## 3 Results and Discussions
### Modeling of Layer Growth Kinetics
That is the reason why a needle-like TiB layer is formed beneath the layer. In order to study the acceleration kinetics of the growth of the boride layer near the transition temperature, in addition to knowing the diffusion rate of B in the boride, the study of the diffusion rate of titanium in B is also necessary. However, due to the limited solid solubility of B in Ti, and Ti and B easily reacting to form compounds, it is very difficult to measure the diffusion coefficient of B in the Ti substrate. This may be the reason why B diffusion data does not exist in the literature. Another question is whether B diffusion in the Ti phase occurs through interstitial or substitution mechanisms near the transition temperature. Assume that there is no interstitial diffusion of B in Ti. In addition, the higher concentration of substitution vacancies in the Ti substrate near the phase transition temperature can promote the increase of B diffusion through the substitution mechanism. In fact, previous studies have also shown [36] that the \(\omega\) phase formed near the transition temperature may be a vacancy supersaturated phase, which tends to promote substitution diffusion near the phase transition temperature. Based on these, it can be reasonably assumed that B atoms diffuse through the substitution sites in Ti at a speed comparable to Ti itself. As we all know, the growth of the interface reaction layer is a thermally activated process, and growth rate (\(k\)) is closely related to temperature. This temperature dependence can be described by the following equation. The specific values are shown in Table 5.  
\[k=k_{0}\exp\left(-\frac{Q^{k}}{2 RT}\right) \tag{9}\]  
where \(k_{0}\) is a constant (a frequency factor), \(Q_{k}\) is activation energy for layer growth, both \(k_{0}\) and \(Q_{k}\) are materials constants, \(T\) is absolute temperature, and \(R\) is the gas constant. These values were determined from the research of [PERSON]. [PERSON] et al. [35]. It can be seen from Table 5 that the growth rate of TiB was higher than that of TiB\({}_{2}\), which is consistent with the results observed in Figure 7 (the thickness of TiB was higher than that of TiB\({}_{2}\)). In addition, it was also found that the diffusion rate of B atoms in the TiB phase along the tip direction was more than 45 times that in the TiB\({}_{2}\) phase. This result is the same as in the research of [PERSON]. [PERSON] et al. [37]. According to the results in Table 5, the diffusion distance was calculated from the diffusion coefficient using Equation (10). Note that \(d\) is diffusion distance, \(D\) is diffusion coefficient, and \(t\) is processing time (it was 7200 s in this research). \[d=\sqrt{Dt} \tag{10}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**T** (\({}^{\circ}\)**C**) & **Diffusivity of B in TiB (m\({}^{2}\)/s)** & **Diffusivity of B in TiB\({}_{2}\) (m\({}^{2}\)/s)** \\ \hline
950 & \(3.23\times 10^{-14}\) & \(6.94\times 10^{-16}\) \\
1050 & \(1.33\times 10^{-13}\) & \(2.79\times 10^{-15}\) \\
1150 & \(4.48\times 10^{-13}\) & \(9.21\times 10^{-15}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Diffusivity data for various phases with different boriding temperatures. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**T** (\({}^{\circ}\)**C**)} & **k (\(\times\)10\({}^{-4}\)) (m/s \({}^{1/2}\))** & **D (\(\times\)10\({}^{-16}\)) (m/s)** & \multirow{2}{*}{\(D_{B}^{\text{TiB}}/D_{B}^{\text{TiB}_{2}}\)} \\ \cline{2-5}  & **TiB** & **TiB\({}_{2}\)** & **TiB** & **TiB\({}_{2}\)** & **DiB\({}_{B}^{\text{TiB}}/D_{B}^{\text{TiB}_{2}}\)** \\ \hline
950 & 14.73 & 2.34 & 322.7 & 6.94 & 46.5 \\
1050 & 29.88 & 4.70 & 1329 & 27.9 & 47.63 \\
1150 & 54.88 & 8.55 & 4485 & 92.1 & 48.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Growth rates and the estimated diffusivity data for B in TiB and TiB\({}_{2}\) with different boriding temperatures. According to the results in Table 6, the thickness of TiB was much greater than that of TiB\({}_{2}\).