Mutual dependence of oxygen and vacancy diffusion in bcc Fe and dilute iron alloys

A combination of density functional theory (DFT) and an efficient calculation method based on atomistic kinetic Monte Carlo simulations (AKMC) is used to investigate the interdependence of oxygen (O) and vacancy (v) diffusion in bcc Fe and in dilute iron alloys with the substitutional atoms Y and Ti. Both O and v are considered as mobile while the substitutional atoms are assumed to be immobile. DFT is applied to determine the binding energy between O and v for different distances, the migration barriers for O in the environment of v, and the corresponding barriers of v in the vicinity of O. In agreement with previous work O and v have a very strong binding at the first-neighbor distance. On the other hand, the calculations show that the Ov pair at the sixth-neighbor distance is instable. The simultaneous jumps of both O and v compensate the lack of jump paths that would occur due to this instability. The DFT results are employed to determine the diffusion coefficient of O and v using the AKMC-based calculation method. At first a model system with fixed O and v concentrations is studied. It is found that even a small v content of some parts per million can lead to a strong reduction of the O diffusivity. A similar effect is obtained for v diffusion under the influence of O. Furthermore, investigations on the interdependence of O and v diffusion in the first phase of thermal processing of oxide dispersion strengthened iron alloys are performed, and the influence of the substitutional atoms Y and Ti is studied. A simple thermodynamic model is employed to determine the concentration of O, Y, and Ti monomers as well as the total v concentration, for a typical total content of O, Y, and Ti. These results are used in calculations of the diffusion coefficients of O and v. Not only a strong mutual dependence but also a significant influence of Y on O diffusion is found. Finally, O and v diffusivities in a system with an O content close to the thermal solubility are calculated, where the monomer and total concentrations are determined by two different thermodynamic models. Even for such a low amount of O in the alloy the diffusion coefficients differ strongly from those in perfect bcc Fe.

Figure 1

(a) Examples for octahedral interstitial sites of oxygen (red) in the neighborhood of a vacancy (green). Neighbors inside the interaction region considered in DFT calculations are marked by black numbers. The gray spheres represent Fe atoms. (b) Examples for bcc sites of the vacancy (green) in the neighborhood of an oxygen atom (red). The meaning of the black numbers is the same as in (a).

Figure 2

Minimum energy path of the cage jump of oxygen. The large green sphere represents the vacancy.

Figure 3

Graphical representation of possible jumps of oxygen and the vacancy between different neighbor positions (connectivity plots). Potential jumps (O: red; v: green) including the sixth-neighbor position are depicted in (a). However, the instability of the Ov pair at the sixth-neighbor distance makes jumps marked by dotted lines impossible. For other reasons (see text) the direct vacancy jump marked by the dashed line is not possible. (b) illustrates the really relevant neighbor positions and jumps. Jump barriers for forward and backward jumps are given in eV, e.g., for oxygen jump from 2 to 5 the barrier is 0.769 eV, while it is 0.198 eV for the jump from 5 to 2. In both figures the binding energy of the Ov pair at a given distance is marked by blue color. The black italic numbers in (b) show the binding energies and migration barriers modified due to the rule of detailed balance (see text).

Figure 4

Atomic configurations illustrating the simultaneous (or coupled) jumps between the first- and the fifth-neighbor position (a) and between the neighbor positions 1 and 9a (b). The scale on the ordinate concerns the binding energy of the Ov pair (see Table 1).

Figure 5

The time ratios $t_{\mathrm{free}}/t_{\mathrm{total}}$ (a) and $t_{\mathrm{inter}}/t_{\mathrm{total}}$ (b) for oxygen, in dependence on temperature and vacancy concentration.

Figure 6

The total diffusion coefficient of oxygen (a) for given concentrations of vacancies, and the total diffusion coefficient of the vacancy (b) for different oxygen concentrations. The corresponding data of $D_{\mathrm{free}}$ (cyan) and $D_{\mathrm{inter}}$ (violet) are also shown.

Figure 7

Diffusion coefficient of the Ov pair inside the interaction region, determined by AKMC simulations, with one mobile oxygen and one mobile vacancy.

Figure 8

Minimum energy path for the migration of the Ov pair. The scale concerns the binding energy of the Ov pair (see Table 1).

Figure 9

Temperature dependence of monomer concentrations $C_{\mathrm{v}},C_{\mathrm{O}},C_{\mathrm{Y}},C_{\mathrm{Ti}}$, and of the total vacancy concentration $C_{\mathrm{v}}^{\mathrm{total}}$ (a) as well as the concentration ratios vs temperature (b). The data were obtained by the solution of Eqs. (13)–(16), for the following total concentrations of solutes: 0.18 at.% O, 0.12 at.% Y, and 1.05 at.% Ti. The solid and dashed lines were determined using $C_{\mathrm{v}}$ from Eqs. (17) and (19), respectively.

Figure 10

Total diffusion coefficients of O (a) and v (b) determined for the monomer or the total concentrations shown in Fig. 9 and given in the text. The concentrations used in the calculations are written close to the corresponding curves for the diffusion coefficient using the same color. These curves were either obtained using $C_{\mathrm{v}}^{}$ from Eq. (17) (solid lines) or $C_{\mathrm{v}}^{}$ from Eq. (19) (dashed lines).

Figure 11

Total diffusion coefficients of O (a) and v (b) for the case that the oxygen concentration is equal to the thermal solubility. The black and blue curves were determined using monomer and total concentrations, respectively, for O and v. These concentrations were calculated by the method described in Sec. 4. The red and magenta curves were obtained by a modified procedure where not only the O-v interaction (as in Sec. 4) but also the O-Ov (or v-Ov) interactions are taken into account (see Supplemental Material [44]). The red and magenta curves were calculated for monomer and total v or O concentrations, respectively. Note that in (b) the blue and the magenta curves as well as the black and red curves are nearly identical. The oxygen diffusion data determined by internal oxidation experiments are marked by S [5], T [6, 7, 8], and F [9]. In the case of S and T the thick lines show the temperature range in which the measurements were performed. Note that the data of Swisher and Turkdogan [5] are based on permeability measurements for δ-iron [4] and are therefore only approximately valid for bcc Fe. The brown dashed line was obtained using the activation energy given by Frank et al. [9] and the preexponential factor from Eq. (9).