Ab initio study of H-vacancy interactions in fcc metals: Implications for the formation of superabundant vacancies

Hydrogen solubility and interaction with vacancies and divacancies are investigated in 12 fcc metals by density functional theory. We show that in all studied fcc metals, vacancies trap H very efficiently and multiple H trapping is possible. H is stronger trapped by divacancies and even stronger by surfaces. We derive a condition for the maximum number of trapped H atoms as a function of the H chemical potential. Based on this criterion, the possibility of a dramatic increase of vacancy concentration (superabundant vacancy formation) in the studied metals is discussed.

Figure 1

Schematic diagram introducing the key quantities/parameters used in our formalism. Shown are the Gibbs energies of formation of a vacancy (black line), of H-vacancy complexes (blue line), and of interstitial hydrogen (green line) as a function of H chemical potential. The formation energy of thermodynamically stable H-vacancy complexes is shown by the red line. ${\mu }_{\mathrm{H}}^{\mathrm{SAV}}$ is the H chemical potential of superabundant vacancy formation. ${\mu }_{\mathrm{H}}^{\mathrm{int},0}$ is the chemical potential for which the formation energy of an interstitial H becomes zero [$G_{\mathrm{f}}^{\mathrm{int}}\left({\mu }_{\mathrm{H}}^{\mathrm{int},0}\right)=0$]. $E_{\mathrm{b}}$ is the binding energy of a single H atom to a vacancy.

Figure 2

Stereographic projection of the potential energy surface (PES) of a H atom on the internal surface of a vacancy in nonmagnetic fcc Fe. Only the irreducible part of the PES is shown.

Figure 3

PES for H in OS in Au (a), in OS in Pb (b), in OS in Pt (c), and in TS in Ag (d) in different directions (blue line, $\langle 100\rangle$; red line, $\langle 110\rangle$; green line, $\langle 111\rangle$) are shown by solid lines. The probability of finding a H atom is shown in the $\langle 111\rangle$ direction by the dashed green line with the filling to the value of zero-point energy (shown by horizontal black dashed line). It is not shown for Au as the zero-point energy for H in OS in Au is higher than the barrier to escape and therefore H is completely unstable there.

Figure 4

Visualization of relevant interstitial sites within a vacancy (top) and a divacancy (bottom) as well as the adjacent OS (brown) and TS (blue), host atoms from the first shell in red and in blue and from the second shell in magenta. The number of nearest-neighbor host metal atoms for atoms shown in magenta is 12, in red 11, in blue 10. H positions in 3f sites shown by green triangles, in 4f by red squares, in top by a yellow circle. The trigonal position in the {111} plane is shown by a white dot and centers of OS or TS are marked by a black dot.

Figure 5

H-adjusted energy of formation [see Eq. (6)] in bulk OS and 4f vacancy positions (indicated by squares), TS and 3f positions (triangles), top vacancy positions (circles), and the formation of H${}_2$ molecules (crosses). Results for H outside a vacancy are shown for bulk interstitial sites (blue/dark symbols) and interstitial sites close to an empty vacancy (light blue). Results for H inside a vacancy are shown with the number of incorporated H atoms. Additionally, stable H-vacancy complexes for a fixed number of H atoms with only one H layer (solid line) and with a second layer (dashed line) are indicated. The color of lines and symbols (triangles or squares) shows the type of filling in the first layer: green, H in 3f sites; red, H in 4f sites. The filling of vacancies by H${}_2$ molecules is indicated for Ag, Au, and Pb by blue solid lines. The vacancy formation energy is shown by dashed black lines. The most stable H-vacancy complexes are the vertices on the convex hull indicated by magenta lines.

Figure 6

H energy in a vacancy in Pd when displaced in different directions from its center: red line, $\langle 100\rangle$ direction; yellow line, $\langle 110\rangle$; green line, $\langle 111\rangle$.

Figure 7

Defect formation energies of interstitial H (green lines), thermodynamically stable H-vacancy complexes (red lines), and the number of H atoms in them (blue lines) as a function of H chemical potential (the corresponding H partial pressure at 300 K is given in bars). The phase stability regions at 0 K are shown at the bottom: light blue, superabundant vacancy formation; yellow, NaCl type of hydride; green, other types of hydrides (not based on the fcc structure).