where εr is the relative permittivi

where ε
r is the relative permittivity of porous materials, Vi and
ε
ri are the volume fraction and relative permittivity of the ith
material, respectively and α is a constant [20]. The value of the
constant α is determined to be α ¼ −1 and α ¼ 1 for serial and
parallel mixing models, respectively. As shown in Fig. 4, the
measured relative permittivity of porous PZT ceramics fell
between the predictions of the serial and parallel mixing
models. According to Zhang et al. [21], the parallel model
predicts the εr of porous PZT ceramics with closed cells, while
the serial model is suitable for the high open-porosity
ceramics. Obviously, both of the two models express two
extreme cases of the mixing of two components. However, the
measured value of ε
r in low porosity was closer to that of the
serial model, which did not match the empirical equation very
well. We believe that besides the porosity, the measured εr is
also sensitive to the shape of pores in this study. This
observation is consistent with the SEM micrographs in
Fig. 3 that shows the irregular shape at low porosity.