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Fractal Dimension of Grain BoundaryDuring the grain growth process, the grain area and grain perimeter in twodimension can be obtained and with fractal dimension, the volume of grain and the surface of grain boundary in three dimension can also be obtained [19, 20]. Fig. 4 a SEM micrographs of TiO_{2} sintered at (a) 300 °C, (b) 400 °C, (c) 500 °C, (d) 600 °C, (e) 700 °C, and (f) 800 °C for 2 h. b TiO_{2} grain growth during heat up at 300800 °C for 2 h soaking time. c Variation of fractal dimension of nano structure TiO_{2} at different temperatures for 2 h
Here, P_{G} is the perimeter of grain boundary and A_{G} is the area of grain;
where A_{G} is the surface area of grain boundary and V_{G} is the volume of grain. They can be transformed and simplified into logarithmic forms,
And
Fractal dimension (D) is calculated by the method of boxcounting after preprocessing SEM images. Grain Growth During Isothermal HoldingsIn order to understand the entire grain growth process, isothermal sintering experiments were carried out at 700 °C, following heatup at 3 °C/min. Figure 5a shows the isothermal grain growth curve of grain size versus time at 700 °C for all samples with same initial grain sizes. It demonstrates that the grain growth behavior of isothermal hold follows a more gradual process than that of heatup. 
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