Diffusion of interstitial species (H and O atoms) in fcc systems (Al, Cu, Co, Ni and Pd): Contribution of first and second order transition states
Introduction
The diffusion mechanisms of atoms in interstitial position are now commonly studied in solid-state physics, as the growing amount of work [[1], [2], [3], [4], [5]] on the matter indicates. If the possibility of forming clusters with vacancies, the interactions with defects or the interfaces of the network are ignored, the atomic process of diffusion of interstitial species is controlled, in first-order approximation, by the energy landscape defined by the network. Atoms then diffuse from one stable site to another. However, these stable sites are not necessarily equivalent from a geometrical standpoint. In the case of fcc systems, where tetrahedral and octahedral sites are often the most stable sites, direct diffusion between t and o sites is (almost) systematically used to describe and predict atomic-scale diffusion. Wimmer et al. [2] proposed an explicit diffusion coefficient formula taking into account these two stable sites with a single jump, using the jump rates from o to t sites, “”. In the literature, various authors have considered that direct transitions between first-nearest neighboring octahedral or tetrahedral sites should be ignored in the calculation of the diffusion coefficient, due to a high migration energy. As shown by David et al. [4], this transition state is always set in the M site, which is located exactly between two nearest octahedral sites, between two nearest tetrahedral sites and between two fcc sites (see Fig. 1). From a geometrical standpoint, when an interstitial atom is located an M site, it can move directly into four different stable sites.
By studying the vibrational properties of interstitial atoms in different positions, results have shown that, in Al [4], these calculated transition states necessarily present two imaginary branches associated with the moving atom. These mechanisms should therefore be included in the transition theory in addition to the direct o-t path. However, most solid-state physics works do not discuss the possibility of interstitial diffusion through these configurations. Most expressions of diffusion equations generally do not include this option.
In this paper, we look back at the often overlooked question of taking these paths into account. We compare the diffusion mechanisms between t and o sites found in two proven cases (H and O atoms) by taking into account the first-order transition state (direct diffusion path between o and t) but also second-order transition states (2TS) by taking into account M site or not. The case of H atoms in Al, Cu, Co, Ni and Pd, and O atom in Ni, Cu, Co and Pd are suitable cases to illustrate our point. This study also entails the opportunity of a detailed discussion on H and O diffusion mechanisms in various fcc metals, which, in some cases, has never been carried out. Our study was conducted by examining the mechanisms of diffusion at the atomic scale using first-principles calculations, and using the Eyring theory, with explicit equation of diffusion coefficients. We hereby study the effect of including 2TS in the atomistic process of diffusion.
Section snippets
Methodology
Calculations were performed using the Vienna ab initio simulation package (VASP) [6]. Self-consistent Kohn-Sham equations were solved using the projector augmented wave (PAW) pseudo-potentials [7]. We used the Perdew-Burke-Ernzerhof [8] exchange and correlation functional. The plane-wave energy cut-off was set to 600 eV and -centered Monkhorst-Pack meshes [9] were used to sample the first Brillouin zone (equivalent to 202020 for cubic cells, i.e. with 4 atoms). A super-cell approach was used
Results
To illustrate the impact of these new jumps, we will present results on different fcc-systems (Al, Co, Cu, Ni and Pd) and for two types of atoms (H and O). In Table 1, formation energies (, in eV) and enthalpy energies, , including the zero-point energy () corresponding to the different sites (o, t and M) are summarized. The formation energy corresponds to:where are the DFT energy of fcc-systems with and without an impurity X. corresponds to the
Conclusion
We showed that at low temperature the atomistic process is clearly controlled by direct jumps, this was an expected result. The jump rates and are thus significantly higher than jump rates of second-order transition states. By including higher-order transition states, the diffusion of species can be increased two-fold in some cases. The effect of second-order transition states on the diffusion coefficient is thus limited in our case. If high-order transition states can be ignored in
Acknowledgments
This work was performed using HPC resources from CALMIP (Grant 2018-p0749) and GENCI-CINES (Grant A0020907722). Authors acknowledge Pr. C. Raynaud (Institute Charles Gerhardt of Montpellier) and D. Tanguy (ILM Lyon) for their comments and advices.
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