- Email: [email protected]
Relationship between non-isothermal transformation curves and isothermal and non-isothermal kinetics
Acta Materialia 53 (2005) 4893–4901 www.actamat-journals.com
Relationship between non-isothermal transformation curves and isothermal and non-isothermal kinetics P.R. Rios
*
Universidade Federal Fluminense, Escola de Engenharia Industrial Metalu´rgica de Volta Redonda, Av. dos Trabalhadores, 420, Volta Redonda, RJ, 27255-125, Brazil Received 22 February 2005; received in revised form 24 June 2005; accepted 8 July 2005 Available online 19 August 2005
Abstract A mathematical method is presented to obtain an isothermal transformation curve from a continuous cooling transformation (CCT) curve. It can also be used for transformations that occur during continuous heating rather than continuous cooling. The method is applied to nucleation and growth transformations and to grain growth. Furthermore, a new additivity rule based on non-isothermal transformation data, analogous to the traditional additivity rule based on isothermal transformation data, is introduced. Its use enables the calculation of the kinetics along an arbitrary cooling path relying solely on a CCT curve. Ó 2005 Published by Elsevier Ltd on behalf of Acta Materialia Inc. Keywords: Kinetics; Phase transformations kinetics; Grain growth; Analytical methods; Additivity rule
1. Introduction Generally, non-isothermal kinetics occurs from a thermal history specified by a function T(t), where T is temperature and t is time. In the most general cases the present state of a structure depends on its entire thermal history, and sometimes only on recent history, e.g., since the last time the specimen was molten or austenitized. The present state and the present kinetics are functionals [1,2], defined as functions of this function T(t). A question that naturally arises is if and how data taken for one thermal path, T(t), can be used to predict results from another path. That includes isothermal data and TTT (time–temperature–transformation) diagrams and special non-isothermal data from which CCT (continuous cooling transformation) diagrams can be constructed, but is not limited to them. All kinds of kinetic processes can be formally dealt with: phase trans*
Tel.: +55 24 3344 3029. E-mail address: [email protected].
formations, coarsening, grain growth, crystallization, recrystallization among others. Many of these processes are routinely followed during non-isothermal transformation and non-isothermal data are collected, for example, by differential scanning calorimetry (DSC). In order to describe the kinetics it is necessary to specify an internal variable of the system, X, for monitoring the extent of microstructural change. X can be any convenient property related to the microstructural change. It can be stereological, such as, volume fraction or grain size. It could also be a physical property such as heat evolution (calorimetry), volume change (dilatometry), magnetic properties, hardness and many others. X should be easy to measure at the transformation temperature or at a reference temperature and be single valued and monotonic. In this paper, X will be referred to as fraction transformed except when otherwise indicated. For some processes, dX/dt can be approximated as a function of just X and current temperature, T [3], dX ¼ f ðX ; T Þ. dt
1359-6454/$30.00 Ó 2005 Published by Elsevier Ltd on behalf of Acta Materialia Inc. doi:10.1016/j.actamat.2005.07.005
ð1Þ
4894
P.R. Rios / Acta Materialia 53 (2005) 4893–4901
This for example, might be valid in annealing when X is the degree of recovery. The kinetics is then only a function of X, which results from thermal history, but is independent of all other aspects of the thermal history. For even simpler cases, isokinetic reactions, a factoring of f(X, T) = u(X)k(T) may be valid. After inserting into Eq. (1) and integrating Z U ðX Þ ¼ kðT Þ dt; ð2Þ where U(X) is a function of X and k(T) is a function of temperature only, such as growth rate, nucleation rate or diffusion constant. The validity of these relations can be experimentally tested by comparing the kinetics in specimens with different T(t), for example, step ageing to the same X, and continuing at the same T. In the general case, however, f(X, T(t)) is not a function but a functional. To turn the functional f(X, T(t)) into a function it is necessary to specify the time– temperature path. A family of these paths can be expressed by a convenient parameterization. The easiest parameterization is to consider isothermal transformations, T(t) = T = constant. The ‘‘parameter’’ is T. For isothermal processes, f(X, T), is a function of X, only. Another important case is when T(t) is limited to constant heating or cooling rates, q, then T ðtÞ ¼ T i þ qt;
ð3Þ
where Ti is the initial temperature. In this case, f(X, T) can be written as a function of q, so dX/dt becomes dX ¼ gðX ; qÞ; dt
ð4Þ
dX/dt is still a functional of the initial microstructure, so f and g become functions only for a fixed initial microstructure. The two parameterizations shown above correspond to the more common experimental paths used in the study of transformations: the TTT and CCT curves. Both TTT and CCT curves give important information. TTT curves are often a better choice from a fundamental point of view as the isothermal kinetics are judged easier to model and interpret. On the other hand, in engineering practice most processes involve heating and cooling steps thus requiring the knowledge of CCT curves. A well-known problem of formal kinetics is the conversion of isothermal kinetic data into predictions about kinetics for various non-isothermal heating and cooling routines used in real processes, such as, for example, continuous cooling, thermal cycling, and step ageing. For many processes the isothermal kinetic data are depicted by TTT curves, and their conversion into nonisothermal kinetic data, often depicted by CCT curves, has been a problem of long standing interest. This is
often accomplished with the help of the additivity rule [3–5] that follows from Eq. (1) Z t dt ¼ 1; ð5Þ sðX 0; T Þ 0 where s(X0, T) represents the isothermal transformation time for X = X0 at a temperature, T, and t is the total transformation time. Cahn [3] showed that Eq. (1) is a sufficient condition for Eq. (5). The additivity principle has been widely used to obtain continuous cooling information from isothermal transformation data. Metallic alloys in general and steels in particular have been the subject of many such studies. Austenite to pearlite transformation has attracted a great deal of attention and has been examined by Umemoto et al. [6] who also studied austenite to bainite transformation [7], by Verdi and Visitin [8], by Homberg [9,10], by Brimacombe and coworkers [11– 14] and more recently by Ye et al. [15]. Non-isothermal precipitation [16] and dissolution kinetics [17,18] have also used related concepts. Rios [19,20] used the additivity principle to model the galvannealing process in zinc coated steels. These papers are just a small sample of a large number of papers that used the additivity principle in a variety of metallic materials transformations. In spite of its widespread usage some authors have recognized that the additivity principle has certain limitations. For instance, Leblond and Devaux [21], studying the specific case of austenite decomposition, have pointed out that the additivity fails when the reaction does not go to completion but stops at an equilibrium volume fraction that depends on transformation temperature. They proposed a new rate law to solve this difficulty. They also proposed a method for the case in which more than one phase forms during continuous cooling and that takes the influence of austenite grain size into account. Other authors [22,23] studied the relationship between the rate law, Eq. (1), and the additivity integral, Eq. (5). The question of the effect of incubation time has also been discussed [24,25]. Whereas Reti and Felde [26] proposed a generalization of the additivity principle for rate laws more complex than Eq. (1). It is also worth mentioning the series of papers by Mittemeijer and coworkers [27–30] who developed a formal mathematical theory for isothermal and non-isothermal transformation kinetics using the concept of a path variable that is compatible with the additivity principle. The inverse problem: to extract isothermal information from non-isothermal kinetics has received less attention in the context of metallic alloys [31,32]. On the other hand in the context of polymer or metallic glass crystallization studies using DSC technique it has received considerable attention. The available models [33–35] use an equation based on the work of Johnson and Mehl [36], Avrami [37–39] and Kolmogorov [40], often called the JMAK theory
P.R. Rios / Acta Materialia 53 (2005) 4893–4901
X ðs; T Þ ¼ 1 expðkðT ÞsnðT Þ Þ;
ð6Þ
where k(T) and n(T) may be temperature dependent parameters but n is often assumed to be temperature independent. These models are normally used with success for glass crystallization during heating that is simpler to model than continuous cooling transformations [32]. The reason for that is the fact that on heating both nucleation and growth rates monotonically increase whereas during continuous cooling nucleation and growth rate often have a ‘‘C’’ or Gaussian like shape when plotted against temperature [31,32]. Pont et al. [31] proposed an algorithm to extract k(T) and n, supposed to be temperature independent, from nonisothermal transformation data. Their method is specifically directed at phase transformations during continuous cooling and they assume a Gaussian dependence of k(T) with temperature. By including the work of Leblond and Devaux [21], Pont et al.Õs methodology was able to handle the formation of more than one phase during continuous cooling. In a more recent work, Malinov et al. [32] studied b ! a transformation in a Ti–6Al–4V alloys during continuous cooling using DSC. They proposed an algorithm to extract k(T) and n from their non-isothermal data that is in its essence similar to the previous work of Pont et al. [31]. As shown above, in spite of the large number of papers on the additivity principle there is no methodology available to solve the inverse problem, namely, how to extract isothermal data from a set of continuous cooling/heating experiments, without resorting to specific assumptions concerning the mathematical form of the isothermal transformation kinetics. Yet such a method could be useful in situations in which isothermal experiments are difficult to perform, for example, when the transformation is fast or when the available experimental technique is better suited to determining non-isothermal kinetics. Moreover, there is no theory for the important situation in which a general, not constant, cooling/heating path, T(t) is employed and only the CCT curve is available. In other words, there is no methodology by which a CCT curve can be used to generate the transformation kinetics for an arbitrary cooling path. This paper focus on the relationship of CCT curves and other time–temperature paths; first the problem of how to obtain TTT curves from CCT curves is addressed; next a general methodology is proposed to use the CCT curve to obtain the transformation kinetics for an arbitrary time–temperature cooling/heating path.
4895
temperature, supposed constant in an isothermal transformation, and s is the transformation time needed to reach a fraction transformed X at temperature T. Instead of representing X = X(s, T) in a three-dimensional graph it is more convenient to represent it as a contour plot on the (s, T) plane. This plot is the wellknown TTT curve shown in Fig. 1. In contrast with isothermal transformations, nonisothermal transformations occur during continuous heating or cooling. When the cooling/heating rate is constant, i.e., independent of time or temperature, X can be written as a function of the cooling/heating rate, q, and transformation temperature, T, X = X(q, T). This function can be represented as a contour plot on the (q, T) plane. The cooling rate is here taken to be a negative number, say, 10 K/s, whereas the heating rate is taken to be a positive number, say, +10 K/s. A schematic CCT curve is shown in Fig. 2 plotted on the (q, T) plane. The second quadrant is used because q < 0. In this work, the initial conditions will be X = 0 at the initial temperature Ti. The initial temperature is above the transformation temperature for cooling and is supposed to be the same for all cooling rates; for heating Ti is obviously below the transformation temperature. In Fig. 2, each curve represents a fixed value of X(q, T); X(q, T) = 0.01 and X(q, T) = 0.05 are shown to illustrate this point. The tangent lines are discussed later in this paper. Notice that in this representation the cooling curves are vertical lines, q = constant, starting at the initial cooling temperature, Ti and extending down to the transformation temperature, T. The transformation curves involve three variables: X, T and q for CCT or X, T and s for TTT. Either variable can be written as a function of the other two. Therefore, it is a matter of choice which one will be the independent variable. When one controls T(t) it is perhaps more logical to choose X as
2. TTT and CCT curves When a transformation proceeds isothermally the fraction transformed, X, can be described by a function of two variables: X = X(s, T), where T is the transformation
Fig. 1. Schematic TTT curve. The TTT curve is a contour plot of X(s, T). Each curve represents a fixed value of X(s, T); X(s, T) = 0.01 and X(s, T) = 0.05 are shown to illustrate this point. The partial derivatives on the time axis are explained in the text.
4896
P.R. Rios / Acta Materialia 53 (2005) 4893–4901
Fig. 2. Schematic CCT curve represented as a contour plot of X(q, T). Each curve represents a fixed value of X(q, T); X(q, T) = 0.01 and X(q, T) = 0.05 are shown to illustrate this point. The tangent lines are discussed in the text.
the independent variable. Nonetheless, one may describe the CCT curve as X(q, T) = 0.01 or T = T(0.01, q) or even q(0.01, T) and the TTT curve as X(s,T) = 0.01 or s(0.01, T). In what follows the latter representations will be employed whenever it is more convenient to do so. An analogous situation would be that in which the reaction occurs upon continuous heating from an initially low temperature, Ti, to a final, higher temperature, T. In this case the fraction transformed, X, is reached upon heating not cooling. Fig. 3 shows a similar curve drawn for continuous heating transformations, a CHT curve. For CHT curves the heating rate is positive, q > 0, and the curve is represented in the first quadrant of the (q, T) plane. Notice that in this representation the heating curves are vertical lines, q = constant, starting at the initial cooling temperature, Ti, which is now below the transformation temperature.
Fig. 4. Schematic CCT curve represented in the traditional way on the (t, T) plane. The curves still represent X(q, T) = constant mapped onto the (t, T) plane by a transformation of the horizontal axis. The vertical lines shown in Fig. 2 are now straight lines passing through (0,Ti). In this schematic diagram a linear time scale is used, in practice a log scale is more common. The total cooling times are indicated on the time axis.
The representation of the CCT curve shown in Fig. 2 is not usually adopted in engineering practice. The usual representation prefers to use the (t, T) plane that is familiar from the TTT curve. A CCT curve plotted on the (t, T) plane is shown in Fig. 4. Fig. 4 can be obtained from Fig. 2 by mapping the (q, T) plane onto the (t, T) plane. This mapping preserves the temperature axis but the time axis is transformed according to Eq. (3). For cooling, the initial temperature is always the highest temperature, so q is negative in Eq. (3) and t is positive as it should be. The family of vertical q = constant lines from the (q, T) plane becomes a family of lines passing through the point (t = 0, Ti).
3. Mathematical method to transform a CCT into a TTT curve As described in Section 1, the additivity rule [3–5] is a method to transform X(s, T) = constant, say 0.01, into X(q, T) = 0.01. Supposing that the time to reach a certain fraction transformed X isothermally at a temperature T is s(X, T) one can find the time to reach the same volume fraction transformed X after continuous cooling the specimen by using the additivity principle, Eq. (5), for a constant cooling rate: q = dh/dt, where h is used for the temperature variable to avoid confusing it with the transformation temperature, T = T(X, q), Z T dh q¼ . ð7Þ sðX ; hÞ Ti Fig. 3. Schematic CHT curve represented as a contour plot of X(q, T). Each curve represents a fixed value of X(q, T); X(q, T) = 0.01 and X(q, T) = 0.05 are shown to illustrate this point. The tangent lines are discussed in the text.
Taking the partial derivative on both sides of Eq. (7) [4] oT sðX ; T Þ ¼ . ð8Þ oq X
P.R. Rios / Acta Materialia 53 (2005) 4893–4901
This derivative is indicated in Fig. 2, s(X, T) is the slope of the X(q, T) = constant curve. For continuous heating transformations Eq. (8) is exactly the same, the only change needed is to replace the cooling rate (q < 0) by the heating rate (q > 0). Eq. (8) depends on the additivity rule, Eq. (5), being valid. Fig. 2 illustrates how it is possible to extract isothermal kinetics data from X(q, T) = constant curves. Consider the X(q, T) = 0.01 curve in Fig. 2. The slope of a tangent line to the X(q, T) = 0.01 curve, Eq. (8), gives the isothermal transformation time to reach X = 0.01 at the transformation temperature T. For example, at temperature T1 Eq. (8) gives s(0.01, T1). If another tangent line is drawn at a higher temperature, say, T2, another isothermal transformation time, s(0.01, T2) is obtained. So, by drawing tangents to the X(q, T) = 0.01, values of s(0.01, T) are obtained and the associated TTT curve is generated. Another possibility would be to draw tangents at a constant temperature, say, T1. Now successive tangent lines will result in isothermal times for increasing fraction transformed: s(0.01, T1) and s(0.05, T1) are exemplified. These times are indicated on the time axis of the TTT curve shown in Fig. 1. An identical procedure can be followed to extract isothermal information from Fig. 3 in which tangent lines similar to those drawn in Fig. 2 are indicated. It is important to comment on the way Figs. 1 and 2 were drawn. TTT curves in phase transformations may have a ‘‘C’’ shape. In Fig. 1 only the upper part of the ‘‘C’’ was drawn. In the upper part the transformation time increases with increasing temperature. That is why Fig. 2 was drawn with a positive concavity; the derivative (oT/oq)0.01, that is s(0.01, T), increases with increasing temperature matching the TTT curve in Fig. 1. If the full ‘‘C’’ curve were drawn in Fig. 1, in the lower part of the ‘‘C’’ the transformation time would increase with decreasing temperature. This would demand a modification of Fig. 2. Fig. 2 would have to be extended towards q ! 1 to include a section with a negative concavity so that (oT/oq)0.01, that is s(0.01, T), increased with decreasing temperature. As a result, the complete curve would have an inflexion point related to the minimum value of the isothermal transformation time at the ‘‘nose’’ of the ‘‘C’’ curve. The complete TTT and CCT curves were not drawn in order to avoid making the figures too complex and overloaded. Notice that not all TTT curves have a ‘‘C’’ shape, for example, recrystallization curves do not have a ‘‘C’’ shape because the driving force comes from cold work deformation and does not change with annealing temperature. Isokinetic reactions were defined by Eq. (2) in the introduction. When the reaction is isokinetic specific mathematical results can be derived that significantly simplify the CCT to TTT conversion problem. From
4897
Eq. (2), the isothermal transformation time, s, for a fraction transformed, X0, is given by sðX 0 ; T Þ ¼
U ðX 0 Þ . kðT Þ
ð9Þ
For an arbitrary X the isothermal time is sðX ; T Þ ¼
U ðX Þ U ðX Þ ¼ sðX 0 ; T Þ. kðT Þ U ðX 0 Þ
ð10Þ
This is only valid for isokinetic reactions. Inserting Eq. (10) into Eq. (7) Z U ðX 0 Þ T dh ; ð11Þ qðX ; T Þ ¼ U ðX Þ T i sðX 0 ; hÞ where q(X, T) is the cooling or heating rate necessary to reach a fraction transformed equal to X at a transformation temperature equal to T. See, for example, q1 = q(0.01, T1) or q2 = q(0.05, T1) in Fig. 2. For X = X0 Z T dh . ð12Þ qðX 0 ; T Þ ¼ T i sðX 0 ; hÞ Combining Eq. (10) with Eqs. (12) and (13) yields a particularly simple result sðX ; T ÞqðX ; T Þ ¼ sðX 0 ; T ÞqðX 0 ; T Þ. Using Eq. (8) finally gives " # oT 1 . sðX ; T Þ ¼ qðX 0 ; T Þ oq X 0 qðX ; T Þ
ð13Þ
ð14Þ
The quantity q(X, T) is the constant cooling rate that results in a fraction transformed X at the transformation temperature T. Eq. (14), a direct consequence of the reaction being isokinetic, allows the isothermal transformation time for a given temperature T, s(X, T) to be found from the cooling rate q(X, T). The derivative in the square brackets needs to be evaluated only at a single value X = X0. With the help of Eq. (14) the whole isothermal kinetics may be evaluated without the need of evaluating the derivative at every point. This is better illustrated with the help of Fig. 2. Considering the temperature T = T1 and X0 = 0.01 in Fig. 2, q1 = q(0.01, T1) and the derivative at the point (q1, T1) is (oT/oq)0.01. s(0.05, T1) can either be found directly from the derivative at (q2, T1), (oT/oq)0.05, or by means of Eq. (14), noticing that q2 = q(0.05, T1): s(0.05, T1) = (oT/oq)0.01 q1/q2. Eq. (8) is valid for continuous heating curves bearing in mind that Ti < T and q > 0. 4. Applicability of the method in some special cases In engineering practice, Eq. (5) has been applied and by extension Eq. (8) will be applied even when their application cannot be fully justified. This can be useful as a first approximation in the absence of data or when the transformation is too complex. But from a
4898
P.R. Rios / Acta Materialia 53 (2005) 4893–4901
fundamental point of view such a situation is not entirely satisfactory. It is desirable to consider the details of the reaction and examine whether for a particular case it is possible to have full confidence in its results. In this section this will be done for some special cases. 4.1. Nucleation and growth transformations
1 ln 1 X ðs; T Þ
¼
pIðT ÞG3 ðT Þ 4 s ðT Þ. 3
ð19Þ
In order to transform X(s, T) into X(q, T) one starts by writing X(s, T) for the isothermal transformation Z 1 4p s 3 ln IðT ÞG3 ðT Þðs tÞ dt ð20Þ ¼ 1 X ðs; T Þ 3 0 or
Classical JMAK theory [36–40] considers two particular cases for which there are exact solutions: (a) when nucleation is site-saturated, all nuclei, NV, are already present at t = 0; (b) when the nucleation rate, I, is constant but depends on temperature. Other assumptions are: nuclei are randomly located in space, growth velocity, G, is constant and the growing regions are spherical in shape. An important concept in JMAK theory is that of the extended volume fraction transformed. The extended volume fraction transformed, XE, is the volume fraction transformed calculated supposing that all nuclei could grow freely. That is, all transformed regions can nucleate and grow without interference from the other regions. In real transformation this of course does not happen owing to the impingement. JMAK were able to find a geometrical relationship between the extended volume fraction transformed, XE, and the real volume fraction transformed, X, 1 X E ¼ ln . ð15Þ 1X
q4 ¼ 3 ln
4p 1 1X ðq;T Þ
Z
T
IðhÞG3 ðhÞðT hÞ3 dh.
ð21Þ
Ti
It is now clear that Eq. (8) fails because the integrand depends on the upper integration limit and 4 Z T oq 4p 2 ¼ IðhÞG3 ðhÞðT hÞ dh. ð22Þ 1 oT X ln Ti 1X ðq;T Þ Successive differentiation eventually yields 4 4 14 1 3 ln 1X ðq;T Þ 1 oq sðT Þ ¼ ¼ . 24 oT 4 X 24pIðT ÞG3 ðT Þ
ð23Þ
Even in this case isothermal information can be obtained from continuous cooling data. Unfortunately, this requires a fourth derivative that is nearly impossible to achieve from experimental data. 4.2. General JMAK kinetics
For small volume fractions, say X < 0.1, X XE. So in what follows, ln(1/(1 X)) can be replaced by X for small values of X. When site saturation [41] occurs it is known that the additivity rule is valid [3,5]. The isothermal JMAK result is 1 4pN V 3 ln G ðT Þs3 ðT Þ; ð16Þ ¼ 1 X ðs; T Þ 3
One might adopt a phenomenological approach to reactions that can be formally described by a general JMAK kinetics during an isothermal process, see Eq. (6). From Eq. (6), the isothermal transformation time can be written as 1nðT1 Þ 0 1 ln 1X ðs;T Þ oT A . sðX ; T Þ ¼ ¼@ ð24Þ oq X kðT Þ
where NV is the number of nuclei per unit of volume. One can transform X(s, T) into X(q, T) Z T 3 1 4pN V ln GðhÞ dh . ð17Þ ¼ 1 X ðq; T Þ 3q3 Ti
Although Eq. (6) formally satisfies Eq. (1) for isothermal processes, the above examples show that Eq. (24) may not always work. This happens when f(X, T) is not a function but a functional f(X, T(t)), that is, it depends not only on the current temperature but on the thermal history, T(t). In those cases, the additivity principle may not be valid and Eq. (24) that depends on the validity of the additivity principle may not work as well. In summary, how good or bad this formal approach is depends on how well a particular reaction follows the additivity principle. Bearing these considerations in mind, one can devise a simple method to find k and n when Eq. (6) is a good description of isothermal kinetics. Using Eq. (24) and taking the logarithms 1 oT ln ln ¼ lnðkðT ÞÞ þ nðT Þ ln . ð25Þ 1X oq X
Therefore for X(q, T) = constant, using Eq. (8), Eq. (16) is recovered 0 113 oT 4pN V A GðT Þ1 . ¼ sðT Þ ¼ @ ð18Þ 1 oq X 3 ln 1X ðq;T Þ
So, when the reaction is site-saturated it is possible to directly extract the temperature dependence of the growth rate from continuous cooling data. For temperature dependent nucleation rate, one would not expect the additivity rule to be valid. The isothermal JMAK expression is
P.R. Rios / Acta Materialia 53 (2005) 4893–4901
For a fixed temperature, T = T1, varying X generates a straight line. The slope of this straight line is equal to n(T1). The intercept is equal to ln(k(T1)). By repeating this process for different temperatures n and k are obtained as a function of temperature. If n is a constant, independent of temperature, the reaction becomes isokinetic and the above procedure can be greatly simplified with the help of the results from Section 3. First Eq. (14) can be used to yield !n ! 1 oT ln ln ¼ ln kðT Þ jqðX 0 ; T Þj 1X oq X 0 n ln ðjqðX ; T ÞjÞ;
ð26Þ
where the absolute value of the cooling rate, |q(X, T)|, is taken to avoid taking the logarithm of a negative number, since q < 0 for cooling in this paper. Now in order to find out n it suffices to plot ln ln (1/(1 X)) as a function of ln(|q(X, T)|), for a fixed temperature, T1. In order to determine k(T) one can set X = X0 in Eq. (26) 1 ln 1X 0 kðT Þ ¼ n . ð27Þ oT oq
X0
The derivative needs to be calculated for a fixed volume fraction, X0, as a function of the transformation temperature. The above equation generates k(T), assuming one already obtained n from the previous equation. Both Eqs. (26) and (27) are only valid when n is independent of temperature otherwise the more tedious procedure using Eq. (25) must be used. It is important to stress that it is not necessary to assume any particular functional form for k(T) [31]. The latter may monotonically increase with temperature as it typically does for glass crystallization during continuous heating or have a ‘‘C’’ or ‘‘Gaussian’’ shape when plotted against temperature [31,32]. So, in principle, the present methodology could be used either for continuous heating or continuous cooling data. The advantage of present methodology over previous work is that it can extract isothermal data from nonisothermal data without any mathematical assumptions regarding the functional form of the isothermal transformation curve. One can find out n and k by assuming that isothermal transformation follows Eq. (6) but one might as well assume another convenient isothermal transformation function.
4899
distribution of the heat affected zone during fusion welding [42] or during processing of nanomaterials [43]. For grain growth, the grain size, D, is the natural choice for X. Grain growth kinetics are often described by [44,45] Dn Dni ¼ kðT Þt;
ð28Þ
where X = D is the grain size or the mean intercept length, k is a kinetic temperature dependent factor and Di is the initial grain size. Theory gives n = 2 for pure, texture free materials, but in practice n may be different from the theoretical value and can even depend on temperature [46]. When n is independent of temperature in Eq. (28) the reaction is obviously isokinetic: U ðX ¼ DÞ ¼ Dn Dni in Eq. (2), and the results of Section 3 can be directly applied. Recalling Eqs. (10) and (13) and using Eq. (28) one obtains Dn Dni ¼ ðDn0 Dni Þ
qðD0 ; T Þ ; qðD; T Þ
ð29Þ
where q(D, T) is the cooling rate that results in a grain size D after cooling from an initial temperature Ti down to a temperature T. Eq. (29) gives the grain size as a function of cooling rate for a constant temperature T. The temperature dependent constant k(T) can be found from Dn Dn kðT Þ ¼ i ; dT dq
ð30Þ
D
where D can be arbitrarily chosen. Eqs. (29) and (30) can be used for continuous heating as well. Eqs. (29) and (30) are simpler than the previously proposed methodology for grain growth during heating and cooling [43]. The present methodology has the advantage that it is not necessary to find Eq. (28) from isothermal kinetics. Both n and k can be found directly from non-isothermal, either cooling or heating, experiments with the help of Eqs. (29) and (30). Grain growth is in fact a good example of a reaction in which the kinetics should be independent of thermal history. Indeed, for normal grain growth, it does not matter how a certain grain size was achieved, the grain growth rate depends only on the current grain size and temperature. It is fair to say that other effects may interfere with this, for example, the presence of a strong texture or particles.
4.3. Grain growth
5. Generalized additivity rule
Grain growth is an important phenomenon in polycrystalline materials. The study of grain growth in non-isothermal conditions is of considerable practical relevance. For example, in understanding the grain size
The additivity rule makes use of isothermal kinetics to find out the transformation kinetics when the specimen is cooled or heated with an arbitrary cooling rate. This approach can be generalized for the situation in
4900
P.R. Rios / Acta Materialia 53 (2005) 4893–4901
Dt1 ¼
Tf h . q1
ð34Þ
Using the additivity property Dt1 Dt2 þ ¼ 1; sc1 sc2 substituting Eqs. (31)–(34) into Eq. (35) Tf h h Ti þ ¼ 1. T1 Ti T2 Ti
Fig. 5. Schematic CCT curve represented as a contour plot of X(q, T). It shows the cooling path of a specimen cooled in two steps: (a) cooled from Ti to h with a cooling rate q2 then (b) cooled from h to Tf with a cooling rate q1. The transformation temperature Tf, after this particular two step cooling path, is higher than T1. Depending on the cooling path Tf could be higher or lower than the transformation temperature for the final cooling rate q1.
which a TTT curve is not available but a CCT curve is. In this section, this idea is first illustrated for the simple case of a two-step transformation and then generalized for an arbitrary continuous cooling curve. Suppose that a sample is cooled with a cooling rate that is not constant but depends on the cooling temperature. One wishes to find out the transformation temperature for a temperature dependent cooling rate using CCT data only. The principle is illustrated in Fig. 5. Fig. 5 shows the thermal path of a sample that is cooled from an initial temperature Ti with a constant cooling rate q2 down to an intermediate temperature h . At h the cooling rate instantaneously changes to q1 and the sample is cooled further until it reaches the transformation temperature Tf. One wishes to find out the temperature Tf so that when the sample reaches Tf it will have the desired fraction transformed, say, X = 0.01. This can be done using the same time transformed proportion of the additivity rule for isothermal transformations as follows. If the sample had been cooled with a constant rate q2 < 0 down to the transformation temperature T2 its cooling time sc2 would be sc2 ¼
T2 Ti ; q2
ð31Þ
sc1 ¼
T1 Ti . q1
ð32Þ
The cooling time, Dt2, from Ti to h with q2 is Dt2 ¼
h Ti q2
and the cooling time, Dt1, from h to Tf with q1 is
ð33Þ
ð36Þ
From Eq. (36) Tf can be found for a given h. Notice that the transformation temperature is a function of the cooling rate, T(X, q), in the constant cooling rate CCT curve. The above approach can be generalized for a temperature dependent cooling rate. From Eq. (4): Z X0 dX ; ð37Þ sc ðX 0 ; qÞ ¼ gðX ; qÞ 0 Z X0 Z tc dX dt ¼ ¼ 1; ð38Þ s s ðX ; qÞgðX ; qÞ ðX c 0 c 0 ; qÞ 0 0 where sc(X0, q) is the cooling time necessary to transform to X0 at a constant cooling rate, q, that corresponds to a transformation temperature T(X0, q), as illustrated in Figs. 2 and 5; tc is the total cooling time. Writing the cooling rate as a function of temperature, q(h) Z Tf dh ¼ 1. ð39Þ T ðX 0 ; qðhÞÞ T i Ti T(X0, q) can be found from the experimentally determined CCT curves at constant cooling rate, see Figs. 2 and 5. The integral is carried out so that it equals one at the transformation temperature Tf. Eq. (38) or (39) is the continuous cooling version of the additivity rule. This might be named the nonisothermal additivity rule. It is straightforward to adapt Eq. (39) for continuous heating. Finally, it is worth noting that the result in this section can be combined with the result of Section 3 when the cooling path involves isothermal steps. For example, a specimen is cooled from Ti to h with a cooling rate q(h), then held at h for a time Dt and further cooled from h to Tf with a cooling rate q(h). The transformation temperature Tf for X0 would be Z Tf dh Dt þ ¼ 1; ð40Þ oT T ðX 0 ; qðhÞÞ T i Ti oq
similarly for q1
ð35Þ
X0
where the derivative is evaluated at temperature h. Thus, the methodology presented here allows the determination of the transformation temperature when the cooling rate is not constant, using only data from the CCT curve. The method can be used to calculate the fraction transformed along an arbitrary path involving continuous cooling and isothermal holding steps, relying solely on data from the CCT curve.
P.R. Rios / Acta Materialia 53 (2005) 4893–4901
6. Summary and conclusions In summary, a mathematical method was presented to allow the conversion of continuous cooling transformation data into isothermal transformation data under the restrictive assumption of a constant cooling rate. The method can be applied if the additivity principle is valid. The advantage of the present methodology over previous work is that it can extract isothermal data from non-isothermal data without any mathematical assumptions regarding the functional form of the isothermal transformation curve. The method can be also used for transformations that occur during continuous heating rather than continuous cooling. Special cases in nucleation and growth transformations and grain growth were examined. This analysis showed that in order to be able to fully assess the validity of the method it is necessary to examine the situations in detail. Nevertheless, in engineering practice it is likely that formal methods are going to be applied even if not all preconditions are fulfilled. This approach could be useful to a first approximation. Furthermore, a new additivity rule for non-isothermal transformation, analogous to the additivity rule for isothermal transformations has been introduced. Its use allows the calculation of the fraction transformed along an arbitrary cooling path relying solely on data from the CCT curve. Acknowledgments This work was supported by Conselho Nacional de Desenvolvimento Cientı´fico e Tecnolo´gico, CNPq, Coordenac¸a˜o de Aperfeic¸oamento de Pessoal de Nı´vel Superior, CAPES, and Fundac¸a˜o de Amparo a` Pesquisa do Estado do Rio de Janeiro, FAPERJ. Thanks go to Professor John W. Cahn for helpful discussions on this subject.
References [1] Volterra V. Theory of functionals and of integral and integrodifferential equations. New York (NY): Dover; 1959. [2] Cahn JW. Private communication; 2004. [3] Cahn JW. Acta Metall 1956;4:572. [4] Scheil E. Archiv fu¨r das Eisenhu¨ttenwesen 1935;8:565. [5] Christian JW. The theory of transformations in metals and alloys. 2nd ed. Oxford: Pergamon Press; 1975. p. 545–46.
4901
[6] Umemoto M, Horiuchi K, Tamura I. Trans Iron Steel Inst Jpn 1983;23:690. [7] Umemoto M, Horiuchi K, Tamura I. Trans Iron Steel Inst Jpn 1982;22:854. [8] Verdi C, Visintin A. Acta Metall 1987;35:2711. [9] Homberg DA. IMA J Appl Math 1995;54:31. [10] Homberg DA. Acta Mater 1996;44:4375. [11] Hawbolt EB, Chau B, Brimacombe JK. Metall Trans A 1983;14A:1803. [12] Hawbolt EB, Chau B, Brimacombe JK. Metall Trans A 1985;16A:565. [13] Kuban MB, Jayaraman R, Hawbolt EB, Brimacombe JK. Metall Trans A 1986;17A:1493. [14] Kamat RG, Hawbolt EB, Brown LC, Brimacombe JK. Metall Trans A 1992;23A:2469. [15] Ye JS, Chang HB, Hsu TY. Metall Mater Trans A 2003;34A:1259. [16] Grong O, Myhr OR. Acta Mater 2000;48:445. [17] Bjorneklett BI, Grong O, Myhr OR, Kluken AO. Acta Mater 1998;46:6257. [18] Nordbakke MW, Ryum N, Hunderi O. Philos Mag A 2002;82:2695. [19] Rios PR. Metall Mater Trans A 1996;27A:1132. [20] Seixas UR, Rios PR. ISIJ Int 1997;37:1040. [21] Leblond JB, Devaux JA. Acta Metall 1984;32:137. [22] Lusk M, Jou HJ. Metall Mater Trans A 1997;28A:287. [23] Todinov MT. Metall Mater Trans B 1998;29B:269. [24] Zhu YT, Lowe TC. Metall Mater Trans B 2000;31B:675. [25] Wierszyllowski IA. Metall Trans A 1991;22A:993. [26] Reti T, Felde I. Comput Mater Sci 1999;15:466. [27] Liu F, Sommer F, Mittemeijer EJ. J Mater Sci 2004;39:1621. [28] Kempem ATW, Sommer F, Mittemeijer EJ. J Mater Sci 2002;37:1321. [29] Mittemeijer EJ. J Mater Sci 1992;27:3977. [30] Mittemeijer EJ, Cheng L, van der Schaaf PJ, Brakman CM, Korevaar BM. Metall Trans A 1988;19A:925. [31] Pont D, Bergheau JM, Rochette M, Fortunier R. In: Bui HD, editor. Inverse problems in engineering mechanics: 2nd international symposium. Leiden: A.A. Balkema; 1994. p. 151. [32] Zhu Y, Ye JS, Chang HB, Hsu TY. Metall Mater Trans A 2003;34A:1264. [33] Vasquez J, Wagner C, Villares P, Jimenez-Garay R. Acta Mater 1996;44:4807. [34] Borrego A, Gonzalez-Doncel G. Mater Sci Eng A 1998;245A:10. [35] Graydon JW, Thorpe SJ, Kirk DW. J Non-Cryst Solids 1994;175:31. [36] Johnson WA, Mehl RF. Trans AIME 1939;13:416. [37] Avrami MJ. Chem Phys 1939;17:1103. [38] Avrami MJ. Chem Phys 1940;8:214. [39] Avrami MJ. Chem Phys 1941;9:177. [40] Kolmogorov AN. Izv Akad Nauk USSR – Ser Matemat 1937;1:355. [41] Cahn JW. Acta Metall 1956;4:449. [42] Ashby MF, Easterling KE. Acta Metall 1982;30:1969. [43] Bourell DL, Kaysser W. Acta Metall Mater 1993;4:2933. [44] Burke JE. Trans Metall Soc AIME 1949;180:73. [45] Burke JE, Turnbull D. Prog Metal Phys 1952;3:220. [46] Rios PR. Metall Mater Trans A 1994;25A:221.
Relationship between non-isothermal transformation curves and isothermal and non-isothermal kinetics P.R. Rios
*
Universidade Federal Fluminense, Escola de Engenharia Industrial Metalu´rgica de Volta Redonda, Av. dos Trabalhadores, 420, Volta Redonda, RJ, 27255-125, Brazil Received 22 February 2005; received in revised form 24 June 2005; accepted 8 July 2005 Available online 19 August 2005
Abstract A mathematical method is presented to obtain an isothermal transformation curve from a continuous cooling transformation (CCT) curve. It can also be used for transformations that occur during continuous heating rather than continuous cooling. The method is applied to nucleation and growth transformations and to grain growth. Furthermore, a new additivity rule based on non-isothermal transformation data, analogous to the traditional additivity rule based on isothermal transformation data, is introduced. Its use enables the calculation of the kinetics along an arbitrary cooling path relying solely on a CCT curve. Ó 2005 Published by Elsevier Ltd on behalf of Acta Materialia Inc. Keywords: Kinetics; Phase transformations kinetics; Grain growth; Analytical methods; Additivity rule
1. Introduction Generally, non-isothermal kinetics occurs from a thermal history specified by a function T(t), where T is temperature and t is time. In the most general cases the present state of a structure depends on its entire thermal history, and sometimes only on recent history, e.g., since the last time the specimen was molten or austenitized. The present state and the present kinetics are functionals [1,2], defined as functions of this function T(t). A question that naturally arises is if and how data taken for one thermal path, T(t), can be used to predict results from another path. That includes isothermal data and TTT (time–temperature–transformation) diagrams and special non-isothermal data from which CCT (continuous cooling transformation) diagrams can be constructed, but is not limited to them. All kinds of kinetic processes can be formally dealt with: phase trans*
Tel.: +55 24 3344 3029. E-mail address: [email protected].
formations, coarsening, grain growth, crystallization, recrystallization among others. Many of these processes are routinely followed during non-isothermal transformation and non-isothermal data are collected, for example, by differential scanning calorimetry (DSC). In order to describe the kinetics it is necessary to specify an internal variable of the system, X, for monitoring the extent of microstructural change. X can be any convenient property related to the microstructural change. It can be stereological, such as, volume fraction or grain size. It could also be a physical property such as heat evolution (calorimetry), volume change (dilatometry), magnetic properties, hardness and many others. X should be easy to measure at the transformation temperature or at a reference temperature and be single valued and monotonic. In this paper, X will be referred to as fraction transformed except when otherwise indicated. For some processes, dX/dt can be approximated as a function of just X and current temperature, T [3], dX ¼ f ðX ; T Þ. dt
1359-6454/$30.00 Ó 2005 Published by Elsevier Ltd on behalf of Acta Materialia Inc. doi:10.1016/j.actamat.2005.07.005
ð1Þ
4894
P.R. Rios / Acta Materialia 53 (2005) 4893–4901
This for example, might be valid in annealing when X is the degree of recovery. The kinetics is then only a function of X, which results from thermal history, but is independent of all other aspects of the thermal history. For even simpler cases, isokinetic reactions, a factoring of f(X, T) = u(X)k(T) may be valid. After inserting into Eq. (1) and integrating Z U ðX Þ ¼ kðT Þ dt; ð2Þ where U(X) is a function of X and k(T) is a function of temperature only, such as growth rate, nucleation rate or diffusion constant. The validity of these relations can be experimentally tested by comparing the kinetics in specimens with different T(t), for example, step ageing to the same X, and continuing at the same T. In the general case, however, f(X, T(t)) is not a function but a functional. To turn the functional f(X, T(t)) into a function it is necessary to specify the time– temperature path. A family of these paths can be expressed by a convenient parameterization. The easiest parameterization is to consider isothermal transformations, T(t) = T = constant. The ‘‘parameter’’ is T. For isothermal processes, f(X, T), is a function of X, only. Another important case is when T(t) is limited to constant heating or cooling rates, q, then T ðtÞ ¼ T i þ qt;
ð3Þ
where Ti is the initial temperature. In this case, f(X, T) can be written as a function of q, so dX/dt becomes dX ¼ gðX ; qÞ; dt
ð4Þ
dX/dt is still a functional of the initial microstructure, so f and g become functions only for a fixed initial microstructure. The two parameterizations shown above correspond to the more common experimental paths used in the study of transformations: the TTT and CCT curves. Both TTT and CCT curves give important information. TTT curves are often a better choice from a fundamental point of view as the isothermal kinetics are judged easier to model and interpret. On the other hand, in engineering practice most processes involve heating and cooling steps thus requiring the knowledge of CCT curves. A well-known problem of formal kinetics is the conversion of isothermal kinetic data into predictions about kinetics for various non-isothermal heating and cooling routines used in real processes, such as, for example, continuous cooling, thermal cycling, and step ageing. For many processes the isothermal kinetic data are depicted by TTT curves, and their conversion into nonisothermal kinetic data, often depicted by CCT curves, has been a problem of long standing interest. This is
often accomplished with the help of the additivity rule [3–5] that follows from Eq. (1) Z t dt ¼ 1; ð5Þ sðX 0; T Þ 0 where s(X0, T) represents the isothermal transformation time for X = X0 at a temperature, T, and t is the total transformation time. Cahn [3] showed that Eq. (1) is a sufficient condition for Eq. (5). The additivity principle has been widely used to obtain continuous cooling information from isothermal transformation data. Metallic alloys in general and steels in particular have been the subject of many such studies. Austenite to pearlite transformation has attracted a great deal of attention and has been examined by Umemoto et al. [6] who also studied austenite to bainite transformation [7], by Verdi and Visitin [8], by Homberg [9,10], by Brimacombe and coworkers [11– 14] and more recently by Ye et al. [15]. Non-isothermal precipitation [16] and dissolution kinetics [17,18] have also used related concepts. Rios [19,20] used the additivity principle to model the galvannealing process in zinc coated steels. These papers are just a small sample of a large number of papers that used the additivity principle in a variety of metallic materials transformations. In spite of its widespread usage some authors have recognized that the additivity principle has certain limitations. For instance, Leblond and Devaux [21], studying the specific case of austenite decomposition, have pointed out that the additivity fails when the reaction does not go to completion but stops at an equilibrium volume fraction that depends on transformation temperature. They proposed a new rate law to solve this difficulty. They also proposed a method for the case in which more than one phase forms during continuous cooling and that takes the influence of austenite grain size into account. Other authors [22,23] studied the relationship between the rate law, Eq. (1), and the additivity integral, Eq. (5). The question of the effect of incubation time has also been discussed [24,25]. Whereas Reti and Felde [26] proposed a generalization of the additivity principle for rate laws more complex than Eq. (1). It is also worth mentioning the series of papers by Mittemeijer and coworkers [27–30] who developed a formal mathematical theory for isothermal and non-isothermal transformation kinetics using the concept of a path variable that is compatible with the additivity principle. The inverse problem: to extract isothermal information from non-isothermal kinetics has received less attention in the context of metallic alloys [31,32]. On the other hand in the context of polymer or metallic glass crystallization studies using DSC technique it has received considerable attention. The available models [33–35] use an equation based on the work of Johnson and Mehl [36], Avrami [37–39] and Kolmogorov [40], often called the JMAK theory
P.R. Rios / Acta Materialia 53 (2005) 4893–4901
X ðs; T Þ ¼ 1 expðkðT ÞsnðT Þ Þ;
ð6Þ
where k(T) and n(T) may be temperature dependent parameters but n is often assumed to be temperature independent. These models are normally used with success for glass crystallization during heating that is simpler to model than continuous cooling transformations [32]. The reason for that is the fact that on heating both nucleation and growth rates monotonically increase whereas during continuous cooling nucleation and growth rate often have a ‘‘C’’ or Gaussian like shape when plotted against temperature [31,32]. Pont et al. [31] proposed an algorithm to extract k(T) and n, supposed to be temperature independent, from nonisothermal transformation data. Their method is specifically directed at phase transformations during continuous cooling and they assume a Gaussian dependence of k(T) with temperature. By including the work of Leblond and Devaux [21], Pont et al.Õs methodology was able to handle the formation of more than one phase during continuous cooling. In a more recent work, Malinov et al. [32] studied b ! a transformation in a Ti–6Al–4V alloys during continuous cooling using DSC. They proposed an algorithm to extract k(T) and n from their non-isothermal data that is in its essence similar to the previous work of Pont et al. [31]. As shown above, in spite of the large number of papers on the additivity principle there is no methodology available to solve the inverse problem, namely, how to extract isothermal data from a set of continuous cooling/heating experiments, without resorting to specific assumptions concerning the mathematical form of the isothermal transformation kinetics. Yet such a method could be useful in situations in which isothermal experiments are difficult to perform, for example, when the transformation is fast or when the available experimental technique is better suited to determining non-isothermal kinetics. Moreover, there is no theory for the important situation in which a general, not constant, cooling/heating path, T(t) is employed and only the CCT curve is available. In other words, there is no methodology by which a CCT curve can be used to generate the transformation kinetics for an arbitrary cooling path. This paper focus on the relationship of CCT curves and other time–temperature paths; first the problem of how to obtain TTT curves from CCT curves is addressed; next a general methodology is proposed to use the CCT curve to obtain the transformation kinetics for an arbitrary time–temperature cooling/heating path.
4895
temperature, supposed constant in an isothermal transformation, and s is the transformation time needed to reach a fraction transformed X at temperature T. Instead of representing X = X(s, T) in a three-dimensional graph it is more convenient to represent it as a contour plot on the (s, T) plane. This plot is the wellknown TTT curve shown in Fig. 1. In contrast with isothermal transformations, nonisothermal transformations occur during continuous heating or cooling. When the cooling/heating rate is constant, i.e., independent of time or temperature, X can be written as a function of the cooling/heating rate, q, and transformation temperature, T, X = X(q, T). This function can be represented as a contour plot on the (q, T) plane. The cooling rate is here taken to be a negative number, say, 10 K/s, whereas the heating rate is taken to be a positive number, say, +10 K/s. A schematic CCT curve is shown in Fig. 2 plotted on the (q, T) plane. The second quadrant is used because q < 0. In this work, the initial conditions will be X = 0 at the initial temperature Ti. The initial temperature is above the transformation temperature for cooling and is supposed to be the same for all cooling rates; for heating Ti is obviously below the transformation temperature. In Fig. 2, each curve represents a fixed value of X(q, T); X(q, T) = 0.01 and X(q, T) = 0.05 are shown to illustrate this point. The tangent lines are discussed later in this paper. Notice that in this representation the cooling curves are vertical lines, q = constant, starting at the initial cooling temperature, Ti and extending down to the transformation temperature, T. The transformation curves involve three variables: X, T and q for CCT or X, T and s for TTT. Either variable can be written as a function of the other two. Therefore, it is a matter of choice which one will be the independent variable. When one controls T(t) it is perhaps more logical to choose X as
2. TTT and CCT curves When a transformation proceeds isothermally the fraction transformed, X, can be described by a function of two variables: X = X(s, T), where T is the transformation
Fig. 1. Schematic TTT curve. The TTT curve is a contour plot of X(s, T). Each curve represents a fixed value of X(s, T); X(s, T) = 0.01 and X(s, T) = 0.05 are shown to illustrate this point. The partial derivatives on the time axis are explained in the text.
4896
P.R. Rios / Acta Materialia 53 (2005) 4893–4901
Fig. 2. Schematic CCT curve represented as a contour plot of X(q, T). Each curve represents a fixed value of X(q, T); X(q, T) = 0.01 and X(q, T) = 0.05 are shown to illustrate this point. The tangent lines are discussed in the text.
the independent variable. Nonetheless, one may describe the CCT curve as X(q, T) = 0.01 or T = T(0.01, q) or even q(0.01, T) and the TTT curve as X(s,T) = 0.01 or s(0.01, T). In what follows the latter representations will be employed whenever it is more convenient to do so. An analogous situation would be that in which the reaction occurs upon continuous heating from an initially low temperature, Ti, to a final, higher temperature, T. In this case the fraction transformed, X, is reached upon heating not cooling. Fig. 3 shows a similar curve drawn for continuous heating transformations, a CHT curve. For CHT curves the heating rate is positive, q > 0, and the curve is represented in the first quadrant of the (q, T) plane. Notice that in this representation the heating curves are vertical lines, q = constant, starting at the initial cooling temperature, Ti, which is now below the transformation temperature.
Fig. 4. Schematic CCT curve represented in the traditional way on the (t, T) plane. The curves still represent X(q, T) = constant mapped onto the (t, T) plane by a transformation of the horizontal axis. The vertical lines shown in Fig. 2 are now straight lines passing through (0,Ti). In this schematic diagram a linear time scale is used, in practice a log scale is more common. The total cooling times are indicated on the time axis.
The representation of the CCT curve shown in Fig. 2 is not usually adopted in engineering practice. The usual representation prefers to use the (t, T) plane that is familiar from the TTT curve. A CCT curve plotted on the (t, T) plane is shown in Fig. 4. Fig. 4 can be obtained from Fig. 2 by mapping the (q, T) plane onto the (t, T) plane. This mapping preserves the temperature axis but the time axis is transformed according to Eq. (3). For cooling, the initial temperature is always the highest temperature, so q is negative in Eq. (3) and t is positive as it should be. The family of vertical q = constant lines from the (q, T) plane becomes a family of lines passing through the point (t = 0, Ti).
3. Mathematical method to transform a CCT into a TTT curve As described in Section 1, the additivity rule [3–5] is a method to transform X(s, T) = constant, say 0.01, into X(q, T) = 0.01. Supposing that the time to reach a certain fraction transformed X isothermally at a temperature T is s(X, T) one can find the time to reach the same volume fraction transformed X after continuous cooling the specimen by using the additivity principle, Eq. (5), for a constant cooling rate: q = dh/dt, where h is used for the temperature variable to avoid confusing it with the transformation temperature, T = T(X, q), Z T dh q¼ . ð7Þ sðX ; hÞ Ti Fig. 3. Schematic CHT curve represented as a contour plot of X(q, T). Each curve represents a fixed value of X(q, T); X(q, T) = 0.01 and X(q, T) = 0.05 are shown to illustrate this point. The tangent lines are discussed in the text.
Taking the partial derivative on both sides of Eq. (7) [4] oT sðX ; T Þ ¼ . ð8Þ oq X
P.R. Rios / Acta Materialia 53 (2005) 4893–4901
This derivative is indicated in Fig. 2, s(X, T) is the slope of the X(q, T) = constant curve. For continuous heating transformations Eq. (8) is exactly the same, the only change needed is to replace the cooling rate (q < 0) by the heating rate (q > 0). Eq. (8) depends on the additivity rule, Eq. (5), being valid. Fig. 2 illustrates how it is possible to extract isothermal kinetics data from X(q, T) = constant curves. Consider the X(q, T) = 0.01 curve in Fig. 2. The slope of a tangent line to the X(q, T) = 0.01 curve, Eq. (8), gives the isothermal transformation time to reach X = 0.01 at the transformation temperature T. For example, at temperature T1 Eq. (8) gives s(0.01, T1). If another tangent line is drawn at a higher temperature, say, T2, another isothermal transformation time, s(0.01, T2) is obtained. So, by drawing tangents to the X(q, T) = 0.01, values of s(0.01, T) are obtained and the associated TTT curve is generated. Another possibility would be to draw tangents at a constant temperature, say, T1. Now successive tangent lines will result in isothermal times for increasing fraction transformed: s(0.01, T1) and s(0.05, T1) are exemplified. These times are indicated on the time axis of the TTT curve shown in Fig. 1. An identical procedure can be followed to extract isothermal information from Fig. 3 in which tangent lines similar to those drawn in Fig. 2 are indicated. It is important to comment on the way Figs. 1 and 2 were drawn. TTT curves in phase transformations may have a ‘‘C’’ shape. In Fig. 1 only the upper part of the ‘‘C’’ was drawn. In the upper part the transformation time increases with increasing temperature. That is why Fig. 2 was drawn with a positive concavity; the derivative (oT/oq)0.01, that is s(0.01, T), increases with increasing temperature matching the TTT curve in Fig. 1. If the full ‘‘C’’ curve were drawn in Fig. 1, in the lower part of the ‘‘C’’ the transformation time would increase with decreasing temperature. This would demand a modification of Fig. 2. Fig. 2 would have to be extended towards q ! 1 to include a section with a negative concavity so that (oT/oq)0.01, that is s(0.01, T), increased with decreasing temperature. As a result, the complete curve would have an inflexion point related to the minimum value of the isothermal transformation time at the ‘‘nose’’ of the ‘‘C’’ curve. The complete TTT and CCT curves were not drawn in order to avoid making the figures too complex and overloaded. Notice that not all TTT curves have a ‘‘C’’ shape, for example, recrystallization curves do not have a ‘‘C’’ shape because the driving force comes from cold work deformation and does not change with annealing temperature. Isokinetic reactions were defined by Eq. (2) in the introduction. When the reaction is isokinetic specific mathematical results can be derived that significantly simplify the CCT to TTT conversion problem. From
4897
Eq. (2), the isothermal transformation time, s, for a fraction transformed, X0, is given by sðX 0 ; T Þ ¼
U ðX 0 Þ . kðT Þ
ð9Þ
For an arbitrary X the isothermal time is sðX ; T Þ ¼
U ðX Þ U ðX Þ ¼ sðX 0 ; T Þ. kðT Þ U ðX 0 Þ
ð10Þ
This is only valid for isokinetic reactions. Inserting Eq. (10) into Eq. (7) Z U ðX 0 Þ T dh ; ð11Þ qðX ; T Þ ¼ U ðX Þ T i sðX 0 ; hÞ where q(X, T) is the cooling or heating rate necessary to reach a fraction transformed equal to X at a transformation temperature equal to T. See, for example, q1 = q(0.01, T1) or q2 = q(0.05, T1) in Fig. 2. For X = X0 Z T dh . ð12Þ qðX 0 ; T Þ ¼ T i sðX 0 ; hÞ Combining Eq. (10) with Eqs. (12) and (13) yields a particularly simple result sðX ; T ÞqðX ; T Þ ¼ sðX 0 ; T ÞqðX 0 ; T Þ. Using Eq. (8) finally gives " # oT 1 . sðX ; T Þ ¼ qðX 0 ; T Þ oq X 0 qðX ; T Þ
ð13Þ
ð14Þ
The quantity q(X, T) is the constant cooling rate that results in a fraction transformed X at the transformation temperature T. Eq. (14), a direct consequence of the reaction being isokinetic, allows the isothermal transformation time for a given temperature T, s(X, T) to be found from the cooling rate q(X, T). The derivative in the square brackets needs to be evaluated only at a single value X = X0. With the help of Eq. (14) the whole isothermal kinetics may be evaluated without the need of evaluating the derivative at every point. This is better illustrated with the help of Fig. 2. Considering the temperature T = T1 and X0 = 0.01 in Fig. 2, q1 = q(0.01, T1) and the derivative at the point (q1, T1) is (oT/oq)0.01. s(0.05, T1) can either be found directly from the derivative at (q2, T1), (oT/oq)0.05, or by means of Eq. (14), noticing that q2 = q(0.05, T1): s(0.05, T1) = (oT/oq)0.01 q1/q2. Eq. (8) is valid for continuous heating curves bearing in mind that Ti < T and q > 0. 4. Applicability of the method in some special cases In engineering practice, Eq. (5) has been applied and by extension Eq. (8) will be applied even when their application cannot be fully justified. This can be useful as a first approximation in the absence of data or when the transformation is too complex. But from a
4898
P.R. Rios / Acta Materialia 53 (2005) 4893–4901
fundamental point of view such a situation is not entirely satisfactory. It is desirable to consider the details of the reaction and examine whether for a particular case it is possible to have full confidence in its results. In this section this will be done for some special cases. 4.1. Nucleation and growth transformations
1 ln 1 X ðs; T Þ
¼
pIðT ÞG3 ðT Þ 4 s ðT Þ. 3
ð19Þ
In order to transform X(s, T) into X(q, T) one starts by writing X(s, T) for the isothermal transformation Z 1 4p s 3 ln IðT ÞG3 ðT Þðs tÞ dt ð20Þ ¼ 1 X ðs; T Þ 3 0 or
Classical JMAK theory [36–40] considers two particular cases for which there are exact solutions: (a) when nucleation is site-saturated, all nuclei, NV, are already present at t = 0; (b) when the nucleation rate, I, is constant but depends on temperature. Other assumptions are: nuclei are randomly located in space, growth velocity, G, is constant and the growing regions are spherical in shape. An important concept in JMAK theory is that of the extended volume fraction transformed. The extended volume fraction transformed, XE, is the volume fraction transformed calculated supposing that all nuclei could grow freely. That is, all transformed regions can nucleate and grow without interference from the other regions. In real transformation this of course does not happen owing to the impingement. JMAK were able to find a geometrical relationship between the extended volume fraction transformed, XE, and the real volume fraction transformed, X, 1 X E ¼ ln . ð15Þ 1X
q4 ¼ 3 ln
4p 1 1X ðq;T Þ
Z
T
IðhÞG3 ðhÞðT hÞ3 dh.
ð21Þ
Ti
It is now clear that Eq. (8) fails because the integrand depends on the upper integration limit and 4 Z T oq 4p 2 ¼ IðhÞG3 ðhÞðT hÞ dh. ð22Þ 1 oT X ln Ti 1X ðq;T Þ Successive differentiation eventually yields 4 4 14 1 3 ln 1X ðq;T Þ 1 oq sðT Þ ¼ ¼ . 24 oT 4 X 24pIðT ÞG3 ðT Þ
ð23Þ
Even in this case isothermal information can be obtained from continuous cooling data. Unfortunately, this requires a fourth derivative that is nearly impossible to achieve from experimental data. 4.2. General JMAK kinetics
For small volume fractions, say X < 0.1, X XE. So in what follows, ln(1/(1 X)) can be replaced by X for small values of X. When site saturation [41] occurs it is known that the additivity rule is valid [3,5]. The isothermal JMAK result is 1 4pN V 3 ln G ðT Þs3 ðT Þ; ð16Þ ¼ 1 X ðs; T Þ 3
One might adopt a phenomenological approach to reactions that can be formally described by a general JMAK kinetics during an isothermal process, see Eq. (6). From Eq. (6), the isothermal transformation time can be written as 1nðT1 Þ 0 1 ln 1X ðs;T Þ oT A . sðX ; T Þ ¼ ¼@ ð24Þ oq X kðT Þ
where NV is the number of nuclei per unit of volume. One can transform X(s, T) into X(q, T) Z T 3 1 4pN V ln GðhÞ dh . ð17Þ ¼ 1 X ðq; T Þ 3q3 Ti
Although Eq. (6) formally satisfies Eq. (1) for isothermal processes, the above examples show that Eq. (24) may not always work. This happens when f(X, T) is not a function but a functional f(X, T(t)), that is, it depends not only on the current temperature but on the thermal history, T(t). In those cases, the additivity principle may not be valid and Eq. (24) that depends on the validity of the additivity principle may not work as well. In summary, how good or bad this formal approach is depends on how well a particular reaction follows the additivity principle. Bearing these considerations in mind, one can devise a simple method to find k and n when Eq. (6) is a good description of isothermal kinetics. Using Eq. (24) and taking the logarithms 1 oT ln ln ¼ lnðkðT ÞÞ þ nðT Þ ln . ð25Þ 1X oq X
Therefore for X(q, T) = constant, using Eq. (8), Eq. (16) is recovered 0 113 oT 4pN V A GðT Þ1 . ¼ sðT Þ ¼ @ ð18Þ 1 oq X 3 ln 1X ðq;T Þ
So, when the reaction is site-saturated it is possible to directly extract the temperature dependence of the growth rate from continuous cooling data. For temperature dependent nucleation rate, one would not expect the additivity rule to be valid. The isothermal JMAK expression is
P.R. Rios / Acta Materialia 53 (2005) 4893–4901
For a fixed temperature, T = T1, varying X generates a straight line. The slope of this straight line is equal to n(T1). The intercept is equal to ln(k(T1)). By repeating this process for different temperatures n and k are obtained as a function of temperature. If n is a constant, independent of temperature, the reaction becomes isokinetic and the above procedure can be greatly simplified with the help of the results from Section 3. First Eq. (14) can be used to yield !n ! 1 oT ln ln ¼ ln kðT Þ jqðX 0 ; T Þj 1X oq X 0 n ln ðjqðX ; T ÞjÞ;
ð26Þ
where the absolute value of the cooling rate, |q(X, T)|, is taken to avoid taking the logarithm of a negative number, since q < 0 for cooling in this paper. Now in order to find out n it suffices to plot ln ln (1/(1 X)) as a function of ln(|q(X, T)|), for a fixed temperature, T1. In order to determine k(T) one can set X = X0 in Eq. (26) 1 ln 1X 0 kðT Þ ¼ n . ð27Þ oT oq
X0
The derivative needs to be calculated for a fixed volume fraction, X0, as a function of the transformation temperature. The above equation generates k(T), assuming one already obtained n from the previous equation. Both Eqs. (26) and (27) are only valid when n is independent of temperature otherwise the more tedious procedure using Eq. (25) must be used. It is important to stress that it is not necessary to assume any particular functional form for k(T) [31]. The latter may monotonically increase with temperature as it typically does for glass crystallization during continuous heating or have a ‘‘C’’ or ‘‘Gaussian’’ shape when plotted against temperature [31,32]. So, in principle, the present methodology could be used either for continuous heating or continuous cooling data. The advantage of present methodology over previous work is that it can extract isothermal data from nonisothermal data without any mathematical assumptions regarding the functional form of the isothermal transformation curve. One can find out n and k by assuming that isothermal transformation follows Eq. (6) but one might as well assume another convenient isothermal transformation function.
4899
distribution of the heat affected zone during fusion welding [42] or during processing of nanomaterials [43]. For grain growth, the grain size, D, is the natural choice for X. Grain growth kinetics are often described by [44,45] Dn Dni ¼ kðT Þt;
ð28Þ
where X = D is the grain size or the mean intercept length, k is a kinetic temperature dependent factor and Di is the initial grain size. Theory gives n = 2 for pure, texture free materials, but in practice n may be different from the theoretical value and can even depend on temperature [46]. When n is independent of temperature in Eq. (28) the reaction is obviously isokinetic: U ðX ¼ DÞ ¼ Dn Dni in Eq. (2), and the results of Section 3 can be directly applied. Recalling Eqs. (10) and (13) and using Eq. (28) one obtains Dn Dni ¼ ðDn0 Dni Þ
qðD0 ; T Þ ; qðD; T Þ
ð29Þ
where q(D, T) is the cooling rate that results in a grain size D after cooling from an initial temperature Ti down to a temperature T. Eq. (29) gives the grain size as a function of cooling rate for a constant temperature T. The temperature dependent constant k(T) can be found from Dn Dn kðT Þ ¼ i ; dT dq
ð30Þ
D
where D can be arbitrarily chosen. Eqs. (29) and (30) can be used for continuous heating as well. Eqs. (29) and (30) are simpler than the previously proposed methodology for grain growth during heating and cooling [43]. The present methodology has the advantage that it is not necessary to find Eq. (28) from isothermal kinetics. Both n and k can be found directly from non-isothermal, either cooling or heating, experiments with the help of Eqs. (29) and (30). Grain growth is in fact a good example of a reaction in which the kinetics should be independent of thermal history. Indeed, for normal grain growth, it does not matter how a certain grain size was achieved, the grain growth rate depends only on the current grain size and temperature. It is fair to say that other effects may interfere with this, for example, the presence of a strong texture or particles.
4.3. Grain growth
5. Generalized additivity rule
Grain growth is an important phenomenon in polycrystalline materials. The study of grain growth in non-isothermal conditions is of considerable practical relevance. For example, in understanding the grain size
The additivity rule makes use of isothermal kinetics to find out the transformation kinetics when the specimen is cooled or heated with an arbitrary cooling rate. This approach can be generalized for the situation in
4900
P.R. Rios / Acta Materialia 53 (2005) 4893–4901
Dt1 ¼
Tf h . q1
ð34Þ
Using the additivity property Dt1 Dt2 þ ¼ 1; sc1 sc2 substituting Eqs. (31)–(34) into Eq. (35) Tf h h Ti þ ¼ 1. T1 Ti T2 Ti
Fig. 5. Schematic CCT curve represented as a contour plot of X(q, T). It shows the cooling path of a specimen cooled in two steps: (a) cooled from Ti to h with a cooling rate q2 then (b) cooled from h to Tf with a cooling rate q1. The transformation temperature Tf, after this particular two step cooling path, is higher than T1. Depending on the cooling path Tf could be higher or lower than the transformation temperature for the final cooling rate q1.
which a TTT curve is not available but a CCT curve is. In this section, this idea is first illustrated for the simple case of a two-step transformation and then generalized for an arbitrary continuous cooling curve. Suppose that a sample is cooled with a cooling rate that is not constant but depends on the cooling temperature. One wishes to find out the transformation temperature for a temperature dependent cooling rate using CCT data only. The principle is illustrated in Fig. 5. Fig. 5 shows the thermal path of a sample that is cooled from an initial temperature Ti with a constant cooling rate q2 down to an intermediate temperature h . At h the cooling rate instantaneously changes to q1 and the sample is cooled further until it reaches the transformation temperature Tf. One wishes to find out the temperature Tf so that when the sample reaches Tf it will have the desired fraction transformed, say, X = 0.01. This can be done using the same time transformed proportion of the additivity rule for isothermal transformations as follows. If the sample had been cooled with a constant rate q2 < 0 down to the transformation temperature T2 its cooling time sc2 would be sc2 ¼
T2 Ti ; q2
ð31Þ
sc1 ¼
T1 Ti . q1
ð32Þ
The cooling time, Dt2, from Ti to h with q2 is Dt2 ¼
h Ti q2
and the cooling time, Dt1, from h to Tf with q1 is
ð33Þ
ð36Þ
From Eq. (36) Tf can be found for a given h. Notice that the transformation temperature is a function of the cooling rate, T(X, q), in the constant cooling rate CCT curve. The above approach can be generalized for a temperature dependent cooling rate. From Eq. (4): Z X0 dX ; ð37Þ sc ðX 0 ; qÞ ¼ gðX ; qÞ 0 Z X0 Z tc dX dt ¼ ¼ 1; ð38Þ s s ðX ; qÞgðX ; qÞ ðX c 0 c 0 ; qÞ 0 0 where sc(X0, q) is the cooling time necessary to transform to X0 at a constant cooling rate, q, that corresponds to a transformation temperature T(X0, q), as illustrated in Figs. 2 and 5; tc is the total cooling time. Writing the cooling rate as a function of temperature, q(h) Z Tf dh ¼ 1. ð39Þ T ðX 0 ; qðhÞÞ T i Ti T(X0, q) can be found from the experimentally determined CCT curves at constant cooling rate, see Figs. 2 and 5. The integral is carried out so that it equals one at the transformation temperature Tf. Eq. (38) or (39) is the continuous cooling version of the additivity rule. This might be named the nonisothermal additivity rule. It is straightforward to adapt Eq. (39) for continuous heating. Finally, it is worth noting that the result in this section can be combined with the result of Section 3 when the cooling path involves isothermal steps. For example, a specimen is cooled from Ti to h with a cooling rate q(h), then held at h for a time Dt and further cooled from h to Tf with a cooling rate q(h). The transformation temperature Tf for X0 would be Z Tf dh Dt þ ¼ 1; ð40Þ oT T ðX 0 ; qðhÞÞ T i Ti oq
similarly for q1
ð35Þ
X0
where the derivative is evaluated at temperature h. Thus, the methodology presented here allows the determination of the transformation temperature when the cooling rate is not constant, using only data from the CCT curve. The method can be used to calculate the fraction transformed along an arbitrary path involving continuous cooling and isothermal holding steps, relying solely on data from the CCT curve.
P.R. Rios / Acta Materialia 53 (2005) 4893–4901
6. Summary and conclusions In summary, a mathematical method was presented to allow the conversion of continuous cooling transformation data into isothermal transformation data under the restrictive assumption of a constant cooling rate. The method can be applied if the additivity principle is valid. The advantage of the present methodology over previous work is that it can extract isothermal data from non-isothermal data without any mathematical assumptions regarding the functional form of the isothermal transformation curve. The method can be also used for transformations that occur during continuous heating rather than continuous cooling. Special cases in nucleation and growth transformations and grain growth were examined. This analysis showed that in order to be able to fully assess the validity of the method it is necessary to examine the situations in detail. Nevertheless, in engineering practice it is likely that formal methods are going to be applied even if not all preconditions are fulfilled. This approach could be useful to a first approximation. Furthermore, a new additivity rule for non-isothermal transformation, analogous to the additivity rule for isothermal transformations has been introduced. Its use allows the calculation of the fraction transformed along an arbitrary cooling path relying solely on data from the CCT curve. Acknowledgments This work was supported by Conselho Nacional de Desenvolvimento Cientı´fico e Tecnolo´gico, CNPq, Coordenac¸a˜o de Aperfeic¸oamento de Pessoal de Nı´vel Superior, CAPES, and Fundac¸a˜o de Amparo a` Pesquisa do Estado do Rio de Janeiro, FAPERJ. Thanks go to Professor John W. Cahn for helpful discussions on this subject.
References [1] Volterra V. Theory of functionals and of integral and integrodifferential equations. New York (NY): Dover; 1959. [2] Cahn JW. Private communication; 2004. [3] Cahn JW. Acta Metall 1956;4:572. [4] Scheil E. Archiv fu¨r das Eisenhu¨ttenwesen 1935;8:565. [5] Christian JW. The theory of transformations in metals and alloys. 2nd ed. Oxford: Pergamon Press; 1975. p. 545–46.
4901
[6] Umemoto M, Horiuchi K, Tamura I. Trans Iron Steel Inst Jpn 1983;23:690. [7] Umemoto M, Horiuchi K, Tamura I. Trans Iron Steel Inst Jpn 1982;22:854. [8] Verdi C, Visintin A. Acta Metall 1987;35:2711. [9] Homberg DA. IMA J Appl Math 1995;54:31. [10] Homberg DA. Acta Mater 1996;44:4375. [11] Hawbolt EB, Chau B, Brimacombe JK. Metall Trans A 1983;14A:1803. [12] Hawbolt EB, Chau B, Brimacombe JK. Metall Trans A 1985;16A:565. [13] Kuban MB, Jayaraman R, Hawbolt EB, Brimacombe JK. Metall Trans A 1986;17A:1493. [14] Kamat RG, Hawbolt EB, Brown LC, Brimacombe JK. Metall Trans A 1992;23A:2469. [15] Ye JS, Chang HB, Hsu TY. Metall Mater Trans A 2003;34A:1259. [16] Grong O, Myhr OR. Acta Mater 2000;48:445. [17] Bjorneklett BI, Grong O, Myhr OR, Kluken AO. Acta Mater 1998;46:6257. [18] Nordbakke MW, Ryum N, Hunderi O. Philos Mag A 2002;82:2695. [19] Rios PR. Metall Mater Trans A 1996;27A:1132. [20] Seixas UR, Rios PR. ISIJ Int 1997;37:1040. [21] Leblond JB, Devaux JA. Acta Metall 1984;32:137. [22] Lusk M, Jou HJ. Metall Mater Trans A 1997;28A:287. [23] Todinov MT. Metall Mater Trans B 1998;29B:269. [24] Zhu YT, Lowe TC. Metall Mater Trans B 2000;31B:675. [25] Wierszyllowski IA. Metall Trans A 1991;22A:993. [26] Reti T, Felde I. Comput Mater Sci 1999;15:466. [27] Liu F, Sommer F, Mittemeijer EJ. J Mater Sci 2004;39:1621. [28] Kempem ATW, Sommer F, Mittemeijer EJ. J Mater Sci 2002;37:1321. [29] Mittemeijer EJ. J Mater Sci 1992;27:3977. [30] Mittemeijer EJ, Cheng L, van der Schaaf PJ, Brakman CM, Korevaar BM. Metall Trans A 1988;19A:925. [31] Pont D, Bergheau JM, Rochette M, Fortunier R. In: Bui HD, editor. Inverse problems in engineering mechanics: 2nd international symposium. Leiden: A.A. Balkema; 1994. p. 151. [32] Zhu Y, Ye JS, Chang HB, Hsu TY. Metall Mater Trans A 2003;34A:1264. [33] Vasquez J, Wagner C, Villares P, Jimenez-Garay R. Acta Mater 1996;44:4807. [34] Borrego A, Gonzalez-Doncel G. Mater Sci Eng A 1998;245A:10. [35] Graydon JW, Thorpe SJ, Kirk DW. J Non-Cryst Solids 1994;175:31. [36] Johnson WA, Mehl RF. Trans AIME 1939;13:416. [37] Avrami MJ. Chem Phys 1939;17:1103. [38] Avrami MJ. Chem Phys 1940;8:214. [39] Avrami MJ. Chem Phys 1941;9:177. [40] Kolmogorov AN. Izv Akad Nauk USSR – Ser Matemat 1937;1:355. [41] Cahn JW. Acta Metall 1956;4:449. [42] Ashby MF, Easterling KE. Acta Metall 1982;30:1969. [43] Bourell DL, Kaysser W. Acta Metall Mater 1993;4:2933. [44] Burke JE. Trans Metall Soc AIME 1949;180:73. [45] Burke JE, Turnbull D. Prog Metal Phys 1952;3:220. [46] Rios PR. Metall Mater Trans A 1994;25A:221.