First-principles study of the thermodynamics of hydrogen-vacancy interaction in fcc iron

The interaction of vacancies and hydrogen in an fcc iron bulk crystal was studied combining thermodynamic concepts with ab initio calculations and considering various magnetic structures. We show that up to six H atoms can be trapped by a monovacancy. All of the studied point defects (single vacancy, H in interstitial positions, and H-vacancy complexes) cause an anisotropic elastic field in antiferromagnetic fcc iron and significantly change the local and total magnetization of the system. The proposed thermodynamical model allows the determination of the equilibrium vacancy concentration and the concentration of dissolved hydrogen for a given temperature and H chemical potential in the reservoir. For H-rich conditions a dramatic increase in the vacancy concentration in the crystal is found.

Figure 1

(Color online) Potential energy surface of interstitial H in the (001) plane in nonmagnetic fcc Fe. The vacancy is in the center. The energetically preferred places for H are situated in the $⟨100⟩$ directions from the vacancy center. The local minima in the bulk like regions correspond to the octahedral site.

Figure 2

(Color online) Different atomic shells around (a) a vacancy and (b) an H atom in an octahedral site in an AFMD magnetic structure. The supercell is a $2\times 2\times 2$ repetition of the conventional fcc unit cell. The labeling of planes includes the differentiation between the first and the second plane with the same spin direction, and the direction of the individual spins. Possible positions of H in a vacancy are shown by the red (lightest) balls. The position of H in octahedral interstitial site (marked by dashed octahedron) is displaced from its center. Two variants of tetrahedral sites are marked by dashed tetrahedrons.

Figure 3

(Color online) Change in magnetization and displacements of atoms around (a) a vacancy (b) an interstitial H atom in an octahedral site of AFMD Fe at 0 K. Relative changes in the local magnetic moment are determined using the defect and perfect bulk calculation by the following formula: $\left({\mu }_{\text{def}}-{\mu }_{\text{bulk}}\right)/\left|{\mu }_{\text{bulk}}\right|$. A relative displacement of individual ions is determined by $\left({\mathbf{r}}_{\text{def}}-{\mathbf{r}}_{\text{bulk}}\right)\cdot {\mathbf{r}}_{\text{def}}/{\left|{\mathbf{r}}_{\text{def}}\right|}^2$ using the relaxed position vectors in the defect and the perfect bulk calculation.

Figure 4

(Color online) Formation energies of point defects at 0 K for different H chemical potentials. Dashed line—for H in OS and solid lines—for H-vacancy complexes with the number of H shown. The dominant H-vacancy complexes are denoted by labels with vertical direction. The yellow (gray) region for ${\mu }_{\text{H}}>0.14\text{eV}$ shows the thermodynamically forbidden region where the six-H-vacancy formation energy becomes negative, i.e., the crystal is unstable against vacancy formation even at $T=0\text{K}$.

Figure 5

(Color online) Vacancy concentration as a function of temperature and hydrogen chemical potential for AFMD fcc Fe. The dominant H-vacancy complexes are separated by dashed lines.

Figure 6

(Color online) Vacancy concentration obtained from the thermodynamic model derived in Sec. 2B assuming a NM (dashed lines) and an AFMD (solid lines) Fe crystal. Blue (dark) lines for ${\mu }_{\text{H}}=-\infty$ and green lines for ${\mu }_{\text{H}}=0.09\text{eV}=\text{const}$.

Figure 7

(Color online) Concentration of hydrogen dissolved in AFMD fcc Fe. Dashed lines—levels of the ratio between H dissolved in vacancies and H dissolved in OS.