Cluster growth model for treelike structures

Havlin, Shlomo; Nossal, Ralph; Trus, Benes L.

doi:10.1103/physreva.32.3829

Cited by 14 publications

(9 citation statements)

References 11 publications

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“…4e,f (see Methods). As far as we know, this control over the fractal dimension of the clusters and the range it spans, as well as over the morphology of the clusters, has not been obtained before under any other related scheme of tunable fractality 36 37 38 39 or morphology 40 41 42 43 .…”

Section: Resultsmentioning

confidence: 76%

Fractality à la carte: a general particle aggregation model

Nicolás-Carlock

Carrillo-Estrada

Dossetti

2016

Sci Rep

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In nature, fractal structures emerge in a wide variety of systems as a local optimization of entropic and energetic distributions. The fractality of these systems determines many of their physical, chemical and/or biological properties. Thus, to comprehend the mechanisms that originate and control the fractality is highly relevant in many areas of science and technology. In studying clusters grown by aggregation phenomena, simple models have contributed to unveil some of the basic elements that give origin to fractality, however, the specific contribution from each of these elements to fractality has remained hidden in the complex dynamics. Here, we propose a simple and versatile model of particle aggregation that is, on the one hand, able to reveal the specific entropic and energetic contributions to the clusters’ fractality and morphology, and, on the other, capable to generate an ample assortment of rich natural-looking aggregates with any prescribed fractal dimension.

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Section: Resultsmentioning

confidence: 76%

Fractality à la carte: a general particle aggregation model

Nicolás-Carlock

Carrillo-Estrada

Dossetti

2016

Sci Rep

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“…Nevertheless, according to Havlin et al, [17] it can be embedded in an infinite Euclidean space g $ (R/l) D f /D x . Equation (2) can then be given by Equation (1).…”

Section: Determination Of the Fractal Characteristic Of Nanofiber-netmentioning

confidence: 99%

Determination of the Fractal Characteristic of Nanofiber-Network Formation in Supramolecular Materials

Liu

Sawant

2002

ChemPhysChem

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“…Three essential factors—the length between branch points, growth site activity, and each branching rate (z)—determine the type of Caley tree type and alter its geometric and physical properties (e.g., skeleton dimension) [ 51 , 60 ]. Networks characterized by Caley tree fractals maintain a constant branching rate (z), and newly formed ‘daughter’ segments maintain identical dimensions to the precursor ‘mother segment’, whereby perpetual recurrence of growth cycles forms the supramolecular hierarchical network [ 51 , 60 , 61 ]. DBS/PEG gels form trimmed Cayley Trees as they contain dead-ends, the branching rate, z, and segment length ζ, are constant throughout the crystal growth process, and branching occurs at all sites simultaneously [ 3 , 51 , 60 ].…”

Section: Resultsmentioning

confidence: 99%

“…When fiber branching occurs at the tip of the growing 1D fiber Cayley treelike, fractal networks are observed [ 3 , 67 , 68 , 69 ]. Cayley Trees are embedded in infinite Euclidian space; the total number of branching sites increases exponentially with R, the distance from the origin to the gth tree shell [ 51 , 61 , 70 ]. The exponent, D f of Equation (2), represents the tree fractal dimension in the Euclidean space.…”

Section: Resultsmentioning

confidence: 99%

“…The mass of the fractal (M) scales over chemical distance (ξ), the shortest vector between two branching sites and the chemical distance is not necessarily equal to the covering shells’ Euclidean distance represented by Equation (3) [ 60 , 61 ]:

where D ξ is the fractal dimension in the chemical space and the fractal gyration radius (R) is obtained by the combination of Equations (2) and (3) [ 60 , 61 ]:

…”

Section: Resultsmentioning

confidence: 99%

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Supramolecular Fractal Growth of Self-Assembled Fibrillar Networks

Nasr

Leung

Auzanneau

et al. 2021

Gels

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Complex morphologies, as is the case in self-assembled fibrillar networks (SAFiNs) of 1,3:2,4-Dibenzylidene sorbitol (DBS), are often characterized by their Fractal dimension and not Euclidean. Self-similarity presents for DBS-polyethylene glycol (PEG) SAFiNs in the Cayley Tree branching pattern, similar box-counting fractal dimensions across length scales, and fractals derived from the Avrami model. Irrespective of the crystallization temperature, fractal values corresponded to limited diffusion aggregation and not ballistic particle–cluster aggregation. Additionally, the fractal dimension of the SAFiN was affected more by changes in solvent viscosity (e.g., PEG200 compared to PEG600) than crystallization temperature. Most surprising was the evidence of Cayley branching not only for the radial fibers within the spherulitic but also on the fiber surfaces.

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Cluster growth model for treelike structures

Cited by 14 publications

References 11 publications

Fractality à la carte: a general particle aggregation model

Fractality à la carte: a general particle aggregation model

Determination of the Fractal Characteristic of Nanofiber-Network Formation in Supramolecular Materials

Supramolecular Fractal Growth of Self-Assembled Fibrillar Networks

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