Fractal Correlation Dimensions Analysis of Al–Si Dendrites

Abstract

Solidification dendrites in metal alloys have intricate shapes with fractal elements, making fractal dimension an attractive potential method for more completely quantifying these important shapes. However, fractal analysis as usually applied to dendrites has some pitfalls, which we demonstrate here on micrographs of dendrites from two different aluminum–silicon alloys with different solidification velocities. Box dimension and a more rigorous analysis of correlation dimension were calculated for each sample, but both measures exhibited non-uniform scaling. Further analysis using the Takens–Theiler estimator of correlation dimension revealed a transition between two distinct scaling regimes: a small-scale exponent associated with dendrite boundary texture and the large-scale with overall dendrite morphology. The scaling transition between regimes may prove useful as an automated alternative to secondary dendrite arm spacing. Overall, the investigations give further insight into the quantitative description of complex dendritic structures and clarify the ability of fractal analysis to describe this important solidification morphology.