Experimental Evidence for Spatio-Temporal Chaos in Diffusion-Limited Growth Phenomena

Abstract

Interfacial growth processes are generally fascinating nonequilibrium phenomena; physical, chemical, biological and geophysical systems exhibit similar rich morphologies^1–4. The main characteristic feature of these systems is the existence of a diffusion field that drives them far from equilibrium. In this context, the diffusion limited aggregation (DLA) model introduced by Witten and Sander⁵ in 1981 has played a major role since it has stimulated a variety of renewed experiments and numerical simulations^1–4. But, despite the apparent simplicity of the DLA model, there is still no rigourous theory for diffusion-limited growth processes. Many important theoretical questions remain unanswered; in particular, it is still an open question whether the geometrical complexity of DLA clusters is a product of the randomness in the growth process^6,7 or the result of a proliferation of deterministic tip-splitting instablities^8–11. Indeed, most previous works^5,12–21 have mainly focused on the geometrical properties of growing aggregates. The fractal geometry of DLA clusters has been analyzed using powerful mathematical techniques such as the computation of the generalized fractal dimensions and f(α) spectrum^22,23 and very recently, the application of the wavelet transform^24,25. In a previous work, we have succeeded in elucidating the conjectured self-similarity^23–25 of DLA clusters. These aggregates are homogeneous fractals; that is, on a range of physical scales, the mass locally behaves as a power-law of the length scale with a unique scaling exponent α ~ 1.60, which is independent of the point chosen on the cluster. DLA clusters are thus statistically self-similar: the generalized fractal dimensions are all equal D _q = 1.60 ± 0.02. In an experimental study^23,25 we have shown that electrodeposition clusters, grown in the limit of weak current and small concentrations of electroactive species are also statistically self-similar. Within the experimental uncertainty their fractal dimensions D _q = 1.63 ± 0.03 are in very good agreement with D _q for DLA clusters. Moreover, it has been recently realized that this geometrical self-similarity of diffusion—controlled aggregates is intimately related to the non-homogeneous distribution of the velocity field along the cluster boundary. The same mathematical tools have been applied to characterize the multifractal properties of the growth probability distribution (or sticking probability) of DLA clusters^11,26–32; its f(α)-spectrum is non-trivial, and the dimensions D _q decrease with increasing q. Experimental evidence for the multifractal nature of the growth probability distribution has been also reported in Refs 33, 34.