Effect of the stress field of an edge dislocation on carbon diffusion in $\alpha$-iron: Coupling molecular statics and atomistic kinetic Monte Carlo

Carbon diffusion near the core of a $\left[111\right]\left(\overline{1}01\right)$ edge dislocation in $\alpha$-iron has been investigated by means of an atomistic model that brings together molecular statics and atomistic kinetic Monte Carlo (AKMC). Molecular statics simulations with a recently developed embedded atom method potential have been carried out in order to obtain atomic configurations, carbon-dislocation binding energies, and the activation energies required for carbon hops in the neighborhood of the line defect. Using information gathered from molecular statics, on-lattice AKMC simulations have been performed for temperatures in the 300–600 K range, so as to study the behavior of a carbon atom as it interacts with the edge dislocation stress field. This model can be seen as a very first step toward the modeling of the kinetics of carbon Cottrell atmosphere formation in iron during the static aging process.

Figure 1

The $K\text{th}$ transition, whose probability per unit of time is $w_K$, will be selected since its assigned value of $s\left(k\right)$ intercepts $r_2\cdot Z$.

Figure 2

(Color online) (a) An edge dislocation in a cubic lattice (chosen for the sake of simplicity in the representation). The dislocation line is found in the locus where the extra half-plane ends and is represented by the symbol $⊺$. (b) Sketch of the cylindrical simulation box with an edge dislocation right in the center (radius $\approx 150\text{Å}$). The Burgers vector is in the [111] direction and the dislocation line is in the $\left[1\overline{2}1\right]$ direction (perpendicular to the plane of the page). The atomic positions in the inner cylinder were allowed to fully relax during molecular statics simulations whereas the positions in the outer ring were kept fixed to the values defined by anisotropic elasticity theory. Periodic boundary conditions were only applied along the dislocation line.

Figure 3

(Color online) Centro-symmetry deviation mapping in the vicinity of an edge dislocation in $\alpha \text{-Fe}$: [green (mid gray) balls] $\text{CSD}<0.005{\text{Å}}^2$, [yellow (light gray) balls] $0.005\le \text{CSD}<0.05{\text{Å}}^2$, [blue (dark gray) balls] $0.05\le \text{CSD}<0.5{\text{Å}}^2$, and (white balls) $\text{CSD}\ge 0.5{\text{Å}}^2$. In the inset, the iron atoms in the dislocation core (side view), i.e., atoms with a CSD parameter larger than $0.5{\text{Å}}^2$.

Figure 4

(Color online) The three nonzero (out of the dislocation core) components of the edge dislocation stress field (plot area: $120\times 120{\text{Å}}^2$), as calculated by molecular statics.

Figure 5

(Color online) (a) [100], [010], and [001] directions (taken with respect to the dislocation line) where saddle-point search with the modified Drag method has been carried out. The dotted line indicates the glide plane, and the dashed circle (whose radius is $100\text{Å}$) encompasses the region where octahedral and tetrahedral sites have been mapped. (b) Local energy minima (filled circles) and saddle points (empty circles) along the [100], [010], and [001] directions, as obtained with the modified Drag method. The energy reference is the total energy of the o site in the corresponding line end. The dashed line indicates the position of the dislocation line.

Figure 6

(Color online) Mean energy barriers inside the molecular statics simulation box. The dispersion (standard deviation) about the mean is in parentheses. The activation energy at $T=0\text{K}$ in a nonstrained bcc iron lattice obtained with the with the current EAM potential is 0.81 eV.

Figure 7

(Color online) Mapping of carbon-dislocation binding energies per octahedral site type in the neighborhood of an edge dislocation (plot area: $120\times 120{\text{Å}}^2$).

Figure 8

(Color online) Region containing sites for which $\left|\Delta E^b\right|>k_{B}T$, for different temperatures. The dotted line indicates the glide plane, and the dashed circle (whose radius is $100\text{Å}$) indicates the boundary of the region of the simulation box where octahedral and tetrahedral sites have been mapped.

Figure 9

Average vectors $⟨\overrightarrow{ϵ}⟩$ in a $20\times 20$ mesh $\left(\Delta x=\Delta y=10\text{Å}\right)$, for $T=300\text{K}$. Only the components perpendicular to the dislocation line are shown, in order to characterize how carbon diffusion is biased with respect to the line defect. Arrows have been rescaled to make visualization easier $\left(10\text{Å}\equiv a0/2\right)$. The longest the arrow, the strongest the bias on carbon diffusion.

Figure 10

Mean diffusion coefficient as a function of volume (delimited by the radius $r$, see Eq. (7)) compared to bulk diffusivity. Considering the whole volume of the AKMC simulation box, the mean diffusion coefficient tends do the bulk value for all temperatures.

Figure 11

(Color online) Fraction of total carbon trajectories that (a) terminated in the core of the edge dislocation or (b) left the AKMC simulation box through its open boundaries. The dashed (dotted) line indicates the fraction of trajectories that should arrive to the dislocation core (simulation box boundary) if bias on carbon diffusion is not considered.

Figure 12

(Color online) (Empty circles) Fraction of dislocation-trapped carbon trajectories starting in either the tensile or the compressive half. (Filled circles) Fraction of dislocation-trapped carbon trajectories that arrived to the dislocation core coming from either the tensile or the compressive half.