Abstract
Classical site percolation is studied on square pore networks generated through the dual site–bond model (DSBM). This model allows different strengths of correlations among the lattice elements in such a way that patches of sites and bonds with similar sizes are structured as correlations are setup. The typical size of these patches depends on the strength of correlations. The aim of this paper is to find whether the global patchy structure is responsible for the patterns found in classical percolation structures. To this end, we measure different static quantities of the correlated networks, such as correlation length, and fractal quantities of the spanning clusters. We observe a significant gap from the reported behavior. For instance, our results indicate that the fractal dimension of the spanning cluster is nonuniversal and changes with the network correlation. The DSBM is good to produce classical percolation clusters with different fractal dimension and different lacunarity behavior. Comparison with other authors’ recent findings is presented.
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Abbreviations
- \(a\) :
-
Slope of lacunarity behavior
- \(b\) :
-
Intercept of lacunarity behavior
- \(d_\mathrm{f}\) :
-
Fractal dimension
- DSBM:
-
Dual site–bond model
- \(F_\mathrm{B}(R_\mathrm{B})\) :
-
Probability density functions by bonds
- \(F_\mathrm{S}(R_\mathrm{S})\) :
-
Probability density functions by sites
- \(L(r)\) :
-
Feature lacunarity \(L\) for a box size \(r\)
- l.u.:
-
A linking unit
- \(m\) :
-
Number of pixels
- \(M\) :
-
Number of sites
- \(n\) :
-
Integer
- \(N(r)\) :
-
Number of boxes \(N\) of size \(r\)
- \(P\) :
-
Probability
- \(R\) :
-
Represent the “metric”
- REV:
-
Representative elementary volume
- \(s_{i}\) :
-
Threshold
- \({\mathbb {Z}}\) :
-
Integers
- \(\mu \) :
-
Mean value of sites distribution
- \(\xi \) :
-
Correlation length
- \(\rho \) :
-
Percolation threshold
- \(\rho _\mathrm{C}\) :
-
Fraction occupied by the spanning cluster
- \(\Omega \) :
-
Overlap between distribution functions site–bond
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Acknowledgments
The authors wish to thank the Ministry of Education of Mexico (SEP-PROMEP) for the support awarded to the Academic Network “Diseño nanoscópico y textural de materiales avanzados”. Thanks are also due to the National Science and Technology Council of Mexico (CONACyT) for the economic support awarded under Project “Apoyo para el desarrollo del programa de fortalecimiento académico del posgrado de alta calidad, COFON 914009.” The use of computer facilities of Laboratorio de Supercómputo y Visualización en Paralelo (Cluster Aitzaloa) of UAM-Iztapalapa is gratefully acknowledged.
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Hidalgo-Olguín, D.R., Cruz-Vázquez, R.O., Alas-Guardado, S.J. et al. Lacunarity of Classical Site Percolation Spanning Clusters Built on Correlated Square Lattices. Transp Porous Med 107, 717–729 (2015). https://doi.org/10.1007/s11242-015-0463-3
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DOI: https://doi.org/10.1007/s11242-015-0463-3