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Fractal Correlation Dimensions Analysis of Al–Si Dendrites

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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.

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Acknowledgements

Al–Si samples were originally created by Sonja Steinbach and Lorenz Ratke in the Institute for Materials Physics in Space at German Aerospace Centre (DLR) in Cologne.

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Authors and Affiliations

  1. UAB IT Research Computing, University of Alabama at Birmingham, 1700 University Blvd, LHL 404, Birmingham, AL, 35294, USA

    William E. Warriner

  2. Department of Materials Science and Engineering, University of Alabama at Birmingham, 1720 2nd Ave. S, Birmingham, AL, 35294, USA

    Jessica Friess & Amber Genau

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  1. William E. Warriner
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  2. Jessica Friess
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  3. Amber Genau
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Correspondence to Amber Genau.

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Warriner, W.E., Friess, J. & Genau, A. Fractal Correlation Dimensions Analysis of Al–Si Dendrites. Metallogr. Microstruct. Anal. 9, 561–569 (2020). https://doi.org/10.1007/s13632-020-00672-z

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  • DOI: https://doi.org/10.1007/s13632-020-00672-z

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