Diffusion Limited Aggregation via Python: Dendritic Structures and Algorithmic Art

Year 2024, , 99 - 112, 29.09.2024

Çağdaş Allahverdi , Yıldız Allahverdi

https://doi.org/10.59313/jsr-a.1454389

Abstract

Diffusion limited aggregation (DLA) has attracted much attention due to its simplicity and broad applications in physics such as nano and microparticle aggregations. In this study, the algorithm of DLA is written with Python. Python's Turtle library is used to plot the aggregate on the computer monitor as it grows. The algorithm is run on the Raspberry Pi. A cheap and portable medium is created for DLA simulation. Two different options are placed in the algorithm. The first path does not allow the primary particle to turn outside of the aggregate after the collision. However, the second one allows the percolation of the primary particle both inside and outside of the aggregate. The spherical dendritic structures consisting of 500-2000 primary particles are obtained by the algorithm. The fractal dimension of these structures is around 1.68. Their porosity is found below 50%. Their gyration radii are also calculated. Beyond scientific investigation, examples of algorithmic art using these dendritic structures are also given.

Keywords

Diffusion limited aggregation, Random walk, Fractal dimension, Porosity, Raspberry Pi, Algorithmic art

Supporting Institution

Toros University

References

• [1] J. Nan, M. Yao, T. Chen, Z. Wang, Q. Li and D. Zhan, “Experimental and numerical characterization of floc morphology: role of changing hydraulic retention time under flocculation mechanisms”, Environ. Sci. Pollut. Res., vol. 23, pp. 3596-3608, 2016, doi: 10.1007/s11356-015-5539-7.
• [2] B. Lim and Y. Xia, “Metal nanocrystals with highly branched morphologies”, Angew. Chem. Int. Ed., vol. 50, no. 1, pp. 76-85, 2011, doi: 10.1002/anie.201002024.
• [3] H. Tronnolone et al., “Difusion-limited growth of microbial colonies”, Sci. Rep., vol. 8, pp. 1-11, 2018, doi: 10.1038/s41598-018-23649-z.
• [4] J. Zhang, J. Luo and Z. Liu, “DLA simulation with sticking probability for viscous fingering”, in 2011 International Conference on Consumer Electronics, Communications and Networks (CECNet), Xianning, China, 2011, pp. 4044-4047, doi: 10.1109/CECNET.2011.5768540.
• [5] I. M. Irurzun, P. Bergero, V. Mola, M. C. Cordero, J. L. Vicente and E. E. Mola, “Dielectric breakdown in solids modeled by DBM and DLA”, Chaos, Solitons and Fractals, vol. 13, no. 6, pp. 1333-1343, 2002, doi: 10.1016/S0960-0779(01)00142-4.
• [6] T. A. Witten and L. M. Sander, “Diffusion-limited aggregation”, Phys. Rev. B, vol. 27, no. 9, pp. 5686-5697, 1983, doi: 10.1103/PhysRevB.27.5686.
• [7] M. Wozniak, F. R. A. Onofri, S. Barbosa, J. Yon and J. Mroczka, “Comparison of methods to derive morphological parameters of multi-fractal samples of particle aggregates from TEM images”, J. Aerosol Sci., vol. 47, pp. 12-26, 2012, doi: 10.1016/j.jaerosci.2011.12.008.
• [8] J. Mroczka, M. Woźniak and F. R. A. Onofri, “Algorithms and methods for analysis of the optical structure factor of fractal aggregates”, Metrol. Meas. Syst., vol. XIX, no. 3, pp. 459-470, 2012, doi: 10.2478/v10178-012-0039-2.
• [9] D. Liu, W. Zhou, Z. Qiu and X. Song, “Fractal simulation of flocculation processes using a diffusion-limited aggregation model”, Fractal Fract., vol. 1, pp. 1-13, 2017, doi: 10.3390/fractalfract1010012.
• [10] R. Wang, A. K. Singh, S. R. Kolan and E. Tsotsas, “Fractal analysis of aggregates: correlation between the 2D and 3D box-counting fractal dimension and power law fractal dimension”, Chaos, Solitons and Fractals, vol. 160, pp. 1-13, 2022, doi: 10.1016/j.chaos.2022.112246.
• [11] A. M. Zsaki, “Hardware-accelerated generation of 3D diffusion-limited aggregation structures”, J. Parallel Distrib. Comput., vol. 97, pp. 24-34, 2016, doi: 10.1016/j.jpdc.2016.06.009.
• [12] N. H. Borzęcka, B. Nowak, R. Pakuła, R. Przewodzki and J. M. Gac, “Cellular automata modeling of silica aerogel condensation kinetics”, Gels, vol. 7, pp. 1-12, 2021, doi: 10.3390/gels7020050.
• [13] M. Polimeno, C. Kim and F. Blanchette, “Toward a realistic model of diffusion-limited aggregation: rotation, size-dependent diffusivities, and settling”, ACS Omega, vol. 7, no. 45, pp. 40826-40835, 2022, doi: 10.1021/acsomega.2c03547.
• [14] O. Tomchuk, “Models for simulation of fractal-like particle clusters with prescribed fractal dimension”, Fractal Fract., vol. 7, pp. 1-25, 2023, doi: 10.3390/fractalfract7120866.
• [15] S. J. Johnston and S. J. Cox, “The Raspberry Pi: a technology disrupter, and the enabler of dreams”, Electronics, vol. 6, pp. 1-7, 2017, doi: 10.3390/electronics6030051.
• [16] N. Liu et al., “Dynamic mechanism of dendrite formation in Zhoukoudian, China”, Appl. Sci., vol. 13, pp. 1-10, 2023, doi: 10.3390/app13042049.
• [17] D. D. Ruzhitskaya, S. B. Ryzhikov and Y. V. Ryzhikova, “Features of self-organization of objects with a fractal structure of dendritic geometry”, Moscow Univ. Phys., vol. 76, no. 5, pp. 253-263, 2021, doi: 10.3103/S0027134921050143.
• [18] Y. Pang et al., “Quantifying the fractal dimension and morphology of individual atmospheric soot aggregates”, J. Geophys. Res. Atmos., vol. 127, no. 5, pp. 1-11, 2022, doi: 10.1029/2021JD036055.
• [19] A. M. Brasil, T. L. Farias and M. G. Carvalho, “Evaluation of the fractal properties of cluster-cluster aggregates”, Aerosol Sci. Tech., vol. 33, no. 5, pp. 440-454, 2000, doi: 10.1080/02786820050204682.
• [20] Ü. Ö. Köylü, G. M. Faeth, T. L. Farias and M. G. Carvalho, “Fractal and projected structure properties of soot aggregates”, Combustion and Flame, vol. 100, no. 4, pp. 621-633, 1995, doi: 10.1016/0010-2180(94)00147-K.
• [21] N. Doner and F. Liu, “Impact of morphology on the radiative properties of fractal soot aggregates”, J. Quant. Spectrosc. Radiat. Transf., vol. 187, pp. 10-19, 2017, doi: 10.1016/j.jqsrt.2016.09.005.
• [22] Z. Merdan and M. Bayirli, “Computation of the fractal pattern in manganese dendrites”, Chinese Phys. Lett., vol. 22, no. 8, pp. 2112-2115, 2005, doi: 10.1088/0256-307X/22/8/080.
• [23] V. Ceric, “Algorithmic art: technology, mathematics and art”, in ITI 2008, 30th International Conference on Information Technology Interfaces, Cavtat, Croatia, 2008, pp. 75-82, doi: 10.1109/ITI.2008.4588386.
• [24] A. Daudrich, “Algorithmic art and its art-historical relationships”, Journal of Science and Technology of The Arts, vol. 8, no. 1, pp. 37-44, 2016, doi: 10.7559/citarj.v8i1.220.