During the grain growth process, the grain area and grain perimeter in two-dimension 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 300-800 °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 box-counting after preprocessing SEM images.

Grain Growth During Isothermal Holdings

In order to understand the entire grain growth process, isothermal sintering experiments were carried out at 700 °C, following heat-up 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 heat-up.