Diffusivity and derivatives for interstitial solutes: activation energy, volume, and elastodiffusion tensors

Abstract

Computational atomic-scale methods continue to provide new information about geometry, energetics and transition states for interstitial elements in crystalline lattices. This data can be used to determine the diffusivity of interstitials by finding steady-state solutions to the master equation. In addition, atomic-scale computations can provide not just the site energy, but also the stress in the cell due to the introduction of the defect to compute the elastic dipole. We derive a general expression for the fully anistropic diffusivity tensor from site and transition state energies, and three derivatives of the diffusivity: the elastodiffusion tensor (derivative of diffusivity with respect to strain), the activation barrier tensor (logarithmic derivative of diffusivity with respect to inverse temperature) and activation volume tensor (logarithmic derivative of diffusivity with respect to pressure). Computation of these quantities takes advantage of crystalline symmetry, and we provide an open-source implementation of the algorithm. We provide analytic results for octahedral–tetrahedral networks in face-centred cubic, body-centred cubic, hexagonal closed-packed lattices, and conclude with numerical results for C in Fe.

Keywords:

• Interstitial diffusion
• activation barrier
• elastodiffusion tensor
• automated computation

Acknowledgements

The author thanks Maylise Nastar and Pascal Bellon for helpful conversations.

Notes

The author has no conflicts of interest.

1 The construction is a second rank tensor such that for any vector .

2 The double contraction sums over both indices of the two second-rank tensors: .

3 The tetrahedral sites are not required to sit at a c / 8 distance away from the basal plane—only that their positions be symmetric under a mirror through the basal plane. Hence, the general case would introduce the Wyckoff parameter z to character this position in the 4f site. However, in the case of diffusion, only the long-range contribution matters, so that change in jump vectors due to z exactly cancels out and we use for simplicity.

Additional information

Funding

Research supported in part by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering [Award #DE-FG02-05ER46217], through the Frederick Seitz Materials Research Laboratory, in part by the Office of Naval Research [grant number N000141210752], and in part by the National Science Foundation [Award 1411106]. Part of this research was performed while the author was visiting the Institute for Pure and Applied Mathematics (IPAM) at UCLA, which is supported by the National Science Foundation (NSF).