Fractal Nanotechnology

Cerofolini, G. F.; Narducci, Dario; Amato, P.; Romano, E.

doi:10.1007/s11671-008-9170-0

Cited by 26 publications

(25 citation statements)

References 25 publications

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“…r r 2 sin θdrdθdϕ: (16) Note that since balls represent here the basic units of the fractal, we have chosen the notation F 0 (q) instead of F(q), in spirit of Eq. (9).…”

Section: Scattering From a Ball And From A Trianglementioning

confidence: 99%

“…In particular, models based on deterministic or exact self-similar fractals (i.e., fractals that are self-similar at every point, such as the Koch snowflake, Cantor set, or Mandelbrot cube) have been frequently used, since this type of fractals allows an analytical representation of various geometrical parameters (radius of gyration) or of the scattering intensity spectrum. Although for most fractals generated by natural processes, this is only an approximation, in the case of deterministic nano-and micro-materials obtained recently such as 2D Sierpinski gaskets [15] and Cantor sets [16], or 3D Menger sponge [17] and octahedral structures [18], this approximation becomes exact.…”

Section: Introductionmentioning

confidence: 99%

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Small-Angle Scattering from Mass and Surface Fractals

Anitas¹

2018

Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

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The concepts of mass and surface fractals are introduced, and the corresponding smallangle scattering (SAS; X-rays, neutrons) intensities are computed. It is shown how to resolve the fractal structure of various complex systems from experimental scattering measurements, and how obtained data are related to specific features of the fractal models. We present and discuss various mass and surface fractal structures, including fractals generated from iterated function systems and cellular automata. In addition to the fractal dimension and the overall fractal size, the suggested analysis allows us to obtain the iteration number, the number of basic units which form the fractal and the scaling factor.

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“…r r 2 sin θdrdθdϕ: (16) Note that since balls represent here the basic units of the fractal, we have chosen the notation F 0 (q) instead of F(q), in spirit of Eq. (9).…”

Section: Scattering From a Ball And From A Trianglementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Small-Angle Scattering from Mass and Surface Fractals

Anitas¹

2018

Complexity in Biological and Physical Systems - Bifurcations, Solitons and Fractals

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“…An intimate link between structural fractal properties of designed, nanotextured materials and functional advantages (e.g., detection sensitivity) has been demonstrated 5 , and synthetic fractal materials are finding applications in sensing, molecular electronics, high-performance filtration, sunlight collection, surface charge storage, and catalysis, among myriad other uses 7,8 . Many fractal fabrication efforts have relied on top-down patterning of surfaces 9 . The bottom-up design of supramolecular fractal topologies – both deterministic (e.g., Sierpinski’s triangles) 10,11 and stochastic fractals (e.g., arborols) 12,13 – has been performed with small molecule building blocks such as inorganic metal-ligand complexes or synthetic dendritic polymers utilizing co-ordinate or covalent bonds, respectively.…”

Section: Main Textmentioning

confidence: 99%

Stimulus-responsive Self-Assembly of Protein-Based Fractals by Computational Design

Hernandez

Hansen

Zhu

et al. 2018

Preprint

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23 †Contributed equally to this work 24 Main Text: 1 Fractional-dimensional (fractal) geometrya property of shapes that are invariant or nearly 2 invariant to scale magnification or contraction across many length scalesis a common feature of 3 many natural objects 1,2 . Fractal forms are ubiquitous in geology, e.g., in the architecture of 4 mountain ranges, coastlines, snowflakes, and in physiology, e.g., neuronal and capillary networks, 5 and nasal membranes, where highly efficient molecular exchange occurs due to a fractal-induced 6 high surface area:volume ratio 3 . Fabrication of fractal-like nanomaterials affords high physical 7 connectivity within patterned objects 4 , ultrasensitive detection of target binding moieties by 8

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“…Given the contrast, concentration of fractals, and the absolute value of intensity, the fractal volume can be determined from Eq. (8).…”

Section: General Remarks On Small-angle Scatteringmentioning

confidence: 99%

Deterministic fractals: Extracting additional information from small-angle scattering data

et al. 2011

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The small-angle scattering curves of deterministic mass fractals are studied and analyzed in momentum space. In the fractal region, the curve I(q)q(D) is found to be log-periodic with good accuracy, and the period is equal to the scaling factor of the fractal. Here, D and I(q) are the fractal dimension and the scattering intensity, respectively. The number of periods of this curve coincides with the number of fractal iterations. We show that the log-periodicity of I(q)q(D) in the momentum space is related to the log-periodicity of the quantity g(r)r(3-D) in the real space, where g(r) is the pair distribution function. The minima and maxima positions of the scattering intensity are estimated explicitly by relating them to the pair distance distribution in real space. It is shown that the minima and maxima are damped with increasing polydispersity of the fractal sets; however, they remain quite pronounced even at sufficiently large values of polydispersity. A generalized self-similar Vicsek fractal with controllable fractal dimension is introduced, and its scattering properties are studied to illustrate the above findings. In contrast with the usual methods, the present analysis allows us to obtain not only the fractal dimension and the edges of the fractal region, but also the fractal iteration number, the scaling factor, and the number of structural units from which the fractal is composed.

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Fractal Nanotechnology

Cited by 26 publications

References 25 publications

Small-Angle Scattering from Mass and Surface Fractals

Small-Angle Scattering from Mass and Surface Fractals

Stimulus-responsive Self-Assembly of Protein-Based Fractals by Computational Design

Deterministic fractals: Extracting additional information from small-angle scattering data

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