Wavelet analysis of the self-similarity of diffusion-limited aggregates and electrodeposition clusters.

Abstract

We present the wavelet transform as a natural tool for characterizing the geometrical complexity of numerical and experimental two-dimensional fractal aggregates. We illustrate the efficiency of this ``mathematical microscope'' to reveal the construction rule of self-similar snowflake fractals and to capture the local scaling properties of multifractal aggregates through the determination of local pointwise dimensions \ensuremath{\alpha}(x). We apply the wavelet transform to small-mass (M\ensuremath{\lesssim}5\ifmmode\times\else\texttimes\fi{}${10}^{4}$ particles) Witten and Sander diffusion-limited aggregates that are found to be globally self-similar with a unique scaling exponent \ensuremath{\alpha}(x)=1.60\ifmmode\pm\else\textpm\fi{}0.02. We reproduce this analysis for experimental two-dimensional copper electrodeposition clusters; in the limit of small ionic concentration and small current, these clusters are globally self-similar with a unique scaling exponent \ensuremath{\alpha}(x)=1.63\ifmmode\pm\else\textpm\fi{}0.03. These results strongly suggest that in this limit the electrodeposition growth mechanism is governed by the two-dimensional diffusion-limited aggregation process.