The classical nucleation theory describes the steady state nucleation rate for each nucleation type and can be expressed as [
66,
67]:
where
j denominates the type of nucleation site;
Nn is the number density of potential nucleation sites
;
Z is the Zeldovich factor representing the thermally activated growth of subcritical nuclei, which is assumed to be constant;
T is the temperature
;
k is the Boltzmann constant
;
h is the Planck constant
; and
QD is the activation energy for the iron self-diffusion in ferrite (
[
18]). The last parameter is the activation energy for nucleation
, which can be calculated by [
67,
68,
69]:
where the parameter
ψ contains information about interface energies and the shape of the nucleus and is estimated to be
for austenite nucleation, with the estimated number of special orientation relationships as between austenite and ferrite [
68];
is the Gibbs free energy for the newly formed phase
; and
is misfit strain energy between the nuclei and matrix
. The influence of the chemical composition on the nucleation can be assessed by
and
. The nucleation is dependent on the nucleation activation energy and temperature as follows:
where the total number of activated nuclei is proportional to the driving force and misfit strain energy.
3.3.3. Strain Energy of Austenite Nucleation
The next step in evaluating the influence of Si and Mn on the nucleation is the strain energy (
). The elastic strain energy can be calculated depending on the shape of the nuclei based on equations from Refs. [
75,
77,
78]. The equation describing the case of ellipsoidal precipitates is as follows:
where μ
mx is the shear modulus of the matrix
; δ is the misfit due to the volumetric difference between precipitate and matrix; and E(c/a) is a function for elastic energy of the precipitate dependent on its shape, with c being polar and an equatorial diameter. The volumetric misfit can be calculated from the atomic volumes of the precipitate and matrix:
where
Vppt and
Vmx are the atomic volumes of the precipitate and matrix, respectively. In the present study, the precipitate phase is austenite (γ) and there are two matrix phases: ferrite (α) and cementite (θ). Hence, we calculate misfits γ/α and γ/θ separately. The atomic volumes for all three phases are calculated, taking into account both the individual compositions and thermal expansion as well. The parameters for calculating ferrite and austenite atomic volumes are taken from Refs. [
79,
80,
81]:
where
is atomic fraction of element
i with
i = (C,Mn,Si); and T is the temperature in K. The data for the influence of substitutional elements on the lattice parameters of cementite are limited. Some experimental data are available showing the changes due to Mn [
82,
83,
84,
85], but there are no experimental data for the Si substitution of Fe. Since, in the case of high-Mn alloys, the concentration of Si in cementite is significant, we estimate the effect of Si from the first-principle calculations [
86,
87,
88,
89,
90]. Si and Mn affect all three lattice parameters differently, leading to a combined equation for the atomic volume:
The C concentrations of both cementite and ferrite are known. We consider two border cases for the C concentration in the austenite nuclei, the first indicated by the maximum driving force and the second by the average carbon content of pearlite. As a first step in calculating the volumes of individual phases, we calculate the atomic fractions of each phase needed for the formation of an austenite nucleus, with a given chemical composition. We use, in this case, mass conservation:
where A and B are the atomic volume fractions of cementite and ferrite to be transformed for the austenite nucleus to form with a given C content; and
are carbon concentrations of the phases
j = (α,θ,γ)
. With the given atomic fractions and using the atomic volumes, we can calculate the actual proportions of the volumes of each phase transformed by:
where the transformation factor (α/θ
trans) is calculated as the ratio between the atomic volume fractions of ferrite (A) and cementite (B) defined in Equation (14). The results are shown in
Figure 13.
The average Si concentration in the alloy affects the transformation factor between ferrite and cementite (see
Figure 13). With the exception of the alloys 0.1Si2Mn and 0.4Si2Mn, the cementite fraction forming the austenite nuclei is higher than expected from the ferrite/cementite ratio of pearlite (α/θ = 7). The alloys 1.5Si2Mn and 1.9Si2Mn show values below 1, meaning that more cementite than ferrite transforms to austenite nuclei. This can be partly explained by the concentration of Si in cementite, which makes it less thermodynamically stable [
89,
91,
92,
93]. As shown by the composition analysis, the concentration of Si in cementite in the high-Mn alloys, although lower than in the matrix, can still be significant (see
Table 5). A slight increase in the Si concentration in ferrite, at the same time, makes ferrite more stable [
77,
79,
80,
81,
82,
94,
95,
96].
As the consequence of the increase in the carbon concentration in austenite nuclei as well as of the ratios of dissolved matrix phases, volumetric changes depend on the alloy. The results of the misfit calculations with Equation (10) are shown in
Figure 14. In the case of high-Mn alloys, with the increase in the overall Si content, the misfit between ferrite and austenite nuclei becomes positive, meaning that the atomic volume of austenite becomes larger than that of ferrite. At the same time, the atomic volume of cementite is much larger than that of austenite. The misfit for the low-Mn alloys, on the other hand, does not depend strongly on the Si concentration due to an almost constant C content in austenite nuclei. What is different from the case of high-Mn alloys is that δ
γ/α is much smaller and has values close to 0, meaning that there is only a small difference in the atomic volume between ferrite and austenite, especially for the 1.5Si0.1Mn alloy.
The models for calculating the elastic properties of ferrite and austenite at elevated temperatures in the current work are taken from Ref. [
80]. The model parameters are adjusted by fitting the model using orthogonal distance regression (Ref. [
97]) to the experimental results published in Refs. [
98,
99,
100,
101,
102,
103]. The influence of the temperature and Mn on the lattice parameters of cementite are taken from Ref. [
104]. There are no experimental data on the influence of Si on the elastic properties of cementite nor on their temperature dependence. The effect of Si on the elastic properties are therefore based on the first principle calculation results given in Refs. [
87,
90]. As for the effect of temperature, we consider the results presented in Ref. [
104]. The authors show that, such as for properties of ferrite, there is a shift due to change in magnetic properties of cementite [
80,
101] occurring at the Curie temperature (
TC). In the case of cementite, Mn decreases the
TC by ~13 °C/at% [
83,
104]. The results shown in Ref. [
105] indicate that Si has a similar influence on
TC, both in terms of direction and magnitude. Therefore, we assume that the Si effect on the elastic properties of cementite will be the same as for Mn in terms of the temperature dependence.
Using the presented parameters, we calculate the γ/α and γ/θ strain energies with the assumption that
E(
c/
a) = 1, which is valid for spherical precipitates (
Figure 15). There are some variations in the γ/α strain energies ranging between 10
6 and 10
8 J/m
3, which are in the correct order of magnitude when compared to reported values in the literature [
68,
106]. The influence of the chemical composition on the strain energies seems to be clear. High-Mn alloys show a minimum in the case of alloys containing between 0.4 to 1.0 wt% of Si (see
Figure 15a–c). The alloys with 0.1 wt% of Mn show a constant decrease in the strain energy with the increase of Si (see
Figure 15d–f). The overall values are an order of magnitude lower when compared to high-Mn alloys. As with the driving force, there are small differences between the homogenized and nonhomogenized materials. In low-Mn alloys, the γ/θ strain energy is rather constant and much higher than the strain energy between ferrite and austenite. The values for γ/θ can be 100–300× higher than those for γ/α. The strain energy between the cementite and austenite nuclei in the case of high-Mn alloys is decreasing with increasing overall Si content, which can be directly correlated with the increase in the C concentration of austenite nuclei. In the case of alloys with Si concentrations of 0.1–1.0 wt%, the strain energy of γ/θ is still much higher than the strain energy of γ/α, which can be between 30–100× higher. In the case of alloys with 1.5 and 1.9 wt% of Si, the differences between the strain energy of austenite with the two matrix phases decreases to 1.5–5×. With the exceptions of these two alloys, the strain energy causes a very high barrier for the cementite-to-austenite transformation considering the driving force magnitude (see
Figure 12 and
Figure 15). The main reason for these high strain energies are the calculated misfits (see
Figure 14). The volumetric misfits of austenite and ferrite are rather small compared with misfits of austenite and cementite. All considered, it is plausible that the change in the austenite volume would rather be accommodated by ferrite than by cementite due to lower energetic barriers. Therefore, to evaluate the extreme values, we consider two scenarios: (1)
due to δ
γ/α=0 and the volume change from θ→γ transformation which is accommodated by ferrite; (2) the strain is dependent on the fraction of each phase dissolved as calculated by Equation (15) and individual misfits between the γ and matrix phases. The misfits in α under conditions of case 1 are shown in
Figure 14 as δ
γ/α exc. The change in the volume due to transformation is always negative in ferrite, although it becomes quite small with an increasing Si concentration. It influences the resulting strain energy, which in case 1 is
and in case 2 is
. The comparison of the two extremes of strain energy and driving force for nucleation is shown in
Figure 16.
The condition that needs to be met for nucleation to take place is
. The values shown in
Figure 16 indicate that this is the case for high-Mn alloys with an Si content higher than 1.0 wt% for all heating rates. In the case of alloy 0.4Si2Mn, the nucleation can take place if we assume the volume change is accommodated by ferrite, minimizing the strain energy. The same condition makes the nucleation possible for low-Mn alloys, where the
values are similar to the
values considering their uncertainty. For the alloys with a strain energy higher than the driving force for nucleation, we reconsider our assumptions for the parameters used here: (1) the shape of the nuclei, and therefore, the value of
; (2) possible difference in the carbon content of austenite nuclei.
Previously, we assumed the nuclei shape to be spherical, for which
= 1. As shown in previous studies, the value of
strongly depends on the properties of the nuclei versus the matrix [
75,
78,
107,
108,
109]. In order to evaluate which alloys in our case are more sensitive to the nuclei shape and its influence on strain energy, we first calculate the minimum value of
for the following condition to be met:
The results are shown in
Figure 17. For all heating rates, the low-Mn alloys have values in the range of 0.02–0.15, but considering large uncertainties, it is difficult to estimate the nuclei shape. The values of
for the high-Mn alloys with 0.1, 0.4 and 1.0 wt% of Si are below or close to 1, indicating that a disk shape is preferable in those cases. The nucleus shape in the alloy 1.0Si2Mn should be more disk-like if we consider the mix of the strain energy. In the case of volume change being accommodated by ferrite, the GS is small enough to allow for any shape of the nuclei.
Recent work by Böhm et al. [
108,
109] defines a shape factor
H(
c/
a) based on the work of Eshelby [
110], which in essence is the
function in our work. Using the methodology presented in Ref. [
108], we calculate the values of this function for γ/α and γ/θ using the elastic properties of all phases at elevated temperature. The resulting values are shown in
Figure 18. In all cases, the
E(
c/a) function for γ/α indicates a slight preference for disk-shaped nuclei, unlike γ/θ (
Figure 18d,e). The shape of the nuclei formed from cementite shows a shift from spherical to disk shaped (
Figure 18e). This shift occurs at an overall 1.0 wt% of Si content, where the value of the function for any shape is close to 1. The increase in the overall Si content seems to be decreasing the minima of the
E(
c/a) when ferrite is the matrix phase (
Figure 18d). The minimum decrease from 0.97 for the 0.1Si2Mn alloy to 0.87 for the 1.9Si2Mn alloy. When compared with the values that are needed for the nucleation to start as shown in
Figure 17, the
E(c/a) calculated for our alloys are not enough to explain the occurrence of the nucleation, especially in the alloy 0.1Si2Mn.
The second scenario we are testing to explain the occurrence of the nucleation is the variation of the C concentration in the newly formed austenite nuclei. The C content of austenite nuclei is initially derived from the maximum of the chemical driving force (
). It influences the lattice parameter of austenite and its elastic properties. Therefore, in the case of high-Mn alloys with 0.1–1.0 wt% of Si heated at 0.1 °C/s, by varying the C content in austenite, we determine if the nucleation is possible with given conditions. The results are shown in
Figure 19.
The changes in the chemical driving force and strain energy due to change in the nuclei C content in the alloys 0.1Si2Mn and 0.4Si2Mn at a 0.1 °C/s heating rate are still insufficient to explain the start of transformation. In contrast, the energies calculated for the alloy 1.0Si2Mn shows the nucleation is possible to start with a relatively large range of C concentrations (0.02–2.45 wt% of C). Since the C concentration influences elastic properties, the only change in this case is the change in the shape of the γ nuclei forming from θ from disk to spherical, which occurs at a C concentration of 0.90 wt% of C in austenite nuclei.