Generally, grain growth occurs in polycrystalline materials to decrease the free energy of the system by decreasing the total grain boundary energy. The grain growth behavior of polycrystalline materials was reported to obey the equation: The rate of grain growth, is inversely proportional to grain size:

where, d is the average grain size at time t, d_{o} is the initial average grain size, n the grain growth exponential reflecting grain growth behavior, and c is the temperature-dependent rate constant can be expressed by the Arrhenius equation

Fig. 5 a SEM micrograph of TiO_{2} film sintered at 700 °C for (a) 1 min, (b) 3 min, (c) 60 min, (d) 100 min, (e) 300 min, (f) 600 min. b Grain area versus growth times TiO_{2} film at 700 °C. c Variation of fractal Dimension of nano structure TiO_{2} sintered at different socking time

s is the pre-exponential factor, E is the activation energy, R is the universal gas constant. Since d_{o} is generally much smaller than d Eq. (5), Eq. (6) can be derived as follows:

k is the kinetic exponent of grain Growth, A is the average grain area, and C_{3} is a constant. K can be obtained from the slope of the linear regressive line of log A versus log t.

As all of grain areas can be collected during the sintering process, average grain areas in different grains can be calculated. Therefore, the average grain area versus growth time can be plotted in a logarithmic scale. The kinetic exponents of grain growth correspond inversely to the slope of curve. They are about 0.24 at the early stages of the sintering process and then increase to 0.40 at the later stages, as shown in Fig. 5b. It can be seen that the kinetic exponents of grain growth are actually a function of the time. At the early stage of sintering has smaller activation energy than those at the later stage of process.

According to Fig. 5c, the variation of fractal dimension of nano structure TiO_{2 }sintered at different socking times, the fractal dimension decrease with the time of grain growth. This means that the surfaces of grain boundaries become smoother with growth time.