# Growth of Zr/ZrO\({}_{2}\) Core-Shell Structures by Fast Thermal Oxidation
## 2 Experimental Method  
The experimental setup used to grow the samples is shown in Figure 1a. A wire made of metal that is to be oxidized is suspended between two electrodes connected to a current source. The full system was kept in an ambient atmosphere. As a consequence of Joule heating, a temperature gradient was established along the wire, promoting the oxidation of the metal and yielding core-shell structures, in which the core is the native metal. In order to determine the thickness of the oxide layer, which was around 10 \(\upmu\)m, we took the average of some radii of the circular crown formed by the oxide layer. To study the influence of the growth parameters (current and time), different sets of samples were prepared. The starting material is a commercial Zr wire (Goodfellows ZR00-WR-000130), as drawn, with a nominal purity of 99.2%. According to the provider's specifications, the main impurity content corresponds to hafnium (2500 ppm). Other transition metals such as chromium or iron were also present but at much lower concentrations (200 ppm). The wire length (distance between electrodes) is 10 cm, and the diameter is 0.25 mm. According to the temperature profile established in the wire, three different zones are defined for characterization purposes. Figure 1 shows a thermal image of the wire (b) and the zones defined (c). Thermal images were recorded using an FLIR E8-XT thermographic camera. The maximum temperatures reached were between 430 and 500 \({}^{\circ}\)C. Temperatures were measured using an optical pyrometer Infrathemr IGA 12-S. There are some analytical models of the stationary distribution of temperature of the wire. A detailed explanation of the derivation of the differential equation that governs the temperature profile because of Joule heating can be found in [21], whose notation is kept below:  
\[\kappa A_{cr}T^{\prime\prime}\left(x\right)-\left(h\pi d-J^{2}A_{cr}\rho_{0} \xi\right)T(x)=J^{2}A_{cr}\rho_{0}(\xi T_{0}-1)-h\pi dT_{0}\]  
The general behavior of the temperature profile can be written in terms of real exponentials (hyperbolic functions) or complex exponentials (sines and cosines), independent of the initial conditions. Furthermore, this behavior depends solely on the sign of the zeroth derivative temperature term. The sign of this term can be positive or negative at will by modifying physical parameters such as intensity, diameter and resistivity (44 \(\upmu\)lcm at 20 \({}^{\circ}\)C for zirconium). If this term vanished, a parabolic profile would be observed. In our case, the model does not adjust for experimental data, even including corrective terms, such as radiative heat loss in the prior equation. We believe that there might be size factors related to the wire diameter (0.25 mm) or physical processes that should not be disregarded in a first approach, which would influence the accuracy of the model. A complementary, but extremely complex, model should account for the time evolution of the oxide shell. Nevertheless, this is out of our scope, since it would require detailed knowledge of the growth kinetics, the defect structure at each growth step and their influence on parameters such as the resistivity and thermal properties of zirconium oxide. In this work, we have prepared two different sets of samples. The first set consists of five wires subjected to currents from 2.2 to 2.8 A for 10 s.  
Figure 1: (**a**) Experimental setup for resistive heating growth. (**b**) Thermographic image of a wire. The cross indicates the point at which the maximum temperature is measured. (**c**) Sketch of the three zones established for the characterization of the core–shell structures grown. For the second set of samples, the maximum intensity was 2.2 A in all cases, but the current-time profiles and the times during which the wires were held at the maximum current were varied. Figure 2 sketches the current-time profile used. The characterization was carried out by means of SEM-based techniques and optical spectroscopy. Emissive mode images were recorded with an FEI Inspect SEM.