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Communication

High Dielectric Design of Polymer Composites by Using Artificial Neural Network

by
Sungyub Ji
1,2,
Dae-Yong Jeong
2,
Cheolhee Kim
1,3,* and
Sung Yi
1,*
1
Department of Mechanical and Materials Engineering, Portland State University, Portland, OR 97227, USA
2
Department of Materials Science and Engineering, Inha University, Incheon 22212, Republic of Korea
3
Adavanced Joining & Additive Manufacturing R&D Department, Korea Institute of Industrial Technology, Incheon 21999, Republic of Korea
*
Authors to whom correspondence should be addressed.
Appl. Sci. 2022, 12(24), 12592; https://doi.org/10.3390/app122412592
Submission received: 23 November 2022 / Revised: 5 December 2022 / Accepted: 6 December 2022 / Published: 8 December 2022
(This article belongs to the Section Mechanical Engineering)
Browse Figures
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Abstract

:
Polymer-based composites with a high dielectric property have shown great potential in electrical energy storage applications. It is important to predict the dielectric constant in designing polymer composites, but it is costly and time consuming. In this study, dielectric properties of various polymer composites have been predicted by using an artificial neural network (ANN) model trained with hundreds of experimentally measured data. Eight variables such as the dielectric constant of matrix, filler, and shell, the diameter of filler, the volume fraction of filler, the dimension of filler, the thickness of shell, and the frequency were considered. To improve the prediction accuracy, hyper parameters of the ANN model were optimized through the hyperband method. Using the ANN model, we demonstrated the correlation between the dielectric constant of polymer composites and the variables. The ANN model predicted the dielectric constant with a coefficient of determination (R2) of 0.97. Furthermore, the ANN model shows good performance to predict dielectric constant at various frequencies (spanning from 100 Hz to 100 kHz). Hence, we present that the AI-based prediction model using ANN method can be helpful in designing the polymer composites with desired properties.

1. Introduction

Dielectric capacitors have been used in pulsed power weapons, electrical power systems and hybrid electric vehicles (HEVs) due to their capability of ultrafast charging-discharging rate and ultrahigh power density [1,2]. In particular, dielectric polymers such as polyvinylidene fluoride (PVDF) are one of the most promising material candidates for high-density energy storage applications because of their high breakdown strength, low dielectric loss, facile fabrication, low cost, and flexibility [3,4]. Dielectric materials store electrostatic energy through reversible orientation polarization under applied electric field. High discharged energy density can be attained by high dielectric property. However, most dielectric polymers have a low dielectric constant, which limits their application to energy storage devices. In order to improve their dielectric properties, many studies have been conducted by introducing high dielectric constant ceramic fillers such as BaTiO3 [5], BaSrTiO3 [6], and Pb(Zr,Ti)O3 [7] as fillers into polymer matrices to improve their dielectric properties.
It is important to predict the dielectric constant of the polymer composites to achieve a desirable energy density. However, it is difficult to predict the dielectric constant of the polymer composites because of nonlinearity in the increase of a dielectric constant and diverse factors such as filler’s morphology, post processing of filler, and frequency [8,9,10,11,12]. In order to predict the dielectric constant of a polymer composite, the methods of using theoretical prediction models or directly measuring through experiments has been used. Luo et al. [13] introduced modified Rother–Lichtenecker, Maxwell–Wagner, and Jayasundere–Smith models to compare experimental data and prediction models. Zak et al. [14] used Maxwell, Furukawa, and Rayleigh prediction models to compare which models showed high accuracy with experimental data. However, these theoretical prediction models have limitations in predicting the dielectric constant of composites. It is difficult to consider all factors, so the theoretical models approximate many factors except for main variables. These models use only a dielectric constant of a matrix and filler, and filler’s volume fraction excluding other conditions such as filler’s morphology and size, frequency, and filler’s post processing. Therefore, the theoretical models can predict the dielectric properties in only simple composite models.
With the advent of machine learning (ML) technology based on experimental and computational data, the machine learning method has been widely applied as a popular and powerful alternative for material and structural design or for predicting electrical properties. Shen et al. [15] introduced a ML model to study the effect of nanoparticles physical properties on the breakdown strength of poly(vinylidene fluoride)-co-hexafluoropropylene [P(VDF-HFP)]-based nanocomposites, which was performed on a dataset from the high-throughput phase-filed simulations. Yi et al. [16] applied a ML model to investigate the influence of the polymer’s key molecular descriptors on the dielectric constant. Although these studies opened up chances for ML and data-driven methods in dielectric polymer-based composite, comprehensive research on a more practical predictive model considering various important factors to predict the dielectric constant has not yet been considered.
In this study, an ML model based on an artificial neural network is developed to predict the dielectric property of polymer composites using hundreds of experimentally measured data from existing references in consideration of various polymer matrices and ceramic nanofillers. An artificial neural network (ANN) network is trained with collected data considering the dielectric constant of matrix, filler, and shell, volume fraction, diameter and morphology of filler, operating frequency, and shell thickness in core–shell structures. In order to improve the model prediction performance, the model structure is optimized by a hyper parameter method [17]. In addition, the main factors dominating the dielectric constant of the polymer composite is studied by using the ANN model and a desirable polymer composite model with a high dielectric constant is presented.

2. Materials and Methods

2.1. Data Collection and Model Variables Description

The experimental data for various polymer-based composites were collected to train the ANN model for predicting dielectric constants from the references [18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49]. Table 1 shows the kinds and dielectric constants of all polymer matrices and ceramic nanofillers used in this work [50,51].
From the data points, the four groups of input parameters were considered: (i) the properties of matrix and filler (dielectric constant, diameter of filler, and volume fraction), (ii) the morphology of filler (particle [0D], nanowire [1D], and nanosheet [2D]), (iii) the properties of shell (dielectric constant and thickness), and (iv) operating parameter (frequency), as shown in Figure 1.
The total experimental datasets available for modeling was 744. The data used in this study show the distribution of a normalized ε r ε c o m p o s i t e / ε m a t r i x ) with a range from 0.25–5.75 in Figure 2. The normalized ε r of most composites exceeds 1 with an average value of 1.27, indicating that the dielectric constant is enhanced by adding ceramic nanofillers due to their higher dielectric constant. In other hands, some data show values below 1, which is the case with nanofillers with a lower dielectric constant than polymers such as boron nitride (BN). Model training mainly concentrated on minimizing a mean squared error (MSE) to estimate the accuracy of the prediction model given as
MSE = 1 N ∑ i = 1 N y i − y i ^ 2
where y i is measured output data from the result in previous research, y i ^ is a calculated output data from the prediction model, and N is a total number of datapoints.

2.2. Development and Optimization the ANN Model

ANN consists of interrelated adaptive network components inspired by the regulation of connected neurons in the human brain and can perform large-scale parallel computations for data processing [52]. Typically, the network components of ANN are composed of layer, node, and connection. Figure 3 shows the structural design of the ANN model with input, hidden, and output layers. The MATLAB version R2021a was used to build the ANN model structure. To perform model training, the 774 experimental databases were divided into training, validation, and testing datasets with 70%, 15%, and 15%, respectively.
Before model training, it is important to optimize the number of hidden layers and nodes to improve a prediction accuracy. The hyper parameter optimization was conducted using Python 3 through a hyperband method [17]. The model was evaluated by increasing the number of layers from 2 to 5 and the number of nodes from 16 to 64 by 16 steps. The rectified linear unit (ReLU) was used to optimize the model as an activation function. The Adam was selected for optimizer. The results of optimization are shown in Table 2, and the best performance model had 4 hidden layers, 48, 32, 32, and 16 nodes at the hidden layers, and a learning rate of 0.001.

3. Results & Discussions

3.1. Training of the ANN Model

Figure 4 shows the MSE drops of the proposed ANN model until 200 epochs when the network learns as expected for a well-trained ANN. This is a good indication of the network’s learning process. The blue line and red line represent the decreasing error of the training and validation data, respectively. It measures the network generalization ability and stops training as soon as the generalization does not improve. As shown in Figure 4, the training error was smaller than the validation error as expected, and the error lines were continuously decreased without overfitting.
The results of predicted ε r and experimental ε r of composites are shown in Figure 5. The ANN model had high accuracy in the prediction of composite’s dielectric constant. Figure 5a presents the prediction results were trained by all 744 databases with a high coefficient of determination (R2) of 0.95. The three red circles indicate relatively larger errors between the predicted ε r and experimental ε r than the other predicted values. Although the prediction error of some data points are relatively large, the ANN model shows a pretty high prediction accuracy. Data points in red circles were collected from the reference [18] using P(VDF-TrFE-CFE) as the matrix and a relatively high frequency of 106 Hz. Ferroelectric matrices with high dielectric constant can abruptly decrease dielectric constant at high frequencies, and under these conditions, there were only a limited number of data points in this study. Figure 5b shows the accuracy of an ANN model trained without the data in red circles. As expected, the prediction accuracy is higher than Figure 5a with the R2 of 0.97.

3.2. Correlation Analysis of Variables

The Pearson coefficients between various input parameters and composite’s ε r were calculated to discover the key factors, as shown in Figure 6. The Pearson coefficient ranges from −1 to 1 with −1 representing the strongest negative correlation and 1 representing the strongest positive correlation [53]. The largest positive correlation to ε r is dominated by ε r m and filler’s volume fraction, while the most negative correlation comes from shell’s thickness. The ε r f has also a positive correlation to ε r , but considering that the correlation coefficient of ε r m is higher than that of ε r f , it can be seen that the dielectric constant of the polymer matrix is more dominant to increase the dielectric constant of the composite. On the other hand, the operating frequency and thickness of the shell are negatively correlated with ε r . The shell plays a role to compensate a dielectric mismatch between polymer matrices and ceramic fillers in core–shell structure. It shows that the shell is important for the purpose of increasing breakdown strength, but it is consistent with the results of previous studies that it plays a negative role in terms of increasing dielectric constant [54].
In polymer-based composites, the dielectric constant of the composites primarily depends on the dielectric constant of the matrix and filler, and filler’s volume fraction as shown in Figure 6. The combined effect of the dielectric constant of matrix and filler, and the volume fraction of filler on composite’s dielectric constant was studied using the ANN as shown in Figure 7a–c. When we set the x-axis and y-axis with the main variables, the other variables are fixed based on the most overlapping data, with a filler diameter of 200 nm, a filler dimension of 1D nanowire, a frequency of 1 kHz, and without a core–shell structure. Figure 7a,b show the combined effect of the dielectric constant of matrix and filler, and the volume fraction of filler, respectively. As we can estimate, the dielectric constant of the composite shows a typically proportional tendency to increase as the dielectric constant of the matrix and the volume fraction of the filler increase. In Figure 7b, there are some distortion contour lines. It is derived from insufficient data with the dielectric constants of filler between about 500 and 800. Despite the lack of data, it shows the tendency of the composite’s dielectric constant to generally increase as the dielectric constant and the volume fraction of the filler increase. When matrix and filler’s dielectric constant are considered together in Figure 7c, it shows the same tendency as in Figure 7a,b that composite’s dielectric constant increases as matrix and filler’s dielectric constant increase. Furthermore, it is presented to discuss the effect of minor variables such as filler’s diameter and shell’s thickness in Figure 7d,e. When it comes to filler’s diameter, the dielectric constant of the composite represents a general tendency to decrease as the filler’s diameter decreases. This is mainly attributed to the lowered dielectric constant of filler caused by decreasing particle size. The dielectric constant of nanofillers decreases with decreasing particle size due to depolarization fields, surface properties, electrical boundary conditions, and defects [55]. As for the effect of the shell’s thickness, it shows the tendency of the composite’s dielectric constant to decrease as the shell’s thickness increases. This is because in the core–shell structure, the shell reduces the high electrical mismatch between the filler and the matrix. Therefore, as the shell’s thickness increases, the electrical mismatch and the effect of filler introduction to increase the composite’s dielectric constant are reduced at the same time [56]. It can be seen in Figure 7a–e that the dielectric constant of the polymer composites through the ANN model can be predicted without a theoretical model or empirical measuring method regarding the major and minor parameters.

3.3. Influence of Variables on Dielectric Constant

3.3.1. Effect of Filler Dimension

Figure 8 presents the predicted dielectric constant depends on three types of different filler’s dimensions (0D, 1D, and 2D) as a function of filler volume fraction. The predicted ε r of 0D nanoparticle and 1D nanowire increases with the volume fraction, while 2D nanosheet decreases with the volume fraction. It is derived from the different ceramic filler’s shape (0D, 1D, and 2D). Compared to spherical nanoparticles (0D), nanofibers or nanowires (1D) can induce higher dielectric constants at much lower concentrations due to their large dipole moments as a result of their high aspect ratio [54]. On the other hand, BN( ε r ~ 4) used in this study as the 2D nanosheet filler, has a lower dielectric constant than P(VDF-HFP) ( ε r ~ 8.4), used as a matrix, showing a tendency to decrease as the volume fraction increases.

3.3.2. Effect of Operating Frequency

In general, the dielectric constant of a dielectrics comes from the result of complex interactions of a total of four polarizations: (i) the electronic polarization between the nucleus and electrons of an atom, (ii) the ionic polarization between a cation and an anion, (iii) the directional polarization according to the orientation of the permanent dipoles, and (iv) the space charge polarization caused by the movement of charge carriers (electron, hole, ion, and vacancy) to form a space charge [54]. For these polarizations to appear, a displacement of the charged particle must occur. When the applied electric field is alternating current, polarization can occur only when the displacement of the particle follows the change in the direction of the electric field. Therefore, as the charged particles are heavier, they cannot respond to high-frequency electric fields, and the dielectric constant decreases gradually. Although it is important to predict the dielectric constant with increasing frequency because of these characteristics, it has not been studied much [55,56].
To further confirm the accuracy of the ANN model with increasing frequency, three polymer composites were considered: P(VDF-TrFE-CFE)/BaTiO3 nanoparticle, P(VDF-HFP)/TiO2 nanowire, and P(VDF-HFP)/BN nanosheet. The normalized ε r as a function of volume fraction with the increasing frequency range from 100–100 kHz is shown in Figure 9a–d. It can be seen that the trends of the volume fraction and frequency-dependent ε r for the three composites are predicted fairly well. The normalized ε r of P(VDF-TrFE-CFE)/BaTiO3 nanoparticle and P(VDF-HFP)/TiO2 nanowire increases with the volume fraction, while P(VDF-HFP)/BN nanosheet decrease with the volume fraction, which agree with Figure 8. When we compared the experimental and predicted values, there is no big difference with increasing frequency until 100 kHz. From the above, the ANN models can well show high accuracy between predicted ε r and experimental ε r at various frequency ranges.

4. Conclusions

In this work, a machine learning driven an ANN model was developed to rationally design the desired dielectric constant of polymer composites, using hundreds of experimentally measured data. The hyper parameters were optimized by a hyperband method to improve the prediction performance. The designed ANN model can represent the correlations between 8 input variables and a dielectric constant of polymer composite with an accuracy of 97%. The main conclusions obtained from this ANN model are listed as the follows. Firstly, when considering the polymer matrix and the ceramic filler, it was found that the dielectric constant of the matrix was more dominant in increasing the dielectric constant of the composite than that of the filler. Secondly, regarding the dimension of the filler, it was shown to increase the dielectric constant of the composite in the order of 1D and 0D. As for 2D, additional studies are required because other 2D fillers except BN were not considered in this study. Lastly, the ANN model shows a good prediction performance for a dielectric constant at various frequencies (spanning from 100 Hz to 100 kHz). Therefore, we demonstrated the AI-based ANN model can design the dielectric constant of polymer composites for a high energy density in electrical energy storage applications.

Author Contributions

Conceptualization, S.J. and S.Y.; methodology, S.J., D.-Y.J. and C.K.; investigation, S.J.; resources, D.-Y.J. and S.Y.; data curation, S.J.; writing—original draft preparation, S.J.; writing—review and editing, S.Y.; supervision, C.K. and S.Y.; funding acquisition, D.-Y.J. and S.Y. All authors have read and agreed to the published version of the manuscript.

Funding

This research was supported in part by the MOTIE (Ministry of Trade, Industry, and Energy) in Korea, under the Fostering Global Talents for Innovative Growth Program (P0017303) supervised by the Korea Institute for Advancement of Technology (KIAT); and in part by the Korea Institute of Industrial Technology, under the project “Development of root technology of multi-product flexible production (KITECH-EO-22-0006)”.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

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Applsci 12 12592 g001 550
Figure 1. Schematic of the input parameters in the machine learning.
Figure 1. Schematic of the input parameters in the machine learning.
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Figure 2. Distribution of normalized εr for all samples in the dataset.
Figure 2. Distribution of normalized εr for all samples in the dataset.
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Figure 3. Schematic of the ANN model structure with input, hidden, and output layers.
Figure 3. Schematic of the ANN model structure with input, hidden, and output layers.
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Figure 4. Training and validation loss curves of the ANN over epochs.
Figure 4. Training and validation loss curves of the ANN over epochs.
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Figure 5. Dielectric constant prediction accuracy between ML prediction vs. experimental values from (a) original dataset and (b) modified dataset.
Figure 5. Dielectric constant prediction accuracy between ML prediction vs. experimental values from (a) original dataset and (b) modified dataset.
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Figure 6. Pearson coefficients between the eight input variables and composite’s dielectric constant.
Figure 6. Pearson coefficients between the eight input variables and composite’s dielectric constant.
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Applsci 12 12592 g007a 550Applsci 12 12592 g007b 550
Figure 7. 2D contour plots showing combined effect of (a) matrix’s dielectric constant, (b) filler’s dielectric constant and filler’s volume fraction, (c) matrix and filler’s dielectric constant, (d) filler’s diameter, (e) shell’s thickness and filler’s dielectric constant.
Figure 7. 2D contour plots showing combined effect of (a) matrix’s dielectric constant, (b) filler’s dielectric constant and filler’s volume fraction, (c) matrix and filler’s dielectric constant, (d) filler’s diameter, (e) shell’s thickness and filler’s dielectric constant.
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Figure 8. Predicted dielectric constant depends on different filler’s dimensions as a function of filler volume fraction.
Figure 8. Predicted dielectric constant depends on different filler’s dimensions as a function of filler volume fraction.
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Figure 9. Comparison of experimental and predicted dielectric constant as a function of filler volume fraction depends on increasing frequency: (a) 100 Hz, (b) 1 kHz, (c) 10 kHz, and (d) 100 kHz.
Figure 9. Comparison of experimental and predicted dielectric constant as a function of filler volume fraction depends on increasing frequency: (a) 100 Hz, (b) 1 kHz, (c) 10 kHz, and (d) 100 kHz.
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Table
Table 1. The dielectric constant values for different polymer matrixes and nano ceramic fillers.
Table 1. The dielectric constant values for different polymer matrixes and nano ceramic fillers.
MatrixDielectric ConstantFillerDielectric Constant
Polypropylene, (PP)2.1BNNS4
Polyimide, (PI)3.2Al2O39.4
Poly(vinylidene fluoride-co-hexafluoropropylene), P(VDF-HFP)8.4TiO2110
Polyvinylidene fluoride,
(PVDF)		8.6NaNbO3200
Poly(vinylidene fluoride-trifluoroethylene),
P(VDF-TrFE)		12SrTiO3200
Poly(vinylidene fluoride-co -chlorotrifluoroethylene,
P(VDF-CTFE)		12.3BaSrTiO3300
Poly(vinylidene fluoride-trifluoroethylene-chlorofluoroethylene),
P(VDF-TrFE-CFE)		41BaTiO31000
Poly(vinylidene fluoride-trifluoroethylene-chlorotrifluoroethylene),
P(VDF-TrFE-CTFE)		42
Table
Table 2. Top 5 model structures in the hyperband optimization.
Table 2. Top 5 model structures in the hyperband optimization.
Ranking12345
No. of hidden layers43243
No. of nodesLayer 14864324848
Layer 23264324832
Layer 33248 3232
Layer 416 16
Learning rate0.0010.0010.0010.010.001
Mean squared error0.13720.13840.14160.14290.1431
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Ji, S.; Jeong, D.-Y.; Kim, C.; Yi, S. High Dielectric Design of Polymer Composites by Using Artificial Neural Network. Appl. Sci. 2022, 12, 12592. https://doi.org/10.3390/app122412592

AMA Style

Ji S, Jeong D-Y, Kim C, Yi S. High Dielectric Design of Polymer Composites by Using Artificial Neural Network. Applied Sciences. 2022; 12(24):12592. https://doi.org/10.3390/app122412592

Chicago/Turabian Style

Ji, Sungyub, Dae-Yong Jeong, Cheolhee Kim, and Sung Yi. 2022. "High Dielectric Design of Polymer Composites by Using Artificial Neural Network" Applied Sciences 12, no. 24: 12592. https://doi.org/10.3390/app122412592

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