Mathematical Modeling of Layered Nanocomposite of Fractal Structure

Abstract

A model of a layered hierarchically constructed composite is presented, the structure of which demonstrates the properties of similarity at different scales. For the proposed model of the composite, fractal analysis was carried out, including an assessment of the permissible range of scales, calculation of fractal capacity, Hausdorff and Minkovsky dimensions, calculation of the Hurst exponent. The maximum and minimum sizes at which fractal properties are observed are investigated, and a quantitative assessment of the complexity of the proposed model is carried out. A software package is developed that allows calculating the fractal characteristics of hierarchically constructed composite media. A qualitative analysis of the calculated fractal characteristics is carried out.

1. Introduction

One of the modern trends in materials science is the design of surfaces and volumes with a repeating structure at arbitrary scales [1,2,3]. Sets that demonstrate similarity properties at different scales of resolution of their geometric structure, in a strict or approximate sense, are called fractals [4,5]. Many physical laws, applying to fractal objects, lose their universality, which are shown in [6,7,8]. Among the material world objects, self-similarity is very widely represented, including the problems of materials science. The same phenomena in Euclidean space and on fractal objects are different. Thus, fractal objects are of high interest for research due to their wide practical application and contribution to the solution of several urgent problems facing modern society. Fractal materials are used in many applied problems, for example, to reduce the size of device elements [9,10], increase the broadband parameters of radio equipment [11,12], develop new absorbing and reflective materials [13,14], develop biocompatible materials with desired properties [15,16], in devices for the physical provision of information security [17,18]. In addition, self-similarity is very widely represented among objects of the material world [19,20].

The analysis of scientific sources demonstrates that most of the works on the physics of fractals are devoted to computer modeling of the growth processes of fractal clusters under certain conditions. There is extensive literature on this topic [20,21,22]. However, it is important to analyze fractal models that can serve to describe the physical properties of real objects. The purpose of this study is to carry out a fractal analysis of a layered hierarchically constructed composite model. We will consider a hierarchically constructed nanocomposite based on silicon and aluminum (Si-Al). Such nanocomposites are of wide interest for various applications in modern optics and radiophysics [23,24,25]. Elements based on such hierarchically constructed nanocomposites are used in the creation of new innovative antenna designs [26,27], broadband and nonlinear radar devices [28,29], the creation of means for localization and tracing of mobile objects [30,31], the synthesis of selective and absorbing materials [32,33], as well as in the creation of intelligent applications for the physical protection of information [34,35].

However, the models developed in the course of this study are not limited to the use of specific materials and can be used to study the electrodynamic properties of a wide range of nanocomposite media. As a result of fractal analysis of the models, it can be said that layered nanocomposites with a hierarchical structure have electromagnetic properties (in particular, complex dielectric constant, electrical conductivity) that differ significantly from the properties of homogeneous materials.

The uniqueness of structural materials lies in the fact that it is possible to design such materials in advance in such a way as to give the product the properties necessary for solving a specific applied problem. The design of a material can be one of the key factors affecting its functional properties (for example, mechanical, electromagnetic, thermal).

As a result of research developed; frequency dependences of the real and imaginary parts of the complex dielectric elements based on such objects are used in the creation of new innovative designs of antennas, broadband, and nonlinear radar devices, the creation of means for localization and tracing of mobile objects, the synthesis of selective and absorbing materials, as well as in the creation of intelligent applications, in particular in the protection of information [36,37]. For the nanocomposite of the fractal structure considered in our study, we propose a new modeling technique based on fractal analysis, including assessing the allowable range of scales; calculation of fractal capacity; Hausdorff, and Minkowski dimensions; calculation of the Hurst exponent. The proposed technique makes it possible to simulate a fractal nanocomposite and obtain information about the material, which is difficult to obtain by traditional methods.

2. Materials and Methods

Currently, several methods for modeling nanocomposites have been proposed. An important role in modeling nanocomposites is played by the so-called effective medium model [38,39]. The essence of the model is that the set of clusters that make up the nanocomposite is considered as a kind of new medium with the same level of polarization. Thus, knowing the parameters of each of the components of the composite, their geometric shape, and concentration, it is possible to determine the characteristics of the resulting composite medium as a whole.

The advantage of this approach is that to analyze the propagation of radiation in a composite medium, there is no need to solve Maxwell’s equations at each point in space. The most widely used models of the effective medium are the models of Rayleigh, Maxwell, Maxwell-Garnett, Bruggeman, etc., [40,41].

Several restrictions are imposed on the effective medium model related to the size of inclusions and the distance between them in comparison with the length of the electromagnetic wave in the medium [42].

There is also a quantum mechanical method for modeling nanocomposites, which makes it possible to describe the physical characteristics of materials from the first principles [43]. However, for systems involving a large number of particles, high computational power is required to solve this equation.

In many cases, to simulate a nanocomposite medium, the method of equivalent circuits is used [44], in which the electrical, mechanical, and magnetic components of the composite are represented in the form of electrical equivalents and the classical equations of electrodynamics. The macroscopic approach to modeling the electrical properties of nanocomposites is reduced to solving the classical equations of Maxwell’s electrodynamics. However, restrictions are imposed on the use of the macro theory in modeling the interaction of an electromagnetic field with nanocomposites [45]. When the radiation wavelength becomes comparable to the size of the molecules, then the use of this theory is incorrect.

To quantify the complexity of the fractal model of the composite, the coefficient of change in the part with a change in scale is calculated. B. Mandelbrot in his work [46] calls such a coefficient the fractal dimension (capacity). In this paper, when speaking about fractal dimension, we will be guided by the specified definition. In addition, it should be noted that there are two fundamentally different approaches related to the concept of dimension. In the classical definition, dimension is the number of dimensions of a geometric figure [47] (for a line, the dimension is equal to one, for a plane—two, for a volume—three). Since this dimension is a topological invariant (that is, it is preserved under a one-to-one mapping that is continuous in both directions), it is called the topological dimension and is denoted by DT [48]. In this study, the dimension is considered from the point of view of a different approach, this is the number D, which expresses the relationship between the natural measure of a geometric figure (for example, length, area, or volume) with the value underlying the original metric system. If a metric standard of such a value, taken as a unit, is increased (decreased) by a factor of k, then the indicated measure will decrease (increase) by a factor of k^D.

Methods of the fractal theory were used to describe the qualitative properties of the object under study, and regression analysis was carried out using the least-squares method. Calculations and calculations were performed in the software package Wolfram Mathematica 12.1, a 3D model of a composite of a fractal structure was developed in the SketchUp environment.

3. Results

The object of research is a model of a layered hierarchically constructed composite with fractional metric dimensions (Figure 1).

Since the developed models can serve to describe the physical properties of real objects, we are dealing with an object that has similar properties in a limited range of scales. The region of existence is superimposed on the fractal—max and min sizes, at which fractal properties are observed [49]. From above, this interval is limited by the size of the object. From below, as a limitation of the region of existence, in the case under consideration, the van der Waals radius is used, since when atoms approach at a distance less than the sum of their van der Waals radii, strong interatomic repulsion occurs, therefore it is the van der Waals radii that characterize minimum admissible contacts of atoms belonging to different molecules. Thus, the above fractal model of the composite has a physical meaning only under the conditions $r\le \frac{d}{2};d\ge R_{V-d-V}$, where is r—object size, d—composite layer size, $R_{V-d-V}$—the van der Waals radius of the chemical element that makes up the composite layer. As an example, the Si-Al composite is considered and the characteristics necessary to assess the limitations of the scale interval.

Table 1. Geometric characteristics of elements Si, Al.

Item Name	Atomic Radius	Van der Waals Radius
Si	117.6 pm	210 pm
Al	143 pm	210 pm

shows the geometric characteristics of elements Si and Al.

Let us estimate the permissible scale interval of the layered hierarchically constructed Si-Al composite (Figure 1). Let us take an object 2 cm in size, respectively d of one layer = 1 cm. With an increase in the level of fractality, the layer size will change according to

Table 2. Dependence of the layer size on the level of fractality.

Layer Size	d	d/4	d/16	d/64	…	d/4k−1
Fractality Level	1	2	3	4	…	k

.

Table 3. Calculation results.

Fractality Level	Layer Thickness, m
0	10−2
1	2.5·10−3
2	6.25·10−4
3	1.5625·10−4
4	3.90625·10−5
5	9.76563·10−6
6	2.44141·10−6
7	6.10352·10−7
8	1.52588·10−7
9	3.8147·10−8
10	9.53674·10−9
11	2.38419·10−9
12 (max)	5.96046·10−10
13	1.49012·10−10 $<2.1·{10}^{-10}={\mathrm{R}}_{\mathrm{B}-д-\mathrm{B}}\left(\mathrm{Si},\mathrm{Al}\right)$

shows the results of calculations performed in the software package Wolfram Mathematica 12.1.

Thus, for fractal models (Figure 1), in the case of a Si-Al composite, the allowable scale interval, with an initial layer size of 1 cm, is limited from above by the 12th fractality level.

To quantitatively estimate the complexity of the fractal model of the composite, we calculate the fractal dimension (capacity). For the case of a compact set, the dimension will be determined by the expression proposed in the work [50]:

$$D=-\underset{d\to 0 }{\lim }\frac{\ln K\left(d\right)}{\ln d},$$

(1)

where $K\left(d\right)$—is the number of elements (segments, cells, cubes, etc.) with a geometric parameter (for example, linear size) d, which determines the approximation of the original set. Let us carry out calculations of the fractal dimension for the above model of a hierarchically constructed composite.

Since with an increase in the level of fractality, two materials that make up the composite evolve, the fractal dimension will have the form:

$$D=-\underset{d_1\to 0 }{\lim }\frac{\ln K_1\left(d_1\right)}{\ln d_1}-\underset{d\to 0 }{\lim }\frac{\ln K_2\left(d_2\right)}{\ln d_2},$$

(2)

If the cell size is d = 1/16, then the number of coating cells for the two materials is $K_1$ = $K_2$ = 8. At the next step, when the cells are reduced by 16 times, we get d = 1/256, $K_1$ = 64, $K_2$ = 192, and so on (

Table 4. Evolution of the hierarchically constructed composite model.

Fractality Level, n	Picture	Material 1		Material 2
		K	d	K	d
1		8	1/16	8	1/16
2		64	1/256	192	1/256
3		512	1/4096	3584	1/4096

).

Thus, the level of fractality is associated with the coverage of cells of size $d={\left(\frac{1}{16}\right)}^n$ in quantity $K\left(d\right)=8^n$ for the first material, and $d={\left(\frac{1}{16}\right)}^n$, $K\left(d\right)={16}^n-8^n$ respectively.

Applying Formula (2) we get:

$$D=-\underset{n\to \infty }{\lim }\frac{\ln 8^n}{\ln {16}^{-n}}-\underset{n\to \infty }{\lim }\frac{\ln ({16}^n-8^n)}{\ln {16}^{-n}}$$

For the Si-Al composite considered as an example:

$$D=-\underset{\mathrm{n}\to 12 }{\lim }\frac{\ln 8^{\mathrm{n}}}{\ln {16}^{-\mathrm{n}}}-\underset{\mathrm{n}\to 12 }{\lim }\frac{\ln ({16}^{\mathrm{n}}-8^{\mathrm{n}})}{\ln {16}^{-\mathrm{n}}}$$

The calculation results and the corresponding regression lines constructed using the least-squares method are shown in Figure 2 and Figure 3.

The Hausdorff dimension is a quantitative measure of how densely a fractal set fills the surrounding Euclidean space [51,52]. Unlike the capacity calculated earlier, the Hausdorff dimension remains unchanged when passing to an arbitrary metric.

For self-similar sets, such as the hierarchically constructed composite under study, the Hausdorff dimension can be calculated explicitly. We split the set A into n parts with the coefficients m₁, m₂, m₃,… m_n (Figure 4).

The Hausdorff dimension H of the set A is a solution to the equation $m_1^H+m_2^H+\cdots +m_2^H=1$. For the object investigated in this paper, the Hausdorff dimension will be $8·{\left(\frac{1}{4}\right)}^H=1$ for z = 1 и $4^{z+1}{\left(\frac{1}{4^z}\right)}^H=1$ for $z>1$ (similarity coefficient ¼), where z—fractality level. Let us carry out the calculations for the composite with $z_{max}=12$. In this case, the Hausdorff dimension will be calculated by the following system of equations:

$$\{\begin{array}{c}8·{\left(\frac{1}{4}\right)}^H=1;\\ \{\begin{array}{c}{4}^3·{\left(\frac{1}{4^2}\right)}^H=1;\\ 4^4·{\left(\frac{1}{4^3}\right)}^H=1;\\ \dots \\ 4^{13}·{\left(\frac{1}{4^{12}}\right)}^H=1\end{array}\end{array}$$

The Minkowski dimension M (in some sources referred to as the “rough dimension”) of the set A is determined by the expression $M=\underset{\epsilon \to 0}{\lim }\frac{\ln \left(N_{\epsilon }\right)}{-\ln \left(\epsilon \right)}$ [53], where $N_{\epsilon }$—is the minimum number of sets of diameter ε that can cover the set A. In the case of an infinite set, the upper and lower limits are considered, and the concepts of upper and lower Minkowski dimensions are introduced, respectively. The Minkowski dimension is a close concept to the Hausdorff dimension and in many cases, they coincide, but there are sets for which these dimensions have different meanings [54]. For composites of morphologies (Figure 1) with the maximum level of fractality $z_{max}$ = 12, the Minkowski dimension will be $M=\underset{\epsilon \to {16}^{-12}}{\lim }\frac{\ln \left(N_{\epsilon }\right)}{-\ln \left(\epsilon \right)}$.

To assess the dynamics of the fractal dimension values, we calculate the Hurst exponent. In terms of the asymptotic behavior of the scaled range of the function of a set of points of a series of dynamics, the Hurst exponent is determined by the expression:

$$\frac{R_n}{S_n}={\left(n/2\right)}^{\chi }, n\to \infty ;\hspace{1em}\chi =\underset{n\to \infty }{\lim }\frac{\ln \frac{R_n}{S_n}}{\ln \frac{n}{2}},$$

where $R_n$—is the range of the first n values of the series, $S_n$—is the standard deviation, n—is the number of points in the time series segment. If the limit exists, the equality χ = 2 − H.

Let us carry out calculations for a hierarchically constructed composite model with $Z_{max}=12$. The calculation results are shown in Figure 5.

4. Discussion

From the calculations of the fractal capacity, it follows that the minimum deviation from the regression line for material 1 corresponds to the 7th level of fractality, for material 2—to the 6th and 7th levels. The use of the least-squares method makes it possible to calculate the expected values of the dependences of the size of the cells covering the object of study on their number for an arbitrary level of fractality.

In this study, the calculations of the Hurst exponent for a hierarchically constructed composite model are obtained for the first time. The results obtained make it possible to determine the probability of the appearance of such materials in nature.

Based on a large empirical material, it was found that the value of the Hurst exponent of natural processes is grouped near the values of 0.73 ± 0.09 [19], 0.72 ± 0.08 [55]. The question of why this so, remains open. For a random process with independent increments and finite variance, it was rigorously proved in [56] that the exponent $\chi$ is 0.5. Sequences for which $\chi$ > 0.5 are considered persistent—they retain the existing trend [57], that is, an increase in the past is more likely to lead to an increase in the future, and vice versa. At a value of 0.5, there is no clear tendency, and at lower values, the process is characterized by antipersistence—any tendency tends to be replaced by the opposite one [57].

5. Conclusions

In the present work, a hierarchically constructed composite model is investigated. The study found:

• For composites of fractal morphologies, the Minkowski dimension does not coincide with the Hausdorff dimension.
• The first and second levels of fractality of composites of morphologies xx demonstrate the presence of a finite variance of a random process, the absence of an obvious tendency to change several dynamics, the fractal properties of the object are weakly expressed.
• The fourth level of fractality demonstrates the closest value to the Hurst exponent of natural processes. Thus, the likelihood of the appearance of such a fractal in nature is higher in comparison with composites of the indicated morphologies of other levels of fractality.
• The hierarchically constructed composite of fractality level 3–12 has the property of persistence (the observed trend is supported).

Thus, it is shown that fractal analysis in the field of materials science is an effective tool for obtaining high-quality results and additional information about the object of research, which is difficult to obtain with traditional research methods.