Introduction

The rate of any chemical reaction involving two and more reactants relies on their encountering. In addition, if the sources of the reactants are a priori separated in space, the reaction rate becomes controlled by the transport properties of the interface between the sources. The very build-up of reaction product at this interface complicates further encounter between reactants. The protection of structural alloys from corrosion when being exposed to oxidizing conditions relies on the corresponding build-up of a durable oxide scale. Rather than being static, this scale is maintained by its growth, its ability to heal, as well as its ability to dominate the competing degradation processes. Competing processes may include scale dissolution at the metal/oxide interface, pore and crack formation in the oxide, scale re-crystallization, and even evaporation. Somewhat paradoxically, the increasing impacts of the said competing processes are indeed heralded by the ideal parabolic or sub-parabolic growth of the barrier oxide itself. This is because of the slow-down in the rate of growth owing to a decreasing chemical potential gradient. The thermodynamic drive for product formation is represented by the chemical potential difference between the two interfaces. While the driving force for the corrosion process reflects a generic thickness dependence of the chemical potential gradient, the rate of the oxide growth is determined in addition by the mobility of reactants through the barrier oxide.

At steady state, the net reaction during oxide growth is simply represented by metal and oxygen forming metal oxide. Typically, the oxide grows at the inner and outer interfaces, i.e., between metal and oxide and between oxide and corrosive environment, respectively. Often, channels for product formation involve vacancy creation at the interfaces, their migration thorough the oxide scale, and their annihilation at the opposite interfaces. Thus, oxygen vacancies formed at the metal/oxide interface diffuse through the oxide scale and become annihilated at the interface between oxide and oxidizing environment. Equivalently, metal vacancies form at the environment/oxide interface, diffuse through the oxide scale, and become annihilated at the interface between oxide and metal. Mobility of neutral such vacancies is seldom high. Instead, separation into mobile electrons and charged vacancies becomes an efficient way to maintain the oxide growth. This channel for oxide growth is captured by Wagner theory [1–4]. The annihilation of mobile entities may be separated in time causing a steady-state separation of charge, which in turn may impact the oxide growth itself, i.e., space charge effects. In what follows, inwards oxide growth is considered. It becomes tempting to designate anodic properties to the oxygen vacancy VO generation at the metal/oxide interface owing to subsequent dissociation into to V 2+O and 2e−. Correspondingly, the interface between oxide and oxidizing surrounding would take the role of cathode, e.g., due to oxygen reduction by ½ O2 + 2e−→ O2− or water reduction H2O + 2e−→ O2− + H2 preceding the annihilation of the charged vacancy by V 2+O  + O2−→ OO, i.e., the external oxygen ion occupying the charged oxygen vacancy.

Owing to its industrial importance, detailed understanding of zirconium oxidation by water has been acquired from first principles. Rather than atomistic simulation, the purpose of this effort was to build understanding by deconstructing the corrosion phenomenon into computationally accessible and at the same time experimentally relevant quantum chemical modules describing sub-processes relevant to anode [5, 6], cathode [6–8], and transport properties [9]. In autoclave, cyclic episodes of sub-parabolic oxide growth and subsequent breakdown of the protective zirconia is observed on Zr-base alloys [10, 11]. In [9], we proposed that this breakdown is caused by a particular dissolution process whereby oxygen, a priori residing in the oxide scale, is dissolved in the Zr matrix. Breakdown occurs when the rates of migration and annihilation of oxygen vacancies generated at the metal/oxide interface become slower than the said dissolution process.

The purpose of the present study is to clarify further the nature of the effective anode during oxidation of zirconium as to the role of the metal. The objective is indeed to separate the conducting electrons in the alloy from the actual anodic property of the metal/oxide interface. A “pre-anodic” property of the oxygen vacancy is arrived at, preceding the V 2+O /2e− separation.

Computational Details

Conceptual insight is sought by means of first principles calculations using density functional theory (DFT) [12, 13]. The CASTEP program package [14] within the Material Studios framework [15] was utilized and the PBE GGA functional [16] was employed for all calculations. Core electrons were described by ultrasoft pseudopotentials [17] in conjunction with 340 eV cut-off energy.

The present study utilizes a periodically repeating interface between t-ZrO2 and α-Zr, while to some extent ambiguous, it was designed to satisfy approximate epitaxy by matching the 100 plane of t-ZrO2 to the 010 plane of α-Zr. The resulting supercell, consisting of 120 atoms, was geometry optimized including optimization of the lattice parameters, see Fig. 1. The lattice parameters were held constant in the subsequent calculations. The k-point sampling of the Brillouin zone was made by means of the Monkhorst–Pack scheme [18] with a 2 × 2 × 1 k-points mesh. Neutral oxygen vacancies were created by moving one oxygen atom from the oxide to an interstitial position in the α-Zr matrix. The lattice parameters were kept constant in order to determine lower bounds to the oxygen dissolution. The vacancy positions are indicated in Fig. 1. The interstitial oxygen in the Zr matrix is positioned at the interface in order to capture the instantaneous local drive for dissolution. Bulk properties are reported elsewhere [5]. The complete linear synchronous transit/quadratic synchronous transit (LST/QST) method [19] was employed to estimate transition states for oxygen vacancy diffusion. The reference calculations on bulk t-ZrO2 were done on a supercell with 48 atoms.

Fig. 1
figure 1

Supercell model [20] of metal/oxide interface on which calculations were performed. The positions where an oxygen vacancy was created in the comparative study are indicated with 1–8. The oxygen atom in the metal is indicated with X. Red balls refer to oxygen and light blue to zirconium. Lattice parameters of the box are a = 11.03 Å, b = 6.89 Å, c = 24.02 Å, α = 90.0°, β = 79.8°, γ = 88.7°

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Result and Discussion

A useful definition of an anodic site is that this is where oxidation takes place. However, owing to the requirement of electro-neutrality of chemical reactions, any oxidation process must be associated with a reduction reaction. Here, though, we address microscopic processes at the nominal anode. In doing so, properties preceding the actual oxidation reaction are addressed.

This study concerns particular aspects of the continued oxidation process of zirconium alloys by water. The process is known to display inwards oxide growth. Here, this is taken to imply initial oxygen vacancy creation at the metal/oxide interface. This process is driven by the high solubility of oxygen in zirconium, i.e., ~30 at% O [6, 21]. Crucial is the distinction between neutral and oxidized oxygen vacancies, i.e., VO and V 2+O , respectively. The former is present prior to oxidation. It results from zirconium alloy dissolving only neutral oxygen atoms. By doing so, oxygen nominally oxidizes the metal, while the metal ions at the oxide site where the oxygen emanated from are reduced. The latter is because the remaining electrons initially associated with the oxygen atom are forced to stay in the oxide. We say that VO formation offers a preparatory step in the forming of the actual anode. This is in contrast to the anodic process which involves the dissociation of VO into V 2+O and 2e−.

In what follows, a representative model interface between metal and oxide is validated. In paragraph III.a, the relevance of the model is justified by the small energies associated with the formation of the VO sites upon transferring a lattice oxygen into an interstitial site in the Zr matrix. In paragraph III.b, the band structure in bulk zirconia is mapped onto the local density of states in the oxide scale. In paragraph III.c, electronic signatures and mobility of an oxygen vacancy in bulk zirconia are reproduced by the interface. In section “Emerging Understanding”, the results are put in context of a novel theory for oxidation of zirconium by water.

Energetics for Oxygen Vacancy Formation

The thermodynamically preferred dissolution of ZrO2 in Zr is taken here to result from inter-diffusion of oxygen atoms into interstitial sites in the Zr alloy as accompanied by the corresponding formation of oxygen vacancies in the zirconia. The enthalpy of formation of a neutral oxygen vacancy (V O) in the oxide at a metal/oxide interface by incorporating oxygen at interstitial site in the Zr matrix, for the various oxygen origins indicated in Fig. 1, is listed in Table 1.

Table 1 Enthalpies of formation of oxygen vacancy at the metal/oxide interface, created by transfer to an interstitial site in the metal, see Fig. 1
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While there are local variations, in particular the immediate interface clearly expresses the chemical incompatibility of the two phases. When comparing for example sites 6 and 8, the terminal role of the former as compared to the integrated position of the latter is taken to explain the increased stability on dissolution of O at site 6. It is interesting however, that already less than 1 nm into the oxide, the drive for dissolution stabilizes, cf. sites 1–4. The corresponding enthalpies of formation of neutral vacancies in bulk come out endothermic by approximately 0.5 eV in the periodic supercell calculations for both the tetragonal and monoclinic phases [9]. This articulates both similarity and dissimilarity of thermodynamic phases and the corrosion process, i.e., emphasizing the impact of proximity. This is further emphasized by the electronic signatures as reflected in the build-up of the band gap including impurity state owing to oxygen vacancy, in the oxide adjacent to the metal/oxide interface, vide infra.

Oxygen Vacancy Formation at Metal/Oxide Interface

The deconvolution of the total density of state (DOS) into its separate local metal and local oxide contributions is provided in Fig. 2a, b.

Fig. 2
figure 2

a Density of states (DOS) for metal and oxide when an oxygen vacancy is created at position 1, see Fig. 1. b Oxide contribution to DOS, dashed rectangle in Fig. 1. The state associated with the oxygen vacancy is found ~1 eV below the Fermi energy (dashed) rendering it neutral. c DOS of t-ZrO2-δ bulk, containing 3.1 at% O vacancies for comparison. Blue represents O 2s states, red O 2p states, and green Zr 4d states

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By comparing the latter contribution to that of the DOS in bulk ZrO2, it is striking to note how rapidly convergence is achieved. A further manifestation of the similarity of bulk ZrO2 electronic properties to the oxide at less than 1 nm from the metal/oxide interface is obtained by considering the characteristics of the oxygen vacancy, cf. Figure 2b, c. Indeed, also the position of the vacancy band in the local band gap reproduces that is observed in bulk, see Fig. 2 again. Looking closer into the decomposition of the total DOS at the interface and the Fermi energy (E F) of the metal part, it is tempting to ascribe the metal as an infinite source of electrons, and in particular so as the V O state lays below the E F of the metal. It may be erroneously inferred that V O is always replenished with electrons upon oxidation, owing to the cathode processes. This is however not the case, as it would imply net charging of the metal. This was understood by Wagner. It is argued here that a better way to understand the role of the metal, is it acting oxygen sink. The electrons of the VO site become accessible to the cathode process by dissociating from the said site and diffusing in a background of equivalent V 2+O sites. This implies a steady state build-up of space charge determined by the diffusion of V 2+O to the cathode where they become annihilated by external O2− ions, i.e., the products of the reduction process at the nominal cathode. This is discussed further in section “Emerging Understanding”.

Oxygen Vacancy Mobility

The activation energy for oxygen vacancy V O diffusion is associated with the rising and setting of the vacancy band on passing the transition state. Again, semi-quantitative agreement between a property at metal/oxide interface and that of bulk ZrO2 is found. This finding puts to rest hypotheses based on spontaneous transient charging of the vacancy as a predominant means to lower the transition state, i.e., by transient injection of vacancy electrons into the conduction band of the metal. Rather, such easy paths should be deemed relevant only at low coverages. Indeed, the activation energy (E A) for oxygen diffusion in bulk was found to depend sensitively on the occupation of the VO band. Thus, activation energies of neutral VO sites came out in vicinity of 2 eV and charged sites display EA < 0.5 eV [9]. Thus, the energetics serve to further support the consistency in the comparison between local oxide properties metal/oxide interface (see Table 2) and those of ZrO2 bulk.

Table 2 Effect of vicinity to metal on the activation energy (EA) for vacancy diffusion
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Emerging Understanding

Here, we summarize our comprehensive understanding [5–9] and put the present findings in III.a–c in that context. In autoclave studies on water oxidation of zirconium alloys, cyclic episodes of sub-parabolic oxide growth and subsequent breakdown of the protective zirconia are observed. In [9], it was indeed proposed that this breakdown is caused by a particular dissolution process whereby oxygen, a priori residing in the oxide scale, is dissolved in the Zr matrix. This was explicitly addressed in paragraphs III.a–c. The breakdown of the barrier oxide is taken to initiate when the rates of migration and annihilation of oxygen vacancies generated at the metal/oxide interface becomes slower than the said dissolution process. Two channels were found to contribute to the oxidation process, one according to classical oxidation theory involving hydrogen evolution at the effective cathode (see Fig. 3), and one which allows inwards transport of protons against the electric field, i.e., hydrogen pick-up. The two channels were associated with charged V 2+O and uncharged oxygen vacancy VO formations at the metal/oxide interface. The first channel, i.e., classical oxidation theory, is summarized in Fig. 3. The sub-division of oxide layers into t-ZrO2 at metal oxide interface in conjunction with an outer m-ZrO2 is inspired by the study by Garner et al. [22]. Consistency of this understanding is provided in [9] based on transport properties of oxygen vacancies.

Fig. 3
figure 3

a Hydroxylation of m-ZrO2 grain boundaries at m-ZrO2/“H2O” interface. b The formation of neutral vacancies by oxygen dissolution in α-Zr at the t-ZrO2/metal interface. c Charging of oxygen vacancies, i.e., anode reaction. d Hydrogen evolution at m-ZrO2/t-ZrO2 interface, i.e., cathode reaction. e Annihilation of charged oxygen vacancies at m-ZrO2/t-ZrO2 interface

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The barrier oxide of zirconia is taken to be subdivided into a defect-rich tetragonal phase providing both electron and ion conductivity, and a defect-free monoclinic phase permeating water in the form of hydroxides resulting from the hydrolysis products of O2− and H2O. The cathode process, occurring at the interface between tetragonal and monoclinic phases, comprised hydrogen evolution in conjunction with annihilation of charged oxygen vacancies in the tetragonal phase. The oxygen vacancies is formed by the dissolution process of oxygen in the Zr alloy, and their transient charging is the result of the defective tetragonal phase offering pathways for electron tunneling connecting preexisting charged oxygen vacancies. The diffusion of charged oxygen vacancies is similarly associated with very small activation energies.

From the formalism developed in [9], we have a thickness-dependent activation energy for diffusion owing to a constant bulk contribution and that originates from charging. The effective diffusivity of oxygen vacancies of charge q \(D_{\text{eff}}^{q} \left( X \right)\) given by

$$D_{\text{eff}}^{q} (X) = D_{\text{bulk}}^{q} \exp \left[ { - \frac{{q^{2} C_{{V_{\text{O}} }}^{q + } }}{{8\varepsilon \varepsilon_{0} k_{\text{B}} T}}X^{2} } \right],$$
(1)

where X is the oxide thickness, \(D_{\text{bulk}}^{q}\) is the diffusivity constant for vacancy of charge q in bulk, \(C_{{V_{\text{O}} }}^{q + }\) is the concentration of charged oxygen vacancies, ɛ is the relative permittivity, ɛ 0 is the vacuum permittivity, k B is Boltzmann constant, and T is the temperature. Given a constant concentration of charged oxygen vacancies, the activation energy increase for the formation of transient charged oxygen vacancies has a quadratic thickness dependence. From Eq. 1, a sub-parabolic oxide thickness curve was derived,

$$X = \left\{ {\lambda_{q}^{2} \ln \left( {1 + \frac{{2K_{p}^{q} }}{{\lambda_{q}^{2} }}t} \right)} \right\}^{{\frac{1}{2}}}$$
(2)

where t is time, Δμ is electro-chemical potential,

$$K_{p}^{q} = D_{\text{bulk}}^{q} \frac{\Delta \mu }{{k_{\text{B}} T}}$$
(3)

and

$$\lambda_{q}^{2} = \frac{{8\varepsilon \varepsilon_{0} }}{{q^{2} C_{{V_{O}^{q + } }} }}k_{\text{B}} T.$$
(4)

The total activation energy W for transport of oxygen vacancies by utilizing channels, which propagate electrons and charged vacancies separately, becomes

$$W = W_{\text{bulk}}^{q} + \frac{{k_{\text{B}} T}}{{\lambda_{q}^{2} }}X^{2} .$$
(5)

The first term in Eq. 5 represents the activation energy for bulk diffusion, while the second term is the thickness-dependent activation energy. It is associated with the forming charged vacancies. This second term causes sub-parabolic oxide growth in classical Wagnerian oxidation theory with H2 evolution. In case of \(W_{\text{bulk}}^{q}\) large, the thickness-dependent term will not impact the scale growth. If on the other hand \(W_{\text{bulk}}^{q}\) is small, then the sub-parabolic behavior will dominate the mass gain curve. Relevant here is that charged oxygen vacancies in t-ZrO2 display ~0.2 eV activation energies for diffusion, i.e., five times smaller than those for uncharged vacancies. Equation 5 says that for constant density of charged vacancies, the cost for the formation of additional charged vacancies increases with thickness of barrier oxide to the power of two. The hydrogen pick-up channel opens owing to the slowing down of the Wagnerian channel, which in turn precedes the transition to the next cycle. We propose that the charging of the oxide causes the build-up of VO concentration close to the metal/oxide interface owing to the dissolution of oxygen into the alloy. A symptom of the oxide dissolution process is neutral oxygen vacancies accumulation, super-saturation, and possible hydrogen-assisted nucleation late in each cycle. The formation of pores was taken to offer easy paths for short-circuiting hydrogen transport through the oxide [9, 23], resulting in the avalanching hydrogen pick-up [24] reported to precede the breakdown of the barrier oxide, the latter resulting from increased loss of mechanical integrity of the barrier oxide.

Conclusions

It was inferred in [9] that the rapid sudden increase in hydrogen pick-up fraction (HPUF) on approaching transition [24, 25] is associated with the formation of pores that build from the metal/oxide interface. Thus, the avalanche in hydrogen pick-up fraction was taken to reflect such pore formation, a symptom rather than the cause of barrier oxide breakdown. This study contributes the local atomistic perspective on the pre-nucleation of neutral oxygen vacancies at the metal/oxide interface. The drive to dissolve oxygen in the alloy is reflected in the results. Moreover, the rapid convergence of electronic properties to those of bulk oxide less than 1 nm into the oxide in spite of the local structural variability provides new insight as to the nature of an amorphous metal/oxide interface. New light is shed on the anode process during metal oxidation. Rather than the metal acting electron source, oxygen dissolution in the alloy matrix produces oxygen vacancies VO. Connection to Wagner theory is made if these become local anodes, i.e., by dissociating into V q+O  + qe −. The role of V q+O is dual. Displaying enhanced mobility relative to VO, the presence of charged oxygen vacancies at steady state offers percolation pathways to the cathode by consecutive electron tunneling processes. This understanding of zirconia formers serves a conceptual platform including also alumina formers [26, 27].