Systematic electronic-structure investigation of substitutional impurity diffusion and flux coupling in bcc iron

The diffusion properties of a wide range of impurities (transition metals and Al, Si, and P) in ferritic alloys are here investigated by means of a combined ab initio–atomic diffusion theory approach. The flux-coupling mechanisms and the solute-diffusion coefficients are inferred from electronic-structure calculations of solute-defect interactions and microscopic jump frequencies. All properties except the second-nearest-neighbor binding energy are found to have a characteristic bell shape as a function of the $d$-band filling for the $4d$ and $5d$ series, and an M shape for the $3d$ row because of the out-of-trend behavior of Mn. The solute jump frequencies are governed by compressibility, which makes diffusion of large solutes faster, although this effect is partially compensated for by lower attempt frequencies and larger correlations with the vacancy. Diffusion coefficients are predicted in a wide temperature range, far below the experimentally accessible temperatures. In accordance with experiments, Co is found to be a slow diffuser in iron, and the same behavior is predicted for Re, Os, and Ir impurities. Finally, flux-coupling phenomena depend on the iron jump frequencies next to a solute atom, which are mainly controlled by similar electronic interactions to those determining the binding energies. Vacancy drag and solute enrichment at sinks systematically arise below a solute-dependent temperature threshold, directly correlated with the electronic-level interactions at the equilibrium and the saddle-point states. Early transition metals with repulsive second-nearest-neighbor interactions also diffuse via vacancy drag, although they show a lower temperature threshold than the late metals. This confirms that drag is the most common solute-vacancy coupling mechanism in iron at low temperatures, and this is likely to be confirmed as well for impurity diffusion in other transition metals.

Figure 1

Nine-frequency model enforced in this work for the calculation of the transport coefficients, in a 2nn thermodynamic model. The numbers in the circles mark the possible nearest-neighbor positions of the vacancy before the jump. The background iron jump frequency (${\omega }_0$) outside the solute LAE is not shown.

Figure 2

(Blue squares) Correcting factor $exp\left[\alpha H\right(T\left)\right]$ to self-diffusion and solute-diffusion coefficients due to the finite-temperature magnetic transition. (Red circles) Relative excess magnetic enthalpy obtained with the Hillert-Jarl model [38]. $T_{\mathrm{C}}=1043$ K marks the Curie temperature in iron.

Figure 3

DFT-calculated solute-vacancy interaction energies in iron (full symbols), and computed by neglecting the relaxation of the cell due to the solute presence (open symbols). Negative values stand for attractive interactions.

Figure 4

DFT-computed solute migration barriers and attempt frequencies (${\omega }_2$). In the above-left panel, experimental bulk moduli of the pure elements [58]. The dashed lines represent the values in pure Fe.

Figure 5

DFT-computed iron migration barriers ${\omega }_{ij}$ in the vicinity of a solute atom, according to the jump-frequency nomenclature of Fig. 1. The dashed lines mark the values in pure Fe (${\nu }_{\mathrm{Fe}}$). In the middle panels, the saddle-point contribution to each migration barrier, obtained by removing the initial-state binding energy and the pure-Fe vacancy migration barrier ($E_{ij}^{\mathrm{SP}}=E_{ji}^{\mathrm{SP}}$).

Figure 6

Meyer-Neldel plot of solute migration energies versus attempt frequencies (jump ${\omega }_2$) for $3d$ and $4d$ metals (above), and $5d$ metals and $3p$ elements (below). The fitting lines for the $4d$ and $5d$ elements are obtained with the least-square method.

Figure 7

(a) Solute-vacancy drag tendencies as functions of temperature for the Fe(x) dilute binary alloys. Drag occurs for positive values of the wind factor. (b) Ratio of solute-to-iron partial diffusion coefficients (PDC) as defined in Eq. (3). Solute enrichment (via vacancy mechanism) occurs for values of this ratio smaller than unity (the solute is slower than Fe) or negative (vacancy drag). The lines of Ir and Os are perfectly superposed.

Figure 8

Group trends of (a) the wind factor at 600 K, (b) the solute-diffusion correlation factor at 600 K, (d) the solute-diffusion activation energy, (e) prefactors corresponding to a least-square fitting between 400 and 800 K, (c) the transition temperature from drag to nondrag tendencies, and (f) from RIS enrichment to depletion tendencies. In the latter, the elements with a temperature-independent enrichment tendency are not shown. The dashed lines represent the values for pure Fe.

Figure 9

Solute-diffusion correlation factors (full symbols) and ratio of average 1nn dissociation frequency ${\omega }_{ij}$ to solute jump frequency ${\omega }_2$ (open symbols) at 600 K. The correlation factor cannot assume values larger than 1. The dashed line marks the correlation factor value in pure Fe ($f_0=0.727$).

Figure 10

Tracer self-diffusion coefficient calculated in this work, and comparison to experimental measurements [76].

Figure 11

Solute tracer diffusion coefficients calculated in this work, and comparison to experimental measurements [32, 75, 76] and previous calculations [21, 22, 23, 24]. The dashed lines mark the Fe tracer self-diffusion coefficient, as shown in Fig. 10.

Figure 12

Solute tracer diffusion coefficients at 1100 K, as calculated in this work (full symbols) and from experimental measurements (open symbols) [32, 75, 76].