Fractal Characterization of Microstructure of Materials and Correlation with Their Properties on the Basis of Digital Materials Science Concept

Abstract

The concept of Digital Materials Science supposes that materials are designed, fabricated, tested, studied, characterized, and optimized on the basis of digital technologies, including the analysis of fractal parameters (fractal dimension, lacunarity, scale invariance, Voronoi entropy, etc.) of materials’ microstructure. Many classes of materials may be considered as composites: polymer composites with inorganic fillers, alloys containing nonmetallic inclusions (oxides, carbides, nitrides, intermetallic ones, etc.), ceramic materials with pores and sintering additives, etc. The analysis of composition-technology-structure-properties relationships for such non-ordered composite materials requires the development of numerical tools for the characterization of their structure, including the interposition of phases. This chapter presents several examples of the implementation of this concept, including the study of filler distributions in dielectric composites, interposition of phases in special ceramic materials, distribution of nonmetallic inclusions in additively manufactured stainless steel, and structural features of tungsten oxide-based electrochromic materials. Based on the analysis of such characteristics as lacunarity and surface functionality, interrelations are established between technical properties of the studied materials and their structure providing approaches to the prediction and optimization of their target performances.

1. Introduction

“Industry 4.0” (4th industrial revolution) is a concept of design, fabrication, testing, characterization, and utilization of products on the basis of digital technologies and Artificial Intelligence (AI). Originated from mechanical engineering and machine building industry, this concept involves the implementation of advanced techniques for manufacturing products via processing of materials possessing a set of required properties. In turn, target properties of materials are determined by their composition, structure, preparation, and pretreatment conditions. To take into account these numerous factors, the Industry 4.0 paradigm should be further supported and supplemented by the concept of Digital Materials Science, including the development of automated methods for the design and synthesis of materials, as well as computational approaches to the design of novel generations of advanced materials and digital analysis of the composition-structure-properties relationships [1].

Digital Materials Science includes but is not limited to the computational materials science [2]. The development of materials with adjustable target properties requires the proper design, synthesis of the required precursors, fabrication of the material in the desired form, as well as its study and characterization. Particularly, this concept includes the following aspects:

1. Design of materials at the atomic level of the structure, involving the crystal structure [3, 4].

2. Modeling the processes of the synthesis of materials and formation of a given composition and microstructure. For example, a periodic microstructure in inorganic materials can be tailored via adjustable reaction-diffusion transformations [5].

3. Modeling the macrostructure and properties of materials including biomorphic design [6, 7].

4. Digital methods for the description of the composition and structure of material [8].

5. Digital methods for the analysis of data on the composition-structure-properties relationships using artificial intellect [9, 10, 11, 12].

6. Digital methods for the design of equipment for the manufacturing of substances (materials) with the desired properties [13, 14, 15].

7. Digital technologies for the fabrication of substances and materials, including the application of advanced robotic techniques [16, 17].

8. Study of materials using modern digital approaches, including databases, 3D scanning, computed tomography, etc. [18, 19, 20].

9. Digital quality control and management including robust design and manufacturing [21, 22, 23].

The Industry 4.0 and Digital Materials Science concepts are implemented at the Department of Theory of Materials Science and Department of Chemistry, Physics and Biology of the Nanoscale State (based on the Institute of Silicate Chemistry of the Russian Academy of Sciences (ISC RAS), head of the Department—Academician of the Russian Academy of Sciences Vladimir Ya. Shevchenko) of the Saint-Petersburg State Institute of Technology in a number of educational and research programs, including a new bachelor degree program “Design, synthesis and application of nanomaterials”, master degree programs “Nanomaterials for Industry 4.0” and “Materials and technologies of additive manufacturing”, and double-degree program for PhD students with the University of Shizuoka (Japan).

Presently, numerous methods are developed for the design, fabrication, and testing of materials. However, positions (4) and (5) in the above list are difficult to fulfill because of the lack of methods to describe the structure of non-ordered materials and thus to derive composition-structure-properties relationships. The properties of substances are determined not only by their chemical composition but also by their structure, i.e., by the mutual spatial arrangement of structural elements (atoms, ions, functional groups, etc.). For the crystal structure, such nanoscale arrangement is well described by the crystal lattice parameters with a strictly ordered translation of the unit cell to the entire volume of the material. However, on a larger scale level relating to the microstructure of composite systems such as the distribution of filler particles in various composites, phases (intermetallic, carbide, etc.) in alloys, and ceramic particles in cermets, the structural order at the microlevel is not so definite, since such parameters as the number of the nearest structural elements (coordination number) and the distance between particles or fragments of a certain phase are essentially statistical. Therefore, the order and regularity of the arrangement of structural elements in such systems can be established on a relatively large scale using statistical approaches.

Certain parameters characterizing the microstructure are required because the properties of materials largely depend on the features of the structural elements arrangement, including the uniformity of their distribution, mutual positioning, and distances between them. Hence, it is necessary to determine the corresponding quantitative characteristics and their relationship with the composition of the material, conditions for its formation, and target properties. A possible approach to this problem is based on the statistical analysis of the distribution of structural elements of a material between its individual volume fragments with the determination of fractal parameters characterizing the structural order, spatial uniformity of certain structural elements, and degree of self-similarity (scale invariance) of the system with an increase in the scale level.

In the recent decades, this approach involving the determination of such parameters as fractal dimension and lacunarity has been applied to the characterization of various materials. Particularly, in [24, 25, 26], these parameters were determined for fractured surfaces in steels and aluminum alloys depending on rupture modes with different crack propagation rates and heat treatment conditions. The fractal dimension was found to almost linearly correlate with the plain strain fracture toughness of steel and regarded as a measure of this parameter [24]. It was generally established that the fractal dimension negatively correlates with the dynamic tear energy and the critical crack extension force would rise faster than Hall-Petch’s 1/√d law with the decrease of grain size due to the increase of the true area of the irregular fracture surface. In [26], the relationships among the fracture regimes and surface roughness index were analyzed in terms of fractal parameters.

More recently, fractal dimension values were determined and analyzed for the formation of dislocations and their organization into grains or subgrains in correlation with the mechanical behavior of metals and alloys, particularly plasticity characteristics [27].

In [28], the analysis of fractal parameters was shown to provide an effective approach to the comparative characterization of porous structure in stainless steels. Particularly, the fractal dimension was found to correlate with the porous structure of such steels growing with the increase of porosity.

In [29], fractal dimension and lacunarity were determined as parameters characterizing the porosity, pore size distribution, and other textural characteristics of carbon samples by the analysis of scanning electron microscopy (SEM) images and mercury porosimetry data. The obtained results provided an effective approach to the characterization of structural uniformity or heterogeneity of porous materials.

The considered parameters can be also applied to the study of surface characteristics of materials. In [30], lacunarity parameters derived from SEM images of femtosecond laser-irradiated metals and characterizing the irregularity of mutual arrangement of certain areas were shown to successfully quantify the spatial texture of the studied images, thus providing a convenient means of reporting changes in surface topography with respect to changes in processing parameters. Furthermore, since lacunarity plots are highly sensitive to the different length scales present within a hierarchical structure, the lacunarity analysis can be considered as a powerful tool for quantitative characterization of multi-scale hierarchical surface topographies.

A similar approach was also applied in [31] to comparatively analyze the surface smoothness and roughness as a result of metal coating deposition onto brass plates. The fractal dimension and lacunarity of the surface SEM image were established to be relevant parameters depicting the quality of coating surface at significance level of 0.05.

Hence, an essential goal is the development of a general concept as well as reliable and reproducible procedures applicable to the study of fractal parameters and surface characteristics for different kinds of materials in order to predict and adjust their target properties and performances. Particularly, an important objective in this field relates to quantitative characterization of various composites with respect to a precise analysis and modification of their microstructure, taking into account a number of factors including the uniformity of filler distribution in the matrix as well as the characteristics of the filler surface responsible for specific filler-binder and filler-filler interactions.

This chapter presents the results obtained by our research group in the development, modification, and implementation of the considered approaches to polymeric, metallic and ceramic materials, organic and inorganic films, etc. Since all these kinds of materials can be considered as composites with certain inclusions like filler particles, pores, and nonmetallic compounds, microstructure of such composites sufficiently influences their electrical, mechanical, optical, and other characteristics. A general approach to the study of the considered materials in this respect involves the characterization of their microstructure followed by the determination of such fractal characteristics as lacunarity, Voronoi entropy in correlation with target properties. In addition to these fractal characteristics, we also introduced the scale invariance as a new parameter reflecting a self-similarity of the material microstructure at various scale levels, determined by a comparative analysis of lacunarity or other fractal parameter measured at the studied object division into segments of different size.

The developed approach can be applied for various classes of materials in order to analyze all kinds of their nonuniformity, particularly with respect to the chemical composition according to the data of various mapping techniques, microstructure studied by microscopy (e.g., optical and electron of atomic force) or tomography, crystal size, and other characteristics.

2. Methodology

The determination of microstructural order characteristics of the studied materials is based on a modified box-counting method involving the analysis of SEM images (obtained using a TESCAN Vega 3 installation) of the studied materials using ImageJ software package and involves the following steps:

1. Isolation of certain structural elements (e.g., filler particles or phase inclusions) and determination of the coordinates of their centers of mass.

2. Partition of the analyzed fragments of SEM images into square cells with certain side size x and counting the number of centers of mass in each of these cells. This operation was performed in the Excel 2016 spreadsheet editor using the “AND” function, which checks the correspondence of the coordinates of the centers of mass to the coordinates of the segments.

3. Statistical treatment of the obtained results, including the calculation of the average value, range, standard deviation, and lacunarity Λ (parameter characterizing the inhomogeneity of filling the space with the considered structural elements):

$$\mathrm{\Lambda }={\left(\frac{\sigma }{\mu }\right)}^2$$E1

where σ and μ are the standard deviation and the average number of centers of mass of the studied objects in the segments under consideration, respectively.

A decrease in the value of this parameter corresponds to a decrease in the proportion of areas free of the filler particles and, accordingly, to an increase in the uniformity of filling the space [32, 33]. Eq. (1) is essentially the ratio of the deviation of the number of particles in a cell and the average number of particles in a cell, and the smaller the deviation, the smaller the differences in the number of particles in the cells, and the more uniformly they are located.

Lacunarity is a parameter characterizing the deviation of a geometric object, such as a fractal, from translational invariance, i.e., indicating how parts from different regions of a geometric object are similar to each other at a given scale. Geometric objects with low lacunarity are homogeneous and translation-invariant, since all dimensions of the gaps (distances between the studied particles, for example) are the same. In contrast, objects with a wide range of gap sizes are inhomogeneous, not translation invariant and have a high lacunarity.

In other words, lacunarity is a characteristic of the uniformity of the distribution of any objects (particles) in space, i.e., the lower its value, the more uniformly the particles are distributed [33].

The considered procedure can be repeated with the variation of the cell size x with a certain step in order to obtain the lacunarity spectrum, i.e., the dependence Λ vs x. Based on thus studied effect of the cell size (scale factor) on the lacunarity, the scale invariance of the system can be determined as a parameter characterizing the sensitivity of lacunarity to the change of scale. Generally, the lacunarity decreases with an increase in the cell size providing a greater “averaging” of the structural elements distribution [33].

The resulting dependences for 3D objects are almost ideally approximated by a power law according to the following general expression:

$$\mathrm{\Lambda }=\mathrm{А}\bullet {\mathrm{х}}^{\mathrm{n}-3}$$E2

or in the logarithmic form:

$$\mathrm{ln\Lambda }=\mathrm{lnА}+\left(\mathrm{n}-3\right)\mathrm{lnx}$$E3

where A is the pre-power coefficient and (n – 3) is the exponent factor characterizing the intensity of the lacunarity change with the cell size. This value represents the rate of change in the heterogeneity of the system with the change in the scale level (cell size). Generally, the less significantly the properties of the system change with the scale changes, the more scale-invariant this system is and the higher its ability to maintain its properties when the scale of study changes. Obviously, the most homogeneous system features the least prominent changes with a change in the scale. Thus, the factor n can be considered as a characteristic of the scale invariance of the system indicating its homogeneity. Since n value is commonly below 3, its growth corresponds to less negative power factors (n – 3) and consequently less prominent lacunarity drop with the increase of cell size reflecting higher scale invariance, self-similarity, and structural order of a material.

Furthermore, the properties of materials are determined not only by the composition, structure, properties, and mutual arrangement of their components but also by their surface properties and interfacial interactions which also should be optimized to improve the target performances. According to the concept of the scientific school created by the corresponding member of the Russian Academy of Sciences V.B. Aleskovsky and subsequently developed by professors S.I. Koltsov, V.G. Korsakov, A.A. Malygin, V.M. Smirnov, and A.P. Nechiporenko et al., a solid substance or material can be considered as a combination of a carcass and surface active centers of various nature, particularly featuring different acid-base or donor-acceptor properties. This factor should be taken into account at the consideration of target properties of materials (including electrophysical and electro-optical performances), since surface centers reflect the imperfection of the structure of the surface of a solid and largely determine its interaction with the environment (particularly including interfacial interactions in composites). Therefore, certain relationships can be established between the nature of such centers, their acid-base (donor-acceptor) properties, concentration on the surface, and various properties of materials sensitive to the state of the surface (including the characteristics of filler distribution in composites affecting their target performances) in order to extend the possibilities for the prediction and adjustment of the properties of functional materials by methods of solid state chemistry. Then the considered problem can be addressed by the adjustment of the surface functional composition, particularly toward the enhancement of certain interfacial interactions, in order to prevent or promote aggregation processes, improve the compatibility of the components, adjust their mutual distribution, and eventually improve the target performances of materials.

The development of this research direction requires the study of quantitative characteristics of the solid surface using digital methods followed by the search and analysis of correlations between the obtained data and target properties using digital materials approaches.

The suggested approach to study the composition-structure-properties relationships generally involves:

• analysis of the considered microstructural order parameters (lacunarity and scale invariance) and/or surface characteristics for a series of similar materials differing in their composition (e.g., content of a certain component or additive);

• measurement of target performances for the studied series of materials;

• analysis of correlations between the microstructural/surface characteristics and target performances in order to reveal the effect of the structural order features upon the properties of materials and provide the background for their prediction and improvement.

The applications of the considered approach to different classes of materials are summarized below.

3. Application examples

3.1 Polymer-inorganic composites

In a series of our recent studies [8, 34, 35], the considered approach was applied to the characterization, modeling, and prediction of the properties of polymer-inorganic dielectric composites. Such materials featuring a combination of a high dielectric permittivity (k) and flexibility are applied in various electronic devices involving a flexible dielectric layer, such as flexible electroluminescent panels, displays, film capacitors, capacitive energy sources, and other devices and systems of the new generation based on “smart” materials [36]. Although the dielectric permittivity of the most of conventional polymers does not exceed 10, the presence of polar groups (cyanoethyl, carbonyl, and hydroxyl) in the polymer backbone can significantly increase this value without any deterioration in other performances. One of the most promising polymers in this respect is cyanoethyl ester of polyvinyl alcohol (CEPVA) involving all the considered functional groups [37, 38]:

In order to provide the highest k values, composites based on such polymers should include various high-k disperse ferroelectric fillers, the most often used among which is barium titanate featuring one of the highest k values (more than 4000 for submicron particles).

In our previous studies [34, 35, 39], the filler surface modification with various nanolayers (“core-shell” structures) bearing specific Broensted acidic surface centers responsible for the interaction with the polymer binder via an acid-base mechanism was shown to provide a network of filler-polymer bonds, thus improving the components compatibility, preventing from the filler particles aggregation, and promoting the enhancement of dielectric performances. A similar effect can be achieved by the decoration of the filler surface with nanoparticles possessing the required active surface centers promoting the enhancement of acid-base interactions between the ferroelectric filler and polymer binder to provide the improvement of their compatibility and filler dispersion uniformity in the polymer.

In this study, barium titanate (Fuji Titanium, Japan; particle size about 0.5 μm, specific surface area 1.43 ± 0.07 m²/g, dielectric permittivity more than 4000) was modified by the deposition of graphene nanoplates (RUSGRAPHEN, Ltd.) from 0.5%wt. colloid solutions in distilled water followed by evaporation, drying, and grinding to obtain BaTiO₃ samples containing graphene on the surface in the amounts from 0.2 to 6 mg per 1 g of barium titanate.

The surface functional composition of thus modified fillers was characterized by the adsorption of acid-base indicators selectively adsorbed on surface centers with certain pKa values corresponding to the transition between the acidic and basic forms of the indicator (HInd ↔ H⁺ + Ind⁻) with the change of its color (optical density at certain wavelengths). Spectrophotometric measurements of the change in optical density after the indicator adsorption as a result of the interaction with the studied material surface provide the determination of the contents of different adsorption centers on its surface [38, 39, 40, 41].

The prepared modifies fillers were dispersed in 3 mL of CEPVA (PB paste, Shanghai Keyan Phosphor Technology Co, Ltd., Chine, k ≈ 25–30) in the amount corresponding to the filler content 20%vol. in the dry composite. Then the obtained mixtures were processed in an ultrasonic bath within 5 min to disperse the filler, followed by the deposition onto aluminum substrates via jets and drying at 70°C for 4 h, resulting in the formation of composite layers of about 50 μm average thickness.

Then electrodes comprising a silver-containing electrically conducting glue Contactol were deposited onto the composite layers, followed by measuring their capacitance and dielectric loss tangent (tgδ) using E7-20 immittance meter (MNIPI, Belarus) at frequency 1 kHz. The dielectric permittivity was calculated as:

$$k=\frac{\mathrm{С}·d}{{\epsilon }_0·S},$$E4

where C is the measured capacitance, and d and S are the composite layer thickness and Contactol electrode surface, respectively; ε₀ = 8.85 × 10⁻¹² F/m is the dielectric constant. The errors of k and tgδ measurements according to the averaged data for 3–5 Contactol electrodes deposited onto the surface of each studied sample do not exceed 15% and 10%, respectively.

Examples of SEM images for the prepared composites with graphene contents 0, 0.6, 0.8, and 4.6 mg/g clearly illustrating the structural differences of the considered materials are presented in Figure 1. These data show that the addition of 0.6 mg/g (Figure 1b) graphene provides a certain improvement of the composite uniformity compared with a similar material without this nanocarbon additive (Figure 1a). On the contrary, the increase of deposited graphene amount to 0.8 mg/g BaTiO₃ (Figure 1c) significantly deteriorates the filler distribution uniformity and in the case of 4.6 mg/g (Figure 1d) graphene content a less prominent negative effect is observed. These results indicate a very high sensitivity of the composite structure to the additive content and suggest the necessity in a precise optimization of the composite composition to achieve the target performances.

Figure 1.

SEM images of CEPVA-BaTiO₃-graphene composites with graphene additive contents 0 (a), 0.6 (b), 0.8 (c), and 4.6 mg/g BaTiO₃ (d).

The considered structural features clearly correlate with lacunarity calculated according to the above procedure and shown in Figure 2 as a function of the SEM image division cell size. At relatively small (1–3 μm) cell sizes, the lowest lacunarity value is observed for the mostly structurally ordered composite with 0.6 mg/g. For the less uniform sample with graphene content 4.6 mg/g, a certain increase of lacunarity is observed and the mostly disordered composite with graphene content 0.8 mg features the highest lacunarity. At higher cell sizes (4–6 μm), the applied method becomes less sensitive to the composite structure and lacunarity values drop to about 0.1.

Figure 2.

Lacunarity of CEPVA-BaTiO₃-graphene composites with graphene additive contents 0.6 (1), 0.8 (2), and 4.6 mg/g BaTiO₃ (3) as a function SEM image division cell size.

The lacunarity vs cell size dependencies were analyzed for a whole series of composites with graphene content from 0.2 to 6 mg/g BaTiO₃ to determine the corresponding scale invariance parameters and compare the obtained results with the target performances of the synthesized materials. It was revealed that dielectric permittivity of the composites prominently grows with the increase (approaching zero) of the scale invariance parameter n (Figure 3).

Figure 3.

Dielectric permittivity of CEPVA-BaTiO₃-graphene composites as a function of scale invariance parameter n.

The data in Figure 3 suggest that the increase of permittivity in the range from less than 50 for graphene-free materials to about 125 for the composite with the optimal graphene content corresponds to the composite structure invariance to the scale factor, i.e., independence on the size of cells or fragments analyzed at the lacunarity determination.

The highest dielectric performances for CEPVA-BaTiO₃ composites (about 400) in couple with an almost 25% decrease in dielectric loss tangent (tgδ) was achieved in the case of barium titanate modification with fullerenol C₆₀(OH)₄₂ in the optimized amount of 3.2 mg/g BaTiO₃.

The analysis of corresponding SEM images (Figure 4) processed according to the considered approach indicated that the filler modification with optimal amount of fullerenol resulted in an almost double decrease in lacunarity from 0.15 to 0.08, suggesting an increased uniformity of the mutual arrangement of the filler particles without their aggregation [8].

Figure 4.

SEM images (a), their binarized representation with marked centers of mass of particles (b), and the distributions of centers of mass of particles over 3 × 3 μm square cells for composites with the additive-free (left) and fullerenol (3.2 mg/g) modified BaTiO₃ filler (right).

This improvement of structural uniformity and permittivity of modified composites is determined by the presence of numerous hydroxyl functional groups in fullerenol capable of chemical interaction with OH-groups of CEPVA binder, thus preventing from the filler particles aggregation (Figure 5).

Figure 5.

Interaction of polymer binder R-OH with multiple OH-groups of fullerenol additive.

The involvement of surface functional groups in the observed enhancement of dielectric permittivity is confirmed by its prominent correlation with the content of Broensted acidic centers with pKa 1.3 supposedly corresponding to OH-groups in fullerenol (Figure 6). The optimal fullerenol amount provides the highest concentration of these centers on the fillers surface, while excessive additive content probably results in aggregation of fullerenol molecules via the condensation of OH-groups, thus preventing from their effective interaction with BaTiO₃ particles.

Figure 6.

Dielectric permittivity (^_____) of CEPVA-BaTiO₃-fullerenol composites and content of Broensted acidic centers with pKa 1.3 on the modified BaTiO₃ surface (----) as a function of fullerenol content.

It should be particularly noted that the increase of fullerenol content above the established optimal level 3.2 mg/g results in a drastic decrease of permittivity probably as a result of undesirable interactions between the filler particles resulting in their aggregation.

Generally, the considered results demonstrate an efficient approach to precise optimization and prediction of the target performances of composites on the basis of quantitative relationships between the contents of their components, their microstructural features (characterizing their mutual arrangements) as well as their surface characteristics (particularly content of specific surface centers or functional groups) responsible for their interactions and compatibility.

3.2 Structure of composite ceramic materials

In [42], the developed methods were applied to a novel ceramic composite “Ideal” comprising diamonds distributed in a silicon carbide matrix and featuring outstanding properties (e.g., modulus of elasticity 754 GPa significantly exceeding the available counterparts) due to both the composition and regular interconnected microstructure (triply periodic structures of silicon carbide on diamond particles formed during the reaction-diffusion synthesis involving filling the pore space between the diamond particles with silicon carbide, thus providing dense diamond-silicon carbide composite). The structure of this novel material is illustrated in Figure 7, and processing of its SEM image is shown in Figure 8.

Figure 7.

Interpenetrating phase structure image (a) and schematic representation (b) for diamond-SiC nanocomposites (SiC is on the left, diamond—on the right side of the image, area of interpenetrating phases highlighted in yellow).

Figure 8.

4600 × 4600 μm-sized SEM image (140× magnification) of “ideal” composite (a), its division into 200 μm square cells including the centers of mass of the selected particles (b), and enlarged area highlighted in yellow square (c).

In addition to the lacunarity analysis, this system was also studied by the determination of Voronoi entropy characterizing the probability Pi for the appearance of a system in the state i. Based on this concept, SEM images of the studied composites were used to plot Voronoi diagrams involving centers of mass of the filler particles as a finite set of points S.

Furthermore, since the studied materials consists of two fractions of particles greatly differing in size, it was decided to analyze both lacunarity and Voronoi entropy separately for the large (about 200–250 μm) or only small (below 28 μm) particle fractions in order to evaluate their contributions to the overall order and uniformity of the system.

The image illustrating the triangulation used for Voronoi entropy determination is shown in Figure 9.

Figure 9.

Visualization of the Voronoi diagram (the sides of the Voronoi polygons are indicated as dashed lines) and triangulation (straight lines connecting particles) for the system of large particles.

Then the analyzed image was subjected to a partition into fragments involving a set of points located closer to one of the points of the overall set S than to any other point of the set [43]. After constructing the Voronoi diagram, we obtain data on the number of sides of the polygon built around each point; thus, we can subdivide the image into different classes of cells according to the number of sides (equal to the number of neighbors). The information entropy is determined as:

$$\mathrm{S}\left(\mathrm{\epsilon }\right)=-\sum {\mathrm{p}}_{\mathrm{i}}\bullet {\mathrm{lnp}}_{\mathrm{i}},$$E5

where p_i is the probability that the considered Voronoi polygon has a certain number of sides. Generally, the Voronoi entropy indicates the amount of information required to determine the location of objects relative to each other and grows with the increase of the system disorder.

In this study, Voronoi diagrams were plotted using MATLAB software [44] involving the “voronoi” function. Also, using the “delaunayTriangulation” function, a Delaunay triangulation was arranged for a given set of points on a plane, with all the circles described around any triangle contain inside no points except the triangle vertices. Based on this triangulation arrangement, Voronoi entropy was determined according to the procedure described in [45] MATLAB displaying the coordinates of the vertices of all Voronoi cells for each center of mass, indicating number of sides in the corresponding polygon is displayed as well as the coordinates of the start and end points of all triangulation lines. Furthermore, the applied software allows the determination of the number of nearest neighbors (coordination number) for each particle as the number of vertices of the sides in the Voronoi cell corresponding to this particle.

The obtained results summarized in Table 1 indicate that the largest lacunarity is observed for the system of large particles (2.58), most likely due to a small number of such particles that cannot occupy all the cells, thus leaving many empty cells free of the particles, in turn causing a significant inhomogeneity. Probably, to study only large particles, it is necessary to consider images with a much larger coverage in order to obtain more reliable statistical information.

Fraction	Lacunarity	Voronoi entropy
All particles	0.36	1.65
Only small (below 28 μm) particles	0.38	1.67
Only large (200–250 μm) particles	2.58	1.48

Table 1.

Lacunarity values calculated for different particle fractions.

For the system involving only small particles, the lacunarity is much lower (0.38) indicating a more uniform distribution. Despite the presence of inhomogeneous regions in the place of eliminated large particles, the number of empty cells is significantly reduced, and the difference in particle filling density between different cells is less prominent.

The system involving both large and small particles features the lowest lacunarity (0.36) slightly below the value for the system of small particles that reflects filling of some empty cells with large particles.

The lowest Voronoi entropy is observed for the system of large particles, the highest value is calculated for small particles, and the system involving all the particles features a little lower entropy. The entropy minimum in the case of only large particles is determined by the decreased number of Voronoi polygon variants in terms of the number of sides in the absence of small particles. The increase in entropy in the case of a system including only small particles can be caused by an increase in the number of sides of the polygons associated with particles located next to the voids that were occupied by excluded large particles. The appearance of a point in the center of a large void forms a Voronoi cell and the “extra” sides from the nearest small cells are “cut off.”

Thus, the Voronoi entropy can be used for a comparative characterization of systems differing in the content of a filler (discrete phase distributed in a continuous phase) or structural order.

The number of nearest neighboring particles calculated for the systems of large and small particles is summarized in Figure 10 as a histogram indicating the distribution of the number of particles with certain number of neighbors.

Figure 10.

Distribution of particles by the number of nearest neighbors for different fractions.

The maximum in this distribution for all particles and for the small fraction is 6 corresponding to the closest package. For the system of only large particles, the most probable number of neighbors is 5 and no values larger than 8 is observed.

Thus, the determination of Voronoi entropy provides an additional information such as the number of neighbors for each particle and distances between the neighboring particles that is promising for the analysis of the structure of various materials.

With respect to the target performances, the considered material features a clear correlation of the porosity and elasticity modulus with the scale invariance (Figures 11 and 12). These data suggest that the increase (approaching zero) of the scale invariance parameter n indicating the reduction of structural changes in the material with scale changes (i.e., increased scale invariance and structural order) corresponds to the reduction of porosity and enhancement of strength performances.

Figure 11.

Scale invariance factor n of “ideal” composites as a function of overall porosity.

Figure 12.

Elasticity modulus of the “ideal” composite as a function of scale invariance factor n for medium and small fractions.

Hence, textural and mechanical characteristics of materials can be predicted on the basis of their microstructure analysis.

3.3 Tungsten oxide-based electrochromic materials and devices

In this case, microstructural order parameters were determined for tungsten oxide (WO₃) electrochromic layers deposited onto glass supports with fluorinated tin oxide (FTO) electrically conducting layers by magnetron sputtering using a novel high-throughput technique based on a periodic modulation of the deposition angle (PMMDA) [46, 47]. This method involves the rotation of the substrate around the axis perpendicular to the plasma flow direction to provide a variable angle of the sputtered substance incidence and periodic exit of the substrate out of the flow. The key parameter responsible for the improvement of the target performances of electrochromic devices based on thus deposited layers (Figure 13) is the support rotation rate which determines an adjustable modification of the surface morphology and consequent increase of the contact area between the electrochromic layer and electrolyte (lithium perchlorate LiClO₄ solution) to improve the adjustable coloration and bleaching upon the application of electric current due to a reversible red-ox reaction:

Figure 13.

Scheme of WO₃-based electrochromic device.

$${\mathrm{WO}}_3\left(\text{colorless}\right)+\mathrm{x}{e}^{-}+{\mathrm{xLi}}^{+}\leftrightarrow {\mathrm{Li}}_{\mathrm{x}}{\mathrm{WO}}_3\left(\text{blue}\right)$$E6

The increase of the support rotation speed up to 3.6 rpm resulted in a linear growth of the main target parameter—electrochromic coloration efficiency (Figure 14) determined as:

Figure 14.

Coloration efficiency of tungsten oxide-based electrochromic devices as a function of the substrate rotation speed at WO₃ layer magnetron deposition.

$${\eta }_c=\frac{\left(D_{c2}-D_{b1}\right)\ast S}{\int I\left(\tau \right)\mathit{d\tau }}=lg\left(\frac{T_{b1}}{T_{c2}}\right)\ast \frac{S}{\int I\left(\tau \right)\mathit{d\tau }}$$E7

where T_b1, Т_c2, and Т_b2 (D_b1, D_c2, and D_b)—transmission (optical density) after bleaching at the 1st (previous) coloration-bleaching cycle, coloration at the 2nd (next) cycle, and bleaching at the 2nd (next) cycle, respectively, S is the sample surface, cm², and I is the current values measured every 1/3 s during the process time τ.

The surface morphology of tungsten oxide layers deposited in a stationary mode (without support rotation) and at different support rotation speeds was studied by atomic force microscopy (AFM) using a Ntegra Aura (NT-MDT, Russia) microscope with a scanning area of 5 × 5 μm. The obtained images (Figure 15) were processed by the box-counting method by division into 25 square 1 × 1 μm fragments and calculating the number of grains in each fragment followed by the calculation of lacunarity as described above.

Figure 15.

AFM images of the samples deposited in a stationary mode (a) and onto the substrates rotating with the speeds 1.2 (b), 2,4 (c), and 3.6 rpm (d).

The comparative analysis of the obtained results indicated that for the samples involving WO₃ layers deposited onto the rotating support, electrochromic efficiency almost linearly grows with the decrease in lacunarity (Figure 16).

Figure 16.

Coloration efficiency of electrochromic devices involving WO₃ layers deposited with (•) and without (ο) the support rotation as a function of lacunarity.

3.4 Microstructure-properties relationship for stainless steels

The considered digital approach was applied to characterize the effect of heat treatment onto microstructural features of austenitic stainless steel X6CrNiTi18-10 samples produced via additive manufacturing (3D printing) technique by a selective laser melting method (SLM) using EOS apparatus [48].

Three types of samples were studied:

1. Initial steel without heat treatment as reference samples;

2. Samples subjected to heat treatment at 300°C within 60 min followed by cooling in air;

3. Samples subjected to heat treatment at 1050°C within 30 min followed by cooling in air.

The applied heat treatment regimes are conventional for X6CrNiTi18-10 steel.

In order to investigate the influence of heat treatment on the steel microstructure, SEM images were obtained for cross sections of the studied 3D-printed samples.

The chemical composition of the obtained samples was studied using an atomic emission spectrometer DFS-50, X-ray fluorescence spectrometer LabCenter XRF-1800, as well as a carbon and sulfur analyzer CS-600 (LECO) in accordance with Russian Standards 12344-2003 and 12345-2001.

The quantity of nonmetallic inclusions was investigated according to Russian Standard 1778-70. The determination of these inclusions and analysis of their types was performed visually via microscopy.

The effect of heat treatment on mechanical performances of the studied steel samples was characterized by measuring their microhardness.

The elemental composition of the studied steel samples is shown in Table 2.

Steel	Content of elements, % wt.
	C	Cr	Ni	Ti	P	S	Si	Mn
Steel sample obtained by 3D printing	0.026	16.90	9.00	0.08	<0.01	0.015	0.31	0.58

Table 2.

Chemical composition of the original workpiece and steel sample after 3D printing.

Microhardness of the obtained 3D steel samples is comparatively illustrated in Table 3.

Sample	Microhardness, HV0.1, GPa
Initial 3D-printed steel without heat treatment	3.29 ± 0.16
Annealing at Т = 300°C, 60 min	2.82 ± 0.21
Annealing at 1050°C, 30 min	2.57 ± 0.11

Table 3.

Microhardness of 3D-printed X6CrNiTi18-10 steel samples depending on heat treatment conditions.

Since chemical composition of steel remains the same, the observed changes of mechanical properties obviously result from changes in the steel microstructure. The most significant influence on steel hardness has the content of nonmetallic inclusions (oxides, carbides, and nitrides) as their hardness is much larger than that of metal matrix. The amount of nonmetallic inclusions in the studied steel samples estimated in accordance with standard procedure is summarized in Table 4.

Sample	Point oxides	String oxides	Silicates	Sulfides	Nitrides and carbides	Total
Initial 3D-printed steel without heat treatment	1.9	0.9	0	0	0	2.8
Annealing at Т = 300°C, 60 min	1.7	1.1	0	0	0	2.8
Annealing at 1050°C, 30 min	2.3	0.6	0	0	0	2.9

Table 4.

Content of nonmetallic inclusions determined according to a standard procedure.

However, it is difficult to directly correlate the content of such inclusions with the steel hardness, since in the course of 3D printing a laser beam selectively melts steel particles one by one, thus forming melt bath with nonmetallic phases distributed along edges of the melted zone. After solidification, a 3D network of nonmetallic inclusions is formed, preventing from the movement of dislocations and mechanical deformations and thus providing increased mechanical properties. Therefore, the design of 3D network of nonmetallic inclusions is a promising approach to the enhancement of steel mechanical properties, and a detailed study of the achieved effects requires the application of more exquisite method to characterize the microstructure of 3D-printed steel.

SEM images of thin sections of samples annealed at different temperatures with distinguished centers of mass are shown in Figure 17. It was found that in 3D-printed samples nonmetallic inclusions are mainly concentrated in the space between crystallized particles of the remelted powder. The chemical composition of the inclusions suggests that they predominantly consist of titanium oxides.

Figure 17.

SEM images of 3D-printed (SLM) samples from steel X6CrNiTi18-10 without heat treatment (a), annealed at 300°C (b) and 1050°C (c) with nonmetallic inclusions marked in red.

The analysis of these data revealed that the increase in the annealing temperature leads to the increase in the lacunarity of nonmetallic inclusions (i.e., a growth of their distribution inhomogeneity), in couple with a decrease in their number and in the scale invariance factor n (Figure 18), reflecting a decrease in the statistical nature (stability) of the system.

Figure 18.

Lacunarity (a) and scale invariance factor (b) as a function of the annealing temperature.

The relationship of microhardness with the uniformity of the mutual arrangement of nonmetallic inclusions was also analyzed. As shown in Figure 19, microhardness of the studied 3D-printed steel grows with the decrease of lacunarity, i.e., with the increase of no-metallic inclusions distribution homogeneity.

Figure 19.

Microhardness of steel samples as a function of lacunarity.

4. Conclusion

The presented examples demonstrate that such fractal parameters as lacunarity and its scale invariance characterizing the uniformity and self-similarity of the structure at different scale levels are essential factors prominently correlating with different properties of various materials. The described methods can be applied to a wide range of functional materials such as alloys, composites, and ceramics in order to describe their microstructural features on the level of mutual arrangement of their components and predict their properties. Particularly, common trends toward an improvement of target characteristics with a decrease of lacunarity and increase of scale invariance are revealed for different materials and their properties, including mechanical characteristics of steels and special ceramics, dielectric properties of polymer-inorganic composites, and ion transport in tungsten oxide-based electrochromic devices. The considered effects are determined by the uniformity of chemical composition and microstructure elements (fillers, phases, inclusions, etc.). The analysis of correlations between the chemical composition of materials, their fabrication and processing methods, microstructural characteristics (involving the above fractal parameters) and properties is an essential goal of Digital Materials Science providing an exquisite approach to the selection of optimal synthetic procedures, prediction, and enhancement of their target characteristics.

Acknowledgments

The study was supported by a grant from the Russian Science Foundation (project no. 21-73-30019). The equipment of the Center for Collective Use “Composition, structure, properties of structural and functional materials” of the National Research Center “Kurchatov Institute”—Central Research Institute of Structural |Materials “Prometey” was used with financial support from the Ministry of Education and Science of the Russian Federation under the agreement No. 13.TsKP.21.0014 (unique identifier RF 2296.61321X0014).