© P.N. Alekseev and A.L. Shimkevich, published by EDP Sciences, 2017

1 Introduction

Zirconium alloys are used for fuel cladding in pressured-water reactors (PWR), thanks to a low capture cross-section of thermal neutrons, the good corrosion resistance in water at high temperatures and to the mechanical properties [1]. However, the mechanism of zirconium oxidation is not understood so far despite the great number of experiments carried out during the last 40 years over studying the oxidation of zirconium alloys in the aqueous coolant [2–7]. There is no consensus so far over mechanisms of oxidation of metals in water [8] though this information is very important for developing a new cladding material of fuel elements for PWR.

2 Electrons in ZrO_2−x

The stoichiometric zirconium dioxide (x = 0) without impurities is a single stable oxide of zirconium with ionic bond of atoms which has ε_g ∼ 4.0 eV, and χ_o = 4.0 eV [9,10].

The oxidation of zirconium in PWR coolant according to electrochemical reactions $$\text{Zr}\to {\text{Zr}}^{4+}+4{\text{e}}^{-}\text{,}$$(1) $$4{\text{e}}^{-}+2{\text{H}}_2\text{O}\to 2{\text{O}}^{2-}+2{\text{H}}_2\uparrow \text{,}$$(2) $${\text{Zr}}^{4+}+2{\text{O}}^{2-}\to {\text{ZrO}}_2\text{,}$$(3) is carried out by generating electrons on the interface, Zr(O)/ZrO_2−x, over the reaction (1), by their passing through oxide scale to the interface, ZrO₂/H₂O, for disintegrating water over the reaction (2), and by diffusion of oxygen anion back to the metal through two different oxide layers (see Fig. 1) for oxidizing zirconium over the reaction (3). Then, the total reaction is $$\text{Zr}+2{\text{H}}_2\text{O}\to {\text{ZrO}}_2+2{\text{H}}_2\uparrow \text{.}$$(4)

One can see that the first layer, ZrO, is the source of electrons for the second, ZrO_2−x, due to disintegrating zirconia over the reaction: $$0\to {\text{V}}_{\text{O}}^{2+}+2{\text{e}}^{-}+(1/2){\text{O}}_2\uparrow \text{,}$$(5) when the oxidation potential, , is expressed by a correspondent partial oxygen pressure of zirconia dissociation [12] $$\ln P_{{\mathrm{\text{O}}}_2(\mathrm{\text{Zr}})}=-11.3/k_{\mathrm{\text{B}}}T+19.9\text{.}$$(6)

The oxide scale on zirconium has relatively high density of electrons as well as the density of oxygen vacancies. However, the electronic conductivity in ZrO_2−x film exceeds the anionic one which allows oxidizing of zirconium by oxygen anions diffusing over zirconia vacancies [13].

Thermodynamics of the reaction (5) can be expressed by the following dependence [14]: $${\epsilon }_{\mathrm{\text{F}}}={\mu }_{\mathrm{\text{o}}}-{\mu }_{\mathrm{\text{v}}}(x)/2-(k_{\mathrm{\text{B}}}T/4)\cdot \ln P_{{\mathrm{\text{O}}}_2}\text{,}$$(7) where the electrochemical potential of oxygen vacancies is expressed by the equation of ideal solution $${\mu }_{\mathrm{\text{v}}}(x)=k_{\mathrm{\text{B}}}T\cdot \ln [x/(1-x)]\text{,}$$(8)and [15,16] $${\mu }_{\mathrm{\text{o}}}(T)=-5.10+0.29k_{\mathrm{\text{B}}}T\text{.}$$(9)

Then, it is easy to define Fermi level, ε_F, in zirconia on the interface Zr/ZrO_2−x using the equations (6)–(9). This level is shown in Figure 2.

One can see that in zirconia contacting zirconium, Fermi level is shifted off the middle of band-gap to the conduction band where quasi-free electrons (full blue line in Fig. 2) appear [17]. Their molar concentration [e⁻] is given by Fermi-Dirac statistics, which can be simplified to Maxwell-Boltzmann one [18]: $$[{\text{e}}^{-}]=N_{\mathrm{\text{A}}}\cdot \exp [({\epsilon }_{\mathrm{\text{F}}}-{\epsilon }_{\mathrm{\text{c}}})/k_{\mathrm{\text{B}}}T]\text{.}$$(10)

It follows that $$x=x_{\mathrm{\text{o}}}+(1/2)\cdot \exp [({\epsilon }_{\mathrm{\text{F}}}-{\epsilon }_{\mathrm{\text{c}}})/k_{\mathrm{\text{B}}}T]\text{,}$$(11) and $$[{\mathrm{\text{V}}}_{\mathrm{\text{O}}}^{2+}]=N_{\mathrm{\text{A}}}\cdot x_{\mathrm{\text{o}}}+[{\mathrm{\text{e}}}^{-}]/2\text{.}$$(12)

Substituting (6), (8), (9), and (11) in (7), we find the first boundary condition $${\epsilon }_{\mathrm{\text{Fz}}}=-2.19-2.87k_{\mathrm{\text{B}}}T\text{,}$$(13) in the oxide scale contacting zirconium cladding and the second $${\epsilon }_{\mathrm{\text{Fw}}}=-3.72+0.92k_{\mathrm{\text{B}}}T\text{,}$$(14)in the oxide scale contacting water whose oxidation potential can be expressed by the equivalent oxygen pressure [11] $$\ln P_{{\mathrm{\text{O}}}_2(\mathrm{\text{w}})}=-5.53/k_{\mathrm{\text{B}}}T+20.7\text{.}$$(15)

Equations (14) and (15) define the variation of the concentrations (10) and (12) in the dense part of oxide scale from 10²¹ to 10¹⁰ mol⁻¹ for electrons and from 5 × 10²⁰ to 6 × 10¹⁸ mol⁻¹ for oxygen vacancies at 650 K.

Fig. 1 The composition profile of dense part of the oxide scale (≤2 mkm [11]) on the surface of Zircaloy-4 (SRA) taken across the metal/oxide interface after 90-day testing in liquid water heated up to 360 °C [1]; the dotted vertical lines separate “black zircon”, ZrO, from zirconium and dense hypo-stoichiometric zirconia, ZrO2−x.

Fig. 2 The electronic band structure of ZrO2−x with free electrons in the conduction band (full blue line) and Fermi level, εFz, (red line) expressed by equation (13) for 650 K.

3 The location of black zircon

One can see in equation (11) and Figure 2 that Fermi level in ZrO_2−x is equal to −2.31 eV in contact with zirconium at 650 K. Thus, χ = 2.31 eV for the dense part of oxide scale (DPOS) is less than χ_z = 4.0 eV for the metal [19]. Therefore, electrons pass in zirconium from ZrO_2−x, recharging the metal negatively and enriching DPOS interface region by oxygen vacancies, , due to the positive Δ_dl: $${\mathrm{\Delta }}_{\mathrm{\text{dl}}}=({\chi }_{\mathrm{\text{z}}}-\chi )/\mathrm{\text{e}}=1.69\mathrm{V}\text{.}$$(16)

The [] distribution in a diffusive part of the double electric layer (dl) is described by equation [20] $${\mu }_{\mathrm{\text{v}}}(\phi )-2\mathrm{\text{e}}\phi ={\mu }_{\mathrm{\text{v}}}(0)\text{.}$$(17)

Substituting (8) in (17), we obtain x_dl at the oxide side of Zr/ZrO_2−x interface: $$x_{\mathrm{\text{dl}}}={[1+x_{\mathrm{\text{o}}}^{-1}\exp (-{\mathrm{\Delta }}_{\mathrm{\text{dl}}}/k_{\mathrm{\text{B}}}T)]}^{-1}\sim 1\text{,}$$(18) i.e. the “black zircon”, ZrO shown in Figure 1 [1]. The thickness of this layer is defined by Debye length L_D = (k_BT/4πe²n_v)^1/2 which is less than 10 nm [13].

The investigation of DPOS on zirconium cladding by analytic tools [2,3] has disclosed ZrO phase between metal and ZrO_2−x layer at the initial oxidation of zirconium alloys by the aqueous coolant. They have shown that the properties of “black zircon” are more similar to zirconium than to zirconia [4]. It means that the last grows at the ZrO/ZrO₂ interface.

4 Growing DPOS

The oxidation rate of zirconium in water over the reaction (2) is characterized by the following dependence on temperature [21] $$R=151\exp (-1.47/k_{\mathrm{\text{B}}}T)\mathrm{.}$$(19)

Obviously, the interface of DPOS and water is the vacancies and electrons sink that arise on the other side of DPOS on contacting zirconium. Then, the sum of their specific currents [13,22] $$j_{\mathrm{\text{e}}}=u_{\mathrm{\text{e}}}{n}_{\mathrm{\text{e}}}\frac{d{\epsilon }_{\mathrm{\text{F}}}}{dy}\text{,}$$(20) $$j_{\mathrm{\text{v}}}=-u_{\mathrm{\text{v}}}{n}_{\mathrm{\text{v}}}\left(\frac{d{\mu }_{\mathrm{v}}}{dy}-2\frac{d{\epsilon }_{\mathrm{\text{c}}}}{dy}\right)\text{,}$$(21) is equal to zero in the range of 0 ≤ y ≤ h when u_e ≫ u_v under conditions $$\frac{d{j}_{\mathrm{\text{e}}}}{dy}=\frac{d{j}_{\mathrm{\text{v}}}}{dy}=0\text{,}$$(22) $$\frac{d^2{\epsilon }_{\mathrm{\text{c}}}}{d{y}^2}=-\alpha [(2n_{\mathrm{\text{v}}}-n_{\mathrm{\text{e}}})/K{N}_{\mathrm{\text{A}}}-2x_{\mathrm{\text{o}}}]\text{,}$$(23) $${\epsilon }_{\mathrm{\text{c}}|y=0}={\epsilon }_{\mathrm{\text{c}}}(0)\text{,}$$(24) $${\epsilon }_{\mathrm{\text{F}}|y=0}={\epsilon }_{\mathrm{\text{Fz}}}\text{,}$$(25) $${\epsilon }_{\mathrm{\text{F}}|y=h}={\epsilon }_{\mathrm{\text{Fw}}}\text{,}$$(26) $$x_{|y=h}=x_{\mathrm{\text{o}}}\text{.}$$(27)

Substituting (20), (21), and (23) in (22), we obtain the steady-state boundary task for three functions: x(y), ε_c(y), and η(y) = n_e/KN_A under boundary conditions (24)–(27).

For the strong inequality: γx_o ≫ 1 where γ ≡ αh²/k_BT, we simplify the task (22)–(27) and find its solution in the form of power series: $$\eta (\xi )={\displaystyle \sum _{k=0}^{\infty }}{a}_k{\xi }^k\text{,}$$(28) $${\epsilon }_{\mathrm{\text{c}}}(\xi )=k_{\mathrm{\text{B}}}T{\displaystyle \sum _{k=0}^{\infty }}{c}_k{\xi }^k\text{,}$$(29) $$x(\xi )={\displaystyle \sum _{k=0}^{\infty }}{b}_k{\xi }^k\text{,}$$(30) with ξ = y/h. This solution implementing the equality of the specific currents (20) and (21) at any y in the ZrO_2−x layer gives the expression for R in the final form $$R=-M_{\mathrm{\text{o}}}K{k}_{\mathrm{\text{B}}}T{u}_{\mathrm{\text{e}}}(a_1+a_0{c}_1)/8h\mathrm{e}\mathrm{.}$$(31)

Substituting (28)–(30) in (22) and (23) under boundary condition (24)–(27), we obtain $$a_1=-a_0(1+c_1)/2\text{,}$$(32) $$c_1=a_0(2\upsilon -1)/[4x_{\mathrm{\text{o}}}+a_0(\upsilon +1)]\text{,}$$(33) where a₀ = 0.057 exp(−0.19/k_BT) and υ = u_e/u_v.

In presenting u_e and u_v by equation [23] $$u_{\mathrm{\text{i}}}=3\mathrm{e}{D}_{\mathrm{\text{i}}}/2k_{\mathrm{\text{B}}}T\text{,}$$(34) we are transforming (31) to $$R\sim 9M_{\mathrm{\text{o}}}K{D}_{\mathrm{\text{v}}}{a}_0/32h\text{,}$$(35)where D_v is presented by [24] as the temperature dependence $$D_{\mathrm{\text{v}}}=1.50\times {10}^{-6}\exp (-1.28/k_{\mathrm{\text{B}}}T)\text{,}$$(36)does (35) equal to (19) for h = 1 mkm.

Thus, growing the oxide scale on the zirconium cladding of fuel elements is being defined by the product of electronic density on the ZrO/ZrO₂ interface and the mobility of oxygen vacancies in the dense ZrO_2−x layer. Decreasing any of them we will inhibit the oxide corrosion of zirconium cladding.

5 Discussion of results

After reaching the critical thickness of 2 mkm, DPOS breaks off from the zirconium cladding surface and the rate of metal corrosion dramatically increases as shown in Figure 3 [5].

This process is known as the “breakaway” oxidation [5] due to opening the unprotected zirconium surface for oxidizers that increases the oxidative corrosion as shown in Figure 3. At the same time, the mechanism of such the breakaway so far is under debate in the scientific literature and the effect of additives on this process is not understood.

Since the oxidation rate (35) depends on the maximal electronic density in DPOS (at ε_Fz) and the mobility of oxygen vacancies there, it is necessary to inhibit the electronic conductivity in the oxide scale and to decrease the mobility of oxygen vacancies. It can be practiced by adding a metal of lesser valence than zirconium [8]. Such the addition stabilizes a high-temperature cubic phase of zirconia as the solid electrolyte with electronic conductivity practically equal to zero [13,25].

For yttrium-stabilized zirconia (YSZ) at its addition of ≤9 mol%, there is no positive effect because the band gap is the same (∼4.0 eV) and the molar density of electrons in the oxide scale is in the same range of 10²¹ to 10¹⁰ mol⁻¹ (see above) at 650 K but the molar density of oxygen vacancies is very high ∼10²² mol⁻¹ [26].

In contrast, ε_g of calcium-stabilized zirconia (CSZ) is equal to ∼ 5.6 eV [25] at ≤15 mol% of the additive that inhibits the electronic conductivity in the oxide scale for the same μ_o(T) (9) and the dimensionless content x_s ∼ 0.1 of oxygen vacancies because ε_Fz (13) becomes appreciably lower than ε_c = −1.2 eV (see Fig. 4 in comparison with Fig. 2): $${\epsilon }_{\mathrm{\text{Fz}}}=-2.28+1.08k_{\mathrm{\text{B}}}T\text{.}$$(37)

One can find from (10) and (37) that at 650 K, the maximal density of electrons in CSZ is less than 10¹⁶ mol⁻¹.

Then, one can find a₀ = 2.94 exp(−1.08/k_BT) by using equations (10) and (32)–(37). For the ratio a₀(υ + 1) ≫ 4x_o, we will obtain R(T) at zirconium oxidation via CSZ layer of 1 mkm in the form $$R=7.03\times {10}^4\exp (-2.36/k_{\mathrm{\text{B}}}T)\text{.}$$(38)

By comparing this equation with (19), one can conclude that the oxidation rate of zirconium cladding with the surface thin layer of the alloy, Zr–Ca(15%), on a few orders of magnitude less than the usual zirconium oxidation. Then, the oxide scale on such the cladding of fuel elements in PWR will grow up to 2 mkm during 10⁵ h instead of 10³ h for the up-to-date cladding.

Obviously, this should be checked by a corrosion test of such cladding.

Fig. 3 The zirconium oxidation; the blue line shows the weight gain that would be expected for a material with a protective barrier layer which breaks down and cyclic oxidation is characterized by the overall linear growth [5].

Fig. 4 The electronic band structure of CSZ with Fermi level, εFz, (red line) expressed by equation (37) for 650 K.

6 Conclusions

The electronic model of the oxide scale on zirconium cladding of the fuel elements in PWR is developed for studying the role of electrons in the zirconium oxidation by the aqueous coolant.

The concentrations correlation of electrons and oxygen vacancies in forming the hypo-stoichiometric zirconia on the zirconium cladding transforms zirconia into a mixed conductor. However, the higher mobility of electrons in this conductor does their concentration by the dominant factor in zirconium oxidation.

The two-layer oxide scale is the result of the action of double electric layer in the Zr/ZrO_2−x interface which enriches ZrO_2−x by oxygen vacancies up to forming the black zircon, ZrO, and facilitates the penetration of zirconium atoms into this layer.

It is possible that the oxidation rate may be inhibited by decreasing the electronic conductivity in the oxide scale. For this, calcium should be implanted into the near-surface layer of zirconium cladding for forming the calcium-stabilized zirconia on its surface.

Nomenclature

e⁻: electron charge (e)

[e⁻]: the molar concentration of electrons in ZrO_2−x (mol⁻¹)

h: the thickness of ZrO_2−x in the dense part of oxide scale (m)

j_i: the specific current of i-particles (e/m² s)

K: the dimensional unit (4.61 × 10⁴ mol/m³)

k_B: Boltzmann constant (8.62 × 10⁻⁵ eV/K)

L_D: Debye length of oxygen vacancies (nm)

M_o: the zirconia molar mass (0.123 kg/mol)

N_A: Avogadro number (6.02 × 10²³ mol⁻¹)

n_e: the volume concentration of electrons in ZrO_2−x equal to K[e⁻] (m⁻³)

n_v: the volume concentration of oxygen vacancies in ZrO_2−x equal  (m⁻³)

: the equivalent oxygen pressure (MPa)

: the same for zirconia dissociation

R: the oxidation rate of zirconium in water (kg/m² s)

T: Kelvin temperature (K)

u_i: the mobility of i-particle (m²/s V)

: the charged oxygen vacancy in hypo-stoichiometric zirconia, ZrO_2−x (e)

: the molar concentration of oxygen vacancies in ZrO_2−x (mol⁻¹)

x: the dimensionless degree of ZrO_2−x non-stoichiometry: (x < 0) for hyper-stoichiometric state and (x > 0) for the hypo-stoichiometric one

x_o: a background non-stoichiometry (∼10⁻⁵)

x_s: the dimensionless content of oxygen vacancies off the metal additive

y: the coordinate in the layer of ZrO_2−x (m)

α: the dielectric parameter of zirconia (1.74 eV/nm²)

ε_c: the bottom of conduction band (eV)

ε_F: Fermi level in the band gap of non-stoichiometric dioxide (eV)

ε_g: the band gap of dioxide (eV)

ε_v: the top of valence band (eV)

Δ_dl: the potential of double electric layer (V)

μ_o(T): the electrochemical potential of stoichiometric zirconia (eV)

μ_v(x): the electrochemical potential of oxygen vacancies in hypo-stoichiometric ZrO_2−x as a function of x (eV)

φ: the electric potential in double layer (V)

χ_o: the work function of stoichiometric dioxide (eV)

χ: the work function of non-stoichiometric dioxide (eV)

Acknowledgments

The authors would like to thank their colleagues for active discussion on the aspects of electronic model in growing the oxide scale on zirconium cladding of fuel elements for PWR.

References

All Figures

	Fig. 1 The composition profile of dense part of the oxide scale (≤2 mkm [11]) on the surface of Zircaloy-4 (SRA) taken across the metal/oxide interface after 90-day testing in liquid water heated up to 360 °C [1]; the dotted vertical lines separate “black zircon”, ZrO, from zirconium and dense hypo-stoichiometric zirconia, ZrO2−x.
In the text

	Fig. 2 The electronic band structure of ZrO2−x with free electrons in the conduction band (full blue line) and Fermi level, εFz, (red line) expressed by equation (13) for 650 K.
In the text

	Fig. 3 The zirconium oxidation; the blue line shows the weight gain that would be expected for a material with a protective barrier layer which breaks down and cyclic oxidation is characterized by the overall linear growth [5].
In the text

	Fig. 4 The electronic band structure of CSZ with Fermi level, εFz, (red line) expressed by equation (37) for 650 K.
In the text