## 3 Results and Discussion
This cannot be explained with only the solute drag or Zener effects as the grain size should decrease due to these effects. According to the two-dimensional nucleation grain growth theory, abnormal grain growth can occur when the grain growth rate or migration of the interface is non-linear with \(\Delta g\) and is governed by the interface reaction [14; 15; 20; 36; 37; 38; 39]. In previous studies, the abnormal grain growth phenomenon was observed in hexaferrites and the results and analyses support the theory of a non-linear grain growth rate [6; 23; 24; 25; 26; 27]. During the sintering of inorganic materials, grain growth occurs due to the difference in pressure induced by interface curvature and a decrease in the total interface energy [20]. The driving force for the growth of a grain is given by [20; 22; 39]:  
\[\Delta g=2\gamma V_{m}\bigg{(}\frac{1}{r^{*}}-\frac{1}{r}\bigg{)} \tag{1}\]  
where \(\gamma\) = interface energy, \(V_{m}\) = molar volume, \(r\) = radius of the grain of interest, and \(r^{*}\) = critical grain radius or the radius of a grain that does not grow or shrink, and is similar to the average grain radius. Each grain has its own \(\Delta g\) and the largest grain has a \(\Delta g_{max}\) as schematically shown in Figure 4. \(\Delta g_{c}\) is the value that determines whether appreciable grain growth occurs in two-dimensional nucleation grain growth. The critical driving force for growth can be expressed by [22; 39]  
\[\Delta g_{c}=\bigg{(}\frac{\pi\sigma^{2}}{KTh}\bigg{)}\ln K^{-1}\propto\frac {\sigma^{2}}{h} \tag{2}\]  
where \(\sigma\) = step free energy, \(h\) = height of the two-dimension nucleus, \(k\) = Boltzmann constant, \(T\) = absolute temperature, and \(K\) is a constant that accounts for the diffusion coefficient and the number of nuclei per unit area. According to Equation (2), \(\Delta g_{c}\) decreases as the sintering temperature increases and can be changed by altering the step free energy. The step free energy can be explained as the energy needed to form one layer and can be changed by the interface energy and structure [36]. \(\Delta g_{c}\) can be \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Sintering Temperature** & **1 wt\% SiO\({}_{2}\) Addition** & \(H_{c}\)**, Oe** & \(M_{s}\)**, emu/g** \\ \hline \multirow{4}{*}{1200 \({}^{\circ}\)C} & no-Si & 2272 & 68.01 \\  & post-Si & 3560 & 67.04 \\  & pre-Si & 3235 & 62.48 \\  & no-Si & 2216 & 69.09 \\
1225 \({}^{\circ}\)C & post-Si & 3581 & 65.37 \\  & pre-Si & 3293 & 63.26 \\  & no-Si & 2298 & 68.16 \\
1250 \({}^{\circ}\)C & post-Si & 3400 & 64.98 \\  & pre-Si & 3351 & 64.85 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: \(M_{s}\) and \(H_{c}\) of sintered samples measured by vibrating sample magnetometer (VSM). altered by controlling the interface energy and structure among grains via changing temperature [15; 23], adding a dopant [14; 38], or changing the sintering atmosphere [16; 17]. Figure 4 depicts the grain growth rate as a function of the \(\Delta g\) for diffusion (dashed line) and interface reactions (solid line) according the Equations (1) and (2). In the case of the no-Si sample sintered at 1250 \({}^{\circ}\)C, most grains can grow since the \(\Delta g\) of the grains is larger than the \(\Delta g_{c}\) for no-Si (\(\Delta g_{c}\)\({}^{0\text{Si}}\)), as shown in Figure 4a. As the grain growth behavior changes with the addition of SiO\({}_{2}\), the \(\Delta g_{c}\) for appreciable grain growth increases when adding SiO\({}_{2}\). This is similar to the phenomenon observed when TiO\({}_{2}\) is added to sodium bismuth titanate [14]. As seen in Figure 4a, when 0.5 wt% SiO\({}_{2}\) was added using the pre-Si method (pre-0.5 Si), \(\Delta g_{c}\)\({}^{0.5\text{Si}}\) increased but is still lower than the \(\Delta g_{max}\). Therefore, some grains can grow rapidly and encroach on one another and Ostwald ripening can occur during the initial stage of sintering, where small grains are dissolved and absorbed into larger grains. Finally, grain growth stops in samples of pre-0.5 Si.