An analytical description of overall phase transformation kinetics applied to the recrystallization of pure iron

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Acta Materialia 59 (2011) 7538–7545 www.elsevier.com/locate/actamat

An analytical description of overall phase transformation kinetics applied to the recrystallization of pure iron B.B. Rath ⇑, C.S. Pande US Naval Research Laboratory, Washington, DC 20375-5341, USA Received 12 July 2011; received in revised form 16 September 2011; accepted 19 September 2011 Available online 15 October 2011

Abstract Experimental studies of recrystallization in deformed single crystals of pure iron are described. The results are used to analyze various parameters associated with the time evolution of the volume fraction of the growing phase during phase transformations in this system associated with the phenomena of nucleation and growth. In addition, using the experimental results, the phenomenon has been modeled by a new approach which may provide a different, and possibly more precise, description of the kinetics of the process. The proposed analytical approach uses easily measured metallographic parameters, obtained from a systematic two-dimensional surface examination, to provide a detailed description on the time dependence of nucleation, nucleation rate, growth rate and interfacial migration and compared with the classical approach based on Kolomogrov formalism. Published by Elsevier Ltd. on behalf of Acta Materialia Inc. Keywords: Annealing; Nucleation of recrystallization; Iron; Kinetics; Deformation structure

1. Introduction Thermally activated phase transformations are widely studied in metallurgy. They are characterized either by interface migration (e.g. recrystallization and grain growth) or by long-range (“diffusive”) transport (e.g. precipitation and dissolution). Their kinetics can be best described by a nucleation and growth process. Based on early microstructural observations, it has been found convenient to describe such a transformation in time by two main parameters: the frequency of the nucleation of the new phase and its subsequent rate of growth at the expense of the disappearing matrix. In order to describe accurately the kinetics of such a process, impingement between the growing phases must be taken into account. After approximate early models [1–3], the first major development, which provided an understanding of the kinetics of such transformations, resulting from a series of careful observations by various ⇑ Corresponding author. Tel.: +1 202 767 3566; fax: +1 202 404 1207.

E-mail address: [email protected] (B.B. Rath).

investigators on the isothermal transformation behavior in different single- and multicomponent systems, was due to Kolmogorov [4], Avrami [5] and Johnson and Mehl [6] (the so-called KJMA model). The well-known isothermal time–transformation relationship proposed by Avrami states that: X V ¼ 1  expðBtk Þ;

ð1Þ

where the XV is the volume fraction of isothermally transformed phase at time t, B is a temperature-dependent parameter related to growth rate, nucleation frequency and a shape factor, and k, which generally varies between 1 and 4 [7], is a number that is expected to be dependent on the dimensionality of growth of the nuclei. The dependence of k on the detailed nature of the nucleation process has been treated by Cahn [8], who showed that, in addition to morphology, the rate of nucleation has a significant effect on k. Although the time independence of growth rate, G, is implicit in Eq. (1), this equation has been extensively used since 1939 in all transformation kinetics studies, regardless of whether the growth rate is constant or not. Eq. (1) therefore should be considered semi-empirical in

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nature. If growth rate and nucleation rates are not constant it is more appropriate to use the Kolmogorov relation, which will be described now. It should be noted that Kolmogorov’s model of kinetics of transformation precedes the work of Avrami [5] and Johnson and Mehl [6]. The relationships between the transformed volume fraction and time at a given temperature describe the process of atomic reorganization by defining that (i) the newly formed phase nuclei are randomly distributed throughout the volume in a statistical sense, and (ii) the real transformed volume is related to an extended volume fraction, XVex, defined as the volume fraction that would result if no account were taken of impingement between the growing grains or phase. In differential form, one therefore gets: dX V ¼ 1  XV ; dX Vex

ð2Þ

where XV is true volume fraction transformed and XVex is the is the volume transformed of a grain or nucleus if allowed to grow unimpeded through other nuclei. Furthermore, defining two transformation parameters such as nucleation frequency, N_ , and interface migration rate, G, the authors showed that XVex (extended volume fraction) can be described as: Z t N_ ðnÞ/ðt  nÞdn; ð3aÞ X Vex ¼ 0

where /(t  n) is the volume of the new grain after its nucleation at time t = n. Assuming that the initial growth is spherical (isotropic): 4 3 /ðt  nÞ ¼ pG3 ðt  nÞ½ðt  nÞ : 3

ð3bÞ

Although Eq. (3a) is an exact geometric description, one needs to know the time dependence of N_ and growth rate G to determine the kinetics of the process. On the other hand, if G or N_ or both are assumed constant, the resulting equation for XV is not expected to be either realistic or universally applicable. Substituting Eq. (3a) in Eq. (2) we obtain the equation developed by Kolmogorov that allows for a general growth rate dependence instead of the Avrami expression (1). The relationship developed by Kolmogorov for the fraction transformed at any time t thus is:  Z t  _ X v ¼ 1  exp  N ðnÞ/ðt  nÞdn : ð4Þ 0

Eq. (4) is an exact result but not very useful in practice since it requires a knowledge of nucleation rate which is not usually available. (A time-dependent value for number of transformed nuclei cannot be directly obtained from a two-dimensional microstructural analysis.) We next consider an alternative description of the kinetics [9] which also provides an exact relationship between average growth rate, G, and volume transformed, XV, as a function of time. Its compatibility with the Kolmogorov and Johnson– Mehl–Avrami formalisms will also be discussed. In order

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to provide experimental underpinning to the concepts discussed, we will illustrate our theoretical discussion with our experimental results on recrystallization studies in pure and deformed iron single crystals. On annealing such systems, primary recrystallization takes place due to a thermally activated microstructural evolution whereby a new population of strain free grains are nucleated and grow, eventually replacing the deformed microstructure. 2. A model based on the evolution of interface area The method is based on the exact relationship between the average growth rate, G, the volume of the transformed phase, XV, and the area of the migrating interface per unit volume, SV [10], and is given as:   1 dX V G¼ : ð5Þ SV dt However, before Eq. (5) can be used to derive the relationship between XV and t, the dependence of the two variables G and SV on XV or t has to be established. The growth rates, often determined from experimental observations, such as recrystallization and other phase transformations [11,12], have been shown to be often time dependent. In order to establish the dependence of the second variable in Eq. (5), SV, with time or XV, an empirical relation between these parameters was first proposed by Speich and Fisher [13] as: S V ¼ K S X V ð1  X V Þ;

ð6Þ

where KS is a constant. This form of the equation indicates that, during the early stages of transformation (XV  0), the shape of the growing grain is the dominant factor, whereas, in the late stages (XV  1), it is the shape of the shrinking matrix, (1  Xv), that defines the equation. Although this quadratic form describes the interfacial area dependence on XV experimentally evaluated from the recrystallization study in 60% cold-worked polycrystalline steel with 3.25% Si, it has two major problems. The first, pointed out by Cahn [14], is that substituting SV from Eq. (6) into Eq. (5) leads to improbable values for growth rates, i.e. G ! 1 as X V ! 1 or 0. Secondly, the above relation with exponent 1 for both XV and (1  XV) suggests that the shapes for both growing nuclei in its early stage and shrinking matrix in the late stage of recrystallization are cylindrical in shape. Cahn [14] has suggested exponents of 2/3. Rath [9] found that the exponents for XV and (1  XV) are different from 1, and from each other. On this basis, he suggested the following general form for SV: S V ¼ K S ðX V Þn ð1  X V Þm ;

ð7Þ

where KS is a constant, n and m are parameters that characterize the growth process and are positive and <1 and, in general, n – m. Substituting this form of SV in Eq. (5) leads to: Z t Z XV dX V Gdt ¼ ð8Þ n m: K S ðX V Þ ð1  X V Þ 0 0

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The right side of Eq. (8) is related to the regularized incomplete beta function, Ix(p, q) defined as: Z X 1 ðq1Þ I X ðp; qÞ ¼ S ðp1Þ ð1  SÞ dS; ð9Þ Bðp; qÞ 0 where p and q are constants, and B(p, q) is the complete beta function. The beta function can be expressed in terms of the more familiar gamma function (C), where: Bðp; qÞ ¼

CðpÞCðqÞ : Cðp þ qÞ

ð10Þ

Series expansion for this function for X 6 0.5 is given as: " # kX ¼1 X p ð1  X Þq Bðp þ 1; k þ 1Þ kþ1 X IX ¼ 1þ ð11Þ Bðp þ q; k þ 1Þ pBðp; qÞ k¼0 for 0 6 X 6 1. For larger X (1.0 P X P 0.5) the following property of the incomplete beta function can be used: I X ðp; qÞ ¼ 1  I ð1X Þ ðp; qÞ:

ð12Þ

Transposing to the microstructural parameters, where X corresponds to XV, p = 1  n, and q = 1  m, it can be shown that: Z 0

t

Gdt ¼

" # kX ¼1 ðX V Þ1n ð1  X V Þ1m Bð2  n; k þ 1Þ : 1þ X kþ1 ð1  nÞK S Bð2  n  m; k þ 1Þ V k¼0 ð13Þ

For large XV (XV P 0.5), it is appropriate to use Eq. (12): Z t 1n 1m 1 ðX V Þ ð1  X V Þ Gdt ¼  KS ð1  mÞK S 0 " # kX ¼1 Bð2  m; k þ 1Þ kþ1 ð1  X V Þ 1þ : ð14Þ Bð2  n  m; k þ 1Þ k¼0 It is easily seen from Eqs. (13) and (14) that n and m must be bounded between 0 and 1, consistent with the earlier statement. 3. Comparison of Kolmogorov and Rath models We utilize the formulation developed by Kolmogorov discussed before, which allows for a general growth-rate dependence instead of the Avrami expression. The relationship developed by Kolmogorov for the fraction transformed at any time t is given by Eq. (4). The specific area, Sv, is given in terms of the growth and fraction transformed rates as shown by Cahn and Hagel (Eq. (5)). Substituting Eq. (4) into (5) leads to: Z t ð1  X v Þ 2 Sv ¼ 4pN_ ðnÞG2 ðt  nÞ½ðt  nÞ GðtÞ 0    dGðt  nÞ þ Gðt  nÞ dn : ð15Þ ðt  nÞ dt

Eq. (15) will be examined for physically realizable nucleation and growth-rate conditions and subsequently compared to the formulation developed by Rath (see Eq. (7)). In many growth phenomena, all of the potential nuclei are present instantaneously upon completion of the deformation or transformation process under appropriate thermodynamic conditions. Mathematically, this sitesaturation of nuclei is given as: N_ ðnÞ ¼ N o dðnÞ;

ð16Þ

where No is the total number of nuclei per unit volume and d(n) is the Dirac delta function. Thus, Eq. (15) becomes: 1=3

S v ¼ ð36pN o Þ ð1   2=3  1 t dG  X v Þ ‘n 1þ : 1  Xv GðtÞ dt

ð17Þ

Experimentally, it has been shown that the growth rate can be described (see Section 4) as: GðtÞ ¼

1 ; a þ bt

ð18Þ

where a and b are constants. Using Eq. (18) in (17), for the case of site-saturated nucleation:  2=3 1 1=3 S v ¼ð36pN o Þ ð1  X v Þ ‘n 1  Xv (  1=3  1=3 ) 3 1 1b ‘n : ð19Þ 4pN o 1  Xv Some of the parameters appearing Eq. (7) can now be compared and correlated with Eq. (19). For Xv  1, Ks = (36 p No)1/3 and n = 2/3 which is reflected in the initial growth being equiaxial in shape. The exponent, m, is more difficult to determine due to the logarithmic singularity as Xv approaches 1, and such cannot be explicitly determined from any simple limiting process. It generally has to be determined by other means (see below). Eq. (19) is rescaled to be compatible with Eq. (7). Using:  1=3 3 1=3 K s ¼ ð36pN o Þ and c ¼ b ; ð20Þ 4pN o  2=3 (  1=3 ) 1 1 S v ¼ K s ð1  X v Þ ‘n 1  c ‘n ; 1  Xv 1  Xv ð21Þ where c is a measure of the time-dependent growth rate. Shown in Fig. 1a–c are comparisons of Eq. (7), labeled the “Rath formulation”, and Eq. (21), labeled “Kolmogorov”, for several values of c. For each value of c, the values for m and n were obtained by a least-squares fit with Eq. (12). Since Ks appears as a constant in both Eqs. (7) and (21), it was set to 1 for comparison purposes. Recall that an increasing c will result in a greater time dependence on the growth rate. In Fig. 1a–c, the agreement between

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Fig. 2. Correspondence between the m and n in the Rath formulation with that of c from the Kolmogorov approach.

the two formulations is good even when the growth rate is a function of time. Fig. 2 shows the dependence of n and m from Eq. (7) on c. These values increase smoothly and linearly as c increases. For large values of c > 0, it should be noted that Eq. (21) will become negative, yielding a non-physical result. Also even for a small c, Sv in Eq. (21) becomes slightly negative when Xv approaches 1; however, Eq. (7) does not present any of these difficulties. As an indicator of this problem, the agreement worsens (see Fig. 1a–c) as c increases from 0.01 to 0.1 and then 0.25, but is still acceptable for most experimental observations. The kinetics of the transformation process can be examined using Eq. (7) in the Cahn–Hagel expression (Eq. (5)) and comparing this to the Eq. (4) under the assumptions of site saturation of nuclei and growth law given by Eq. (18). When scaled using the values in Eq. (20), Eq. (4) becomes: (  3 ) s b X v ¼ 1  exp  ð22Þ withs ¼ t; cð1 þ sÞ a where a and b appear in the growth rate, Eq. (18). From the Cahn–Hagel expression, Eq. (5), we find that s is given in terms of the volume fraction: hc i s ¼ exp bðX v ; 1  n; 1  mÞ  1; ð23Þ 3 where bðX v ; 1  n; 1  mÞ ¼

Z

Xv

m

wn ð1  wÞ dw;

ð24Þ

0

Fig. 1. (a–c) Comparison of Eq. (7) with the Kolmogorov formulation. Listed are the best-fit values of m and n for three values of c: 0.01, 0.10 and 0.25.

is the generalized incomplete beta function. Fig. 3a–c show Xv, the volume transformed as function of t, the rescaled time, using the same values for n, m and c as in Fig. 1a– c. From these figures, the agreement between the two formulations is excellent. The formulation presented by Rath

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Fig. 4. Inverse growth rates as a function of scaled time as determined from experimental data (a) for several temperatures. Normalized plot of inverse growth rates as a function of scaled time. (b) The constants a and b obtained from of the data at each temperature from (a) are utilized for this purpose.

is an attractive alternative to the formalism developed by Kolmogorov and the empiricism inherent in the Avrami approach. 4. Experimental

Fig. 3. (a–c) Fraction of volume transformed as function of scaled time. Comparison between the Rath and Kolmogorov methods for three values of c: 0.01, 0.10 and 0.25.

The formalism for the overall kinetics of transformation, as presented here, has been experimentally verified through a systematic study using decarburized FerrovacE, seed-oriented iron single crystal. The experimental results show excellent agreement with the developed formalism and provide an appropriate function describing the time dependence of growth rate, G. As an illustration, the applicability of the formulation is examined using the

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experimental data on iron. The growth rates, evaluated at each temperature, were monitored over time. From Fig. 4a, which shows a fit of the inverse of G for each temperature as a function of time, a and b of Eq. (18) were determined and then plotted for a series of temperatures as shown in Fig. 4b. It has been shown from extensive studies of the rolling texture in polycrystalline iron [15–23] that one of the dominant components of the rolled texture corresponds to ð1 1 1Þ½ 1 1 2 orientation, which refers to a major fraction of the grains having (1 1 1) parallel to the rolling plane and ½ 1 12 parallel to the rolling direction. Polycrystalline iron samples were repeatedly cold rolled and annealed and chemically etched to final dimensions of strips 2.5 cm wide, 30 cm long and 0.15 cm thick to produce an uniform grain size of approximately 35 lm. X-ray pole figure measurements confirmed the presence of ð1 1 1Þ½ 1 1 2, as a major texture component in the deformed samples. The strips were then critically stretched to 1.8% to introduce a small strain. In order to grow seed-oriented single crystals, with an orientation of (1 1 1) parallel to the plane of the strip, a furnace was designed and used with a sharp temperature gradient from 25 to 890 °C (below the transformation temperature). The procedure used was identical to the method developed by Dunn and Nonker [24]. These single crystals were cold rolled in the [1 1 2] rolling direction to 70% thickness reduction. Because the crystal was rolled along a stable orientation, it was expected to exhibit minimal lattice rotation and strain inhomogeneity. X-ray pole figures confirmed the stability of the crystal orientation with minimal spread around ð1 1 1Þ½ 1 1 2. The strain homogeneity was also verified by optical and transmission electron microscopy. The deformed single-crystal matrix provided random nucleation of recrystallized grains, growing equiaxially into the matrices during subsequent annealing [9]. Samples were isothermally annealed at six different temperatures between 450 and 600 °C for various time intervals. Topological parameters such as volume fraction of recrystallization, XV, interfacial areas for migrating interfaces, SV, and the rate of growth (to obtain G) of the largest unimpinged grain were measured by quantitative metallography, using method applied by Hillard and Cahn [25]. In order to obtain a statistically representative data set, a high-resolution traveling sample stage coupled to a digital acquisition system was built to evaluate all parameters and to distinguish the migrating interfaces from the impinged ones. All measurements were conducted on the center plane of the samples. The relation between XV and SV measured as a function of temperature and time is shown in Fig. 5. It should be noted that an unambiguous determination of nucleation rate is usually very difficult to obtain from a two-dimensional metallographic analysis. An approximate method, which uses two-dimensional stereological relationships, is sometimes used for measurements carried out on the plane of polishing. However, this requires knowledge of the shape of the particles [26]. Gokhale and

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Fig. 5. Experimentally determined Sv and Xv data for a range of temperatures. Shown are the formulations of Rath and Kolmogorov.

DeHoff [27], and Vandermeer and Rath [28–30] used a mathematical procedure based on Laplace transform techniques to obtain an expression for both nucleation and growth rates. This procedure depends on the parameters of the so-called extended structure and also requires assumptions about the growth geometry and shape. It also assumes that the nucleation is homogeneous. It should also be noted that the global equations used by Gokhale and DeHoff [27] for extended structures are strictly valid only for a random distribution of nuclei, all of which were indeed the case for recrystallization in deformed single crystals of iron [28–30]. The experimental work presented here is a continuation of experiments described in Ref. [28,29]. Random nucleation or equiaxial growth of recrystallized grains growing into the deformed matrix is not commonly observed in most experiments. Along with experimental data, the formulations developed by Rath (Eq. (7)) and by Kolmogorov (Eq. (21)) are given based on the assumption that the Ks term is the same in both cases. The description of the experimental data is quite adequate for both formulations, indicating the validity of the more phenomenological approach given in Eq. (7). The method proposed has wide applicability and is quite general. However, it does have a few drawbacks. The exponents n and m have to be determined from experiments by measuring SV as a function of XV. In addition, as stated earlier, the functional dependence of G with time needs to be independently evaluated from experiments. 5. Discussion There is no doubt that KJMA theory and its subsequent extensions [32–41] have found widespread usage. Although initially applied to the systems undergoing first-order transitions, KJMA theory has been more or less has been suc-

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cessfully applied to a variety of transformations [42–48]. Even in cases where nucleation and growth retained only a purely operational meaning or where some of the assumptions of the Kolomogrov model are marginally or grossly violated. KJMA essentially deals with transformations taking place in the interior of very large samples, so that the effect of the specimen external surfaces is not taken into account. Cahn [31] introduced the concept of the time cone or causal cone to deal with such cases. Furthermore, KJMA is usually limited to very simple and traditional approaches to microstructural characterization. A key assumption of the KJMA theory is that nuclei are uniformly randomly located in space. However, there are experimental cases where nuclei are not homogeneously distributed in space but are located in clusters or in an inhomogeneous fashion. It is now realized that a need to couple not only the shape of microstructural features, but also their crystallography may sometimes be critical. Also, grain boundary plane orientation and plane topology along with the classical issues of misorientation need to be coupled into the KJMA formalism. In addition, recrystallization phenomena in textured body-centered cubic materials is heavily governed by interactions between grain boundary mesotexture, microtexture, coincident site lattice boundary decomposition, and boundary migration issues, all of which play an important role in the recrystallization process, and we certainly do not have an experimental system with a uniform boundary model. All this was a precursor to present day 3-D electron backscatter diffraction (EBSD) techniques and time-consuming serial sectioning process. How 3-D shape and crystallographic information can be coupled or captured in the KJMA formulation remains a major challenge. In recent years there have been several attempts to take these factors into account. For example Rios et al. [49] used recent developments in stochastic geometry [50–52] to revisit the classical KJMA theory and have generalized the Kolomogrov model to situations in which nucleation took place not only for homogeneous but also for inhomogeneous Poisson point processes, i.e. in which nuclei are located in space in either homogeneous or inhomogeneous fashion. In such a situation the familiar relationships between mean volume density and extended volume density are not valid. Needless to say, these improvements to the KJMA theory add complexity and may need additional experimental information not easily available. In such cases the Rath approach presented here may provide an equivalent, or in some cases better, alternative to understanding the phenomenon of nucleation and growth in the sense that these complexities are all included in the factors n and m. 6. Summary This paper has two aims. The first is to show that there is an alternative mathematical treatment for an unambiguous description of the overall kinetics of the nucleation and growth processes using experimentally observed behavior

during recrystallization in addition to that provided by Kolmogorov and also by many subsequent workers popularly known as the KJMA model. A key assumption of the KJMA theory is that nuclei are uniformly randomly located in transforming volume, and the growth rate and nucleation rate are known. We show that under these conditions the treatment based on the Rath formulation is equivalent. We illustrate this for experimentally observed parameters. The two approaches presented here provide all relevant information on the kinetics of the process from the aforementioned measured quantities in the study of the kinetics of a nucleation process. Furthermore, it is shown that nucleation rate, which has thus far been elusive, can be easily evaluated from simple microstructural parameters without resorting to a highly time-consuming serial sectioning process. During a study of the solid-state transformation process, experimental researchers are therefore encouraged to evaluate the proposed variables, which are simply obtained from a two-dimensional surface using well-known quantitative metallographic methods. The second purpose is to extend the method to situations where the assumptions used in the Kolmogorov treatment are not satisfied. We show that such cases can also be dealt with by using Rath formulation if the exponents m and n can be obtained by conventional quantitative metallography by measuring the time dependence of the transformed volume, XV, the area of migrating interfaces, SV, and the size of the largest nucleus. This is possible since no assumption is made about the nucleation process in the Rath formulation. Acknowledgements The authors wish to acknowledge many valuable discussions with Dr. Robert Masumura, Dr. Khershed Cooper, and Mr. Jeffrey Wolla. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

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