengthening as well as on thickening kinetics of allotriomorphs in the same alloy will be reported, and ~ will be computed on the assumption that allotriomorphs can be modeled as oblate ellipsoids. Atkinson's ~4 numerical analysis of the growth kinetics of the ellipsoidal morphology will be used for this calculation. If the calculated values of are still smaller than an appreciable proportion of the experimental ones, an attempt will then be made to an- alyze the difference between them on the basis of the interfacial diffusion-aided growth mechanism. If not, then the diffusivities of carbon along austenite:ferrite boundaries and austenite grain boundaries needed to make this mechanism a significant contributor to the growth kinetics of ferrite allotriomorphs will be calculated and compared with an independent estimate of the boundary diffusivity of carbon in an effort to explain why this mechanism did not play a detectable role.

A special effort will be made to provide useful additional details for the operational aspects of both analyses. Particular emphasis will be placed upon the techniques for taking into account the variation of D

Slice Number, N I 2 3 N

I I II{ 4 - - Preci pi ta te ~ M otr', x ~ <

I I Position Number, r 0 I I 2 3 .

I / 2 N - I / 2 Fig. 1--Notation used in the Atkinson analyses to describe the locations of slices of thickness 5c and of their interfaces.

with compos i t ion , and upon p rob l ems which a r o s e in compu te r p r o g r a m m i n g . It is hoped that these supp le - m en t a ry de ta i l s wi l l al low other w o r k e r s to use these r a t h e r complex ana lyses m o r e r ead i l y , not only for F e - C a l loys but a l so for other s y s t e m s in which D in the m a t r i x phase v a r i e s s igni f icant ly with compos i t ion .

ANALYSIS OF THE MIGRATION KINETICS OF AN INFINITE PLANAR BOUNDARY 13

Adaptation to the Specific Problem of Ferrite Allotriomorph Thickening

This analysis is a finite difference method for solving Fick's Second Law under the condition of a concentration-dependent diffusivity in the matrix phase. The analysis is begun by dividing the solute concentration gradient in the matrix into N slices with equal intervals of concentration, 6C. Denoting the mole fraction of solute in the matrix at the interphase boundary as Co and that at the distance where this concentration is the same as that prior to transformation as CN, 5c = (Co - CN)/N. (The notation employed earlier T M is used throughout this paper.) Information on the concentration dependence of the diffusivity in the matrix is incorporated into the analysis through a set of ratios. These ratios are based upon averages of the diffusivity within specified ranges of concentration. The variation of the diffusivity of carbon with the mole fraction of carbon in the aus ten i t e , C, is r e p r e s e n t e d by: la

D = O.5 e ~ [ - 3 0 C ] e ~ [ - ( 3 a 3 0 0 - 1.9

x i0 ~ c + 5.5 x i0 ~ C2)/RT] [I]

The various ratios are:

a) Fr§ = ratio of the average diffusivity in the N'th slice (extending from r to r + 1, as illustrated in Fig. 1) to D at Co.

b) FN-~ = ratio of the average diffusivity in the last slice to D at Co.

c) F ' = ratio of the first moment of the average diffusivity over the entire concentration range to D at Co.

Beginning with Fr§ these ratios are derived from Eq. [1] as follows. This equation is first rewritten as:

D = B exp[-(Ao + AIC + A2C2)] [2]

whe re B = 0.5, ,40 = 38300/RT, AI = (30RT - 1.9 x IOS)/RT and A2 = 5.5 x IOS/RT. T h i s r e l a t ionsh ip is in turn r e a r r a n g e d as:

D = B exp[-(Ao + n , Co + A2C~)]

[ e [-Cao + A,c + A2C2)]] [al • L e [-(Ao + A, Co + A2c )]J

The defining equation for Pr*~n is:

c,

f ac Fr+I/2 - C,+I c, [4]

D f dC Cr+ 1

where D = Do at Co, and C r and Cr§ = carbon concentration at each side of the N'th slice. Noting from Eq. [3] that D{C}/Do - F{C}, i t is convenien t to continue the deve lopmen t of Fr.~/2 in t e r m s of:

784-VOLUME 4, MARCH 1973 METALLURGICAL TRANSACTIONS

F * =

: e ~~ f Cr+ l

,G

C r

f F{C}dC Cr+ 1

C r 2 C f exp[-[4"A~C + (,4x/24"~)] z - A o + (do/ /4A2)]d C 2

Cr+I exp[-(Ao + A~Co + A2 o)]

Cr exp[-[4"-A~C + (A~/24-A~)] ~ ] d ( v ~ C )

2 4 - N ,

e ~~ v~-;Cr§ ( A J 2 CA-~]

- er f [ -~c~ -(A,/24~)]} [5]

where ~o = A, Co + AzC~ + (A~/4A2). The negat ive s igns of the a rgumen t s of the e r r o r funct ions were in t roduced through a re~i r rangement in making the las t s tep. This was r equ i r ed because in the p r e s e n t c i r c u m s t a n c e s lA~/2d-~zl > I AV-A~-2CI and A~ is negat ive. Combining Eqs~ [4] and [5]:

- F* _ NF* [6 ] Fr§ - Cr - Cr§ Co - C~

Proceed ing next to F' , the defining equation for this quant i ty is:

F' 2 Cfo D { C } ( C - C N ) d C [7] =D--~- c ~ = ( C o - C ~ ) ~

Co eCF '*= f F { C } ( C - C N ) dC

c~ co

= er f exp[-[CX;c + (A,/2~S)V] c~

x [ v r -~C+ ( A ~ / / 2 v ~ ) - C N - ~ - ~ ] d C [8]

(set t ing u = 4 -~C + ( A ~ / 2 v ~ )

= e ~~ f e u(du/Az)

- e~~ [CN + (A , /2nz ) ]e -u~ du 4-A7~

In tegra t ing , and r e a r r a n g i n g to change the sign of the a r g u m e n t s of the e r r o r funct ions ,

1 {exp[-(4"~C/v + A J 2 ,/-~)~] F'* = 2A2

- exp[-(J-A-~Co + A~/2 ~-A~)=]}

r ((~;c~ + A' ) - 2 A 2

• { e ~ / [ - 4 - ~ c ~ - (A,/2 r ] - ery[-~Co - (A,/24-s [0]

Eq. [7] thus becomes :

F' = 2F'* exp[A,Co + A2C2o + (.42/4.42)] [I0] (Co ~ C N ) 2

In defining ff/V-~n, the (obviously reasonable ) a s s u m p - t ion is made that D v a r i e s so slowly with composi t ion in the region of the las t s l ice that it can be safely con- s ide red as a s imple average :

CAr- 1

f DdC

Do6C (DN-1 + DN) (CNq - CIV)

2Do6C

(DN-~ + DN) 2Do

[11]

Before beginning the numerical analysis, an initial estimate of ol must be made. This can be done with adequate accuracy by using the Dub~ 2-Zener s equation, with D = DoF' = D:

( C o - C N ) ( ~ ) ~ / 2 ~ e ~ 2 / 4 ~ e r f c ( ~ / 2 v r ~ _ ) [12] (Co X l ) = 2-

where X~ = mole fraction of carbon in ferrite at the c~/(c~ + y) phase boundary.

The numerical analysis for the planar case has been previously described in considerable detail, x3 It is therefore useful in the present paper to offer only a few brief comments on the methodology of the analysis, the first three dealing with computer programming and the fourth with a technique for extrapolating the computed values of ~ to obtain a more accurate final value~

a) Once the first estimate of I~/2 has been obtained (from Atkinson's '3 equation number [32] - [AI-32] rather than from Eq. [AI-31]) and 7/0 has been set equal to zero, 77r and Ir+~/2 are successively calculated in iterative fashion.* Using, for example,

*Eqs. [A1-2] and [A1-3]: 7? = [x- a(Dot) 1/~]/2(Dot) In, where r/o defines the position of the interface, i.e. x = a(Dot) ~ n, the distance the interphase boundary has moved in time t. Eqs. [AI-15] and [AI-16]:

Cr Ir + 1/2 =Fr + l/2l(r?r + 1 - ~Tr) = ll ~c f CN 2~?dc " nr + aN- r - 1/2).

J =- r + 1 as an indexing p a r a m e t e r (at l eas t in F o r t r a n IV, the p r o g r a m m i n g language used in the p r e s e n t s tudy, a subsc r ip t of ze ro is not allowed), the i t e ra t ion should proceed f rom J = 2 to J = N.

b) P a r t i c u l a r l y if the in i t i a l e s t ima te of a is insuf - f ic ient ly accu ra t e , Ir+l/~ may become negat ive . This is i nadmis s ib l e ; see Eq. [Al-15] , where ~?r+~ > ~r and Fr§ is pos i t ive . Immedia t e ly af ter each value of It+2 is computed, t he re fo re , a tes t should be made to a s c e r t a i n whether or not this has occur red . If any Ir+~/2 is found to be negat ive , the value of I~/2 should be i nc rea sed by an a r b i t r a r y amount , say 0.5 or 1 and the i t e ra t ive ca lcula t ion of 77r and I r § begun anew.

c) In the ca lcula t ion of I~v-~/2 (from Eqs . [.4.1-27] and [A1-28]) , evaluat ion of er fc (x) becomes a p rob lem when x is large .* The p rocedure employed in the

[~r ls an independently and accurately calculated value of IN_l/2. When the difference between these quantities is less than a suitably small amount, this signals that I,/2 and rh have been sufficiently refined so that it has become worthwhile to calculate a.

p r e s e n t stuay was to use the re la t ionsh ip : 19

erJ(x) = 1 - 1.1283792 e-X2y (0.2258365 - y (0.25212867 = y (1.2596951 - y (1.2878225 - 0.94064607y)))) [13]

where y = 1/(1 + 0.3275911x) when x -< 3, and to r e - place Eq. [A1-28] in i ts en t i re ty with: 2~

A{x} = 1 - 2 / z + lO/z 2 - 74/z 3 + 706/z 4 - 8.62/z 5 + I08830/z 8 [14]

where z = 2x ~ when x > 3~ d) After a had been i t e ra t ive ly evaluated at s eve ra l

N ' s , a more accura te value of a was obtained by p lo t -

METALLURGICAL TRANSACTIONS VOLUME 4, MARCH 1973-785

,.,eo~~"'e2L!''' I'''' I'''' I ' 1

' '" / o - - 3.30 /

g - / -

3.24

3.22

3.20

3.18 I I .001 .003 .01 .015

I/N Fig. 2--Typical extrapolations of c~ vs 1IN (where N = number of sl ices used in a part icular calculation of a) to 1IN = 0 for the Atkinson 13 planar analysis.

t ing ot a s a funct ion of 1 / N and ex t r apo l a t i ng to 1 I N = O. F o u r va lue s of N were u sed , r ang ing f r o m 60 to 200.

R e s u l t s of the P l a n a r A n a l y s i s and D i s c u s s i o n

F ig . 2 shows typ ica l e x t r a p o l a t i o n s of a vs 1/N. The p lo t s a r e r e a s o n a b l y l i n e a r and the amount of the ex t r apo l a t i on r e q u i r e d is seen to have i n c r e a s e d with d e c r e a s i n g r e a c t i o n t e m p e r a t u r e . F ig . 3 shows the K insman and A a r o n s o n I da t a on a , t oge the r with the c~ vs t e m p e r a t u r e c u r v e s c a l cu l a t ed f rom the Dubfi- Z e n e r equat ion with D = Do (the W a g n e r 5 a p p r o x i - mat ion) , the Dubf i -Zener equat ion with D = DoF' , and the p l a n a r Atk inson a n a l y s i s . In a l l c a s e s , the va lues of Co and X~ w e r e obta ined f rom Eqs . [10] and [24] of Ref . 21. Although the t h ree ca l cu l a t ed c u r v e s a r e qui te c l o s e toge the r a t the h igher r e a c t i o n t e m p e r a - t u r e s , they d ive rge i n c r e a s i n g l y as the t e m p e r a t u r e is d e c r e a s e d . C l e a r l y , s u c c e s s i v e i m p r o v e m e n t s in the m a t h e m a t i c s of the growth of a p l a n a r , d i s o r d e r e d boundary of inf ini te ex ten t p rov ide a p r o g r e s s i v e l y p o o r e r account ing of the e x p e r i m e n t a l m e a s u r e m e n t s on the th ickening k ine t i c s of g r a i n boundary f e r r i t e a l l o t r i o m o r p h s in Fe -0 .11 pc t C.

ANALYSIS OF THE GROWTH OF AN OBLATE ELLIPSOID 14

Adapta t ion to the Growth of F e r r i t e A U o t r i o m o r p h s

More than in the c a s e of the p l a n a r a n a l y s i s , i t i s i m p o r t a n t to p rov ide an a c c u r a t e in i t i a l e s t i m a t e of the r a t e cons tan t p r i o r to employ ing the n u m e r i c a l a n a l y s i s for the growth of an e l l i p so id under the condi t ion of D v a r y i n g with so lu te concen t r a t i on in the m a t r i x phase . The in i t i a l e s t i m a t e can be obtained f rom the work of Horvay and Cahn 22 on the growth of e l l i p s o i d s (and o ther shapes ) when D is independent of concen t r a t ion . T h e i r g e n e r a l equat ion for the growth of an e l l i p so id i s :

X2 2 Z2 _ _ + Y___Z____ + = 4Dot [15] a l + COo (12 + COo a3 + COo

w h e r e x, y, and z a r e C a r t e s i a n c o o r d i n a t e s , coo = r a t e cons t an t a t the s u r f a c e of the e l l i p so id and ai = o b l a t e n e s s p a r a m e t e r s . In the c a s e of an obla te

7 8 6 - V O L U M E 4, M A R C H 1973

e l l i p s o i d , x = y, aa = 0, and a~ = a2 = coo(1 - K~) / /~ , whe re K = the a s p e c t r a t i o , a quant i ty which m u s t be e x p e r i m e n t a l l y d e t e r m i n e d . The so lu t ion to th is e q u a - tion i s :

Co - CN L~o - X ~ - e~~176 (a + coo) ~ G{coo, a} [16]

In an unpubl ished r e p o r t , 2a Horvay and Cahn gave an e x p r e s s i o n for G{coo,a} when a/COo is s m a l l , i . e . , w h e n K is n e a r uni ty:

4a2 / 1 3 \ -] + --f~-~X-~-~o + "4-~o) J

- 24"~ e r f c ( f~o) 1 ~ - + ~ [16a]

and when a/coo is l a r g e (K < ~ 1/3):

G {coo,a} = r~ ea e r f e (-I-a)

- - 7 1 +

[ ( 3 / 2 ) + w o ] C~~ e -~~ 1 + 2 a a

[16b]

Rewr i t i ng Eq. [15] for the p a r t i c u l a r c a s e of an Oblate e l l i p so id , le t t ing R = ~ and f r o m the Appendix to a p a p e r by G o l d m a n e t al. 9 (wherein ~2 is used i n - s t ead of coo) showing that a + coo = wo /K 2,

R ~ Z 2 + - - - 1 [17]

(4woDot)/K z 4cooDot

F r o m the equat ion for an e l l i p s e , the s e m i m i n o r ax i s of the e l l i p s o i d i s :

zo = 2 ~ = otz t w2 [17a]

and the s e m i m a j o r ax is i s :

Ro = (2/ / / ) ~04--~--~nnnnnnnnnn~T = /3t ~/2 [175]

where ot z = 2 ~ and /3 = (2/K)q'cooDo. a z and /3 may thus be d e s c r i b e d as the p a r a b o l i c r a t e cons tan t s for the th ickening and the lengthening of an obla te e l l ipso id .*

*Eqs. [17a] and [17b] replace eqs. [4] and [5] and the line of equations im- mediately above eq. [4] in the Appendix to Ref. 9. Also, the second line above Eq. [62] in the Appendix should state that r~ = z2/4Dt.

The a n a l y s i s 14 for the growth of an e l l i p s o i d emp loys two r a t i o s not u t i l i zed in the p l a n a r a n a l y s i s . T h e s e r a t i o s , ne i t he r of which is an a v e r a g e , a r e de s igna t ed F r and FN-1. The f o r m e r is the r a t i o of the d i f fus iv i ty a t pos i t ion r f r o m the i n t e r f ace to that a t the i n t e r f a c e , i . e . , a t r = 0.* The l a t t e r i s the r a t i o of the d i f fus iv i ty

*As in the planar analysis, the concentration interval C o to C N is divided into N equal steps, wherein C = C - r5 c.

at pos i t ion r = N - 1 to tha t a t r = 0. F o r the d i f fu s iv - i ty of ca rbon in aus ten i t e :

F r = exp \ "R-T 30 ( C r - C o t

(5.5xfffl0S// (C~ - C ~ ) 1 [18]

r (Co - [ 1 8 a ] C r = Co N

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

Both the initial es t imate of /1/2 and the evaluation of I~v-1/2 a re apprec iab ly more complex than in the planar ana lys i s . As or iginal ly published, the equations for these quanti t ies included complicated in tegra ls . An approximate method of evaluating these in tegra ls will be provided here .

The re la t ionship for obtaining an initial es t imate of I~12 is: 14

1 1 1 2 _ N A 2 1 F ' ( l _ _ + 1 + 1 _ _ - - - 7 - - - ) [ 1 9 ] A3 2 4 at + coo a2 + COo a3 COo

(The f i r s t t e r m on the R.H.S. was inadvertent ly wri t ten as As/As in Ref. 14.)

A 2 =

1 + ~ - al + co a2 + co a3 co exp ( -w/F')dco ---7---

wo {(al + cot (as + co) (a3 + 09)} 112 [19a]

exp ( - w / F ' ) dw A3 = fo {(al + '-~ (--~ + w-)~aa + co)} 1/2 [19b]

Letting u = (co - coo)/F', a* = (al + coo)/F', a* = (a2 + coo)/F' and a* = (a3 + coo)/F',

A2 A2* - [20a]

A3 A~'

where :

0

,C1 1 1)} g -a~-t*+---u ~ + ~ e - U d u

[(a* + u) (ag + u) (a~ + u)] ~/2 [20b]

T e - U du A~ = J (a2 + u) (a* + u) ] o [(a* + U ) ~ - - - - 1 /2

[20c]

These in tegra ls can be evaluated by means of the G a u s s - L a g u e r r e quadra ture formula : 24

m

F{u} e - u du = ~ H m ' F { b m } [21] 0 ? n = l

In Eqs. [20a] and [20b], r e spec t ive ly ,

1 ( 1 1 1 ) l + g a* + bm + a* + bm + a* + bm

F{bm} [ (a* + bm) (a*2 + b m) (a~ + bin) ]112

Fibre } = 1 [(a* + bm) (a* + bm) (a* + bin)] 1/a

[22a]

[22b]

exp (-w / F N-112) dco [23b] L2 = 7 [ (al . . . . . . . 1/2

~N- , + co) (a~ + co) (a, + co)]

Letting u = (w -COn-I)/FN-112, a* = (al + colV~ a* = (aa + CON-1 ) /FN- l l S and a* = (a3 + OON~

L1 = 7 al + co

L_L = L~* L2 L~

where

L.,= f 0

L..= f 0

Table I l is ts the values of bm and H m f rom m = 1 to 15.

The equation for I~=112 is: 14

I~V-112 - L1 1 FN-1 L2 2 2

• + + [23] al + CON-I a2 + CON-I a3 + CON-I

a2 + co a3 co- exp(-co/FN-ln)

C~ l

1

[24a]

+ g + a* +-------u + e - U d u

[(a* + u) (a* + u) (a* + u) ]112 [24b]

e - u du [(a,* + u)(a* + u)(a* + u)] in [24c]

[(al + co)(a2+ co)(a3 + co)] *Is

Noting that the fo rms of Eqs . [23a] and [23b] a re the same as those of Eqs . [20a] and [20b], r e spec t ive ly , the G a u s s - L a g u e r r e substi tut ions of Eqs. [22a] and [22b] a re again applicable.

Because the i terat ion p rocedure for the el l ipsoidal ana lys is is both more compl ica ted and was more br ief ly desc r ibed in the original publication than that for the p lanar ana lys i s , the essen t ia l s teps in this p rocedure will be s u m m a r i z e d here .

1) Calculate the f i r s t value of wo f rom Eq. [16], the Horvay-Cahn re la t ionship.

2) With r sti l l equal to ze ro , obtain wl f rom:

[A2-29] cot§ = wr + Pr§ Ir+l/a

( " A 2 " r e f e r s to Ref. 14; " 2 9 " is the equation number in that re fe rence . )

3) Inc reas ing r by one, compute Ir§ f rom:

[A2-31] I t + l / 2 = I r_ l /2

- 1 - - - 7 - - - - + - - + - - az cor a2 + cor a3 + (d r

4) Recycle to Eq. [A2-29]. Continue until r = N - 1, i .e . , until J = N.

5) Compare IN-l /2 with/~-1/2. If the di f ference b e - tween them is g rea t e r than the des i red a c c u r a c y (10 -4 was used in the p resen t investigation), obtain an i m - proved es t imate of /1/2, (/~/2)1, f rom:

[A2-16] (I1/2)1 = Ii12 -- 0.5(IN-1/2 --/~-112)

and r ecyc l e to step no. 2. If this d i f ference is l ess than that requi red and the calculat ion being made of wo by the Atkinson el l ipsoidal analys is is the f i r s t , the new value of wo is obtained through rewri t ing Eq. [A2-32] a s :

Co - c N [25] co'~ = co(~ - g ( C o - z l )

where i = number of the calculat ion of too and coo ~/'1) was computed in step no. 1. When i > 2, coo is computed f r o m the re la t ionship obtained by applying Newton 's method to Eq. [A2-32], as desc r ibed in Eqs. [A2-17a, b, c]:

dco

[23a]

METALLURGICAL TRANSACTIONS VOLUME 4, MARCH 1973-787

, I , i , i , i , I , A

-~ A A A

o 3 A A - -

A

0 i I I I , I i l , I , 7 0 0 7 2 0 7 4 0 7 6 0 7 8 0 8 0 0 8 2 0

Reaction Tempero ture , *C Fig. 3 - - I n d i v i d u a l da t a po in t s r e p r e s e n t da t a of K i n s m a n and A a r o n s o n 1 on the p a r a b o l i c r a t e cons t an t , (~, f o r the t h i c k e n - ing of f e r r i t e a l l o t r i o m o r p h s in Fe -0 .11 pc t C. C u r v e s a r e c a l c u l a t e d f r o m the D u b 6 2 - Z e n e r 3 equa t ion , u s i n g the c o n s t a n t d i f f u s i v i t i e s noted, and f r o m the A t k i n s o n 13 p l a n a r a n a l y s i s .

(i+1) ( i+t) Co - CN (i+2) ( i§ Wo - oJ1 + N(Co - X I )

wo = w0 - [26] r i§ ) _ W(1 i )

1 - - -

(i+1) Oj(o i ) Uj 0

The ca lcula t ion is recyc led through step no. 2 unt i l the di f ference between t0o (~+~) and Wo (*) becomes less than the des i red value; in the p r e s e n t s tudy, 10-4wo (~) was se lec ted as the c r i t e r i o n for effective convergence , a z was then ca lcula ted f rom Eq. [17a]o

Table I. Coefficients for the Gauss-Laguerre Quadrature Formula (Eq. [21] )

M b m H m

1 0.093307812017 0.218234885940 2 0.492691740302 0.342210177923 3 1.215595412071 0.263027577942 4 2.269949526204 O. 126425818106 5 3.667622721751 0.402068649210 • 1 O- ] 6 5.425336627414 0.856387780361 X 10 -2 7 7.565916226613 0.121243614721 • 10 -2 8 10.120228568019 0.111674392344 • 10 -3 9 13.1:30282482176 0.645992676202 • 10 -5

10 16.654407708330 0.222631690710• 10 -6 11 20.776478899449 0.422743038498 • 10 -a 12 24.623894226729 0,392189726704 • 10 -l~ 13 31.407519169754 0.145651526407 • 10 -]2 14 38.530683306486 O. 148302705111 • 10 -]5 15 48.026085572686 0.160059490621 • 10 -]9

2 4 - -

22

2 0 - -

18 -

Ir 16 2

~c 14

12 o " I0

2 8

8 0 0 "C

tJ. S..', x 1(~4c,

�9 " I . O x 1()4Cl

K ".15

4 ~ ' / ( Z o )

0 I 0 I 2 3 4 5 6

(Growth T i m e ) l / Z s e c l / 2 Fig. 4- - -Typical p l o t s of a l l o t r i o m o r p h h a l f - l e n g t h , R 0, and of a l l o t r i o m o r p h h a l f - t h i c k n e s s , Z 0, a s a func t ion of the s q u a r e r o o t of the g rowth t i m e .

Fig. 5---Data obtained during this investigation for the parabolic rate constants for the thickening (~z) and the lengthening 03) of grain boundary ferrite allotriomorphs in Fe-0.11 pet C.

0 4

U O Il l

E r

N

10

9

8

7

6

5

4

:5

2

I

0

J i l l ,g9 0

I ' 1 ' 1 ' 1 ' 1 ' 1 ' 1

�9 - a z ( e x p t l , , p resen t invest igat ion )

~ - a z ( e x p t l . , K i n s m a n - Aaronson ) _

o - a z ( c o l c . , A tk inson e l l ipso id )

_ _ m

- - D D A t

.se D o k D 4 Horvay- Cohn m

6 9 0 z �9 F ' K "

- - k - 0 3 , 0 . ~ - , . , , ~ - . - - �9 Az A4 a ~ ' A t k i n l o n Planar

a

�9 1 , [ t I i I 1 1 , I , I , I , l 7 0 0 7 2 0 7 4 0 7 6 0 7 8 0 8 0 0 8 2 0 8 4 0 8 6 0

R e a c t i o n T e m p e r a t u r e , =C

Fig. 6 - - T r i a n g l e s and s q u a r e s a r e e x p e r i m e n t a l da ta po in t s on ot z f r o m t h i s i n v e s t i g a t i o n and f r o m tha t of K i n s m a n and A a r o n s o n , l r e s p e c t i v e l y . C i r c l e s a r e c a l c u l a t e d f r o m the A t k i n s o n 14 a n a l y s i s for an ob la t e e l l i p s o i d . C u r v e s a r e c a l c u - l a t e d f r o m the A tk inson 13 p l a n a r a n a l y s i s ; and wi th K = 0.3, f r o m the H o r v a y - C a h n ~2 a n a l y s i s wi th the c o n s t a n t d i f f u s i v i t i e s i n d i c a t e d and f r o m the A t k i n s o n 14 e l l i p s o i d a l a n a l y s i s .

788-VOLUME 4, MARCH 1973 METALLURGICAL TRANSACTIONS

New Expe r imen ta l Data on Lengthening and Thickening Kine t ics

These data were acqui red f rom motion p ic ture f i lms taken dur ing the o r ig ina l the rmion ic e m i s s i o n m i c r o - scopy s tudies of the thickening k ine t ics of f e r r i t e a l lo - t r i omorphs in Fe-0 .11 pct C.1 The r e a d e r is r e f e r r e d to Ref. 1 for the expe r imen ta l de ta i l s . The only d i f fe r - ence with r e spec t to the f i r s t se t of m e a s u r e m e n t s made on these f i lms is that only those a l lo t r iomorphs were m e a s u r e d for which lengthening as wel l as thickening data could be obtained. Typica l plots of half length, Ro, and of half th ickness , Zo, as a function of the square root of the growth t ime a re shown in Fig . 4. All of the (new) data on the parabol ic ra te cons tan ts for l eng then- ing (/3) and thickening (az) a re plotted as a funct ion of r eac t ion t e m p e r a t u r e in Fig . 5. The data on az a re replot ted on an en la rged v e r t i c a l sca le in Fig . 6, toge ther with those of K insman and Aaronson 1 on ot in the same al loy. The la t te r data a re seen to l ie within the outer l imi t s of the f o r m e r . Although the K i n s m a n - Aaronson data do tend to fall on the average , at s o m e - what higher va lues of a , the s ca t t e r involved is suf - f ic ient so that it s e ems fa i r to say that the two sets of data a re e s sen t i a l l y the same .

The sca t t e r in the va lues of az or a is seen to be comparab le in the two inves t iga t ions . Fig . 4 shows typical ly that this sca t t e r exceeds by far the unce r t a in ty in the individual va lues of the ra te cons tan ts . A detai led cons ide ra t ion of this p rob lem in Ref. 1 led to the con- c lus ion that the p redominan t source of the s ca t t e r is the p re sence of d is locat ion facets at the in te rphase boundar ies of the a l lo t r iomorphs in p ropor t ions which va ry with the angle between the habit plane of the nu - c leus and the plane of the gra in boundary . Accordingly , the sca t t e r , though ce r t a in ly es the t ica l ly not p leas ing , is thus both r ea l and unavoidable . However, these data a re s t i l l t r ea tab le , the p rob lem being conver ted f rom one of seeking a g r e e m e n t between the calcula ted and m e a s u r e d va lues of a z to a t tempt ing to demons t r a t e that ~z(calcula ted) is always equal to or g r ea t e r than otz(experimental) .

Resu l t s of the El l ipso ida l Ana lys i s and Discuss ion

As in the p lanar ana lys i s , f inal va lues of the r a t e cons tan ts a re obtained by ext rapola t ion to 1/ /N = O. Since a(exper imenta l ) / / f l (exper imenta l ) = k = a (theoretical)// /3(theoretical), i t follows that a ( e x p e r i - menta l ) / /a ( theore t ica l ) = ~(exper imental ) / /3( theoret ical ) . It is thus useful to r epo r t only one se t of ra te cons tan t s . Fo r the sake of compar i son with the p lanar ana lys i s , the ra te cons tant chosen was a z .

Fig . 7 shows typical plots of a z vs 1/ /N , extrapolated to 1 I N = O. These plots were found to be more a c c u r - ately l i nea r than those obtained f rom the p l ana r ana ly - s i s and to yield ext rapola ted va lues which did not differ as much f rom those calcula ted in the range N = 60-200.

In addit ion to the expe r imen ta l data p rev ious ly noted, Fig . 6 conta ins s eve ra l ca lcula ted curves of az vs r eac t ion t e m p e r a t u r e . These include curves f rom the Horvay-Cahn ana lys i s (Eqo [16])with D = Do (the Wagner 5 approximat ion) and with D = D o F ' , the ana lys i s for an oblate e l l ipsoid , and for the purpose of compa r i son , the p l ana r ana lys i s . Since the locat ion of the f i r s t three cu rves v a r i e s with K, a constant value (K = 0.3) was

METALLURGICAL TRANSACTIONS

chosen. In pa r t , this f igure is r e p r e s e n t a t i v e of many of the lower va lues of K. It was also found, however, that the Horvay-Cahn approximat ion of G {Wo, a} for l a r g e r va lues of K (Eq. [16b]), used in this study when 0.5 < K < 1, is insuff ic ient ly accura t e . In this range , az ca lcula ted f rom the Horvay-Cahn approx imat ion with D = Do yie lds va lues l e s s than those obtained f rom the Atkinson elUpsoidal a na l y s i s , a c l ea r ly imposs ib le r e s u l t in view of the i n c r e a s e in D with carbon content . However, as a check, Wo was de te rmined (at a l l va lues of K considered) f rom a n u m e r i c a l solut ion of Eq. [16] with a n u m e r i c a l evaluat ion of the in tegra l which Eqs . [16a] and [16b] approx imate . These r e su l t s agreed with those of the e l l ipso ida l ana lys i s pe r fo rmed under the condit ion of D independent of composi t ion. F ina l ly , the c i r c l e s in Fig. 6 a r e the va lues of a z ca lcula ted f rom the Atkinson e l l ipos ida l ana lys i s us ing expe r i - menta l ly de t e rmined va lues of K; these values a re indicated on the left hand side of the c i r c l e s . (The n u m - be r s on the r ight hand side of the c i r c l e s d is t inguish among d i f fe rent se ts of data obtained at the same r e - act ion t empera tu re . )

As in the case of the p lanar Atkinson ana lys i s , at a l l t e m p e r a t u r e s examined in the Fe-0 .11 pct C alloy the va lues of az ca lcula ted f rom the cons tan t -D ana lys i s (here that of Horvay and Cahn) with D = Do a re highest , those obtained f rom this ana lys i s with D = D o F ' a re in t e rmed ia t e and the r e su l t s of the Atkinson ana lys i s a re the lowest . The d i rec t ion of the di f ference between the two Horvay -Cahn-based r e su l t s is obviously to be expected. However, the di f ference between the Horvay - Cahn a z ' s with D = D o F ' and the Atkinson e l l ipso ida l va lues is not read i ly fo reseen ; it is a lso not obvious that this r e s u l t would be repeated in another al loy s y s - t em with a d i f fe rent dependence of D upon compos i t ion , even though D i n c r e a s e s with solute content in both.

Examina t ion of the locat ions of the c i rc le at 800~ (K = 0.15) and of c i rc le no. 1 at 760~ (K = 0.11) r e l a - tive to the curve calcula ted f rom the Atkinson p lanar ana ly s i s shows that as K approaches zero , az computed f rom the e l l ipso ida l ana lys i s approaches a obtained f rom the p l ana r ana lys i s . As Atkinson 14 has pointed out, this is a useful tes t of p r o g r a m m i n g ac - cu racy , s ince the two r e su l t s mus t become es sen t i a l ly ident ica l when K is suff ic ient ly sma l l . Compar i son of the curves for the Atkinson p lana r and e l l ipsoidal a n a l - yses shows that even when K is 0.3 the di f ference in the thickening k ine t ics of the two morphologies is not l a rge . The 770 ~ and 790~ r e su l t s indicate , however , that when K - 1//2 the difference is apprec iable ; and the 730~ point , ca lcula ted for K = 0.99, shows that as the spher ica l morphology is approached the d i f fe rence becomes quite l a rge .

Compar i son , in Fig . 6, of the individual expe r imen ta l va lues of a z obtained dur ing this inves t igat ion with the i r co r re spond ing va lues calculated f rom the e l l i p - soidal ana lys i s shows that in 13 out of 15 cases the ca lcula ted value is g r e a t e r than that measu red e x p e r i - menta l ly , i . e . , the calcula ted growth k ine t ics a re more rapid than those exper imen ta l ly observed . (For the sake of c l a r i t y , the di f ference between the ca lcula ted and m e a s u r e d va lues of az is shown as a function of r e - act ion t e mpe r a t u r e in Fig. 8.) This finding is quite adequately cons i s t en t with the or ig ina l expectat ion 1 noted in the In t roduct ion , i . e . , that a z should have e i ther the value pe rmi t t ed by volume d i f fus ion-con t ro l ,

VOLUME 4, MARCH 1973-789

4.32 l ' ' ' l ' ' ' ' l ' ' ' ' l ' l

4.50 - - 700 "~C

4.4e _

4 . 4 6

4.44

u E 4.4~'

VO 4,40 a~

,oo. 3.20

3"e[-I i i , [ , , l , l l , , , I , I I .oo, .o03 .o, .am

I / N

Fig. 7--Typical extrapolations of ~z vs 1/N to 1 /N = 0 for the Atkinson 14 ellipsoidal analysis.

or a l e s s e r value as a r e s u l t of i n t e r f e r ence with growth by d is loca t ion face ts . This r e s u l t r e p r e s e n t s a con- s ide rab le i m p r o v e m e n t with r e spec t to that obtained f rom the p lanar a n a l y s i s , where in only 8 of 15 e xpe r i - menta l va lues a re s m a l l e r than those ca lcula ted . It mus t thus be concluded that the aspec t ra t io of f e r r i t e a l lo t r iomorphs is often too high to just i fy use of the inf ini te plane a s sumpt ion . When the growth of the a l lo - t r i omorphs is analyzed on the bas i s of the Atkinson e l l ipso ida l ana lys i s , however , the t r a n s f o r m a t i o n me c h - a n i s m can be sa t i s f ac to r i ly de sc r ibed as one bas ica l ly control led by the volume diffusion of carbon in aus ten i te , but in te r fe red with to a modera te , though va r i ab le de - gree by the p r e s e n c e of a p ropor t ion of pa r t i a l ly co- he ren t facets along one of the broad faces . M e a s u r e - ments of the thickening k ine t ics of f e r r i t e s idepla tes demons t r a t e that such facets on proeutectoid f e r r i t e c r y s t a l s a re d isp laced by means of the ledge me c ha n - i s m , at r a t e s usua l ly s igni f icant ly less than those a l - lowed by volume d i f fus ion-cont ro l~ 25 The lack of a d i s - ce rn ib le pa t t e rn to the K data sugges t s , in view of the wel l known i r r e g u l a r i t y c h a r a c t e r i s t i c of the operat ion

�9 26 of ledge m e c h a m s m s , which is p a r t i c u l a r l y p r o - nounced in the case of proeutectoid ferrite,25 that the lengthening k ine t ics of f e r r i t e a l lo t r iomorphs a re s i m i l a r l y inf luenced.

The two data points f rom the p r e s e n t inves t iga t ion which fel l above the i r ca lcula ted va lues (indicated by a r rowheads in Fig . 6) should probably be asc r ibed to fa i lure of the aus teni te g ra in boundar ies involved to be nea r ly pe rpend icu la r to the plane of pol ish. A t i l t of approximate ly 45 deg would explain each of these d i s - c r epanc i e s . It should be emphas ized , however , that meta l lographic examina t ion has es tab l i shed that such devia t ions f rom pe rpend icu l a r i t y a re of quite inf requent

1 occu r r ence .

THE INTERFACIAL DIFFUSION-AIDED MECHANISM OF ALLOTRIOMORPHIC GROWTH

Although the or ig ina l e x p e r i m e n t a l data plotted in F ig . 6 tend to fal l at h igher va lues than the p r e se n t data , the lack of in fo rmat ion on K for the f o r m e r r e - su l t s and the c i r c u m s t a n c e that they do lie within the envelope of the p r e s e n t expe r imen ta l data indicate that they cannot be used as evidence for in t e r fac ia l d i f fus ion-

7 9 0 - V O L U M E 4 , M A R C H 1 9 7 3

---_. ~+4

~

0

x 0

<3 -I

V . N ~

OI ~

15 .3 o 5 o 2

- 0 2

- - o i

t A a = a c a l c - a e x p t l .

I I I I I ] I I 700 720 740 760

Tempera tu re , * C

e 2 O l �9

I I I I 780 8 0 0

Fig. 8--Differences between c~z (calculated) and (~z (experimental) as a function of reaction temperature, derived from Fig. 6. The numbers to the right of the data points at a given temperature correspond to data so identified in Fig. 6.

aided growth. The fa i lu re of such growth to occur can be explained on the bas i s of ana lyses which have been developed of this mechan i sm. 7'26'e7 In view of the s ca t t e r in the ra te cons tan t data, it is l ikely that detect ion of a cont r ibut ion to growth f rom in te r fac ia l diffusion would r equ i re that this mechan i sm produce va lues of c~z and /3 as la rge as those developed by volume d i f fus ion- cont ro l led growth. The magni tudes of the diffusivi ty of carbon along aus ten i te g ra in boundar i e s , D ~ ~, and along aus teni te : f e r r i t e boundar i e s , Dc~v, needed to obtain such ra te cons tants wil l now be calcula ted on this ba s i s . These ca lcula t ions wil l be under taken only at 770~ the midpoint in the t e mpe r a t u r e range of the expe r imen ta l data obtained dur ing this inves t iga t ion , s ince the sca t t e r in the ra te cons tant data p reven t s evaluat ion of the t empera tu re -dependence of the r e - quired boundary di f fus iv i t ies . The obviously crude but none the less n e c e s s a r y a s sumpt ion wil l have to be made in these ca lcula t ions that growth by the in te r fac ia l d i f fus ion -as s i s t ed me c ha n i sm on one hand and by vo l - ume diffusion d i rec t ly away f rom the a l lo t r iomorphs on the other proceed independent ly of each other .

An approximate re la t ionsh ip for /3, the parabol ic ra te constant for lengthening, is :26,27

fl : 2Do~ o~. ( C o { r } - C N ) [27]

r'(Co{r} - xl) [In - ~ + o.366] where X = grain boundary (or austenite: ferrite boundary) width, Co{r} = capillarity-corrected value of Co, D is here taken to be Do, R = radius of the ellipsoid at a given growth time, Xz = mole fraction of carbon in ferrite at the (~/((~ + V) phase boundary and r ' = radius of the allotriomorph adjacent to its junction with the grain boundary. The capillarity correction is made as outlined by Hillert 28 and Townsend and Kirkaldy. 29 From electron micrographs, 3~ r ' ~ 2 x 10 -5 cm; the energy of a disordered austenite: ferrite boundary was taken to be 750 erg//cm 2. Employing a mean value of /3 = 7 • 10 -4 c m / s I/2, D ~ = 5.6 • 10 -5 cm2//s. An approximate re la t ionsh ip for ~z is:7

4XDa a(C0 {r} - Co) [28] z = pk ~ (Co - X~)

METALLURGICAL TRANSACTIONS

where p = (1 - 4-272) and k = the lengthening ra te con- s tant computed on the bas i s of a t "/4 growth law. Using average va lues of k = 9.6 • 10 -4 c m / s ~/4 and a z = 2.5 • 10 -4 c m / s 1/2, Eq. [27] yields D~ v = 3.8 • 10 -4 cm2/s.

In a current critique of the literature on the growth kinetics of pearlite, Kirkaldy and Puls ~2 estimated the likely range of austenite:pearlite boundary diffusivities on the bas i s of diffusion k ine t ics in l iquids . These e s t i - mates indicate that at 770~ Db ~ 1-2 • 10 -6 cm2/s , l ikely appl icable to both D a n and Dory . Thus the boundary di f fus ivi t ies appear to be much too sm a l l to allow the in t e r rac ia l d i f fus ion -as s i s t ed growth mechan- i sm to make a detectable cont r ibut ion in the t e m p e r a - ture range employed. Kirkaldy and Pu l s concluded that i n t e r f ac i a l diffusion does not play a s igni f icant role in the growth of pea r l i t e at 700~ but does so at 644~ The rapidly dec reas ing in tens i ty of the rmionic e m i s s i o n and the swiftly i nc reas ing t r a n s f o r m a t i o n kinet ics with dec rea s ing t e m p e r a t u r e prevented any data on the growth k ine t ics of f e r r i t e a l l o t r i omorphs in Fe-0 .11 pct C f rom being obtained below 700~ and made such m e a s u r e m e n t s ve ry difficult below ca . 730~ It appea r s poss ib le , however, that in t e r rac ia l d i f fus ion-a ided growth of f e r r i t e a l lo t r iomorphs may be observed at lower reac t ion t e m p e r a t u r e s provided that a) another expe r imen ta l technique is used , b) high nuclea t ion r a t e s do not p reven t m e a s u r e m e n t s of lengthening k ine t ics and, c) the rapid evolut ion of secondary s idepla tes does not s e r ious ly inhibi t the gather ing of data on the k ine t - ics of thickening. The use of an F e - C - X alloy could a l lev ia te the la t te r two p r o b l e m s , but would r equ i r e that a genera l ly acceptable accounting be made of the inf luence of the al loying e l emen t upon Co and that the a l loying e l emen t not o therwise affect the growth p roces s ( e . g . , molybdenum and probably other s t rong c a r b i d e - fo rming al loying e l emen t s r e t a r d growth through a solute d r ag - l i ke mechanism31).

SUMMARY

This inves t iga t ion developed out of the e a r l i e r f ind- ing 1 that the parabol ic ra te cons tant , a , for the th ickening of 'g ra in boundary a l lo t r iomorphs of proeutectoid f e r r i t e in an Fe-0 .11 pct C alloy often exceeded that ca lcula ted on the a s sumpt ion of volume d i f fus ion-con t ro l . The calculated value of ot was obtained f rom the Dub~ 2- Z e n e r 3 equation on the a s sumpt ions that the broad faces of g ra in boundary a l lo t r iomorphs may be t rea ted as d i so rde red , p lanar boundar ies of inf ini te extent and that the operat ive diffusivi ty is that of carbon in aus ten i te at the carbon content of the A e 3 or Y/(a + Y) phase bound- a ry . In view of the p r e s e n c e of a sma l l , though va r i ab le propor t ion of d is locat ion facets a t one of the broad faces of a l l o t r i omorphs , K insman and Aaronson had a n t i c i - pated that the expe r imen ta l value of a would usua l ly be somewhat lower - -no t h igher - - than that ca lcula ted . A three pa r t approach was employed here in the s ea rch for the explanat ion for this anamalous r e su l t .

1) Reta in ing the infini te plane a s sumpt ion , a was reca lcu la ted by means of Atk inson ' s ~3 n u m e r i c a l a na l - ys i s of this growth geomet ry . This ana lys i s takes full account of the va r i a t ion with composi t ion of the d i f - fusivi ty in the ma t r ix phase . The incorpora t ion of in fo r - mat ion on the diffusivi ty of ca rbon in aus ten i te into the

Atkinson p lanar ana lys i s was desc r ibed and some aspec ts of computer p r o g r a m m i n g this ana lys i s were d i scussed . Calcula t ion of ot f rom this ana ly s i s was found, however , to i nc r ea se the d i sc repancy between the calcula ted and measu red va lues .

2) Assuming next that the aspect ra t io of f e r r i t e a l lo t r iomorphs is la rge enough so that these p r e c i p i - ta tes a re more accura t e ly t rea ted as oblate e l l ipso ids , data on the lengthening as well as the thickening k ine t - ics of f e r r i t e a l lo t r iomorphs were obtained f rom the motion p ic ture f i lms produced by means of the rmion ic e m i s s i o n mic roscopy dur ing the or ig ina l inves t iga t ion . Addit ional deta i ls of A tk inson ' s 14 n u m e r i c a l ana lys i s of the growth k ine t ics of oblate e l l ipsoids under the condit ion of a concen t ra t ion-dependen t diffusivity were p resen ted . This ana lys i s was then used to ca lcula te the k ine t ics of thickening (and of lengthening) of f e r r i t e a l l o t r i omorphs . Near ly al l of the caleula ted va lues of

now lay above those measu red expe r imen ta l ly , in much improved a g r e e m e n t with the or ig ina l expectat ion. It was concluded that, in the t empera tu re range studied in this and in the prev ious inves t iga t ion , 700 ~ to 810~ the growth k ine t ics of f e r r i t e a l lo t r iomorphs a re b a s i c - a l ly cont ro l led by the volume diffusion of carbon in aus ten i te , with a l imi ted amount of i n t e r f e r ence being offered by the p re sence of a sma l l p ropor t ion of p a r - t ia l ly coheren t facets at one of the broad faces .

3) Volume diffusion appears to be the p redominan t me c ha n i sm for the growth of g ra in boundary a l l o t r i o - morphs in subs t i tu t iona l al loys in which the m a t r i x is fee only at t e m p e r a t u r e s above c a . 0.9 T m.9'32 At lower t e m p e r a t u r e s , growth is control led by an i n t e r - facia l d i f fus ion-a ided m e c h a n i s m ] '9 Using the theory of this mechanism,7'26'27 the average d i f fus iv i t ies of carbon along aus teni te gra in boundar i e s , Do~a , and along aus teni te : f e r r i t e boundar i e s , D a y , needed to make this me c ha n i sm a s igni f icant con t r ibu tor to the growth of f e r r i t e a l lo t r iomorphs in the t e mpe r a t u r e range studied (centered at c a . 0.6 T i n ) were ca lcula ted . These d i f fus iv i t ies , Dac~ = 5.6 • 10 -5 cm2/s and D~7 -- 3.8 x 10 -~ cm2/s , a re subs tan t ia l ly above the boundary di f - fusivi ty of carbon , which Kirkaldy and Pu l s ~2 es t ima te to be c a . 10 -6 emZ/s in this t e mpe r a t u r e range . It was concluded that apprec iab ly lower reac t ion t e m p e r a t u r e s would be r equ i r ed in o rder to allow this mechan i sm to play an impor t an t ro le in the growth k ine t ics of f e r r i t e a l l o t r i omorphs .

REFERENCES

1. K. R. Kinsman and H. I. Aaronson: Transformation andHardenability in Steels, p. 39, Climax Molybdenum Co., Ann Arbor, Mich., 1967.

2. C. A. Dub& Ph.D. Thesis, Carnegie Institute of Technology, 1948. 3. C. Zener: J. Appl. Phys., 1949, vol. 20, p. 950. 4. C. Wells, W. Batz, and R. F. Mehl: A1ME Trans., 1950, vol. 188, p. 553. 5. C. Wagner: AIME Trans., 1952, vol. 194, p. 91. 6. H. I. Aaronson: Decomposition of Austenite by DiffusionalProeesses, p. 387,

Interscience Publishers, New York, 1962. 7. H. B. Aaron and H. I. Aaronson: AetaMet., 1968, vol. 16, p. 789. 8. H. I. Aaronson: The Mechanism of Phase Transformations in Metals, p. 270,

Institute of Metals, London, 1955. 9. J. Goldman, H. I. Aaronson, and H. B. Aaron: Met. Trans., 1970, vol. 1, p.

1805. 10. J. W. Cahn and W. C. Hagel: Decomposition of Austenite by Diffusional Pro-

cesses, p. 131, Interscience Publishers, New York, 1962. 11. G. F. Bolling and R. H. Richman: Met. Trans., 1970, vol. 1, p. 2095. 12. J. S. Kirkaldy and M. Puls: Met. Trans., in press. 13. C. Atkinson: AetaMet., 1968, vol. 16, p. 1019.

METALLURGICAL TRANSACTIONS VOLUME 4, MARCH 1973 791

14. C. Atkinson: Trans. TMS-AIME, 1969, vol. 245, p. 801. 15. G. R. Purdy, D. H. Weichert and J. S. Kirkaldy: Trans. TMS-AIME, 1963, vol.

227, p. 1255. 16. H. 1. Aaronson, H. A. Domian, and G. M. Pound: Trans. TMS-AIME, 1966,

vol. 236, p. 768. 171M. Hillert: The Mechanism of Phase Transformations in Crystalline Solids, p.

231, Institute of Metals, London, 1969. 18. L. Kaufman, S. V. Radcliffe, and M. Cohen: Decomposition of Austenite by

Diffusional Processes, p. 313, l nterscience Publishers, New York, 1962. 19. M. Abramowitz and 1. A. Stegun: Handbook of Mathematical Functions, p.

299, Dover, London, 1955. 20. J. R. Philip: Trans. Faraday Soc., 1955, vol. 51, p. 885. 21. H I. Aaronson, H. A. Domian, and G. M. Pound: Trans. TMS-AIME, 1966,

vol. 236, p. 753. 22. G. Horvay and J. W. Cahn: ActaMet., 1961, vol. 9, p. 695.

23. G. Horvay and J. W. Cahn: G. E. Research Report No. 60-RL-2561M, General Electric Research and Development Center, Schenectady, N.Y., 1960.

24. Z. Kopal: NumericalAnalysis, p. 564, John Wiley and Sons, New York, N.Y., 1955.

25. K. R. Kinsman, E. Eichen, and H. I. Aaronson: Ford Motor Co., Dearborn, Mich., unpublished research, 1972.

26. H. I. Aaronson, C. Laird and K. R. Kinsman: Phase Transformations, p. 313, A.S.M., Metals Park, Ohio, 1970.

27. A. D. Brailsford and H. B. Aaron: J. Appl. Phys., 1968, vol. 40, p. 1702. 28. M. Hillert: Jernkontorets Ann., 1957, vol. 141, p. 757. 29. R. D. Townsend and J. S. Kirkaldy: Trans. ASM, 1968, vol. 61, p. 605. 30. H. 1. Aaronson: Ph.D. Thesis, Carnegie Institute of Technology, 1954. 31. P. Boswell, K. R. Kinsman, and H. 1. Aaronson: Ford Motor Co., Dearborn,

Mich., unpublished research, 1972. 32. E. B. Hawbolt and L. C. Brown: Trans. TMS-A1ME, 1967, vol. 239, p. 1916;

E. B. Hawbolt, Ph.D. Thesis, Univ. of British Columbia, 1967.

7 9 2 - V O L U M E 4, MARCH 1973 M E T A L L U R G I C A L TRANSACTIONS

on the growth kinetics of grain boundary ferrite allotriomorphs

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