## 3 Results and Discussion
Based on these microstructural observations, and according to the three-dimensional critical grain size model proposed by [PERSON], the tetragonal-to-monoclinic phase transformation is governed by the relationship between the grain size and the critical grain size (r\({}_{\rm c}\)). \[r_{c}=\frac{2\gamma}{\Delta G_{\rm v}}\]  
where r\({}_{\rm c}\) is the critical grain size (radius), \(\gamma\) is the interfacial energy, and \(\Delta\)G\({}_{\rm v}\) is the difference in volumetric free energy. MnO\({}_{2}\) doping not only significantly increased the average grain size (from 2 to 11 \(\mu\)m to 36-50 \(\mu\)m), but also dramatically reduced the monoclinic phase fraction (from 32.6% to 2.5-6.5%). This indicates that MnO\({}_{2}\) doping increases the critical grain size, thereby stabilizing the tetragonal phase even at much larger grain sizes. The observed decrease in monoclinic phase fraction with increasing MnO\({}_{2}\) content is therefore attributed to the increase in r\({}_{\rm c}\) rather than a reduction in grain size. \begin{table}
\begin{tabular}{c c c c c c} \hline  & & **9-CSZ** & **2 MnCSZ** & **3 MnCSZ** & **4 MnCSZ** \\ \hline \multirow{6}{*}{m-ZrO\({}_{2}\)} & volume \% & 32.6 & 3.8 & 6.5 & 2.5 \\ \cline{2-6}  & a (Å) & 5.149 & 5.137 & 5.471 & 5.223 \\ \cline{2-6}  & b (Å) & 5.203 & 5.234 & 5.662 & 5.154 \\ \cline{2-6}  & c (Å) & 5.319 & 5.269 & 5.143 & 5.219 \\ \cline{2-6}  & \(\alpha=\gamma\) & 90\({}^{\circ}\) & 90\({}^{\circ}\) & 90\({}^{\circ}\) & 90\({}^{\circ}\) \\ \cline{2-6}  & \(\beta\) & 99.198 & 99.093 & 99.439 & 99.156 \\ \cline{2-6}  & lattice volume (Å\({}^{3}\)) & 140.6650712 & 139.8875853 & 157.1567065 & 138.7020022 \\ \cline{2-6}  & chemical composition & ZrO1.987 \(\pm\) 0.003 & ZrO1.982 \(\pm\) 0.018 & ZrO1.978 \(\pm\) 0.006 & ZrO1.976 \(\pm\) 0.010 \\ \hline \multirow{6}{*}{t-ZrO\({}_{2}\)} & volume \% & 58.2 & 88.2 & 90.1 & 90.3 \\ \cline{2-6}  & a = b (Å) & 3.626 & 3.613 & 3.616 & 3.616 \\ \cline{2-6}  & c (Å) & 5.122 & 5.129 & 5.126 & 5.111 \\ \cline{2-6}  & \(\alpha=\beta=\gamma\) & 90\({}^{\circ}\) & 90\({}^{\circ}\) & 90\({}^{\circ}\) & 90\({}^{\circ}\) \\ \cline{2-6}  & lattice volume (Å\({}^{3}\)) & 67.34342087 & 66.9527812 & 67.02478746 & 66.82865562 \\ \cline{2-6}  & chemical composition & ZrO1.973 \(\pm\) 0.005 & ZrO1.966 \(\pm\) 0.018 & ZrO1.966 \(\pm\) 0.003 & ZrO1.964 \(\pm\) 0.010 \\ \hline \multirow{6}{*}{c-ZrO\({}_{2}\)} & volume \% & 9.2 & 8 & 3.4 & 7.2 \\ \cline{2-6}  & a = b = c (Å) & 5.134 & 5.115 & 5.113 & 5.094 \\ \cline{2-6}  & \(\alpha=\beta=\gamma\) & 90\({}^{\circ}\) & 90\({}^{\circ}\) & 90\({}^{\circ}\) & 90\({}^{\circ}\) \\ \cline{2-6}  & lattice volume (Å\({}^{3}\)) & 135.3217461 & 133.8248959 & 133.6679779 & 132.1833706 \\ \cline{2-6}  & chemical composition & ZrO1.995 \(\pm\) 0.007 & ZrO1.992 \(\pm\) 0.014 & ZrO1.985 \(\pm\) 0.009 & ZrO1.980 \(\pm\) 0.010 \\ \hline MnO\({}_{2}\) & volume \% & - & 0 & 0 & 0 \\ \hline \multirow{6}{*}{R\({}_{\rm wp}\)} & 22.474 & 18.498 & 17.342 & 16.886 \\ \cline{2-6}  & R\({}_{\rm p}\) & 18.458 & 14.549 & 13.594 & 13.564 \\ \cline{1-1} \cline{2-6}  & \(\chi^{2}\) & 4.0256 & 2.604 & 2.191 & 1.958 \\ \hline \end{tabular}
\end{table} 
Table 2: Rietveld results of 9-CSZ powder calinated at 1200 \({}^{\circ}\)C.  
Figure 2: Mn 2p\({}_{3/2}\) XPS spectrum and deconvoluted curves for Mn\({}^{2+}\), Mn\({}^{3+}\), and Mn\({}^{4+}\) after background subtraction of **(a)** 2 Mn_CSZ, **(b)** 3 Mn_CSZ, **(c)** 4 Mn_CSZ. The MnO\({}_{2}\)-doped CSZ samples were subjected to thermal shock treatment by rapid cooling from 1300 \({}^{\circ}\)C to 900 \({}^{\circ}\)C, corresponding to a temperature difference of 400 \({}^{\circ}\)C. The phase analysis results obtained after this treatment are shown in Figure 4 and Table 3. After thermal shock, the monoclinic phase fraction was 30.7% for 9-CSZ, whereas it was significantly lower for 2-4 mol% MnO\({}_{2}\)-doped CSZ, ranging from 1.1% to 2.9%. This trend was consistent with the results observed prior to thermal shock.