DiffusionControlled Lengthening Rates of Bainitic Ferrite a Part of the Steel Genome
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Abstract
As a step in the further development of models and databases to support design of new steels, i.e., the “steel genome”, the growth of bainitic ferrite plates is accounted for by a thermodynamic and kinetic approach. The thermodynamic aspects are represented by CALPHAD databases and a Gibbs energy barrier for growth B_{m}. Experimental information on ferriteplate growth rates for a number of FeC alloys, some of highpurity, are analyzed in terms of a modified Zener–Hillert model and the barrier as well as some kinetic parameters are evaluated. It is found that the barrier varies in a smooth way with carbon content and lengthening rate. In order to improve the agreement with the experimental information it was necessary to adjust the diffusion coefficient of carbon in austenite at low temperatures. It is concluded that the representation of the experimental data is satisfactory.
1 Introduction
The materials genome is the combination of models and databases that supports efficient materials design,[1] the aim being to decrease the development time and costs for materials and processes. The CALPHAD type of models and databases for thermochemical and diffusional data may be seen as role models for materials genomic databases. The purpose of the present work is to provide a practical formalism based on the CALPHAD methodology and databases for predicting the lengthening rate of bainitic ferrite, an important part of the bainite transformation. The approach is inspired by the remarkable success of CALPHAD in materials science and engineering.
The bainite transformation in steels is now considered because it has received a lot of attention over the last decade. Such steels hold promises of excellent engineering properties, e.g., high hardness and strength, good wear resistance and toughness, to mention a few. In order to optimize product and performance, models are increasingly being used to facilitate tailoring of material properties.
In this paper we shall not discuss the mechanism of the bainite transformation as such discussions can be found elsewhere, e.g., in References 2 and 3. We will rather base our approach on the hypothesis that the primary ferritic component in bainite grows by the same diffusional mechanism as Widmanstätten ferrite over the whole temperature range,[4] i.e., the transformation rate is mainly controlled by carbon diffusion in austenite. The approach is developed through a combined thermodynamic and phenomenological model and the parameters in the model will be evaluated from experimental data on starting temperatures and growth rates.
2 Growth of Acicular Ferrite
2.1 Modified Zener–Hillert Model
Equations [1], [5], and [7] constitutes three nonlinear equations from which the same number of unknowns may be calculated. At a given temperature, alloy composition \(u_C^0\) and a given value of \(\rho \) we may thus calculate \(u^{\gamma /\alpha }\), \(u_C^{\alpha }\) and v.
2.2 Critical Curvature and Maximum Growth Rate
In accordance with Zener we shall take this maximum growth rate and the corresponding radius of curvature as the values that should be compared with experiments.
In the classical Zener–Hillert model the maximum growth rate occurs at \(2\rho _{cr}\). However, Leach et al.[7] showed that the growth rate of their modified model has a rather different dependence of the curvature than the original Zener–Hillert model. In fact the maximum occurs at \(2\rho _{cr}\) only at very high temperatures where the dilute solution approximation is valid. In general the maximum could occur at much higher values and has to be calculated for each particular case.
3 Evaluating the Barrier \(B_m^0\) in FeC
As mentioned it is observed experimentally that the growth of Widmanstätten and bainitic ferrite occurs below a critical temperature referred to as \(WB_s\) which depends on the carbon content of the initial austenite, i.e.\(u_C^0\). In the following we shall first consider cases where the influence of alloy content on the barrier can be neglected and the treatment will then hold for binary FeC alloys and approximately for lowalloy steels.
4 Evaluating the Unknown Parameters from Experimental Growth Rate Information in FeC
It is clear that Eq. [4] can represent the onset of growth of bainitic ferrite. We shall now investigate to what extent the present model, see Section II–B, is capable of representing the experimental lengthening rate data. The following quantities are needed to calculate the lengthening rate; the thermodynamic properties of austenite and ferrite, the diffusion coefficient \(D_C^{\gamma }\) for carbon in austenite, the ferrite/austenite interfacial energy \(\sigma \), the barrier \(B_m^0\) and the kinetic transfer coefficient \(L_C\). The information on thermodynamics available in CALPHAD databases will be accepted and the barrier \(B_m^0\) was discussed in Section III.
4.1 Diffusion in Austenite
In principle the diffusion coefficient \(D_C^{\gamma }\) is available in the CALPHAD databases and it is well established that it increases not only with temperature but also drastically with carbon content. However, in Eq. [1] only a single value enters for a given temperature and alloy composition. This means that some kind of effective value must be used. A simple choice would be the arithmetic mean between the maximum value close to the edge, where the carbon content is highest, and the value far away in the austenite. This average value was chosen in the present work. Nevertheless, it should be kept in mind that the diffusion coefficient in the databases is evaluated from experimental information above the eutectoid temperature where austenite is stable whereas the present application represents a long extrapolation both in temperature and carbon content. Consequently the database value will thus be very uncertain at the lower temperatures where the carbon contents \(u_C^{\gamma /\alpha }\) becomes very high.
In view of these uncertainties we have chosen to accept the database values of \(D_C^{\gamma }\) above 1000 K and take the average value as discussed above. Below 1000 K we have introduced a correction factor \(f_D=e^{\frac{Q_{corr}}{R}(1/T1/1000)}\) which is unity at 1000 K and \(Q_{corr}\) is an adjustable parameter.
4.2 Experimental Data from Literature
The experimental lengthening rates of bainitic ferrite in binary FeC alloys were collected from literature as the absence of additional alloy elements allowed for a simpler comparison with the model. These data were therefore used as a basis for the evaluation of the unknown parameters. The microstructures of interest were selected as Widmanstätten ferrite, acicular ferrite, lower bainite, or upper bainite. Ferrite is the leading phase and in principle the formation of carbides is a secondary reaction that occurs after plate lengthening. The presence of cementite in the experimental data used is therefore not thought to have any effect on the lengthening rate unless it formed on the tip or close to the tip of the plate, in which case the diffusion field of carbon would be strongly affected. Much of the data fall into one of two categories in terms of the methods used to measure lengthening rates. In situ measurements were done by References 11 and 12 with hot stage metallography. Another method used was to measure the longest plate on a polished plane after isothermal heat treatment from optical micrographs.[8,13,14] For in situ measurements it is not necessarily the longest plate that was observed but rather a typical plate. In certain instances[12] distinct surface and bulk lengthening rates were measured with reference to the spatial location of the acicular plates in the sample. In such instances surface lengthening rates were higher and there was better agreement with the model than with bulk lengthening rates.
We now only use binary FeC alloys with the carbon contents of the binary alloys are 0.24, 0.33, 0.43, 0.50, 0.70, 0.90, 0.96, 1.16, and 1.43 given in mass percent carbon from References 11 through 14. The aim was to be able to predict Ccurves for growth rates of bainite.
4.3 Optimization Procedure
The calculation method outlined in Section II was now coded in MATLAB[15] using the ThermoCalc MATLAB interface[16] and the optimization procedure in MATLAB. The unknown parameters, i.e., \(L_C\), \(\sigma \) and the correction to the diffusivity were then fitted to the experimental growth rates as discussed in Section IV–B.
4.4 Results of Optimization

\(f_{D}=\exp \underline{[}3.53(1000/T1)\underline{]}\)

\(\sigma =0.098\, {\text{Jm}}^{2}\)

\(L_C=2.344\, 10^{5}\, {\text{mol}}^2{\text{J}}^{1}{\text{m}}^{2}{\text{s}}^{1}\)
The fitted diffusivity (dashed line) is compared with the database value calculated for \(u_C=0.03\) (solid line) in Figure 6. The optimized interfacial energy is somewhat lower than the value \(0.23\, {\text{Jm}}^{2}\) suggested by Hillert.[8]
5 Discussion
The present approach gives a reasonable representation of the experimental data for binary FeC alloys. It should be emphasized that in some cases data from different authors are not consistent probably due to experimental uncertainties.
In Figures 2 and 3, calculated growth rate curves are shown with their respective experimental FeC growth rates for alloy data used in the optimization. In Figure 2 there is good agreement with the Fe0.5C and Fe0.7C alloys though towards lower temperatures, the growth rate data decrease less with temperature. There is a notable disparity in the temperature dependence of growth rates below 630 K (357 \(^{\circ }\)C) for the Fe0.7C alloy compared to the alloys with 0.90C, 0.96C, 1.16C, and 1.43C, which are in agreement with the calculations. The low carbon alloys at high temperatures[13] in Figure 3 also indicate a weaker decrease in growth rate than is predicted by the model. This characteristic could not be reproduced by the present model and in recent work[14] it was tested with composition dependence of diffusivity. The use of a maximum value, \(D(u_C^{\gamma /\alpha },T)\) broadens the curve but the change was not sufficient to capture the effect that would cause a flatter curve. It was proposed that such an effect may be related to an increased supersaturation of carbon in the ferrite, in which case some fundamental adjustments would be required to adequately represent the curves. Figure 4 shows comparison between calculated growth rates and experimental data for low alloyed steels not used in the optimization. The high and intermediate temperature data are in fairly good agreement with the calculated lines with the exception of some of the low carbon data[14] at lower temperatures. In spite of this characteristic, the experimental growth rates are satisfactorily represented by the calculated curves. The overall result of all the data examined is compared in Figure 5 and the largest discrepancies correspond to the lower temperature velocity range of the low carbon alloys and of the 0.7C alloy.
At lower temperatures the model critically depends on long extrapolations of thermodynamic and kinetic properties. We have chosen to rely on established CALPHAD databases for the thermodynamic properties but to adjust the diffusivity. In Figure 6 it is seen that the fitted value of the diffusivity at 500 K is more than 30 times higher than the database value. The magnitude of velocity can thus be reduced to fit the data by an adaptation of the fitted diffusivity. Evidently, the type of data used bears influence on the optimized parameters and applicability to other types of alloy data. The data used here were binary FeC and for the strong ferrite stabilizers Mo and Cr, the parameters may require due adjustment. Future improvements to the predictive capacity can be made by careful selection of experimental information over a wide temperature range with preference to binary FeC alloys.
6 Conclusion
A model for the formation of bainite has been developed on the basis of an energy dissipation approach taking into account existing kinetic and thermodynamic aspects. Important developments are introduced to mitigate the uncertainty of extrapolated diffusivity and interfacial energy. Both parameters are optimized to experimental data and the calculated result is sufficiently satisfactory. Discrepancies observed were in part due to experimental accuracy though the peculiar temperature dependency for low carbon steels would require further investigation in another context. Further work is required to give generic fitting.
Notes
Acknowledgments
The authors wish to thank Professor em Mats Hillert for valuable discussions. The work was performed within the Competence Centre Herom 2i financed by VINNOVA, the Swedish Governmental Agency for Innovation Systems, Swedish Industry and KTH Royal Institute of Technology.
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