Modeling of soft impingement effect during solid-state partitioning phase transformations in binary alloys

Figure 1a and b are the schematics which illustrate the evolution of diffusion fields as the interface migrates from the α-phase into the β-phase for the diffusion-controlled growth model and the mixed-mode model, respectively. The main difference between the diffusion-controlled growth model and the mixed-mode model is that the solute concentration at the interface in the diffusion-controlled growth model is assumed to be fixed during the entire phase transformation, while it should change physically in the mixed-mode model. Referring to Fig. 1, we define t ₂ as the time after which the diffusion fields start to overlap, x ₀ as the interface position, L as the length of diffusion field, \( C_{\text{eq}}^{\beta \alpha } \) and \( C_{\text{eq}}^{\alpha \beta } \) as the equilibrium concentration in the β- and α-phase, C ₀ as the bulk concentration, C _m as the carbon concentration at the center of the β-phase, and 2X as the thickness of the parent phase.

In this study, the Murray–Landis method [15, 16] is chosen for the numerical solution, and the finite difference equation is written aswhere \( c_{i}^{t} \) is the concentration at grid point i at time t, \( c_{i}^{t + \Updelta t} \) is the concentration at grid point i at time t + Δt. The second term on the right side of Eq. 33 accounts for the time dependence of the grid points. The diffusion-controlled growth model and the mixed-mode model will be combined with the Murray–Landis method to simulate the partitioning phase transformation here.

To illustrate the effect of soft impingement here, the austenite to ferrite transformation in the binary Fe-1.0 at.%C alloys at 1050 K is investigated here. At the given temperature, Thermo-Calc gives χ = 110 J/(at.%), and the equilibrium carbon concentration in the austenite phase and ferrite phase is 2.05 and 0.09 at.%, respectively. The diffusion coefficient of carbon in austenite is \( 1.14 \times 10^{ - 12} \;{\text{m}}^{2} / {\text{s}} \), and the interface mobility M is taken to be \( 5.4 \times 10^{ - 8} \;{\text{mol m/Js}}. \) In order to consider the phase transformation in a finite medium, the finite thickness of the austenite phase 2X is assumed to be 20 μm, and the specific volumes of both the phases are taken equal.

In Fig. 2a, the thickness of ferrite phase is calculated as a function of time by the mixed-mode model with linear diffusion field and quadratic diffusion field. The solid line in the figure is the modeling results with soft impingement correction, and the dotted line is the results without soft impingement correction. It is shown that the mixed-mode model with soft impingement correction predicts the interface migration stops when the fraction of ferrite reaches the equilibrium value, while the mixed-mode model without soft impingement correction shows that the interface migration will not stop in finite time. Furthermore, the figure indicates that the mixed-mode model with linear diffusion field predicts a slower kinetics than the model with quadratic diffusion field does. Figure 2b shows the results predicted by the diffusion-controlled growth model, which indicates that the diffusion-controlled growth model with soft impingement correction also predicts the partitioning phase transformation to finish as the thermodynamic equilibrium is approached.

Figure 3 shows the fraction of ferrite as a function of time obtained by the analytical models and the fully numerical solutions. It shows that both the analytical mixed-mode model and the analytical diffusion-controlled model with a quadratic diffusion field predict the kinetics more precisely than those with linear diffusion field assumption. Comparing the present mixed-mode model assuming a quadratic diffusion profile with the mixed-mode model assuming an exponential diffusion profile in [14], the derived equation for solute concentration at the interface during the non-overlapping diffusion field stage is the same in these two reformulated mixed-mode models. However, as mentioned above, the exponential diffusion profile is only valid in the infinite medium, while the quadratic diffusion profile can also be applied in the overlapping diffusion field stage, as shown in “Model”. It has to be mentioned here that the quadratic diffusion profile is just one case of polynomial diffusion profiles; in general, the exact diffusion profile can be approximated as

Equation 34 is a (n − 1) order polynomial diffusion profile, in which n parameters have to be determined, which means n boundary conditions are needed to solve the problem. Except Eqs. 2 and 3, (n − 2) extra boundary conditions can be written asBased on the above boundary conditions, the parameters in Eq. 34 can be obtained. Although the accuracy of the model could possibly be increased increasing the order of the diffusion profile, it also make the case more complicated, that is why the linear diffusion profile [3, 7, 13] has been widely applied in the past. As shown in Fig. 3, the accuracy of the quadratic diffusion profile is quite close to that of the numerical solution, which means the accuracy will not be further improved significantly by increasing the order of the diffusion profile. Therefore, the quadratic diffusion field approximation would be applied to the analytical models in all the following calculations.

Considering the soft impingement effect at the later stage, the solute concentration at the center of the austenite phase and diffusion length, which are the two key parameters in the overlapping diffusion stage, is calculated as a function of time by the mixed-mode model and diffusion-controlled growth model in Fig. 4a and b, respectively. The vertical dotted line in the figure indicates the critical point at which the diffusion fields in the neighboring grains start to overlap. Before the critical point is reached, the diffusion length extends as the interface migrates into the austenite phase, and the solute concentration at the center of the austenite phase is not affected by diffusion and is fixed at the bulk concentration of the Fe–C alloys. After the critical point, the diffusion length would starts to shrink, and the solute concentration at the center of the austenite phase would begin to increase toward the equilibrium concentration. In the model considering the soft impingement, both the solute concentration at the center of the austenite phase and the diffusion length can be used to estimate the soft impingement starting point, while only the diffusion length can be the effective factor for detecting soft impingement in the model without soft impingement correction.

As discussed in the original mixed-mode model [3], the bulk concentration C ₀ affects the evolution of diffusion length during the austenite to ferrite phase transformation at a certain temperature, and it is concluded that the smaller the bulk concentration C ₀, the shorter the diffusion length. However, at different temperatures, the equilibrium concentrations in the austenite and ferrite phase are different, which also affects the diffusion length evolution. Therefore, it is necessary and meaningful to summarize all the effect factors into one factor to obtain a general law. In Fig. 5, the diffusion lengths as a function of the thickness of ferrite phase for different degrees of super-saturation are calculated by the diffusion-controlled growth model. The value of diffusion length is only affected by the degree of super-saturation, in which the bulk concentration and the equilibrium concentration in both the austenite and the ferrite phase are included. It shows that the magnitude of the diffusion length decreases with increasing the degree of super-saturation. Actually, decreasing the bulk concentration at a certain temperature discussed in the original mixed-mode model [3] is just one specific case of increasing the degree of super-saturation according to Eq. 13, and Fig. 5 can be considered as a master curve for estimating the diffusion length.

In order to investigate whether the overlapping diffusion stage or the non-overlapping diffusion stage dominates the transformation kinetics, the ratio of ferrite transformed during the overlapping diffusion stage and the total equilibrium fraction of ferrite as a function of the degree of super-saturation is calculated by the diffusion-controlled growth model in Fig. 6. It is indicated that the ferrite transformed during the overlapping diffusion stage decreases with increasing the degree of super-saturation. This can be easily understood in this way: as the super-saturation increases, the solute concentration difference between the growing ferrite phase and the bulk concentration would be smaller, which means less carbon has to be rejected from the ferrite phase into the austenite phase as the interface migrates into the austenite phase and thus less carbon would pile up in front of the interface, so the overlapping diffusion stage would be shorter.

As shown in Figs. 5 and 6, the diffusion length in the diffusion-controlled growth model is a simple function of the degree of super-saturation. The magnitude of diffusion length in the mixed-mode model is not just determined by the degree of super-saturation but also by the diffusion coefficient and the interface mobility. As discussed in recent study [17], the ratio of the interface mobility and the diffusion coefficient has a strong effect on the kinetics of partitioning phase transformation, thus there is no doubt that the soft impingement effect during the partitioning phase transformation is also determined by this ratio. Compared with the diffusion coefficient, the value of interface mobility during partitioning phase transformation is still not known accurately [18‐20]. Therefore, fixing the value of diffusion coefficient and varying the value of interface mobility, in Fig. 7, the diffusion length and size of ferrite phase during the austenite to ferrite phase transformation at 1050 K in Fe-1.0 at.%C alloys as a function of time are predicted by the mixed-mode model and the diffusion-controlled growth model. As the temperature and bulk concentration of the Fe–C alloys are fixed, the degree of super-saturation is a constant, thus only the effect of interface mobility and diffusion coefficient on the soft impingement is investigated in Fig. 7. It is indicated that the increase in the interface mobility has a small effect on the diffusion length as a function of time but this leads to a significant increase in the interface migration rate, which would make the soft impingement start earlier. Also, increasing the interface mobility, the kinetics predicted by the mixed-mode model would become closer to that by the diffusion-controlled growth model, since the transformation kinetics is more and more controlled by the solutes diffusion, more discussion about this point can be found in [17]. In Fig. 8, the diffusion length as a function of the fraction of ferrite phase is calculated by the mixed-mode model with different interface mobilities and diffusion-controlled growth model. Although the soft impingement starts earlier with increasing interface mobility as shown in Fig. 7, the ferrite transformed during the soft impingement stage decreases as shown in Fig. 8, which means the dominance of the soft impingement effect on the overall transformation kinetics reduces. In order to indicate the effect of interface mobility on the carbon profile evolution, in Fig. 9, the carbon profile in the austenite phase is calculated by the diffusion-controlled growth model and the mixed-mode model with different interface mobilities.

The carbon concentration at the interface in the austenite phase, which determines the magnitude of driving force for interface migration, is a key physical parameter in the mixed mode, thus it is calculated by the mixed-mode model in Fig. 10a. It shows that the interface concentration increases as the interface migrates, and the driving force would decrease to 0 and interface stops migrating when the equilibrium concentration is approached. The interface concentration predicted by the mixed-mode model without soft impingement correction [14] would not be equal to equilibrium concentration even when the equilibrium fraction is approached. It should be pointed out the interface concentration equals C ₀ for all values of M ₀ when the ferrite fraction is zero. However, in the initial stage, the interfacial carbon concentration increase extremely fast, this steep increase is not properly reflected in Fig. 10a. In Fig. 10b, the newly defined growth mode parameter \( H\left( { = {\frac{{C_{\text{eq}}^{\beta \alpha } - C^{\beta } }}{{C_{\text{eq}}^{\beta \alpha } - C_{\text{eq}}^{\alpha \beta } }}}} \right) \)[17] as a function of fraction of ferrite phase for different interface mobilities is presented. For a certain interface mobility, the H decreases as the phase transformation proceeds, which indicates that the growth mode become more diffusion controlled. As the soft impingement effect is corrected in the present mixed-mode model, the growth mode parameter would decrease to 0 when the thermodynamic equilibrium is established. Also, the growth mode is changed correspondingly with the variation of interface mobility, and the growth mode is approaching to the pure diffusion-controlled growth with increasing the ratio of interface mobility and the diffusion coefficient. In other words, it can also be stated that the dominance of soft impingement on the transformation kinetics is determined by the growth mode if the degree of super-saturation is fixed.