1. Introduction
Solutes with an atomic size smaller than that of the host atoms migrate via the interstitial mechanism [
1]. In this case the most stable site of the solute and the saddle point for the migration are highly symmetric interstitial positions in the lattice, e.g., octahedral and tetrahedral sites. Foreign atoms with atomic sizes similar to or larger than those of the solvent occupy substitutional sites. The presence of native defects such as vacancies and self-interstitial atoms is generally a prerequisite for their migration. The diffusion via the vacancy and the interstitialcy (or indirect interstitial) mechanism is much slower than that of the interstitial solutes, since the concentration of the native point defects is rather low under the conditions of thermal equilibrium. Therefore, in calculations of the diffusion coefficient of interstitial solutes in dilute alloys, pre-existing substitutional solutes can be assumed to be immobile. By definition, in a dilute alloy the migration of the diffusing interstitial atom cannot be influenced at the same time by more than one substitutional solute.
In this work, efficient methods for calculating the diffusion coefficient of an interstitial solute in dilute ferritic iron alloys under the influence of substitutional foreign atoms are presented. Typical interstitial diffuser in bcc Fe are C, N, O, He, and H. Present investigations are focused on oxygen diffusion. In pure bcc Fe the most stable site of oxygen is the octahedral site and the most probable migration path is a first-neighbor jump between two octahedral sites, with the saddle point at a tetrahedral position [
2]. Recently, the effect of various substitutional solutes on oxygen diffusion in bcc Fe was investigated using a combination of Density Functional Theory (DFT) calculations and atomistic kinetic Monte Carlo (AKMC) simulations on a rigid lattice [
2]. DFT was applied to determine both the binding energy between oxygen and different substitutional atoms and the respective migration barriers in the vicinity of those solutes. Although the migration barriers are often rather different to that in pure bcc Fe, in most cases the relevant migration paths are first-neighbor jumps between modified (nearest neighbor) octahedral sites with modified tetrahedral sites as saddle points. The oxygen diffusion coefficient, as function of temperature and concentration of the substitutional solute, was obtained from AKMC simulations with the migration barriers from DFT calculations as input data. It was found that the deviation of the diffusion coefficient of oxygen in the dilute alloy from that in pure bcc Fe becomes more pronounced with increasing attraction between oxygen and the substitutional atom. Such deviations can be significant, even at concentrations of the substitutional solute below 0.1%.
2. Calculation Methods
In the present work the DFT data for binding energies and migration barriers determined in Reference [
2] are used. In that paper spin-polarized DFT calculations were performed using the Vienna ab initio simulation package (VASP) [
3,
4,
5] with a plane-wave cutoff energy of 500 eV. Electron-core interaction was described by projector-augmented wave (PAW) pseudopotentials [
6,
7] and exchange and correlation effects were treated within the framework of the generalized gradient approximation (GGA) using the Perdew-Burke-Ernzerhof (PBE) functional [
8]. A supercell with 128 bcc lattice sites was considered, and Brillouin-zone sampling was performed using a 3 × 3 × 3 k point grid within the Monkhorst-Pack scheme [
9]. The integration in reciprocal space was done with a Methfessel-Paxton smearing width of 0.2 eV [
10].
Figure 1 shows how many oxygen octahedral interstitial sites were considered in the environment of a substitutional foreign atom. The notation of the neighbor positions is according to the scheme for a simple cubic lattice (see [
11]), which consists of the bcc lattice sites and the octahedral interstitial sites of the bcc lattice. Within the framework of this scheme oxygen cannot reside on third, fourth, seventh, eighth, etc. neighbor positions of the substitutional solute, since these sites are already occupied by iron atoms, and there are two different ninth neighbor sites (9a and 9b). After introduction of an oxygen and a foreign atom on the respective sites the positions of all atoms and the volume and shape of the supercell were relaxed, until the residual force on each atom became lower than 10
−2 eV/Å and the change of total energy during self-consistent energy minimization for given atomic positions fell below 10
−5 eV.
Figure 2 illustrates three characteristic results of Reference [
2]: (i) In the environment of Ti, strong attractive states exist for oxygen at the 1st and 2nd neighbor distance, and in this region the barriers are relatively high compared to those in perfect bcc Fe (0.512 eV). (ii) In the interaction region with Cr the attraction is weaker and the barriers are somewhat lower. (iii) Very weak attraction and repulsion dominates in the region near a Si atom and the migration barriers are rather different. In
Figure 2 the interaction between oxygen and the substitutional solute extends up to the 10th neighbor shell. For use in AKMC simulations the original DFT data for barriers were modified according to the rule of detailed balance since the binding energy at neighbors 9a, 10, and at sites outside the 10th neighbor shell must be zero. Furthermore, the binding energy at neighbor 9b was also set to zero and the detailed balance was applied to change the respective barrier. Details of the AKMC simulations which are based on the residence time algorithm were described in Reference [
2].
In all cases considered in that paper the AKMC simulation cell contained one diffusing oxygen atom and one substitutional solute. In order to model the concentration of substitutional foreign atoms, in Reference [
2] separate AKMC simulations were performed for different cell sizes. In the following it is shown that in general AKMC simulations are only required for one concentration (or cell size), and the obtained results can be employed to obtain data for other concentrations in a very efficient manner.
At a given temperature and concentration of substitutional foreign atoms the diffusion coefficient of oxygen (or of another interstitial solute) can be written as
with
where
is the diffusion coefficient of oxygen in pure bcc Fe, i.e., outside the region of influence by the substitutional solute, and
is the diffusion coefficient of oxygen inside the interaction region. The quantities
and
correspond to the sum of the time periods for diffusion outside or inside the region of influence, respectively, and
is the total diffusion time. The value of
is given by the known analytical formula
with the lattice constant
, as well as the migration barrier
and the attempt frequency
in pure bcc Fe, with
Å,
, and
[
2]. The values of
,
, and
may be obtained from AKMC simulations as performed in Reference [
2] for different solute concentrations or cell sizes. However, the time ratios in Equation (1a) can be also expressed by analytical relations containing terms with probabilities for a certain interaction of oxygen with the substitutional solute.
These relations are based on the Gibbs distribution of the probability to find the system with the oxygen atom and the substitutional solute in a particular state (see Reference [
12]). The quantity
denotes the binding energy of the pair at the
th neighbor distance, see
Figure 1. The quantity
is the concentration of the substitutional solute, and
is the possible number of substitutional solute sites in the
th neighborhood of oxygen. In calculations of the time ratios it is taken into account that at neighbors 9a, 9b, 10, and at sites outside the 10th neighbor shell (see
Figure 1) the binding energy has to be zero. It can be assumed that
is nearly independent of the concentration of the substitutional solute (or the size of the AKMC simulation cell) since this quantity is only determined by migration paths inside the region of influence by the substitutional solute. In this case AKMC simulation needs to be used only once, i.e., for a certain concentration of the substitutional foreign atom, and the total diffusion coefficient
can be then determined for the other concentrations using Equations (1)–(3). It is quite clear that such a method is much more efficient than performing separate AKMC simulations for different concentrations or cell sizes as done in Reference [
2].