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

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 α(x). We apply the wavelet transform to small-mass (M≲5×${10}^4$ particles) Witten and Sander diffusion-limited aggregates that are found to be globally self-similar with a unique scaling exponent α(x)=1.60±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 α(x)=1.63±0.03. These results strongly suggest that in this limit the electrodeposition growth mechanism is governed by the two-dimensional diffusion-limited aggregation process.