Information thermodynamics of ternary fractal material texture: Application to BaTiO₃/β-Si₃N₄/polyvinylidene fluoride composites

In the self-organized textures within composite matrices, network formation through interactions between filler particles is crucial for expressing material functionality. To capture the microstructural characteristics of the binary system, we employed multifractal analysis,^1–5 a technique widely used in physics, chemistry, medicine, and engineering, in conjunction with the analysis of composite material structures, aiming to establish material design indices. As reported in prior studies on fractal material texture organization,^6–11 this self-organization/self-assembly construction involves two primary processes: (1) the formation of particle groups from fillers dispersed in the matrix and (2) the growth of particle group networks through particle connections. In particular, by combining the fractal material design indices^6–10 obtained for composite materials with complex network analysis of filler particle groups,^11–15 we were able to capture the characteristics of material microstructures in binary systems from the perspectives of quantitative morphology and stereology. In this manner, the self-assembled/self-organized material structure is a composite material with fractal properties and complex network characteristics.¹¹ However, a sufficient understanding of filler distribution in composite materials remains challenging. Additionally, a detailed theoretical understanding through complex network models^12–15 and other methods has not yet been fully realized. Traditionally, material research has focused on enhancing material performance and multifunctionality.^16–18 To achieve multifunctionality, multiple fillers have been added to polyvinylidene fluoride (PVDF), a piezoelectric polymer, enabling the coexistence of multiple properties.^16–18 In these composites, the microstructure features a uniform dispersion of fillers with different morphologies^16,17 or hybrid particles.¹⁸ Conversely, when multiple fillers are kneaded simultaneously, they tend to mix, making it challenging to achieve multifunctionality through aggregated fillers in composite materials.

Typically, two fillers become mixed within the particle group, which complicates the realization of dual functionalities. To address this, we synthesized a sample using the lamination kneading process (LKP), which preserves the particle group. However, recent studies have shown that even a simple kneading process (SKP) can form functionally distinct particle groups, allowing dual functionalities in a single composite material system.¹⁹ This demonstrates that the filler/additive dispersion state in self-assembled/self-organized composites can be controlled, and optimized fabrication processes can potentially realize multifunctional materials.

In this study, we leveraged prior knowledge of BaTiO₃/polyvinylidene fluoride (BT/PVDF) and β-Si₃N₄/polyvinylidene fluoride (SN/PVDF) composites to fabricate a BaTiO₃/β-Si₃N₄/polyvinylidene fluoride (BT/SN/PVDF) composite. This study aims to clarify the compatibility of high dielectric properties with high thermal conductivity and to analyze the unique composite texture. To investigate the relationship between materialographic characteristics and properties (dielectric properties and thermal conductivity) of these three-component self-organized fractal materials, we analyzed the filler particle distribution using multifractal analysis, developed for evaluating material structures. Additionally, focusing on the information dimension (information entropy) from the analysis results, we calculated the mutual information (MI)^20–22 commonly used in chemical reaction network systems²³ to assess the interaction between two filler types and consider the effects of different preparation processes on filler interactions.

BT/SN/PVDF composites were prepared by melting PVDF powder (KYNAR^® 711, Arkema S.A.) at 483 K. BT powder (Fuji Titanium Industry Co., Ltd.) and SN powder (Denka Co., Ltd.) were then each added at a volume fraction of 20 vol. % to the PVDF melt to form BT/PVDF and SN/PVDF mixtures, respectively. Upon stirring each mixture with a kneader at 10 rpm and 483 K for 30 min, they were cooled to 298 K in air to obtain the BT/PVDF and SN/PVDF composites. To improve filler dispersion, polyethylene glycol (PEG 1000, FUJIFILM Wako Pure Chemical Corporation) was added at approximately 4 wt. % of the BT or SN powder. Each BT/PVDF or SN/PVDF melt mixture was poured into dies to produce sheets approximately 0.4 mm thick (details in Refs. 9 and 10). The BT/PVDF and SN/PVDF sheets were laminated, stretched to 0.4 mm, cut in half, stacked, and stretched again to a final thickness of approximately 0.4 mm. This lamination kneading process (LKP) was repeated 4, 8, and 12 times to produce films with 2⁴, 2⁸, and 2¹² layers, respectively.

For comparison, a sample was prepared by premixing the fillers and adding them to PVDF, followed by kneading (simple kneading process, SKP). In this method, BT and SN powders were added to the PVDF melt at 10 vol. % each to create a BT/SN/PVDF blend. After stirring at 10 rpm and 483 K for 30 min, the mixture was cooled in the air to 298 K, yielding the BT/SN/PVDF composite. Approximately 4 wt. % PEG, relative to the BT and SN powders, improved the mixed filler dispersibility. All BT/SN/PVDF composites obtained had a volume fraction of BT:SN:PVDF of 10:10:80. The relative density of the samples prepared, calculated using the Archimedes method, exceeded 96% of the theoretical density.

Cross-sectional images of the composites were obtained using a scanning electron microscope (SEM; TM3000, Hitachi Ltd.) and a field-emission scanning electron microscope (FE-SEM; S-4100, Hitachi Co., Japan). These images were converted into ternary images to estimate the average area of the SN and BT aggregates in each sample. Ternary images were created based on the brightness values in the SEM images, assuming homogeneous aggregate formation. Given the atomic number brightness hierarchy—BT (Ba, Ti) >> SN (Si) > PVDF (C, F)—the white area was assigned to BT, while the gray area was assigned to SN, excluding gray dots surrounding BT.²⁴ The distribution of SN, indicated in orange in the ternary image, was also confirmed in the energy dispersive x-ray spectroscopy (EDS) mapping image for the Si element in the same field of view. While the SN distribution may contain minor noise from BT, this noise is minimal compared to the average particle size of SN and is expected to have a negligible effect on the SN particle group distribution. The average secondary particle area (S) was calculated by dividing the total number of black or gray pixels by the number of particle groups in the ternary image. The measurement error of S was less than ±0.1 μm² (details of the measurements are provided in Refs. 6, 9, and 10).

The ε′ and loss tangent (tan δ) of the BT/SN/PVDF composite in the frequency range of 100–110 MHz were measured at 298 K using an Agilent 4294 A precision impedance analyzer with platinum lead wires. Each sample had a rectangular shape with a surface area of 70 mm² and a thickness of 0.3 mm. The electrodes were made of gold with square faces.

In the material texture of the current experimental system, the information dimensions D₁ and D₋₁ of the distribution of filler particles can be determined using the multifractal analysis described earlier. Since the information dimension in multifractal analysis can be expressed through probability, as shown in Eq. (7), the MI can be represented by the information dimension D_±1. In general, MI measures the reduction in uncertainty about one state (x) when the other state (y) is known, and vice versa. Physically, this means that MI, I, can be determined by quantifying the degree to which being in state x is correlated with being in state y and vice versa. The information dimension (or information entropy) in this case relates to the configuration entropy of particles and particle groups.

From these equations, the goal is to obtain the degree of interaction between the BT and SN particles, as well as between the particle groups, by calculating the marginal entropy, conditional entropy, and joint entropy from the information dimensions D₁ and D₋₁ related to the BT and SN particle groups obtained by image processing.

Figure 1 illustrates the SEM photographs (backscattered electron image) and ternary images of the dispersions of the BT/SN/PVDF composites after mixing in a kneader at 10 rpm for 30 min. Figures 1(a)–1(d) present the typical cross-sectional SEM images of samples fabricated by different processes: LKP [repeated 4, 8, and 12 times, respectively, denoted as LKP (4), LKP (8), and LKP (12)] and SKP. Figures 1(e)–1(h) illustrate the distributions corresponding to the aggregates of BT and SN particles, respectively, obtained by image processing techniques. In these images, the black regions of the ternary image correspond to BT particle groups, the red regions represent SN particle groups, and the white regions represent PVDF.

As illustrated in Fig. 1, the BT and SN particles form secondary particle groups through a self-organization process within the PVDF matrix, which connect and grow during the lamination and kneading process. Since the growth of these particle groups is stochastic, the distribution of BT and SN particles is non-uniform yet self-similar. During lamination and kneading, when the number of layers is small, a sheet forms with alternate layers of each particle group, where alignment and connection occur for each particle group. In contrast, when the number of layers increases, the interaction between the BT and SN particle groups becomes more intense, similar to the simple kneading process, and particle mixing becomes more evident.¹⁹ 

These SEM images of the sample obtained by repeating the LK process 12 times [LKP(12)] and the sample obtained by the simple kneading process (SKP) show no significant difference in the particle dispersion state. Furthermore, the coexistence of BT and SN particles is rarely observed, meaning that the same particles aggregate and bond together (attractive interactions), while different particles do not aggregate (repulsive interactions). In the process of increasing the number of layers in the lamination kneading process and the simple kneading process, it is typically thought that the two filler components are simply arranged in a uniform three-dimensional distribution. However, as described in previous reports^9,10 on the BT/PVDF and SN/PVDF systems, the material structure obtained demonstrates that the BT and SN particles retain their respective properties, and each particle group is distributed and arranged independently. These characteristics align with previous multifractal analysis results,^9,10 which demonstrated that BT particle groups have a narrow range of influence between particles and are distributed as dispersed and independent particle groups, whereas SN particle groups have a strong influence between particles and tend to form networks.

The material texture obtained through these processes is thought to reflect the characteristics of the original particle groups and to have constructed a material texture with multiple functions. In general, if BT and SN powder are added to ethanol at the same time and stirred, in addition to the formation of BT particle groups and SN particle groups, it is expected that particle groups will form where BT and SN particles are mixed, SN particle groups wrapped in BT particles, and BT particle groups wrapped in SN particles. Therefore, the reason it was difficult to confirm the mixture of BT and SN particles in this case is likely due to the influence of the formation state of the PVDF/filler interface, which is related to the polarization properties of PVDF.

To evaluate the material properties of the obtained samples, the thermal conductivity and dielectric constant of each sample were measured. The thermal conductivity and dielectric constant were measured perpendicular to the surface of the molded sheet. The real part ε′ of the complex dielectric constants at 1 kHz for each sample and the thermal conductivity are illustrated in Fig. 2. For comparison, the measured results for PVDF and BT (10 vol. %)/PVDF in dielectric constant⁹ and the values for PVDF and SN (10 vol. %)/PVDF in thermal conductivity¹⁰ are also included in Fig. 2. The dielectric constant of the added filler is approximately 1000 for BT and 8–10 for SN, which is similar to that of PVDF. Additionally, the thermal conductivity of the fillers is approximately 6 W/mK for BT and about 70 W/mK for SN. Regardless of the addition process, the thermal conductivity and dielectric constant of the BT/SN/PVDF composites were equal to or greater than those of the fillers added individually. In our previous report,²⁵ we confirmed that the material texture of SN and BT contributes significantly to thermal conductivity and dielectric properties, respectively, and has a small influence on the other properties. This is thought to be due to the low mixture of BT and SN particles, as aforementioned, and is, therefore, reflected in the material properties of each particle group. BT particles exhibit high dielectric properties as the particle groups grow, while the formation of particle groups and the networking of SN particles leads to high thermal conductivity. The results of this study correspond to the SEM observations, suggesting that each particle group is distributed independently and exerts its respective function.

To characterize the obtained material texture, a multifractal analysis was performed on the ternary images of each sample, using the respective generalized dimension, D_q, and scaling exponent, τ_q, determined from the results in Fig. 1. The plots of D_q vs q and τ_q vs q are illustrated in Figs. 3 and 4, respectively, with measurement errors for D_q and τ_q being less than ±0.005. The fractal dimension of each sample was obtained by averaging multiple SEM images at various field sizes. For the BT and SN distributions in PVDF, the results for each sample are listed in Table I: D₋₁ (BT + SN) from the overall distribution of BT and SN particles, D₋₁ (BT) from the distribution of BT particles, and D₋₁ (SN) from the distribution of SN particles in the q = −1 (global region (g): q < 0). Similarly, D₁ (BT + SN) from the overall distribution of BT and SN particles, D₁ (BT) from the distribution of BT particles, and D₁ (SN) from the distribution of SN particles in q = 1 (local region (l): q > 0) are also listed. For comparison, the values of D₋₁ (BT) and D₋₁ (SN) obtained from the BT/PVDF and SN/PVDF systems in the global region (q < 0) and D₁ (BT) and D₁ (SN) obtained in the local region (q > 0) are also listed in Table I.

Both the global and local regions demonstrated results extremely similar to those obtained when the filler was added independently, suggesting that, despite the mixture of BT and SN particles, they exist independently as particle groups due to their interactions with each other. To analyze the interactions of particle groups, we calculated the MI I(BT;SN) using Eq. (11), with X as BT and Y as SN. We also calculated the conditional probabilities H(BT|SN) and H(SN|BT). The global region (q < 0) and local region (q > 0) are distinguished by subscripts. In correspondence with information theory, in the global region, D₋₁ (BT + SN), D₋₁ (BT), and D₋₁ (SN) can be interpreted as Hg(BT, SN), H_g(BT), and H_g(SN), and in the local region, D₁ (BT + SN), D₁ (BT), and D₁ (SN) as H_l (BT, SN), H_l (BT), and H_l (SN). These calculation results are summarized in Table II.

In general, in information theory, MI refers to the amount of influence two pieces of information have on each other. The larger the value, the greater the influence the two pieces of information exert on one another. In this study, the I(BT;SN), H(BT|SN) and H(SN|BT) values listed in Table II represent the entropy related to the particle arrangement. In other words, a large I(BT;SN) value indicates that the interaction between particles/particle groups is greater, and particle groups with strong correlations are highly dispersed. However, the conditional probabilities H(BT|SN) and H(SN|BT) are thought to relate to the arrangement of different particle/particle groups.

The first important observation is that I(BT;SN) is much larger than H(BT|SN) and H(SN|BT), meaning that most of the particle groups in BT and SN have strong correlations and interact with each other. Additionally, different particles are considered to have almost no effect on each other. The distributions of BT and SN in the BT/SN/PVDF sample closely resemble those of BT/PVDF and SN/PVDF in previous studies.^9,10 The results demonstrate that the two particle groups exist separately, each resulting from interactions within their respective groups.

Here, S_int denotes the entropy change related to the coalescence and division of the particle groups, and E_int denotes the interaction (network formation) energy between the aggregates. From this expression, it is clear that S_int and E_int were calculated separately for the contributions of BT and SN, BT only, and SN only and are summarized in Table IV for each sample. For comparison, in the BT/PVDF and SN/PVDF systems, S_int (BT) obtained from the distribution of BT particles, S_int (SN) obtained from the distribution of SN, E_int (BT) obtained from the distribution of BT particles, and E_int (SN) obtained from the SN distribution are shown. The increase in both S_int and E_int terms even in the BT and SN indistinguishable (BT + SN) component indicates that the particle groups interact with each other and form a network throughout the material texture, suggesting a self-organization process. From Tables III and IV, it can be observed that the interactions between the BT and SN particles and the particle groups in the BT/SN/PVDF sample are extremely close to those of BT/PVDF⁹ and SN/PVDF¹⁰ in previous studies, respectively, which indicates that the two particle groups behave differently as a result of the interactions between the particle groups, as listed in Tables I and II. In this case, I (BT;SN) is larger than H(BT|SN) and H(SN|BT), indicating that the effects of the interactions between the particle groups extend to the entire material texture. Therefore, although the amount of SN added with respect to the thermal conductivity did not reach a clear percolating concentration, the formation of a network of particle groups was thought to be a self-organization process rather than a self-assembly process.

Thus, it was demonstrated that by adding fillers with two different functions, both functions can be achieved without significantly impairing their individual properties. If the two fillers are mixed within the same particle population, they will not be able to form a network of identical particles, complicating the realization of both functions. Therefore, in this study, we aimed to synthesize samples using a process called layered mixing to preserve the integrity of the particle groups. However, in practice, the formation of particle groups was maintained even with simple kneading, clearly showing that both functions can be achieved in a single material system. Cross-sectional FE-SEM images of BT and SN aggregates in the BT/SN/PVDF composites fabricated using the SK process are shown in Fig. 7 . This figure shows a backscattered FE-SEM image of a typical material texture obtained as a result of interactions between particles and particle groups, in which BT and SN particle groups are in close proximity. A schematic diagram of this material texture formation is shown in Fig. 8. The PVDF matrix wraps around the filler particles, forming a filler/PVDF heterointerface. The particle groups grow through this heterointerface. In other words, the interfacial energy is thought to be the driving force behind the formation of the particle groups. As a result, the PVDF sandwiched between SN particles, which have strong particle cohesion and tend to form large particle groups, becomes thinner and has lower thermal resistance.¹⁰  On the other hand, the PVDF sandwiched between BT particles forms an electronic n/p/n junction at the interface, allowing charge to accumulate and contributing to an improvement in the dielectric constant.⁹  It is considered that the characteristics of this heterointerface are reflected in the material properties. Therefore, the fractal material texture shown in Figs. 7 and 8 would be constructed by the influence of the properties of these interfaces on the interaction (connection or repulsion) between particles. However, the details of the texture structure formation mechanism are not fully understood, and further investigation is required.

Conversely, in this system, as shown in Table II, I(BT;SN) is considerably larger than the conditional entropy H(BT/SN) and H(SN/BT) in both the global and local regions. In general, I is essentially entropy and reflects the strength of correlation. In other words, the strong correlation between different particles affects the configuration entropy of each coexisting particle group. It is thought that the same type of particle group grows together, while different types of particle groups repel each other, forming a complex network. Consequently, as illustrated in Fig. 2, it is believed that the growth of the BT particle group enhances the dielectric properties, while the network growth of the SN particle group contributes to high thermal conductivity properties.

To analyze the information thermodynamic characteristics of the material textures produced by the self-assembled/self-organized process, ternary BT/SN/PVDF composites exhibiting both good thermal conductivity and dielectric properties were fabricated using two processes. In the LK process, BT/PVDF and SN/PVDF sheets were laminated to create a laminate film. In the SK process, BT and SN particles were premixed and kneaded with PVDF. The results obtained for the four samples prepared by these processes are as follows:

1. The mutual information I(BT;SN) calculated from the information dimension D_±1 obtained from the particle distribution was larger than the conditional probabilities H(BT|SN) and H(SN|BT) in both the local (q > 0) and global (q < 0) regions, indicating a strong correlation between particles in this system.

2. The strong correlation between particles suggests that similar particles interact strongly to form aggregates. This strong particle correlation is thought to be responsible for the development of a material texture with both good dielectric properties and thermal conductivity.

3. The thermodynamic parameters (entropy S_int and internal energy changes E_int) associated with interactions, derived from the D_q–q and τ_q–q plots based on image analysis, showed that the BT/SN/PVDF system strongly reflected the characteristics of the BT/PVDF and SN/PVDF systems.

4. The obtained fractal material texture exhibited complex network properties, and the γ/α plot suggested that the network formation of each BT and SN particle group was distinct, even within the ternary BT/SN/PVDF system.

These results indicate that BT particles and SN particles form independent groups, which allows for the imparting of two distinct functions: high dielectric properties and high thermal conductivity. Furthermore, although the amount of SN added to the material did not reach a clear percolation concentration in terms of enhancing thermal conductivity, the formation of the particle network is believed to be a self-organization process rather than a self-assembly process. This observation suggests that it is possible to independently control different properties even in metal alloys containing second phase components or composites made from different ceramics, facilitating the development of multifunctional materials through the use of composite structures.

This work was partially supported by the TOKYO CITY UNIVERSITY Interdisciplinary Research Center for Nano Science and Technology for Instrumental Analysis.

The authors have no conflicts to disclose.

Fumio Munakata: Conceptualization (lead); Formal analysis (equal); Methodology (lead); Project administration (lead); Writing – original draft (lead). Taito Ogiya: Data curation (equal); Software (supporting); Validation (equal); Visualization (equal). Yoshihiro Sato: Data curation (equal); Methodology (equal); Software (lead); Writing – review & editing (equal). Suguru Kitani: Data curation (equal); Formal analysis (equal). Hitoshi Kawaji: Formal analysis (equal); Methodology (equal); Resources (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.