On coarsening of cementite during tempering of martensitic steels

ABSTRACT

The coarsening of cementite in a martensitic Fe–1C–1Cr (wt-%) alloy upon tempering at 700°C is investigated. When considering that the main location of cementite is at grain boundaries, classical coarsening theory can accurately predict the mean size evolution, while the predicted size distribution evolution disagrees with the experimentally observed log-normal distribution maintained throughout the whole tempering (5000 h). We conclude that classical theory of coarsening, as given by Lifshitz–Slyozov–Wagner and included in the Langer–Schwartz Kampmann–Wagner numerical approach for modelling precipitation reactions, is not fully adequate to simulate coarsening of cementite for tempering in practice.

KEYWORDS:

• Martensitic steel
• tempering
• precipitation
• coarsening
• particle size distribution

Introduction

In order to increase toughness, martensitic steels are in practice often tempered [1]. During tempering, carbides such as cementite (M₃C, M = Fe, X, where X is a substitutional element) are precipitated from the supersaturated matrix, contributing to the final properties of the steel [2,3]. Tempering of steels leading to the precipitation of cementite has been studied extensively, both experimentally and theoretically for binary Fe–C [4–6], ternary Fe–C–X and higher-order systems [6–17].

In general, a precipitation reaction entails nucleation, growth and coarsening of the precipitates, and a thorough understanding of each of these, partly concomitant, processes is important in order to be able to predict the microstructure evolution during tempering [18–24]. First, nuclei form from a supersaturated metastable matrix if the driving force is sufficient [25,26]; second, nuclei larger than the supercritical size will grow [26], during growth, the superstation in the matrix continuously decreases while the size and volume fraction of precipitates increase. Finally, coarsening of the precipitates will occur where the particle size increases at maintained equilibrium volume fraction. In the real alloy system, these three stages overlap with each other [27]. In the present study, we focus on the stage when the equilibrium volume fraction has been reached and the number density of precipitates is decreasing significantly, i.e. the assumed coarsening stage. Coarsening or Ostwald ripening is the process where the excess energy due to curved interfaces drives the growth of larger precipitates on the expense of smaller ones, which dissolves [28,29]. Although the driving force for coarsening in principle is present during the whole precipitation reaction, it is dominating after long tempering times [29,30]. The theoretical understanding of coarsening stems from the seminal works of Lifshitz and Slyozov [26] and Wagner [31], forming the LSW theory of coarsening. From the LSW theory, the asymptotic analysis leads to the invariant shape of the particle size distribution (PSD) where the maximum particle size is 1.5 times the average particle size. An important assumption in the LSW theory is that of zero volume fraction of the coarsening phase, and much of the literature following the work by LSW has concentrated on relaxing this assumption by deriving modified equations for finite volume fractions of the coarsening phase, see e.g. Voorhees [32]. In a review paper by Jayanth and Nash [33], no less than 10 additional factors, not accounted for in the original LSW theory, were listed and discussed.

Coarsening theory should not only be capable of predicting the evolution of the mean size of precipitates but also their PSD. The PSD is important since it, together with the volume fraction of precipitates, determines the strengthening contribution coming from the precipitates [34–36]. Numerous works have studied the coarsening of cementite in steels [4–17], but few have carefully investigated long-term tempering of martensitic steels and compared it with detailed experimental information and classical coarsening theory [16,17]. Deb and Chaturvedi [16] found that the experimental maximum of the PSD for a 10B30 steel is significantly smaller than the predicted one by the LSW theory for all tempering temperatures between 630 and 690°C up to 25 h. They suggested that this difference is due to an encounter mechanism and/or the effect of boron on the coarsening rate of cementite. The encounter mechanism is generally explained as two growing particles that coalesce as a result of increasing volume fraction of precipitates, by which it will not enhance the growth rate to any significant extent but it does broaden the PSD [16]. Di Nunzio [17] proposed that the broad and right-skewed stationary size distributions in coarsening experiments on cementite in a ferritic matrix during tempering at 700°C up to 100 h could plausibly be explained by direct interactions among particles.

The purpose of the present work is to study the coarsening of M₃C during tempering of a martensitic Fe–1C–1Cr alloy (wt-%) at 700°C. Careful examination of the PSD evolution is compared with state-of-the-art modelling using the diffusion and precipitation modules DICTRA and TC-PRISMA in Thermo-Calc with databases TCFE8 and MOBFE3 [37,38].

Experimental and simulation

The investigated alloy has the chemical composition (wt-%): C 0.95, Cr 1.06, Si 0.01, Mn 0.07, Al 0.03, Cu 0.01, Fe bal. Specimens of dimensions 10 × 10 × 1 mm³ were austenitised at 1100°C for 10 min followed by quenching in brine, tempered at 700°C for up to 5000 h, followed by quenching in brine. Structural characterization was performed using a Scanning Electron Microscope (SEM) JEOL JSM-7800F, equipped with an electron backscatter diffraction (EBSD) detector, operating at 15 kV, and a Scanning Transmission Electron Microscope (STEM) JEOL JEM-2100 operating at 200 kV. Details on alloy and sample preparation as well as SEM and STEM microscopy can be found in [20,21]. Only the coarsening stage, where the number density of precipitates continuously decreases and its volume fraction is close to the calculated equilibrium value, i.e. 0.14 M₃C in the present case, is in focus here. TEM-EDS analysis carried out on the M₃C has shown, with the stoichiometry defined as (Fe_1−x, Cr_x)₃C, that the chemical composition is x = 0.074 ± 0.002, no matter the particle size. These results are in agreement with thermodynamic calculations carried out considering the ternary system Fe–C–Cr. Further details for the simulations can be found in Ref. [39]. Quantitative information, i.e. size and subsequently the PSD, of M₃C precipitates were determined by studying 600–1000 random particles.

Results and discussion

It was found that the precipitates were located at high-angle grain boundaries after 5 h tempering at 700°C, see Figure 1(a,b). Without accounting for grain boundary diffusion, it was not possible to predict the experimentally observed evolution of the mean radius of cementite during coarsening using DICTRA. The effect of grain boundary diffusion can be taken into consideration by evaluating the pre-factor, giving the best fit, of the activation energy for diffusion, i.e. the new diffusional mobility of Cr is given as [40] (1) $M_{\text{new}}=\delta /d\cdot M^{gb}+(1-\delta /d)M^{\text{bulk}}$(1) where (2) $M^{gb}={M_0}^{\text{bulk}}\exp \left(\frac{F_{\text{redGB}}{Q}^{\text{bulk}}}{RT}\right)$(2) $\delta$ is the grain boundary thickness, set as 5 × 10⁻¹⁰ m, d is the grain size as a function of time and temperature. ${M_0}^{\text{bulk}}$ is the frequency-factor for volume diffusion, $Q^{\text{bulk}}$ is the activation energy for volume diffusion, $F_{\text{redGB}}$ is the pre-factor for grain boundary diffusion activation energy. By setting F_redGB= 0.7, and a reasonable interfacial energy of 0.4 J/m², the predictions showed good agreement with the experimental data, see Figure 1(c). It should be noted that both F_redGB and the interfacial energy are used as calibration factors to fit simulations to the experiments, but the values of these parameters are also physically reasonable. The slope for the experimental and simulated lines is equal to 1/3, in accordance with the LSW theory, see Figure 1(c).

Figure 1. STEM-Bright Field micrograph (a) and EBSD phase map overlayed with boundaries of misorientation angle >10° shown by black lines (b) after tempering for 5 h. The predicted mean radius from DICTRA (c) considering different pre-factors $\left(F_{\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{G}\mathrm{B}}\right)$ for activation energy for diffusion to account for grain boundary diffusion, and interfacial energies (σ).

Turning the attention towards the PSD of M₃C precipitates, experimental as well as TC-PRISMA simulated PSDs are shown in Figure 2. The mean size of M₃C precipitates and the standard deviation of the fitted log-normal PSDs together with the skewness are given in Table 1. The experimental PSDs can be described by a log-normal distribution throughout the whole tempering process, which is in agreement with earlier works [17,41], see Figure 2(a). The skewness is a statistical factor indicating the degree of asymmetry of a distribution around its mean. If the skewness value is positive, then the distribution has an asymmetric tail extending towards more positive values. If the skewness value is negative, the distribution has an asymmetric tail extending towards more negative values [42]. The skewness of the experimental PSDs is constant at 0.70 throughout the whole tempering time. On the other hand, the modelling predicts the classical LSW-type distribution from ∼30 s of tempering and onwards, see Figure 2(b). The PSD’s from the experiments are also much broader than the predictions from the modelling.

Figure 2. Evolution of the M₃C PSDs for the indicated tempering times at 700°C, both from experiments (a-1 to a-6) and from TC-PRISMA (b). Each PSD is scaled such that the area under the distribution is equal to unity, i.e. the following conditions were satisfied: ${\int }_0^{\mathrm{\infty }}\rho f\left(\rho \right)\text{d}\rho =1$ and $\rho =r/\overline{r}$, the bin width is $0.025r_{max}/\overline{r}$. Log-normal distributions are indicated by the solid lines in (a-1–6). Note that the timescale for the PSD between the experiments and the calculation cannot be compared directly due to the deviation of coarsening onset between the modelling and the experiments.

Table 1. Mean equivalent radius of M₃C, standard deviation of fitted log-normal PSD’s and their skewness as a function of tempering time

		PSD log-normal fitting
Time, h	Mean radius, nm	Standard deviation	Skewness
5 h	157 ± 11	0.43 ± 0.07	0.74 ± 0.5
100	279 ± 18	0.37 ± 0.03	0.70 ± 0.2
500	434 ± 31	0.40 ± 0.04	0.71 ± 0.2
1000	524 ± 45	0.38 ± 0.02	0.69 ± 0.2
2500	757 ± 48	0.41 ± 0.04	0.70 ± 0.2
5000	842 ± 45	0.37 ± 0.02	0.70 ± 0.2

According to the existing LSW theory, all the PSDs of the second phase should reach a self-similar shape during the long-term coarsening stage [26,31]. In order to compare the experimental and modelling results specifically for coarsening, we first need to identify the coarsening stage. This was performed by carefully measuring the number density and volume fraction of the M₃C precipitates. It was found that the start of coarsening, without interference from nucleation and growth, is approximately after 5 h of tempering, since from thereon, the volume fraction of M₃C is almost constant, see Figure 3. The slight difference in volume fraction between the calculated equilibrium and experiments for the studied system can in part be related to, for example, the assumption of a perfectly spherical shaped particle in the experimental evaluation while in fact faceted particles are commonly observed in the experiments, and also to uncertainties in the thermodynamic description of the Fe–C–Cr system. Before 5 h of tempering, it can be seen that the number density of M₃C is decreasing at the same time as the volume fraction of M₃C is increasing. This stage is identified as overlapping nucleation, growth and coarsening, and it should be noted that nucleation is completed already after 5 s [39].

Figure 3. Experimental volume fraction and number density evolution of M₃C during tempering, including data from [34]. Equilibrium volume fraction of M₃C at 700°C, calculated by Thermo-Calc is shown. The red dotted line roughly divides the regions of mixed nucleation + growth + coarsening and pure coarsening.

As shown in Figure 2, the experimental PSDs are completely different from the model predictions of coarsening. This difference between model and experiments has been shown a few times before. It should here be emphasised that the PSD from the original LSW theory is a steady-state solution to the governing equations, and it is possible that the discrepancies found between the PSD from LSW and experiments are due to that the coarsening has not reached the steady state but is in a transient stage [43–45]. Steady-state coarsening can be described as the stage when scaling laws apply, i.e. when the PSD is overlapping when normalised by the mean radius [43]. In the present work, we see that the PSD shape does not evolve throughout the coarsening process up to 5000 h, which is indicating some kind of steady state, but not as described by classical theory. Brown [46] suggested that it is very unlikely that a PSD showing a tail to the right during transient coarsening (non-steady-state) should suddenly develop into the very asymmetric PSD, with an abrupt end of the distribution at roughly 1.5 times the average size, as given by LSW. However, Brown’s work has been frequently debated in the literature, e.g. Chen and Voorhees showed numerically, allowing for non-steady-state solutions, that for all initial PSDs studied, the LSW-shaped PSD is the most preferred one after extensive coarsening times [43].

As already mentioned, the effect of a finite volume fraction of the precipitate phase has been invoked to explain the discrepancies between experimental and simulated PSDs [32]. Unfortunately, systematic studies on the effect of volume fraction of cementite on the coarsening kinetics during tempering of martensite seems to be lacking in the literature, but extensive studies on coarsening of γ′ precipitates in Ni-based alloys have been performed [47–49]. Although these studies are not directly comparable to cementite precipitation, they show that it is difficult to experimentally observe any differences in the coarsening rate constant with small volume fractions (<0.2). In addition, a recent study has also shown that in the intermediate range of volume fractions from ∼0.1 to roughly 0.3, the coarsening kinetics does not depend strongly on volume fraction [41]. In the present case, the equilibrium volume fraction of cementite is ∼0.14, and we, therefore, believe that this factor is of lesser importance.

When modelling coarsening, it is often tacitly assumed that the original three-dimensional LSW theory directly can be used and compared to experimental situations where the diffusion fields have other dimensionalities. As found in the present work, diffusion along grain boundaries should have an important effect on the coarsening of cementite since most of the cementite particles are found there. The cementite particles are three-dimensional, but the diffusion field in the grain boundary is two-dimensional, and it has been suggested from numerical experiments that 3D/2D systems will not develop strict self-similarity as predicted by LSW [50].

Conclusions

Despite the possibility that our experiments are not considering steady-state coarsening, and that the simulations neglect the influence of a finite volume fraction of the precipitates, we can conclude that for practical tempering of martensitic steels, the classical theory of coarsening, as given by Lifshitz–Slyozov–Wagner and included in the Langer–Schwartz Kampmann–Wagner numerical approach for modelling precipitation reactions [51,52], is not fully adequate to predict coarsening of the cementite.

Acknowledgements

The authors are grateful to Drs Fredrik Lindberg and Niklas Pettersson at Swerea KIMAB for experimental assistance. The suggestions from Professor Peter Voorhees at Northwestern University on this manuscript are gratefully acknowledged.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Ziyong Hou http://orcid.org/0000-0003-4825-7430

R. Prasath Babu http://orcid.org/0000-0003-4351-3132

Peter Hedström http://orcid.org/0000-0003-1102-4342

Joakim Odqvist http://orcid.org/0000-0003-3598-2465

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Additional information

Funding

This work was supported by the VINN Excellence Center Hero-m, financed by VINNOVA, the Swedish Governmental Agency for Innovation Systems, Swedish industry and KTH Royal Institute of Technology.