Equilibrium Vacancy Concentration Driven by Undetectable Impurities

By means of ab initio calculations combined to statistical mechanics, we provide new evidence that an experimentally undetectable tiny amount of impurities can be responsible for drastic changes in vacancy concentrations ([$V$]), inducing large deviations from an Arrhenius law even at low temperature. It is the case of O and N in $\alpha \text{−}\mathrm{Fe}$. The present finding is fully compatible with existing experiments, and changes the previous common vision that C has the dominant effect. This study provides a route for bridging the longstanding theoretical-experimental gap on the prediction of [$V$] in metals.

Figure 1

Most stable cluster configurations. Gray spheres, gray squares, and blue spheres represent Fe atoms, $V$, and solutes, respectively. $\tilde{v}$ is the Voronoï volume of $V$ (free space inside $V$), normalized by the Voronoï volume of the isolated $V$ ($10.9{\AA }^3$).

Figure 2

Effective $V$ formation free enthalpy as a function of temperature: the solid green line is $G^f{(V)}_{|\mathrm{Fe}}$. The black, blue, and red lines represent $G^f{(V)}_{|\mathrm{Fe}-X}$ when $[X]={[X]}_T^{\mathrm{sol}}$, $X$ being, respectively, C, N, and O. For these three, dashed lines are obtained using only $VX$ pairs in our calculations, while solid lines are obtained considering $VX$ as well as $V{X}_2$ clusters.

Figure 3

Calculated $G^f{(V)}_{|\mathrm{Fe}\text{−}\mathrm{CNO}}$ as a function of temperature with various nominal impurity contents (in appm). The solute content in the solid solution is $\min ([X];{[X]}_T^{\mathrm{sol}})$. The black dashed line corresponds to an estimated PAS detection threshold, $G^f{(V)}_{\min }$ [47].