The microstructure of lithium-ion battery (LIB) electrodes significantly impacts their performance, influencing effective electronic and ionic transport, electrochemical kinetics, and mechanical properties. High-rate charging, crucial for automotive LIBs, necessitates electrode designs that overcome transport limitations. Physics-based models can accelerate optimization by linking microstructure and performance, but accurate input parameters are crucial. Porous electrode structures increase the specific interfacial area, enhancing active material capacity at high rates. Most battery models use a macroscopic treatment, employing homogenized parameters like porosity (ε), specific active surface area (ASA, *a*), and tortuosity factor (τ) to represent the geometry. However, these parameters can be inhomogeneous and anisotropic. Porosity can be measured directly (3D imaging) or indirectly (helium pycnometry). Surface area is often determined using adsorption techniques (BET). Tortuosity factor determination is more challenging, requiring transport experiments or simulations. This paper focuses on determining tortuosity factors relevant to LIB electrodes, using the standard definition: τ = PeffKoDo / PoKeffDeff = NM (where NM is the MacMullin number; Po, Ko, and Do are intrinsic electrolyte properties; and Peff, Keff, and Deff are observed effective values). The paper will compare two experimental techniques: the Restricted Diffusion Method (RDM) and the Symmetric Cell Method (SCM).

The literature review discusses existing methods for determining tortuosity in porous materials, focusing on the Restricted Diffusion Method (RDM) and the Symmetric Cell Method (SCM). RDM, a time-domain approach, measures diffusion by applying a bias to generate a concentration gradient and monitoring the relaxation step after bias removal. SCM, a frequency-domain technique, measures the high-frequency impedance response of a symmetric cell to calculate the effective ionic conductivity. The limitations of these methods for electronically conducting porous materials like battery electrodes are highlighted. The paper then introduces adapted versions of these methods: the eRDM (developed by Thorat et al.) and eSCM (developed by Landesfeind et al. and Malifarge et al.). These adapted methods account for the electronic conductivity of battery electrodes and their limitations are discussed. Finally, the role of 3D imaging techniques like X-ray computed tomography (XCT) and focused ion beam scanning electron microscopy (FIB-SEM) in capturing electrode geometry and extracting microstructural properties is discussed, mentioning the TauFactor platform as an open-source tool for analyzing tomographic data and calculating tortuosity factors. The limitations of these methods in capturing fine features and different phases are acknowledged.

This study develops a new frequency-domain solver within the TauFactor framework to replicate the eSCM approach and compare it with the eRDM approach. The analysis involves simulating both eRDM and eSCM concepts to extract tortuosity factors (τ for eRDM and τe for eSCM) and MacMullin numbers (NM and NMe) from various 2D and 3D microstructures. The simulations use numerically generated microstructures. The conventional tortuosity factor (τ) considers the steady-state flux through "through-pores", while the electrode tortuosity factor (τe) accounts for the contribution of "dead-end" pores to the transport processes relevant to battery electrodes. Two-dimensional simulations illustrate the difference between eRDM and eSCM for microstructures with only through-pores or only dead-end pores. Further 2D simulations explore microstructures with both through and dead-end pores, varying the morphology of the dead-end pores to assess their impact on τ and τe. Three-dimensional simulations are performed on microstructures with varying particle sizes and porosities to investigate more realistic electrode geometries. The impedance spectra simulated by eSCM are analyzed to assess deviations from the Transmission Line Model (TLM) response, offering insights into the homogeneity of the microstructure and the applicability of porous electrode theory. The TauFactor platform is used for simulations, and the methodology is detailed in the paper, including equations for calculating the tortuosity factors and analyzing impedance spectra. The parameters used in eSCM simulations are provided in Table 1.

The key findings demonstrate a significant discrepancy between the conventional tortuosity factor (τ) obtained from eRDM and the electrode tortuosity factor (τe) obtained from eSCM, particularly for microstructures with a substantial fraction of dead-end pores. In simple 2D cases, eRDM yields infinite τ for dead-end pores, while eSCM provides τe < 1. In more complex 2D and 3D microstructures, the presence of dead-end pores causes a significant difference between τ and τe. The eRDM approach, based on steady-state flow through the entire structure, fails to capture the contribution of dead-end pores to ion transport in real battery electrodes where electrochemical reactions occur at the interfacial area of all pores, including dead-end pores. The electrode tortuosity factor (τe), reflecting the ease of access to the solid/liquid interface, is shown to be more relevant for predicting battery electrode performance. Simulations reveal that dead-end pores can significantly affect τe, highlighting the importance of considering pore morphology in electrode design. Analysis of simulated impedance spectra using eSCM shows deviations from the idealized TLM response when the control volume is too small or the microstructure is inhomogeneous. The study suggests that a graded porosity design (higher porosity near the separator) could improve electrode capacity. The results indicate that the electrode tortuosity factor (τe) should be preferred over the conventional tortuosity factor (τ) in models such as Newman's P2D model for accurate parametrization. Finally, the study shows how impedance spectroscopy can quantitatively assess deviations from porous electrode theory and provide insights into the homogeneity of electrode microstructures.

The study highlights a critical discrepancy between two widely used methods for determining tortuosity in porous battery electrodes. The conventional tortuosity factor (τ), based on eRDM, focuses on steady-state flow through the entire structure, ignoring the crucial role of dead-end pores in ion transport to the active surface area. The newly proposed electrode tortuosity factor (τe), derived from eSCM, more accurately reflects the access to the electrochemically active surface area, making it a more suitable parameter for battery models. The discrepancy between τ and τe emphasizes the need for a more nuanced understanding of transport processes in porous battery electrodes, especially regarding the influence of dead-end pores. The findings have significant implications for optimizing electrode design and enhancing battery performance. This work provides a strong argument for replacing the conventional tortuosity factor with the electrode tortuosity factor in electrochemical models, leading to more accurate predictions of battery behavior.

This study reveals that the conventional "flow through" tortuosity factor (τ) is inadequate for characterizing porous Li-ion battery electrodes. The introduced electrode tortuosity factor (τe) more accurately represents transport in such electrodes. The importance of considering dead-end pores in electrode design is highlighted. The developed eSCM simulation tool, integrated into the open-source TauFactor platform, offers a valuable tool for the battery community. Future work should focus on applying these concepts to real electrode microstructural data to further validate findings and explore optimal electrode designs.

The study relies on numerically generated microstructures, which may not fully capture the complexity of real electrode structures. The accuracy of the results depends on the accuracy of the numerical models and assumptions made in the simulations, such as the assumptions made in the TLM for the eSCM analysis. The analysis of impedance spectra assumes simplified conditions such as a pure capacitance at the double-layer, which may not always be true in real systems. Future studies should involve experimental validation using real electrode materials and consider broader ranges of microstructural parameters and operating conditions.