The bonding of H in Zr under strain

Accurate computer simulation is important for understanding the role of irradiation-induced defects in zirconium alloys found in nuclear reactors. Of particular interest is the distribution and trapping of hydrogen, and the formation of zirconium hydride. These simulations require an accurate representation of Zr-H bonding in order to predict the behaviour of H around atomic-scale defects, dislocation lines, and dislocation loops. Here we explore the bonding of H in Zr under strain, how well it is represented by state-of-the-art Embedded Atom Method (EAM) potentials, and what physics is needed for an accurate representation in a Linear Combination of Atomic Orbitals (LCAO) Density Functional Theory (DFT) framework. For H in dilute solution under hydrostatic strain in the range -10% to +10%, solution energies and Zr-H bond lengths computed using EAM potentials are found to be in poor agreement with plane-wave DFT results. We note that the bond lengths are in a poor agreement even in equilibrium. LCAO basis sets are used to explore the importance of electron distribution around H atoms, and the transfer of electrons between H and Zr. The electron distribution around H atoms is found to be important to the explanation of the difference between octahedral and tetrahedral interstitial sites for H, with H in a tetrahedral site having very similar bonding to H in zirconium hydrides. The interatomic electron transfer has a smaller impact but is needed for maximum accuracy

Systematic development of ab initio tight-binding models for hexagonal metals

A systematic method for building an extensible tight-binding model from ab initio calculations has been developed and tested on two hexagonal metals: Zr and Mg. The errors introduced at each level of approximation are discussed and quantified. For bulk materials, using a limited basis set of s p d orbitals is shown to be sufficient to reproduce with high accuracy bulk energy versus volume curves for fcc, bcc, and hcp lattice structures, as well as the electronic density of states. However, the two-center approximation introduces errors of several tenths of eV in the pair potential, crystal-field terms, and hopping integrals. Environmentally dependent corrections to the former two have been implemented, significantly improving the accuracy. Two-center hopping integrals were corrected by taking many-center hopping integrals for a set of structures of interest, rotating them into the bond reference frame, and then fitting a smooth function through these values. Finally, a pair potential was fitted to correct remaining errors. However, this procedure is not sufficient to ensure transferability of the model, especially when point defects are introduced. In particular, it is shown to be problematic when interstitial elements are added to the model, as demonstrated in the case of octahedral self-interstitial atoms

Beyond two-center tight binding: Models for Mg and Zr

We describe a systematic approach to building ab initio tight-binding models and apply this to hexagonal metals Mg and Zr. Our models contain three approximations to plane-wave density functional theory (DFT): (i) we use a small basis set, (ii) we approximate self-consistency, and (iii) we approximate many-center exchange and correlation effects. We test a range of approximations for many-center exchange-correlation and self-consistency to gauge the accuracy of each in isolation. This systematic approach also allows us to combine multiple approximations in the optimal manner for our final tight-binding models. Furthermore, the breakdown of errors into those from individual approximations is expected to be a useful guide for which approximations to include in other tight-binding models. We attempt to correct any remaining errors in our models by fitting a pair potential. Our final tight-binding model for Mg shows excellent agreement with plane-wave results for a wide range of properties (e.g., errors below 10% for self-interstitial formation energies and below 3% for equilibrium volumes) and is expected to be highly transferable due to the minimal amount of fitting. Calculations with our Zr model also show good agreement with plane-wave results (e.g., errors below 2% for equilibrium volumes) except for properties where self-consistency is important, such as self-interstitial formation energies. However, for these properties we are able to generate a tight-binding model which shows excellent agreement with non-self-consistent DFT with a small basis set (i.e., many-center effects are captured accurately by our approximations). As we understand the source of remaining errors in our Zr model we are able to outline the methods required to build upon it to describe the remaining properties with greater accuracy

Atomistic modelling of iodine-oxygen interactions in strained sub-oxides of zirconium

In water reactors, iodine stress corrosion cracking is considered the cause of pellet-cladding interaction failures, but the mechanism and chemistry are debated and the protective effect of oxygen is not understood. Density functional theory calculations were used to investigate the interaction of iodine and oxygen with bulk and surface Zr under applied hydrostatic strain (2% to +3%) to simulate crack tip conditions in Zr to ZrO, using a variety of intermediate suboxides (ZrO, ZrO, ZrO and ZrO). The formation energy of an iodine octahedral interstitial in Zr was found to decrease with increasing hydrostatic strain, whilst the energy of an iodine substitutional defect was found to be relatively insensitive to strain. As the oxygen content increased, the formation energy of an iodine interstitial increased from 1.03 eV to 8.61 eV supporting the idea that oxygen has a protective effect. At the same time, a +3% tensile hydrostatic strain caused the iodine interstitial formation energy to decrease more in structures with higher oxygen content: 4.56 eV decrease in ZrO compared to 1.47 eV decrease for pure Zr. Comparison of the substitutional and interstitial energies of iodine, to the adsorption energy of iodine, in the presence of oxygen, shows the substitutional energy of iodine onto a Zr site is more favourable for all strains and even interstitial iodine is favourable between strains of +1-5%. Although substitutional defects are preferred to octahedral interstitial defects, in the ordered suboxides, a 3% tensile strain significantly narrows the energy gap and higher strains could cause interstitial defects to form

Transferability of Zr-Zr interatomic potentials

Tens of Zr inter-atomic potentials (force fields) have been developed to enable atomic-scale simulations of Zr alloys. These can provide critical insight in the in-reactor behaviour of nuclear fuel cladding and structural components exposed, but the results are strongly sensitive to the choice of potential. We provide a comprehensive comparison of 13 popular Zr potentials, and assess their ability to reproduce key physical, mechanical, structural and thermodynamic properties of Zr. We assess the lattice parameters, thermal expansion, melting point, volume-energy response, allotropic phase stability, elastic properties, and point defect energies, and compare them to experimental and ab-initio values. No potential was found to outperform all others on all aspects, but for every metric considered here, at least one potential was found to provide reliable results. Older embedded-atom method (EAM) potentials tend to excel in 2-3 metrics each, but at the cost of poorer transferability. The two highest-performing potentials overall, with complementary strengths and weaknesses, were the 2021 angular-dependent potential of Smirnova and Starikov (Comp. Mater. Sci. 197, 110581) and the 2019 embedded-atom method potential of Wimmer et al (J. Nucl. Mater. 532, 152055). All potentials trained through machine learning algorithms proved to have lower overall accuracy, and less transferability, than simpler and computationally faster potentials available. Point defect structures and energies is where the greatest divergence and least accuracy is observed. We created maps that will help modellers select the most suitable potential for a specific application, and which may help identify areas of improvement in future potentials

Atomistic modelling of iodine-oxygen interactions in strained sub-oxides of zirconium

In water reactors, iodine stress corrosion cracking is considered the cause of pellet-cladding interaction failures, but the mechanism and chemistry are debated and the protective effect of oxygen is not understood. Density functional theory calculations were used to investigate the interaction of iodine and oxygen with bulk and surface Zr under applied hydrostatic strain (-2 % to +3 %) to simulate crack tip conditions in Zr to ZrO, using a variety of intermediate suboxides (ZrO, ZrO, ZrO and ZrO). The formation energy of an iodine octahedral interstitial in Zr was found to decrease with increasing hydrostatic strain, whilst the energy of an iodine substitutional defect was found to be relatively insensitive to strain. As the oxygen content increased, the formation energy of an iodine interstitial increased from 1.03 eV to 8.61 eV supporting the idea that oxygen has a protective effect. At the same time, a +3 % tensile hydrostatic strain caused the iodine interstitial formation energy to decrease more in structures with higher oxygen content: 4.56 eV decrease in ZrO compared to 1.47 eV decrease for pure Zr. Comparison of the substitutional and interstitial energies of iodine, to the adsorption energy of iodine, in the presence of oxygen, shows the substitutional energy of iodine onto a Zr site is more favourable for all strains and even interstitial iodine is favourable between strains of +1-5%. Although substitutional defects are preferred to octahedral interstitial defects, in the ordered suboxides, a 3 % tensile strain significantly narrows the energy gap and higher strains could cause interstitial defects to form