# Interdiffusion in Zr-Mo/W Intermetallics
## 3 Results and Discussion
### Growth Kinetics of Mo\({}_{2}\)Zr and W\({}_{2}\)Zr  
Assuming that the growth of the Mo\({}_{2}\)Zr and W\({}_{2}\)Zr phase was controlled by diffusion mechanism, the growth constants of the Mo\({}_{2}\)Zr and W\({}_{2}\)Zr phases could be calculated based on the thickness of the Mo\({}_{2}\)Zr and W\({}_{2}\)Zr layers at different temperatures combined with Equation (1) [25; 26]:  
\[K_{p}=\frac{x^{2}}{t} \tag{1}\]  
where \(K_{p}\) is the parabolic growth constant, \(x\) is the thickness of the phase, and \(t\) is the diffused time. The calculated results are shown in Figure 6. At 1300 \({}^{\circ}\)C, 1400 \({}^{\circ}\)C, and 1500 \({}^{\circ}\)C, the growth constants of Mo\({}_{2}\)Zr were 1.115 \(\times\) 10\({}^{-15}\) m\({}^{2}\)/s, 1.755 \(\times\) 10\({}^{-15}\) m\({}^{2}\)/s and 3.736 \(\times\) 10\({}^{-15}\) m\({}^{2}\)/s, respectively, while the growth constants of W\({}_{2}\)Zr were 9.56 \(\times\) 10\({}^{-16}\) m\({}^{2}\)/s, 3.142 \(\times\) 10\({}^{-15}\) m\({}^{2}\)/s, and 11.233 \(\times\) 10\({}^{-15}\) m\({}^{2}\)/s, respectively. Mo\({}_{2}\)Zr and W\({}_{2}\)Zr had approximate growth constants at 1300 \({}^{\circ}\)C, while at 1400 \({}^{\circ}\)C and 1500 \({}^{\circ}\)C, W\({}_{2}\)Zr had greater growth constants than Mo\({}_{2}\)Zr.  
To quantitatively calculate the activation energy for growth of the Mo\({}_{2}\)Zr and W\({}_{2}\)Zr phases, the growth constant versus temperature was obtained as shown in Figure 7, with \(lnK_{p}\) as the vertical coordinate and 10,000/T as the horizontal coordinate. The goodness of fit (r\({}^{2}\)) in both Figure 7a,b were over 0.99, which indicates that the values of the growth constants at each temperature were in accordance with the Arrhenius relationship [20; 27]:  
\[lnK_{p}=lnK_{0}-\frac{Q}{RT} \tag{2}\]  
Figure 5: Schematic diagrams of the (**a**) Zr-Mo and (**b**) Zr-W diffusion region after diffusion at 1300–1500 \({}^{\circ}\)C.  
Figure 6: Growth constants of Mo\({}_{2}\)Zr and W\({}_{2}\)Zr (\(\times\)10\({}^{-15}\) m\({}^{2}\)/s).  
where \(Q\) is the activation energy of growth, \(T\) is the annealing temperature in Kelvin, \(R\) is the ideal gas constant, and \(K_{0}\) is known as the pre-exponential factor. The intercept of the fitted line was \(lnK_{0}\), and the slope was \(-Q/R\). According to the slope and intercept of the fitted line in Figure 7, the pre-exponential factor and activation energy for the growth of the Mo\({}_{2}\)Zr phase and W\({}_{2}\)Zr phase can be calculated. The quantitatively calculated pre-exponential factors for the growth of Mo\({}_{2}\)Zr and W\({}_{2}\)Zr phases are 4.9 \(\times\) 10\({}^{-12}\) m\({}^{2}\)/s and 2.7 \(\times\) 10\({}^{-6}\) m\({}^{2}\)/s, respectively. The activation energies for the growth of the Mo\({}_{2}\)Zr and W\({}_{2}\)Zr phases are 109 kJ/mol and 285 kJ/mol, respectively.  
Although the activation energy for the growth of the W\({}_{2}\)Zr phase was greater than that of the Mo\({}_{2}\)Zr phase, the Mo\({}_{2}\)Zr layer was thinner than that of the W\({}_{2}\)Zr at 1400 \({}^{\circ}\)C and at 1500 \({}^{\circ}\)C, and the growth constant of the Mo\({}_{2}\)Zr phase was less than that of W\({}_{2}\)Zr phase. The reason behind this phenomenon can be explained as follows: (i) it is well known that the driving force of diffusion is the chemical potential gradient; (ii) Mo\({}_{2}\)Zr was formed between pure Zr and pure Mo in the initial stage of diffusion, and as the diffusion proceeded, the (Zr, Mo) solid solution formed between pure Zr and Mo\({}_{2}\)Zr, Mo\({}_{2}\)Zr grew between the (Zr, Mo) solid solution and pure Mo in the late stage of diffusion; consequently, (iii) the chemical potential of the formation of the Mo\({}_{2}\)Zr phase was reduced. In brief, (Zr, Mo) solid solution was formed between Mo\({}_{2}\)Zr and pure Zr as diffusion proceeded, resulting in a decreased chemical potential for the formation of Mo\({}_{2}\)Zr. Quantitative calculation of the kinetic parameters of diffusion in the (Zr, Mo) solid solution phase will be presented in the next section.