# Controlled-Atmosphere Sintering of KNbO\({}_{3}\)
## 4 Discussion  
During solid state sintering of a polycrystalline ceramic, grain growth takes place due to grain boundary curvature-induced differences in pressure and atom chemical potential between grains of differing size [52; 53; 54]. Driving force \(\Delta G\) for the growth of a grain is given by [54; 55; 56]:  
\[\Delta G=2\gamma V_{m}\Big{(}\frac{1}{r\star}-\frac{1}{r}\Big{)} \tag{1}\]  
where \(\gamma=\) grain boundary energy, \(V_{m}=\) molar volume, \(r\star\) = critical grain radius (radius of a grain that does not grow or shrink, usually considered to be the mean grain radius), and \(r=\) radius of the grain of interest. For a faceted grain, \(r=\) perpendicular distance from the grain center to the surface and \(\gamma=\) specific surface energy [54]. The chemical potential (and hence, \(\Delta G\)) of atoms in a grain with faceted surfaces also depends on grain size [57; 58]. The grain growth rate also depends on the grain boundary structure, as well as the driving force. Grain boundaries have disordered (rough) or ordered (faceted) structures on an atomic scale [59; 60]. For grains with disordered grain boundaries, there are many possible atom attachment sites on the grain surface. Grain growth is limited by atom diffusion across the grain boundary [54; 56]. The grain growth rate shows a linear increase with \(\Delta G\). Grains with \(r>r\star\) have positive values of \(\Delta G\) and can grow (the black dotted line in Figure 11), whereas grains with \(r<r\star\) have negative values of \(\Delta G\) and will shrink [54]. Grain size distribution remains unimodal with sintering time, and abnormal grain growth will not occur [54]. If the grain boundaries are ordered, atoms attaching to the surface of a grain are unstable due to their many broken bonds. They will quickly detach from the grain unless they can migrate and attach \begin{table}
\begin{tabular}{l c c c} \hline \hline  & **Orthorhombic-Tetragonal** & **Tetragonal-Cubic Phase** & **Cubic-Tetragonal Phase** \\
**Sintering Atmosphere** & **Phase Transition** & **Transition Temperature** & **Transition Temperature** \\  & **Temperature (\({}^{\circ}\)C)** & **(\({}^{\circ}\)C)** & **(\({}^{\circ}\)C)** \\ \hline O\({}_{2}\) & 233 & 407 & 381 \\ Air & 234 & 408 & 383 \\ N\({}_{2}\) & 227 & 406 & 383 \\ N\({}_{2}\)-5\% H\({}_{2}\) & 159 & 403 & 376 \\ H\({}_{2}\) & - & 381 & 358 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: DTA phase transition temperatures of KNbO\({}_{3}\) samples sintered at 1040 \({}^{\circ}\)C for 3 h in different atmospheres. Figure 11: Schematic of growth rate vs. driving force for samples sintered in different atmospheres (2 DNG = 2D nucleation-controlled growth). to low-energy kink sites such as 2D nuclei, screw dislocations, steps, and re-entrant edges [54; 60; 61]. Grain growth is now limited by the rate of atom attachment at such kink sites (interface-reaction controlled growth). For a system in which grain growth is controlled by 2D nucleation and growth, the rate at which 2D nuclei form varies exponentially with \(\Delta G\)[62; 63]. Below a critical driving force \(\Delta G_{c}\), the 2D nuclei form at a very slow rate, and atom attachment to the grain is difficult. At \(\Delta G=\Delta G_{c}\), the rate at which 2D nuclei form increases exponentially. For \(\Delta G>\Delta G_{c}\), kinetic roughening takes place [64; 60; 65]. Many 2D nuclei form, and atom attachment to the grain becomes easy. Grain growth becomes diffusion-limited, as in the case of a rough grain boundary. Hence, the grain growth rate is negligible at \(\Delta G<\Delta G_{c}\), increases exponentially at \(\Delta G\approx\Delta G_{c}\) (the dashed lines in Figure 11), and then becomes a linear function of \(\Delta G\) (the solid lines in Figure 11) [66]. Grains with negative values of \(\Delta G\) will shrink as a linear function of \(\Delta G\), as the energy requirement for atoms to detach from grain corners and edges is low [54; 56; 67].