Forced velocity pearlite in high purity Fe-C alloys: Part II. Theoretical
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I.
INTRODUCTION
BUILDING on the work of Brandt ~and Zener, 2 Hillert3 has presented a fairly complete theoretical treatment of pearlite growth. Pearlite growth is very similar to eutectic growth, and the theoretical treatment of eutectic growth by Jackson and Hunt4 is essentially equivalent to Hillert's treatment and it will be used here to present the Zener-Hillert model for pearlite growth. Starting with the assumption of a flat transformation interface, Jackson and Hunt solve the diffusion problem and obtain an expression for the average composition in the 7 phase ahead of the a plates, X~/~, and ahead of the cementite plates, X rjcm. If one combines these expressions (their Eqs. [5a] and [5b]), the velocity of the pearlite front becomes
900
/Ac m u
~n ....
X~"'~
L
5
~
I
/ !
~'Tr
~
Lines )
\
o
~ •
600
-~Equ,d,ffusivity (xl08 cm2/sec
9 9
,oo-LI *~"
/
~.,,~l-
\~
~
X
~'.XX
5O0
where D is the diffusion coefficient of C in the 7 phase, f~ andf~m are the volume fractions of a and Cm, respectively, AX is the composition difference between a and Cm given by the phase diagram at the eutectoid temperature, P is a series term,
0
i I
i 2
i 3
i 4
i 5 ATOM
i 6
I 7
i 8
i
i
9
I0
i It
12
%C
Fig. 1 --Extension of A3 and A,m lines plus lines of equidiffusivity of carbon in au~tenite superimposed on Fe-C phase diagram9
:r
p = ~ sin2(nzrf~)/(nTr) 3, n=l
and S is the pearlite plate spacing. The quantity (X ~/~ - Xv/cm) is the average composition difference which produces the lateral diffusion between the eutectoid plates. Its value is estimated in the Appendix and it is shown as AXd on the phase diagram of Figure 1. The total undercooling at the growth front, ATr, produces a free energy per volume, AGr, which may be approximated as AS 9 ATr where AS is the entropy change per volume for the pearlite reaction. This total free energy is partitioned into a part used for diffusion, AGd, and a part used for generating a / c m interfaces, AG~. A minimum possible plate spacing, J.D. VERHOEVEN is a Professor in the Department of Materials Science and Engineering at Iowa State University, Ames, IA 50011. D. D. PEARSON is a Research Scientist with United Technologies Research Center, Silver Lane, East Hartford, CT 061089 This paper is based on a presentation made at the symposium "Establishment of Microstructural Spacing during Dendritic and Cooperative Growth" held at the annual meeting of the AIME in Atlanta, Georgia on March 7, 1983 under the joint sponsorship of the ASM-MSD Phase Transformations Committee and the TMS-AIME Solidification Committee.
METALLURGICALTRANSACTIONS A
Sin, is defined as the spacing when all free energy is used to generate interfaces, so that a free energy balance gives AGi = A G r ( S m / S ) and one then obtains ATr = ATd + ATT(Sm/S )
[2]
where ATd is the undercooling associated with the free energy consumed by the diffusion process, AGd = AS 9 ATd. If one assumes the extrapolated phase boundary lines A3, and A,m have constant slopes, ms and m,m, then the composit
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