# Structural Properties of Vicsek-like Deterministic Multifractals

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## Abstract

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## 1. Introduction

## 2. Theoretical Background

#### 2.1. Multifractals

#### 2.2. Small-Angle Scattering

- $F\left(\mathit{q}\right)\to F\left(\beta \mathit{q}\right)$ if the particle’s length is scaled as $L\to \beta L$,
- $F\left(\mathit{q}\right)\to F\left(\mathit{q}\right){e}^{\u2013i\mathit{q}\xb7\mathit{a}}$ if the particle is translated by the vector $\mathit{r}\to \mathit{r}+\mathit{a}$,
- $F\left(\mathit{q}\right)=\left[{V}_{I}{F}_{I}\left(\mathit{q}\right)+{V}_{II}{F}_{II}\left(\mathit{q}\right)\right]/\left({V}_{I}+{V}_{II}\right)$, if the particle can be decomposed as a union of two non-overlapping subsets I and $II$.

## 3. Results and Discussions

#### 3.1. Construction of the Multifractal Model

#### 3.2. Dimension Spectra

#### 3.3. Pair Distance Distribution Function

#### 3.4. Small-Angle Scattering Form Factor

## 4. Conclusions

- If ${\beta}_{\mathrm{s}1}<<{\beta}_{\mathrm{s}2}$, the system is highly heterogeneous and structural parameters are more clearly visible in pddf (see Figure 3a), since the mass fractal region of the scattering intensity is very short Figure 4a). The scaling factor ${\beta}_{\mathrm{s}1}$ is extracted from the periodicity of large groups of distances, while ${\beta}_{\mathrm{s}2}$ can be extracted in a relatively good approximation, from the periodicity of smaller groups found inside larger ones. The number of fractal iterations coincide with the number of large distinct groups in pddf.
- If ${\beta}_{\mathrm{s}1}\lesssim {\beta}_{\mathrm{s}2}$, separation of pddf in distinct groups of distance is not very clear since the values of distances arising from each of the scaling factors begin to mix with each other (see Figure 3b,c), and thus extracting exact values of the scaling factors can become a very difficult task. However, in the reciprocal space, the corresponding mass fractal region of scattering intensity is characterized by a succession of maxima and minima on a power-law decay (generalized power-law decay) and the value of the largest scaling factor can be clearly estimated from the periodicity of minima. In addition, the fractal dimension can be obtained from the scattering exponent of this power-law decay while the fractal iteration number can be obtained from the number of the minima.
- If ${\beta}_{\mathrm{s}1}={\beta}_{\mathrm{s}2}$, the system reduces to a single scale fractal. Structural properties of such systems have been studied elsewhere (see Reference [30]).

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Acosta, K.L.; Wilkie, W.K.; Inman, D.J. Characterizing the pyroelectric coefficient for macro-fiber composites. Smart Mater. Struct.
**2018**, 27, 115001. [Google Scholar] [CrossRef] - Bica, I.; Anitas, E.M.; Lu, Q.; Choi, H.J. Effect of magnetic field intensity and γ-Fe2O3 nanoparticle additive on electrical conductivity and viscosity of magnetorheological carbonyl iron suspension-based membranes. Smart Mater. Struct.
**2018**, 27, 095021. [Google Scholar] [CrossRef] - Theerthagiri, J.; Durai, G.; Karuppasamy, K.; Arunachalam, P.; Elakkiya, V.; Kuppusami, P.; Maiyalagan, T.; Kim, H.S. Recent advances in 2-D nanostructured metal nitrides, carbides, and phosphides electrodes for electrochemical supercapacitors—A brief review. J. Ind. Eng. Chem.
**2018**, 67, 12–27. [Google Scholar] [CrossRef] - Kim, K.W.; Ji, S.H.; Park, B.S.; Yun, J.S. High surface area flexible zeolite fibers based on a core-shell structure by a polymer surface wet etching process. Mater. Des.
**2018**, 158, 98–105. [Google Scholar] [CrossRef] - Semitekolos, D.; Kainourgios, P.; Jones, C.; Rana, A.; Koumoulos, E.P.; Charitidis, C.A. Advanced carbon fibre composites via poly methacrylic acid surface treatment; surface analysis and mechanical properties investigation. Compos. Part B—Eng.
**2018**, 155, 237–243. [Google Scholar] [CrossRef] - Anagnostou, D.; Chatzigeorgiou, G.; Chemisky, Y.; Meraghni, F. Hierarchical micromechanical modeling of the viscoelastic behavior coupled to damage in SMC and SMC-hybrid composites. Compos. Part B—Eng.
**2018**, 151, 8–24. [Google Scholar] [CrossRef] [Green Version] - Bica, I.; Anitas, E.M. Magnetic flux density effect on electrical properties and visco-elastic state of magnetoactive tissues. Compos. Part B—Eng.
**2019**, 159, 13–19. [Google Scholar] [CrossRef] - El Naschie, M.S. Nanotechnology for the developing world. Chaos Soliton Fract.
**2006**, 30, 769. [Google Scholar] [CrossRef] - He, J.H.; Wan, Y.Q.; Xu, L. Nano-effects, quantum-like properties in electrospun nanofibers. Chaos Soliton Fract.
**2007**, 33, 26–37. [Google Scholar] [CrossRef] - Zhang, Z.; Gao, C.; Wu, Z.; Han, W.; Wang, Y.; Fu, W.; Li, X.; Xie, E. Toward efficient photoelectrochemical water-splitting by using screw-like SnO
_{2}nanostructures as photoanode after being decorated with CdS quantum dots. Nano Energy**2016**, 19, 318–327. [Google Scholar] [CrossRef] - Sichert, J.A.; Tong, Y.; Mutz, N.; Vollmer, M.; Fischer, S.; Milowska, K.Z.; García Cortadella, R.; Nickel, B.; Cardenas-Daw, C.; Stolarczyk, J.K.; et al. Quantum Size Effect in Organometal Halide Perovskite Nanoplatelets. Nano Lett.
**2015**, 15, 6521–6527. [Google Scholar] [CrossRef] [PubMed] - Zhu, W.; Esteban, R.; Borisov, A.G.; Baumberg, J.J.; Nordlander, P.; Lezec, H.J.; Aizpurua, J.; Crozier, K.B. Quantum mechanical effects in plasmonic structures with subnanometre gaps. Nat. Commun.
**2016**, 7, 11495. [Google Scholar] [CrossRef] [PubMed] - Newkome, G.R.; Wang, P.; Moorefield, C.N.; Cho, T.J.; Mohapatra, P.P.; Li, S.; Hwang, S.H.; Lukoyanova, O.; Echegoyen, L.; Palagallo, J.A.; et al. Nanoassembly of a fractal polymer: A molecular “Sierpinski hexagonal gasket”. Science
**2006**, 312, 1782–1785. [Google Scholar] [CrossRef] [PubMed] - Cerofolini, G.F.; Narducci, D.; Amato, P.; Romano, E. Fractal Nanotechnology. Nanoscale Res. Lett.
**2008**, 3, 381–385. [Google Scholar] [CrossRef] [Green Version] - Berenschot, E.J.W.; Jansen, H.V.; Tas, N.R. Fabrication of 3D fractal structures using nanoscale anisotropic etching of single crystalline silicon. J. Micromech. Microeng.
**2013**, 23, 055024. [Google Scholar] [CrossRef] - Li, C.; Zhang, X.; Li, N.; Wang, Y.; Yang, J.; Gu, G.; Zhang, Y.; Hou, S.; Peng, L.; Wu, K.; et al. Construction of Sierpiński Triangles up to the Fifth Order. J. Am. Chem. Soc.
**2017**, 139, 13749–13753. [Google Scholar] [CrossRef] [PubMed] - Li, N.; Zhang, X.; Gu, G.C.; Wang, H.; Nieckarz, D.; Szabelski, P.; He, Y.; Wang, Y.; Lü, J.T.; Tang, H.; et al. Sierpiński-triangle fractal crystals with the C3v point group. Chin. Chem. Lett.
**2015**, 26, 1198–1202. [Google Scholar] [CrossRef] - Zhang, X.; Li, R.; Li, N.; Gu, G.; Zhang, Y.; Hou, S.; Wang, Y. Sierpiński triangles formed by molecules with linear backbones on Au(111). Chin. Chem. Lett.
**2018**, 29, 967–969. [Google Scholar] [CrossRef] - Fan, J.A.; Yeo, W.H.; Su, Y.; Hattori, Y.; Lee, W.; Jung, S.Y.; Zhang, Y.; Liu, Z.; Cheng, H.; Falgout, L.; et al. Fractal design concepts for stretchable electronics. Nat. Commun.
**2014**, 5, 3266. [Google Scholar] [CrossRef] [Green Version] - Mandelbrot, B.B. The Fractal Geometry of Nature; W.H. Freeman: New York, NY, USA, 1982; p. 460. [Google Scholar]
- Filoche, M.; Sapoval, B. Transfer Across Random versus Deterministic Fractal Interfaces. Phys. Rev. Lett.
**2000**, 84, 5776–5779. [Google Scholar] [CrossRef] [Green Version] - Khoshhesab, M.M.; Li, Y. Mechanical behavior of 3D printed biomimetic Koch fractal contact and interlocking. Extrem. Mech. Lett.
**2018**, 24, 58–65. [Google Scholar] [CrossRef] - Nikbakht, M. Radiative heat transfer in fractal structures. Phys. Rev. B
**2017**, 96, 125436. [Google Scholar] [CrossRef] [Green Version] - Zhang, B.; Xiang, M.; Zhang, Q.; Zhang, Q. Preparation and characterization of bioinspired three-dimensional architecture of zirconia on ceramic surface. Compos. Part B—Eng.
**2018**, 155, 77–82. [Google Scholar] [CrossRef] - Mayama, H.; Tsujii, K. Menger sponge-like fractal body created by a novel template method. J. Chem. Phys.
**2006**, 125, 124706. [Google Scholar] [CrossRef] [PubMed] - Feigin, L.A.; Svergun, D.I. Structure Analysis by Small-Angle X-ray and Neutron Scattering; Springer: Boston, MA, USA, 1987; p. 335. [Google Scholar]
- Gille, W. Particle and Particle Systems Characterization: Small-Angle Scattering (SAS) Applications, 1st ed.; CRC Press: Boca Raton, FL, USA, 2013; p. 336. [Google Scholar]
- Teixeira, J. Small-angle scattering by fractal systems. J. Appl. Cryst.
**1988**, 21, 781–785. [Google Scholar] [CrossRef] [Green Version] - Schmidt, P.W.; Dacai, X. Calculation of the small-angle x-ray and neutron scattering from nonrandom (regular) fractals. Phys. Rev. A
**1986**, 33, 560–566. [Google Scholar] [CrossRef] - Cherny, A.Y.; Anitas, E.M.; Osipov, V.A.; Kuklin, A.I. Deterministic fractals: Extracting additional information from small-angle scattering data. Phys. Rev. E
**2011**, 84, 036203. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Anitas, E.M.; Slyamov, A.M. Small-angle scattering from deterministic mass and surface fractal systems. Proc. Rom. Acad. A
**2018**, 19, 353–360. [Google Scholar] - Cherny, A.Y.; Anitas, E.M.; Osipov, V.A.; Kuklin, A.I. The structure of deterministic mass and surface fractals: Theory and methods of analyzing small-angle scattering data. Phys. Chem. Chem. Phys.
**2019**. [Google Scholar] [CrossRef] - Anitas, E.M.; Slyamov, A.; Todoran, R.; Szakacs, Z. Small-Angle Scattering from Nanoscale Fat Fractals. Nanoscale Res. Lett.
**2017**, 12, 389. [Google Scholar] [CrossRef] - Anitas, E.M.; Slyamov, A. Structural characterization of chaos game fractals using small-angle scattering analysis. PLoS ONE
**2017**, 12, e0181385. [Google Scholar] [CrossRef] [PubMed] - Vicsek, T. Fractal Growth Phenomena, 2nd ed.; World Scientific: Singapore, 1992; p. 528. [Google Scholar]
- Gouyet, J.F. Physics and Fractal Structures; Elsevier: Masson, UK, 1996; p. 234. [Google Scholar]
- Arneodo, A.; Decoster, N.; Roux, S. A wavelet-based method for multifractal image analysis. I. Methodology and test applications on isotropic and anisotropic random rough surfaces. Eur. Phys. J. B
**2000**, 15, 567–600. [Google Scholar] [CrossRef] - Decoster, N.; Roux, S.; Arnéodo, A. A wavelet-based method for multifractal image analysis. II. Applications to synthetic multifractal rough surfaces. Eur. Phys. J. B
**2000**, 15, 739–764. [Google Scholar] [CrossRef] - Chhabra, A.; Jensen, R.V. Direct Determination of the f (alpha) Singularity Spectrum. Phys. Rev. Lett.
**1989**, 62, 1327. [Google Scholar] [CrossRef] [PubMed] - Kantelhardt, J.W.; Zschiegner, S.A.; Koscielny-Bunde, E.; Havlin, S.; Bunde, A.; Stanley, H. Multifractal detrended fluctuation analysis of nonstationary time series. Phys. A
**2002**, 316, 87–114. [Google Scholar] [CrossRef] [Green Version] - Muzy, J.F.; Bacry, E.; Arneodo, A. Multifractal formalism for fractal signals: The structure-function approach versus the wavelet-transform modulus-maxima method. Phys. Rev. E
**1993**, 47, 875–884. [Google Scholar] [CrossRef] [Green Version] - Stuhrmann, H.B. Small-angle scattering and its interplay with crystallography, contrast variation in SAXS and SANS. Acta Cryst.
**2008**, 64, 181–191. [Google Scholar] [CrossRef] - Izumi, A.; Shudo, Y.; Nakao, T.; Shibayama, M. Cross-link inhomogeneity in phenolic resins at the initial stage of curing studied by 1H-pulse NMR spectroscopy and complementary SAXS/WAXS and SANS/WANS with a solvent-swelling technique. Polymer
**2016**, 103, 152–162. [Google Scholar] [CrossRef] - Dierolf, M.; Menzel, A.; Thibault, P.; Schneider, P.; Kewish, C.M.; Wepf, R.; Bunk, O.; Pfeiffer, F. Ptychographic X-ray computed tomography at the nanoscale. Nature
**2010**, 467, 436–439. [Google Scholar] [CrossRef] - Martin, J.E.; Hurd, A.J. Scattering from fractals. J. Appl. Cryst.
**1987**, 20, 61–78. [Google Scholar] [CrossRef] - Schmidt, P.W. Small-angle scattering studies of disordered, porous and fractal systems. J. Appl. Cryst.
**1991**, 24, 414–435. [Google Scholar] [CrossRef]

**Figure 1.**(Color online) First three iterations of the two-scale multifractal models. Upper row: ${\beta}_{{\mathrm{s}}_{1}}=0.1$ and ${\beta}_{{\mathrm{s}}_{2}}=0.8$ (Model M1). Note that for $m=3$ the disks of radii ${l}_{0}{\beta}_{{\mathrm{s}}_{1}}^{3}/2=0.0005{l}_{0}$ are too small to be seen in the figure (at the given size). Middle row: ${\beta}_{{\mathrm{s}}_{1}}=0.2$ and ${\beta}_{{\mathrm{s}}_{2}}=0.6$ (Model M2). Lower row: ${\beta}_{{\mathrm{s}}_{1}}=0.3$ and ${\beta}_{{\mathrm{s}}_{2}}=0.4$ (Model M3). Black, orange and green colors denote the disks generated at iterations $m=1$, $m=2$, and respectively at $m=3$.

**Figure 2.**(Color online) Dimension spectra ${D}_{\mathrm{s}}$ for the three multifractal models: M1 (black), M2 (red), M3 (green). The intersection of the vertical line with each horizontal (dashed) line gives the box-counting dimension ${D}_{0}$.

**Figure 3.**(Color online) The coefficients ${C}_{p}$ (orange dots) in Equation (13) for the pair distribution function of the considered multifractal models at $m=4$. (

**a**) Model M1; (

**b**) Model M2; (

**c**) Model M3. For a better visualization of pddf grouping the vertical line (blue) for each distance is shown.

**Figure 4.**(Color online) Scattering form factor (Equation (21)) for monodisperse (black) and polydisperse (red) multifractal models at $m=4$. (

**a**) Model M1; (

**b**) Model M2; (

**c**) Model M3. Vertical lines indicate the lower and upper edges of mass fractal region.

**Figure 5.**(Color online) The quantity $I\left(q\right){q}^{{\mathrm{D}}_{0}}$, where ${D}_{0}$ is the box-counting fractal dimension for monodisperse (black) and polydisperse (red) multifractal models at $m=4$. (

**a**) Model M1; (

**b**) Model M2; (

**c**) Model M3. Vertical lines indicate the lower and upper edges of the mass fractal region.

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**MDPI and ACS Style**

Anitas, E.M.; Marcelli, G.; Szakacs, Z.; Todoran, R.; Todoran, D.
Structural Properties of Vicsek-like Deterministic Multifractals. *Symmetry* **2019**, *11*, 806.
https://doi.org/10.3390/sym11060806

**AMA Style**

Anitas EM, Marcelli G, Szakacs Z, Todoran R, Todoran D.
Structural Properties of Vicsek-like Deterministic Multifractals. *Symmetry*. 2019; 11(6):806.
https://doi.org/10.3390/sym11060806

**Chicago/Turabian Style**

Anitas, Eugen Mircea, Giorgia Marcelli, Zsolt Szakacs, Radu Todoran, and Daniela Todoran.
2019. "Structural Properties of Vicsek-like Deterministic Multifractals" *Symmetry* 11, no. 6: 806.
https://doi.org/10.3390/sym11060806