Journal of Interfaces, Thin Films, and Low dimensional systems

Fractal analysis of ballistic deposition model with power-law distributed noise

Document Type : Original Article

Authors

1 Faculty of Physics, North Tehran Branch, Islamic Azad University, Tehran, Iran

2 Faculty of Physics, East Tehran Branch, Islamic Azad University, Iran

3 Faculty of Condensed Matter Physics, Department of Physics, Alzahra University, Tehran, Iran

Abstract

The ballistic deposition model with power-law distributed noise (BD-PLN) has been simulated and investigated. Analysis of scaling exponents and statistical features seems essential in understanding the mechanism of noise in the phenomena. In the BD-PLN model, heterogeneous particles with rod-like shapes are deposited during growth time and lead to the forming of porous structures. By using the Hoshen-Kopelman algorithm, porous structures are converted to contour loops, and the fractal properties of the loops are considered. The fractal dimension of each loop, D_f, the fractal dimension of the contour set, d, the generalized dimensions, D_q, and the mass function, τ_q are calculated. The fractal dimension, d, increases as d = a + bμ^c versus μ exponent, and remains constant for μ >"μ" _c=3, where μ is the decay of the noise amplitude. The results indicate that augmentation of μ exponent and conspicuity of the Gaussian ballistic deposition model prepare to decrease in structure porosity and multi-affinity, and also increase in contour loops area and perimeter.

Keywords

Main Subjects

Article Title [Persian]

تحلیل فرکتالی مدل رشد بالستیک با نوفه توزیعی قانون توان

Authors [Persian]

  • معصومه رحیمی 1
  • سکینه حسین آبادی 2
  • انیر علی مسعودی 3
  • لاله فرهنگ متین 1

1 گروه فیزیک، دانشگاه آزاد اسلامی واحد تهران شمال، تهران، ایران.

2 گروه فیزیک، دانشگاه آزاد اسلامی واحد تهران شرق، تهران شرق، ایران.

3 گروه ماده چکال، دانشکده فیزیک، دانشگاه الزهرا، تهران، ایران

[1] T. Shitara, D. D. Vvedensky, M. R. Wilby, J. Zhang, J. H. Neave, and B. A. Joyce, “Step-density variations and reflection high-energy electrondiffraction intensity oscillations during epitaxial growth on vicinal GaAs(001).” Physical Review B, 46 (1992) 6815.
[2] J. W. Evans, P. A. Thiel, and M. C. Bartelt, “Morphological evolution during epitaxial thin film growth: Formation of 2D islands and 3D mounds.” Surface Science Report, 61 (2006) 1.
[3] D. A. Mirabella and C. M. Aldao, “Surface growth by random deposition of rigid and wetting clusters.” Surface Science, 646 (2016) 282-287.
[4] F. L. Forgerini and W. Figueiredo, “Random deposition of particles of different sizes.” Physical Review E, 79 (2009) 041602. [5] Y. Pellegrini and R. Jullien, “Ballistic deposition of clusters.” Physica A: Statistical Mechanics and its Applications, 165 (1990) 19-30.
[6] R. C. Buceta, D. Hansmann, and B. von Haeften, “Revisiting random deposition with surface relaxation: approaches from growth rules to the Edwards-Wilkinson equation.” Journal of Statistical Mechanics: Theory and Experiment, 2014 (2014) 12028.
[7] S. Hosseinabadi, Z. Karimi, and A. A. Masoudi, “Random deposition with surface relaxation model accompanied by long-range correlated noise.” Physica A: Statistical Mechanics and its Applications, 560 (2020) 125130.
[8] S. Hosseinabadi, A. A. Masoudi, and M. S. Movahed, “Solid-on-solid model for surface growth in 2+1 dimensions.” Physica B: Condensed Matter, 405 (2010) 8.
[9] H. F. El-Nashar, W. Wang, and H. A. Cerdeira, “Surface growth kinetics and morphological structural transition in a (2 + 1)-dimensional deposition model.” Journal of Physics Condensed Matter, 8 (1996) 3271.
[10] S. K. Das, D. Banerjee, and J. N. Roy, “Particle Shape-Induced Correlation Effect in Random Deposition in 1+1 Dimension and Related Effect in Ballistic Deposition.” Surface Review and Letters, 28 (2021) 2050043.
[11] I. Sumirat, Y. Ando, and S. Shimamura, “Theoretical consideration of the effect of porosity on thermal conductivity of porous materials.” Journal of Porous Materials, 13 (2006) 439–443.
[12] R. Dasgupta, S. Roy, and S. Tarafdar, “Correlation between porosity, conductivity and permeability of sedimentary rocks |a ballistic deposition model.” Physical Review A, 275 (2000) 22-32.
[13] Z. Ebrahiminejad, H. Hamzehpour, and S. F. Masoud, “Electrical conductivity of thin films grown by deposition of random clusters of particles.” Journal of Material Science: Materials in Electronics, 27 (2020) 1-10.
[14] A. Mortezaali, S. Ramezani Sani, and F. Javani Jooni, “Correlation Between Porosity of Porous Silicon and Optoelectronic Properties.” Journal of Non-Oxide Glasses, 1 (2009) 293 – 299.
[15] J. M. Kim and H. Choi, “Depinning Transition of the Quenched Edwards-Wilkinson Equation.” Journal of the Korean Physical Society, 48 (2006) 241-244.
[16] S. Hosseinabadi and A. A. Masoudi, “Random deposition with a power-law noise model: Multiaffine analysis.” Physical Review E, 99 (2019) 012130.
[17] M. A. Rubio, C. A. Edwards, A. Dougherty, and J. P. Gollub, “Self-affine fractal interfaces from immiscible displacement in porous media.” Physical Review Letters, 63 (1989) 1685.
[18] F. Gerges, X. Geng, H. Nassif, and M. Boufadel, “Anisotropic Multifractal Scaling of Mount Lebanon Topography: Approximate Conditioning.” Fractals, 29 (2021) 2150112.
[19] S. Laurent, “Fractals and Multifractals in Ecology and Aquatic Science.” CRC Press, 2009, Ch. 7.
[20] R. Mulligan, “Fractal analysis of highly volatile markets: an application to technology equities.” The Quarterly Review of Economics andFinance, Elsevier, 44 (2004)155–179.
[21] S. Kamenshchikov, “Transport Catastrophe Analysis as an Alternative to a Monofractal Description: Theory and Application to Financial Crisis Time Series.” Journal of Chaos, 2014 (2014) 346743.
[22] F. Zappasodi, E. Olejarczyk, L. Marzetti, and G. Assenza, “Fractal Dimension of EEG Activity Senses Neuronal Impairment in Acute Stroke.” PLOS ONE, 9 (2014) 100199.
[23] E. E. Mozo Luis, F. A. Oliveira, and T. A. de Assis, “Accessibility of the surface fractal dimension during film growth.” Physical Review E, 107, (2023) 034802.
[24] S. Qureshi, M. A. Akanbi, A. A. Shaikh, A. S. Wusu, O. M. Ogunlaran, W. Mahmoud, and M. S. Osman, “A new adaptive nonlinear numerical method for singular and stiff differential problems.” Alexandria Engineering Journal, 74 (2023) 585-597.
[25] A. Padder, L. Almutairi, S. Qureshi, A. Soomro, A. Afroz, E. Hincal, and A.Tassaddiq, “Dynamical Analysis of Generalized Tumor Model with Caputo Fractional-Order Derivative.” fractal and fractional, 7 (2023) 7030258.
[26] B. B. Mandelbrot, “The fractal geometry of nature.”, Echo Point Books and Media, LLC., 2021, ch. 2. [27] D. Sornette, “Critical Phenomena in Natural Sciences.” Chaos, Fractals, Self-organization and Disorder: Concepts and Tools SpringerVerlag, Heidelberg, 2000, 123-160.
[28] S. Hosseinabadi, M. A. Rajabpour, M. Sadegh Movahed, and S. M. Vaez Allaei, “Geometrical exponents of contour loops on synthetic multifractal rough surfaces: Multiplicative hierarchical cascade p model.” Physical Review E, 85 (2012) 031113.
[29] S. Hosseinabadi and M. Rajabi, “Roughness kinetic and multiaffinity of anisotropic etched silicon.” Superlattices and Microstructures, 102 (2017) 180-188.
[30] W. Bing, W. Yan, and W. Ziqin, “Multifractal behavior of solid-on-solid growth.” Solid State Communications, 96 (1995) 69-72.
[31] W. G. Hanan and D. M. Heffernan, “Multifractal analysis of the branch structure of diffusionlimited aggregates.” Physical Review E, 85 (2012) 021407.
[32] A. Chaudhari, C. C. S. Yan, and S. L. Lee, “Multifractal analysis of growing surfaces.” Applied Surface Science, 238 (2004) 513-517.
[33] S. Hosseinabadi, S. M. S. Movahed, M. A. Rajabpour, and S. M. V. Allaei, “Dynamical and geometrical exponents of self-affine rough surfaces on regular and random lattices.” Journal of Statistical Mechanics: Theory and Experiment, 12 (2014) 12023.
[34] J. Kondev, and C. L. Henley, “Geometrical Exponents of Contour Loops on Random Gaussian Surfaces.” Physical Review Letters, 74 (1995) 4580.
[35] A. A. Saberi, M. D. Niry, S. M. Fazeli, M. R. Rahimi Tabar, and S. Rouhani, “Conformal invariance of isoheight lines in a twodimensional Kardar-Parisi-Zhang surface.” Physical Review E, 77 (2008) 051607.
[36] S. Hosseinabadi, “Iso-height lines of multifractal etched silicon rough surfaces.” Materials Science in Semiconductor Processing, 88 (2018) 79–85. [37] M. A. Rajabpour, and S. M. Vaez Allaei, “Scaling relations for contour lines of rough surfaces.” Physical Review E, 80 (2009) 011115.
[38] I. Giordanelli, N. Posé, M. Mendoza, and H. J. Herrmann, “Conformal Invariance of Graphene Sheets.” Scientific Reports, 6 (2016) 22949.
[39] J. Hoshen and R. Kopelman, “Percolation and cluster distribution. I. Cluster multiple labeling technique and critical concentration algorithm.” Physical Review B, 14 (1976) 3438.
[40] M. Bouda, J. S. Caplan, and J. E. Saiers, “BoxCounting Dimension Revisited: Presenting an Efficient Method of Minimizing Quantization Error and an Assessment of the Self-Similarity of Structural Root Systems.” Frontiers in Plant Science, Sec. Technical Advances in Plant Science, 7 (2016) 149.
[41] J. Z. Liu, L. D. Zhang, and Y. H. Guang, “Fractal Dimension in Human Cerebellum Measured by Magnetic Resonance Imaging.” Biophysical Journal, 85 (2003) 4041-4046.
[42] T. Babadagli and K. Develi, “On the application of methods used to calculate the fractal dimension of fracture surfaces.” Fractals, 9 (2001) 105-128.
[43] C. K. Peng, S. Buldyrev, A. Goldberger, S. Havlin, F. Sciortino, M. Simons, and H. E. Stanley, “Long-range correlations in nucleotide sequences.” Nature (London), 356 (1992) 168- 170.
[44] Z. R. Struzik, and A. P. J. M. Siebes, “Wavelet transform based multifractal formalism in outlier detection and localisation for financial time series.” Physica A, 309 (2002) 388.
[45] J. Zhang, C. Wei, X. Chu, V. Vandeginste, and W. Ju, “Classification of Pore–fracture Combination Types in Tectonic Coal Based on Mercury Intrusion Porosimetry and Nuclear Magnetic Resonance.” ACS Omega, 5 (2020) 19385-19401.
[46] A. Dathe, A. M. Tarquis, and E. Perrier, “Multifractal analysis of the pore- and solidphases in binary two-dimensional images of natural porous structures.” Geoderma, 134 (2006) 318–326.
[47] S. Hosseinabadi and A. A. Masoudi, “Random deposition with a power-law noise model: Multiaffine analysis.” Physical Review E, 99 (2019) 012130.
[48] H. Chen, X. Sun, H. Chen, Z. Wu, and B. Wang, “Some problems in multifractal spectrum computation using a statistical method.” New Journal of Physics, 6 (2004) 84.
[49] A. Chaudhari, C. C. Sanders Yan and S. L. Lee, “Eley–Rideal diffusion limited reactions over rough surface.” Physical Chemistry Chemical Physics, 4 (2002) 5330–5334.
Journal of Interfaces, Thin Films, and Low dimensional systems
Volume 7, Issue 2
July 2024
Pages 747-755