Significance of Self-Trapping on Hydrogen Diffusion

The diffusion rate of hydrogen in Nb was calculated using ab initio molecular dynamics simulations. At low temperatures the hydrogen is strongly trapped in a local strain field which is caused by the elastic response of the lattice. At elevated temperatures, the residence time ($\tau$) of hydrogen in an interstitial site is not sufficient for fully developing the local strain field. This unbinding of the interstitial hydrogen and the strain field increases the hopping rate ($1/\tau$) at elevated temperatures ($>400\mathrm{K}$). These results call for a revision of the conceptual framework of diffusion of hydrogen in transition metals at elevated temperatures.

Figure 1

The self-trapping lowers the hydrogen-polaron quasiparticle energy level $E_S$ with respect to the unstrained lattice. $E_{\mathrm{I}}$ is the difference in energy when moving the hydrogen to the closest tetrahedral site, keeping the position of the strain field fixed with respect to the lattice. $E_{\mathrm{IIa}}$ and $E_{\mathrm{IIb}}$ are similarly the energies for moving the hydrogen to the next nearest tetrahedral sites. The geometries of the jumps are shown in the bottom panel.

Figure 2

An isosurface of the hydrogen density in bcc Nb at 550 K. Notice how the points cluster around the tetrahedral sites in excellent agreement with experimental results [2]. The lines connecting the tetrahedral sites show the $4T$ and $6T$ rings. The gray spheres are Nb atoms.

Figure 3

Arrhenius plot of the calculated and experimental diffusion coefficients. The black circles and squares show our calculated diffusion rates. The experimentally obtained rates are drawn in red (solid, 873–1390 K) [20] and blue (dashed) [26]. The blue triangles are results obtained from classical MD [23].

Figure 4

The average distance between hydrogen and the nearest neighbor Nb as a function of temperature (red circles). The probability of residence, $P(\tau )$, as a function of residence time ($\tau$), at 500 K, is illustrated as an inset. The green squares show the H-Nb distance for residence times smaller than the average residence time, $⟨\tau ⟩$.

Figure 5

Two-dimensional mapping of the jump probability as a function of residence time and temperature, showing the probability that the proton has jumped to the next tetrahedral site.