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A method for obtaining quantitative dilatometric data from alloys undergoing a phase transformation
229
Metallography
A M e t h o d for O b t a i n i n g Q u a n t i t a t i v e D i l a t o m e t r i c Data from Alloys U n d e r g o i n g a Phase T r a n s f o r m a t i o n
G. L. FISHER AND R. H. GEILS" The International Nickel Company, Inc., Paul D. Merica Research Laboratory, Sterling Forest, Suffern, New York
Since most metals and alloys undergo a dimensional change during a phase transformation, dilatometry has long been used to determine the critical temperatures for these reactions. T h e technique has found wide use for determining the transformation temperatures in steels during heating and cooling, and during isothermal transformation. Dilatometry is not limited to determining the reaction temperatures. Using a technique described below, one can quantitatively determine the amount of transformed phase formed as a function of temperature. T h e derivation and an example of the method follow. Assume that a metal or alloy undergoes a transformation from a high-temperature phase (y) to a lower-temperature phase (c,) over a range of temperatures during cooling. Assume further that both phases have the cubic crystal system. A schematic dilatometer curve for the transformation appears in Fig. I.
I Z
s~ d i s/SS
To I"IG.
1.
TEM~RATURE
Schematic of a length-temperature curve obtained from a dilatometer.
" Graduate student at the University of Illinois, Urbana, Illinois. :ldetallography, 3 (1970) 229-233
Copyright @ 1970 by American Elsevier Publishing Company, Inc.
230
G. L . Fisher a n d R . H . Geils
T h e specimen volume, V , , at some temperature, T, where both phases are present, is given by Vs = n,~a~ 3 ~- n ~ a ~ ~
(1)
where nu ~ = number of unit cells of ~. nu ~ = n u m b e r of unit cells of 7. a~ and a, are the lattice parameters of the two phases at temperature T. Consider that the specimen is cylindrical in shape, and that it is isotropie. Therefore its shape remains unchanged during the transformation, so that r = rol/lo, where r 0 and l0 are the radius and length of the specimen at some reference temperature T O , r and l are the radius and length at temperature T.
where A 0 is the cross-sectional area of the specimen at T O . In general, n~a" = A o ( ! ] 2 l
(3)
and nu - -
nA
(4)
where nA = total number of atoms in the alloy. = n u m b e r of atoms per unit cell in the phase. Then a" - -
A°q!/°)2l " a
(5)
nA
Assuming the rod to be composed entirely of ~, we have a~~ =
Ao~(l~/lo~)2l~ " c~ nA
Similarly, assuming the rod to be composed entirely of 7, we have ,, 3 =
3o,(Uto,)
~l, . ~
nA
Since the shape of the specimen is unchanged, roy = roa
(6)
Quantitative Dilatometric Data
231
and
l' Io,
'i '
so that a,f
Ao~(UZo,)~z, .,,
-
(7)
nA
Substituting equations 2, 4, 6, and 7 into equation I to obtain the volume of the specimen at any temperature within the range of the transformation, we have
\10!
O'c~
//A
O'v
/'/A
where na= and nay are the total number of atoms on c~ or Y crystal lattice sites, respectively. But at the reference temperature T o , Ao
Ao~
and
I.
10~
T h e n equation 8 reduces to l~3 = N~l~ 3 +
(1
-
-
N,) l,,3
where N , and N~ are the fractions of" the total atoms on c~or Y crystal lattice sites
l~a -- lva Since 1 . ~ - - l . ~ l and l ~ - - l . ~ at any temperature T, X~ ,-~
[(l' [(l~ -
(10)
1, and since ( A l - k / ) a - - I a ~ 3 F A l ,
l,) + Z] ~ -- U iv) + L,p - U
.,~ --
l~ l, -
l., t,,
then
(~ l)
T h u s one can obtain the atom fraction of the c~ or Y phase at any temperature by applying a "lever rule" to the dilatometer curve. It is often desirable to obtain the volume fraction, 1~, of a phase from the dilatometer curve: ~ffa _.=
nA~,
n.4,~ ~ na,,(,~l%)(a,ja~)a
(12)
Knowing the lattice parameters and the number of atoms per unit cell for the two phases, we can plot atom fraction versus volume fraction. This can then be used to determine the volume fraction of either phase. One of the most common alloy systems studied by dilatometry is the iron carbon system. In a hypoeutectoid plain carbon steel, the high-temperature
232
G. L. Fisher and R. H. Geils
f.c.c, austenite transforms to b.c.c, ferrite and pearlite in the temperature range 600°C to 800°C. If the carbon level is not too high ( 4 0 . 4 0 weight °/o), the predominant phase is ferrite. For the case of austenite transforming to ferrite % / ~ ~ 0.5 and (ao~/ao~) 3 ~ 1.986 [a0~(900°C) = 3.639 (ref. 1) and a0~(800°C) ~ 2.894 (ref. 1)]. Therefore equation 12, which gives the volume fraction of ferrite, Vf ~, reduces to •
Vf~ =
nA~
=N~--
I7 /l~. , -/- ~
(13)
nA~ @ nA.~
T h u s the volume fraction of ferrite can be determined directly from the dilatometer curve. Equation 13 can also be used for the transformation of austenite to martensite (b.c.t.) in hypoeutectoid steels, since the c/a ratio is nearly equal to 1.0 (1.036 for a 0.8 weight O//ocarbon steel; ref. 2) and the change in (ao~/ao~) ~ is slight over the temperature range 20°C to 800°C (ref. 3). T h e procedure for obtaining quantitative phase transformation data from a dilatometer is quite simple. Temperature intervals of 50C ° are marked on the dilatometer cooling curve. When using a Bollenrath dilatometer this can be done by momentarily increasing the intensity of the light spot tracing the curve. T o obtain the volume fraction of ferrite as a function of temperature, the straightline portions of the austenite and ferrite curves are first extended, as shown schematically in Fig. 1 by the dashed lines. Lines parallel to the length axis are then drawn through each of the temperature points and also at 10C ° intervals. At each temperature interval measurements are made of the distance between the austenite and ferrite lines (lr~ -- l ~ in Fig. 1) and the specimen curve and the austenite line (l~ -- l ~ in Fig. l). T h e volume fraction of ferrite is then calculated from equation 13.
STEEL COMPOSITION IN WEIGHT P E R C E N T 0.023C 0.9ON/ 1 2 0 C u 0 4 6 M n O 30Si ioc
u_
~ z
uJ
8o
6o
40
20
>~ 800
775
750
725 700 675 650 625 6 0 0 575 TEMPERATURE- DEGREES CENTIGRADE
F I G . 2. E f f e c t o f c o o l i n g r a t e o n t h e 7 - a t r a n s f o r m a t i o n austenitized at 1068°C for one hour.
550
525
500
in a nickel-copper
steel
Quantitative Dilatometric Data
233
An example of the use of this technique is shown in Fig. 2. T h e effect of cooling rate on the austenite-to-ferrite transformation in a low-carbon nickel-copper steel was studied. At the slow cooling rate the transformation curve is typical of equiaxed ferrite. At the faster cooling rate the transformation is extended over a much wider temperature range. T h e r e is an inflection in the curve at about 600°C due to the initiation of a bainitic ferrite transformation.
I. \V. B. Pearson, .4 Handbook of Lattice Spacings arid Structures ~/" Metals am/ .I//oys, Pergamon Press, New York, 1958, pp. 626-627. 2. Ibid., p. 923. 3. Ibid., p. 631.
_4ccepted February 14, 1970
Metallography
A M e t h o d for O b t a i n i n g Q u a n t i t a t i v e D i l a t o m e t r i c Data from Alloys U n d e r g o i n g a Phase T r a n s f o r m a t i o n
G. L. FISHER AND R. H. GEILS" The International Nickel Company, Inc., Paul D. Merica Research Laboratory, Sterling Forest, Suffern, New York
Since most metals and alloys undergo a dimensional change during a phase transformation, dilatometry has long been used to determine the critical temperatures for these reactions. T h e technique has found wide use for determining the transformation temperatures in steels during heating and cooling, and during isothermal transformation. Dilatometry is not limited to determining the reaction temperatures. Using a technique described below, one can quantitatively determine the amount of transformed phase formed as a function of temperature. T h e derivation and an example of the method follow. Assume that a metal or alloy undergoes a transformation from a high-temperature phase (y) to a lower-temperature phase (c,) over a range of temperatures during cooling. Assume further that both phases have the cubic crystal system. A schematic dilatometer curve for the transformation appears in Fig. I.
I Z
s~ d i s/SS
To I"IG.
1.
TEM~RATURE
Schematic of a length-temperature curve obtained from a dilatometer.
" Graduate student at the University of Illinois, Urbana, Illinois. :ldetallography, 3 (1970) 229-233
Copyright @ 1970 by American Elsevier Publishing Company, Inc.
230
G. L . Fisher a n d R . H . Geils
T h e specimen volume, V , , at some temperature, T, where both phases are present, is given by Vs = n,~a~ 3 ~- n ~ a ~ ~
(1)
where nu ~ = number of unit cells of ~. nu ~ = n u m b e r of unit cells of 7. a~ and a, are the lattice parameters of the two phases at temperature T. Consider that the specimen is cylindrical in shape, and that it is isotropie. Therefore its shape remains unchanged during the transformation, so that r = rol/lo, where r 0 and l0 are the radius and length of the specimen at some reference temperature T O , r and l are the radius and length at temperature T.
where A 0 is the cross-sectional area of the specimen at T O . In general, n~a" = A o ( ! ] 2 l
(3)
and nu - -
nA
(4)
where nA = total number of atoms in the alloy. = n u m b e r of atoms per unit cell in the phase. Then a" - -
A°q!/°)2l " a
(5)
nA
Assuming the rod to be composed entirely of ~, we have a~~ =
Ao~(l~/lo~)2l~ " c~ nA
Similarly, assuming the rod to be composed entirely of 7, we have ,, 3 =
3o,(Uto,)
~l, . ~
nA
Since the shape of the specimen is unchanged, roy = roa
(6)
Quantitative Dilatometric Data
231
and
l' Io,
'i '
so that a,f
Ao~(UZo,)~z, .,,
-
(7)
nA
Substituting equations 2, 4, 6, and 7 into equation I to obtain the volume of the specimen at any temperature within the range of the transformation, we have
\10!
O'c~
//A
O'v
/'/A
where na= and nay are the total number of atoms on c~ or Y crystal lattice sites, respectively. But at the reference temperature T o , Ao
Ao~
and
I.
10~
T h e n equation 8 reduces to l~3 = N~l~ 3 +
(1
-
-
N,) l,,3
where N , and N~ are the fractions of" the total atoms on c~or Y crystal lattice sites
l~a -- lva Since 1 . ~ - - l . ~ l and l ~ - - l . ~ at any temperature T, X~ ,-~
[(l' [(l~ -
(10)
1, and since ( A l - k / ) a - - I a ~ 3 F A l ,
l,) + Z] ~ -- U iv) + L,p - U
.,~ --
l~ l, -
l., t,,
then
(~ l)
T h u s one can obtain the atom fraction of the c~ or Y phase at any temperature by applying a "lever rule" to the dilatometer curve. It is often desirable to obtain the volume fraction, 1~, of a phase from the dilatometer curve: ~ffa _.=
nA~,
n.4,~ ~ na,,(,~l%)(a,ja~)a
(12)
Knowing the lattice parameters and the number of atoms per unit cell for the two phases, we can plot atom fraction versus volume fraction. This can then be used to determine the volume fraction of either phase. One of the most common alloy systems studied by dilatometry is the iron carbon system. In a hypoeutectoid plain carbon steel, the high-temperature
232
G. L. Fisher and R. H. Geils
f.c.c, austenite transforms to b.c.c, ferrite and pearlite in the temperature range 600°C to 800°C. If the carbon level is not too high ( 4 0 . 4 0 weight °/o), the predominant phase is ferrite. For the case of austenite transforming to ferrite % / ~ ~ 0.5 and (ao~/ao~) 3 ~ 1.986 [a0~(900°C) = 3.639 (ref. 1) and a0~(800°C) ~ 2.894 (ref. 1)]. Therefore equation 12, which gives the volume fraction of ferrite, Vf ~, reduces to •
Vf~ =
nA~
=N~--
I7 /l~. , -/- ~
(13)
nA~ @ nA.~
T h u s the volume fraction of ferrite can be determined directly from the dilatometer curve. Equation 13 can also be used for the transformation of austenite to martensite (b.c.t.) in hypoeutectoid steels, since the c/a ratio is nearly equal to 1.0 (1.036 for a 0.8 weight O//ocarbon steel; ref. 2) and the change in (ao~/ao~) ~ is slight over the temperature range 20°C to 800°C (ref. 3). T h e procedure for obtaining quantitative phase transformation data from a dilatometer is quite simple. Temperature intervals of 50C ° are marked on the dilatometer cooling curve. When using a Bollenrath dilatometer this can be done by momentarily increasing the intensity of the light spot tracing the curve. T o obtain the volume fraction of ferrite as a function of temperature, the straightline portions of the austenite and ferrite curves are first extended, as shown schematically in Fig. 1 by the dashed lines. Lines parallel to the length axis are then drawn through each of the temperature points and also at 10C ° intervals. At each temperature interval measurements are made of the distance between the austenite and ferrite lines (lr~ -- l ~ in Fig. 1) and the specimen curve and the austenite line (l~ -- l ~ in Fig. l). T h e volume fraction of ferrite is then calculated from equation 13.
STEEL COMPOSITION IN WEIGHT P E R C E N T 0.023C 0.9ON/ 1 2 0 C u 0 4 6 M n O 30Si ioc
u_
~ z
uJ
8o
6o
40
20
>~ 800
775
750
725 700 675 650 625 6 0 0 575 TEMPERATURE- DEGREES CENTIGRADE
F I G . 2. E f f e c t o f c o o l i n g r a t e o n t h e 7 - a t r a n s f o r m a t i o n austenitized at 1068°C for one hour.
550
525
500
in a nickel-copper
steel
Quantitative Dilatometric Data
233
An example of the use of this technique is shown in Fig. 2. T h e effect of cooling rate on the austenite-to-ferrite transformation in a low-carbon nickel-copper steel was studied. At the slow cooling rate the transformation curve is typical of equiaxed ferrite. At the faster cooling rate the transformation is extended over a much wider temperature range. T h e r e is an inflection in the curve at about 600°C due to the initiation of a bainitic ferrite transformation.
I. \V. B. Pearson, .4 Handbook of Lattice Spacings arid Structures ~/" Metals am/ .I//oys, Pergamon Press, New York, 1958, pp. 626-627. 2. Ibid., p. 923. 3. Ibid., p. 631.
_4ccepted February 14, 1970