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Fractal Dimension of Grain Boundary

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].

a SEM micrographs of TiO sintered at

Fig. 4 a SEM micrographs of TiO2 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 TiO2 grain growth during heat up at 300-800 °C for 2 h soaking time. c Variation of fractal dimension of nano structure TiO2 at different temperatures for 2 h

Here, PG is the perimeter of grain boundary and AG is the area of grain;

where AG is the surface area of grain boundary and VG is the volume of grain. They can be transformed and simplified into logarithmic forms,


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.

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