α ↔ γ phase transformation in iron: comparative study of the influence of the interatomic interaction potential

We use the method of the metric scaling [31, 32] for calculating the difference in free energy, Δf_bcc−fcc, between an atom in fcc and in bcc structure. This is done by transforming an fcc crystal via a linear transformation into the bcc structure along the Bain path, see below. We choose the Bain path here for its simplicity, even though it is known that several other transformation paths are relevant in the α ↔ γ phase transformation in iron and steel, such as the Nishiyama–Wassermann path [33, 34] and the Kurdjumov–Sachs path [35].

This transformation happens over 50 scaling steps. For each scaling step the change of the free-energy change is given by the work done [36, 37]:

$$\begin{equation}{\Delta}F=-{P}_{xx}{A}_{x}\mathrm{d}x-{P}_{yy}{A}_{y}\mathrm{d}y-{P}_{zz}{A}_{z}\mathrm{d}z.\end{equation} \tag{ 1 }$$

Here the P_αα (α = x, y, z) are the stress components along the coordination axes, A_α are the perpendicular cross-sectional areas in the scaling step and dα the change of the length along the α axis to the previous scaling step.

The Bain transformation [38] contracts an fcc crystal by 21% along one ⟨100⟩ direction while expanding it by 12% along the two other ⟨100⟩ directions and thus transforms it to an equal-volume bcc crystal. The orientation relationship between the fcc and bcc crystals is

$$\begin{equation}{\left(001\right)}_{\text{fcc}}\parallel {\left(001\right)}_{\text{bcc}}\quad \mathrm{a}\mathrm{n}\mathrm{d}\quad {\left[110\right]}_{\text{fcc}}\parallel {\left[100\right]}_{\text{bcc}}.\end{equation} \tag{ 2 }$$

For the simulation we use cubic fcc samples with a size of (36.37 Å)³ and containing N = 4000 atoms. The x, y and z axes correspond to the ⟨100⟩ crystal directions. Since at zero pressure, the volumes of the fcc and bcc units cells are not equal, the values of the expansion and contraction have to be slightly modified to transform the fcc unit cell to a bcc unit cell, which is again at zero pressure. We illustrate the scaling process in figure 1.

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The simulation is performed in the NVT ensemble by controlling the temperature with the Nosé–Hoover thermostat [39, 40]. Every 50 ps a scaling step is performed so that the total transformation from the fcc crystal into the bcc crystal takes 2.5 ns. Due to thermal fluctuations, the quantities P_αα and A_βγ are averaged over 1000 data points taken during these 50 ps. The change of the free energy of the entire crystal between the original fcc structure and the structure at the scaling step i is:

$$\begin{equation}{\Delta}{F}_{0\to i}=\sum _{j=1}^{i}{\Delta}{F}_{j-1\to j},\end{equation} \tag{ 3 }$$

where i = 0 refers to the unscaled fcc crystal and i = 50 to the completely transformed crystal in the bcc phase. For the change of the free energy for one atom we have:

$$\begin{equation}{\Delta}{f}_{0\to i}=\frac{{\Delta}{F}_{0\to i}}{N},\end{equation} \tag{ 4 }$$

where N is the number of atoms in the simulation box. For the complete Bain path the free energy is given as

$$\begin{equation}{\Delta}{f}_{\text{bcc}-\text{fcc}}={\Delta}{f}_{0\to 50},\end{equation} \tag{ 5 }$$

and denotes the free-energy difference between fcc and bcc structure.

For studying the dynamics of the phase transformation, we use a simulation volume with periodic boundary conditions. For the study of the bcc → fcc transition, our bcc sample has a size of 32.0 × 64.1 × 14.3 ų containing approximately 2500 atoms. For the martensitic fcc → bcc transformation, we use an fcc sample of size 73.3 × 146.5 × 14.3 with approximately 13 000 atoms. The bcc sample was chosen so small because otherwise—besides for the Meyer–Entel potential—no transformation would occur. It has been known for long [3] that defects help the phase transformation to nucleate; without the presence of any defects, the nucleation process may take longer than reasonable MD simulation times. We therefore include a grain boundary in the crystal; we use here a Σ5 (210) grain boundary whose influence on the transformation was studied in detail in [10]. It is a tilt grain boundary around the [001] axis, which we denote as the z direction. The x direction is the direction orthogonal to z in the GB plane, and y measures the distance perpendicular to the GB. Both crystallites are shown in figure 2. In both cases, periodic boundary conditions are applied in all Cartesian directions, such that our system actually corresponds to a multi-layered nanocrystalline material.

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The simulation systems containing the grain boundaries are first relaxed by energy minimization using the conjugate-gradient method at 0 K and equilibrated at 10 K for 50 ps in an NPT ensemble, keeping the pressure at zero and controlling the temperature by a Nosé–Hoover thermostat [39, 40]. From this initial state, we study both the austenitic and the martensitic transformation. For the austenitic transformation, we heat the bcc sample with a heating rate of 1 K ps⁻¹ up to 2000 K, while for the martensitic transformation, we leave the fcc sample at 10 K, since the metastable or unstable fcc phase decays at this temperature. Both these simulations are performed in an NPT ensemble at zero pressure. Temperatures above 2000 K are not relevant, since then the crystals will melt.

All calculations are performed with the open-source LAMMPS code [41]. The local lattice structure is determined by the common neighbor analysis (CNA) [42, 43]. The free software tool OVITO [44] is used for visualization of the local lattice structure.

We first consider the potential properties and compare them with experimental data. In table 1, several quantities of the fcc and bcc crystals as calculated for the potentials used here—the lattice constant a, the atomic volume Ω, the cohesive energy E_coh and the elastic constants C₁₁, C₁₂ and C₄₄—are compared to experimental data. The data for the free-energy difference and the energy barrier between the phases, as well as the transformation temperature will be discussed in section 3.2 below.

For the bcc phase, the calculated data are in good agreement with the experiment. This is not surprising since these data have been used to fit the potential. However, for the fcc phase, larger deviations are found, in particular for the elastic constants. An important issue is the atomic volume Ω, which in experiment is by 3.4 % smaller in the fcc than in the bcc phase, while it is predicted to be larger in all potentials with the exception of the Lee potential.

The energy difference between the bcc and the fcc phase at 0 K, ΔE_bcc−fcc, is negative for all potentials considered here—in agreement with experiment—meaning that the bcc phase is stable at 0 K. We also include the energy difference between the fcc and the hcp phase at 0 K, ΔE_fcc−hcp. Interestingly it has a positive sign for many potentials which thus predict the hcp phase to be more stable than the fcc phase at 0 K such that the order of stability is bcc < hcp < fcc. Only the Lee potential differs as it puts the energy of the fcc lattice lower than that of the hcp lattice. These findings will be relevant when discussing the relative amounts of hcp and fcc phase found in the austenitic transitions.

The free-energy data for the potentials investigated here are assembled in figures 3–7. We start the discussion with the Meyer–Entel potential, figure 3. Figure 3(a) shows the evolution of the free energy along the Bain path. A strong temperature dependence is visible. While at low temperatures, the bcc phase is energetically more favorable, at high temperatures, the fcc phase has lower free energy. The free-energy difference between the two phases as a function of temperature is plotted in figure 3. The phase transition temperature T_c, at which both phases have equal free energy, is close to 500 K; a more exact calculation gives 487 K [49], see table 1. We also assemble in table 1 the value of the energy difference between the fcc and the bcc phase at 0 K, ΔE_bcc−fcc.

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A noteworthy feature of the free energy evolution along the Bain path is the occurrence of a maximum between the two phases. It denotes an energy barrier that hinders the phase transformation. Its value at the phase transition temperature, E_barr, is 6.5 meV; here, no data from experiment or ab initio calculations appear to be available.

For the Meyer–Entel potential we see that the free-energy curve along the Bain path, figure 3(a), has a very low energy barrier at low temperatures. This is the reason why we can observe a martensitic phase transformation at low temperatures.

The Lee and Müller potentials feature the bcc–fcc transformation and their free energies are displayed in figures 4 and 5, respectively. For these two potentials, we do not show data below 400 K, since at small temperatures parts of the simulation volume on the Bain path spontaneously start phase transforming to the bcc phase. These potentials have a higher T_c than the Meyer–Entel, which is in better agreement with experiment (1184 K). Our value of T_c for the Lee potential is T_c ∼ 700 K, and thus definitely smaller than the value of T_c = 1070 K given in [16]. We presume that the use of the Gibbs–Helmholtz relation for thermodynamic integration used in [16] to determine T_c overestimates the value of T_c, as we previously observed in [32]. For the Müller potential, our value of T_c ∼ 1000 K is in good agreement with the published value of T_c = 1030 K [17].

In contrast to the Meyer–Entel potential, the Lee and Müller potentials exhibit a high energy barrier at low temperatures. This barrier will inhibit a martensitic phase transformation occurring at low temperatures. Also at the phase transition temperature, E_barr is higher than for the Meyer–Entel potential, see table 1.

The free-energy difference, Δf_bcc−fcc, has a maximum at around 1400–1500 K for both potentials. This is an indication of the tendency of Fe to return at even higher temperatures to the bcc phase. In experiment, the re-entrant bcc phase is denoted as the δ phase and it occurs at temperatures above 1665 K.

The Lee potential shows very small values of Δf_bcc–fcc in a wide range of temperatures around T_c, which are less than 3 meV in the temperature range of 400–2000 K. This is in good agreement with experimental data for temperatures higher than 1000 K [51].

The Zhou and Mendelev potentials, displayed in figures 6 and 7, respectively, feature a metal that does not phase transform: the bcc phase is stable up to 2000 K. In these potentials, the cohesive energy of the bcc phase is approximately 100 meV larger than for the fcc phase, in contrast to experiment. This results in the very negative values of Δf_bcc–fcc at small temperatures.

Δf_bcc–fcc increases with temperature, but the fcc phase is not stabilized. Even at 2000 K, Δf_bcc−fcc is around −40 ⋯ −30 meV.

We study separately the α → γ transition, which we denote as the austenitic transformation, and the γ → α transition (martensitic transformation). In both cases we use a system including a Σ5–(210)⟨100⟩ tilt grain boundary in order to help induce the transformation.

For studying the austenitic transformation, we heat the bcc sample up from 10–2000 K with a heating rate of 1 K ps⁻¹. We note that—in contrast to the Meyer–Entel potential—the Lee and the Müller potential did not show phase transformations for larger simulation systems. It is known [10] that the transition temperature for the austenitic transformation increases with the size of the simulation system, d, which is in our case the distance between grain boundaries. In other words, since the transformation nucleates at the grain boundaries, large distances between the grain boundaries and correspondingly small defect energy densities let the new phase nucleate only late, after substantial superheating. The higher energy barrier between fcc and bcc phase, E_barr, table 1, prevents large systems to phase transform in the Lee and the Müller potential.

Figure 8 shows the sequence of processes occurring in the Meyer–Entel potential. The phase transformation begins with the nucleation of the new phase at the grain boundary, figure 8(a). The transformation starts at a temperature of 510 K, which is above the equilibrium transition temperature of 487 K, table 1. This is plausible, as the free-energy barrier between the fcc and bcc phase, see figure 3(a), needs some overheating to be passed.

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After the nucleation of the new phase, it starts expanding over the entire sample. Figure 8(b) shows that two variants of the new phase have nucleated at the two grain boundaries bounding each grain and start coalescing. The final state at figure 8(c) shows in the upper variant a lamellar structure of mainly hcp phase including stacking fault planes, while in the lower variant fcc dominates. Between the two growing variants, a new grain boundary is established. The entire phase transformation took only 9 ps to occur.

The details of this phase transformation were already described in [10]. There a larger simulation system was used and the final phase contained only a single grain with dominant hcp fraction, in agreement with the upper variant seen in figure 8(c).

In total, we find 45.6% hcp and 37.4% fcc atoms in the transformed phase, so that in total the hcp phase is dominating. The high fraction of hcp atoms in the new phase is reasonable since we know [49] that in the Meyer–Entel potential, the energy of the hcp phase is lower than the energy of the fcc phase, see also table 1.

For the Lee potential, figure 9(a) shows that the transformation starts at a higher temperature, 969 K, in agreement with the higher transition temperature of the Lee potential, table 1. Phase detection is more inefficient at high temperatures and hence more unidentified atoms are seen in this transformation. Note that initially a high fraction of hcp atoms are produced, figure 9(b); however, they eventually tend to transfrom to a majority of fcc, figure 9(c). In the end we have 49.2% fcc and 21.5% hcp atoms. This is in agreement with the Lee potential being the only one to favor the fcc over the hcp phase, see table 1.

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In contrast to the Meyer–Entel potential the new phase is contained in a single grain, so that at the end of the phase transformation the system is again in a bi-crystalline structure, figure 9(c). A further analysis shows that the new grain boundary is a Σ7(451)–⟨111⟩_fcc tilt grain boundary.

The phase transformation in the Müller potential, figure 10, occurs at an even higher temperature, 1054 K. Here the nucleation of the new phase occurs only on one side of the grain, figure 10(a). The new phase proceeds over the entire sample, figure 10(b), until the sample is completely transformed, figure 10(c).

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The resulting microstructure is similar to that of the upper variant in the Meyer–Entel potential, figure 8(c), and is characterized by a high hcp content: 50% of hcp and only 26% of fcc phase. This is in agreement with the high energy difference between fcc and hcp phase, ΔE_fcc−hcp, see table 1, which is similar to that in the Meyer–Entel potential.

After the phase transformation, the grain boundary can be classified as a symmetrical Σ13(341)–⟨111⟩_fcc—respectively a Σ39(2570)–⟨0001⟩_hcp—tilt grain boundary.

We conclude that the energy barrier between fcc and bcc phase, E_barr, table 1, determines the system size that can be used in MD simulations of the austenitic transformation. It requires small system sizes for the Lee and Müller potentials. In addition, apart from the different transformation temperatures, also the microstructures of the newly formed phases differ. This applies in particular to the fcc/hcp content of the new phase, which is correlated to the energy difference between fcc and hcp phase, ΔE_fcc−hcp.

In previous studies of the martensitic transformation using the Meyer–Entel potential [11], a cooling process was applied to an fcc crystallite in analogy to the heating process for studying the austenitic transformation. In detail, the sample was cooled from 400 K with a rate of 0.333 K ps⁻¹; the bcc phase nucleated at 82 K, the entire transformation needed 34 ps for completion. However, when attempting to apply this scenario for the Lee or Müller potentials, the fcc phase did not transform, even when reducing the size of the simulation volume. We argue that it is the high energy barrier between fcc and bcc phase, visible in figures 4(a) and 5(a), which stay constant or even increase at small temperatures, that is responsible for impeding the transformation. Note that for the Meyer–Entel potential, this barrier decreases with temperature, figure 3(a), and hence allows the martensitic transformation to occur within a reasonable time in a MD simulation.

In the Mendelev and Zhou potentials, the fcc phase is unstable, such that it decays immediately even without a cooling process. To set all simulations of the martensitic transformation on an equal footing, we therefore set up an fcc bi-crystal containing a Σ5–(210)⟨100⟩ tilt grain boundary at 10 K, and study the transformation of this system at constant temperature. Again, for the Lee and Müller potentials, no transformation occurred within 1 ns simulation time.

For the Meyer–Entel potential, the fcc phase decays within 3.45 ps, see figure 11. The new phase nucleates in a plate structure, both at the grain boundaries and in the bulk; after growth of the plates, these coalesce and finally the bcc phase is defect-free and fills the entire grain. The grain boundary stays in place in the new bcc phase and can be described as a Σ44(774)–⟨110⟩_bcc tilt grain boundary with a tilt angle of 44°. This transformation has already been described previously [11]; there, however, a larger simulation crystallite was studied, such that the final bcc phase grew in two variants, separated by twin grain boundaries.

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In the Zhou potential, figure 12, the transformation proceeds even faster, within 1.25 ps. The new phase nucleates at the grain boundary, and quickly expands throughout the entire crystal. The intermediate stage, figure 12(b), looks quite irregular with a high fraction of unidentified atoms. However, the final microstructure, figure 12(c), shows the complete transformation of the fcc phase into a rather regular array of twinned bcc variants. The original grain boundary adopts in the bcc phase the structure of a Σ9(221)–⟨110⟩_bcc tilt grain boundary with a tilt angle of 38.94°.

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Also in the Mendelev potential, figure 13, the fcc phase nucleates at the grain boundaries and quickly expands into a rather irregular mixture of transformed bcc and unidentified atoms. However, the final bcc phase is defect-free and spans the entire grain in a single variant, figure 13(c). The grain boundary in the bcc phase is again a tilt boundary; it is now characterized as a Σ27(552)–⟨110⟩_bcc boundary with a tilt angle of 31.59°. The entire transformation takes 2.55 ps.

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We conclude that again the energy barrier between the fcc and the bcc phase determines whether a martensitic transformation happens at all; here, now the barrier at low temperatures is relevant. For the Müller and Lee potentials the barrier is too high at low temperatures so that the metastable fcc phase does not decay at all. In contrast, for the Zhou and Mendelev potentials, there is no barrier at all; the fcc phase is unstable. The Meyer–Entel potential is intermediate, since here the barrier has a finite height at higher temperatures, but decreases toward zero at low temperatures such that at low temperatures, the martensitic transformation is possible.

While the new phase nucleates mostly from the grain boundaries—with the exception of the Meyer–Entel potential, where a small fraction of homogeneous nucleation in the bulk phase is also observed—the resulting bcc phase may have quite different microstructures, ranging from a single variant to a fine lamellar nanotwin structure for the Zhou potential. Even more astonishing is that, while the transformation starts from the same Σ5–(210)⟨100⟩ grain boundary in the fcc phase, the resulting grain boundaries in the bcc phase assume tilt angles of 31.6°, 39° and 44°.