Issue 
EPJ Nuclear Sci. Technol.
Volume 3, 2017



Article Number  19  
Number of page(s)  5  
DOI  https://doi.org/10.1051/epjn/2017014  
Published online  16 June 2017 
https://doi.org/10.1051/epjn/2017014
Regular Article
A role of electrons in zirconium oxidation
NRC Kurchatov Institute,
1, Kurchatov Sq.,
Moscow
123182, Russia
^{*} email: shimkevich_al@nrcki.ru
Received:
22
April
2016
Received in final form:
26
April
2017
Accepted:
29
May
2017
Published online: 16 June 2017
Growing the oxide scale on the zirconium cladding of fuel elements in pressuredwater reactors (PWR) is caused by the current of oxygen anions off the waterside to the metal through the layer of zirconia and by the strictly equal inverse electronic current. This process periodically speeds up the corrosion of the zirconium cladding in the aqueous coolant due to the breakaway of the dense part of oxide scale when its thickness reaches 2 mkm. It is shown that the electronic resistivity of zirconia is not limiting the zirconium oxidation at working temperatures. For gaining this limitation, a metal of lesser valence than zirconium has to be added to this oxide scale up to 15%. Then, oxygen vacancies arise in the complex zirconia, increase its bandgap, and thus, sharply decrease the electronic conductivity and form the solid oxide electrolyte whose growth is inhibited in contact with water at working temperatures of PWR.
© P.N. Alekseev and A.L. Shimkevich, published by EDP Sciences, 2017
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Zirconium alloys are used for fuel cladding in pressuredwater reactors (PWR), thanks to a low capture crosssection 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, ZrO_{2}/H_{2}O, 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] $$\mathrm{ln}\text{\hspace{0.17em}}{P}_{{\mathrm{\text{O}}}_{2}(\mathrm{\text{Zr}})}=11.3/{k}_{\mathrm{\text{B}}}T+19.9\text{.}$$(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]: $${\epsilon}_{\mathrm{\text{F}}}={\mu}_{\mathrm{\text{o}}}{\mu}_{\mathrm{\text{v}}}(x)/2({k}_{\mathrm{\text{B}}}T/4)\cdot \mathrm{ln}\text{\hspace{0.17em}}{P}_{{\mathrm{\text{O}}}_{2}}\text{,}$$(7) where the electrochemical potential of oxygen vacancies is expressed by the equation of ideal solution $${\mu}_{\mathrm{\text{v}}}(x)={k}_{\mathrm{\text{B}}}T\cdot \mathrm{ln}\text{\hspace{0.17em}}[x/(1x)]\text{,}$$(8)and [15,16] $${\mu}_{\mathrm{\text{o}}}(T)=5.10+0.29{k}_{\mathrm{\text{B}}}T\text{.}$$(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 bandgap to the conduction band where quasifree electrons (full blue line in Fig. 2) appear [17]. Their molar concentration [e^{−}] is given by FermiDirac statistics, which can be simplified to MaxwellBoltzmann one [18]: $$[{\text{e}}^{}]={N}_{\mathrm{\text{A}}}\cdot \mathrm{exp}[({\epsilon}_{\mathrm{\text{F}}}{\epsilon}_{\mathrm{\text{c}}})/{k}_{\mathrm{\text{B}}}T]\text{.}$$(10)
It follows that $$x={x}_{\mathrm{\text{o}}}+(1/2)\cdot \mathrm{exp}[({\epsilon}_{\mathrm{\text{F}}}{\epsilon}_{\mathrm{\text{c}}})/{k}_{\mathrm{\text{B}}}T]\text{,}$$(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 $${\epsilon}_{\mathrm{\text{Fz}}}=2.192.87{k}_{\mathrm{\text{B}}}T\text{,}$$(13) in the oxide scale contacting zirconium cladding and the second $${\epsilon}_{\mathrm{\text{Fw}}}=3.72+0.92{k}_{\mathrm{\text{B}}}T\text{,}$$(14)in the oxide scale contacting water whose oxidation potential can be expressed by the equivalent oxygen pressure [11] $$\mathrm{ln}\text{\hspace{0.17em}}{P}_{{\mathrm{\text{O}}}_{2}(\mathrm{\text{w}})}=5.53/{k}_{\mathrm{\text{B}}}T+20.7\text{.}$$(15)
Equations (14) and (15) define the variation of the concentrations (10) and (12) in the dense part of oxide scale from 10^{21} to 10^{10} mol^{−1} for electrons and from 5 × 10^{20} to 6 × 10^{18} 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 Zircaloy4 (SRA) taken across the metal/oxide interface after 90day testing in liquid water heated up to 360 °C [1]; the dotted vertical lines separate “black zircon”, ZrO, from zirconium and dense hypostoichiometric zirconia, ZrO_{2−x}. 
Fig. 2 The electronic band structure of ZrO_{2−x} with free electrons in the conduction band (full blue line) and Fermi level, ε_{Fz}, (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}: $${\mathrm{\Delta}}_{\mathrm{\text{dl}}}=({\chi}_{\mathrm{\text{z}}}\chi )/\mathrm{\text{e}}=1.69\text{\hspace{0.17em}}\mathrm{V}\text{.}$$(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: $${x}_{\mathrm{\text{dl}}}={[1+{x}_{\mathrm{\text{o}}}^{1}\mathrm{exp}({\mathrm{\Delta}}_{\mathrm{\text{dl}}}/{k}_{\mathrm{\text{B}}}T)]}^{1}\sim 1\text{,}$$(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_{B}T/4πe^{2}n_{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/ZrO_{2} interface.
4 Growing DPOS
The oxidation rate of zirconium in water over the reaction (2) is characterized by the following dependence on temperature [21] $$R=151\text{\hspace{0.17em}}\mathrm{exp}(1.47/{k}_{\mathrm{\text{B}}}T)\mathrm{.}$$(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) $$\frac{{d}^{2}{\epsilon}_{\mathrm{\text{c}}}}{d{y}^{2}}=\alpha [(2{n}_{\mathrm{\text{v}}}{n}_{\mathrm{\text{e}}})/K{N}_{\mathrm{\text{A}}}2{x}_{\mathrm{\text{o}}}]\text{,}$$(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 steadystate 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 ≡ αh^{2}/k_{B}T, 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) $${c}_{1}={a}_{0}(2\upsilon 1)/[4{x}_{\mathrm{\text{o}}}+{a}_{0}(\upsilon +1)]\text{,}$$(33) where a_{0} = 0.057 exp(−0.19/k_{B}T) and υ = u_{e}/u_{v}.
In presenting u_{e} and u_{v} by equation [23] $${u}_{\mathrm{\text{i}}}=3\mathrm{e}{D}_{\mathrm{\text{i}}}/2{k}_{\mathrm{\text{B}}}T\text{,}$$(34) we are transforming (31) to $$R\sim 9{M}_{\mathrm{\text{o}}}K{D}_{\mathrm{\text{v}}}{a}_{0}/32h\text{,}$$(35)where D_{v} is presented by [24] as the temperature dependence $${D}_{\mathrm{\text{v}}}=1.50\times {10}^{6}\mathrm{exp}(1.28/{k}_{\mathrm{\text{B}}}T)\text{,}$$(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/ZrO_{2} 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 hightemperature cubic phase of zirconia as the solid electrolyte with electronic conductivity practically equal to zero [13,25].
For yttriumstabilized 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 10^{21} to 10^{10} mol^{−1} (see above) at 650 K but the molar density of oxygen vacancies is very high ∼10^{22} mol^{−1} [26].
In contrast, ε_{g} of calciumstabilized 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): $${\epsilon}_{\mathrm{\text{Fz}}}=2.28+1.08{k}_{\mathrm{\text{B}}}T\text{.}$$(37)
One can find from (10) and (37) that at 650 K, the maximal density of electrons in CSZ is less than 10^{16} mol^{−1}.
Then, one can find a_{0} = 2.94 exp(−1.08/k_{B}T) by using equations (10) and (32)–(37). For the ratio a_{0}(υ + 1) ≫ 4x_{o}, we will obtain R(T) at zirconium oxidation via CSZ layer of 1 mkm in the form $$R=7.03\times {10}^{4}\text{\hspace{0.17em}}\mathrm{exp}(2.36/{k}_{\mathrm{\text{B}}}T)\text{.}$$(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 10^{5} h instead of 10^{3} h for the uptodate 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, ε_{Fz}, (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 hypostoichiometric 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 twolayer 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 nearsurface layer of zirconium cladding for forming the calciumstabilized zirconia on its surface.
Nomenclature
[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 iparticles (e/m^{2} s)
K: the dimensional unit (4.61 × 10^{4} mol/m^{3})
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 × 10^{23} 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/m^{2} s)
u_{i}: the mobility of iparticle (m^{2}/s V)
: the charged oxygen vacancy in hypostoichiometric 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} nonstoichiometry: (x < 0) for hyperstoichiometric state and (x > 0) for the hypostoichiometric one
x_{o}: a background nonstoichiometry (∼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/nm^{2})
ε_{c}: the bottom of conduction band (eV)
ε_{F}: Fermi level in the band gap of nonstoichiometric 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 hypostoichiometric 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 nonstoichiometric 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.
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Cite this article as: Pavel N. Alekseev, Alexander L. Shimkevich, A role of electrons in zirconium oxidation, EPJ Nuclear Sci. Technol. 3, 19 (2017)
All Figures
Fig. 1 The composition profile of dense part of the oxide scale (≤2 mkm [11]) on the surface of Zircaloy4 (SRA) taken across the metal/oxide interface after 90day testing in liquid water heated up to 360 °C [1]; the dotted vertical lines separate “black zircon”, ZrO, from zirconium and dense hypostoichiometric zirconia, ZrO_{2−x}. 

In the text 
Fig. 2 The electronic band structure of ZrO_{2−x} with free electrons in the conduction band (full blue line) and Fermi level, ε_{Fz}, (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, ε_{Fz}, (red line) expressed by equation (37) for 650 K. 

In the text 
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