# Controlled-Atmosphere Sintering of KNbO\({}_{3}\)
## 4 Discussion
If the grains contain screw dislocations, the grain growth rate increases parabolically with \(\Delta G\) until \(\Delta G_{c}\) is reached, at which point grain growth becomes diffusion-limited again. According to the relative values of \(\Delta G_{c}\) and \(\Delta G_{max}\) (where \(\Delta G_{max}\) is the value of \(\Delta G\) for the largest grain), different grain growth behaviors (pseudo-normal, abnormal, and stagnant) can be observed [54; 56; 67; 68]. This grain growth theory, called the mixed control theory, was originally developed to describe liquid phase sintering, but similar behavior also takes place in single phase systems with faceted grain boundaries in which the formation and lateral spreading of steps governs grain boundary migration [54; 66; 67; 69; 70; 71; 72; 73; 74; 75; 76; 77; 78]. The critical driving force \(\Delta G_{c}\) is dependent on the step or edge free energy \(\varepsilon\), which is the excess energy of the edge of the kink site [54; 56; 67; 70; 73]. The value of \(\varepsilon\) can vary depending on sintering atmosphere, sintering temperature, or dopant addition [59; 60; 79]. As \(\varepsilon\) decreases, \(\Delta G_{c}\) also decreases. Grain morphology is a qualitative indication of the value of \(\varepsilon\)[55; 68; 80]. The cubic morphology of the KNbO\({}_{3}\) grains (Figure 3) with faceted faces, sharp edges, and corners (Figure 7) indicates a high value of \(\varepsilon\). During sintering of KNbO\({}_{3}\), evaporation of potassium can take place. However, the potassium vapor pressure over KNbO\({}_{3}\) is much lower than that of lead oxide over PZT [81]. Oxygen vacancies will also form via the following defect reaction:  
\[2O_{O}^{x}\rightarrow\uparrow O_{2}(g)+2V_{O}^{\bullet\bullet}+4e^{\prime} \tag{2}\]  
The formation of potassium and oxygen vacancies increases the configurational entropy of the system, which in turn reduces step free energy \(\varepsilon\) and critical driving force \(\Delta G_{c}\)[82; 83; 84]. The decreased orthorhombic distortion in the unit cell of the H\({}_{2}\)-sintered sample (Figures 8 and 9) and the decrease in temperature of the orthorhombic to tetragonal and tetragonal to cubic phase transitions in the samples sintered in N\({}_{2}\)-5%H\({}_{2}\) and H\({}_{2}\) (Figure 10) are caused by the increased concentration of oxygen vacancies [45; 85; 86; 87; 88; 89; 90]. The appearance of secondary phases in the H\({}_{2}\)-sintered samples also indicates increased evaporation of potassium (Figures 4 and 8). By considering the rate of formation of oxygen vacancies in different sintering atmospheres, the grain growth behavior of KNbO\({}_{3}\) can be explained. For O\({}_{2}\)- and air-sintered samples, the number of oxygen vacancies is relatively low. The values of \(\varepsilon\) and hence \(\Delta G_{c}\) are high. The value of \(\Delta G_{c}\) is higher than \(\Delta G_{max}\), the driving force of the largest grain (Figure 11). No grains in the samples had \(\Delta G\geq\Delta G_{c}\), and so, no grains could grow appreciably. This caused stagnant grain growth behavior. For the N\({}_{2}\)-sintered sample, the lower \(P_{O_{2}}\) would push Equation (2) to the right and create more oxygen vacancies, increasing configurational entropy, lowering \(\varepsilon\) and \(\Delta G_{c}\). A few grains now had \(\Delta G\approx\Delta G_{c}\). These grains should be able to grow more rapidly than the surrounding matrix grains and become abnormal grains (Figures 6 and 11). However, there was little size difference between the abnormal grains and matrix grains. For values of \(\Delta G\) very close to \(\Delta G_{c}\), the grain growth rate does not increase dramatically with small increases in \(\Delta G\) (the region of the blue curve in Figure 11 which is beginning to curve upwards). With increasing sintering time, \(\Delta G\) of the abnormal grains should slowly increase as they were slowly growing, while the matrix grains (which control the value of the critical radius \(r^{\bullet}\)) were barely growing [Equation (1)].