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. 2024 Jan 22;10(1):83.
doi: 10.3390/gels10010083.

Acoustic Fractional Propagation in Terms of Porous Xerogel and Fractal Parameters

Maria-Alexandra Paun  1 , Vladimir-Alexandru Paun  2 , Viorel-Puiu Paun  3   4
Affiliations

Acoustic Fractional Propagation in Terms of Porous Xerogel and Fractal Parameters

Maria-Alexandra Paun et al. Gels. .

Abstract

This article portrays solid xerogel-type materials, based on chitosan, TEGylated phenothiazine, and TEG (tri-ethylene glycol), dotted with a large number of pores, that are effectively represented in their constitutive structure. They were assumed to be fractal geometrical entities and adjudged as such. The acoustic fractional propagation equation in a fractal porous media was successfully applied and solved with the help of Bessel functions. In addition, the fractal character was demonstrated by the produced fractal analysis, and it has been proven on the evaluated scanning electron microscopy (SEM) pictures of porous xerogel compounds. The fractal parameters (more precisely, the fractal dimension), the lacunarity, and the Hurst index were calculated with great accuracy.

Keywords: SEM images; acoustic equation; fractal parameters; fractional propagation; porous xerogel.

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Conflict of interest statement

The authors declare no conflicts of interest.

Figures

Figure 1
Figure 1
Wave-front velocity in the fractal media as: (a) a fractal dimension function for various L (2D); (b) a function of fractal dimension and L (3D).
Figure 2
Figure 2
The Bessel functions of zero order.
Figure 3
Figure 3
The polynomial solution y1x of differential Equation (28).
Figure 4
Figure 4
Graphic representation of J0x. Asymptotic approximation.
Figure 5
Figure 5
(a) Original SEM image with the material pores evidenced; (b) Indoor shot enlarged 5 times compared to the original.
Figure 6
Figure 6
(a) Simple xerogel FTIR spectra; (b) Statistical histogram to pore size distribution of L SEM picture.
Figure 7
Figure 7
(a) The filtered version of the SEM image; (b) The processed grayscale image.
Figure 8
Figure 8
(a) The binary version of the picture; (b) The mask made for the lacunarity calculation.
Figure 9
Figure 9
Voxels representation within the evaluated image.
Figure 10
Figure 10
Local fractal dimension evaluated by the box-counting method.
Figure 11
Figure 11
Verification of the results using the HarFA program.
See this image and copyright information in PMC

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