# Interdiffusion in Zr-Mo/W Intermetallics
## 3 Results and Discussion
### Interdiffusion Coefficient and Activation Energy of Diffusion in (Zr, Mo) Solid Solution Phase  
Since a concentration gradient existed in the (Zr, Mo) solid solution, the interdiffusion coefficient could be calculated by solving [PERSON]'s second law. The concentration C at each part of the diffused region is related to the distance \(x\) and time \(t\). [PERSON]'s second law revealed the relationship between \(C\), \(x\), and \(t\)[28]:  
\[\frac{\partial C}{\partial t}=\frac{\partial}{\partial x}\left(\widetilde{D} \frac{\partial C}{\partial x}\right) \tag{3}\]  
[PERSON] and [PERSON] successively proposed methods for calculating the interdiffusion coefficient by solving [PERSON]'s second law with experimentally measured concentration distribution curves and concentration gradients. However, this method caused some inaccuracies because it neglected the change in molar volume at different concentrations, so it should be modified. In this study, the den Broeder method based on the Boltzmann  
Figure 7: Temperature-dependence of growth constants calculated for Mo\({}_{2}\)Zr and W\({}_{2}\)Zr: (a) Mo\({}_{2}\)Zr; (b) W\({}_{2}\)Zr. Matano model was used to calculate the interdiffusion coefficients. This method avoids the calculation error caused by the introduction of the Matano plane and also takes into account the change of the molar volume \(V_{m}\) owing to the change in the concentration of the components during the diffusion process [29]. The interdiffusion coefficient in the (Zr, Mo) solid solution phase was calculated based on the concentration distribution curve in Figure 2. The interdiffusion coefficient at the atomic concentration \(c^{*}\) (abscissa \(x^{*}\)) was expressed as [30,31]:  
\[\widetilde{D}(c^{*})=\frac{V_{m}}{2t}\left(\frac{d_{x}}{d_{c}}\right)_{c^{*}} \left[\left(\frac{c_{2}-c^{*}}{c_{2}-c_{1}}\right)\int_{-\infty}^{x^{*}}\frac{ (c-c_{1})}{V_{m}}dx+\left(\frac{c^{*}-c_{1}}{c_{2}-c_{1}}\right)\int_{x^{*}}^{ +\infty}\frac{(c_{2}-c)}{V_{m}}dx\right] \tag{4}\]  
where \(t\) is the diffusion time, \(c_{1}\) and \(c_{2}\) are respectively the atomic concentrations on both sides of the diffused region, and c represents the atomic concentration of Mo at position \(x\). \(V_{m}\) is the molar volume of the concentration of the specific component, and since the Zr-Mo system is in the bcc phase region at experimental conditions in this work, \(V_{m}\) can be calculated as:  
\[V_{m}(c_{m})=N_{0}\left[c_{m}\frac{a_{m}^{3}}{2}+(1-c_{m})\frac{a_{z}^{3}}{2}\right] \tag{5}\]  
where \(N_{0}\) is Avogadro's constant as 6.02 \(\times\) 10\({}^{23}\); \(c_{m}\) is the molar fraction of the element Mo; \(a_{m}\) and \(a_{Z}\) are lattice parameters of the components Mo and Zr with bcc structure, respectively. The concentration-dependent interdiffusion coefficients calculated from Equation (4) for the (Zr, Mo) solid solution phase in the range of 1300-1500 \({}^{\circ}\)C are shown in Figure 8. The interdiffusion coefficient of the (Zr, Mo) solid solution was between 1.5 \(\times\) 10\({}^{-13}\) m\({}^{2}\)/s and 1.4 \(\times\) 10\({}^{-12}\) m\({}^{2}\)/s at 1300-1500 \({}^{\circ}\)C, which was two magnitudes larger compared to the growth constant of the Mo\({}_{2}\)Zr phase. Furthermore, the interdiffusion coefficient of the Zr-Mo system with the solid solution phase decreased with the increase of Mo content. At 1300 \({}^{\circ}\)C and Mo content of 0-10 at. %, the interdiffusion coefficient of (Zr, Mo) solid solution ranged from 1.5 \(\times\) 10\({}^{-13}\) m\({}^{2}\)/s to 4 \(\times\) 10\({}^{-13}\) m\({}^{2}\)/s; at 1400 \({}^{\circ}\)C and Mo content of 0-10 at. %, it ranged from 3.0 \(\times\) 10\({}^{-13}\) m\({}^{2}\)/s to 8.0 \(\times\) 10\({}^{-13}\) m\({}^{2}\)/s; at 1500 \({}^{\circ}\)C and Mo content of 0-10 at. %, it ranged from 2.8 \(\times\) 10\({}^{-13}\) m\({}^{2}\)/s to 1.4 \(\times\) 10\({}^{-12}\) m\({}^{2}\)/s. In general, the interdiffusion coefficient showed concentration dependence.