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Diffusion of carbon during the formation of low-carbon martensite
Scripta
METALLURGICA
Vol. 17, pp. 1 2 8 5 - 1 2 8 8 , 1983 P r i n t e d in the U . S . A .
P e r g a m o n P r e s s Ltd. All rights reserved
DIFFUSION OF CARBON DURING THE FORMATION OF LOW-CARBON MARTENSITE T. Y. Hsu (Xu Zuyao) and Li Xuemin Department of Materials Science and Engineering Shanghai Jiao Tong University, Shanghai, China ( R e c e i v e d M a y 18, 1 9 8 3 ) ( R e v i s e d A u g u s t 29, 1 9 8 3 )
Introduction Rao and Thomas (1) found from high resolution TEM that in quenched steel with low or medium carbon the retained austenite is trapped between the non-twin related lath martensite. In addition, the lattice image indicated substantial carbon enrichment of the retained austenite (from 0.27% to 0.4-1.04%) and the presence of interfacial ledges of several unit cells at lath martensite/austenite interphase boundary. Recent work of Thomas and Sarikaya (2) has confirmed carbon partitioning during the formation of lath martensite by field-ion microscopy-atom probe studies. Consequently, they suggested that it may be more appropriate to describe lowcarbon martensite in terms of bainite. Our previous work (3) has pointed out that the diffusion of carbon atoms may occur concomitantly and keep pace with the formation of lath martensite, however, the formation mechanism of lath martensite is not identical with that of bainite by comparison of the morphology of the a/y interphase boundary in lath martensite and upper bainite in a 0.12 C-low-Ni-Cr steel and of the thermodynamical driving force between them. The present work attempts to verify the above conclusion through the calculation of the time required for carbon diffusion during the formation of the lath martensite. Diffusion in the case of uniform initial distribution and surface concentration constant. The initial distribution of carbon both in martensite (M) and retained austenite (y) is uniform during quenching and carbon will diffuse from M to y owing to the different solubility in each phase. According to the results of the work by Thomas et al. (1,2), the carbon concentration profile can be shown as Fig. i. In case of diffusion in a plane sheet with uniform initial distribution and surface concentration being equal, ie: C = CO C CI
as t = O, t = O,
-i < x < 1 x = 1
the solution of Fick's law becomes (4): C -C O
4
Cl_C 0
~
(-I) n
i - ~ n~O ~n 7 1
e
-D(2n+l)2~2t/4~ 2 cos (2n+l)~x 2
(I)
I f Mt d e n o t e s t h e t o t a l amount o f d i f f u s i n g s u b s t a n c e w h i c h h a s e n t e r e d and M~ c o r r e s p o n d i n g q u a n t i t y a f t e r i n f i n i t e time, then Mt K
8
= I - n~O
(2n+I)2~ 2
the sheet at time t,
-D(2n+l)2~2t/4~ 2
(2)
e
The corresponding solutions useful for small times are C-C O CI-C0
® n~ O (_l)nerfc
(2n+l)~-x 2~(Dt) + n~ O (-i) n erfc
(2n+l)~+x 2/(Dt)
(3)
and
Mt K
Dt ½ n£ = 2 (7) {~-½ + 2 n~ I (-i) n i e r f c T - ~
(4)
}
1285 0036-9748/83 $ 3 . 0 0 + .00 Copyright (c) 1 9 8 3 P e r g a m o n Press
Ltd.
1286
DIFFUSION
OF
CARBON
t2 ) tt ) 0
C%,
Vol.
17,
t4 ) t3) 0 t't 4
1.04
M t=O
0.27 0.22
t--tz
I
I
X
Fig. i
From the graph of
C - CO
Sketch of carbon concentration profile in martensite and retained austenite.
V
x S
~
as
,
C1 - CO C - CO CI _ CO
i (97%),
X ~
= O,
we get Dt £2
- -
=
(5)
1.5
Following
the data from Thomas et al. (1,2), in the case of martensite,
2£ 1 = 15 x 10 -6 cm, T = 700K. Let D = D~ c substitute
=
0.02 exp (
) cm2/s
(5) and
in equ. (5); we thus obtain:
t
1.5 £2 D
1.5 £~ D~ I c
In case of austenite,
let D = D~c
and take C = Substituting t
-20100 RT
= 7.25 x 10 -3 sec.
take 2 £2 = I x 10 -6 cm, T = 700K.
.-31400. = (0.06 + 0.07% C) exp t ~ )
CO + CI 2
= 0.27 + 1.04 2
these in equ. 1.5 Z2
D
(5), we get:
1.5£~ =
D~
=
2
x
10 -2 sec
em2/s,
(5)
take
No.
ii
Vol.
17,
No.
ii
DIFFUSION
OF
CARBON
1287
Carbon diffusion to interphase boundary dislocation during the srowth of martensite According to the growth model of martensite suggested by Owen et al. (6) carbon diffuses by means of boundary dislocations during the formation of lath martensite. Utilizing the equation given by Cottrell (7):
N (t)
N =
ADt (K--T-)
3 (2) 1/3 o
2/3 (6)
in which N (t) is the number of atoms on a unit length of dislocation (in no. atoms/cm), N o -initial concentration (in no. atoms/cm3). A -- a constant; we may take A = 3 x 10 -25 N cm2-(8). K -- Boltzmann constant = 1.38 x 10 -23 J/K, D = D~, t -- diffusion time. Let p be the dislocation density; we have: t = (N(t)p)3/2
1
3/2
1/2 (7) AD~
Take T = 700K,
Note
N (t) p NO
p
= CI
1012/cm 2, 1.04 0.27
D~e
=
0.02 exp (-20100.~) cm2/s.
in which C O is the carbon content of the initial structure,
CO
and C I is that of the enriched austenite. Substituting in equ. (7), we get the time required for diffusion of carbon to enrich the boundary to 1.04% C, t = 3.2 x 10 -4 sec. In extreme cases, i.e. when the dislocations distributed in martensite are beneficial to carbon diffusion, the activation energy of diffusion is assumed to be 1/2 of that in the normal condition, i.e. D c = 0.02 exp (- 10050 R-----~) then, the time required for diffusion t = 2.45 x 10 -7 sec. The srowth rate of martensite as it STOWS analo$ous to banite Assume that the growth of low-carbon martensite is analogous to that of incompletely transformed upper bainite, i.e. y (0.27% C) ÷ a + Y' Following the results given by Thomas et al. (1,2), the width of a phase d~ = 15 x 10 -6 cm, the width of interlath retained austenite dy = i x 10 -6 cm. As concentration of y' is 1.04, 0.4 1/2 (1.04 + 0.4) = 0.665%, the corresponding concentration of ~ will be 0.2187, 0.261 and 0.24 respectively. Let C Y/~ and C e/Y be the equilibrium concentration at the interphase boundary of y and a respectively. The growth rate of ~ phase is given as: dX
D Y (C Y / ~ c
_ C0 )
G = d-~ = £ (cYl n _ ca/y ) in which
Z
is the diffusion range in nonuniform y and assume £ = 50 nm.
(8) Taking D Y = 0.06
CO cYla c + 0.07 (. + , cm2/s, we get the growth rate of ~ phase G = 3.6 x 10 -4 , 2 ) exp (,-31400 ~ ) 2.84 x 10 -4 and 3.16 x 10 -4 cm/s as the carbon concentration of y' is 1.04, 0.4 and 0.665% respectively. Discussion The time required for carbon diffusion to enrich the retained austenite during the formation of low-carbon martensite is 7 x 10 -3 -- 3 x 10 -4 sec as calculated. From the available experimental data of the growth rate of low-carbon martensite (9-12), i.e. as high as 7 x 103 -- 4 x 104 em/s (9), i0 cm/s (i0) and as low as 1 x 10 -2 cm/s (ii), we take the growth rate I0 -- 10 -2 cm/s noting the width of lath Ca 15 x 10 -6 cm. Then, the time of the formation of a martensite lath will be 10 -3 -- 10 -6 sec. In comparison with the experimental
1288
DIFFUSION
OF
CARBON
Vol.
17,
No.
Ii
data, the result of the calculation implies that carbon diffusion may keep pace with, or slightly lag behind, the formation of lath martensite, The calculated time required for equalization of enriched retained austenite is 2 x 10 -2 sec, at least one order of magnitude lower than the formation of lath martenslte; this implies that carbon diffusion may occur, however, it is not a main or required process in lath martensite transformation. Suppose that the growth of lath martensite is analogous to that of upper balnite, we can obtain the growth rate of lath martensite as only 3 x 10 -4 cm/s, at least two orders of magnitude lower than the available experimental data. The present work empasizes that, contrary to the suggestion of Thomas et al., the formation mechanism of low-carbon martensite differs from that of upper bainite although both reactions may involve the diffusion of carbon. Acknowledgement The authors are in debt to Professor H. I. Aaronson for his beneficial discussion. References i.
B.V.
Rao and G. Thomas, Proc. ICOMAT-79, p.12 Massachusetts Institute of Technology,
2.
G. Thomas and M. Sarikaya, Proc. Inter. Conf. on Solid-Solid Phase Transformations, 1981, p. 999, ed. H. I. Aaronson, D. E.-Laughlin, R. F. Sekerka and C. M. Wayman, TMS-AIME (1982). T. Y. Hsu (Xu Zuyao) and Li Xuemin, Acta Metall, Sinica, 19, A83 (1983). J. Crank, The Mathematics of Diffusion, p.45-46, Oxford University Press, (1956). C. Wells and R. F. Mehl, Trans. AIME, 140, 279 (1940). W. S. Owen, F. J. Sehoen and G. R. Srinivasan. Phase Transformations, p. 157, AMS. (1970). A. H. Cottrell, Dislocations and Plastic Flow in Crystals, p. 149, Oxford Press, (1953). ibid, p. 134; 147-149. J. M. Marder and A. R. Marder, Trans, ASM., 62, i (1969). T. Honma, Electric Furnace Steel, Japan, 29, 261 (1958). R. B. G. Yeo, Trans, ASM., 57, 48 (1964). T. Y. Hsu, (Xu Zuyao), Martensitic Transformation and Martensite, p. 328-329. Science Press, Beijir~(1980, 1981).
(1979). 3. 4. 5. 6. 7. 8. 9. i0. ii. 12.
METALLURGICA
Vol. 17, pp. 1 2 8 5 - 1 2 8 8 , 1983 P r i n t e d in the U . S . A .
P e r g a m o n P r e s s Ltd. All rights reserved
DIFFUSION OF CARBON DURING THE FORMATION OF LOW-CARBON MARTENSITE T. Y. Hsu (Xu Zuyao) and Li Xuemin Department of Materials Science and Engineering Shanghai Jiao Tong University, Shanghai, China ( R e c e i v e d M a y 18, 1 9 8 3 ) ( R e v i s e d A u g u s t 29, 1 9 8 3 )
Introduction Rao and Thomas (1) found from high resolution TEM that in quenched steel with low or medium carbon the retained austenite is trapped between the non-twin related lath martensite. In addition, the lattice image indicated substantial carbon enrichment of the retained austenite (from 0.27% to 0.4-1.04%) and the presence of interfacial ledges of several unit cells at lath martensite/austenite interphase boundary. Recent work of Thomas and Sarikaya (2) has confirmed carbon partitioning during the formation of lath martensite by field-ion microscopy-atom probe studies. Consequently, they suggested that it may be more appropriate to describe lowcarbon martensite in terms of bainite. Our previous work (3) has pointed out that the diffusion of carbon atoms may occur concomitantly and keep pace with the formation of lath martensite, however, the formation mechanism of lath martensite is not identical with that of bainite by comparison of the morphology of the a/y interphase boundary in lath martensite and upper bainite in a 0.12 C-low-Ni-Cr steel and of the thermodynamical driving force between them. The present work attempts to verify the above conclusion through the calculation of the time required for carbon diffusion during the formation of the lath martensite. Diffusion in the case of uniform initial distribution and surface concentration constant. The initial distribution of carbon both in martensite (M) and retained austenite (y) is uniform during quenching and carbon will diffuse from M to y owing to the different solubility in each phase. According to the results of the work by Thomas et al. (1,2), the carbon concentration profile can be shown as Fig. i. In case of diffusion in a plane sheet with uniform initial distribution and surface concentration being equal, ie: C = CO C CI
as t = O, t = O,
-i < x < 1 x = 1
the solution of Fick's law becomes (4): C -C O
4
Cl_C 0
~
(-I) n
i - ~ n~O ~n 7 1
e
-D(2n+l)2~2t/4~ 2 cos (2n+l)~x 2
(I)
I f Mt d e n o t e s t h e t o t a l amount o f d i f f u s i n g s u b s t a n c e w h i c h h a s e n t e r e d and M~ c o r r e s p o n d i n g q u a n t i t y a f t e r i n f i n i t e time, then Mt K
8
= I - n~O
(2n+I)2~ 2
the sheet at time t,
-D(2n+l)2~2t/4~ 2
(2)
e
The corresponding solutions useful for small times are C-C O CI-C0
® n~ O (_l)nerfc
(2n+l)~-x 2~(Dt) + n~ O (-i) n erfc
(2n+l)~+x 2/(Dt)
(3)
and
Mt K
Dt ½ n£ = 2 (7) {~-½ + 2 n~ I (-i) n i e r f c T - ~
(4)
}
1285 0036-9748/83 $ 3 . 0 0 + .00 Copyright (c) 1 9 8 3 P e r g a m o n Press
Ltd.
1286
DIFFUSION
OF
CARBON
t2 ) tt ) 0
C%,
Vol.
17,
t4 ) t3) 0 t't 4
1.04
M t=O
0.27 0.22
t--tz
I
I
X
Fig. i
From the graph of
C - CO
Sketch of carbon concentration profile in martensite and retained austenite.
V
x S
~
as
,
C1 - CO C - CO CI _ CO
i (97%),
X ~
= O,
we get Dt £2
- -
=
(5)
1.5
Following
the data from Thomas et al. (1,2), in the case of martensite,
2£ 1 = 15 x 10 -6 cm, T = 700K. Let D = D~ c substitute
=
0.02 exp (
) cm2/s
(5) and
in equ. (5); we thus obtain:
t
1.5 £2 D
1.5 £~ D~ I c
In case of austenite,
let D = D~c
and take C = Substituting t
-20100 RT
= 7.25 x 10 -3 sec.
take 2 £2 = I x 10 -6 cm, T = 700K.
.-31400. = (0.06 + 0.07% C) exp t ~ )
CO + CI 2
= 0.27 + 1.04 2
these in equ. 1.5 Z2
D
(5), we get:
1.5£~ =
D~
=
2
x
10 -2 sec
em2/s,
(5)
take
No.
ii
Vol.
17,
No.
ii
DIFFUSION
OF
CARBON
1287
Carbon diffusion to interphase boundary dislocation during the srowth of martensite According to the growth model of martensite suggested by Owen et al. (6) carbon diffuses by means of boundary dislocations during the formation of lath martensite. Utilizing the equation given by Cottrell (7):
N (t)
N =
ADt (K--T-)
3 (2) 1/3 o
2/3 (6)
in which N (t) is the number of atoms on a unit length of dislocation (in no. atoms/cm), N o -initial concentration (in no. atoms/cm3). A -- a constant; we may take A = 3 x 10 -25 N cm2-(8). K -- Boltzmann constant = 1.38 x 10 -23 J/K, D = D~, t -- diffusion time. Let p be the dislocation density; we have: t = (N(t)p)3/2
1
3/2
1/2 (7) AD~
Take T = 700K,
Note
N (t) p NO
p
= CI
1012/cm 2, 1.04 0.27
D~e
=
0.02 exp (-20100.~) cm2/s.
in which C O is the carbon content of the initial structure,
CO
and C I is that of the enriched austenite. Substituting in equ. (7), we get the time required for diffusion of carbon to enrich the boundary to 1.04% C, t = 3.2 x 10 -4 sec. In extreme cases, i.e. when the dislocations distributed in martensite are beneficial to carbon diffusion, the activation energy of diffusion is assumed to be 1/2 of that in the normal condition, i.e. D c = 0.02 exp (- 10050 R-----~) then, the time required for diffusion t = 2.45 x 10 -7 sec. The srowth rate of martensite as it STOWS analo$ous to banite Assume that the growth of low-carbon martensite is analogous to that of incompletely transformed upper bainite, i.e. y (0.27% C) ÷ a + Y' Following the results given by Thomas et al. (1,2), the width of a phase d~ = 15 x 10 -6 cm, the width of interlath retained austenite dy = i x 10 -6 cm. As concentration of y' is 1.04, 0.4 1/2 (1.04 + 0.4) = 0.665%, the corresponding concentration of ~ will be 0.2187, 0.261 and 0.24 respectively. Let C Y/~ and C e/Y be the equilibrium concentration at the interphase boundary of y and a respectively. The growth rate of ~ phase is given as: dX
D Y (C Y / ~ c
_ C0 )
G = d-~ = £ (cYl n _ ca/y ) in which
Z
is the diffusion range in nonuniform y and assume £ = 50 nm.
(8) Taking D Y = 0.06
CO cYla c + 0.07 (. + , cm2/s, we get the growth rate of ~ phase G = 3.6 x 10 -4 , 2 ) exp (,-31400 ~ ) 2.84 x 10 -4 and 3.16 x 10 -4 cm/s as the carbon concentration of y' is 1.04, 0.4 and 0.665% respectively. Discussion The time required for carbon diffusion to enrich the retained austenite during the formation of low-carbon martensite is 7 x 10 -3 -- 3 x 10 -4 sec as calculated. From the available experimental data of the growth rate of low-carbon martensite (9-12), i.e. as high as 7 x 103 -- 4 x 104 em/s (9), i0 cm/s (i0) and as low as 1 x 10 -2 cm/s (ii), we take the growth rate I0 -- 10 -2 cm/s noting the width of lath Ca 15 x 10 -6 cm. Then, the time of the formation of a martensite lath will be 10 -3 -- 10 -6 sec. In comparison with the experimental
1288
DIFFUSION
OF
CARBON
Vol.
17,
No.
Ii
data, the result of the calculation implies that carbon diffusion may keep pace with, or slightly lag behind, the formation of lath martensite, The calculated time required for equalization of enriched retained austenite is 2 x 10 -2 sec, at least one order of magnitude lower than the formation of lath martenslte; this implies that carbon diffusion may occur, however, it is not a main or required process in lath martensite transformation. Suppose that the growth of lath martensite is analogous to that of upper balnite, we can obtain the growth rate of lath martensite as only 3 x 10 -4 cm/s, at least two orders of magnitude lower than the available experimental data. The present work empasizes that, contrary to the suggestion of Thomas et al., the formation mechanism of low-carbon martensite differs from that of upper bainite although both reactions may involve the diffusion of carbon. Acknowledgement The authors are in debt to Professor H. I. Aaronson for his beneficial discussion. References i.
B.V.
Rao and G. Thomas, Proc. ICOMAT-79, p.12 Massachusetts Institute of Technology,
2.
G. Thomas and M. Sarikaya, Proc. Inter. Conf. on Solid-Solid Phase Transformations, 1981, p. 999, ed. H. I. Aaronson, D. E.-Laughlin, R. F. Sekerka and C. M. Wayman, TMS-AIME (1982). T. Y. Hsu (Xu Zuyao) and Li Xuemin, Acta Metall, Sinica, 19, A83 (1983). J. Crank, The Mathematics of Diffusion, p.45-46, Oxford University Press, (1956). C. Wells and R. F. Mehl, Trans. AIME, 140, 279 (1940). W. S. Owen, F. J. Sehoen and G. R. Srinivasan. Phase Transformations, p. 157, AMS. (1970). A. H. Cottrell, Dislocations and Plastic Flow in Crystals, p. 149, Oxford Press, (1953). ibid, p. 134; 147-149. J. M. Marder and A. R. Marder, Trans, ASM., 62, i (1969). T. Honma, Electric Furnace Steel, Japan, 29, 261 (1958). R. B. G. Yeo, Trans, ASM., 57, 48 (1964). T. Y. Hsu, (Xu Zuyao), Martensitic Transformation and Martensite, p. 328-329. Science Press, Beijir~(1980, 1981).
(1979). 3. 4. 5. 6. 7. 8. 9. i0. ii. 12.