Doping in the Valley of Hydrogen Solubility: A Route to Designing Hydrogen-Resistant Zirconium Alloys

Hydrogen pickup and embrittlement pose a challenging safety limit for structural alloys used in a wide range of infrastructure applications, including zirconium alloys in nuclear reactors. Previous experimental observations guide the empirical design of hydrogen-resistant zirconium alloys, but the underlying mechanisms remain undecipherable. Here, we assess two critical prongs of hydrogen pickup through the ${\mathrm{ZrO}}_2$ passive film that serves as a surface barrier of zirconium alloys; the solubility of hydrogen in it—a detrimental process—and the ease of ${\mathrm{H}}_2$ gas evolution from its surface—a desirable process. By combining statistical thermodynamics and density-functional-theory calculations, we show that hydrogen solubility in ${\mathrm{ZrO}}_2$ exhibits a valley shape as a function of the chemical potential of electrons, ${\mu }_e$. Here, ${\mu }_e$, which is tunable by doping, serves as a physical descriptor of hydrogen resistance based on the electronic structure of ${\mathrm{ZrO}}_2$. For designing zirconium alloys resistant against hydrogen pickup, we target either a dopant that thermodynamically minimizes the solubility of hydrogen in ${\mathrm{ZrO}}_2$ at the bottom of this valley (such as Cr) or a dopant that maximizes ${\mu }_e$ and kinetically accelerates proton reduction and ${\mathrm{H}}_2$ evolution at the surface of ${\mathrm{ZrO}}_2$ (such as Nb, Ta, Mo, W, or P). Maximizing ${\mu }_e$ also promotes the predomination of a less-mobile form of hydrogen defect, which can reduce the flux of hydrogen uptake. The analysis presented here for the case of ${\mathrm{ZrO}}_2$ passive film on Zr alloys serves as a broadly applicable and physically informed framework to uncover doping strategies to mitigate hydrogen embrittlement also in other alloys, such as austenitic steels or nickel alloys, which absorb hydrogen through their surface oxide films.

Figure 1

(a) Calculated Kröger-Vink diagram for the undoped monoclinic ${\mathrm{ZrO}}_2$ at 1200 K. Point defects are represented by the Kröger-Vink notation, $n_c$ denotes conduction-band electrons, and $p_v$ denotes valence-band holes. Only defects with concentrations [$d$] greater than ${10}^{-10}$ per ${\mathrm{ZrO}}_2$ are shown. The dashed lines are guide for the eye showing the slopes of ($1/6$) and ($-1/6$). (b) Calculated chemical potential of electrons that achieves charge neutrality in undoped monoclinic ${\mathrm{ZrO}}_2$ as a function of $T$ and $P_{{\mathrm{O}}_2}$.

Figure 2

Calculated Kröger-Vink diagram for undoped (a) and hydrogen-doped (b) monoclinic ${\mathrm{ZrO}}_2$. Only the defects with concentration [$d$] greater than ${10}^{-16}$ per ${\mathrm{ZrO}}_2$ are shown. Point defects are denoted by the Kröger-Vink notation, and $p_v$ denotes the valence-band holes. Hydrogen defects are shown by dashed-dotted lines. In (b), the results are evaluated at $P_{{\mathrm{H}}_2\mathrm{O}}=1\mathrm{atm}$ and using Eq. (3) for the chemical potential of hydrogen. (c) The relaxed structure of the complex formed between a hydrogen and a zirconium vacancy in the 3- charge state, ${\mathrm{H}}_{\mathrm{Zr}}^{\prime \prime \prime }$. (d) The relaxed structure of the interstitial proton, ${\mathrm{H}}_i^{•}$. In (c) and (d), green (large), red (medium), and blue (small) balls represent zirconium, oxygen, and hydrogen, respectively. The visualizations in (c) and (d) are generated using the software VESTA 51].

Figure 3

(a) The calculated solubility of hydrogen in monoclinic ${\mathrm{ZrO}}_2$ as a function of the electron chemical potential (${\mu }_e$) at 600, 900, and 1200 K. (b) On each solubility curve, we map 11 elements for which we explicitly calculate the value of ${\mu }_e$ that they establish at their solubility limit in monoclinic ${\mathrm{ZrO}}_2$ when they codope it with hydrogen. The case of ${\mathrm{ZrO}}_2$ doped with hydrogen but without any other elements is marked with “Zr” as a reference point. The calculations in (a) and (b) are performed assuming $P_{{\mathrm{O}}_2}=1\mathrm{atm}$ and $P_{{\mathrm{H}}_2\mathrm{O}}=1\mathrm{atm}$, which are representative of the oxygen-rich part of ${\mathrm{ZrO}}_2$ in contact with water. The valley shape of the solubility curve is retained even at lower $P_{{\mathrm{O}}_2}$ conditions as we show in Supplemental Material, Sec. 12 [17]. (c) The formation free energy at 600 K of interstitial hydrogen, ${\mathrm{H}}_i$, and the complex formed between hydrogen and a zirconium vacancy, ${\mathrm{H}}_{\mathrm{Zr}}$, as a function of ${\mu }_e$. Only the charge states of these two defects with the lowest formation free energy are shown. The thick line traces the lowest formation free energy over all hydrogen defects as a function of ${\mu }_e$. The two vertical dashed lines correspond to the range of ${\mu }_e$ accessible by codoping with hydrogen and a 3d transition metal, as spanned in Fig. 4 from 0.54 to 1.23 eV.

Figure 4

(a) Experimentally determined hydrogen pickup fraction in zirconium as a function of alloying across the 3d transition-metal row. The data are based on zirconium alloys containing 1%–5% of a given transition metal and at temperatures between 573 and 773 K. The shading indicates the hydrogen pickup fraction during the oxidation of pure zirconium metal. The figure is adapted from the data in Ref. [14]. (b) The ratio of hydrogen solubility in monoclinic ${\mathrm{ZrO}}_2$ doped with 3d transition metals to that in the undoped oxide. The doping level for each metal is its equilibrium solubility in $M{\mathrm{ZrO}}_2$ under the thermodynamic conditions of interest. These solubilities range between ${10}^{-3}$ and ${10}^{-14}$ per ${\mathrm{ZrO}}_2$ unit formula. (c) The chemical potential of electrons, ${\mu }_e$, in monoclinic ${\mathrm{ZrO}}_2$ codoped with hydrogen and a transition metal. The horizontal dashed line indicates the value of the ${\mu }_e$ in the oxide doped with hydrogen only, without any transition metal. In (b) and (c), the data are evaluated at $T=600\mathrm{K}$, $P_{{\mathrm{O}}_2}=1\mathrm{atm}$, and $P_{{\mathrm{H}}_2\mathrm{O}}=1\mathrm{atm}$. The effect of varying $T$ and $P_{{\mathrm{O}}_2}$ on the volcano shape in (b) is discussed in Supplemental Material, Fig. S28 [17].

Figure 5

(a) The ratio of the electronic conductivity $\sigma (T,{\mu }_e)$ in $M{\mathrm{ZrO}}_2$ to ${\sigma }^{\ast }(T)$. ${\sigma }^{\ast }(T)$ is the electronic conductivity in the same oxide at the electron chemical potential ${\mu }_e$ that minimizes hydrogen solubility [the dip of the valley in Fig. 3for each temperature]. (b) Mapping of five alloying elements on the hydrogen solubility curves that can maximize ${\mu }_e$ in ${\mathrm{ZrO}}_2$. On each isotherm the top data are evaluated at a fixed doping concentration by the alloying element at ${10}^{-4}$ per ${\mathrm{ZrO}}_2$ unit formula, the middle data at ${10}^{-5}$, and finally the bottom data at ${10}^{-6}$. The calculations in (a) and (b) are performed assuming $P_{{\mathrm{O}}_2}=1\mathrm{atm}$ and $P_{{\mathrm{H}}_2\mathrm{O}}=1\mathrm{atm}$, which are representative of the oxygen-rich part of ${\mathrm{ZrO}}_2$ in contact with water.

Figure 6

The calculated hydrogen solubility in monoclinic ${\mathrm{ZrO}}_2$ as a function of electron chemical potentia, ${\mu }_e$ and temperature $T$. The temperature-dependent loci of the minimum hydrogen solubility and maximum attainable ${\mu }_e$ are marked with triangles and circles, respectively. The two dominant hydrogen defects are also marked in the ranges of ${\mu }_e$ over which they predominate. The zones marked by (i), (ii), and (iii) represent the three different doping strategies which are discussed in the text. These calculations are performed assuming $P_{{\mathrm{O}}_2}=1\mathrm{atm}$ and $P_{{\mathrm{H}}_2\mathrm{O}}=1\mathrm{atm}$, which are representative of the oxygen-rich part of ${\mathrm{ZrO}}_2$ in contact with water.