# Design of Polymer Nanodielectrics for Capacitive Energy Storage

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. The Mixed-Variable Nanodielectrics Design Space

#### 2.2. The Design Framework

#### 2.2.1. Design of Experiments

#### 2.2.2. Material Generation

#### Microstructure Characterization and Reconstruction

#### Interfacial Layers

_{2}particles in a Cross-linked polyethylene (XLPE) polymer matrix. We simulate a multiphase system that considers, in addition to particle and matrix, extrinsic, and intrinsic interfacial regions (Figure 2c,d).

#### Extrinsic Interface

#### Intrinsic Interface

#### 2.2.3. Property Evaluation: Physics-Based Simulation Methods

#### Breakdown Strength Calculations

#### First-Principles Predictions of Trap States

#### Permittivity and Loss Calculations

#### 2.2.4. Metamodeling and Multi-Objective Optimization

#### Latent Variable Gaussian Process (LVGP) for Metamodeling

#### Bayesian Optimization

#### 2.2.5. Design Analysis

#### 2.3. Global Sensitivity Analysis for the Mixed-Variable Design Space

## 3. Results

#### 3.1. Initial Design of Experiments (DOE)

#### Global Sensitivity Analysis

#### 3.2. Nanodielectrics Design Optimization

## 4. Discussion

## Supplementary Materials

^{−2}Hz and 1 × 10

^{7}Hz with respect to the mixed-variable design space.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Latent Variable Gaussian Process (LVGP) Modeling

## References

- Zhang, G.; Li, Q.; Allahyarov, E.; Li, Y.; Zhu, L. Challenges and Opportunities of Polymer Nanodielectrics for Capacitive Energy Storage. ACS Appl. Mater. Interfaces
**2021**, 13, 37939–37960. [Google Scholar] [CrossRef] [PubMed] - Wang, S.; Huang, X.; Wang, G.; Wang, Y.; He, J.; Jiang, P. Increasing the Energy Efficiency and Breakdown Strength of High-Energy-Density Polymer Nanocomposites by Engineering the Ba0.7Sr0.3TiO
_{3}Nanowire Surface via Reversible Addition–Fragmentation Chain Transfer Polymerization. J. Phys. Chem. C**2015**, 119, 25307–25318. [Google Scholar] [CrossRef] - Li, Q.; Xue, Q.; Hao, L.; Gao, X.; Zheng, Q. Large dielectric constant of the chemically functionalized carbon nanotube/polymer composites. Compos. Sci. Technol.
**2008**, 68, 2290–2296. [Google Scholar] [CrossRef] - Zhang, X.; Liang, G.; Chang, J.; Gu, A.; Yuan, L.; Zhang, W. The origin of the electric and dielectric behavior of expanded graphite–carbon nanotube/cyanate ester composites with very high dielectric constant and low dielectric loss. Carbon
**2012**, 50, 4995–5007. [Google Scholar] [CrossRef] - Ning, N.; Bai, X.; Yang, D.; Zhang, L.; Lu, Y.; Nishi, T.; Tian, M. Dramatically improved dielectric properties of polymer composites by controlling the alignment of carbon nanotubes in matrix. RSC Adv.
**2014**, 4, 4543–4551. [Google Scholar] [CrossRef] - Grabowski, C.A.; Fillery, S.P.; Koerner, H.; Tchoul, M.; Drummy, L.; Beier, C.W.; Brutchey, R.L.; Durstock, M.F.; Vaia, R.A. Dielectric performance of high permitivity nanocomposites: Impact of polystyrene grafting on BaTiO
_{3}and TiO_{2}. Nanocomposites**2016**, 2, 117–124. [Google Scholar] [CrossRef] - Kim, P.; Jones, S.C.; Hotchkiss, P.J.; Haddock, J.N.; Kippelen, B.; Marder, S.R.; Perry, J.W. Phosphonic Acid-Modified Barium Titanate Polymer Nanocomposites with High Permittivity and Dielectric Strength. Adv. Mater.
**2007**, 19, 1001–1005. [Google Scholar] [CrossRef] - Siddabattuni, S.; Schuman, T.P.; Dogan, F. Dielectric Properties of Polymer–Particle Nanocomposites Influenced by Electronic Nature of Filler Surfaces. ACS Appl. Mater. Interfaces
**2013**, 5, 1917–1927. [Google Scholar] [CrossRef] - Zhang, G.; Brannum, D.; Dong, D.; Tang, L.; Allahyarov, E.; Tang, S.; Kodweis, K.; Lee, J.-K.; Zhu, L. Interfacial Polarization-Induced Loss Mechanisms in Polypropylene/BaTiO
_{3}Nanocomposite Dielectrics. Chem. Mater.**2016**, 28, 4646–4660. [Google Scholar] [CrossRef] - Zdorovets, M.V.; Kozlovskiy, A.L.; Shlimas, D.I.; Borgekov, D.B. Phase transformations in FeCo—Fe
_{2}CoO_{4}/Co_{3}O_{4}-spinel nanostructures as a result of thermal annealing and their practical application. J. Mater. Sci. Mater. Electron.**2021**, 32, 16694–16705. [Google Scholar] [CrossRef] - Trukhanov, A.V.; Trukhanov, S.V.; Panina, L.V.; Kostishyn, V.G.; Chitanov, D.N.; Kazakevich, I.y.S.; Trukhanov, A.V.; Turchenko, V.A.; Salem, M.M. Strong corelation between magnetic and electrical subsystems in diamagnetically substituted hexaferrites ceramics. Ceram. Int.
**2017**, 43, 5635–5641. [Google Scholar] [CrossRef] - Sanida, A.; Stavropoulos, S.G.; Speliotis, T.; Psarras, G.C. Evaluating the multifunctional performance of polymer matrix nanodielectrics incorporating magnetic nanoparticles: A comparative study. Polymer
**2021**, 236, 124311. [Google Scholar] [CrossRef] - Zhou, W.; Cao, G.; Yuan, M.; Zhong, S.; Wang, Y.; Liu, X.; Cao, D.; Peng, W.; Liu, J.; Wang, G.; et al. Core-Shell Engineering of Conductive Fillers toward Enhanced Dielectric Properties: A Universal Polarization Mechanism in Polymer Conductor Composites. Adv. Mater.
**2023**, 35, e2207829. [Google Scholar] [CrossRef] [PubMed] - Zhou, W.; Li, T.; Yuan, M.; Li, B.; Zhong, S.; Li, Z.; Liu, X.; Zhou, J.; Wang, Y.; Cai, H.; et al. Decoupling of inter-particle polarization and intra-particle polarization in core-shell structured nanocomposites towards improved dielectric performance. Energy Storage Mater.
**2021**, 42, 1–11. [Google Scholar] [CrossRef] - Schadler, L.S. 6.3 The Elusive Interphase/Interface in Polymer Nanocomposites. In Comprehensive Composite Materials II; Beaumont, P.W.R., Zweben, C.H., Eds.; Elsevier: Oxford, UK, 2018; pp. 52–72. [Google Scholar]
- Gupta, P.; Schadler, L.S.; Sundararaman, R. Dielectric properties of polymer nanocomposite interphases from electrostatic force microscopy using machine learning. Mater. Charact.
**2021**, 173, 110909. [Google Scholar] [CrossRef] - Zhang, M.; Askar, S.; Torkelson, J.M.; Brinson, L.C. Stiffness Gradients in Glassy Polymer Model Nanocomposites: Comparisons of Quantitative Characterization by Fluorescence Spectroscopy and Atomic Force Microscopy. Macromolecules
**2017**, 50, 5447–5458. [Google Scholar] [CrossRef] - Qiao, R.; Catherine Brinson, L. Simulation of interphase percolation and gradients in polymer nanocomposites. Compos. Sci. Technol.
**2009**, 69, 491–499. [Google Scholar] [CrossRef] - Qiao, R.; Deng, H.; Putz, K.W.; Brinson, L.C. Effect of particle agglomeration and interphase on the glass transition temperature of polymer nanocomposites. J. Polym. Sci. Part B Polym. Phys.
**2011**, 49, 740–748. [Google Scholar] [CrossRef] - Huang, Y.; Krentz, T.M.; Nelson, J.K.; Schadler, L.S.; Li, Y.; Zhao, H.; Brinson, L.C.; Bell, M.; Benicewicz, B.; Wu, K.; et al. Prediction of interface dielectric relaxations in bimodal brush functionalized epoxy nanodielectrics by finite element analysis method. In Proceedings of the 2014 IEEE Conference on Electrical Insulation and Dielectric Phenomena (CEIDP), Des Moines, IA, USA, 19–22 October 2014; pp. 748–751. [Google Scholar]
- Li, X.; Zhang, M.; Wang, Y.; Zhang, M.; Prasad, A.; Chen, W.; Schadler, L.; Brinson, L.C. Rethinking interphase representations for modeling viscoelastic properties for polymer nanocomposites. Materialia
**2019**, 6, 100277. [Google Scholar] [CrossRef] - Wang, Y.; Zhang, Y.; Zhao, H.; Li, X.; Huang, Y.; Schadler, L.S.; Chen, W.; Brinson, L.C. Identifying interphase properties in polymer nanocomposites using adaptive optimization. Compos. Sci. Technol.
**2018**, 162, 146–155. [Google Scholar] [CrossRef] - Zhao, H.; Li, Y.; Brinson, L.C.; Huang, Y.; Krentz, T.M.; Schadler, L.S.; Bell, M.; Benicewicz, B. Dielectric spectroscopy analysis using viscoelasticity-inspired relaxation theory with finite element modeling. IEEE Trans. Dielectr. Electr. Insul.
**2017**, 24, 3776–3785. [Google Scholar] [CrossRef] - Natarajan, B.; Li, Y.; Deng, H.; Brinson, L.C.; Schadler, L.S. Effect of Interfacial Energetics on Dispersion and Glass Transition Temperature in Polymer Nanocomposites. Macromolecules
**2013**, 46, 2833–2841. [Google Scholar] [CrossRef] - Virtanen, S.; Krentz, T.M.; Nelson, J.K.; Schadler, L.S.; Bell, M.; Benicewicz, B.; Hillborg, H.; Zhao, S. Dielectric breakdown strength of epoxy bimodal-polymer-brush-grafted core functionalized silica nanocomposites. IEEE Trans. Dielectr. Electr. Insul.
**2014**, 21, 563–570. [Google Scholar] [CrossRef] - Prasad, A.S.; Wang, Y.; Li, X.; Iyer, A.; Chen, W.; Brinson, L.C.; Schadler, L.S. Investigating the effect of surface modification on the dispersion process of polymer nanocomposites. Nanocomposites
**2020**, 6, 111–124. [Google Scholar] [CrossRef] - Bell, M.; Krentz, T.; Keith Nelson, J.; Schadler, L.; Wu, K.; Breneman, C.; Zhao, S.; Hillborg, H.; Benicewicz, B. Investigation of dielectric breakdown in silica-epoxy nanocomposites using designed interfaces. J. Colloid Interface Sci.
**2017**, 495, 130–139. [Google Scholar] [CrossRef] [PubMed] - Huang, Y.; Wu, K.; Bell, M.; Oakes, A.; Ratcliff, T.; Lanzillo, N.A.; Breneman, C.; Benicewicz, B.C.; Schadler, L.S. The effects of nanoparticles and organic additives with controlled dispersion on dielectric properties of polymers: Charge trapping and impact excitation. J. Appl. Phys.
**2016**, 120, 055102. [Google Scholar] [CrossRef] - Krentz, T.M.; Huang, Y.; Nelson, J.K.; Schadler, L.S.; Bell, M.; Benicewicz, B.; Zhao, S.; Hillborg, H. Enhanced charge trapping in bimodal brush functionalized silica-epoxy nanocomposite dielectrics. In Proceedings of the 2014 IEEE Conference on Electrical Insulation and Dielectric Phenomena (CEIDP), Des Moines, IA, USA, 19–22 October 2014; pp. 643–646. [Google Scholar]
- Krentz, T.; Khani, M.M.; Bell, M.; Benicewicz, B.C.; Nelson, J.K.; Zhao, S.; Hillborg, H.; Schadler, L.S. Morphologically dependent alternating-current and direct-current breakdown strength in silica–polypropylene nanocomposites. J. Appl. Polym. Sci.
**2017**, 134. [Google Scholar] [CrossRef] - Roy, M.; Nelson, J.K.; MacCrone, R.K.; Schadler, L.S. Candidate mechanisms controlling the electrical characteristics of silica/XLPE nanodielectrics. J. Mater. Sci.
**2007**, 42, 3789–3799. [Google Scholar] [CrossRef] - Chen, W.; Schadler, L.; Brinson, C.; Wang, Y.; Zhang, Y.; Prasad, A.; Li, X.; Iyer, A. Materials Informatics and Data System for Polymer Nanocomposites Analysis and Design. In Handbook on Big Data and Machine Learning in the Physical Sciences; World Scientific Publishing: Hackensack, NJ, USA, 2020; pp. 65–125. [Google Scholar]
- Wang, Y.; Zhang, M.; Lin, A.; Iyer, A.; Prasad, A.S.; Li, X.; Zhang, Y.; Schadler, L.S.; Chen, W.; Brinson, L.C. Mining structure–property relationships in polymer nanocomposites using data driven finite element analysis and multi-task convolutional neural networks. Mol. Syst. Des. Eng.
**2020**, 5, 962–975. [Google Scholar] [CrossRef] - Schadler, L.S.; Chen, W.; Brinson, L.C.; Sundararaman, R.; Gupta, P.; Prabhune, P.; Iyer, A.; Wang, Y.; Shandilya, A. A perspective on the data-driven design of polymer nanodielectrics. J. Phys. D Appl. Phys.
**2020**, 53, 333001. [Google Scholar] [CrossRef] - Iyer, A.; Zhang, Y.; Prasad, A.; Gupta, P.; Tao, S.; Wang, Y.; Prabhune, P.; Schadler, L.S.; Brinson, L.C.; Chen, W. Data centric nanocomposites design via mixed-variable Bayesian optimization. Mol. Syst. Des. Eng.
**2020**, 5, 1376–1390. [Google Scholar] [CrossRef] - Schadler, L.S.; Chen, W.; Brinson, L.C.; Sundararaman, R.; Prabhune, P.; Iyer, A. (Invited) Combining Machine Learning, DFT, EFM, and Modeling to Design Nanodielectric Behavior. ECS Trans.
**2022**, 108, 51. [Google Scholar] [CrossRef] - Ba, S.; Myers, W.R.; Brenneman, W.A. Optimal Sliced Latin Hypercube Designs. Technometrics
**2015**, 57, 479–487. [Google Scholar] [CrossRef] - Wang, W.; Wang, J.; Kim, M.-S. An algebraic condition for the separation of two ellipsoids. Comput. Aided Geom. Des.
**2001**, 18, 531–539. [Google Scholar] [CrossRef] - Shandilya, A.; Schadler, L.S.; Sundararaman, R. First-principles identification of localized trap states in polymer nanocomposite interfaces. J. Mater. Res.
**2020**, 35, 931–939. [Google Scholar] [CrossRef] - Sundararaman, R.; Letchworth-Weaver, K.; Schwarz, K.A.; Gunceler, D.; Ozhabes, Y.; Arias, T.A. JDFTx: Software for joint density-functional theory. SoftwareX
**2017**, 6, 278–284. [Google Scholar] [CrossRef] [PubMed] - Perdew, J.P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett.
**1996**, 77, 3865–3868. [Google Scholar] [CrossRef] - Garrity, K.F.; Bennett, J.W.; Rabe, K.M.; Vanderbilt, D. Pseudopotentials for high-throughput DFT calculations. Comput. Mater. Sci.
**2014**, 81, 446–452. [Google Scholar] [CrossRef] - Zhang, Y.; Tao, S.; Chen, W.; Apley, D.W. A Latent Variable Approach to Gaussian Process Modeling with Qualitative and Quantitative Factors. Technometrics
**2020**, 62, 291–302. [Google Scholar] [CrossRef] - Zhang, Y.; Apley, D.W.; Chen, W. Bayesian Optimization for Materials Design with Mixed Quantitative and Qualitative Variables. Sci. Rep.
**2020**, 10, 4924. [Google Scholar] [CrossRef] - Wang, Y.; Iyer, A.; Chen, W.; Rondinelli, J.M. Featureless adaptive optimization accelerates functional electronic materials design. Appl. Phys. Rev.
**2020**, 7, 041403. [Google Scholar] [CrossRef] - Comlek, Y.; Pham, T.D.; Snurr, R.; Chen, W. Rapid Design of Top-Performing Metal-Organic Frameworks with Qualitative Representations of Building Blocks. arXiv
**2023**, arXiv:2302.09184. [Google Scholar] - Sobol′, I.M. Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Math. Comput. Simul.
**2001**, 55, 271–280. [Google Scholar] [CrossRef] - Comlek, Y.; Wang, L.; Chen, W. Mixed-variable global sensitivity analysis with applications to data-driven combinatorial materials design. In International Design Engineering Technical Conferences and Computers and Information in Engineering Conference; Paper Number: IDETC2023-110756; American Society of Mechanical Engineers: New York, NY, USA, 2023. [Google Scholar]
- Yu, K.; Wang, H.; Zhou, Y.; Bai, Y.; Niu, Y. Enhanced dielectric properties of BaTiO3/poly(vinylidene fluoride) nanocomposites for energy storage applications. J. Appl. Phys.
**2013**, 113, 034105. [Google Scholar] [CrossRef] - Li, B.; Randall, C.A.; Manias, E. Polarization Mechanism Underlying Strongly Enhanced Dielectric Permittivity in Polymer Composites with Conductive Fillers. J. Phys. Chem. C
**2022**, 126, 7596–7604. [Google Scholar] [CrossRef] - Carroll, B.; Cheng, S.; Sokolov, A.P. Analyzing the Interfacial Layer Properties in Polymer Nanocomposites by Broadband Dielectric Spectroscopy. Macromolecules
**2017**, 50, 6149–6163. [Google Scholar] [CrossRef] - Holt, A.P.; Griffin, P.J.; Bocharova, V.; Agapov, A.L.; Imel, A.E.; Dadmun, M.D.; Sangoro, J.R.; Sokolov, A.P. Dynamics at the Polymer/Nanoparticle Interface in Poly(2-vinylpyridine)/Silica Nanocomposites. Macromolecules
**2014**, 47, 1837–1843. [Google Scholar] [CrossRef] - Cheng, S.; Mirigian, S.; Carrillo, J.-M.Y.; Bocharova, V.; Sumpter, B.G.; Schweizer, K.S.; Sokolov, A.P. Revealing spatially heterogeneous relaxation in a model nanocomposite. J. Chem. Phys.
**2015**, 143, 194704. [Google Scholar] [CrossRef]

**Figure 2.**Nanodielectrics microstructure with varying characteristics (

**a**,

**b**) and polymer matrix and intrinsic interface properties (

**c**,

**d**). (

**a**) High Aspect Ratio (AR), high Orientation Variation (OV), high Volume Fraction (VF), and low dispersion; (

**b**) Low Aspect Ratio (AR), low Orientation Variation (OV), low Volume Fraction (VF), and high dispersion (

**c**) schematic of extrinsic and intrinsic interface layer arrangement (

**d**) microstructure representative volume element (RVE) where volume fraction, orientation, and aspect ratio of fillers vary.

**Figure 3.**Characteristic structures of the three molecules considered here, attached to an amorphous surface, surrounded by polymer (not shown for clarity). We simulate an ensemble of 15 structures for each molecule to account for structural variations in the amorphous interface and predict trap distributions for estimating dielectric breakdown strength (Section First-Principles Predictions of Trap States).

**Figure 4.**Polymer matrix and intrinsic interface properties (IP) corresponding to levels in Table 1 (

**a**) Loss (

**b**) Permittivity. f denotes frequency in Hz.

**Figure 5.**Ensemble-averaged local density of states (LDOS) of interfaces functionalized with (

**a**) thiophene, (

**b**) terthiophene, and (

**c**) ferrocene, showing the energy and spatial distribution of trap states within the energy gap of these interfaces.

**Figure 6.**Probability distribution of trap states introduced by the three functional groups with inset visualizing the electronic orbital associated with one of the trap states on each molecule.

**Figure 7.**The latent variables of the extrinsic and intrinsic interface design choices were obtained from the LVGP models trained on the initial DOE at 60 Hz in the Z direction. Here, the axis (z1, z2) represents the latent variables obtained from the LVGP model.

**Figure 8.**Nanodielectrics properties in the z direction based on the DOE designs at two frequencies 1000 Hz and 60 Hz with respect to extrinsic interface design choices.

**Figure 9.**GSA of two Z components of design objectives, namely stored energy density and loss with respect to design variables, both quantitative and qualitative.

**Figure 10.**GSA of breakdown strength with respect to design variables, both quantitative and qualitative.

**Figure 11.**The Loss (Z direction) and Stored Energy Density (Z and X direction) properties of 100 DOE designs (black) against the Top 5 (red) and Pareto front (light green) designs were obtained after design optimization at 60 Hz.

**Figure 12.**The Loss (Z direction) and Stored Energy Density (Z and X direction) properties of 100 DOE designs (black) against the Top 5 (red) and Pareto front (light green) designs were obtained after design optimization at 1000 Hz.

**Figure 13.**Pareto front designs are plotted in loss space and SED space. A color bar is used to denote the Aspect Ratio (AR) of the Pareto front designs and shapes are used to denote the intrinsic layer choices on the loss plot (

**left column**) whereas shapes denote extrinsic layer choices in SED plot (

**right column**).

**Table 1.**The Mixed Variable Nanodielectrics Design Space with bounds on the variable ranges. The dispersion parameter refers to the nearest neighbor distance.

Design Variables | Design Choices | |
---|---|---|

Microstructural (Quantitative)
| Volume Fraction (VF) | (1,4)% |

Aspect Ratio (AR) | (1–6) | |

Dispersion (D) | (11–36) nm | |

Orientation Variation (OV) | (0,1) | |

Interfacial (Qualitative)
| Intrinsic Interface | Attractive Lossy, Attractive Non-Lossy, Repulsive Lossy, Repulsive Non-Lossy, No Interface |

Extrinsic Interface | Ferrocene, Terthiophene, Thiophene, No Interface (No Extrinsic Interface) |

**Table 2.**List of the molecule choices for the extrinsic interface and the conductivity used in the modeling.

# | Ligand Molecule | Conductivity $\mathit{\sigma}\left(\mathit{S}/\mathbf{cm}\right)$ |
---|---|---|

1 | Thiophene | 1 × 10^{−10} |

2 | Terthiophene | 1 × 10^{−7} |

3 | Ferrocene | 1 × 10^{−1} |

**Table 3.**List of the intrinsic interface choices and the associated parameters used in the modeling.

# | Intrinsic Interface | ${\mathit{S}}_{\mathit{\beta}}$ | ${\mathit{M}}_{\mathit{\beta}}$ | ${\mathit{S}}_{\mathit{\alpha}}$ | ${\mathit{M}}_{\mathit{\alpha}}$ | $\mathit{C}$ |
---|---|---|---|---|---|---|

1 | Attractive Lossy (AL) | 5.0 | 1.2 | 7.0 | 1.1 | 0 |

2 | Attractive Non-Lossy (ANL) | 5.0 | 0.5 | 7.0 | 0.5 | 0 |

3 | Repulsive Lossy (RL) | 0.05 | 1.2 | 0.07 | 1.1 | 0 |

4 | Repulsive Non-Lossy (RNL) | 0.05 | 0.5 | 0.07 | 0.5 | 0 |

5 | No Intrinsic Interface | 1 | 1 | 1 | 1 | 0 |

Properties/Directions | X | Y | Z |
---|---|---|---|

Loss vs. Breakdown Strength | 0.21 | 0.2 | 0.12 |

Loss vs. Permittivity | 0.78 | 0.7 | 0.93 |

Loss vs. Stored Energy Density | 0.63 | 0.54 | 0.85 |

**Table 5.**Loss and SED components in x, y, and z directions for top 5 designs (visualized in Figure 11) at 60 Hz.

Top Designs/Properties | Loss_x | Loss_y | Loss_z | $\mathbf{SED}\_\mathbf{x}\left(\times {10}^{2}\right)$ | $\mathbf{SED}\_\mathbf{y}\left(\times {10}^{2}\right)$ | $\mathbf{SED}\_\mathbf{z}\left(\times {10}^{2}\right)$ |
---|---|---|---|---|---|---|

1 | 0.068 | 0.041 | 0.309 | 2976 | 2411 | 5060 |

2 | 0.038 | 0.035 | 0.943 | 2364 | 2325 | 7709 |

3 | 0.108 | 0.074 | 0.381 | 2921 | 2709 | 5276 |

4 | 0.194 | 0.090 | 0.460 | 3638 | 2854 | 5459 |

5 | 0.117 | 0.104 | 0.319 | 3043 | 2941 | 4952 |

**Table 6.**Loss and SED components in x, y and z directions for top 5 (visualized in Figure 12) at 1000 Hz.

Top Designs/Properties | Loss_x | Loss_y | Loss_z | $\mathbf{SED}\_\mathbf{x}\left(\times {10}^{2}\right)$ | $\mathbf{SED}\_\mathbf{y}\left(\times {10}^{2}\right)$ | $\mathbf{SED}\_\mathbf{z}\left(\times {10}^{2}\right)$ |
---|---|---|---|---|---|---|

1 | 0.055 | 0.033 | 0.300 | 2759 | 2432 | 4679 |

2 | 0.071 | 0.062 | 0.073 | 2880 | 2882 | 2931 |

3 | 0.179 | 0.074 | 0.376 | 3512 | 2885 | 5150 |

4 | 0.030 | 0.032 | 0.774 | 2316 | 2341 | 7238 |

5 | 0.107 | 0.077 | 0.153 | 3272 | 2828 | 3590 |

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## Share and Cite

**MDPI and ACS Style**

Prabhune, P.; Comlek, Y.; Shandilya, A.; Sundararaman, R.; Schadler, L.S.; Brinson, L.C.; Chen, W.
Design of Polymer Nanodielectrics for Capacitive Energy Storage. *Nanomaterials* **2023**, *13*, 2394.
https://doi.org/10.3390/nano13172394

**AMA Style**

Prabhune P, Comlek Y, Shandilya A, Sundararaman R, Schadler LS, Brinson LC, Chen W.
Design of Polymer Nanodielectrics for Capacitive Energy Storage. *Nanomaterials*. 2023; 13(17):2394.
https://doi.org/10.3390/nano13172394

**Chicago/Turabian Style**

Prabhune, Prajakta, Yigitcan Comlek, Abhishek Shandilya, Ravishankar Sundararaman, Linda S. Schadler, Lynda Catherine Brinson, and Wei Chen.
2023. "Design of Polymer Nanodielectrics for Capacitive Energy Storage" *Nanomaterials* 13, no. 17: 2394.
https://doi.org/10.3390/nano13172394