1 Introduction

The elucidation of the self-organized material textures of filler particles in composite matrices and their physical properties is important from the viewpoints of both complex systems physics and composite materials science. In particular, the interaction of the filler particles in the matrix and the resulting network formation are important for expressing the material functionality. On the other hand, multifractal analysis [1–5], which is widely used in physics, chemistry, medicine, and engineering involving morphology and shape, is being applied to the field of materials science. Recently, this analysis has been used to reveal the particle morphology, arrangement, dispersion, and statistical thermodynamic properties during chemical reaction processes, and has been shown to be effective in clarifying the dissipation state of particles due to self-organized or self-assembled processes in real binary systems (metal/ceramics and polymer/ceramics) [6–10]. In particular, the general dimension Dq (multiple fractal dimension for q-th order moment) and scaling exponent τq obtained from the microstructural morphologies of self-organized or self-assembled materials serve as material design indices for designing composite materials with specific functionality. In fact, in β-Si3N4 (SN)/austenitic stainless steel (SUS) [6, 7] and BaTiO3 (BT)/piezoelectric polyvinylidene fluoride (PVDF) [8–15] systems, the characteristics of the material structure constructed by the process of self-assembly or self-organization have been mapped to material performance such as dielectric properties and thermal conductivity.

On the other hand, the distribution of additives in the matrix has been analyzed and evaluated as a fractal particle size distribution [16, 17]. However, with respect to the particle size and distribution of secondary components, which are important from the perspective of materials science, there has been little research from the perspective of the connections between particles in a complex network. Unlike conventional research on particle size distribution, we attempt to perform an analysis that focuses on interactions and connectivity between particles from the perspective of the Barabasi-Albert (BA) model, [18] which is often studied in complex network systems. It is widely accepted that many complex networks found in fields as diverse as physics, biology, sociology, ecological science, and information engineering possess scale-free properties; that is, the distribution function of degree k obeys a power law [19, 20]. In fact, a common property of gene networks and many large-scale networks is that the connectivity of vertices follows a scale-free power-law distribution. The BA model comprises two general mechanisms.

  1. (i)

    The network continuously expands by adding new vertices.

  2. (ii)

    New vertices preferentially connect to well-connected sites.

This is similar to the agglomeration/aggregation and growth of fillers in a matrix. As previously reported for fractal material texture organization, [6–15] the construction of this self-assembly/self-organization consists of the following two processes:

  1. 1)

    formation of particle groups by fillers/additives dispersed in the matrix,

  2. 2)

    growth of particle group networks through particle connections.

These two processes are similar to the two mechanisms in the BA model. Therefore, for self-assembled or self-organized fractal material textures, it is necessary to analyze the distribution of the filler particles using a power-law distribution.

In fact, in the self-assembled/self-organized material texture that we have been dealing with, the particle size distribution of the filler in the matrix exhibits a power law distribution. Therefore, in this study, we discuss the fractal material texture and complex network properties such as scale-free property and small-world property of the highly fractal composite systems of SN/SUS,6 BT/PVDF,8 and SN/PVDF10 with improved thermal conductivity.

2 Experimental procedure

2.1 Sample preparation

2.1.1 SN/SUS system

SN/SUS composites were prepared from commercially available SN powder (Denka Company Ltd.) and SUS316L powder (Epson Atomix Corp.) as the starting materials. The compositions were 0 vol% SN-100 vol% SUS316L, 5 vol% SN-95 vol% SUS316L, and 10 vol% SN-90 vol% SUS316L. Typically, green compacts are sintered by spark plasma sintering (SPS) at 1173 K and 50 MPa for 10 min and pressureless sintering (PLS) at 1273 K for 3 h in an argon atmosphere (1 atm). (Details of the sample preparation are described in Ref. 6.)

2.1.2 BT/PVDF and SN/PVDF systems

The BT/PVDF and SN/PVDF composites were prepared by melting PVDF powder (KYNAR® 711, Arkema S.A.) at 483 K. Subsequently, BT powder (Fuji Titanium Industry Co., Ltd.) and SN powder (Denka Company Ltd.) were added to the PVDF melt at various volume fractions (5, 10, 15, 20, and 25 vol%) to form mixtures of BT/PVDF and SN/PVDF, respectively. After stirring with a kneader at 10 rpm and 483 K for 30 min, the mixtures were cooled to room temperature in air to obtain the BT/PVDF and SN/PVDF composites. To enhance the dispersion of the BT/PVDF and SN/PVDF composites, polyethylene glycol PEG (PEG 1000, FUJIFILM Wako Pure Chemical Corporation) was added to the PVDF melt containing the BT or SN powder. The amount of PEG in the mixture was approximately 4 wt% relative to the amount of BT or SN powder. The BT/PVDF or SN/PVDF melt mixture was poured into dies to obtain sheets. (Details of the sample preparation are described in Ref. 8,10.)

2.2 B. measurements

Cross-sectional images of the composites were obtained using scanning electron microscopy (SEM; TM3000, Hitachi Ltd.) and field-emission scanning electron microscopy (FE-SEM; Hitachi Co., Japan: S-4100). The images were then converted into binary images to estimate the average area of the SN or BT aggregates in each sample. The average secondary particle area (S) is defined as the number of pixels obtained when the number of black pixels is divided by the number of particle groups in the binary image. The measurement error of S was less than ± 0.1 µm2. (Details of the measurements are provided in Ref. 6,8,10.). The thermal conductivity, κ, of the composite was measured using a standard steady-state method with a customized apparatus under vacuum(Details of the measurements are provided in Ref. 6, 10.).

3 Results and discussion

Figures 1, 2, 3 show the secondary particle area fraction P(s) as a function of the secondary particle area s for the three composite material systems: SN/SUS, SN/PVDF, and BT/PVDF. The secondary particle area fraction (%) was plotted at particle group area of 0.25 µm2. Both distributions are not normal distributions, but are power distributions. Furthermore, SN/SUS (Fig. 1) tended to form a peak at approximately 3 µm2, whereas the SN/PVDF (Fig. 2) tended to form a peak at approximately 4 µm2. In contrast, no clear peaks were observed for BT/PVDF (Fig. 3). As previously reported, in the SN/SUS system6 and SN/PVDF system,10 the particle group grows and has an average secondary particle area of approximately 2.2 µm2, whereas in the BT/PVDF system8 it is small at approximately 1.4 µm2, which corresponds to the remarkable formation of very small particle groups.

Fig. 1
figure 1

Secondary particle area fraction P(s) as a function of the secondary particle area s for SN/SUS systems

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Fig. 2
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Secondary particle area fraction P(s) as a function of the secondary particle area s for SN/PVDF systems

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Fig. 3
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Secondary particle area fraction P(s) as a function of the secondary particle area s for BT/PVDF systems

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Considering the network structure of the BA model, it is expected that a system with scale-free properties will have a power law distribution [18–20]. In a particle network model, assuming that the degree k of a vertex (the number of links) is the number of edges (links) in contact with each particle (node), the area of a particle group reflects the number of links and the distance between particles. Because the degree distribution P(k) corresponding to degree k represents the frequency of the vertices of degree k in these composite materials, degree k and distribution P(k) are determined by the area size s and number N(s) of the particle group, respectively. Figures 4, 5 and 6 show log-log plots of s versus the number of secondary particle area N(s) for three composite material systems: SN/SUS, SN/PVDF, and BT/PVDF. As can be seen from these figures, the particle area size distribution of all composite systems follows a power law:

Fig. 4
figure 4

Log-log plots of secondary particle area s versus the secondary particle area number N(s) for SN/SUS systems. Inset: the semilogarithmic plots of the secondary particle area number N(s)

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Fig. 5
figure 5

Log-log plots of secondary particle area s versus the secondary particle area number N(s) for SN/PVDF systems. Inset: the semilogarithmic plots of the secondary particle area number N(s)

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Fig. 6
figure 6

Log-log plots of secondary particle area s versus the secondary particle area number N(s) for BT/PVDF systems. Inset: the semilogarithmic plots of the secondary particle area number N(s)

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$$N\left(s\right)\propto {\text{s}}^{-\gamma }{,}$$
(1)

here, γ is the scaling index. This indicates that these material systems exhibit scale-free properties, unlike the exponential distribution, which is considered a normal particle-size distribution with a typical average particle size. In the plots for all compositions, the slope γ of the straight line varied in the middle of the plot. Therefore, the slopes in the small and large s regions were determined, and are summarized in Tables I–III for each composition.

As shown in Fig. 4 and Table 1, the log-log plot for the SN/SUS system is relatively linear, and the difference in the slope of each region of the sample produced by the SPS or PLS process is small. In addition, the semi-logarithmic profile of N(s) (Fig. 4 inset) decreases exponentially as s increases. The slope γ decreased with the amount of SN added. In particular, PLS has a greater effect on γ with an increase in the amount of added SN than SPS. This is thought to be due to the longer sintering time required for PLS than for SPS, which reflects the influence of grain growth in the base material. This result also corresponds to the results of a previously reported multifractal analysis [6, 7]. The crossover from the power-law profile to the exponential function of such networks has already been reported as a local world property [21, 22]. This indicates that the fractal network becomes random as the particle group grows larger. It has been pointed out that this local world growth model cannot maintain the scale-free property unless the size of the local world is increased if the preferential selection of connections is performed only within the set of network nodes (within the local world) [22]. In this study, increasing the amount added may contribute to maintaining scale-free properties because the growth of particle groups occurs along with connections in the vicinity of local world (particle groups). In other words, the collection of network nodes and the local world correspond to the formation of particle groups and particle group connectivity. This conclusion suggests that it is necessary to grow the particle group to increase the size of the local world, that is, to promote the coupling of the particle group.

Table 1 The slope γ in the small and large secondary particle area for the SN/SUS system
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A similar crossover was also observed in the SN/PVDF system, as shown in Fig. 5, and was more pronounced than that in the SN/SUS system. Additionally, the bending of N(s), which was slight in the SN/SUS system, can be clearly seen at approximately 2 µm2. In contrast, the crossover behavior of BT/PVDF differs from that of SN/PVDF in the large s region. Whereas, in the small region of s, there is a large bend at approximately 0.9 µm2, similar to the SN/PVDF system. As the amount of BT added increases, the slope γ decreases. Therefore, it is pointed out that γ is related to the interaction between particles and particle groups. In particular, values of γ smaller than 2 are extremely rare [22]. The bend in this plot is considered to be related to the formation of particle groups (micro network formation) and the subsequent connection of particle groups (macro network formation) in these three systems. In fact, from the results of this power distribution and the changes in the γ value, there are regions where the formation of particle groups is dominant (small particle groups) and regions where their growth is dominant (large particle groups). However, in the BT/PVDF system, once a particle group is formed, the interaction between the particle groups is suppressed [8, 9], and the system does not exhibit a power distribution or normal distribution in the region where s is large.

These three systems exhibit a common change in plot slope γ. The before and after bending of the plot in these figures are considered to correspond to the change in behavior at q = 0 in the plot of the general dimension Dq -q, the scaling exponent τq -q and the singularity spectrum f(α)-q [1–5]. In other words, D0 in multifractal analysis can be defined as the boundary between local and global regions. It has been experimentally reported that this boundary D0 is correlated with the average secondary particle area to 1/2 power (S1/2) [10]. Therefore, assuming that the secondary particle area s*, where the plot bends, is the average secondary particle area S, a very good approximate curve can be obtained from the plots in each figure. This shows that this transition point is correlated with D0 and that the regions before and after this point are the local and global regions in the multifractal analysis plot. From the above, it is considered that P(s)-s log-log plot and multifractal plot (Dq-q and τq-q) that represent complementary analysis methods to understand the distribution and interaction of fillers in composite materials will be useful.

As mentioned above, the average secondary particle area S is one of the characteristic parameters of these systems. Figure 7 shows the results of plotting the thermal conductivity \(\kappa\), which is especially affected by network connections, as a function of the average secondary particle area S. These comparisons suggest that changes in the average secondary particle area of filler aggregates have a significant impact on the thermal conductivity of the composites [6, 10, 23]. The results obtained for the SN/SUS system indicate that the formation of SN secondary particle groups through the agglomeration-dispersion reaction between SN ceramics and SUS metal particles and the crystal growth of SUS particles play an important role in improving thermal conductivity. In other words, it was shown that the sintering method (SPS or PLS), which affects the diffusion grain growth of the base material particles, also affects the connection of particle groups due to the arrangement and dispersion of SN particles. As shown in Fig. 7(a), as S increases, a particle group network is formed in the SN/SUS (SPS) system, but in the SN/SUS (PLA) system, the heat path is cut by the SUS grain growth. This effect occurs in the region where s is large and corresponds to a large change in the slope γ value of PLA, as shown in Table 1. That is, as the slope of γ increases, the deviation in particle diameter increases, and it is difficult to form a gentle network. On the other hand, in the PVDF system [8, 10] shown in Fig. 7(b), it has been revealed that SN addition can form percolation, but BT particle groups are isolated. As shown in Tables 2 and 3, this effect is also significant in the region where s is large, and the addition of BT is greater than the addition of SN for any filler addition amount. Therefore, the characteristic that the distribution of large secondary particle diameters behaves exponentially or non-exponentially is thought to correspond to the ease of forming a global network. In fact, the present results show that network formation is important for thermal conductivity, rather than simply growing a thermal path under the growth of particle groups.

Fig. 7
figure 7

Thermal conductivity \(\kappa\) as a function of the average secondary particle area S for the (a) SN/SUS and (b) SN/PVDF and BT/PVDF

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Table 2 The slope γ in the small and large secondary particle area for the SN/PVDF system
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Table 3 The slope γ in the small and large secondary particle area for the BT/PVDF system
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In the BA model, in addition to the power distribution, a typical property [28] is the relationship between the distance L between the nodes in the network and the total number of nodes (filler particles) N:

$$L\propto \frac{\text{l}\text{o}\text{g}N}{\text{l}\text{o}\text{g}\text{l}\text{o}\text{g}N}{.}$$
(2)

As stated in a previous report [10], if we consider the interacting distance (correlation distance) within a particle group to be the distance L between the nodes, the relationship between S and the distance L between the filler particles is as follows:

$$L\propto {S}^{1/2}{.}$$
(3)

Therefore, the relationships between the interparticle distance L obtained from the previously reported experimental data of SN/SUS [6], SN/PVDF,[10] BT/PVDF [8] and the total number of particles N obtained from the actual binarized image are summarized in Fig. 8(a). Furthermore, as is clear from the present results, the average secondary particle area is not the average and is normally treated as a normal distribution. Although there was a large variation in the measured S values, there was a correlation between D0 and S1/2 [8, 10, 29]. Fig. 8(b) shows the plotted results when S1/2 was changed to D0 based on previously reported experimental data [6, 8, 10] for each system, similar to Fig. 8(a). The plot in Fig. 8(b) indicates that these systems follow the BA model, yielding a scale-free network by approximating the average internode distance L using Eq. 3. Therefore, D0 can be considered as a representative parameter that accurately reflects the characteristics of complex networks.

Fig. 8
figure 8

(a) S1/2vs. log N / log (log N) and (b) D0vs. log N / log (log N) for the SN/SUS, SN/PVDF, and BT/PVDF systems

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In recent years, the relationship between the small-world nature of complex networks and fractals has been discussed, particularly with regard to the World Wide Web network, [30] brain networks, [31] and the tree algorithm (family tree) [32] and so on. Most complex networks in the real world exhibit small-world properties. More precisely, the average node pair distance L in these cases is considered to be much smaller than the number of nodes N [30–32]. The application of a modified model that possesses these features will yield an understanding of nature because most of the fractal networks observed in the real world, such as the World Wide Web or metabolic networks, have a scale-free property. [33], [34] Moreover, it has been reported that self-organized critical dynamics, which combine network growth and decline, can be a possible explanation for the emergence of fractal and small-world networks [35], [36]. The small-world property, according to the Watts and Strogatz (WS) model [37] suggests that “it is possible to obtain from a certain node (point) to a certain node by simply passing through a small number of nodes, and nodes connected to a certain node are connected to each other.‘’ For the small-world property, [36] the relationship between the distance L between the filler particles and the total number of nodes (filler particles) N:

$$L\propto logN.$$
(4)

Because D0 is correlated with the average secondary particle area to 1/2 power (S1/2), [8, 10, 29] the actual D0 -\(logN\) variations for each system were plotted when L is changed to D0 (Fig. 9). As shown in Fig. 9, Eq. 4 also holds, it is thought that the small-world property holds in a broad sense. These trends are almost identical to those shown in Fig. 7(b).

Fig. 9
figure 9

D0vs. log N for SN/SUS, SN/PVDF, and BT/PVDF systems

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As described above, a binary composite system can be considered to be consistent with both small-world and scale-free properties. The WS model [36] with small-world properties is concerned with the connectivity of the formed particles and particle groups in the self-assembly/self-organization process involving solids. This connectivity is thought to be closely related to the percolation-like properties of transport phenomena, such as thermal and electrical conduction. The thermal conductivities of the SN/PVDF system [10] show percolation properties. However, in the BA model [18] with scale-free properties, the preferential growth process of nodes (particles) is considered to be the formation of hubs (large particle groups). However, there may be differences in the connectivity between the WS and BA models. That is, even if a network is formed, its properties differ depending on whether connectivity is high or low. As previously reported, this difference is due to the difference in thermal conductivity between the SN/PVDF system [10] and the BT/PVDF system, [23] which is thought to be the difference in the behavior of γ and Eint.

Apart from considering the above BA model, we focused on the relationship between the multifractal and complex network nature of the three particle groups. In the PVDF composite system, [8, 10, 23] research has focused on the dielectric properties and thermal conductivity and has been studied in relation to the distribution state of the added fillers in the self-assembled material structure.

1) Although the addition of SN did not affect the dielectric properties, it significantly improved the thermal conductivity by forming a particle-group network.

2) With the addition of BT, a significant improvement in dielectric properties was observed due to the formation of a heterointerface of particle groups; however, the improvement in thermal conductivity was limited to the Bruggeman’s model [24, 25]. It can be said that the SN particle group has a remarkable network formation that is effective for forming thermal paths, whereas the BT particle group has particles that are strongly connected, but the network formation is insufficient. This is because when estimating the interaction energy (Eint.) between particle groups using multifractal analysis, [7, 10, 26, 27] Eint. between the SN particle groups [10] tends to increase with the addition amount; however, Eint. between the BT particle groups8 tends to decrease. Previous research on multifractals has reported that the information dimension D1 and the correlation dimension D2, which are related to particle connectivity and cohesion, become smaller; that is, particle cohesion is promoted in the SN/SUS [6, 7] and SN/PVDF [10] systems. The trend is considered to correspond to a decrease in γ (the particle group is growing continuously). As the particles agglomerated (or aggregated) and grew, the particle size distribution shifted from finer particles to larger particles (particle groups). The progression of this agglomeration (or aggregation) growth reaction results in a decrease in the distribution of fine particles and an increase in the distribution of coarse particles. In other words, the particle size distribution became smooth and the rate of change in the particle group area (number of particle groups) relative to the particle area decreased. This is thought to correspond to the fact that the connectivity of the BT/PVDF system is lower than that of the SN/SUS and SN/PVDF systems, which have high connectivities, as shown in Tables I–III. These results suggest that the addition of BT makes the formation of a particle-group network more difficult than the addition of SN. This is related to the behavior of the large particle area s of the particle groups, as shown in Figs. 5 and 6. From Fig. 5 and its inset, it can be observed that the SN particle group changed from a fractal network to a random network. However, the BT/PVDF system exhibited different behavior from that of a random network in the region where s was large. As mentioned in a previous report, [10] we confirmed from the D1 vs. Dq plot that the probability density changes owing to higher-order correlated interactions between particles and particle groups. When this plot is applied to the BT/PVDF system, [8] the deviation of the probability density at higher moments is larger than that for the SN/PVDF system [10]. In other words, it can be said that changes in interactions between particle groups affect particle density.

In previous reports [6, 8, 10] have considered the connection of particle groups and their network formation from a multifractal perspective. In the SUS system, [6, 7] the intermolecular force of the filler plays an important role in the formation of SN particle groups. In the case of PVDF, it is thought that the physical and chemical effects of the filler interface have an influence. [12, 14] This difference in the aggregation mechanism of fillers is thought to reflect the network formation energy of the particle groups. The interaction energy Eint. of the particle group mentioned above can be found from the change in α (equivalent to internal energy) in the positive and negative regions of q from the τ-q plot as an index regarding binding energy, as previously reported [6–8, 10]. As mentioned above, the bending point of γ, which reflects the characteristics of the particle size distribution of each system, corresponds to the particle group area s* related to the multifractal dimension D0. That is, for all three systems, the slope γ of the power distribution varies with s*. This suggests that there is a relationship between the multifractal parameter and the slope γ of the power distribution. In particular, D0 corresponds to the inflection point of the spectrum in multifractal analysis [1–5]. With this D0 as the boundary, the scaling exponent τ changes, and the information thermodynamic parameter α (related to internal energy change) regarding the interaction of the particle group is estimated. Figure 10 shows a plot of γ-α between the information thermodynamic parameter α involved in particle group formation and γ reflecting the characteristics of particle size distribution. Here, γsmall is the region where particle group formation is prominent, and it is thought to be related to the binding energy αlocal in the local scale region where particles form particle groups in multifractal analysis. Similarly, γlarge is considered to be related to the particle group network formation energy αglobal, where the coalescent growth of particle groups occurs. The obtained results show that there is a correlation between particle connectivity and particle size distribution. In the SUS system, γ also increases as α increases in both SPS and PLS as shown in Fig. 10(a). In both regions, the rate of change in the plots shows a steeper change in PLS than in SPS. In particular, in the γlarge/αglobal region, the grain growth of PLS is more pronounced than that of SPS, suggesting that the change in grain size distribution is large with respect to changes in composition, and as a result, it is thought that it becomes difficult to form a network. If extreme grain growth progresses, the network is likely to become fragmented and change to a random distribution. Therefore, in general, it is necessary to confirm that the power law follow for each system and to investigate the complex network of particle distribution. In the PVDF system, the behavior in the γsmall/αlocal region differs depending on the added fillers of SN and BT as shown in Fig. 10(b). With SN addition, γ has little dependence on α. With BT addition, γ decreases while α increases. In generally, the internal energy α increases as the number of particles in the system increases. However, γ does not change unless the distribution form changes. In fact, when comparing Figs. 2 and 3, in the region where the particle groups are small, the change in particle size distribution associated with an increase in the number of particles (increase in the added filler concentration) is more pronounced in BT than in SN. Furthermore, in the γlarge/αglobal region, the α dependence of γ is significantly different between SN and BT. In this region, as in the SUS system, a particle group network is formed due to interactions between particle groups formed with the amount of filler added. It is thought that if the particles are connected, the distribution will be smoother and γ will be smaller. If the growth of the particle group network is insufficient and the formation of particle groups is promoted as the amount added is increased, the distribution will change significantly. In other words, as shown in Fig. 10(b), it is thought that the formation of particle groups and network growth are not promoted because BT have higher particle dispersibility than with SN, and the particle size distribution changes with the amount added. Therefore, it is thought that BT did not provide stable bonding of particle groups and did not improve thermal conductivity compared to SN.

Fig. 10
figure 10

Slope γ as a function of the internal energy α for the (a) SN/SUS and (b) SN/PVDF and BT/PVDF

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The binary composite system described above was generated by mixing and diffusing filler particles during sample preparation. In particular, during the mixing process, filler particles are repeatedly aggregated and dispersed to form particle groups and networks, thereby imparting new functions and improving the material performance. In other words, these systems can be considered examples of reaction-diffusion systems. Therefore, the network structure constructed from this reaction-diffusion system is a complex network system with a fractal texture owing to self-assembly/self-organization. The particle group network texture that occurs in reaction-diffusion systems involving solids can be thought of as constructing a type of Turing-pattern-like organization that occurs spontaneously owing to chemical reactions and has no time dependence. Furthermore, the formation of patterns in textures will be an experimental object of information thermodynamics.

4 Conclusion

To investigate the relationship between the histological properties and thermal conductivities of binary self-assembled/self-organized fractal materials, we analyzed the degree distribution of the filler particle group area S (particle group number distribution) in composite systems of SN/SUS, SN/PVDF, and BT/PVDF, and discussed the properties of composite materials as complex networks. In this study, we discussed the relationship between particle distribution and connectivity from the influence of network formation on thermal conductivity.

The results obtained are as follows:

  1. (1)

    In all the three systems, the particle size distribution (order distribution) of the secondary particles followed a power-law distribution until the particle groups became relatively large.

  2. (2)

    In the SN/SUS and SN/PVDF systems, a crossover phenomenon was observed in which the order distribution exhibited a power-law distribution in the region where the particle group area was small, and as the particle group became larger, the order distribution switched to exponential distribution. This indicates that the fractal network became random as the particle group grew larger. On the other hand, the BT/PVDF system deviated from the power law distribution as the particle group area increased but did not show an exponential distribution. Furthermore, the particle distribution characteristics of the three systems were shown to influence the thermal conductivity.

  3. (3)

    The relationship between D0, which is strongly correlated with the distance L between the particle groups, and the total number of filler particle groups showed characteristics corresponding to the BA model. However, because no contradictions occurred in the WS model, it was assumed that the texture of the synthesized composite　material system had both small-world and scale-free properties.

  4. (4)

    Apart from considering the log-log slope γ based on the BA model mentioned above, we focused on the relationship between the multifractal and complex networks of the three particle groups. It was found that the bending point of γ, which reflects the characteristics of the particle size distribution of each system, corresponds to the secondary particle area s*, which is related to the multifractal dimension D0. Furthermore, there was a correlation between the slope γ, which reflects the characteristics of the power distribution, and α obtained from multifractals. In particular, in the γlarge/αglobal region, it was suggested that the larger the α dependence of γ, the more difficult it is to form a particle group network (connection of particle groups).

The binary composite system described above is an example of a reaction-diffusion system in which particles and particle groups form a network during the aggregation and dispersion of filler particles. Furthermore, the constructed network structure is considered a complex network system with a fractal texture owing to self-assembly and self-organization. Our findings will be useful for researchers working not in applied physics and materials science, but also in understanding nature and information sciences because most of the fractal networks observed in the real world exhibit a scale-free property.