Multifractal analysis of the fracture surfaces of foamed polypropylene/polyethylene blends

The two-dimensional multifractal detrended fluctuation analysis is applied to reveal the multifractal properties of the fracture surfaces of foamed polypropylene/polyethylene blends at different temperatures. Nice power-law scaling relationship between the detrended fluctuation function $F_{q}$ and the scale $s$ is observed for different orders $q$ and the scaling exponent $h(q)$ is found to be a nonlinear function of $q$, confirming the presence of multifractality in the fracture surfaces. The multifractal spectra $f(\alpha)$ are obtained numerically through Legendre transform. The shape of the multifractal spectrum of singularities can be well captured by the width of spectrum $\Delta\alpha$ and the difference of dimension $\Delta f$. With the increase of the PE content, the fracture surface becomes more irregular and complex, as is manifested by the facts that $\Delta\alpha$ increases and $\Delta f$ decreases from positive to negative. A qualitative interpretation is provided based on the foaming process.