Abstract
Stochastic growth processes give rise to diverse and intricate structures everywhere in nature, often referred to as fractals. In general, these complex structures reflect the non-trivial competition among the interactions that generate them. In particular, the paradigmatic Laplacian-growth model exhibits a characteristic fractal to non-fractal morphological transition as the non-linear effects of its growth dynamics increase. So far, a complete scaling theory for this type of transitions, as well as a general analytical description for their fractal dimensions have been lacking. In this work, we show that despite the enormous variety of shapes, these morphological transitions have clear universal scaling characteristics. Using a statistical approach to fundamental particle-cluster aggregation, we introduce two non-trivial fractal to non-fractal transitions that capture all the main features of fractal growth. By analyzing the respective clusters, in addition to constructing a dynamical model for their fractal dimension, we show that they are well described by a general dimensionality function regardless of their space symmetry-breaking mechanism, including the Laplacian case itself. Moreover, under the appropriate variable transformation this description is universal, i.e., independent of the transition dynamics, the initial cluster configuration, and the embedding Euclidean space.
Highlights
Found everywhere in nature, the intricate structures generated by fractal growth usually emerge from non-trivial self-organizing and self-assembling pattern formation[1,2,3,4]
In order to clarify these aspects of the Laplacian theory, as well as to establish a possible general framework to analyze more complex morphological transitions in stochastic growth processes, we present a dynamical model that addresses the fractality of these transitions
When an energetic element is introduced in the growth dynamics, the fractal dimension of the clusters decreases; for example, D → 1 as η → ∞ in the Dielectric Breakdown Model (DBM)
Summary
The intricate structures generated by fractal growth usually emerge from non-trivial self-organizing and self-assembling pattern formation[1,2,3,4]. In the absence of long-range interactions, the growth probability at a given point in space, μ, is generated by the spatial variation of a scalar field, φ, i.e., μ ∝ ∇φ One example of such processes is the paradigmatic diffusion-limited aggregation (DLA) model, where particles performing a random walk aggregate one-by-one to form a cluster, starting from a seed particle (see Fig. 1). One of the most challenging aspects of the theory arises when the growth is not purely limited by diffusion, e.g., when it takes place under the presence of long-range attractive interactions, where strong screening and anisotropic effects must be taken into account In this case, the growth probability has been generalized www.nature.com/scientificreports/. One of the best analytical results to describe the fractality of transitions such as BA-DLA and the one associated with the DBM, is the generalized Honda-Toyoki-Matsushita mean-field equation[22,23,24]: DMF(d, dw, η)
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