Ab-initio calculations of hydrogen diffusion coefficient in monoclinic zirconia

Abstract

During the corrosion in primary water of zirconium, hydrogen from the water diffuses through the oxide. To better understand this process, we use Density Functional Theory with hybrid functionals to calculate the energetics of interstitial hydrogen ions in defect-free monoclinic zirconia. While there is only one stable site for hydride ions in zirconia, protons have four different sites. We calculate the migration paths and energies between insertion sites to obtain the diffusion coefficients of hydrogen. We find that protons diffuse orders of magnitude faster than hydride ions, proving that protons are responsible for diffusion of hydrogen in monoclinic zirconia.

Introduction

Hydrogen is usually considered as an undesirable impurity in many materials such as nickel-base alloy, stainless steel, zirconium alloys because hydrogen tends to embrittle the materials even at low concentration by acting on the strength of the chemical binding as well as to act as carrier trap in semi-conductor.

In Pressurized Water Reactors, zirconium alloys are used as fuel rod cladding materials. These claddings form the first containment barrier to fission products and their mechanical integrity has to be ensured during their lifetime in reactor. They are exposed in service to aggressive aqueous environment. In these conditions, the oxidation kinetics of Zircaloy-4 cladding alloy show a significant acceleration above 35 GWd/tU burnup [1]. As a consequence, the corrosion of Zircaloy-4 is one of the main limiting factors of the fuel rod lifetime. According to literature data, this enhanced corrosion rate would be due to the precipitation of hydrides at the metal/oxide interface in the metallic part of the cladding [[2], [3], [4], [5], [6], [7], [8]]. Indeed, during the corrosion of the cladding in primary water, a fraction of the hydrogen from the water involved in the oxidation reaction goes across the oxide layer up to the alloy. Once its solubility limit in the matrix is reached at the irradiation temperature in reactor core, hydrogen precipitates as brittle zirconium hydrides mainly located just under the metal/oxide interface. Another issue related to hydrogen transport through zirconium oxide is the potential release in dihydrogen gas form during the spent fuel transport. Hydrogen trapped within the oxide layer that grew in reactor core is indeed likely to be released from the spent fuel assembly into the transport conditioning. For this reason, there is a potential safety risk for the transport. Hydrogen desorption from the cladding is actually the opposite process of hydrogen absorption occurring during corrosion.

Irrespective of the process, absorption or desorption, hydrogen has to diffuse through the oxide layer towards the metal or the surface of the cladding. It was usually assumed that the diffusion process through the oxide layer was the rate-limiting step of hydrogen pick-up or of the desorption [9,10]. However, some researchers considered that hydrogen pick-up is controlled by the migration of hydrogen under an electric field built in the oxide layer [11].

Anyway, research was initially focused on hydrogen diffusion through the oxide layer. In the past, the diffusion coefficient of hydrogen in the corrosion film was widely investigated either by 1) isotopic tracers, or 2) hydrogen implantation followed by Secondary Ion Mass Spectrometry (SIMS) or nuclear reaction analysis (NRA) [9,[12], [13], [14], [15], [16], [17], [18], [19]]. However, due to the presence of two sub-layers in the oxide with different hydrogen transport properties [12], as well as the two types of hydrogen interaction within the oxide [9] or the polycrystalline structure of the oxide layer [19], apparent diffusion coefficient of hydrogen is difficult to evaluate accurately from measurements. The diffusion coefficients measured in the literature are very scattered for the Zircaloy-4 alloy and to a lesser extent for Zr–Nb alloy. They rank from 10⁻¹⁶ to 10⁻¹³ cm²s⁻¹ at 285 °C.

Sundell et al. [20] showed by Atom Probe Tomography that hydrogen in the oxide layer is mainly located and concentrated in the grain boundaries compared to the volume of the grains. Hydrogen transport in the oxide is thus a mixed diffusion process through the grain boundaries and the grains. Because of the very low diameter (∼35 nm) of the columnar oxide grains [21], mesoscopic numerical calculations of grain boundary and bulk diffusion coefficients from fitting hydrogen diffusion profiles did not lead to accurate values of these coefficients. Finally, the charge state of hydrogen in monoclinic zirconia during corrosion in primary water is not clearly identified. Although recent study suggested that hydrogen could diffuse in the molecular form through nano-pipes in the oxide layer [22], it is usually considered in the corrosion modelling that hydrogen is in the proton form in the oxide layer as it comes from water molecules involved in the corrosion process [8,23,24].

For all these reasons, it is relevant to perform Density Functional Theory (DFT) calculations in order to know the charge state of hydrogen in zirconia and the hydrogen diffusion coefficient in the volume of the monoclinic zirconia grains. The present study is restricted to hydrogen in interstitial positions in pure monoclinic zirconia (m-ZrO₂). Hydrogen is indeed usually found to be an interstitial impurity in many materials [25,26]. Zirconia being an insulator, hydrogen interstitials are expected to introduce defect levels in the insulating gap. It is therefore important to describe correctly this gap. In order to do so, our calculations were performed with the HSE06 [27] Hybrid Functional (HF).

A few studies are available in literature on this subject. They deal with various phases of zirconia (monoclinic, tetragonal, stabilized with an aliovalent cation). Most use Generalized Gradient Approximation (GGA) functionals and thus describe poorly the gap of zirconia and the hydrogen defect levels in this gap [[28], [29], [30], [31], [32], [33], [34], [35]]. Some studies are dedicated to hydrogen diffusion in the zirconia lattice. Some authors have indeed performed calculations of hydrogen diffusion coefficients in tetragonal or monoclinic zirconia using GGA functionals [30,33]. A couple of hybrid functional calculations [36,37] are available but only one [37] considers monoclinic zirconia and it does not deal with the diffusion of hydrogen.

We will discuss our results in view of the previous works in the discussion section (see below).

In the present work, we carry out a systematic search of the insertion sites of hydrogen with various charge states in monoclinic zirconia within hybrid functional DFT in order to describe correctly the gap. We then calculate the migration paths between these sites and integrate the results to obtain the diffusion coefficients of hydrogen in defect-free monoclinic zirconia. The following section presents the technicalities of our calculations, results are then presented and we discuss them in the final part.