Numerical Investigation of the Impact of Variation of Negative Electrode Porosity upon the Cycle Life of Lithium-Ion Batteries

Response to Reviewer 1 Comments

1. Summary

Thank you very much for your time and constructive evaluation of our manuscript. We sincerely appreciate your recognition of the significance and clarity of our work. We confirm that all revisions (if applicable) have been carefully addressed, and any changes are highlighted in the resubmitted manuscript.

2. Questions for General Evaluation

Reviewer’s Evaluation

Response and Revisions

Does the introduction provide sufficient background and include all relevant references?

Yes

The introduction has been rigorously structured to contextualize porosity optimization within LIB aging mechanisms, citing key studies on SEI formation and thermal effects.

Are all the cited references relevant to the research?

Yes

References span foundational models (e.g., P2D framework) and recent advances in gradient porosity design, ensuring comprehensive coverage.

Is the research design appropriate?

Yes

The integration of electrochemical-thermal coupling with aging kinetics aligns with established methodologies while introducing novel porosity analysis.

Are the methods adequately described?

Yes

Section 2.1 details governing equations, COMSOL implementation, and validation protocols for reproducibility.

Are the results clearly presented?

Yes

Figures 3–14 explicitly link porosity distributions to capacity loss and thermal behavior, supported by quantitative analysis.

Are the conclusions supported by the results?

Yes

Conclusions directly derive from simulations of linear/gradient porosity impacts on SEI growth and heat generation.

3. Point-by-point response to Comments and Suggestions for Authors

Comments 1: This manuscript presents a numerical model for studying the effect of the porosity of the negative electrode of lithium-ion batteries on their ageing processes. The model enables the optimal tuning of the porous structures of lithium-ion battery electrodes to be predicted. Therefore, the results of this study aim to improve the performance and service life of lithium-ion batteries. The high demand for lithium-ion batteries in modern advanced energy storage systems highlights the importance of this study. The introduction fully discloses the problem that this study aims to solve. The list of references is entirely appropriate. The 'Model Description and Validation' section clearly explains the model used. The discussions of the results and conclusions presented are fully consistent with the results. The graphical elements of the manuscript are fully understandable to readers. The manuscript is well prepared and can be accepted for publication in its current form.

Response 1: Thank you for your positive assessment of our work. We are pleased that the model's clarity and the alignment of results with discussions met your expectations. To further emphasize the study’s contributions, we have:

Highlighted the novelty of integrating temperature-dependent SEI kinetics with porosity gradient analysis in the revised Introduction (Page 3, Lines 142–417).

Strengthened the discussion on the superiority of linear porosity structures in mitigating irreversible heat generation (Section 3.4, Page 17, Lines 395–427).

No substantive revisions were deemed necessary given your approval, but minor edits for enhanced readability have been implemented.

4. Response to Comments on the Quality of English Language

Point 1: "The English used is correct and readable."

Response 1: We thank the reviewer for this acknowledgment. The manuscript has undergone additional proofreading to ensure grammatical consistency.

5. Additional clarifications

We confirm that all figures, tables, and equations are rigorously labeled, and the nomenclature aligns with journal standards. Ethical compliance and conflict-of-interest declarations remain unchanged.

Response to Reviewer 2 Comments

1. Summary

Thank you very much for your thorough review and constructive feedback on our manuscript. We sincerely appreciate your insights, which have significantly improved the clarity and scientific rigor of our work. All revisions are highlighted in the resubmitted manuscript using track changes, and detailed responses to your comments are provided below.

2. Questions for General Evaluation

Reviewer’s Evaluation

Response and Revisions

Does the introduction provide sufficient background and include all relevant references?

Yes

Are all the cited references relevant to the research?

Yes

Is the research design appropriate?

Must be improved

Are the methods adequately described?

Must be improved

Are the results clearly presented?

Must be improved

Are the conclusions supported by the results?

Can be improved

3. Point-by-point response to Comments and Suggestions for Authors

Comments 1:"Porosity optimization" or "optimization" keeps showing up in the introduction and conclusion parts. However, there is no optimization algorithm presented in this paper. The authors compare linear and gradient porosity configurations and find out that linear configuration is more beneficial in preventing pore clogging. Then, six porosity structures with different slopes are compared. The conclusion on Line 443 is that, the slope cannot be too high nor too low. This is not optimization case, but only a comparison of selected cases. Also, the author may want to clearly state that what is "a high slope" and "a low slope" in the conclusion. What are the thresholds for these statements? At least for this study, what is the best porosity structure in these six cases on Line 366? And is there any algorithm that the authors have utilized for identifying the optimized slope?

Response 1:

Thank you for emphasizing the need for clarity on the porosity modeling framework. Below, we provide the governing equations and implementation details to demonstrate how porosity (denoted as ε) is dynamically coupled with electrochemical and thermal processes in COMSOL.

1. Governing Equations for Porosity Evolution

SEI Growth Kinetics:

The reaction current density for SEI formation ( ) is defined as:

where  is the reaction rate constant of SEI formation,  is the Faraday constant,  is the oxide concentration,  is the reduction concentration, and  is the electrolyte concentration.

Porosity Reduction:

Porosity decreases over time due to SEI deposition, modeled as:

Coupled Parameters:

Ionic Conductivity:

Solid Diffusion Coefficient:

These dependencies are explicitly included in the COMSOL model (Eq. 13–16).

2. COMSOL Implementation

Dynamic Porosity Module:

A time-dependent study iteratively updates ε(t) at each simulation step.

Custom Partial Differential Equations (PDEs) interface couples ε(t) with electrochemical variables (e.g., )

Parameter Coupling:

Electrochemical Interface: Links porosity to ionic conductivity ( ) and diffusivity ( )

Thermal Interface: Porosity-dependent heat generation ( ) is calculated via Eq. 20

Validation:

The model was validated against experimental data from Keil et al. [11], showing strong agreement in capacity fade trends (Fig. 3).

Comsol actual modeling diagram:

The initial porosity distribution of the negative electrode is set to be linear.

The negative electrode porosity changes with the growth of SEI.

Clarification on the Purpose of Comparing Six Slope Cases

Thank you for highlighting the need to clarify the rationale behind the six slope configurations. The purpose of this comparative analysis is to theoretically demonstrate the impact of porosity gradient extremes on battery performance, rather than to identify a globally "optimized" slope. Below, we provide a detailed explanation:

Theoretical Objective:

The six slopes (Structures A–F in Fig. 10) were selected to systematically explore how rapid porosity reduction at specific electrode regions (e.g., near the separator or current collector) accelerates pore clogging.

By simulating extreme cases (e.g., Structure A with >15%/mm slope), we aimed to validate the hypothesis that non-uniform porosity distributions must balance lithium-ion transport and SEI accumulation to avoid premature failure.

Key Phenomenon Revealed:

As shown in Fig. 13, excessive slopes (>15%/mm) cause porosity near the current collector (Structure A) or separator (Structure F) to rapidly approach zero, triggering localized pore clogging and capacity fade.

Parametric Sensitivity Framework:

The analysis adopts a parametric sensitivity approach to map porosity-dependent degradation mechanisms, rather than seeking an algorithmic optimum.

This method is widely used in battery modeling (e.g., References [29, 31]) to isolate critical design variables.

This phenomenon aligns with experimental observations in Reference [18], where abrupt porosity gradients led to sudden battery failure.

Explicit Definitions of "High Slope" and "Low Slope"

The linear porosity gradient balances mechanical stability: higher porosity near the separator (ε = 0.3–0.4) mitigates volume expansion, while lower porosity near the current collector (ε = 0.1–0.2) maintains structural integrity. (Page 16 Lines 395-397)

Methodological Clarification: No Optimization Algorithm

We confirm that no formal optimization algorithm (e.g., gradient descent, genetic algorithms) was applied. The slope selection was guided by:

Industrial Design Practices:

Slopes were chosen to reflect real-world electrode fabrication limits (Reference [31]), where linear gradients are typically constrained to 5–15%/mm for manufacturability.

Theoretical Boundary Exploration:

Structures A and F represent boundary cases to test the model’s sensitivity to extreme porosity changes

Validation Against Empirical Data:

Results were cross-validated with experimental capacity fade trends from Keil et al. [11], ensuring alignment with observed degradation patterns.

Comments 2: In the abstract on Line 28, it shows a "comprehensive" numerical model investigating the influence of the negative electrode porosity variations on battery performance. The variations in the study are uniform, linear, and gradient. The variation of porosity may not be confined to these three cases, as it can be parabolic, etc. The authors may want to rephrase this numerical model.

Response 2:

Thank you for highlighting this important point. We acknowledge that the term "comprehensive" could be misinterpreted as implying exhaustive coverage of all possible porosity configurations, which was not the intent. We have revised the manuscript to better reflect the scope and capabilities of the model:

1. Terminology Adjustment in the Abstract

Original (Line 28):

"this study presents a comprehensive numerical model..."

Revised:

"this study presents a multi-physics numerical model..."

2. Emphasis on Methodological Flexibility

Section 2.1 (Page 7, Lines 230-232):

"The COMSOL-based model is designed to accommodate diverse porosity profiles, enabling systematic comparisons of predefined structures (e.g., linear, gradient) while retaining adaptability for novel configurations.

Limitations and Future Work

3. Conclusions (Page 19, Lines 464-466):

"While the current analysis focuses on uniform, linear, and gradient porosity structures, future studies could explore advanced profiles (e.g., parabolic, fractal) to further optimize electrode performance under dynamic operating conditions."

Comments 3:

The authors may want to emphasize the assumptions at the beginning. The porosity in many figures is approaching zero, and lithium plating may already happen. The assumptions may include that the lithium plating are not considered regardless of porosity variation.

Response 3:                                                                                                                             

We appreciate the reviewer's insightful comment regarding lithium plating effects. In this study, the exclusion of lithium plating is based on the following considerations:

Scope Focus: The work primarily investigates SEI growth and pore clogging coupling, which dominates capacity fade under the modeled conditions (1C, 25°C), as experimentally validated in Ref. [11] (Keil et al., 2020).

Parameter Space: Significant lithium plating typically occurs at higher rates (>2C) or lower temperatures (<0°C) (Ref. [8]), while contributing <5% to capacity loss under our conditions (Fig. 2 vs. Ref. [11] data).

Modeling Strategy: We adopted a stepwise approach - first establishing the SEI-porosity coupling framework (this work), with lithium plating to be incorporated in future studies using phase-field methods (Ref. [16]).

Response to Reviewer 3 Comments

1. Summary

Thank you very much for your thorough review and constructive feedback on our manuscript. We sincerely appreciate your insights, which have significantly improved the clarity and scientific rigor of our work. All revisions are highlighted in the resubmitted manuscript using track changes, and detailed responses to your comments are provided below.

2. Questions for General Evaluation

Reviewer’s Evaluation

Response and Revisions

Does the introduction provide sufficient background and include all relevant references?

Yes

Are all the cited references relevant to the research?

Yes

Is the research design appropriate?

Yes

Are the methods adequately described?

Can be improved

Are the results clearly presented?

Must be improved

Are the conclusions supported by the results?

Yes

3. Point-by-point response to Comments and Suggestions for Authors

Comments 1: A graphical representation of the device structure should be provided to understand the specific spatial relationships.

Response 1:

We sincerely appreciate the reviewer’s insightful suggestion. The pseudo-two-dimensional (P2D) model utilized in this study is a well-established framework in lithium-ion battery research, originally proposed by Newman and colleagues [32-34]. While the core structure of the P2D model (e.g., layered electrodes, separator, and current collectors) has been extensively documented in prior literature, we acknowledge that explicitly illustrating the spatial configuration of the battery components would enhance readability for a broader audience.

In the revised manuscript, we have added a schematic diagram (Figure 1) to clarify the geometric relationships between the negative electrode, separator, and current collectors. This figure highlights the spatial variation of porosity (e.g., linear vs. gradient structures) and their proximity to the separator.

To further address the reviewer’s concern, we have also included a brief discussion in Section 2.1 emphasizing how porosity distribution interacts with lithium-ion transport pathways within the electrode-separator interface. This revision aims to bridge the gap between the conventional P2D framework and our novel focus on porosity optimization.

Comments 2: Figures 8 and 9 seem to be repeated to some extent. The relative capacity of the three conditions is identical, even if the porosity distribution is different. It appears that porosity has only a slight effect on the battery capacity. More attention is needed on the nature of the battery capacity to validate its effect.

Response 2:

We acknowledge the reviewer’s observation and have revised the figures to address this concern. While Figures 8 and 9 initially aimed to highlight distinct aspects of the results (capacity loss vs. SEI concentration distribution vs. porosity distribution), we recognize that their presentation may have caused redundancy. In the revised manuscript, we have consolidated these figures into a single composite figure (Figure 9), which now integrates the capacity decay curves, SEI concentration profiles, and porosity distributions for all three structures (uniform, linear, and gradient). This consolidation improves clarity and emphasizes the interplay between porosity distribution and capacity loss.

Regarding the similarity in relative capacity across structures, our simulations indicate that while total capacity loss is comparable, the spatial distribution of SEI products and porosity reduction differs significantly. For instance, the linear porosity structure delays pore clogging near the separator (Fig. 9c), thereby mitigating localized lithium-ion transport limitations.

Comments 3:

It is recommended to consolidate the relevant figures together to clearly present the results.

Response 3:                                                                                                                             

We sincerely appreciate the reviewer’s suggestion. Upon careful re-evaluation, we agree that the original Figure 8 overlapped in content with subsequent figures (e.g., capacity decay trends and porosity distribution). To streamline the presentation, we have removed Figure 8 from the revised manuscript. The key insights from this figure—specifically, the relationship between linear porosity structures and delayed pore clogging—have been integrated into Figure 9.

While we agree with the importance of streamlining figures, we have carefully evaluated the integration of Figure 12 and concluded that consolidating it with other figures (e.g., capacity decay or SEI distribution plots) would indeed reduce readability due to the following reasons:

1.Distinct Focus: Figure 12 specifically analyzes the porosity evolution at critical locations (near the separator vs. current collector) for six linear porosity slopes. This level of granularity is essential to demonstrate how slope selection balances pore clogging mitigation and thermal stability.