In this work, we investigate the microstructural evolution of chromium steel subjected to a thermomechanical treatment simulating hot compression. The steel, with the composition given in Table 1, has previously been experimentally investigated by Safara et al. (2019) and Safara et al. (2020) by means of a Gleeble thermomechanical simulator, hardness measurements, scanning electron microscopy (SEM), and energy-dispersive spectroscopy (EDS). The non-isothermal thermomechanical treatment applied consists of cooling from the austenitization temperature, 1,250°C, with a rate of 5°C/s down to holding temperatures (900, 1,000, and 1,100°C). The material is maintained for 15 s at each temperature and deformed to reach a strain of 0.7 at different strain rates, 10s⁻¹ and 1s⁻¹. The total holding time is 300 s. The obtained microstructure consisted of a distribution of precipitates embedded in a martensitic matrix. The first work by Safara et al. (2019) included a plastic deformation of the material after 15 s holding exactly as in the Figure 1. The authors determined both the flow stress curve experimentally during this deformation and used a physically based model—consisting of a set of equations for the dislocations evolution, vacancies, recrystallization, and grain growth—to estimate the flow stress numerically. This model is dependent on the volume fraction of precipitates and their mean size. They were, thus, able to adjust these values in order to obtain a good agreement between the experimental measurement and the model predictions of the flow stress. Table 2 gives the evolution of the ratio between the volume fraction and mean size of precipitates, as estimated by Safara et al. (2019). In addition, in a second work, Safara et al. (2020) performed the Gleeble test on the same steel and with the same thermal profile but without plastic deformation. The holding at the holding temperature was stopped after 15 s in order to experimentally characterize the microstructure of the steel right before the deformation in comparison to their first work. Comparing theoretical calculations of the hardness involving the precipitate’s mean size and volume fraction with experimentally measured hardness values gave an estimation of the mean size and volume fraction of the precipitates in the steel. These values are given in Table 2 and used in this work to parametrize the simulations, especially the interfacial energy, even though it was considered that the precipitate’s mean size and volume fraction were not evolving during the deformation.

The developed model requires many input parameters, which can be divided into several classes depending on their kind. The parameters in Table 3 are intrinsic material properties and can be found in handbooks or derived from a thermodynamic database. The parameters related to the micromechanical model that are necessary in Eq. 12, are taken from Safara et al. (2019), Safara et al. (2020), and Safara Nosar (2021). The molar volume and lattice parameter are extracted from a thermodynamic database (TCFE12, 2022). In addition, the number of nucleation sites for bulk precipitation is simply the number of atoms per cubic meter that can be obtained from the molar volume as well. Finally, the interfacial energy was adjusted to reproduce experimental results. A discussion is given about this parameter later on. The modeling tool is coupled with Thermo-Calc using the TCFE12 database (TCFE12, 2022), MOBFE4 database (MOBFE4, 2022), and tc-python library to compute the driving force and the equilibrium composition as a function of the temperature and also to extract the mobility data that are necessary to run the simulations. They are all updated as a function of temperature in the course of the transformation.

Figure 2 presents the equilibrium property diagram of the molar fraction of phases as a function of temperature of the investigated steel as predicted by Thermo-Calc software and the TCFE12 database. At the austenitization temperature, 1,250°C, the matrix is fully austenitic and at the holding temperatures M₇C₃ is expected to precipitate. It is worth noting that although some sulfides were observed experimentally, they are not considered in this modeling work due to their low volume fraction. Table 2 gives the equilibrium volume fraction of M₇C₃ as a function of temperature. It is clear that when the kinetics is not accounted for, it is difficult to understand the obtained volume fraction that is lower than equilibrium.

In order to understand the effect of deformation on the precipitation kinetics, the precipitation model presented in the previous section will be compared with a so-called classical precipitation treatment, in which nucleation takes place only in the bulk and where the dislocation density evolution is not accounted for. In the following, this model will be called “classical,” while the newly developed integrated and coupled model will be called “integrated” scheme.

In Figures 3A,B,C,D, the evolution of the M₇C₃ precipitate volume fraction, number density, mean size, and the evolution of the dislocations density, respectively, as a function of time for the three holding temperatures (900, 1,000, and 1,100°C) and for a strain rate of 10 s⁻¹ are presented. The strain rate was chosen as it was the strain rate applied in hot compression tests performed in the Gleeble thermomechanical simulator to back calculate the f / r m ratio by Safara et al. (2019). Despite that the experimental measurements were made after only 15 s of holding; we perform in this work simulation for 300 s holding in order to investigate the impact of deformation on the overall kinetics. For the treatment at 1,100°C, both the volume fraction and number density of precipitates do not appear in the graphs, since their values were too small for the used log scale.

Deformation takes place after 45, 65, and 85 s for the holding temperatures 1,100, 1,000, and 900°C, respectively, due to different times to reach the holding temperatures upon cooling from the austenitization temperature. This explains the changes, at different times for the different holding temperatures, in the curves of the integrated scheme in Figure 3. The precipitate volume fraction, number density, and mean size (Figures 3A,B,C) for the classical implementation and the newly integrated scheme are overlapping, as they must at the beginning of the treatment before the deformation starts. Indeed, at this stage, nucleation, if happening, takes place classically. At the beginning of the simulation (from 1,250°C), there is no driving force for precipitation, and thus, there is no nucleation. When the temperature has decreased enough, nucleation starts to take place classically in the bulk, and the volume fraction, number density, and mean size start to increase. Once the stress is applied and dislocations are generated, a clear increase is observed in the volume fraction (Figure 3A) for the integrated scheme results, since nucleation increases due to the decrease in the nucleation barrier (Figure 4). At the same time, growth of the already created precipitates is enhanced by the diffusion increase due to the presence of the dislocations. The volume fraction increases to a greater extent for the lower temperature, that is, 900°C, despite the volume fraction being smaller than the one for the 1,000°C treatment right before the deformation of the material. It is worth highlighting that the compression has already stopped when the increase takes place, since at a strain rate equal to 10 s⁻¹, it only takes 0.07 s to deform the material. Nevertheless, until the dislocation density has reduced to its original value or until nucleation has stopped because the volume fraction is close to the equilibrium value, the dislocations assist nucleation and enhance diffusion. The competition between thermodynamics and kinetics can explain why at 1,000°C, the highest volume fraction is obtained right before deformation starts. At a higher temperature, diffusion is faster but the driving force for precipitation lower, compared to lower temperature, where the driving force is higher but diffusion is slower. The maximum volume fraction is thus obtained for the intermediate temperature of this study. Additionally, the lower the holding temperature, the longer the treatment time since for all tests the cooling rate is kept the same. This leads to more time for the intermediate temperature compared to the highest one for forming the first precipitates. All the curves are actually overlapping until the first deformation takes place at 45 s for the highest temperature.

The number density of precipitates (Figure 3B) also exhibits a sharp increase right after deformation. The increase in the number density is higher for the lower temperature, 900°C, than for 1,000°C. This is due to the lower precipitate volume fraction (Figure 3A) compared to the one at 1,000°C when the deformation starts. Therefore, the matrix content at 900°C is still very close to the nominal composition, and the driving force for precipitation is, thus, very high and higher than that at 1,000°C. When nucleation becomes heterogeneous after the application of the stress, this results in a Lower heterogeneous nucleation barrier at 900°C than at 1,000°C (Figure 4) and thus in a higher nucleation rate and, therefore, in the creation of more precipitates. The number density of precipitates decreases for the treatment at 900°C after a while. This is due to coarsening. Indeed, this treatment is the only one for which the volume fraction has reached its equilibrium value. Therefore, to continue to decrease its overall energy, the material eliminates the smaller precipitates and lets the larger ones grow.

The evolution of the mean size is more exotic. This is first due to the fact that while the temperature decreases, the driving force increases and in turn the critical size decreases. It leads to the creation of smaller and smaller size classes of precipitates with time affecting the average size drastically. This decrease in the mean size is visible for the 900°C treatment. Once the holding temperature is reached, the mean size increases again since the newly created precipitates are not smaller than the average value.

The abrupt changes in the different precipitates characteristics are, of course, directly connected to the abrupt change in the dislocation density, as depicted in Figure 3D. The dislocation density for the treatment at 900°C is not presented after 90 s, but this is only due to a modeling aspect. Indeed, nucleation stops very rapidly for the 900°C treatment, since the volume fraction is rapidly approaching the equilibrium value; thus, the number density of dislocations is not computed anymore as it is not needed for calculating the growth and coarsening rates. The classical treatment of precipitation predicts final sizes that are larger than the prediction from the integrated scheme (Figure 3) for the 900 and 1,000°C. The trend as a function of temperature is the same for both schemes with lower temperatures giving rise to a smaller mean size. However, the volume fraction at the end of the treatment is larger in the integrated scheme for the lower temperature. This is not surprising in view of the thermodynamics predicting larger equilibrium volume fraction at the lower temperature and since the kinetics is enhanced with deformation. Finally, the impact of the deformation is more pronounced when the deformation takes place at an early stage of precipitation as at 900°C. Indeed, there are almost three orders of magnitude of difference in the final volume fraction obtained at 900°C between the classical treatment and the integrated scheme.

The ratio f / r m that have been back calculated by Safara et al. (2019) while calibrating their micromechanical model to flow stress measurements are 3.9 x 10⁻⁴, 2.7 x 10⁻⁴, and 5.2 x 10⁻⁵ nm⁻¹ at 900, 1,000, and 1,100°C, respectively (Table 2). It is important to note that these values give information about the ratio during deformation and not about the ratio at the end of the 300 s-long holding. Therefore, the values from the simulation using the integrated scheme to compare with are the ones obtained after 0.7 s of applied stress. They are 4.53 x 10⁻⁴, 6.58 x 10⁻⁶, and 0 nm⁻¹, respectively, (Table 2). Despite that, there is a qualitative agreement, with the highest ratio obtained for the 900°C treatment; the results do not agree quantitatively. Several reasons can be given to explain the mismatch:

1) First, as highlighted in the introduction, the back calculations performed by Safara et al. consider a constant value of the volume fraction of precipitates and the mean size during the overall deformation, which is not what the integrated scheme does.

The volume fractions that have been back calculated from Safara et al. in their second work (Safara et al., 2020) while calibrating their hardness equation to their experimental measurements are 1.3%, 1.6%, and 0.3% at 900, 1,000, and 1,100°C, respectively. These values were obtained without any plastic deformation, but they were determined using the ratio f / r m obtained with plastic deformation. Safara et al. used these values as they assumed that deformation was too short to affect the precipitation kinetics. The integrated scheme predicts 0.0001%, 0.004%, and 0% right before deformation and 0.23%, 0.012%, and 0% right after deformation for 900, 1,000, and 1,100°C, respectively, when the interfacial energy is equal to 0.16 J m⁻². The values show that it is maybe too strong to assume that deformation cannot affect the PSD because of the short stress application time. Finally, when using an interfacial energy equal to 0.14 J m⁻², the obtained volume fractions at the end of the deformation are ∼2.1%, 4.1%, and 0%. The latter are in better agreement with the experimental back-calculated values, but the corresponding ratio f / r m obtained with this interfacial energy and that are given previously is too high. It is important to highlight that in that case, the deformation does not affect the precipitation kinetics integrated scheme results much when compared to the classical treatment results. This is because with such a value of the interfacial energy, the onset of precipitation is very rapid with or without dislocations. It shows that deformation would affect the precipitation kinetics more when the interfacial energy is larger, and thus, when it is more difficult to nucleate in the classical fashion.

The carbides are also supposed to be larger at higher holding temperatures according to Safara’s calculations after 15 s. This is also obtained with both the classical and integrated scheme, see Table 2 and Figure 3. In general, the predictions from the integrated scheme are giving larger precipitates with a lower volume fraction compared to the back-calculation from experiments.

In Figure 4, the evolution of the nucleation barrier as a function of time for the three holding temperatures using the new integrated scheme and the classical implementation is displayed. The curves are all overlapping at the beginning of the kinetics because they all follow the same temperature profile. They are decreasing with time because the lower the temperature, the higher the driving force as illustrated in the equilibrium property diagram (Figure 2). The highest holding temperature stops decreasing earliest and at the highest value of the nucleation barrier for the same reason. Once the stress is applied, the nucleation barrier drops for all treatments since the nucleation is favored as it eliminates part of the dislocations (Eq. 9). Therefore, despite that the number of nucleation sites for heterogenous precipitation on dislocations is significantly lower than the available bulk sites, it becomes much more favorable to nucleate on dislocations because it is thermodynamically easier. However, because of the high strain rate used in these tests (10 s⁻¹), the deformation time is very small and the nucleation barrier goes back to a higher value very rapidly. It is interesting to note that for 900°C, for which it has been shown (Figure 3) that the volume fraction of precipitates right after the deformation is the largest one, the nucleation barrier goes back to a further away value, that is, to a very high nucleation barrier, relative to its initial value before deformation, compared to the result at 1,000°C. This is of course because the supersaturation in the matrix has been more affected during the deformation and the elimination of the dislocations at 900°C. The nucleation barrier value for the 900°C treatment, in addition, reaches a constant value after deformation. This is because the concentration in the matrix has reached its equilibrium value and the material is undergoing coarsening during which the driving force is almost not evolving anymore, as is the nucleation barrier. For the 1,000°C treatment, the behavior is different with a nucleation barrier continuously increasing after the deformation. This is due to the evolution of the material toward the thermodynamic equilibrium. Finally for the 1,100°C treatment, the nucleation barrier is also constant after deformation, and this time, it is because the driving force is also constant since nothing is precipitating into the material. The difference in the nucleation barrier between the classical precipitation modeling tool and the integrated scheme is not negligible and impacts the continuation of the transformation, as shown in Figure 3. Indeed, the critical step during precipitation is usually nucleation and the help given by deformation, even though very short, allows to activate many nucleation sites becoming precipitates that will continue to grow during the rest of the thermal treatment. In addition, one can easily expect that the longer the deformation time, the more significant the impact.

Figure 5 presents the same type of results as the one presented in Figure 3, that is, volume fraction, number density and mean size of the precipitates, and dislocation density as a function of time for the holding temperature 1,000°C, but comparing two different strain rates, d ε / d t = 1 s⁻¹ and 10 s⁻¹. The deformation time is then extended to 0.7 s, and the overall precipitation kinetics is affected from 65 s, the time at which the deformation starts. With this longer deformation time, the volume fraction (Figure 5A) is increasing even faster. The maximum number density of dislocations generated is slightly higher for the largest strain rate, 10s⁻¹; despite that, the deformation time is 10 times shorter. In addition, while dislocations are generated, they are also eliminated, see Eq. 12. Yet, this higher number of dislocations is not positively correlated with the number density of precipitates as shown in Figure 5B. Indeed, for a lower strain rate, the maximum value of the number density of precipitates is higher than for the larger strain rate because the number density of dislocations stays higher for a longer time. The precipitates have had more time to nucleate and thus to be stabilized in view of the critical size for coarsening. Finally, Figure 5C shows that the precipitates are slightly bigger at the end of the treatment for the lower strain rate. This can be explained by a lower nucleation barrier after deformation for the lower strain rate, since the volume fraction of precipitates already reached its equilibrium value and thus, a larger value than for the larger strain rate.

This new integrated modeling tool allows to account for the dislocation density evolution as a function of time and shows a great impact on the overall kinetics of the generation of precipitates. However, it is well-known that nucleation can also occur at grain and subgrain boundaries (Militzer et al., 1994; Trillo and Murr, 1998), and it, thus, impacts the recrystallization process essentially through the pinning effect and that the precipitates can delay the onset of recrystallization (Robson, 2020). In Al alloys, the coupling between precipitation kinetics and plasticity has been intensively investigated (Deschamps et al., 2012; Genevois et al., 2006; Hutchinson et al., 2014). It has for instance been shown that precipitation can be initiated even at room temperature under cyclic loading; regardless of whether a heat treatment is applied during or after deformation, an effect on the precipitation kinetics can be observed especially for temperatures lower than 300°C. At higher temperatures, the morphology of the precipitate can be modified during the plastic deformation.

It is also well-known from experimental observations that larger precipitates than the ones observed in the matrix usually are obtained both at subgrains and grain boundaries (Kwon and DeArdo, 1991; Wang et al., 2018). However, in this work, we do not consider rearrangement of the subgrains taking place during recrystallization and recovery. The coupling with a model describing grain growth during these transformations would be very beneficial even for the micromechanical model as the pinning effect of precipitates on subgrains and grain boundaries is well-known to impede grain growth. It is also known that elastic stresses around a nucleus can affect the nucleation barrier (Bonvalet et al., 2019), which is not yet accounted for. Nevertheless, it is important to highlight that the subgrains formed during recovery and recrystallization are created due to a rearrangement of the dislocations, and the dislocations are already included in this work. From that perspective, precipitation on subgrain boundaries might not be that different as considered in this work. However, considering the fact that the precipitation is homogenous, there is no effect of the specific location within the material. The kinetic difference between nucleation where recovery is already going on or where the density of dislocations is still high is not accounted for. Even if, to the opinion of the authors, the impact on precipitation kinetics compared to classical bulk precipitation, especially its increasing rate, would still exist, the magnitude of this impact might be different from what is presented in this work. In any case, this study demonstrated the importance of accounting for the microstructural changes generated by a deformation in order to understand the precipitation.

The simulations were performed with two different values of the interfacial energy in order to highlight its strong impact on the proposed modeling tool. The high impact is not surprising as the nucleation barrier is proportional to the interfacial energy to the power of three. In addition, the critical size for nucleation and coarsening is also proportional to the interfacial energy. Therefore, all stages of the precipitation are affected by the interfacial energy, which is difficult to estimate properly. In theory, it is a function of the composition on both sides of the interface, temperature, and crystallographic structures. Density functional theory (Lu et al., 2013) can help to calculate the value, but it is computationally expensive to obtain values as a function of temperature, composition, and phases. Broken bounds models (Sonderegger and Kozeschnik, 2009) have also been developed and implemented in mean-field approaches to capture the dependency of the interfacial energy on temperature, composition, and present phases with input from thermodynamic databases. However, large uncertainties still exist because of the simplicity of the approach. It would be extremely valuable for the community if more generic and systematic tools would exist. In the meantime, models such as the one proposed in this study can be adjusted to experimental data in order to extract a valuable function for the interfacial energy, using so-called reverse modeling.

This integrated modeling tool could also be applied to more complex thermomechanical treatments with several deformation cycles, mimicking an industrial process and could be compared to experimental investigations. Nevertheless, in that case, it would also be interesting to treat the precipitation locally. Indeed, the distribution of both stress and heat can be inhomogeneous for larger steel parts affecting the overall precipitation kinetics. LSKW approaches have in the past been coupled with one-dimensional finite elements schemes to account for this non-homogeneous matrix phase (Bardel et al., 2016; Roussel et al., 2018). The same approach could be applied to deformed parts.

Finally, coupling this type of modeling tool with other phase transformation tools such as the austenitic transformation to ferrite or martensite during cooling would allow capturing the initiation and evolution of the microstructural changes of a complete thermomechanical treatment.