Self-trapping of carbon atoms in ${\alpha }^{\prime }$-Fe during the martensitic transformation: A qualitative picture from ab initio calculations

Strain-induced and chemical interactions of interstitial carbon atoms in bcc or $\alpha$-Fe are obtained in first-principles calculations. Subsequent Monte Carlo simulations show that at low temperatures, carbon atoms prefer to occupy at least two different octahedral sublattices, which is due to quite strong attractive interactions of carbon atoms at the corresponding coordination shells. The direct total-energy calculations of one of the obtained ordered structures with composition ${\mathrm{Fe}}_{16}{\mathrm{C}}_2$, show that it is more stable than the predicted earlier structure with the same composition but carbon atoms occupying only one octahedral sublattice. This indicates that the long-existing thermodynamic mean-field theory of ordering of carbon in $\alpha$-Fe assuming strong preference of carbon atoms to occupy only one octahedral sublattice is deficient. It is shown that the presence of carbon atoms only at one octahedral sublattice in the experimentally observed martensitic phase, ${\alpha }^{\prime }$-Fe, is a self-trapping effect. It occurs during a displacive martensitic transformation from $\gamma$- to $\alpha$-Fe, which kinematically transfers the carbon atoms from a single fcc octahedral sublattice to one of three octahedral sublattices, where they appear to be locked by a consequent tetragonal distortion minimizing elastic energy of the phase. The latter creates a strong preference for carbon atoms to be only at one already occupied octahedral sublattice preventing them from further distribution over the other sublattices.

Figure 1

Octahedral sites (small filled circles) in the fcc structure (top) in their positions in the bct structure after a martensitic transformation (bottom left). Three types of octahedral sites in the bcc structure are shown on the right.

Figure 2

Strain-induced (upper panel) and chemical and total (lower panel) carbon-carbon interactions at the first five coordination shells compared with the results of microscopic theory of elasticity (MTE) [3, 9] and embedded atom method (EAM) [21]. Coordination shells are shown in Fig. 3.

Figure 3

First five coordination shells of the octahedral interstitial sites.

Figure 4

Calculated carbon-carbon interactions at the same $\left(V_{11}\right)$ and different $\left(V_{12}\right)$ octahedral sublattices. Filled circles show strain-induced contribution to the carbon-carbon interactions. The ab initio results by Domain et al. [6] are shown by filled triangles.

Figure 5

Positions of carbon atoms (small spheres) in ${\mathrm{Fe}}_{16}{\mathrm{C}}_2$. Larger spheres show the positions of Fe atoms, which form a $4\times 4\times 1$ “supercell” based on the cubic bcc unit cell. It is assumed that the direction perpendicular the topmost surface in the figure is [100]. The four atoms in the corners (suppose they occupy sublattice 1) are at the (140) [or (104)] coordination shell relative to the central atom in the face, which occupies a different sublattice (sublattice 2).

Figure 6

On-site $V_2^{\left(1\right)}(=V_3^{\left(1\right)},V_1^{\left(1\right)}=0)$ interaction at the second octahedral interstitial sublattice as a function of the $c/a$ ratio. The boundary of the existing martensite phase (appearing as a result of displacive transformation) is shown by dashed lines. The on-site interactions are calculated for two cases: for the fixed lattice constant $a$ and volume per atom $\Omega$.