Phase-field modeling and machine learning of electric-thermal-mechanical breakdown of polymer-based dielectrics

Understanding the breakdown mechanisms of polymer-based dielectrics is critical to achieving high-density energy storage. Here a comprehensive phase-field model is developed to investigate the electric, thermal, and mechanical effects in the breakdown process of polymer-based dielectrics. High-throughput simulations are performed for the P(VDF-HFP)-based nanocomposites filled with nanoparticles of different properties. Machine learning is conducted on the database from the high-throughput simulations to produce an analytical expression for the breakdown strength, which is verified by targeted experimental measurements and can be used to semiquantitatively predict the breakdown strength of the P(VDF-HFP)-based nanocomposites. The present work provides fundamental insights to the breakdown mechanisms of polymer nanocomposite dielectrics and establishes a powerful theoretical framework of materials design for optimizing their breakdown strength and thus maximizing their energy storage by screening suitable nanofillers. It can potentially be extended to optimize the performances of other types of materials such as thermoelectrics and solid electrolytes.

The manuscript titled "Electric-Thermal-Mechanical Breakdown of Polymer-based Dielectrics: High-Throughput Phase-Field Simulations and Machine Learning" by Shen et al. reports a phase-field model to explore and study the electric, thermal, and mechanical effects in the breakdown process of polymer-based dielectrics. The developed model is applied to carryout high-throughput 2D simulations of breakdown strengths for the P(VDF-HFP)-based nanocomposites filled with nanoparticles of different dielectric elastic and thermal properties. The data is then analyzed within a simple machine learning based framework to produce an analytical expression for the breakdown strength of polymer nanocomposites. Overall the work is carefully carried out and the paper is well written. The work is undoubtably of high value for those in the community who are working in this specialized field of polymer-dielectric composites for energy storage and related applications. However, in my opinion, neither the research approach nor the insights that come out of the work are novel enough to warrant a publication in Nature Communication.
The phase-field model reported in the present manuscript can be considered an incremental improvement over the recent past works published in the following references: Adv. Mater. 30, 1704380 (2018) and Adv. Energy Mater. 1800509 (2018). The two papers have already reported the details of the electrical and electrothermal phase field simulations. As far as the phase field model development is concerned, the current work can be considered only a minor extension of these works by adding mechanical effects (which seem to play relatively minor role in dictating the breakdown field strength; looking at Fig. 2 of the manuscript, for instance).
The high throughput framework has also been previously reported in Adv. Mater. 30, 1704380 (2018).
The machine learning analysis and the feature selection approach by considering combinatorial functional forms of a pre-selected set of primary features for mining the analytical relationships such as those presented in the paper have also been reported before [ for example, see Chem. Mater. 28, 1304-1311(2016]. In fact, the machine learning analysis presented here appears to be highly inspired from the approach and work flow presented in Chem. Mater. 28, 1304-1311(2016.
Results and findings of the present work do not go any further from confirming the conventional wisdom. Specifically "What new insights have come out as a result of this work?" is not clear.
For the above reasons, the manuscript is more suitable for a specialized journal rather than Nature Communication, which emphasizes on novelty and impact.

Reviewer #2 (Remarks to the Author):
This manuscript presents a continuum, phase-field model for the breakdown of polymer dielectrics, which involves the electric energy, Joule heating, and strain. This model includes significantly more physics than the old Stark-Garton model and is able to reproduce the experimental data well to fairly well. The authors also use machine-learning techniques on a set of ~400 simulations to arrive at an analytical expression that replicates well the results of the phase-field simulations, including data not included in the training set. The final part of the manuscript deals with predictions and experiments on nanocomposites. The phase-field model represents a genuine advance over the Stark-Garton expression of breakdown and thus deserves publication in Nature Communications. However, the presentation and discussion in terms of figures and their descriptions left me wondering about the deeper physical issues and the expected accuracy, which have not been addressed fully. It is also hard to develop deeper physical insight from an examination of a few figures. My suggestions are the following: (i) The number of experimental data points used to compare to the results of phase-field modeling is rather limited, while the phase-field model uses parameters and approximations. The authors should discuss the extent to which their parameters are uniquely determined and the sensitivity of their results to parameter variations. This is particularly important because the physical properties of complex polymers vary between samples and are not known with high accuracy. What accuracy is expected? Are only the general trends expected to be reproduced or are the modeling results truly quantitative? (ii) I would like to see a deeper physical description of the model in the main text, perhaps instead of some figures if space is a problem. (iii) Clearly, the phase-field model uses a continuum description and ignores atomic-scale effects, such as defects and material inhomogeneities. A discussion of these left-out effects would benefit the readers.
Reviewer #3 (Remarks to the Author): This manuscript developed a comprehensive phase-field model to investigate electric-thermalmechanical breakdown of polymer-based dielectrics. Until now, both the theoretical and experimental study of the mechanism of breakdown strength of polymer-based composites are rather insufficient to obtain a definite conclusion compared with other dielectric properties. Due to the various influencing factors, such as the electrode system, measuring environment, the shape of the sample, the intrinsic properties of materials and so on, it is unrealistic to depict the breakdown phenomenon with a unified theory. However, it does not affect the contribution of this paper revealing the internal breakdown mechanism of polymer-based composites. Comparison between the calculation and the experiment results indicates that the incorporation of the phase field method is effective. It is of great significant to guide the experiment design with the conclusions from the phase field calculation and the linear regression. Based on different energies, this model could identify the breakdown mechanisms and predict the breakdown strength of polymer-based dielectrics under different stimulus. Then, high-throughput calculations and machine learning were conducted to produce an analytical expression of breakdown strength. I think this work is interesting and very helpful to design and screen nanocomposites for experiments. I recommend this manuscript to be published after the following comments are addressed. 1. The author mentioned "assuming that the initial breakdown phase is nucleated from the two needle electrodes". Why does the breakdown path start from the electrodes? Please give enough explanation. 2. In this work, the matrix polymer is P(VDF-HFP). Does this model also work in other systems? How about the microstructure of nanocomposites? Does it consider the state of the molecular chain or crystallinity of polymer? 3. For dielectric breakdown, there are so many factors affecting this process. Sometimes, the extrinsic factors, e.g., defects, impurities or air hole, may also be important. 4. Using the analytical expression of dielectric breakdown by machine learning, it has found that fillers like Al2O3 or MgO can improve the breakdown strength of nanocomposites. How can we use the expression to find more new materials? 5. The introduce of fillers like Al2O3 or MgO can improve the breakdown strength of nanocomposites. But those fillers also lead to the decrease of dielectric constant due to the low intrinsic dielectric constant. So if we want to get high energy density of nanocomposites, how to balance these two parameters? 6. Whether the simulation model is 2D or 3D. if it is 2D, when extending to a three-dimensional condition, the circle representing the ceramic particles will become cylinder rather sphere. It will deviate from the actual situations and the well agreement between the calculation and the experiment will be less convincing. 7. When simulating the thermal breakdown, whether the heat dissipation is considered? In other works, whether the temperature(363K) is only used to change the temperature-dependent electrical conductivity or used as the ambient temperature? 8. Supporting Information[line601-604]: The kinetic coefficient L¬0 is not given. 9. Supporting Information[equation 8-9]: Can you give a detailed derivation for eq.8 to eq.9. 10. Supporting Information [equation 11] Please explain the equation more detailed. Where does the expression (η^3 (10-15η+6η^2)) come from?

Reviewer #1
The manuscript titled "Electric-Thermal-Mechanical Breakdown of Polymer-based Dielectrics: High-Throughput Phase-Field Simulations and Machine Learning" by Shen et al. reports a phase-field model to explore and study the electric, thermal, and mechanical effects in the breakdown process of polymer-based dielectrics. The developed model is applied to carryout high-throughput 2D simulations of breakdown strengths for the P(VDF-HFP)-based nanocomposites filled with nanoparticles of different dielectric elastic and thermal properties. The data is then analyzed within a simple machine learning based framework to produce an analytical expression for the breakdown strength of polymer nanocomposites. Overall the work is carefully carried out and the paper is well written. The work is undoubtably of high value for those in the community who are working in this specialized field of polymer-dielectric composites for energy storage and related applications. However, in my opinion, neither the research approach nor the insights that come out of the work are novel enough to warrant a publication in Nature Communication.

Response:
We are thankful to the referee for the careful reading of our manuscript and the positive comments on the value of our work and the writing. We would like to address the concerns by the referee about the novelty and general interest of our work from the following aspects: 1) This is the first time that the electric, thermal, and mechanical effects are simultaneously incorporated into a phase-field model of dielectric breakdown, which represents a breakthrough advance in the theory and computation of dielectric breakdown. Such a model is critical for understanding the transition of breakdown mechanisms under different electric fields and temperatures.
2) Combining the high-throughput calculations of dielectric breakdown and machine learning in this work is another major advance, making it possible to guide the design of materials to achieve enhanced breakdown strength. For example, based 7 on the results, we are able to show that some oxides such as Al2O3, MgO, SiO2, and TiO2 with lower dielectric constant and lower electrical conductivity can be used as nanofillers in polymer nanocomposites to increase the breakdown strength. Such computational guidance is expected to stimulate experimental efforts for validation and further research on high-energy-density polymer nanocomposites.
3) The general framework combining high throughput phase-field simulations of responses of microstructures under external stimuli and machine learning can be extended to the understanding and design of other types of materials systems, e.g., optimizing the "ZT" values of a two-phase thermoelectric system with respect to volume fraction, morphology as well as the electric and thermal conductivities of each individual phase. Therefore, this work is of general interest rather than only to the community of polymer composites.
In summary, we believe both of our comprehensive phase-field model of breakdown incorporating thermal, mechanical and electrical effects and the integration of high throughput phase-field simulations and machine learning are novel, and this computational framework can be generally applied to the understanding and design of many other materials systems and is thus of general interest.
The phase-field model reported in the present manuscript can be considered an incremental improvement over the recent past works published in the following references: Adv. Mater. 30, 1704380 (2018) and Adv. Energy Mater. 1800509 (2018).
The two papers have already reported the details of the electrical and electrothermal phase field simulations. As far as the phase field model development is concerned, the current work can be considered only a minor extension of these works by adding mechanical effects (which seem to play relatively minor role in dictating the breakdown field strength; looking at Fig. 2 of the manuscript, for instance).

Response:
We thank the referee has carefully read our manuscript and previous works.
1) It is true that the comprehensive model in this work is built on our previous results. However, the model published in Adv. Mater. 30, 1704380 (2018) only considers the electrical effect, so it can only be used to study the electrical breakdown at room 8 temperature. Then, we incorporate the thermal effect and published the electrothermal breakdown model in Adv. Energy Mater. 1800509 (2018). It can be used to study the breakdown at different temperatures. However, at high temperatures, particularly above the glass transition temperature, the polymer will become softer, and the mechanical effect cannot be neglected. Therefore, incorporating the mechanical effect is required.
The model presented in this work is a comprehensive model which can be used to study the breakdown under simultaneous electrical, thermal, and mechanical stimuli and help understand under which breakdown mechanism will dominate under a given set of thermoelectromechanical conditions.
2) In the test example of P(VDF-HFP) in Fig not have to assume which breakdown mechanism would dominate as a priori in predicting the breakdown mechanism under a given condition. Therefore, it can be used to analyze the role of each breakdown mechanism in different systems under different stimulus, which is very helpful for providing guidance for the experimentalists to design materials with high breakdown strength, as summarized in Fig.4.
The high throughput framework has also been previously reported in Adv. Mater. 30, 1704380 (2018).

Response:
We thank the referee for this question. Yes, we have done high-throughput 9 calculations in our previous work published in Adv. Mater. 30, 1704380 (2018).
However, in previous work, we used the high-throughput calculations to study the microstructure-property relationship: the dependences of effective dielectric constant, breakdown strength, and energy density on the shape and orientation of the nanofillers.
In this work, we perform high-throughput phase-field simulations to study the effects of material parameters including the electrical conductivity, dielectric constant, and Young's modulus on the breakdown strength and energy contributions. The previous work emphasizes the microstructure, and this work emphasizes the material parameters of the nanofiller. In addition, the high-throughput simulation results in this work are designed for performing the machine learning to obtain the analytical expression of breakdown strength as function of material parameters.
The machine learning analysis and the feature selection approach by considering combinatorial functional forms of a pre-selected set of primary features for mining the analytical relationships such as those presented in the paper have also been reported before [for example, see Chem. Mater. 28, 1304-1311 (2016)]. In fact, the machine learning analysis presented here appears to be highly inspired from the approach and work flow presented in Chem. Mater. 28, 1304Mater. 28, -1311Mater. 28, (2016. Results and findings of the present work do not go any further from confirming the conventional wisdom.
Specifically "What new insights have come out as a result of this work?" is not clear.

Response:
We agree with the referee that the workflow of machine learning in this work is partly inspired from the paper of Chem. Mater. 28, 1304-1311 (2016), as cited in our manuscript. The machine learning in this work is mainly used to obtain an analytical expression of breakdown strength, allowing one to make quick predictions of breakdown strength for nanocomposites with available material parameters.
Therefore, the focus of this part is to use a suitable and effective machine learning to connect with phase-field model rather than to design a new workflow or algorithm of machine learning. Furthermore, some specific improvements have been made in our machine learning, such as multiple rounds of screening and the specific interactions between different fingerprints, as described in the section of methods. Therefore, the novelty of machine learning in this work is producing an analytical expression from the high-throughput phase-field simulation results using a simple but efficient approach, which can be practically very useful.
Furthermore, we also tried another machine learning method, back-propagation neural network (BPNN) to predict the breakdown strength of nanocomposites. More details are described in supporting information of section 5 on page 41-42. As shown in Fig.   S9, the predictive ability of this neural network method is much stronger than the machine learning of LSR, with a higher coefficient of determination R 2 = 0.983. There is no doubt that both novelty and accuracy of BPNN are superior. However, as we have stated above, we want to obtain an analytical expression to help researchers make a quick estimation of breakdown strength for their material systems. To achieve this goal, the machine learning of LSR is a better choice than the neural network and other advanced methods. To clarify our purpose of machine learning in this work, some discussions are added in red text on page 13. "Aside from the LSR, we also tried another machine learning method, the back-propagation neural network (BPNN) with details described in the supporting information. In comparison to LSR, the BPNN exhibits better prediction ability of the breakdown strength. However, the BPNN cannot give an expression of breakdown strength as functions of the dielectric constant, electrical conductivity, and Young's modulus, thereby it is less convenient for experimental researchers to make a quick estimation of the breakdown strength for a new material system." Based on the phase-field simulations and the machine learning, we found that the addition of oxides with lower dielectric constant and electrical conductivity such as Al2O3, MgO, SiO2 and TiO2 in polymer matrix could lead to enhanced breakdown strength. Traditionally, researchers in this field preferred to fill high-dielectric-constant nanofillers into the polymer to improve the energy density by improving the effective dielectric constant. However, it is hard to significantly improve dielectric constant at a low volume fraction of the nanofillers. With the volume fraction of high-dielectricconstant nanofillers increasing, the breakdown strength may be severely decreased. As a result, the energy density may be reduced due to the quadratic relationship between the energy density and the breakdown strength. Therefore, adding oxides with low dielectric constant and electrical conductivity into the polymer to enhance the breakdown strength at the expense of partially sacrificing the effective dielectric constant is one of our new insights. Moreover, this insight was verified by the successful synthesis and characterization of Al2O3/P(VDF-HFP) nanocomposites in this work.
Therefore, the novelties of this work include phase-field model development, highthroughput simulations, machine learning, and experimental verification. In order to make our insights more clear, we have added some sentences in red text to describe our results, as follows: The section of abstract on page 1 "It is found that the addition of oxides with lower dielectric constant and electrical conductivity such as Al2O3, MgO, SiO2 and TiO2 into the P(VDF-HFP) polymer can enhance the breakdown strength. " On pages 12-13 "In general, the Young's modulus of the ceramic nanofillers is much larger than that of the polymer matrix, therefore seeking for nanofillers with lower dielectric constant and lower electrical conductivity will be more critical to improve the breakdown strength of nanocomposites." On page 16 "According to the analytical expression and the mechanism analysis above, if nanofillers with low dielectric constant and low electrical conductivity are added into the polymer, the breakdown strength of nanocomposites can be improved." On pages 16-17 "Specific examples include nanocomposites of P(VDF-HFP) filled with oxides such as Al2O3, SiO2, MgO and TiO2. The machine learning predicts that those nanocomposites should exhibit higher breakdown strength than pure polymer matrix, and this prediction is verified by both calculations and experiments in this work. " From the above reasons, the manuscript is more suitable for a specialized journal rather than Nature Communication, which emphasizes on novelty and impact.

Response:
We are deeply thankful for the referee for her/his insights and comments, which are undoubtedly helpful to improve our manuscript. However, we respectfully disagree with his/her recommendation for a more specialized journal. Here, we would like to briefly summarize our work again as follows: 1) We developed a comprehensive phase-field model of dielectric breakdown which simultaneously incorporates electrical, thermal, and mechanical effects.
2) We employed the developed model to perform high-throughput simulations and machine learning.
3) We produced an analytical expression of the breakdown strength as function of material parameters to quickly screen nanofillers.
4) We proposed to use oxides like Al2O3, SiO2, MgO and TiO2 as nanofillers to enhance the breakdown strength.

5)
We performed targeted experiments to verify our simulation results.
We believe our results and conclusions will be helpful to understand the breakdown mechanisms and screen the nanofillers in future experiments. It will attract broad attention from the material and energy communities. Furthermore, we believe that the combination of high-throughput phase-field simulation, machine learning, and targeted experiments provide a paradigm to achieve the goal of Materials Genome Initiative project. The material system can also be extended to other functional composite materials such as thermoelectrics and solid electrolytes. Therefore, we hope we are able to convince the referee that the work is a significant advance and is of general interest, and the manuscript is suitable for publication in Nature Communications.

Reviewer #2
This manuscript presents a continuum, phase-field model for the breakdown of polymer dielectrics, which involves the electric energy, Joule heating, and strain. This model includes significantly more physics than the old Stark-Garton model and is able to reproduce the experimental data well to fairly well.
The authors also use machine-learning techniques on a set of ~400 simulations to arrive at an analytical expression that replicates well the results of the phase-field simulations, In order to make our conclusions more clear, we have revised some red texts on page 1 and page 11.
On page 1 "It can be used to semiquantitatively predict the breakdown strength of the P(VDF-HFP)-based nanocomposites with a wide variety of candidate nanofillers."

Response:
We thank the reviewer for this enlightening suggestion. We added a schematic diagram on phenomenological energy profile of the breakdown process in Fig. 4(a). In the main text of the revised manuscript, we have added following texts in red on page 9: " Fig. 4a phenomenologically shows the variation of the energy profile for polymers under physical stimuli. Curve 1 describes a double-well energy density as function of the order parameter , and the energy barrier height between η=0 (unbroken phase) and η=1 (broken phase) represents how difficult a local point can be broken down. With the increase of external physical stimuli such as the electric field E or temperature T or both, the energy barrier drops and energy profile tilts, leading to a lower energy state at η=1 than that at η=0, as illustrated by the variation from curve 1 to curve 3 shown in Fig.   4(a). Once the physical stimuli is sufficiently large, the energy barrier vanishes and the breakdown occurs. The height of this energy barrier is also related to the material parameters including the dielectric constant ε, the electrical conductivity σ and the Young's Modulus Y. For a material with higher dielectric constant ε, higher electrical conductivity σ and lower Young's Modulus Y, the energy barrier is lower. Thus, the breakdown strength will be lower. " More captions are added for Fig.4 on page 47. (iii) Clearly, the phase-field model uses a continuum description and ignores atomicscale effects, such as defects and material inhomogeneities. A discussion of these leftout effects would benefit the readers.

Response:
We thank the reviewer for such a constructive suggestion. Yes, it is important to discuss the defects effects for benefiting the readers. We have added brief discussions on page 14 of the revised manuscript: "Many factors such as voids, space charge effects 40-42 and incomplete crystallization 11,13,43,44 of the polymer that are not incorporated into the phase-field model may cause this difference between the phase-field-based machine learning and the experimental results. For example, the existence of voids may cause partial discharge at the void/polymer interfaces where the local electric field is intensified. A high concentration of space charges can cause the increases in the electrical 17 conductivity. The crystallinity of polymers may also affect the breakdown strength, arising from the different transport behaviors of charge carriers in the amorphous phase and crystalline phase. Unfortunately, the introduction of a large number of nanofillers can easily introduce these effects due to the incompatibility of ceramics nanofillers and polymers. Therefore, the preparation of high-quality polymer nanocomposites is extremely important to achieve a high breakdown strength." The referee has provided us very constructive comments, which are helpful to us for improving the readability of our manuscript. We therefore sincerely thank the referee again for her/his encouragement and suggestions.

Reviewer #3
This manuscript developed a comprehensive phase-field model to investigate electricthermal-mechanical breakdown of polymer-based dielectrics. Until now, both the theoretical and experimental study of the mechanism of breakdown strength of polymer-based composites are rather insufficient to obtain a definite conclusion compared with other dielectric properties. Due to the various influencing factors, such as the electrode system, measuring environment, the shape of the sample, the intrinsic properties of materials and so on, it is unrealistic to depict the breakdown phenomenon with a unified theory. However, it does not affect the contribution of this paper revealing the internal breakdown mechanism of polymer-based composites. Comparison between the calculation and the experiment results indicates that the incorporation of the phase field method is effective. It is of great significant to guide the experiment design with the conclusions from the phase field calculation and the linear regression.
Based on different energies, this model could identify the breakdown mechanisms and predict the breakdown strength of polymer-based dielectrics under different stimulus.
Then, high-throughput calculations and machine learning were conducted to produce an analytical expression of breakdown strength. I think this work is interesting and very helpful to design and screen nanocomposites for experiments. I recommend this manuscript to be published after the following comments are addressed.

Response:
We greatly appreciate the referee's highly encouraging comments on our work. Yes, we fully agree with the referee that the dielectric breakdown is rather complicated phenomenon and hardly depicted with a unified theory by considering all factors. This work is aimed at developing a relatively comprehensive model to help us understand the internal breakdown process and provide some theoretical guidance to experiments.
1. The author mentioned "assuming that the initial breakdown phase is nucleated from the two needle electrodes". Why does the breakdown path start from the electrodes?
Please give enough explanation.

Response:
We thank the referee for this question. This assumption is based on two considerations. 1) Due to the huge property contrast between the polymer dielectrics and the metal electrode, the electric field may concentrate at the metal/polymer interface, making this area vulnerable. Therefore, we assume that the initial breakdown phase is nucleated from this area. We have simply simulated the local electric field distribution around a Cu needle electrode deposited at P(VDF-HFP) polymer. As shown in Fig. R1, the electric field at the electrode/polymer interface is much higher than that at other region, rationalizing this assumption. 2) When operating at high voltage and high temperature, charge injection from electrodes into the dielectric may make the area around electrodes hot spots, thereby 19 triggering the nucleation of breakdown at this area. This has been demonstrated by some experiments and simulations (for example, see Advanced Materials, 2017, 29(35): 1701864. andMaterials Letters, 2015, 141: 14-19.).
Therefore, based on above reasons, we assume the initial breakdown phase is nucleated from the two needle electrodes.
2. In this work, the matrix polymer is P(VDF-HFP  (2018)), the microstructure effect on the electrostatic breakdown has been systematically investigated. However, the focus of this work is not on the microstructure effect but the filler material effect. For effects of the molecular chain and the crystallinity of polymers, we didn't include them in this model currently, because we assume the polymer phase a homogeneous phase. If distinguishing the crystalline phase and amorphous phase and taking them as two different phases in the polymer, this model can also be used to simulate the crystallinity effect on the breakdown behavior in pure polymer dielectric. This question gives us a next-step direction to expand this model to explore more interesting research topics, e.g., the microstructure of the pure polymer on the breakdown property. We thank the reviewer again for so nice questions.
3. For dielectric breakdown, there are so many factors affecting this process. Sometimes, the extrinsic factors, e.g., defects, impurities or air hole, may also be important.

Response:
We thank the referee for this great comment. Yes, we totally agree that the breakdown process may be affected by extrinsic factors such as the pinhole, defects and void. All these factors may cause the concentration of local electric field due to the large contrast of dielectric or electrical properties. The region where these defects are located can easily become the hot spot to trigger the sequential breakdown process before the intrinsic electric breakdown occurs. However, in this work, the simulation of breakdown process is performed under an ideal condition to investigate the intrinsic breakdown mechanism, without considering those extrinsic factors considering that they are hard to control in experiments. By incorporating those defects into the microstructure, they effects on breakdown can actually be studied. If necessary, we can design some simulations to investigate effects of those extrinsic factors on the breakdown behavior in the future. We thank the reviewer again for this nice comment.  6. Whether the simulation model is 2D or 3D. if it is 2D, when extending to a threedimensional condition, the circle representing the ceramic particles will become cylinder rather sphere. It will deviate from the actual situations and the well agreement between the calculation and the experiment will be less convincing.

Response:
We thank the referee for raising this important issue. We totally agree that there are some inevitable differences when extending 2D simulation to 3D simulation.
To clarify this question, we discuss several points as follows: 1) In this model, the local energy density is considered as the breakdown criterion. If the energy density at one point exceeds the corresponding critical energy density, the breakdown path will grow. Thus, the local electric field distribution is one of the most important factors, because it is strongly related with the energy density of every point.
In order to compare different conditions, we simulate the electric field distributions of three states of BTO/(PVDF-HFP): 2D circle, 3D sphere, and 3D cylinder, as shown in sphere when compared with 2D circle. However, 3D sphere can cause more severe concentration of local electric field along applied electric field. When considering the 3D distributions as shown in Fig. R2(b) and R2(d), the distribution is more dependent on the shape of nanofillers. 2) In order to verify our analysis above, we perform a set of 2D and 3D simulations to compare their results. In the simulation, 10 random microstructures for 5% Al2O3/P(VDF-HFP) nanocomposites are generated to calculate the corresponding breakdown strength, as shown in Fig. R3. It can be seen that the 3D simulation gives an average value of 1.45 for composite matrix bb / EE , which is slightly larger than the average value of 1.42 from 2D simulation. Although there exist small deviations due to the change of dimensionality, it doesn't affect the capability of semiquantitatively predicting the breakdown strength by phase-field model. It doesn't affect the conclusion 23 that the addition of oxides with lower dielectric constant and electrical conductivity such as Al2O3, MgO, SiO2 and TiO2 into the P(VDF-HFP) polymer can enhance the breakdown strength neither. Therefore, we used 2D high-throughput simulations in this work for reducing the huge computational cost. Of course, if the computational resource is allowed, 3D simulation is definitely the best choice, particularly when modeling polymer nanocomposites filled with nanofibers and nanosheets, Fig. R3 The evolution of breakdown path obtained in (a)-(c) 3D and (d)-(f) 2D simulations for 5% Al2O3/P(VDF-HFP) nanocomposites. 10 random microstructures are generated and used for each set of simulation.
7. When simulating the thermal breakdown, whether the heat dissipation is considered?
In other works, whether the temperature (363K) is only used to change the temperaturedependent electrical conductivity or used as the ambient temperature?
Response: We thank the reviewer for this question. In this simulation, the heat dissipation is not considered. As the thickness of nanocomposite films in experiments is only about ~10 μm, thus the heat can immediately dissipate without causing the increase in temperature according to our thermal steady-state simulation. Here, we exhibit the example of thermal simulation results of two dielectric films when operating at 363K and 200kV/mm by solving The electrical conductivity is set at 10 -11 S/m and the convective heat transfer coefficient is 10 Wm -2 K -1 . As shown in Fig. R4, the temperature in film (a) is almost equal to the ambient temperature. However, when the thickness increases to 10mm, the temperature can reach about 560K. Therefore, in this simulation of thin polymer film, the internal temperature can be regarded as uniform and same as the ambient temperature.

Response:
We thank the reviewer for this kind reminder. Here, the kinetic coefficient L0 is related to the breakdown phase wall mobility and is assigned a value of 1.0 m 2 s 1 N 1 due to the lack of experimental data. We have added it into the manuscript on page 26. 9. Supporting Information [equation 8-9]: Can you give a detailed derivation for eq.8 to eq.9.

Response:
We thank the reviewer for this question. Yes, the derivation from Eq.(8) to