Modeling Two- and Three-Stage Oxygen Tracer Experiments during High-Temperature Oxidation of Metals with a High Oxygen Solubility

Abstract

The numerical tool EKINOX-Zr has been upgraded to simulate oxygen tracer experiments during the high-temperature oxidation of a metal with a high oxygen solubility limit. The penetration of ¹⁸O tracer is calculated during the dynamic evolution of the ZrO_2-x/α_{Zr(O )}/β_zr system. The numerical approach allows to explicitly take into account the variation of the tracer diffusion coefficient through the oxide scale as a function of the vacancy concentration. A classical two-stages ¹⁶O₂/¹⁸O₂ tracer experiment has been simulated. It is shown that a classical fitting procedure on the ¹⁸O concentration profile obtained for short-time experiments leads to the identification of the oxygen chemical diffusion coefficient. The second type of tracer experiment is proposed using a three-stages ¹⁶O₂/¹⁸O₂/¹⁶O₂ oxidation. It allows the direct estimation of the diffusion coefficient from the transport of ¹⁸O peak in the growing oxide scale.

Introduction

Oxygen tracer (O-tracer) experiments are one mode to measure oxygen diffusion coefficients and/or to understand the growth mechanisms of an oxide scale. Nowadays, these experiments consist in a two-stage oxidation. The first stage is generally performed under a natural oxygen atmosphere (majority of ¹⁶O₂) while the second stage is carried out under an ¹⁸O₂-enriched atmosphere. Then, SIMS, RBS or NRA can be used to measure the distribution profile of each isotope and extract the expected information.

In order to analyze the isotopic distribution profiles, an analytical theory of O-tracer penetration in two-stage oxidation experiments has been formalized by Mishin and Borchardt [1]. This theory gives the equations for the O-tracer penetration through both grain boundaries and sublattice diffusion in a growing oxide layer, with both the anionic and the cationic transports. Hence, their work has guided numbers of studies concerning O-tracer experiments to correctly define the time ratio between ¹⁸O₂ and ¹⁶O₂ sequences [2]. Despite this very useful approach, their analytical theory suffers from several restrictions. First, it is not possible to handle neither a temperature transient oxidation nor a triple sequence ¹⁶O₂/¹⁸O₂/¹⁶O₂. Furthermore, it is assumed in their work that the oxide scale growth kinetics is fully controlled by the fluxes in the oxide because oxygen dissolution in the metal can be often neglected.

In this study, the aim was to measure the oxygen diffusion coefficient in specific phases of the zirconium/zirconia system. Zirconium, as titanium and hafnium, has a very high oxygen solubility limit [3]. Thus, in order to analyze the isotope penetration profiles in these specific systems, it was required to develop a numerical model able to calculate ¹⁶O and ¹⁸O penetration for the particular case of the oxygen penetration in the underlying metal. To do that, EKINOX-Zr numerical model [4,5,6] has been upgraded in order to calculate the isotopic penetration. Formerly, the EKINOX-Zr model was developed to simulate the high-temperature (HT) oxidation of zirconium alloys. Hence, this numerical model is able to calculate the growth of both the oxide scale and the α_Zr(O) phase additionally to the penetration of oxygen. To calculate the isotope penetration, ¹⁸O was added to the system of equations to be solved, and it was assumed that ¹⁶O and ¹⁸O were two distinct species that diffuse independently but on the same sublattice and under the same driving forces.

This paper deals with the development of theoretical approach to formalize the flux equations of isotopic tracers in the oxide scale with anionic transport. Then, its appliance with the EKINOX-Zr numerical model was tested.

Formulation of the Tracer Diffusion Equations

Theoretical Approach

In this part, the equations of the fluxes of ¹⁸O²⁻ and ¹⁶O²⁻ species in the oxide scale are detailed. They lead to the numerical model presented in the next part.

The electrochemical potential $\tilde{\mu }_{\text{i}}$ of a given specie $i$ ¹ is linked to its chemical activity $a_{i}$ by the Eq. (1):

$$\forall \space i \in \left\{ {16;18} \right\}, \tilde{\mu }_{\text{i}} = \mu_{i} + \alpha e\phi^{\text{ox}} = \mu_{i}^{0} + RT{ \ln }\left( {a_{i} } \right) + \alpha e\phi^{\text{ox}}$$

(1)

where $\phi^{\text{ox}}$ is the local electrostatic potential, e is the elementary charge, and α is the charge of the specie i. Let us assume the case of an oxygen deficient n-type semiconducting oxide.

In this case, there is a transport of oxygen anions and electrons in the oxide due to the electrochemical potential gradient across the oxide scale. The diffusion of charged particles gives rise to partial current densities of oxygen and electrons that leads to a total current $i_{\text{tot}}$ [7, 8]. Because of the electroneutrality of the overall oxide layer, $i_{\text{tot}} = 0$, the flux of specie $i$ can be expressed as follows:

$$\forall \space i \in \left\{ {16;18} \right\} , J_{i} = -\, \frac{{D_{i} .C_{i} }}{RT}\left[ {\frac{{{\text{d}}\mu_{i} }}{{{\text{d}}x}} + a_{i} e\frac{{{\text{d}}\phi^{\text{ox}} }}{{{\text{d}}x}}} \right]$$

(2)

By using the Wagner model of oxidation (see [8]), with the hypotheses of a semiconducting oxide in which the electronic defects diffuse much faster than the anions and with the assumption of the local equilibrium across the oxide, it is possible to write the flux equations of the diffusing species in the oxide scale as a function of the oxygen activity gradient across the oxide scale:

$$\forall \space i \in \left\{ {16;18} \right\}, J_{i} = -\, D_{O} \cdot \left( {\frac{{{\text{d}}C_{i} }}{{{\text{d}}x}} - C_{i} \cdot \frac{{{\text{d}}lna_{O} }}{{{\text{d}}x}}} \right)$$

(3)

In the following, the case of an oxide scale that grows via anionic transport² only by a vacancy mechanism is considered. At the oxide/gas interface, the oxygen adsorption is possible thanks to the annihilation of anionic vacancies following the Eq. (4) (using Kröger-Vink notations):

$${\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}{\text{O}}_{2} + {\text{V}}_{\text{O}}^{a \bullet } + \alpha {\text{e}}^{ - } = {\text{O}}_{\text{o}}^{ \times }$$

(4)

The transport mechanism for anions can be described as the transport in the opposite way of anionic vacancies ${\text{V}}_{\text{O}}^{a \bullet }$; anionic vacancies are created at the metal/oxide interface and annihilated at the oxide/gas interfaces. It is possible to write the equilibrium constant $K$ of the chemical reaction at the oxide/gas interface (Eq. 5):

$$K = \frac{1}{{a_{\text{O}} n^{\alpha} \left[ V_{\text{O}}^{\alpha \bullet} \right]}}$$

(5)

Expressions of the chemical activities $a_{\text{O}}$, $a_{16}$, and $a_{18}$ can be deduced from the Eq. (5), using the local electroneutrality to remove the electron concentration $n$ from the equation:

$$a_{\text{O}} = \frac{1}{{a^{\alpha} \left[ {V_{\text{O}}^{\alpha \bullet } } \right]^{{\left( {1 + \alpha} \right)}} K}}$$

(6)

$$\forall \space i \in \left\{ {16;18} \right\}, a_{i} = \frac{{C_{i} }}{{C_{\text{O}} }}a_{\text{O}}$$

(7)

Following Eqs. (1), (6), and (7), the expression of the chemical potential $\mu_{16}$ ³ is given by:

$$\mu_{16} = \mu_{16}^{0} + RT\left( {{ \ln }\left( {\frac{{C_{16} }}{{C_{\text{O}} }}} \right) - \left( {1 + \alpha } \right)\left( {{ \ln }\left( {\left[ {V_{\text{O}}^{\alpha \bullet } } \right] - { \ln }\left( {Ka^{\alpha} } \right)} \right)} \right)} \right)$$

(8)

Considering that, in the oxide scale, the anionic vacancies and anions are complement one another in the anionic sublattice: $C_{\text{O}} = 1 - \left[ {V_{\text{O}}^{a \bullet } } \right]$, and that $Ka^{\alpha}$ is a constant across the scale, an expression of $\nabla \mu_{16}$ can be deduced from Eq. (8):

$$\nabla \mu_{16} = RT\left( {\frac{{\nabla C_{16} }}{{C_{16} }} - \left( {\frac{1 + a}{{\left[ {V_{\text{O}}^{\alpha \bullet } } \right]}} - \frac{1}{{1 - \left[ {V_{\text{O}}^{\alpha \bullet } } \right] }}} \right)\nabla C_{\text{O}} } \right)$$

(9)

According to the thermodynamic of irreversible processes (e.g. [9]), the flux $J_{16}$ of ¹⁶O²⁻ and the flux $J_{18}$ of ¹⁸O²⁻ can be expressed with the crossed diffusion coefficients and the chemical potentials of each specie as given in Eq. (10):

$$\forall \space \left\{ {i;j} \right\} \in \left\{ {16;18} \right\} | \space i \ne j, \space J_{i} = -\, L_{ii} \nabla \mu_{i} - L_{ij} \nabla \mu_{j}$$

(10)

Knowing that $\frac{{L_{\text{OO}} }}{{C_{O} }} = \frac{{L_{1616} }}{{C_{16} }}$, it can be assumed that: $D_{\text{O}} = \frac{RT}{{C_{\text{O}} }}L_{\text{OO}} f_{\text{O}}$. It is chosen to fix that $f_{\text{O}} = 1 - \frac{{C_{\text{O}} L_{1618} }}{{C_{18} L_{\text{OO}} }}$ and $L_{1618} = 0$, thus $f_{\text{O}} = 1$. Then, by a combination of the Eqs. (9) and (10), one can obtain an expression of the flux $J_{16}$ and $J_{18}$:

$$\forall \space i \in \left\{ {16;18} \right\}, J_{i} = -\, D_{\text{O}} \nabla C_{i} - D_{\text{O}} C_{i} \left( {\frac{1 + \alpha }{{\left[ {V_{\text{O}}^{\alpha \bullet } } \right]}} - \frac{1}{{1 - \left[ {V_{\text{O}}^{\alpha \bullet } } \right]}}} \right)\nabla C_{\text{O}}$$

(11)

In the flux expression (Eq. 11), we can find two distinct parts in the RHS. The first part, $-\, D_{\text{O}} \nabla C_{a}$, corresponds to the diffusion (Brownian motion) while the second part, $-\, D_{\text{O}} C_{i} \left( {\frac{1 + \alpha}{{\left[ {V_{\text{O}}^{ \alpha \bullet } } \right]}} - \frac{1}{{1 - \left[ {V_{\text{O}}^{\alpha \bullet } } \right]}}} \right)\nabla {\text{C}}_{\text{O}}$, is the transport induced by the gradient of the electrochemical potential. Considering that ${\text{C}}_{\text{O}} {\text{D}}_{\text{O}} = \left[ {{\text{V}}_{\text{O}}^{\alpha \bullet } } \right]{\text{D}}_{{{\text{V}}_{\text{O}}^{\alpha \bullet } }}$ and defining $\gamma = \frac{{\left[ {V_{\text{O}}^{\alpha \bullet } } \right]}}{{\left( {1 + \alpha} \right)\left( {1 - \left[ {V_{\text{O}}^{\alpha \bullet } } \right]} \right)}}$ and $\tau_{i} = \frac{{C_{i} }}{{C_{\text{O}} }}$, one can express the fluxes $J_{16}$ and $J_{18}$ as given by Eq. (17):

$$\forall \space i \in \left\{ {16;18} \right\}, J_{i} = -\, \gamma \left( {1 + \alpha} \right)D_{{{\text{V}}_{\text{O}}^{\alpha \bullet } }} \nabla C_{i} - \tau_{i} \left( {1 - \gamma } \right)\left( {1 + \alpha} \right)D_{{{\text{V}}_{\text{O}}^{\alpha \bullet } }} \nabla \left[ {V_{\text{O}}^{\alpha \bullet } } \right]$$

(12)

In the case where $\left[ {V_{\text{O}}^{\alpha \bullet } } \right] \ll 1 \Rightarrow \gamma \approx \frac{{\left[ {V_{\text{O}}^{\alpha \bullet } } \right]}}{1 + \alpha}$, one can simplify the Eq. (12):

$$J_{18} = -\, \left[ {V_{\text{O}}^{\alpha \bullet } } \right]D_{{{\text{V}}_{\text{O}}^{\alpha \bullet } }} \nabla C_{18} - \tau_{18} \left( {1 + \alpha} \right)D_{{{\text{V}}_{\text{O}}^{\alpha \bullet } }} \nabla \left[ {V_{\text{O}}^{\alpha \bullet } } \right]$$

(13)

It is possible to obtain an expression of $\tilde{D}_{\text{O}}$ (the chemical diffusion coefficient of oxygen) from the Eq. (13):

$$J_{18} = -\, \left[ {V_{\text{O}}^{\alpha \bullet } } \right]D_{{{\text{V}}_{\text{O}}^{\alpha \bullet } }} \nabla C_{18} - \tau_{18} \left( {1 + \alpha} \right)D_{{{\text{V}}_{\text{O}}^{\alpha \bullet } }} \nabla \left[ {V_{\text{O}}^{\alpha \bullet } } \right] = - \tilde{D}_{\text{O}} \nabla C_{18}$$

$$\Rightarrow \tilde{D}_{\text{O}} = \left( {1 + \left( {\frac{1 + \alpha}{{\left[ {V_{\text{O}}^{\alpha \bullet } } \right]}} - \frac{1}{{C_{\text{O}} }}} \right)C_{18} \frac{{\nabla \left[ {V_{\text{O}}^{\alpha \bullet } } \right]}}{{\nabla C_{18} }}} \right)\frac{{\left[ {V_{\text{O}}^{\alpha \bullet } } \right]}}{{C_{\text{O}} }}D_{{{\text{V}}_{\text{O}}^{\alpha \bullet } }}$$

(14)

Knowing that $\left( {1 + \alpha} \right)D_{{{\text{V}}_{\text{O}}^{\alpha \bullet } }} \nabla \left[ {V_{\text{O}}^{\alpha \bullet } } \right] = - J_{{{\text{V}}_{\text{O}}^{\alpha \bullet } }} = J_{\text{O}}$, the Eq. (18) becomes:

$$J_{18} = \tau_{18} J_{\text{O}} - C_{\text{O}} D_{\text{O}} \nabla C_{18}$$

(15)

In [1], Mishin and Borchardt defined the oxygen tracer flux as the following equation (Eq. (10) p. 867 in [1]):

$$J_{\text{O}}^{ *} = cJ_{\text{O}} - C_{\text{O}} D_{\text{O}}^{ *} \nabla c$$

(16)

A one-to-one identification between Eqs. (15) and (16) can be done. $J_{\text{O}}^{ *}$ is the flux of O-tracer, noticed $J_{18}$ in the present work; $c$ is the ratio of O-tracer, corresponding to $\tau_{18}$ in the present work; $\nabla c$ is the gradient of the tracer concentration, that is to say $\nabla C_{18}$ in the present work; and $D_{\text{O}}^{ *} = fD_{\text{O}}$, where the correlation factor $f$ is named $f_{\text{o}}$ and it is equal to the unity.

Thus, when $\left[ {V_{\text{O}}^{z \bullet } } \right] \ll 1$ the Eqs. (15) and (16) are equivalent, this corresponds to the conditions of Mishin and Borchardt’s work.

The isotope fluxes in the metallic matrix are governed by the Brownian’s motion and are given by Eq. (17). In those two equations, the approximation on the activity coefficient $\gamma_{a} = 1$ is assumed.

$$\forall \space i \in \left\{ {16;18} \right\}, J_{i} = -\, D_{\text{O}} \nabla C_{i}$$

(17)

Modifications of EKINOX-Zr Model

Detailed presentations of the EKINOX-Zr model are available in [5, 6]. It is reminded that in this model, the fuel cladding tube is modeled as a one-dimensional planar domain divided into $n$ slabs (Fig. 1). The growth kinetics of both the oxide scale and the α_Zr(O) layer is calculated by EKINOX-Zr model. A numerical resolution of Fick’s equations (thanks to an explicit finite difference algorithm) with boundary conditions on moving interfaces allows determining the penetration profile of oxygen. The interfaces are considered at local equilibrium, and the thermodynamic database Zircobase [10] (CALPHAD formalism) is used to calculate the boundary concentrations.

Fig. 1

Schematic of the EKINOX-Zr architecture [6]

Given that EKINOX-Zr model is already able to calculate the overall oxygen penetration, one can follow the penetration of each isotope by adding only the calculation of the penetration of ¹⁶O and calculating ¹⁸O by the difference. Firstly, it is assumed that the Eq. (18)⁴ is valid in the bulk metal (O diffuses as an interstitial) and that the Eq. (19) is valid in the oxide. Then, the whole penetration profile of ¹⁸O can be deduced from Eqs. (18) and (19) once the O, ¹⁶O, and vacancies concentrations have been calculated by the EKINOX-Zr model.

$$\left( {\text{metal}} \right)\quad \forall \space p \in \mathbb{N}| 1\leq p \leq i, \space X_{\text{O}}^{p} = X_{18}^{p} + X_{16}^{p}$$

(18)

$$\left( {\text{oxide}} \right)\quad \forall \space q \in \mathbb{N}| i+1\leq q \leq n, \space X_{\text{O}}^{q} = 1 - X_{{{\text{V}}_{\text{O}} }}^{q} = X_{18}^{q} + X_{16}^{q}$$

(19)

A discretized expression of Eq. (17) for the flux $J_{16}^{p}$ in the bulk metal is given by the Eq. (20). If one defines the diffusion coefficient $D_{\text{ox}}$ as $D_{\text{ox}} = \left( {1 + a} \right)D_{{{\text{V}}_{\text{O}}^{a \bullet } }}$, a discretized expression of the Eq. (12) for the flux $J_{16}^{q}$ can be provided by Eq. (21).

$$\left( {\text{metal}} \right)\quad\forall \space p \in \mathbb{N}| 1\leq p \leq i, \space J_{16}^{p} = -\, D_{O}^{p} \frac{{X_{16}^{p + 1} - X_{16}^{p} }}{{\frac{{e^{p} + e^{p + 1} }}{2}}}$$

(20)

$$\left( {\text{oxide}} \right)\quad\forall \space q \in \mathbb{N}| i+1\leq q \leq n, \space J_{16}^{q} = - \gamma^{q} D_{\text{ox}}^{q} \frac{{X_{16}^{q + 1} - X_{16}^{q} }}{{\frac{{e^{q} + e^{q + 1} }}{2}}} + \tau_{16}^{q} \left( {1 - \gamma^{q} } \right)D_{\text{ox}}^{q} \frac{{X_{{{\text{V}}_{\text{O}} }}^{q + 1} - X_{{{\text{V}}_{\text{O}} }}^{q} }}{{\frac{{e^{q} + e^{q + 1} }}{2}}}$$

(21)

From the Eqs. (20) and (21), it is possible to obtain the evolution of isotopic concentrations as follows:

$$\forall \space r \in \mathbb{N} \backslash\left\{ {j,j + 1,i,i + 1} \right\}| 2\leq p \leq n-1, \space \frac{{{\text{d}}X_{16}^{r} }}{{{\text{d}}t}} = \Omega^{r} \frac{{J_{16}^{r} - J_{16}^{r + 1} }}{{e^{r} }}$$

(22)

The variations of isotopic concentrations in the slabs around each interface are given by Eqs. (23–28):

$$\dot{X}_{16}^{1} = - \frac{{J_{{16,{\text{met}}}}^{1} }}{{e^{1} }}$$

(23)

$$\dot{X}_{16}^{j} = \frac{{J_{{16,{\text{met}}}}^{j - 1} }}{{e^{j} }}$$

(24)

$$\dot{X}_{16}^{j + 1} = - \frac{{J_{{16,{\text{met}}}}^{j + 1} }}{{e^{j + 1} }}$$

(25)

$$\dot{X}_{16}^{i} = \frac{{J_{{16,{\text{met}}}}^{i - 1} }}{{e^{i} }}$$

(26)

$$\dot{X}_{16}^{i + 1} = - \frac{{J_{{16,{\text{ox}}}}^{i + 1} }}{{e^{i + 1} }}$$

(27)

$$\dot{X}_{16}^{n} = \frac{{J_{{16,{\text{ox}}}}^{n - 1} }}{{e^{n} }}$$

(28)

It is now possible to express the equations of the variation of isotopic concentrations between two time steps:

$$\forall \space r \in \mathbb{N}| 1\leq r \leq n-1, \space X_{16}^{r} \left( {t + \Delta t} \right) = X_{16}^{r} \left( t \right) + \Delta t\frac{{{\text{d}}X_{16}^{r} }}{{{\text{d}}t}}$$

(29)

$$\forall \space r \in \mathbb{N}| 1\leq r \leq n-1, X_{18}^{r} \left( {t + \Delta t} \right) = 1 - X_{\text{O}}^{r} \left( {t + \Delta t} \right) - X_{16}^{n} \left( {t + \Delta t} \right)$$

(30)

At the oxide/gas interface, the proportion of each isotope changes as a function of the incoming flux of mixed ¹⁶O₂/¹⁸O₂ gas. The fraction of ¹⁸O isotope in the ¹⁶O₂/¹⁸O₂ mixed gas is called ${\text{R}}_{18}$. Then, the concentrations of ¹⁶O and ¹⁸O at each time step in the slab $n$ are given by Eqs. (31) and (32):

$$X_{16}^{n} \left( {t + \Delta t} \right) = X_{16}^{n} \left( t \right) + \left( {1 - {\text{R}}_{18} } \right)\Delta t\dot{X}_{16}^{n}$$

(31)

$$X_{18}^{n} \left( {t + \Delta t} \right) = 1 - X_{{V_{O} }}^{n} \left( {t + \Delta t} \right) - X_{16}^{n} \left( {t + \Delta t} \right)$$

(32)

As stated before, the oxygen concentrations at the interface are fixed to the equilibrium values. Then, the isotopic concentrations in the slabs 1, $j$, $j + 1$, $i$, and $i + 1$ are fixed to the boundary concentrations as a function of the ratio of each isotope in the slab. For example, the isotopic concentrations in the slab $i$ (metal/oxide interface in α_Zr(O)) are given by the Eqs. (33) and (34):

$$X_{16}^{i} \left( {t + \Delta t} \right) = \left( {\frac{{X_{16}^{i} \left( t \right)}}{{X_{16}^{i} \left( t \right) + X_{18}^{i} \left( t \right)}}} \right)C_{{\alpha_{\text{Zr}} /{\text{ox}}}}$$

(33)

$$X_{18}^{i} \left( {t + \Delta t} \right) = \left( {1 - \frac{{X_{18}^{i} \left( t \right)}}{{X_{16}^{i} \left( t \right) + X_{18}^{i} \left( t \right)}}} \right)C_{{\alpha_{\text{Zr}} /{\text{ox}}}}$$

(34)

Results and Discussion

Tables 1 and 2 summarize all the thermodynamic and diffusion data used to perform calculations shown in this part. Please note that in the oxide scale, the boundary concentrations of vacancies and oxygen are defined with reference to the stoichiometry of ZrO_2-x . Notice that a constant value for diffusion coefficient of vacancies in the oxide is used as the only input parameter for diffusion in the oxide for EKINOX calculation. Considering the gradient of vacancies in the oxide, this leads to variation of the O intrinsic diffusion coefficient across the oxide. Table 2 also gives in the two last columns the corresponding values for O intrinsic diffusion coefficient at ox/α_Zr interface and oxide/gas interface.

Table 1 Oxygen boundary concentrations calculated with ThermoCalc and the Zircobase [10]

Table 2 Oxygen diffusion coefficients used in EKINOX-Zr calculations [11]

Figure 2 shows a comparison between Mishin and Borchardt study [1] and EKINOX-Zr calculations. Figure 2 presents two isotopic distribution profiles calculated for the oxidation of zirconium during 1000 s at 1200 °C. The first 500 s corresponds to an oxidation under ¹⁶O₂ (natural stage) and the last 500 s are under ¹⁸O₂ (tracer stage). The total oxygen concentration profile is also depicted in the figure. Hence, one can follow both the growth of the zirconia scale and α_Zr(O) layer. The shape of the ¹⁸O concentration profiles corresponds to the typical profiles for systems where diffusion takes place only by lattice diffusion. Notice that the ¹⁸O distribution profile would be very different in the case of short-circuit contribution [12].

Fig. 2

EKINOX-Zr calculation: two sets of distribution profiles of O, ¹⁶O, and ¹⁸O plotted for an oxidation of zirconium alloy during 1000 s at 1200 °C, the first stage (500 s) is performed under ¹⁶O₂ while the second stage (500 s) is performed under ¹⁸O₂. Comparison between the analytical solution “erf” (Eq. 38) and the ¹⁸O penetration profile calculated by EKINOX-Zr

Mishin and Borchardt [1] report that the tracer penetration profile can be fitted by the Eq. (35) where $\overline{{\tilde{D}_{\text{O}} }}$ is the average chemical diffusion coefficient of the tracer:

$$C_{18} \left( x \right) = C_{18}^{{{\text{oxide}}/{\text{gaz}}}} {\text{erf}}\left( {\frac{x}{{2\sqrt {\overline{{\tilde{D}_{\text{O}} }} t} }}} \right)$$

(35)

Following this statement, Fig. 2 presents a comparison between the ¹⁸O penetration profile calculated by EKINOX-Zr and the analytical solution of $C_{18} \left( x \right)$ given by Eq. (35). The parameters of the analytical formula were chosen as followed: $C_{18}^{{{\text{oxide}}/{\text{gaz}}}}$ = 0.988 at (equal to the value of $X_{18}^{n}$ given by EKINOX-Zr calculation), $t$  = 550 s (that corresponds to the calculation duration). With an adjusted value of $\overline{{\tilde{D}_{\text{O}} }} =$ 4.5 × 10⁻¹⁰ cm² s⁻¹ in Eq. (35), a good fit is obtained between the analytical solution and the numerical calculation. Following the Eq. (14), it is possible to calculate the values of $\tilde{D}_{\text{O}}$ close to the oxide/gas interface and at the “end” of the ¹⁸O penetration profile. All the values are summarized in Table 3 that clearly shows that the value of $\overline{{\tilde{D}_{\text{O}} }}$ used in the Eq. (35) is consistent with two values of $\tilde{D}_{\text{O}}$ calculated with the Eq. (14).

Table 3 Comparison between the values of $\tilde{D}_{\text{O}}$ calculated with the Eqs. (14) and (35)

Initially, the Eq. (35) is provided in [1] to measure the self-diffusion coefficient $D_{\text{Oeff}}$ of the oxygen tracer. To do so, authors assumed that the self-diffusion coefficient is constant in the whole oxide scale, even if they clearly mentioned that the self-diffusion coefficient depends strongly on the local vacancy concentration. Finally, they conclude that their assumption is available for short-term oxidation (i.e., in the case when the diffusion is largely predominant over the transport). In the case of zirconium oxidation, if we take a look at the Eq. (14), we can see that close to the oxide/gas interface, because of the very small value of the vacancy concentration; the global diffusion coefficient of the diffusion part (first term in RHS of Eq. 14) is equal to 3.12 × 10⁻¹⁵ cm² s⁻¹, whereas the global diffusion coefficient of the transport part (second term in RHS of Eq. 14) is equal to 9.36 × 10⁻⁷ cm² s⁻¹. Thus, the transport part cannot be neglected even for short-time tracer stage. So, we can say that the Eq. (35) allows us to measure only an average chemical diffusion coefficient and not the self-diffusion coefficient.

The numerical model has also been used to imagine a new type of tracer experiments with a short-time ¹⁸O₂ stage surrounded by two ¹⁶O₂ stages of longer durations. Hence, a calculation has been realized considering that the entire oxidation is performed under the natural stage with 100% ¹⁶O₂ except between 200 and 230 s when the tracer stage is performed under 100% ¹⁸O₂.

Figure 3a shows the time evolution of the concentration of ¹⁸O in the first numerical slab of oxide besides the oxide/gas interface, that is to say the evolution of ¹⁸O concentration at the oxide scale surface for this specific calculation. The evolution of ¹⁸O concentration at the surface results from the Eqs. (31) and (32) which differ from the drastic assumption usually made by imposing a constant surface concentration. Figure 3a shows that the enrichment in ¹⁸O at the oxide surface is rapid and the maximum value is reached 30 s after the introduction of ¹⁸O₂ in the atmosphere. Figure 3b shows the evolution and displacement of the peak of ¹⁸O tracer for three different durations. The evolution of the ¹⁸O distribution profiles illustrates both contributions: diffusion and transport. The transport term induced the inward displacement of the ¹⁸O peak position in the oxide scale while the diffusion term induced the spreading of the O peak with time.

Fig. 3

a Time evolution of ¹⁸O concentration in the first numerical slab of oxide. b Isotope diffusion profiles calculated at 240, 280 and 360 s by EKINOX-Zr for an oxidation of zirconium containing a 30 µm prior-oxide layer. The following sequences have been ordered as follows: 200 s under ¹⁶O₂ → 30 s under ¹⁸O₂ → 970 s under ¹⁶O₂. For a better understanding, we chose to set all the oxide/gas interfaces at the same abscissa

If one follows the abscissa of the summit of the ¹⁸O peak as a function of time, we can obtain values of $\tilde{D}_{\text{O}}$ considering that $\Delta x = \sqrt {\tilde{D}_{\text{O}} \Delta t}$ [13]. The values are summarized in Table 4.

Table 4 Values of $\tilde{D}_{\text{O}}$ measured by following the ¹⁸O peak displacement from Fig. 3. The maximum value of the ¹⁸O at the oxide/gas interface is reached for t = 230 s

Table 4 shows the same evolution of $\tilde{D}_{\text{O}}$ than in Table 3. Tables 3 and 4 clearly show that $\tilde{D}_{\text{O}}$ is increasing from the oxide/gas interface to the metal/oxide interface, as for $D_{\text{O}}$. This is due to the increase of the vacancy concentration.

Figure 4 presents a schematic illustration of the evolution of each diffusion coefficient. We can see that all the values are consistent with each other. Furthermore, Fig. 4 shows that the assumption made in [1] on the constant value of the self-diffusion coefficient is not relevant in the case of zirconium oxidation.

Fig. 4

Schematic illustration of the evolution of each diffusion coefficients along the oxide scale

Conclusions

In this work, our aim was to simulate O-tracer experiments at high temperature taking into account a high oxygen solubility limit of oxygen in the metal. A second objective was to obtain a numerical model able to simulate any kind of tracer diffusion during the oxidation, even in non-stationary regimes. The diffusion equations of ¹⁶O and ¹⁸O were deduced from classical diffusion equations taking into account the electrochemical potential gradient across the oxide scale through the large gradient of anionic vacancy concentration. The numerical implementation of these equations in the existing EKINOX-Zr model enabled direct calculation of isotope fluxes without making the approximation of a constant oxygen intrinsic diffusion coefficient which was done in Mishin and Borchardt’s work [1]. Indeed, the oxygen diffusion coefficient is proportional to the local vacancy concentration; therefore, it is best to use a flux equation expressed as a function of a constant diffusion coefficient of the oxygen vacancies. Two examples of numerical calculations illustrated how the shape of the ¹⁸O-tracer distribution profile is governed by both the diffusion by Brownian motion and the transport under the large gradient of defects through the growing oxide scale.

Firstly, the simulation of a classical two-stage tracer experiment is consistent with the analytical solution given in [1]. The average value of $\overline{{\tilde{D}_{\text{O}} }}$ used in the Eq. (35) is clearly included between the two values of $\tilde{D}_{\text{O}}$ obtained from the Eq. (14). Despite what is reported in [1], in the case of zirconium oxidation, the transport contribution cannot be neglected in the isotope flux expression, even for short-term oxidation.

Secondly, a new type of O-tracer experiment is proposed to evaluate oxygen chemical diffusion and transport contributions from the drift of a ¹⁸O peak. In this case, the values of $\tilde{D}_{\text{O}}$ were calculated using the peak drift. The values of $\tilde{D}_{\text{O}}$ obtained by this way and reported in Table 4 are consistent with the one reported in Table 3. By three different ways of calculation, all the values of $\tilde{D}_{\text{O}}$ are consistent with each other. Thus, we can conclude that the “peak shift” method together with Eq. 14 should be used to determine the values of oxygen diffusion coefficient in growing oxide scales.

Next step to improve the EKINOX-Zr will be to express the flux as a function of the gradient of the local chemical potential gradients calculated using the thermodynamic database Zircobase [11]. This improvement would avoid the strong assumption of the constant value of the coefficient of activity in the metal. For example, thermodynamic calculations show that in α_Zr(O) the activity coefficient evolves over two order of magnitude.