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Continuous cooling (CT) diagrams and critical cooling rates: A direct method of calculation using the concept of additivity
Journal of Non-Crystalline Solids 53 (1982) 61-72 North-Holland Publishing Company
61
C O N T I N U O U S C O O L I N G (CT) D I A G R A M S A N D C R I T I C A L COOLING RATES: A DIRECT METHOD OF CALCULATION USING THE CONCEPT OF ADDITIVITY D.R. M A C F A R L A N E * Department of Chemistry, Purdue University, West Lafayette, Indiana 47907, USA Received 9 November 1981 Revised manuscipt received 31 March 1982
A new method for the calculation of continuous cooling curves and their associated critical cooling rates for glass formation is described. This method is based directly on the theories of nucleation and growth using the concept of additivity. These calculations also yield a master curve describing the amount of crystallinity at any temperature during cooling, the shape of which is independent of cooling rate. Comparison is made between the new method and that of Grange and Kiefer for several systems and the approximations in their method examined.
1. Introduction During the last decade considerable use [1-6] has been made of t i m e - t e m p e r a t u r e - t r a n s f o r m a t i o n (TTT) diagrams and their derivative continuous-cooling (CT) diagrams in assessing the critical cooling rates required for glass formation in any particular system. Critical cooling rates are an integral part of a theory of glass forming ability described by U h l m a n n [7] which views all liquids as glass formers given only a sufficiently rapid quench rate such that all nucleation and subsequent growth is essentially bypassed. The measure of glass forming ability is thus the critical cooling rate and the associated critical thickness of material that can be vitrified at that cooling rate. The critical cooling rate can of course be measured for most marginal glass formers but it can also be calculated for any liquid from the theories of h o m o g e n e o u s nucleation and thermally activated growth. U h l m a n n ' s theoretical approach was to calculate the T I T curve by making several simplifying assumptions in the above theories to obtain
t:
kT
-----S NOf
{1 _ W [ ~ ~ T ~ R ~ T e ]
}3
(1)
where t is the isothermal time required for a liquid to transform to the extent * Lord Rutherford Memorial Research Fellow. 0 0 2 2 - 3 0 9 3 / 8 2 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 7 5 © 1982 N o r t h - H o l l a n d
62
D.R. MacFarlane / Continuous cooling diagram and critical cooling rates
of a given volume fraction X (usually 10 6) of solid phase, ~ is the viscosity, k Boltzmann's constant, T the absolute temperature, a 0 the average atomic diameter, N ° the number of atoms per unit volume, f the fraction of sites on the interface where atoms are preferentially added or removed, A Hf the enthalpy of fusion, and T E the equilibrium melting point. From the T T T diagram thus obtained, the continuous cooling diagram, representing the amount of transformation obtained at each temperature during cooling rather than isothermally, was calculated using the method of Grange and Kiefer [8]; the CT diagram yielding directly the critical cooling rate required. The method of Grange and Kiefer (explained in detail below) is however, only approximate and it is the purpose of this paper to describe a more accurate (accurate to the same level as the T T T diagram) method of calculation based directly on the theory of nucleation and growth. The calculations also yield directly the value of thecritical cooling rate as opposed to the graphical approach used previously which itself is an approximation.
2. Theoretical
In general the theory of nucleation and growth has been largely confined to isothermal transformations, however Christian [9] has shown how the concept of additivity can be used to make the non-isothermal problem tractable and we will outline and extend his approach here. The concept of additivity is based on the assumption that the instantaneous transformation rate is a function only of the amount of transformation and the temperature and not of the thermal history of the transformation. For example, consider an idealized 2 stage heat treatment where the sample is held for a time t I at temperature T~ and then quenched instantaneously to T2. The concept of additivity then implies that the course of the transformation at T2 is exactly the same as if the amount of transformation XI, produced at T~, had been produced at T2. Now the assumptions stated above allow the transformation rate to be written in the form
dX( t)/dt = h(T)//g( X), (2) where the functions h(T) and g ( X ) can be obtained from the classical theories of nucleation and growth and will contain factors such as the diffusion coefficient and the interfacial energy. From this we can obtain
h(T)dt = g( X ) d X = d G ( X ) ,
(3)
this equation being general for any thermal history. In the case of an isothermal transformation this equation becomes
G(X) : h ( T ) t
(4)
and thus the time required, ta, to transform the system to any given extent X a
D.R. MacFarlane / Continuous cooling diagram and critical cooling rates
63
is given by
ta(T ) : G ( X a ) / h ( T ).
(5)
h(T) then ta(T)dX/dt = G( Xa)/g( X )
Substituting in eq. (2) for
yields (6)
which on integration gives
fodt/ta(T'
) = G(Xa)/G (Xa)
=
1
(7)
This, then, is the general statement of the concept of additivity, the fraction dt/ta(T ) being summed at each temperature (infinitesimally incremented) until the sum equals one. At this point the transformation has reached the extent of volume fraction Xa. Let us now specialize the thermal history to that of a constant cooling rate, Q, where t =-0 when T =- T E, hence
l/Qfr~dT'/ta(T' ) =
1
(8)
and ta(T' ) can be calculated, for any given X a, from eq. (1). This will then yield the temperature T, during cooling, at which the transformation reaches the extent X a. An equation of considerably more utility can be obtained, however, by treating X(T) as the variable to be determined for a given T. As such X(T) can be removed from the integral ( T being a constant in the calculation of each integral) to yield
since eq. (1) can be rewritten as
t(r') = [X(T)]'/'K(T'),
(10)
where
kT'
Nor 3
{1 - e x p [ - - A H f ( T E --
T')/RT'TE] }3
This integral can be calculated numerically, given only the parameters required in eq. (1), for a series of temperatures T; the results, when plotted, show how the volume fraction transformed, X(T), develops during cooling. These calculations need only be performed for one cooling rate, e.g. Q = 1, since, as can be seen from the form of eq. (9) the integrand does not contain Q. This fact can be exploited further by observing that on taking logs in eq. (9) we obtain log
X(T)=4
log(frdr'/K(T')}
Thus the shape of a plot of log
X(T)
- 4 log Q.
(11)
versus T will be independent of Q and
64
D.R. MacFarlane / Continuous cooling diagram and critical cooling rates
hence a "master" curve can be generated using the Q = 1 data, which can be used at any cooling rate simply by adjusting the log X scale by an amount 4 log Q. Finally, the results can be transformed into a CT curve by calculating the temperature at which X ( T ) reaches the given degree of transformation (usually 10 6) for each cooling rate. The smallest cooling rate at ~which this degree of crystallinity is never attained corresponds to the critical cooling rate. It is thus an integral part of the CT curve calculated by this method, being the point at which the CT curve terminates. Thus it appears that the assumption of additivity allows the direct calculation of the course of non-isothermal transformations from theoretically obtained rates of nucleation and growth. These theories are, however, formulated for the isothermal case and the assumption of additivity must be examined in this light. Eq. (2) expresses the basic limitation imposed by the assumption that a transformation is additive, i.e. that the rate of the transformation is proportional only to some function of the temperature and to some function of the extent of transformation. In other words, the instantaneous transformation rate must depend only on the state of the system and not on the time-temperature path by which that state was obtained. Thus, for example, a sample which is allowed to attain a certain extent of transformation at one temperature and is then quenched rapidly to a much lower temperature, at which the establishment of steady state nucleation takes a significant amount of time, will not initially continue to transform at the same rate as a sample which had attained the same degree of transformation at the lower temperature. Under such conditions of non steady-state nucleation the concept of additivity can therefore not be applied. Similarly in the case of interface controlled growth the concept of additivity will not be applicable to any system (or temperature range) in which a significant amount of time, relative to that of observation, is required for the establishment of the equilibrium flow of heat away from the interface. The crucial point of the foregoing is, of course, that the non steady-state effects must take place for an appreciable fraction of the total transformation time before the assumption of additivity becomes invalid. Such transients are often related to the transport properties of the system. For example, in the case of non steady-state nucleation, the characteristic relaxation time ~" of the transient is given by
where n* is the number of atoms in the critical nucleus, Ns the number of atoms on its surface and p the frequency of transport at the nucleus-liquid interface. The same properties control the steady-state rates of nucleation and growth, and hence the total transformation time. Thus it is commonly assumed (see for example, ref. 7) that, in the T T T / C T calculations that concern us here, transient effects can be ignored unless the barrier to nucleation is unusually large, a situation which is not expected to be found with any generality. The
D.R. MacFarlane / Continuous cooling diagram and critical cooling rates
65
assumptions, then, that are central to the concept of additivity appear to be in common with those generally made in this area, and which have been found to provide a reasonable description of the actual kinetics. Finally it should be noted that the same direct analysis can equally be obtained from the powerful approach of crystallization statistics, Onorato et al. (1980) [1] the only advantage of the present approach being the conceptual simplicity of the final equations. Undoubtedly, however, further extensions into regimes where non steady-state effects become important will have to employ the crystallization statistics approach.
3. Calculations Numerical integration was performed using a Cautious Adaptive Romberg Extrapolation technique, the final iterations being required to be self consistent over the range of integration to within 0.1%. Subsequent comparisons have shown that the simple Simpson's rule methods available on many pocket calculators can also perform these calculations equally well. The whole integral from T E down to T was not calculated for each T since T
F
TE
T~_
"T~
T l being the upper limit of the previously calculated integral. In all calculations the integral was determined at 1 K intervals (i.e. T 1 - T = 1 K). Also it was found that, as is physically reasonable, the integral was vanishingly small when T was close to T E. In order to obtain the CT curve from this (Q = 1) X(T)
E'--T - -TTcu ' - NRV ,. To
F'r
2
' '
\ ~
', i
', i
~-~
LINEARCOOLING TRAJECTORY
GIVEN
W
(t-to)
to
TIME
Fig. 1. Illustration of Grange and Kiefer's method of calculation of the CT curve.
66
D.R. MacFarlane / Continuous cooling diagram and critical cooling rates
D.R. MacFarlane / Continuous cooling diagram and critical cooling rates
67
versus T data a range of cooling rates was chosen such that approximately 100 points could be calculated on the CT curve before it terminated. For comparison CT curves were also calculated from eq. (1) using the method of Grange and Kiefer. This method involves the assumptions that there is negligible crystallization during cooling to the temperature TO (see fig. 1), regardless of cooling rate, and that on cooling through a limited temperature range from TO to some temperature T that the amount of transformation is substantially equal to the amount indicated by the T T T (i.e. isothermal) diagram at the average temperature ( T o + T)/2 after a time interval t - t 0. This is illustrated in fig. 1. For any given cooling rate TO is determined and successively lower values of T chosen until the amount of transformation attains the critical value (usually X is taken to be 10 6 for just detectable crystallization). The time corresponding to this point in temperature is of course given by ( T E -- T)/Q. As in the direct calculations the T T T curve transformation times were calculated at 1 K intervals and the same range of cooling rates chosen in obtaining the CT curve. Calculations were performed for three systems (Auo.77Ge0.136Si0.094, [5,6] Pd0.82Si0.18 [5,6] and SiO 2 [7]) for which T T T (and in some cases CT) curves have been previously published. The estimated critical cooling rates required to vitrify these liquids range from - - 10 6 K s - l in the case of Auo.77Ge0A36Sio.094 to -- 10 -5 K s - i in the case of SiO 2 and hence these examples should provide a good test of the new calculation method. The parameters (and sources) used in the calculation are listed in table 1 for completeness, although their precise values are relatively unimportant in this application since it is the comparison between the CT curves calculated by the two different methods that is of interest here. In all cases f was taken to be 1. It is also notable that the calculation of the CT curve by Grange and Kiefer's method (including calculation of the T T T curve) took significantly ( ~ 5 times) longer to compute than the CT curve calculated from eq. (10).
4. Results and discussion
Cooling curves calculated by the method of Grange and Kiefer (labeled CTcK ) and by the direct method described in this paper (labeled CTomEcv) are compared, along with their associated T T T curves for the alloys Auo.77Ge0.136Si0.094 and Pd0.82Si0.18 and also for SiO 2 in figs. 2, 3 and 4, respectively. The immediate conclusion that can be drawn is that there is little difference between the approximate and direct methods of calculation of the CT curves in any of these systems. As would be expected, Grange and Kiefer's first approximation causes a slight underestimation of the extent of crystallization at very slow cooling rates (i.e. the high temperature part of the curve). At higher cooling rates the direct calculation drops below that of Grange and Kiefer and at a given temperature the directly calculated transformation time
68
D.R. MacFarlane / Continuous cooling diagram and critical cooling rates
50,
40C
-o15o
-o'45 ~
'
- 0.5 5
Log t (s). Fig. 2. TTT, CTGK , and CTDIRECT
curves
for the system Au0.77Geo.136Si0,094.
is slightly longer. This presumably reflects the inability of Grange and Kiefer's second approximation to accurately describe the complex temperature dependence of the nucleation and growth rates displayed in eq. (1). This relationship between the curves calculated from the two different methods is similar to that obtained by Onorato et al. (1980) in ref. 1, where the crystallization statistics approach mentioned previously was employed. - (For a direct comparison between the two curves in fig. 5(b) of Onorato et al. (1980) [1] the CTGK curve
9oc
//////
85(::
T(K:
~ ,.,:.,
800
"PSC I
I
-z,o
-LS
I
I
i
-t.O
-0.5
0
Log ! (s) Fig. 3. TTT, CTcK and CTDmECT curves for the system Pdo.g2Siojg.
%
D.R. MacFarlane / Continuous cooling diagram and critical cooling rates
69
w o c /~TTT CTDIRECT
T(K 1500
....
..........................
1400
o
~
,b, x 06'cs)
t5
~o
Fig. 4. TTT, CTGK and CTD1RECT curves for the system SiO 2. Here the CTDIRECT curve extends to considerably lower temperatures than does the CTcK curve.
must be shifted along the log time axis by an a m o u n t __ ( 1"25 × 10-7 ) 1 log x l = ] log in order to take account of the different extents of transformation used in the two curves.) The exact nature of this approximation can be elucidated here using the equations developed above. The assumption states that the a m o u n t of transform a t i o n observed over the limited temperature range between TO and T is that observed at ( TO+ T ) / 2 after time t - t 0. If the a m o u n t of isothermal transformation, X(T, t), is written in the general form suggested by Avrami's work [13],
X(T, t) oz F(T)t" so that t = c[ X ( r ) ] '/°/I-l(r), where c is a constant, then the above assumption in terms of eq. (8) a m o u n t s to writing
1/QfToc H ( r ' l d T ' = T
{ X [ ( T 0 + T ) / 2 , t-- t0]} 1/"
This equality is f o u n d to hold only when H ( T ) = a constant or when H ( T ) cc T so that G r a n g e and Kiefer's approximation is an assumption of near constancy or near linearity of the dependence of X 1/" on temperature. Finally it can be seen in all cases that G r a n g e and Kiefer's method, again
70
D.R. MacFarlane / Continuous cooling diagram and critical cooling rates
Table 2 C o m p a r i s o n of the critical cooling rates Re (in K s i) o b t a i n e d from the direct calculations and G r a n g e and Kiefer's a p p r o x i m a t e m e t h o d System
R c (CTGK) previous work *
R c (CTGK) this work
R c (direct)
Auo.77Geo.1365io.094 Pds2Sils SiO
2 X 107 [6] 8 X 10 3 [6] -
2 X 107 8X 10 3 8 x 10 - s
1 X 1() 7 6 X 10 3 6 x 10 s
* These R c values a p p e a r to have been o b t a i n e d from R c = ( T - T n ) / t n where the subscript n refers to the nose Of the CTGK curve. R,: values o b t a i n e d in this work are always the cooling rate at which the curve terminates.
not surprisingly, is unable to take into account the transformation which takes place at temperatures below the nose of the T T T curve. Thus the directly calculated curve extends considerably lower in temperature before it terminates. This behavior is most pronounced in the case of SiO 2 where the CT curve extends some 75 K lower than the termination point of the CT~K curve. It should be notee here that while the T I T and CT~K curves calculated in these three systems are in good agreement with those published previously [6] there is some discrepancy as to the form of the curve at cooling rates approaching R c. Many authors show the CT~K curve tailing off into the T T T curve on the low temperature side of the nose of the curve and displaying no obvious termination point, as there physically should be at the cooling rate Rc (at which the extent of the transformation just barely reaches 10 - 6 during cooling). Grange and Kiefer's original paper on the other hand shows CT curves which do terminate at a distinct point at temperatures significantly below the isothermal curve in the same manner as in the present calculations. Such considerations are, however, far beyond the level of accuracy of the calculations, given the approximations in the theory, and hence have little significance in the estimation of R c. The critical cooling rates calculated from the two different procedures for the three systems are summarized in table 2. The directly calculated critical cooling rates appear to be consistently smaller, despite the fact that the CT curve extends to considerably lower temperatures, than the R~(CTcK) values. The difference is hardly significant, however, in view of the assumptions that must be made to obtain eq.. (1) and which have not been addressed here. Fig. 5 shows how the amount of transformation develops with temperature at a given cooling rate (Q = 1 x 10 - 7 K s-1). This has been plotted in the form suggested by eq. (11), the shape of this curve being, therefore, appropriate for all cooling rates; a change of cooling rate requiring only a shift along the log X axis by an amount 4 log Q. Such diagrams could equally be drawn from the method of Grange and Kiefer, or from the approach of crystallization statistics [14]; however it is not obvious in these analyses how the shape of the curves
D.R. MacFarlane / Continuous cooling diagram and critical cooling rates
71
f
-8
-18
Log X -28
-38
5. o
sbo
T(K)
4go
46o
Fig. 5. Plot of log X versus T showing how the crystallinity develops during cooling in the system Au0.77Geo.136Sio.094 . The curve as drawn is appropriate for Q = I × 10 7 K s - I , but can be transformed for any other cooling rate by adding an amount 4 log Q to the log X scale.
relates to the cooling rate. These curves should be useful in assessing the extent of crystallization during any cooling process, with the limitation that the extent of crystallization does not proceed to the extent that impingement of neighboring growing crystals takes place. Under the latter circumstances the theory from which eq. (1) is obtained fails and a more elaborate analysis is necessary.
5. Summary and conclusions The concept of additivity as described by Christian has been extended to the calculation of CT curves from the theory of nucleation and growth. These exact calculations substantially confirm the adequacy of the method of Grange and Kiefer for the calculation of critical cooling rates. It is believed, however, that the direct calculations are considerably simpler, and results more rapidly obtained, using the method described here and its use is advocated for future CT curve calculations. The true critical cooling rate is an integral part of the CT curve thus obtained being the point at which the curve terminates.
72
D.R. MacFarlane // Continuous cooling diagram and critical cooling rates
The author is indebted to Professor C.A. Angell for his encouragement of this work and the many helpful discussions, and also to the National Science Foundation for support under Grant No. DMR-8007053.
References [1] P.I.K. Onorato and D.R. Uhlmann, J. Non-Crystalline Solids 22 (1976) 367; P.I.K. Onorato, D.R. Uhlmann and R.W. Hopper, J. Non-Crystalline Solids 41 (1980) 189. [2] H.A. Davies, J. Non-Crystalline Solids 17 (1975) 266. [3] M. Naka, Y. Nishi and T. Masumoto, in: Rapidly Quenched Metals II1, Vol. 1, ed., B. Cantor (The Metals Society, 1978) p. 231. [4] H.A. Davies, ibid., p. 1. [5] P. Ramachandraroa, B. Cantor and R.W. Cahn, J. Non-Crystalline Solids 24 (1977) 109. [6] P.M. Anderson, III, J. Steinberg and A.E. Lord, Jr., J. Non-Crystalline Solids 34 (1979) 267. [7] D.R. Uhlmann, J. Non-Crystalline Solids 7 (1972) 337. [8] R.A. Grange and J.M. Kiefer, Trans. AMS 29 (1941) 85. [9] J.W. Christian, The Theory of Transformations in Metals and Alloys (Pergamon, New York, 1965). [10] D.E. Polk, Acta Met. 17 (1969) 1021. [11] E.H. Fontana and W.A. Plumber, Phys. Chem. Glasses 7 (1966) 139. [12] H.A. Davies, Scripta Met. 8 (1974) 1179. [13] M. Avrami, J. Chem. Phys. 7 (1939) 1103; 8 (1940) 212; 9 (1941) 177. [14] H. Yinnon and D.R. Uhlmann, J. Non-crystalline Solids 44 (1981) 37.
61
C O N T I N U O U S C O O L I N G (CT) D I A G R A M S A N D C R I T I C A L COOLING RATES: A DIRECT METHOD OF CALCULATION USING THE CONCEPT OF ADDITIVITY D.R. M A C F A R L A N E * Department of Chemistry, Purdue University, West Lafayette, Indiana 47907, USA Received 9 November 1981 Revised manuscipt received 31 March 1982
A new method for the calculation of continuous cooling curves and their associated critical cooling rates for glass formation is described. This method is based directly on the theories of nucleation and growth using the concept of additivity. These calculations also yield a master curve describing the amount of crystallinity at any temperature during cooling, the shape of which is independent of cooling rate. Comparison is made between the new method and that of Grange and Kiefer for several systems and the approximations in their method examined.
1. Introduction During the last decade considerable use [1-6] has been made of t i m e - t e m p e r a t u r e - t r a n s f o r m a t i o n (TTT) diagrams and their derivative continuous-cooling (CT) diagrams in assessing the critical cooling rates required for glass formation in any particular system. Critical cooling rates are an integral part of a theory of glass forming ability described by U h l m a n n [7] which views all liquids as glass formers given only a sufficiently rapid quench rate such that all nucleation and subsequent growth is essentially bypassed. The measure of glass forming ability is thus the critical cooling rate and the associated critical thickness of material that can be vitrified at that cooling rate. The critical cooling rate can of course be measured for most marginal glass formers but it can also be calculated for any liquid from the theories of h o m o g e n e o u s nucleation and thermally activated growth. U h l m a n n ' s theoretical approach was to calculate the T I T curve by making several simplifying assumptions in the above theories to obtain
t:
kT
-----S NOf
{1 _ W [ ~ ~ T ~ R ~ T e ]
}3
(1)
where t is the isothermal time required for a liquid to transform to the extent * Lord Rutherford Memorial Research Fellow. 0 0 2 2 - 3 0 9 3 / 8 2 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 7 5 © 1982 N o r t h - H o l l a n d
62
D.R. MacFarlane / Continuous cooling diagram and critical cooling rates
of a given volume fraction X (usually 10 6) of solid phase, ~ is the viscosity, k Boltzmann's constant, T the absolute temperature, a 0 the average atomic diameter, N ° the number of atoms per unit volume, f the fraction of sites on the interface where atoms are preferentially added or removed, A Hf the enthalpy of fusion, and T E the equilibrium melting point. From the T T T diagram thus obtained, the continuous cooling diagram, representing the amount of transformation obtained at each temperature during cooling rather than isothermally, was calculated using the method of Grange and Kiefer [8]; the CT diagram yielding directly the critical cooling rate required. The method of Grange and Kiefer (explained in detail below) is however, only approximate and it is the purpose of this paper to describe a more accurate (accurate to the same level as the T T T diagram) method of calculation based directly on the theory of nucleation and growth. The calculations also yield directly the value of thecritical cooling rate as opposed to the graphical approach used previously which itself is an approximation.
2. Theoretical
In general the theory of nucleation and growth has been largely confined to isothermal transformations, however Christian [9] has shown how the concept of additivity can be used to make the non-isothermal problem tractable and we will outline and extend his approach here. The concept of additivity is based on the assumption that the instantaneous transformation rate is a function only of the amount of transformation and the temperature and not of the thermal history of the transformation. For example, consider an idealized 2 stage heat treatment where the sample is held for a time t I at temperature T~ and then quenched instantaneously to T2. The concept of additivity then implies that the course of the transformation at T2 is exactly the same as if the amount of transformation XI, produced at T~, had been produced at T2. Now the assumptions stated above allow the transformation rate to be written in the form
dX( t)/dt = h(T)//g( X), (2) where the functions h(T) and g ( X ) can be obtained from the classical theories of nucleation and growth and will contain factors such as the diffusion coefficient and the interfacial energy. From this we can obtain
h(T)dt = g( X ) d X = d G ( X ) ,
(3)
this equation being general for any thermal history. In the case of an isothermal transformation this equation becomes
G(X) : h ( T ) t
(4)
and thus the time required, ta, to transform the system to any given extent X a
D.R. MacFarlane / Continuous cooling diagram and critical cooling rates
63
is given by
ta(T ) : G ( X a ) / h ( T ).
(5)
h(T) then ta(T)dX/dt = G( Xa)/g( X )
Substituting in eq. (2) for
yields (6)
which on integration gives
fodt/ta(T'
) = G(Xa)/G (Xa)
=
1
(7)
This, then, is the general statement of the concept of additivity, the fraction dt/ta(T ) being summed at each temperature (infinitesimally incremented) until the sum equals one. At this point the transformation has reached the extent of volume fraction Xa. Let us now specialize the thermal history to that of a constant cooling rate, Q, where t =-0 when T =- T E, hence
l/Qfr~dT'/ta(T' ) =
1
(8)
and ta(T' ) can be calculated, for any given X a, from eq. (1). This will then yield the temperature T, during cooling, at which the transformation reaches the extent X a. An equation of considerably more utility can be obtained, however, by treating X(T) as the variable to be determined for a given T. As such X(T) can be removed from the integral ( T being a constant in the calculation of each integral) to yield
since eq. (1) can be rewritten as
t(r') = [X(T)]'/'K(T'),
(10)
where
kT'
Nor 3
{1 - e x p [ - - A H f ( T E --
T')/RT'TE] }3
This integral can be calculated numerically, given only the parameters required in eq. (1), for a series of temperatures T; the results, when plotted, show how the volume fraction transformed, X(T), develops during cooling. These calculations need only be performed for one cooling rate, e.g. Q = 1, since, as can be seen from the form of eq. (9) the integrand does not contain Q. This fact can be exploited further by observing that on taking logs in eq. (9) we obtain log
X(T)=4
log(frdr'/K(T')}
Thus the shape of a plot of log
X(T)
- 4 log Q.
(11)
versus T will be independent of Q and
64
D.R. MacFarlane / Continuous cooling diagram and critical cooling rates
hence a "master" curve can be generated using the Q = 1 data, which can be used at any cooling rate simply by adjusting the log X scale by an amount 4 log Q. Finally, the results can be transformed into a CT curve by calculating the temperature at which X ( T ) reaches the given degree of transformation (usually 10 6) for each cooling rate. The smallest cooling rate at ~which this degree of crystallinity is never attained corresponds to the critical cooling rate. It is thus an integral part of the CT curve calculated by this method, being the point at which the CT curve terminates. Thus it appears that the assumption of additivity allows the direct calculation of the course of non-isothermal transformations from theoretically obtained rates of nucleation and growth. These theories are, however, formulated for the isothermal case and the assumption of additivity must be examined in this light. Eq. (2) expresses the basic limitation imposed by the assumption that a transformation is additive, i.e. that the rate of the transformation is proportional only to some function of the temperature and to some function of the extent of transformation. In other words, the instantaneous transformation rate must depend only on the state of the system and not on the time-temperature path by which that state was obtained. Thus, for example, a sample which is allowed to attain a certain extent of transformation at one temperature and is then quenched rapidly to a much lower temperature, at which the establishment of steady state nucleation takes a significant amount of time, will not initially continue to transform at the same rate as a sample which had attained the same degree of transformation at the lower temperature. Under such conditions of non steady-state nucleation the concept of additivity can therefore not be applied. Similarly in the case of interface controlled growth the concept of additivity will not be applicable to any system (or temperature range) in which a significant amount of time, relative to that of observation, is required for the establishment of the equilibrium flow of heat away from the interface. The crucial point of the foregoing is, of course, that the non steady-state effects must take place for an appreciable fraction of the total transformation time before the assumption of additivity becomes invalid. Such transients are often related to the transport properties of the system. For example, in the case of non steady-state nucleation, the characteristic relaxation time ~" of the transient is given by
where n* is the number of atoms in the critical nucleus, Ns the number of atoms on its surface and p the frequency of transport at the nucleus-liquid interface. The same properties control the steady-state rates of nucleation and growth, and hence the total transformation time. Thus it is commonly assumed (see for example, ref. 7) that, in the T T T / C T calculations that concern us here, transient effects can be ignored unless the barrier to nucleation is unusually large, a situation which is not expected to be found with any generality. The
D.R. MacFarlane / Continuous cooling diagram and critical cooling rates
65
assumptions, then, that are central to the concept of additivity appear to be in common with those generally made in this area, and which have been found to provide a reasonable description of the actual kinetics. Finally it should be noted that the same direct analysis can equally be obtained from the powerful approach of crystallization statistics, Onorato et al. (1980) [1] the only advantage of the present approach being the conceptual simplicity of the final equations. Undoubtedly, however, further extensions into regimes where non steady-state effects become important will have to employ the crystallization statistics approach.
3. Calculations Numerical integration was performed using a Cautious Adaptive Romberg Extrapolation technique, the final iterations being required to be self consistent over the range of integration to within 0.1%. Subsequent comparisons have shown that the simple Simpson's rule methods available on many pocket calculators can also perform these calculations equally well. The whole integral from T E down to T was not calculated for each T since T
F
TE
T~_
"T~
T l being the upper limit of the previously calculated integral. In all calculations the integral was determined at 1 K intervals (i.e. T 1 - T = 1 K). Also it was found that, as is physically reasonable, the integral was vanishingly small when T was close to T E. In order to obtain the CT curve from this (Q = 1) X(T)
E'--T - -TTcu ' - NRV ,. To
F'r
2
' '
\ ~
', i
', i
~-~
LINEARCOOLING TRAJECTORY
GIVEN
W
(t-to)
to
TIME
Fig. 1. Illustration of Grange and Kiefer's method of calculation of the CT curve.
66
D.R. MacFarlane / Continuous cooling diagram and critical cooling rates
D.R. MacFarlane / Continuous cooling diagram and critical cooling rates
67
versus T data a range of cooling rates was chosen such that approximately 100 points could be calculated on the CT curve before it terminated. For comparison CT curves were also calculated from eq. (1) using the method of Grange and Kiefer. This method involves the assumptions that there is negligible crystallization during cooling to the temperature TO (see fig. 1), regardless of cooling rate, and that on cooling through a limited temperature range from TO to some temperature T that the amount of transformation is substantially equal to the amount indicated by the T T T (i.e. isothermal) diagram at the average temperature ( T o + T)/2 after a time interval t - t 0. This is illustrated in fig. 1. For any given cooling rate TO is determined and successively lower values of T chosen until the amount of transformation attains the critical value (usually X is taken to be 10 6 for just detectable crystallization). The time corresponding to this point in temperature is of course given by ( T E -- T)/Q. As in the direct calculations the T T T curve transformation times were calculated at 1 K intervals and the same range of cooling rates chosen in obtaining the CT curve. Calculations were performed for three systems (Auo.77Ge0.136Si0.094, [5,6] Pd0.82Si0.18 [5,6] and SiO 2 [7]) for which T T T (and in some cases CT) curves have been previously published. The estimated critical cooling rates required to vitrify these liquids range from - - 10 6 K s - l in the case of Auo.77Ge0A36Sio.094 to -- 10 -5 K s - i in the case of SiO 2 and hence these examples should provide a good test of the new calculation method. The parameters (and sources) used in the calculation are listed in table 1 for completeness, although their precise values are relatively unimportant in this application since it is the comparison between the CT curves calculated by the two different methods that is of interest here. In all cases f was taken to be 1. It is also notable that the calculation of the CT curve by Grange and Kiefer's method (including calculation of the T T T curve) took significantly ( ~ 5 times) longer to compute than the CT curve calculated from eq. (10).
4. Results and discussion
Cooling curves calculated by the method of Grange and Kiefer (labeled CTcK ) and by the direct method described in this paper (labeled CTomEcv) are compared, along with their associated T T T curves for the alloys Auo.77Ge0.136Si0.094 and Pd0.82Si0.18 and also for SiO 2 in figs. 2, 3 and 4, respectively. The immediate conclusion that can be drawn is that there is little difference between the approximate and direct methods of calculation of the CT curves in any of these systems. As would be expected, Grange and Kiefer's first approximation causes a slight underestimation of the extent of crystallization at very slow cooling rates (i.e. the high temperature part of the curve). At higher cooling rates the direct calculation drops below that of Grange and Kiefer and at a given temperature the directly calculated transformation time
68
D.R. MacFarlane / Continuous cooling diagram and critical cooling rates
50,
40C
-o15o
-o'45 ~
'
- 0.5 5
Log t (s). Fig. 2. TTT, CTGK , and CTDIRECT
curves
for the system Au0.77Geo.136Si0,094.
is slightly longer. This presumably reflects the inability of Grange and Kiefer's second approximation to accurately describe the complex temperature dependence of the nucleation and growth rates displayed in eq. (1). This relationship between the curves calculated from the two different methods is similar to that obtained by Onorato et al. (1980) in ref. 1, where the crystallization statistics approach mentioned previously was employed. - (For a direct comparison between the two curves in fig. 5(b) of Onorato et al. (1980) [1] the CTGK curve
9oc
//////
85(::
T(K:
~ ,.,:.,
800
"PSC I
I
-z,o
-LS
I
I
i
-t.O
-0.5
0
Log ! (s) Fig. 3. TTT, CTcK and CTDmECT curves for the system Pdo.g2Siojg.
%
D.R. MacFarlane / Continuous cooling diagram and critical cooling rates
69
w o c /~TTT CTDIRECT
T(K 1500
....
..........................
1400
o
~
,b, x 06'cs)
t5
~o
Fig. 4. TTT, CTGK and CTD1RECT curves for the system SiO 2. Here the CTDIRECT curve extends to considerably lower temperatures than does the CTcK curve.
must be shifted along the log time axis by an a m o u n t __ ( 1"25 × 10-7 ) 1 log x l = ] log in order to take account of the different extents of transformation used in the two curves.) The exact nature of this approximation can be elucidated here using the equations developed above. The assumption states that the a m o u n t of transform a t i o n observed over the limited temperature range between TO and T is that observed at ( TO+ T ) / 2 after time t - t 0. If the a m o u n t of isothermal transformation, X(T, t), is written in the general form suggested by Avrami's work [13],
X(T, t) oz F(T)t" so that t = c[ X ( r ) ] '/°/I-l(r), where c is a constant, then the above assumption in terms of eq. (8) a m o u n t s to writing
1/QfToc H ( r ' l d T ' = T
{ X [ ( T 0 + T ) / 2 , t-- t0]} 1/"
This equality is f o u n d to hold only when H ( T ) = a constant or when H ( T ) cc T so that G r a n g e and Kiefer's approximation is an assumption of near constancy or near linearity of the dependence of X 1/" on temperature. Finally it can be seen in all cases that G r a n g e and Kiefer's method, again
70
D.R. MacFarlane / Continuous cooling diagram and critical cooling rates
Table 2 C o m p a r i s o n of the critical cooling rates Re (in K s i) o b t a i n e d from the direct calculations and G r a n g e and Kiefer's a p p r o x i m a t e m e t h o d System
R c (CTGK) previous work *
R c (CTGK) this work
R c (direct)
Auo.77Geo.1365io.094 Pds2Sils SiO
2 X 107 [6] 8 X 10 3 [6] -
2 X 107 8X 10 3 8 x 10 - s
1 X 1() 7 6 X 10 3 6 x 10 s
* These R c values a p p e a r to have been o b t a i n e d from R c = ( T - T n ) / t n where the subscript n refers to the nose Of the CTGK curve. R,: values o b t a i n e d in this work are always the cooling rate at which the curve terminates.
not surprisingly, is unable to take into account the transformation which takes place at temperatures below the nose of the T T T curve. Thus the directly calculated curve extends considerably lower in temperature before it terminates. This behavior is most pronounced in the case of SiO 2 where the CT curve extends some 75 K lower than the termination point of the CT~K curve. It should be notee here that while the T I T and CT~K curves calculated in these three systems are in good agreement with those published previously [6] there is some discrepancy as to the form of the curve at cooling rates approaching R c. Many authors show the CT~K curve tailing off into the T T T curve on the low temperature side of the nose of the curve and displaying no obvious termination point, as there physically should be at the cooling rate Rc (at which the extent of the transformation just barely reaches 10 - 6 during cooling). Grange and Kiefer's original paper on the other hand shows CT curves which do terminate at a distinct point at temperatures significantly below the isothermal curve in the same manner as in the present calculations. Such considerations are, however, far beyond the level of accuracy of the calculations, given the approximations in the theory, and hence have little significance in the estimation of R c. The critical cooling rates calculated from the two different procedures for the three systems are summarized in table 2. The directly calculated critical cooling rates appear to be consistently smaller, despite the fact that the CT curve extends to considerably lower temperatures, than the R~(CTcK) values. The difference is hardly significant, however, in view of the assumptions that must be made to obtain eq.. (1) and which have not been addressed here. Fig. 5 shows how the amount of transformation develops with temperature at a given cooling rate (Q = 1 x 10 - 7 K s-1). This has been plotted in the form suggested by eq. (11), the shape of this curve being, therefore, appropriate for all cooling rates; a change of cooling rate requiring only a shift along the log X axis by an amount 4 log Q. Such diagrams could equally be drawn from the method of Grange and Kiefer, or from the approach of crystallization statistics [14]; however it is not obvious in these analyses how the shape of the curves
D.R. MacFarlane / Continuous cooling diagram and critical cooling rates
71
f
-8
-18
Log X -28
-38
5. o
sbo
T(K)
4go
46o
Fig. 5. Plot of log X versus T showing how the crystallinity develops during cooling in the system Au0.77Geo.136Sio.094 . The curve as drawn is appropriate for Q = I × 10 7 K s - I , but can be transformed for any other cooling rate by adding an amount 4 log Q to the log X scale.
relates to the cooling rate. These curves should be useful in assessing the extent of crystallization during any cooling process, with the limitation that the extent of crystallization does not proceed to the extent that impingement of neighboring growing crystals takes place. Under the latter circumstances the theory from which eq. (1) is obtained fails and a more elaborate analysis is necessary.
5. Summary and conclusions The concept of additivity as described by Christian has been extended to the calculation of CT curves from the theory of nucleation and growth. These exact calculations substantially confirm the adequacy of the method of Grange and Kiefer for the calculation of critical cooling rates. It is believed, however, that the direct calculations are considerably simpler, and results more rapidly obtained, using the method described here and its use is advocated for future CT curve calculations. The true critical cooling rate is an integral part of the CT curve thus obtained being the point at which the curve terminates.
72
D.R. MacFarlane // Continuous cooling diagram and critical cooling rates
The author is indebted to Professor C.A. Angell for his encouragement of this work and the many helpful discussions, and also to the National Science Foundation for support under Grant No. DMR-8007053.
References [1] P.I.K. Onorato and D.R. Uhlmann, J. Non-Crystalline Solids 22 (1976) 367; P.I.K. Onorato, D.R. Uhlmann and R.W. Hopper, J. Non-Crystalline Solids 41 (1980) 189. [2] H.A. Davies, J. Non-Crystalline Solids 17 (1975) 266. [3] M. Naka, Y. Nishi and T. Masumoto, in: Rapidly Quenched Metals II1, Vol. 1, ed., B. Cantor (The Metals Society, 1978) p. 231. [4] H.A. Davies, ibid., p. 1. [5] P. Ramachandraroa, B. Cantor and R.W. Cahn, J. Non-Crystalline Solids 24 (1977) 109. [6] P.M. Anderson, III, J. Steinberg and A.E. Lord, Jr., J. Non-Crystalline Solids 34 (1979) 267. [7] D.R. Uhlmann, J. Non-Crystalline Solids 7 (1972) 337. [8] R.A. Grange and J.M. Kiefer, Trans. AMS 29 (1941) 85. [9] J.W. Christian, The Theory of Transformations in Metals and Alloys (Pergamon, New York, 1965). [10] D.E. Polk, Acta Met. 17 (1969) 1021. [11] E.H. Fontana and W.A. Plumber, Phys. Chem. Glasses 7 (1966) 139. [12] H.A. Davies, Scripta Met. 8 (1974) 1179. [13] M. Avrami, J. Chem. Phys. 7 (1939) 1103; 8 (1940) 212; 9 (1941) 177. [14] H. Yinnon and D.R. Uhlmann, J. Non-crystalline Solids 44 (1981) 37.