Carbon diffusion paths and segregation at high-angle tilt grain boundaries in α-Fe studied by using a kinetic activation-relation technique

Carbon diffusion and segregation in iron is fundamental to steel production but is also associated with corrosion. Using the kinetic activation-relaxation technique (k-ART), a kinetic Monte Carlo (KMC) algorithm with an on-the-fly catalog that allows to obtain diffusion properties over large time scales taking into account long-range elastic effects coupled with an EAM force field, we study the motion of a carbon impurity in four Fe systems with high-angle grain boundaries (GB), focusing on the impact of these extended defects on the long-time diffusion of C. Short and long-time stability of the various GBs is first analyzed, which allows us to conclude that the $\mathrm{\Sigma }3\left(111\right)\theta =109.{53}^{\circ }\langle 110\rangle \mathrm{GB}$ is unstable, with Fe migration barriers of ∼0.1 eV or less, and C acts as a pinning center. Focusing on three stable GBs, in all cases, these extended defects trap C in energy states lower than found in the crystal. Yet, contrary to general understanding, we show, through simulations extending to 0.1 s, that even tough C diffusion takes place predominantly in the GB, it is not necessarily faster than in the bulk and can even be slower by one to two orders of magnitude depending on the GB type. Analysis of the energy landscape provided by k-ART also shows that the free cavity volume around the impurity is not a strong predictor of diffusion barrier height. Overall, results show rather complex diffusion kinetics intimately dependent on the local environment.

Figure 1

Details of the GB topologies after 1 ns of MD used for k-ART simulations and at the center, a general sketch of the four boxes. In blue, atoms with different structure from bcc at GB and in yellow, Fe atoms with crystalline structure. The GB rotation axes $\langle 100\rangle$ and $\langle 110\rangle$ are parallel to the $x$ axis out of planes, while GB planes are perpendicular to the $z$ axis. The red circle indicates a large free volume in the 18.93° $\langle 100\rangle$ GB, in which the Fe atoms represented in blue jump back and forth; this is also the ground state of a C atom.

Figure 2

K-ART evolution starting form an ideal 109.53°$\langle 110\rangle$ GB structure without prior MD. The first 50 KMC steps are shown. Red circles represent local minima, black crosses the saddle points and the blue is guide to the eye. All energies are measured with respect to that of the initial configuration.

Figure 3

Long-time evolution of minimum and barrier energies, and square displacement SD for (top) the 18.93°$\langle 100\rangle$ and (bottom) 70.53°$\langle 110\rangle$ GB systems. Blue dots: energy measured from the ground state (initial state, here); green dots: energy barrier; and red line: total square displacement.

Figure 4

Energy vs KMC steps for two intervals. Left, C approaching the GB from the bulk region. Horizontal red lines indicate the GS energy in the bulk, $E_0-E_{\mathrm{GS}}$, equal to 0.86, 0.32, and 0.93 eV from top to bottom. Right, regimes of C diffusion in the GB. The total square displacement SD and square displacement along $x,S{D}_x$, are also given. Pink shaded areas in top and bottom plots are intervals where the C atom moves into the GB plane while the gray band area, middle panel, indicates an interval in which the C atom returns to the crystalline side. Connected open dots indicate the exact trajectory. Filled circles indicate a discontinuity associated with an analytical solution of the flickering states using bac-MRM.

Figure 5

Trajectory paths of a C atom at the GB planes $xy$. The Insets show the $yz$ planes, perpendicular to the GBs planes. Fe atoms outside of bcc environment in the GBs are shown in blue. Note that some motion also takes place along the $x$ direction and that multiples pathways can be projected on top of each other.

Figure 6

Views of the positions visited by the C atom at the GBs (red dots). In top, yellow atoms represent crystalline positions that are removed, in the bottom panel, to simplify the diagram. Atoms in blue are at GBs. In the case of 70.53° $\langle 110\rangle$, orange is used to distinguish the two alternating planes defined by a normal parallel to the rotation tilt axis. Flickering states along $\langle 100\rangle$ planes are indicated by the red dots underlined by a black line.

Figure 7

Time evolution of C diffusion at the GBs with k-ART plotted on a logarithmic scale. Left axis: barrier energies $E_b$ (blue crosses), segregation energy Es (green lines), and the basin threshold energy (horizontal lines in orange), in eV. Right axis: total and partial square displacement SD, $S{D}_x,S{D}_y$, and $S{D}_z$ in ${\AA }^2$. Top left panel: 18.93° $\langle 100\rangle$ GB; top right panel: 70.53° $\langle 110\rangle$ GB; bottom panel: 85.91° $\langle 100\rangle$ GB.

Figure 8

Evolution of the systems as a function of KMC step. Left axis: instantaneous (red line) and average (black line) time step as a function of KMC step. Right axis: total square displacement (SD) (blue). The average time step is computed as moving average using a window of 20 steps. Gray areas correspond to movements into the crystalline side.

Figure 9

(Left) Evolution of the 18.93°$\langle 100\rangle$ system as a function of KMC step near the reorganization in GB2 due to the presence of a C atom in GB1: see events 980–985. (Right) The corresponding Fe-atom displacements to a new ground state (blue dots, right GB).

Figure 10

Evolution of the number of searched topologies and CPU time (in hour) as a function of KMC step for typical run of three GBs. Top: 18.93°$\langle 100\rangle$; middle: 70.53°$\langle 110\rangle$; bottom: 85.91° $\langle 100\rangle$.

Figure 11

Relations between energy barrier $E_b$, energy minimum $E_{\min }$, energy saddle $E_{\mathrm{sadd}}$, and Voronoi volume $V_{\mathrm{voro}}$. To better see physical trends, flickering states and states that produce jumps into the same plane, perpendicular to direction of diffusion are removed. Squares are used for data corresponding to C in the bulk, whereas circles correspond to data obtained when the C atom is in the GB.