Effects of dilute substitutional solutes on interstitial carbon in $\alpha$-Fe: Interactions and associated carbon diffusion from first-principles calculations

By means of first-principles calculations coupled with the kinetic Monte Carlo simulations, we have systematically investigated the effects of dilute substitutional solutes on the behaviors of carbon in $\alpha$-Fe. Our results uncover the following. (i) Without the Fe vacancy the interactions between most solutes and carbon are repulsive due to the strain relief, whereas Mn has a weak attractive interaction with its nearest-neighbor carbon due to the local ferromagnetic coupling effect. (ii) The presence of the Fe vacancy results in attractive interactions of all the solutes with carbon. In particular, the Mn-vacancy pair shows an exceptionally large binding energy of $-$0.81 eV with carbon. (iii) The alloying addition significantly impacts the atomic-scale concentration distributions and chemical potential of carbon in the Fe matrix. Among them, Mn and Cr increase the carbon chemical potential, whereas Al and Si reduce it. (iv) Within the dilute scale of the alloying solution, the solute concentration- and temperature-dependent carbon diffusivities demonstrate that Mn has a little impact on the carbon diffusion, whereas Cr (Al or Si) remarkably retards the carbon diffusion. Our results provide a certain implication for better understanding the experimental observations related with the carbon solubility limit, carbon microsegregation, and carbide precipitations in the ferritic steels.

Figure 1

Solute $M$-C configurations. The C atom's site labeled $i$ represents the first- to the fourth-nearest-neighbor (1$nn$ to 4$nn)$ octahedral site relative to the solute $M$ in the bcc lattice.

Figure 2

Solute $M$-C binding energies for the 3$d$ (top panel), 4$d$ (middle panel), and 5$d$ (bottom panel) elements with C in the 1$nn$ to 4$nn$ octahedral site relative to the solute $M$.

Figure 3

Solute $M$-C binding energies for the 1$nn$ configuration as a function of the solute size factor. (The size factor of solute $M$ in the Fe matrix is defined as ${\Omega }_{sf}^M=\frac{{\Omega }_M-{\Omega }_{\text{Fe}}}{{\Omega }_{\text{Fe}}}$, where ${\Omega }_M$ and ${\Omega }_{\text{Fe}}$ are the volume of supercell with the solute $M$ and defect-free supercell, respectively. See more details in Ref. [41].)

Figure 4

(a),(b) Local density of states calculated in the Fe${}_{53}M$C [$M=\text{Mn}$ (a) and Ti (b)] supercell. (c),(d) Charge density difference map for a single C with (c) Mn and (d) Ti in the 1$nn$ configuration. The blue balls and purple ball represent the Fe atoms and the C atom, respectively. The Mn and Ti atoms are both labeled.

Figure 5

The local magnetic moment of the solute $M$ $(M=\text{Mn}$ and Ti) versus the $M$-C distance in the most energetically stable Fe${}_{53}M$C supercell.

Figure 6

Solute $M$-vacancy-carbon configurations. The value $i$ represents the position of the carbon in the $i$th configuration.

Figure 7

(a) The total binding energies among the solute $M$, carbon, and vacancy. (b) The incremental binding energies between the $M$-vacancy pair and carbon. $M=3d$ (triangles), 4$d$ (squares), and 5$d$ (circles).

Figure 8

Local density of states of the Mn-C interaction (a) and of the Fe4-C interaction with Fe substituting Mn site (b) in the cfg${}^5$ model and their corresponding charge density difference maps: Mn-vacancy-C (c) and Fe-vacancy-C (d). The Mn, Fe, and C atoms are all labeled.

Figure 9

Carbon diffusion energy curves in the presence of a single vacancy or the Mn-vacancy pair in $\alpha$-Fe. The embedded figure shows the corresponding migration pathways. The values near the curves represent the energy barrier.

Figure 10

Solute $M$-C binding energy versus the $M$-C distance in units of the bcc lattice constant. $M=\text{Mn}$ (a), Cr (b), Si (c), and Al (d).

Figure 11

Carbon fraction at octahedral interstices for the first 6$nn$ shells with respect to the solute atom $M$ [$M$ = Si (a), Al (b), Mn (c), and Cr (d)] and beyond (labeled with $\infty )$ for a composition of ${\mathrm{Fe}}_{0.99}$$M$${}_{0.01}$${\mathrm{C}}_{0.001}$ (here ${\mathrm{C}}_{0.001}$ denotes the fraction of carbon atoms occupying interstitial octahedral sites).

Figure 12

Carbon fraction $f_{\infty }$ in the octahedral interstices beyond the solute $M$-C interaction distance as a function of temperature for a composition of Fe${}_{0.99}{M}_{0.01}$${\mathrm{C}}_{0.001}$ $(M=\text{Mn}$, Cr, Si, and Al).

Figure 13

The minimum-energy paths of the carbon migrating from the 1$nn$ to the 5$nn$ $o$ sites with respect to the solute Mn (a) or Si (b) in $\alpha$-Fe. The values with and without square brackets represent the carbon's positions (see Table 2) and the energy barriers, respectively.

Figure 14

Carbon diffusivities in the presence of the solute $M$ $(M=\text{Mn}$, Cr, Al, and Si) relative to that without $M$ versus the solute concentrations at temperatures of 500 K and 1000 K.