Interface mobility in case of the austenite-to-ferrite phase transformation

Computational Materials Science 37 (2006) 94–100 www.elsevier.com/locate/commatsci

Interface mobility in case of the austenite-to-ferrite phase transformation E. Gamsja¨ger

a,*

, M. Militzer b, F. Fazeli b, J. Svoboda c, F.D. Fischer

a,d

a Institut fu¨r Mechanik, Montanuniversita¨t Leoben, Franz Josef Strasse, 18, A-8700 Leoben, Austria The Centre of Metallurgical Process Engineering, University of British Columbia, Vancouver, Canada V6T 1Z4 Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Zˇizˇkova 22, CZ-616 62 Brno, Czech Republic d Erich Schmid Institute of Materials Science, Austrian Academy of Sciences and Institute of Metal Physics, Montanuniversita¨t Leoben, Jahnstr. 12, A-8700 Leoben, Austria b

c

Abstract The kinetics of the austenite(c)-to-ferrite(a) phase transformation in steel grades with a sufficiently small amount of substitutional and interstitial components (e.g., ultra low carbon and interstitial free steels) is assumed to be controlled by the interface reaction only. Longrange diffusion of the components does not influence the kinetics. Experimental data (e.g., from dilatometer tests) provide the time dependence of a quantity related to the volume fraction of ferrite. For the steel grade considered in this work simple, however, realistic geometrical assumptions (e.g., planar growth, spherical growth) are made with respect to the shape of the ferrite phase. The numerical results imply planar growth for the initial transformation stages. Based on this analysis the effective mobility of the a–c interface is determined. Furthermore, reasonable values for the intrinsic, thermally activated mobility are found by considering solute drag.  2005 Elsevier B.V. All rights reserved. Keywords: Ferrite; Austenite; Steel; Phase transformation

1. Introduction The kinetics of the c/a phase transformation in steels can be controlled by the diffusion of the components [1] or by the interfacial reaction [2] or by both mechanisms [3]. It is also possible that the transformation mechanism changes during the transformation process [4,5]. By means of recently developed, physically based sharp interface models the transformation kinetics can be calculated in binary [3], and ternary or higher order systems with immobile substitutional components [6]. In addition, the transformation kinetics has been modelled in iron alloys with substitutional components only [4]. From the well established tracer diffusion coefficients the Onsager coefficients can be determined in a multicomponent substitutional

*

Corresponding author. Tel.: +43 3842 402 4006; fax: +43 3842 46048. E-mail address: [email protected] (E. Gamsja¨ger).

0927-0256/$ - see front matter  2005 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2005.12.011

alloy. Numerical difficulties occur in models taking both the fast interstitial and the much slower substitutional diffusion into account. An attempt to consider both interstitial and substitutional diffusion has been made in Fe–C– X alloys with a particular amount of component X. The kinetics is determined by a finite interface mobility, longrange carbon diffusion and short-range X-diffusion in front of the migrating interface [7]. That means that the site fractions of the substitutional components remain constant in both phases except for the spike piling up in front of the interface and dissipating Gibbs energy. However, in all these models the mobility of the a–c interface serves as a fit parameter. The suitable way to gain information on the interface mobility is to investigate systems under conditions where the kinetics is controlled by the interface reaction. In 1975 Hillert [8] had already estimated the mobility of the interface from grain growth and recrystallization experiments in pure iron, denoting his procedure as a crude

E. Gamsja¨ger et al. / Computational Materials Science 37 (2006) 94–100

attempt. In Fe–X alloys with a sufficiently small amount of substitutional X the kinetics is controlled by the interface mobility. These systems have been examined experimentally, see Krielaart and van der Zwaag [9], Wits et al. [10], Vooijs et al. [11], Kop [12]. However, the values for the interface mobility reported in these studies are orders of magnitude smaller than the value proposed by Hillert [8] and, moreover, they vary strongly from system to system. Sophisticated in situ techniques such as hot-stage transmission electron microscopy can be applied to measure the interface velocity directly and, thereby, for a known driving pressure (for ‘‘driving pressure’’ the synonymous expression ‘‘driving force’’ can frequently be found in the literature), the interface mobility can be obtained [13]. Unfortunately, these experiments are limited to a small observation range. Conventionally, the time-dependent volume fraction of ferrite can be acquired from dilatometer tests or differential thermal analysis, whereas the evolution of the c/a phase arrangement cannot a priori be obtained by these experiments. Thus, a comprehensive description of the situation is still missing. The aim of this work is to predict the effective interface mobility Meff in an ultra low carbon steel as a function of temperature T. Thus, the calculations have to show unambiguously that the assumptions on the shape of the growing a-phase are consistent with the data.

n X

xai ½½li ;

Experimental data observed by dilatometer tests on a continuously cooled ultra low carbon steel [14] are analyzed. It is assumed that nucleation of the new phase ferrite preferably occurs at the austenite grain boundaries and that nucleation is completed at the beginning of the growth process. Then, the experimental data reflect the growth kinetics only, and the nucleation process can be neglected. In addition, the final ferrite grain sizes are larger than the austenite grains. Ferrite nuclei may grow as spheres or if the nuclei coalesce as plates. For non-equilibrium processes close to equilibrium a linear relation between the interface normal velocity v and its driving pressure Df, which can be derived from Onsager’s extremal principle, can be assumed, ð1Þ

as described by Svoboda et al. [4]. In [7] it is shown that for a sufficiently high undercooling and a small amount of Mn the formation of the substitutional spike in front of the advancing interface becomes unrealistic. The half-thickness of these spikes would be in the same range or less than the thickness of the interface. In that case substitutional diffusion does not influence the kinetics. The material is entirely transformed before substitutional spikes develop or, even more unlikely, bulk diffusion can occur. In general the chemical driving pressure Dfchem available as the relevant contribution to Df for the migration of the c/a-interface

ð2Þ

i¼1

is described by [15–17]. The substitutional components are labelled from 1 to n with 1 being the main component iron. The mole fractions in ferrite and the chemical potentials are denoted by xai and li, respectively. The jump of the chemical potentials for component i at the a–c interface is denoted by [[li]]. The exact composition of the investigated steel grade is given in Table 1. Due to microalloying with titanium virtually no interstitials (carbon, nitrogen) remain in solution. As a result, a binary Fe–Mn steel with a mole fraction xMn = 1.12 · 103 is considered for calculating the driving pressure Df. The thermodynamic data for calculating the chemical potentials li in both phases and, thereby, the driving pressure Dfchem are obtained from [18,19]. For immobile substitutional components the ratios Ki of the mole fractions of the solute components to the mole fraction of iron have to remain constant values during transformation in both phases, i.e., x1 ð3Þ K i ¼ ¼ const: xi Following [17,20] the mechanical part of the driving pressure Dfmech can be calculated as Dfmech ¼ hV m i 

2. Theory

v ¼ M eff Df

Dfchem ¼

95

dW ; dV a

ð4Þ

where hVmi is the average value of the molar volumes of both phases. The total mechanical work, denoted by W, has been calculated in [20], the derivative dW/dVa remains almost constant during transformation. Kempen et al. [21] propose that the mechanical driving pressure or more precisely its retarding pressure is in the same order of magnitude as the chemical driving pressure. However, for realistic values of the chemical driving pressures, e.g., Dfchem being in the order of 100 J mol1 or even much higher, the mechanical driving pressure which is about 10 J mol1, is rather small. Furthermore, it has to be noted that the stresses relax very fast by diffusion processes at the temperatures of the c/a phase transformation in ultra low carbon steels and, thus, the mechanical term is neglected here with the consequence that Df is approximated by Dfchem. In the papers of the van der Zwaag group [9–12] Dfmech is also neglected. In this work relatively simple assumptions concerning the shape of the evolving a-phase are made to calculate the interface velocity. The time-dependent volume fraction of ferrite n(t) is determined by analyzing the data observed

Table 1 Chemical composition of the investigated steel C

Mn

P

S

Si

Ti

Nb

Al

N

0.002

0.11

0.01

0.008

0.01

0.059

0.009

0.033

0.0041

Values are mass fractions in %.

96

E. Gamsja¨ger et al. / Computational Materials Science 37 (2006) 94–100

from dilatometer tests. The function n(t) can be described accurately by the following analytical function a ð5Þ nðtÞ ¼ 1 þ b expðctÞ with t being the transformation time and a, b and c are fitparameters which depend on the cooling rate dT/dt. Then, the transformation rate dn/dt is given by dn _ abc expðctÞ ¼n¼ : 2 dt ½1 þ b expðctÞ

ð6Þ

With respect to the c/a phase arrangement three different choices are made as shown in Fig. 1: (i) spherical growth, (ii) planar growth of one plate and (iii) planar growth of two coalescing plates. The last configuration is a simple way to take impingement effects into account. Independent of the geometrical assumptions the transformation rate n_ is obtained from the experimental data via the fit-function Eq. (6) and the driving pressure Df is calculated using Eq. (2). As a result the effective mobility Meff can be calculated for the three different shapes of the growing ferrite.

For planar growth the volume fraction of ferrite can be expressed in terms of the actual interface position s(t) as n = s(t)/d with d being the half grain size. Thus, the interface velocity v(t) is related to the transformation rate by n_ ¼ vðtÞ=d. The interface velocity can be substituted using Eq. (1), which results in the following relation for the mobility as M eff ¼

d _ n: Df

ð7Þ

For spherical growth n = s3/d3 with d being the radius of the final ferrite sphere. From the transformation rate n_ ¼ 3s2 vðtÞ=d 3 the mobility can be calculated, i.e., M eff ¼

d3 _ n: 3s2 Df

ð8Þ

The volume fraction of ferrite in case of two coalescing plates is n = (4sd  s2)/4d2, and again from the transformation rate n_ the mobility is derived as M eff ¼

2d 2 _ n: ð2d  sÞDf

ð9Þ

3. Evaluation of the experimental data Results from dilatometer tests for an ultra low carbon steel are reported in [14]. In dilatometric studies the displacement is measured versus temperature. In this case the volume fraction versus time can be calculated by the lever rule. Data are available for four different cooling conditions. These data are analyzed for the following nominal cooling rates, 1 K s1, 10 K s1, 55 K s1 and 195 K s1, respectively. The experimental data are fitted by the function of Eq. (5), and the fit-parameters are listed in Table 2. During the dilatometer tests the time–temperature data pairs are collected. These data are fitted by polynomial curves with the effect that the noise associated with the measurement of the temperature is eliminated. Fig. 2 shows the experimental volume fraction n as a function of the temperature. The solid curves in Fig. 2 are obtained by using both the derivative of the fit-function (Eq. (6)) and the time–temperature function. For each cooling curve the mobility Meff is calculated for the three different c/a phase arrangements considered here. The c-grain size before phase transformation is 40 lm. The grain sizes of the final a-grains show a decreasing tendency with increasing cooling rate, 111 lm at 1 K s1, 81 lm at 10 K s1, 65 lm at 55 K s1 and 48 lm at 195 K s1 [14].

Table 2 Fit-parameters a, b and c at different nominal cooling rates

Fig. 1. Independent planar growth, planar growth of two coalescing plates and spherical growth of ferrite.

Cooling rate/K s1

a

b

c

1 10 55 195

1.07 1.02 0.97 1.10

90.6 94.5 65.0 40.0

0.16 0.89 3.02 9.72

E. Gamsja¨ger et al. / Computational Materials Science 37 (2006) 94–100

Fig. 2. Volume fraction of ferrite versus temperature at different cooling rates, symbols indicate experimental data from [14].

Since the ferrite grain sizes are all larger than the initial austenite grains the growth of spherical shells from the austenite grain boundaries to the center can be ruled out for modelling the transformation kinetics. It is more likely that plate-like or spherical nuclei grow until the microstructure consists of a-grains with an average grain size of 2d. The effective mobility is plotted versus temperature for the different cooling rates in case of growing a-spheres, see Fig. 3 and for plate-like growth, see Fig. 4. It is obvious that the values for Meff in Fig. 3 depend strongly on the cooling rate. As far as Dfchem only depends on temperature and not on the thermal history, the interface velocity as well as the effective mobility should not depend on the cooling rate. Thus, in this case spherical growth of ferrite appears to be unrealistic. Spherical growth may occur only in the first stages of transformation at small volume fractions of ferrite. As soon as ferrite has formed plates along the grain boundaries the growth proceeds by thickening of these plates.

Fig. 3. Calculation of the effective mobility in case of growing a-spheres for different cooling rates.

97

Fig. 4. Calculation of the effective mobility in case of growing a-plates for different cooling rates.

Wits et al. [10] state that the mobility in their work is also an effective mobility, implicitly accounting for all effects which are not already integrated in their model. Their transformation model is based on the assumption that spherical ferrite grains grow from the corners of a tetrakaidecahedron. They only allow for effective mobilities which are following an Arrhenius relationship. For the same assumption with respect to the ferrite shape a thermally activated effective mobility could also be obtained in the present work. However, for this phase arrangement a physically unrealistic, cooling rate dependent mobility would be obtained as discussed above. It is important to remark that the mobility presented by Wits et al. [10] depends on the cooling rate as well. For the experimental data considered here, the cooling rate dependence of Meff is eliminated when planar growth is considered. As the thickening of ferrite plates proceeds, the plates might coalesce. The beginning of the coalescing process can be modelled by two plates growing together at one corner. The initial kinetics of ferrite formation to volume fractions of about 30% can be described by those simple planar growth models, as can be seen in Fig. 4. The present data permit to obtain mobility information for the temperature range of 830–880 C, where the mobility can be approximated by a single temperature dependent curve, see Fig. 5. Here, an effective mobility is obtained, which decreases with increasing temperature. The decrease of the effective mobility with temperature might be associated with the role substitutional Mn plays in affecting the transformation kinetics. For a discussion of the physics of the problem in more detail normalized velocities are considered. The normalized velocity is a ratio which indicates whether or not the transformation kinetics is influenced by diffusion. Whereas ratios well above 1 suggest a minor influence of diffusion on the kinetics, diffusion plays an important role for normalized velocities in the order of one. Here, two normalized velocities are considered, i.e.,

98

E. Gamsja¨ger et al. / Computational Materials Science 37 (2006) 94–100

taken into account explicitly by a solute drag pressure Dfsd. Then, Eq. (1) is modified as proposed in [23] to v ¼ MðDf  Dfsd Þ:

ð12Þ

The intrinsic mobility M is described as a thermally activated quantity following an Arrhenius relation, i.e.,   QM M ¼ M 0 exp ; ð13Þ RT

Fig. 5. Master curve for the effective mobility as a function of temperature assuming planar growth.

• the quantity vn,c with respect to Mn-bulk diffusion in austenite va vn;c ¼ c ; ð10Þ DMn • the quantity vn,int with respect to Mn-diffusion across the interface vn;int ¼

vd ; Dint

ð11Þ

where DcMn is the Mn-diffusion coefficient in austenite, Dint that across the interface, d is the half-thickness of the interface assumed to be 0.5 nm, and a is the distance between the nearest neighboring atoms to be 0.2 nm. In the temperature range between 830 and 880 C the quantity DcMn varies from 6 · 1018 to 2 · 1017 m2 s1 according to [22]. With these values and the temperature dependent velocity obtained from Eq. (1) vn,c decreases then from 2000 to 14. These large values for vn,c indicate that the diffusion process is slow compared to the interface movement. In the present case, particularly at lower temperatures, the diffusion of an Mn-spike in front of the interface is unlikely to influence the motion of the interface. The half-thickness of such a spike would be in the order of the interface thickness itself. Therefore, substitutional diffusion in the Mn-bulk material is too slow to influence the kinetics, and it is appropriate to assume immobile substitutional components in the c-bulk. A crude estimation for the interface diffusion normalized velocity vn,int yields to a value which is expected to be a hundred to a thousand times lower than the bulk diffusion normalized velocity vn,c based on the assumption of a normalization factor Dint/d that is two to three orders of magnitude times larger than DcMn =a. Then, substitutional diffusion across the interface is expected to occur on the same time scale than the motion of the interface. Thus, it is likely that the transformation kinetics is influenced by solute segregation to the moving interface which can be

where M0 is the pre-exponential factor and QM denotes the activation energy. The activation energy QM is in the range between 135 and 150 J mol1, Hillert [8] stated a value of 147 J mol1, Krielaart and van der Zwaag [9] found 138 J mol1 and 144 J mol1. Whereas the value for the activation energy QM is a tried and proven value of about 140 kJ mol1, the uncertainty of the mobility is brought about by M0. Hillert’s [8] M0-value for recrystallization and grain growth in a-iron marks an upper limit. With an average molar volume Vm = 7.3 · 106 m3 mol1 Hillert’s M0-value equals 4800 mol s kg1 m1. In contrast, Krielaart and van der Zwaag [9] found 0.058 mol s kg1 m1 in an Fe–Mn alloy. The latter, however, appears to be an effective value. According to [24] two dissipative processes occur in the interface, one due to the interface motion, the second due to solute drag. It is shown in [24] that the solute drag term is proportional to the square of the interface velocity for not too high interface velocities. In this case Eq. (12) can be applied. The solute drag pressure Dfsd can be calculated from the effective mobility such that   M eff Dfsd ¼ 1  Df : ð14Þ M Meff has been determined between 830 and 880 C and, thus, the solute drag term can be calculated for a known intrinsic mobility M. A lower limit of M0 can be obtained by assuming that the largest effective mobility, i.e., Meff at 830 C, would also represent the intrinsic mobility at this particular temperature. As a result, the lower limit for M0 in this ultra-low carbon steel is estimated to be 5.8 mol s kg1 m1. This lower limit is two orders of magnitude larger than the M0-value of 0.058 mol s kg1 m1 proposed by Krielaart and van der Zwaag [9]. If the latter were used physically meaningless negative values for Dfsd would be obtained from Eq. (14) in the present case. The upper limit of M0 is the value provided by Hillert [8] as far as grain growth in a-iron is comparable to the transformation process. The value from [8] gives a solute drag pressure which is only insignificantly lower than the driving pressure. For this alloy, solute drag pressures that do not follow the driving pressure lead to a comparatively small M0-range of 6– 15 mol s kg1 m1. As an example, the solute drag pressure is calculated from Eq. (14) with M0 = 6 mol s kg1 m1. The effective driving pressure Df–Dfsd is shown in Fig. 6 together with its two contributions, i.e., Dfsd and Df.

E. Gamsja¨ger et al. / Computational Materials Science 37 (2006) 94–100

99

approaches zero for vn,int ! 0 and for vn,int ! 1, as shown in Fig. 7. This is consistent with the general prediction of solute drag models [26]. In the framework of these models, the actual amount of solute drag pressure depends also on the binding energy Eb of solute to the interface. Adopting the modified solute drag model of Purdy and Bre´chet [25,26] a binding energy of 3RT is estimated for the present data. This value for Eb is similar to values in the order of RT proposed for Mn-segregation in the literature [25–27]. 4. Conclusions and outlook

Fig. 6. Effective driving pressure Df–Dfsd and its contributions (Df, Dfsd) as a function of temperature.

Whereas the effective driving pressure increases with decreasing temperature, the solute drag pressure displays a maximum at an intermediate temperature. The temperature scale can be replaced by a velocity scale since the velocity increases continuously as temperature decreases. In Fig. 7 the solute drag pressure is depicted as a function of the normalized velocity vn,int. The normalization factor Dint/d is related to diffusion across the interface and the associated diffusion coefficient is assumed to obey an Arrhenius relation. The activation energy for Mn-diffusion across the interface is approximated by 132 kJ mol1 which is half the value of the activation energy for bulk diffusion in austenite, as discussed in [25]. The pre-exponential value is selected such that the ratio Dint =DcMn falls between 400 and 700, i.e., the interface diffusivity is about two to three orders of magnitude larger than the bulk diffusivity. The largest values for the solute drag pressure are in an intermediate range of the normalized velocity and

The effective mobility of the a–c interface during phase transformation in an ultra low carbon low alloy steel has been estimated based on data from dilatometer tests. Unlike spherical growth planar growth of ferrite provides a mobility independent of the cooling rate. Thus, ferrite forms plates at low volume fractions and the entire material transforms by a thickening and coalescing process of these plates. Although the mobility is a thermally activated quantity, the effective mobilities found in this work decrease with increasing temperature similarly to that reported in [7]. The assumption of ferrite growing as spheres [9–12] is not confirmed with the present analysis. A value of the bulk diffusion normalized velocity vn,c well above 1 indicates that the substitutional diffusion in the bulk-material is so slow compared to the motion of the interface that it does not influence the kinetics. Thus, it is likely that solute drag of manganese is the reason for a decreasing effective mobility with increasing temperature. The solute drag pressure is estimated by replacing the effective mobility with an effective driving pressure. This estimation depends in a crucial way on the selection of the intrinsic mobility which is essentially an unknown quantity. In particularly the pre-exponential factor M0 of the intrinsic mobility is unknown. However, the evaluation of the experimental data constricts the M0-value to a range, which is narrow compared to results reported previously [8–12]. A solute drag behavior is concluded from the present data which is consistent with the solute drag analyses reported in the literature, e.g., [26,27]. Future research should be focussed on the interface migration and the solute interface interaction to obtain more insight into the underlying physics. An in situ study of this phenomenon is currently beyond experimental capabilities. However, transformation studies on well selected steels and model alloys may be useful to delineate trends of this solute interface interaction for a systematic change in the composition. A further challenge is to establish realistic austenite/ferrite phase arrangements in order to describe the overall transformation kinetics for the entire range of ferrite volume fractions. References

Fig. 7. Solute drag pressure Dfsd as a function of the normalized velocity vn,int.

[1] G. Inden, Computer modelling of diffusion controlled transformations, in: A. Finel et al. (Eds.), Thermodynamics, Microstructure and Plasticity, Kluwer Academic Publishers, 2003, pp. 135–153.

100

E. Gamsja¨ger et al. / Computational Materials Science 37 (2006) 94–100

[2] A. Borgenstam, M. Hillert, Massive transformation in the Fe–Ni system, Acta Mater. 48 (2000) 2765–2775. [3] G.P. Krielaart, J. Sietsma, S. van der Zwaag, Ferrite formation in Fe– C alloys during austenite decomposition under non-equilibrium interface conditions, Mater. Sci. Eng. A237 (1997) 216–223. [4] J. Svoboda, E. Gamsja¨ger, F.D. Fischer, P. Fratzl, Application of the thermodynamical extremal principle to the diffusional phase transformations, Acta Mater. 52 (2004) 959–967. [5] J. Sietsma, S. van der Zwaag, A concise model for mixed-mode phase transformations in the solid state, Acta Mater. 52 (2004) 4143–4152. [6] E. Gamsja¨ger, F.D. Fischer, J. Svoboda, Interaction of phase transformation and diffusion in steels, ASME J. Eng. Mater. Technol. 125 (2003) 22–26. [7] E. Gamsja¨ger, F.D. Fischer, J. Svoboda, Austenite-to-ferrite transformation in low alloyed steels, Comput. Mater. Sci. 32 (2005) 360– 369. [8] M. Hillert, Diffusion and interface control of reactions in alloys, Metall. Trans. 6A (1975) 5–19. [9] G.P. Krielaart, S. van der Zwaag, Kinetics of c ! a phase transformations in Fe–Mn alloys containing low manganese, Mater. Sci. Technol. 14 (1998) 10–18. [10] J.J. Wits, T.A. Kop, Y. van Leeuwen, J. Seitsma, S. van der Zwaag, A study on the austenite-to-ferrite phase transformations in binary substitutional alloys, Mater. Sci. Eng. A 283 (2000) 234–241. [11] S.I. Vooijs, Y. van der Leeuwen, J. Sietsma, S. van der Zwaag, On the mobility of the austenite/ferrite interface in Fe–Co and Fe–Cu, Metall. Mater. Trans. 31A (2000) 379–385. [12] T. Kop, A dilatometric study of the austenite/ferrite interface mobility, Ph.D. thesis. Technische Universiteit Delft, 2000. [13] M. Onink, Decomposition of hypo-eutectoid iron carbon austenites, Ph.D. thesis. Technische Universiteit Delft, 1995. [14] M. Militzer, Austenite decomposition kinetics in advanced low carbon steels, in: M. Koiwa, K. Otsuka, T. Miyazaki (Eds.), Solid–Solid Phase Transformations ’99, JIM, Sendai, 1999, pp. 1521–1524.

[15] J. Svoboda, F.D. Fischer, P. Fratzl, E. Gamsja¨ger, N.K. Simha, Kinetics of interfaces during diffusional transformation, Acta Mater. 49 (2001) 1249–1259. [16] M. Hillert, Overview No. 135: Solute drag, solute trapping and diffusional dissipation of Gibbs energy, Acta Mater. 47 (1999) 4481– 4505. [17] F.D. Fischer, N.K. Simha, J. Svoboda, Interaction of phase transformation and diffusion in steels, ASME J. Eng. Mater. Technol. 125 (2003) 22–26. [18] A.T. Dinsdale, SGTE Data for pure elements, CALPHAD 15 (1991) 317–425. [19] W. Huang, An assessment of the Fe–Mn system, CALPHAD 13 (1989) 243–252. [20] E. Gamsja¨ger, F.D. Fischer, J. Svoboda, Influence of non-metallic inclusions on the austenite-to-ferrite phase transformation, Mater. Sci. Eng. A365 (2004) 291–297. [21] A.T.W. Kempen, F. Sommer, E.J. Mittemeijer, The kinetics of the austenite–ferrite phase transformation of Fe–Mn: differential thermal analysis during cooling, Acta Mater. 50 (2002) 3545–3555. [22] H. Oikawa, Lattice Diffusion in iron—a review, Tetsu-to-Hagane´ 68 (1982) 1489–1497. [23] F. Fazeli, M. Militzer, Towards an austenite decomposition model for TRIP steels, Steel Res. 73 (2002) 242–248. [24] J. Svoboda, F.D. Fischer, E. Gamsja¨ger, Influence of solute segregation and drag on properties of migrating interfaces, Acta Mater. 50 (2002) 967–977. [25] F. Fazeli, M. Militzer, Analysis of ferrite allotriomorph growth in terms of solute drag, in: M. Militzer, W.J. Poole, E. Essadiqi (Eds.), Transformation and Deformation Mechanisms in Advanced High Strength Steels, The Metallurgical Society of CIM, Montreal, PQ, 2003, pp. 203–218. [26] G.R. Purdy, Y.J.M. Bre´chet, A solute drag treatment of the effects of alloying elements on the rate of the proeutectoid ferrite transformation in steels, Acta Metall. Mater. 43 (1995) 3763–3774. [27] M. Enomoto, Influence of solute drag on the growth of proeutectoid ferrite in Fe–C–Mn alloy, Acta Mater. 47 (1999) 3533–3540.