Simulating the carbon distribution in zirconium alloy spent fuel cladding

The diffusivity values of C in pure Zr and in Zr containing dissolved O (Zr(O)) were determined in a temperature range of 623–1023 K for Zr and 923–1123 K for Zr(O) from depth profiles of C obtained by glow-discharge emission spectroscopy. The C diffusivity in Zr(O) decreased as the O concentration in Zr increased and that for Zr containing 15 at % O was two orders of magnitude smaller than that for pure Zr. One-dimensional numerical calculations of Fick’s diffusion equation with a Soret effect indicated various non-uniform distributions of C in a 5-mm-thick Zr matrix under a temperature gradient of 573 to 773 K for 3 years, assuming a heat of transport of − 1.5 to + 1.5 eV. Isothermal annealing at 773 K for 10 years could result in a uniform distribution, whereas dissolution of O in the interstitials of the Zr matrix would hinder C transport through the interstitials. As concentration of O increased in HIP Zr by 15 at%, the diffusivity of C decreased more than two orders of magnitude. For the positive Q∗c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{Q}^{*}}_{c}$$\end{document}, concentration of C slightly segregated at the surface of the cooler side but had maximum peaks at a middle to a higher temperature zone, and depleted at the surface of the hotter side. For the negative Q∗c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{Q}^{*}}_{c}$$\end{document}, concentration of C depleted at the surface of the cooler side and at the middle to the higher temperature zone, and highly segregated at the surface of the hotter side. As concentration of O increased in HIP Zr by 15 at%, the diffusivity of C decreased more than two orders of magnitude. For the positive Q∗c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{Q}^{*}}_{c}$$\end{document}, concentration of C slightly segregated at the surface of the cooler side but had maximum peaks at a middle to a higher temperature zone, and depleted at the surface of the hotter side. For the negative Q∗c\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{Q}^{*}}_{c}$$\end{document}, concentration of C depleted at the surface of the cooler side and at the middle to the higher temperature zone, and highly segregated at the surface of the hotter side.


Introduction
Spent fuel cladding hulls made of Zr alloys will be mechanically compressed into blocks before disposal in a repository in Japan. Therefore, the distribution of radioactive 14 C produced in the Zr matrix is an important factor in estimating the release behavior of 14 C that accompanies the oxidation of the Zr matrix [1]. Based on the conditions in the spent fuel cladding, it is expected that the Zr alloys will be oxidized by coolant water at high temperatures and would absorb O and even H isotopes in the interstitials of the host Zr matrix. 14 C can be produced by neutron irradiation of N impurities [2] and distributed in the Zr cladding matrix by diffusion or precipitation under concentration and temperature gradients during the nuclear reactor operation. Subsequently, 14 C could be relocated by isothermal annealing over time during mid-term storage before hull processing.
In the present study, diffusion and thermotransport data for C, H, N, and O in Zr were revisited and diffusion data for C in Zr containing dissolved oxygen (Zr(O)) were experimentally obtained. Numerical calculations were carried out to simulate distributions of C in the Zr matrix with the transport parameters for C in Zr and Zr(O) with a temperature gradient for various periods.

Diffusion of C in pure Zr and Zr(O)
Five samples were prepared for the diffusion experiments. The first was square plates (12 × 12 × 1.0 mm) of pure commercially available Zr (99.9 wt%), hereinafter called Bulk Zr. Three were disks (∅12 × 1.0 mm) of Zr(O) with oxygen contents of 5, 10, and 15 at%, which were made by a powder metallurgy method with Zr and ZrO 2 powders using a hot isostatic press (HIP) at 175 MPa, 1973 K for 2 h in an Ar atmosphere. These samples are referred to as HIP Zr(5%O), Zr(10%O), and Zr(15%O), respectively. The fifth sample was prepared by using the same method as for the disks but without ZrO 2 powder, and is referred to as HIP Zr. The crystal structures of the HIP samples were confirmed as a single phase of αZr by X-ray diffraction analyses. All sample surfaces were mechanically polished with a fine abrasive paper before diffusion experiments.
The sample surfaces were loaded with C by high-temperature absorption of methane gas at 13.3 kPa and 973 K for 10 min in a glass quartz tube. The sample was annealed in a vacuum (≤ 10 −5 Pa) at a constant temperature ranging 1 3 from 673 to 1173 K for several hours to allow C to diffuse into deeper regions.
After the diffusion experiment, the depth profile of C in the sample was measured by glow-discharge optical emission spectroscopy (GD-OES; GD-Profiler 2, HORIBA, Japan). Argon plasma was discharged at 600 Pa and an output power of 35 W, and the photoelectron-multiplier voltage was set at 700 V. C/Zr and O/Zr were evaluated from intensity of the emission spectra, which were calibrated with reference samples. The depth of the analyzed area (⌀4 mm) was examined separately with the stylus method. The detection limit of C was approximately 0.1% for C/Zr owing to the deviation in the emission spectra of C in Zr.

Thermotransport properties of solute elements in Zr
The flux of solute elements in a host Zr matrix is expressed by Fick's first law [3], where D i (T) is diffusivity at temperature T, i is the concentration of species i in Zr, and Q * i is the heat of transport of species i in Zr. The right-hand term in the brackets on the right-hand side of the equation indicates the transport of species i by diffusion under concentration gradient ∇ i , and the left-hand term describes transport of the species i under temperature gradient ∇T , known as the Soret effect. If Q * > 0 , solute species i moves toward the low temperature side and if Q * < 0 , it moves toward the high temperature side, although this tendency could be prevented by diffusion under the resultant concentration gradient.
The thermotransport properties of the solute elements in various metal matrices are summarized in Table 1. For a Zr matrix, the reported heats of transport for solute species of H, O, and N are positive. Although there are no data available for C in Zr, the heats of transport for C in V (BCC), Fe (BCC), Ni (FCC), and βTi (BCC) are all negative regardless of the matrix crystal structures.

Diffusion and thermotransport simulation
The Tritium Migration Analysis Program, Version 4 (TMAP4) was originally developed to simulate migration of hydrogen isotopes in fusion reactor materials [4]. This program is a one-dimensional solver of Fick's diffusion equation [5], shown in Eq. 2 using space-centered, implicit (time-backward) difference approximations for derivatives, and the method can be applied to wider variety of problems for different solute elements in materials.
∇T For a thermal gradient in Zr alloy cladding, the transport of C in the radial and axial directions in a cross-section could be considered. Because the diffusivity of C in Zr is too small for transport in the axial direction, only transport in the radial direction through a thickness of 0.5 mm was examined under a temperature gradient from 573 K (coolant side) and 773 K (fuel side). The temperature gradient in the radial direction of the cladding was estimated based on typical operation conditions of light water reactors. Assumingf a uniform C concentration through the thickness of the Zr cladding, one-dimensional thermal-and mass-diffusive transport of C in Zr were simulated using diffusion coefficients of C in Zr and Zr(O), and with heat of transport Q * c from − 1.5 to + 1.5 eV. Assuming continuous liberation of C from the carbide region and one-dimensional C penetration in the deeper region by diffusion, a theoretical solution curve was obtained from Fick's diffusion equation (Eq. 2) as [5] ( where 0 is the surface concentration of C. This theoretical solution curve was fitted to the experimental data in the deeper region (solid line in Fig. 1) and the diffusivities of C in the HIP samples were determined. Figure 2 shows the temperature dependence of diffusivity of C in Bulk Zr, HIP Zr, and HIP Zr(5%O), Zr(10%O), and Zr(15%O). The diffusivity of C in Bulk Zr was lower than that obtained by Agarwala and Paul [6], which were determined from the depth profiles of the radioactive 14 C tracer. This difference was attributed to C diffusion through interstitials because it was plausible that C in the grain boundaries was below the limit of detection by GD-OES, but could be detected by the 14 C tracer technique, as noted in Ref. [6]. The diffusivity data of C in HIP Zr agreed well with that in Bulk Zr, indicating the validity of the HIP samples as a host matrix for C diffusion. As the O concentration increased in HIP Zr to 15 at %, the diffusivity of C decreased by more than two orders of magnitude. This was because interstitial dissolution of O in the host Zr matrix could prevent diffusion of C in the interstitials due to lattice expansion or trapping effects.  Figure 3a shows the distribution of C in Bulk Zr through the thickness of the cladding under the temperature gradient from 573 to 773 K for 3 years with Q * c ranging from − 1.5 to + 1.5 eV. For positive Q * c , the C concentration was segregated slightly at the surface of the cooler side but had maximum peaks in the medium-to highertemperature zones, and was low at the surface of the hotter side. In contrast, for negative Q * c , C concentration was low at the surface of the cooler side and in the medium-to higher-temperature zones, and was highly segregated at the surface of the hotter side. These tendencies became more pronounced as the absolute value of Q * c increased. The effects of isothermal annealing for 10 years corresponding to the long-term storage of the Zr cladding were examined by using a non-uniform distribution of C formed by diffusion and thermotransport with Q * c = + 1.0 eV as an initial concentration profile (i), as shown in Fig. 3b. The C distribution did not change much at annealing temperatures below 673 K, whereas it became uniform at annealing temperatures above 773 K.

Simulation of transport of C in Zr and Zr(O)
The diffusivity of C in Zr(O) determined in the present study was from one to two orders of magnitude smaller than that in Zr depending on dissolved O concentration. Based on the lower diffusivity of C in Zr(O), C transport was strongly suppressed, resulting in stabilization of C and O in the Zr matrix whether they were uniform or nonuniform (Fig. 3c).

Conclusions
Diffusion of C in the interstitials of the host Zr matrix could be strongly affected by dissolved O coexisting in the interstitials, resulting in the reduction of diffusivity of C in Zr(15 at %O) by a factor of two. Non-uniform distributions of C through a 5-mm-thick Zr matrix by thermotransport accompanied by diffusion under the temperature gradient were obtained by numerical calculations. Because the sign (positive or negative) and value of the heat of transport, Q * c , govern C segregation by thermotransport, the heat of transport for C in Zr and Zr(O) should be evaluated experimentally in future work. Isothermal annealing at temperatures higher than 773 K for decades could change a non-uniform distribution of C in the Zr matrix to a uniform distribution throughout the thickness. However, the distribution of C in Zr(O) would not be changed or would be preserved due to the lower diffusivity of C in Zr(O).
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