# Analysis on Microstructure–Property Linkages of Filled Rubber Using Machine Learning and Molecular Dynamics Simulations

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## Abstract

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## 1. Introduction

## 2. Problem Setting

## 3. Previous Work

#### 3.1. CNN-Based CGMD Surrogate Model

#### 3.2. Filler Morphology Search Method

## 4. Method

#### 4.1. LR and PH Analyses

#### 4.2. CNN-Based Analysis

## 5. Results and Discussion

#### 5.1. Filler Aggregates Extracted by the PH and CNN Methods

- The dense distribution of filler aggregates reflected the short distances between their surfaces. In addition, the sizes of agglomerates (quadratic aggregates) are small as shown in Figure 19.

#### 5.2. Comparison between the Extracted and Non-Extracted Filler Particles

#### 5.3. Validation Using CGMD Simulations

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Stress distributions obtained by the random sampling and the developed method in our preliminary study [30]. (

**b**)Stress distribution of the morphologies obtained by either method with a filler volume fraction of 20 vol.% and was used as the training data in the present study.

**Figure 2.**RDFs of the filler particles in the positive and negative examples. The blue lines denote the RDFs of the morphologies obtained by random sampling, and the orange lines represent the RDFs of the morphologies obtained in the previous study. The solid and dotted lines show the RDFs of the negative and positive examples, respectively.

**Figure 3.**Flow chart of the developed morphology search method. The circles in Step 1 represent the filler candidate positions. The open circles and filled circles are the polymer and filler particles, respectively. The left figure in Step 2 is the initial state of this step. The right figure shows an updated filler configuration. CGMD simulation was carried out to measure the property in Step 3, because the properties evaluated in both Step 1 and Step 2 were merely predicted values by the CNN-based surrogate model for the efficient search. Thus, Step 3 was required to confirm if the morphologies obtained by the first 2 steps actually show the desired property.

**Figure 4.**Target distributions of each temperature. The blue line, $t={t}_{1}$, is the probability distribution at the lowest temperature. The red line, $t={t}_{4}$, is the probability distribution at the highest temperature.

**Figure 5.**Schematic diagram illustrating the PH. (

**a**) The balls are disconnected when the ball radius $r$ is smaller than ${r}_{1}$. (

**b**) A loop emerges at $r={r}_{1}=\sqrt{b}$. (

**c**) The loop disappears at $r={r}_{2}=\sqrt{d}$. (

**d**) PD representing the results of PH analysis.

**Figure 6.**Schematic illustration of the vectorization procedure. The left plots include the PDs; the center figures are the persistence surfaces; and the right histograms contain the PIs. Two morphologies are vectorized in the upper and lower figure panels. As shown in the upper center figure, the persistence surface is divided by a grid, and representative values of all grid cells are arranged along the green arrows.

**Figure 7.**Prediction accuracies of the validation data with different combinations of parameters. The accuracy was defined as the ratio of the number of morphologies that predicted the correct class to the number of validation data instances. The top row in the horizontal axis is the weight on the persistence surface $\beta $; the middle row is the standard deviation of the Gaussian distribution $h$; and the bottom row is the number of subdomains $g$. The orange bar represents the combination of parameters that produces the highest accuracy.

**Figure 8.**Distribution of the L1 regularization LR coefficient, which predicts a class from the filler morphology. The coefficient is visualized in the PD space. The red region corresponds to the positive coefficients and contributes to the positive example. The blue region corresponds to the negative coefficients and contributes to the negative example.

**Figure 9.**Examples of the loops composed of filler particles, which were extracted by the PH. The green box denotes the modeled space, and the red dots represent the filler particles.

**Figure 10.**CNN architecture utilized in this work. The numbers in parentheses represent the dataset sizes. The input image of $32\times 32\times 32$ pixels is convoluted into the image of $32\times 32\times 32$ pixels under the periodic boundary conditions with a single kernel. The leaky ReLU function and global average pooling were applied to the convoluted image to reduce the dimension. Finally, the probability of the positive example is output using the sigmoid function.

**Figure 11.**Relationships between the epoch number and prediction accuracy determined by applying the validation data. The prediction accuracy was equal to the ratio of the number of morphologies that predicted the correct class to the number of validation data instances. The blue line, orange line, and green line correspond to the prediction accuracies of the kernels with sizes of $5\times 5\times 5$, $9\times 9\times 9$, and $13\times 13\times 13$, respectively.

**Figure 12.**Input and convoluted images obtained using the kernels with sizes of $5\times 5\times 5$, $9\times 9\times 9$, and $13\times 13\times 13$. The upper row represents the superimposed images of input and convoluted images. The white circles and black areas denote the filler particles and rubber regions in the input images, respectively. The red and blue areas are visualized in the convoluted images depicted in the lower row and represent the pixels with positive and negative values, respectively. The red areas contribute to the positive example, and the blue areas contribute to the negative example. The pixels with values ranging from −0.3 to 0.3 are visualized in black to emphasize the areas with large contributions.

**Figure 13.**Ratio of the positive pixels in the convoluted image to the filler particles in the input image, which was defined as an A/B fraction. Here, A is the number of pixels with positive values in the convoluted image that overlap with the filler particles in the input image. B is the total number of pixels with positive values in the convoluted image. The blue, orange, and green colors correspond to the kernel sizes $5\times 5\times 5$, $9\times 9\times 9$, and $13\times 13\times 13$, respectively.

**Figure 14.**Histograms of the radius of gyration, ${R}_{g}$, and the number of filler particles in the loop, $F$, of the filler aggregate extracted by the PH. The blue bars and orange bars denote the aggregates extracted from the positive and negative examples, respectively.

**Figure 15.**(

**a**) Definition of the aggregate size. The red circles and black points represent the filler particles and their centers, respectively. The size in each direction is the difference between the largest and smallest coordinates along this direction. (

**b**) Aggregate size distribution in each direction. The blue and orange bars denote the results obtained for the positive and the negative examples, respectively. Z-direction is the elongation direction of the system, while X- and Y-directions are compressed during deformation.

**Figure 16.**Schematic illustration of the filler particles belonging to multiple loops. The blue and red circles denote the filler particles constituting different loops. The particles surrounded by the orange circle belong to both the i-th and j-th loops connected physically. However, these two loops are considered different aggregates.

**Figure 17.**Example of the aggregate extracted by the CNN and size distributions of filler aggregates. (

**a**) Cross-section of the convoluted image. The aggregate size is defined as the difference between the largest and smallest pixel coordinates. The red region is the set of pixels with positive values that contributes to the positive example. The blue region is the set of pixels with negative values that contributes to the negative example. The black region includes pixels with small values ranging from −0.3 to 0.3 and does not affect the output. The adjacent red pixels are extracted as filler aggregates under the periodic boundary conditions. (

**b**) Histograms of the size of the set of pixels with positive values, namely aggregate sizes, obtained from all morphologies in the positive (blue bars) and negative (orange bars) examples. Z is the elongation direction of the system.

**Figure 18.**Fractions of the filler particles extracted by the PH and CNN that contributed to the positive example. Their values were computed as the ratios of the numbers of pixels of the extracted particles to the number of pixels for all filler particles. The blue bars denote the ratios of the filler particles extracted by CNN; the orange bars represent the PH results; and the green bars denote ratios of the filler particles common to the fillers extracted by CNN and the fillers extracted by PH.

**Figure 19.**Distribution of the LR coefficient of the aggregate extracted by the CNN on the PD, and schematic illustration of the filler aggregates extracted by the PH method. (

**a**) Distribution of the L2 regularized LR coefficient, which reflects the distribution of aggregates. The regression coefficient is positive in the red region and negative in the blue region, which contribute to the positive and negative examples, respectively. (

**b**) Aggregates extracted by the CNN. Each rectangle represents a pixel, and each color denotes a filler aggregate. (

**c**) Centers of the pixels that constitute filler aggregates. (

**d**) ${r}_{1}$ is a representative value of the distance between the filler surfaces because a loop emerges at $r={r}_{1}=\sqrt{Birth}$. (

**e**) ${r}_{2}$ is a representative size of the filler agglomerate (quadratic aggregate) because the loop disappears at $r={r}_{2}=\sqrt{Death}$.

**Figure 20.**Spatial correlations in the YZ plane of $r$-space including $r=\left(0,0,0\right)$ of the CNN-extracted fillers from all morphology examples that contribute to the high stress and that of the non-extracted fillers (upper row figures). The isotropic spatial correlation of the fillers not extracted by the CNN suggests the isotropically shaped filler aggregates. Meanwhile, the anisotropic spatial correlation of the filler particles extracted by the CNN indicates that the probability of filler orientation along the elongation direction is high. Furthermore, bottom two figures show the spatial correlations of the extracted fillers from all positive examples and all negative examples, respectively. The higher orientation probability of the filler particles extracted from the positive example along the elongation direction suggests that the aggregate size measured along the elongation direction in the positive example is larger than the size of the negative example.

**Figure 21.**Distributions of the CGMD-simulated stress at a strain of 0.3. The blue bars denote the results obtained for the CNN-extracted aggregates from the positive example. The Z-direction was the elongation direction. The orange bars represent the elongation of the CNN-extracted filler aggregates along the X- and Y-directions. The elongation direction was perpendicular to the aggregate principal direction (Z) in these cases. The green bars denote the results obtained for the filler configurations, in which the particle positions were randomly determined with the filler volume equal to that for the blue bars. Schematic diagrams of the filler aggregates and elongation direction are shown in the rectangles with the frame colors identical to those of the corresponding bars. The black circles and arrows designate the filler particles and the elongation direction, respectively.

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**MDPI and ACS Style**

Kojima, T.; Washio, T.; Hara, S.; Koishi, M.; Amino, N.
Analysis on Microstructure–Property Linkages of Filled Rubber Using Machine Learning and Molecular Dynamics Simulations. *Polymers* **2021**, *13*, 2683.
https://doi.org/10.3390/polym13162683

**AMA Style**

Kojima T, Washio T, Hara S, Koishi M, Amino N.
Analysis on Microstructure–Property Linkages of Filled Rubber Using Machine Learning and Molecular Dynamics Simulations. *Polymers*. 2021; 13(16):2683.
https://doi.org/10.3390/polym13162683

**Chicago/Turabian Style**

Kojima, Takashi, Takashi Washio, Satoshi Hara, Masataka Koishi, and Naoya Amino.
2021. "Analysis on Microstructure–Property Linkages of Filled Rubber Using Machine Learning and Molecular Dynamics Simulations" *Polymers* 13, no. 16: 2683.
https://doi.org/10.3390/polym13162683