# Simulation-Based and Data-Driven Techniques for Quantifying the Influence of the Carbon Binder Domain on Electrochemical Properties of Li-Ion Batteries

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## Abstract

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## 1. Introduction

## 2. Generation and Characterization of Electrode Microstructures

#### 2.1. Stochastic 3D Microstructure Modeling of the Active Material

#### 2.2. Conductive Additive and Binder Model

#### 2.3. Effective Conductivity and Tortuosity Calculations

#### 2.3.1. Numerical Conductivity Simulations

#### 2.3.2. Weighted Geodesic Tortuosity and Relative Path Length

## 3. Electrochemical Simulations

#### 3.1. Microstructure-Resolved Electrochemical Simulations

#### 3.2. Homogenized Electrochemical Model

#### 3.3. Model for the Effective Electronic Conductivity

#### 3.4. Parameters and Operation Conditions

## 4. Results and Discussion

#### 4.1. Effect of CBD on Ionic and Electronic Conductivity

#### 4.1.1. Effective Electronic Conductivity

#### 4.1.2. Effective Ionic Conductivity

#### 4.2. Effect of CBD on Electrochemical Performance

#### 4.2.1. Discharge Curves

#### 4.2.2. Energy Density

#### 4.3. Effect of CBD in Homogenized Cell Models

#### 4.3.1. Effective Electronic Conductivity of CAM Particles

#### 4.3.2. Effective Electronic Conductivity on the Electrode Scale

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Schematic depiction of current pathways at low (

**a**) and high (

**b**) CBD content; (

**c**) equivalent circuits of the serial and parallel connections modeling the effective solid phase conductivity.

**Figure 3.**(

**a**) Dependence of the electronic conductivity of NMC on the SoC [2]; (

**b**) effective electronic conductivity of an electrode with 50 vol-% CAM content, ${d}_{50}=14\phantom{\rule{3.33333pt}{0ex}}$µm particle diameter and varying CBD content. Solid lines represent numerical conductivity simulations (${\kappa}_{s}^{eff,num}$) and dashed lines are a result of the conductivity model based on the geodesic tortuosity computations (${\kappa}_{s}^{eff,geo}$, Equations (20) and (22)).

**Figure 4.**(

**a**) Schematic depiction of current pathways in an electrode with low CBD content. Weighted geodesic tortuosity (

**b**) and relative path length (

**c**) in an electrode with 50 vol-% CAM and ${d}_{50}=14$ µm.

**Figure 5.**Electronic (solid lines) and ionic conductivity (dashed lines) as a function of CBD content for electrodes with ${d}_{50}=10$ µm. All conductivity values in the graph are obtained by numerical simulations (${\kappa}_{s}^{eff,num}$). Corresponding tabulated data sets can be found as the supporting information in Table S1.

**Figure 6.**Discharge simulations for electrode with 60 vol-% CAM (${d}_{50}=10$ µm) and current densities of 3 mA/cm${}^{2}$ (blue lines) and 12 mA/cm${}^{2}$ (red lines). The line style represents different CBD contents.

**Figure 7.**Normalized lithium concentration (SoC) in the CAM at the end of lithiation simulations with 6 mA/cm${}^{2}$. Left: No CBD. Middle: 10 vol-% CBD. Right: 30 vol-% CBD.

**Figure 8.**Energy density as a function of CBD content. (

**a**) Variation of CAM content (${d}_{50}=10$ µm). (

**b**) Variation of particle size (60 vol-% CAM). Corresponding tabulated data sets can be found as the supporting information in Table S2.

**Figure 9.**Comparison of BEST-simulations with the homogenized models for electrodes with 60 vol-% CAM (${d}_{50}=10$ µm) and no additional CBD. Line styles represent the different models for the calculation of the effective particle conductivity.

**Figure 10.**Comparison of predicted energy densities of 3D microstructure-resolved (solid line) and P2D simulations using the standard (dotted) and extended (dashed) model for the solid phase conductivity. Colors represent different CAM contents. (

**a**) current density of 3 mA/cm${}^{2}$ and (

**b**) current density of 12 mA/cm${}^{2}$.

**Table 1.**List of governing equations used for the spatially resolved electrochemical simulations. The effective transport parameters in the CBD and separator are calculated according to Equation (12).

Domain | Phase | Equation | Flux |
---|---|---|---|

Elyte | e |
$$\frac{\partial {c}_{e}}{\partial t}=-\overrightarrow{\nabla}{\overrightarrow{N}}_{e}^{elyte}$$
| ${\overrightarrow{N}}_{e}^{elyte}=-{D}_{e}^{elyte}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\overrightarrow{\nabla}{c}_{e}+\frac{{t}_{+}}{F}{\overrightarrow{J}}_{e}^{elyte}$ |

e |
$$0=-\overrightarrow{\nabla}{\overrightarrow{J}}_{e}^{elyte}$$
| ${\overrightarrow{J}}_{e}^{elyte}=-{\kappa}_{e}^{elyte}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\overrightarrow{\nabla}{\phi}_{e}+{\kappa}_{e}^{elyte}\frac{1-{t}_{+}}{F}\left(\frac{\partial {\mu}_{e}}{\partial {c}_{e}}\right)\overrightarrow{\nabla}{c}_{e}$ | |

AM | s |
$$\frac{\partial {c}_{s}}{\partial t}=-\overrightarrow{\nabla}{\overrightarrow{N}}_{s}$$
| ${\overrightarrow{N}}_{s}=-{D}_{s}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\overrightarrow{\nabla}{c}_{s}$ |

s |
$$0=-\overrightarrow{\nabla}{\overrightarrow{J}}_{s}^{AM}$$
| ${\overrightarrow{J}}_{s}^{AM}=-{\kappa}_{s}^{AM}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\overrightarrow{\nabla}{\Phi}_{s}$ | |

Sep | e |
$$\frac{\partial {c}_{e}}{\partial t}=-\overrightarrow{\nabla}{\overrightarrow{N}}_{e}^{sep}$$
| ${\overrightarrow{N}}_{e}^{sep}=-{D}_{e}^{sep,eff}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\overrightarrow{\nabla}{c}_{e}+\frac{{t}_{+}}{F}{\overrightarrow{J}}_{e}^{sep}$ |

e |
$$0=-\overrightarrow{\nabla}{\overrightarrow{J}}_{e}^{sep}$$
| ${\overrightarrow{J}}_{e}^{sep}=-{\kappa}_{e}^{sep,eff}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\overrightarrow{\nabla}{\phi}_{e}+{\kappa}_{e}^{sep,eff}\frac{1-{t}_{+}}{F}\left(\frac{\partial {\mu}_{e}}{\partial {c}_{e}}\right)\overrightarrow{\nabla}{c}_{e}$ | |

CBD | e |
$$\frac{\partial {c}_{e}}{\partial t}=-\overrightarrow{\nabla}{\overrightarrow{N}}_{e}^{CBD}$$
| ${\overrightarrow{N}}_{e}^{CBD}=-{D}_{e}^{sep,eff}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\overrightarrow{\nabla}{c}_{e}+\frac{{t}_{+}}{F}{\overrightarrow{J}}_{e}^{sep}$ |

e |
$$0=-\overrightarrow{\nabla}{\overrightarrow{J}}_{e}^{CBD}$$
| ${\overrightarrow{J}}_{e}^{CBD}=-{\kappa}_{e}^{CBD,eff}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\overrightarrow{\nabla}{\phi}_{e}+{\kappa}_{e}^{CBD,eff}\frac{1-{t}_{+}}{F}\left(\frac{\partial {\mu}_{e}}{\partial {c}_{e}}\right)\overrightarrow{\nabla}{c}_{e}$ | |

s |
$$0=-\overrightarrow{\nabla}{\overrightarrow{J}}_{s}^{CBD}$$
| ${\overrightarrow{J}}_{s}^{CBD}=-{\kappa}_{s}^{CBD,eff}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\overrightarrow{\nabla}{\Phi}_{s}$ | |

CC | s |
$$0=-\overrightarrow{\nabla}{\overrightarrow{J}}_{s}^{CC}$$
| ${\overrightarrow{J}}_{s}^{CC}=-{\kappa}_{s}^{CC}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\overrightarrow{\nabla}{\Phi}_{s}$ |

Domain | Phase | Equation | Flux |
---|---|---|---|

Elyte | e |
$$\frac{\partial \left({\epsilon}_{e}{c}_{e}\right)}{\partial t}=-\overrightarrow{\nabla}{\overrightarrow{N}}_{e}^{elyte}+{a}_{v}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\frac{{i}_{inter}}{F}$$
| ${\overrightarrow{N}}_{e}^{elyte}=-{D}_{e}^{elyte,eff}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\overrightarrow{\nabla}{c}_{e}+\frac{{t}_{+}}{F}{\overrightarrow{J}}_{e}^{elyte}$ |

e |
$$0=-\overrightarrow{\nabla}{\overrightarrow{J}}_{e}^{elyte}+{a}_{v}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{i}_{inter}$$
| ${\overrightarrow{J}}_{e}^{elyte}=-{\kappa}_{e}^{elyte,eff}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\overrightarrow{\nabla}{\phi}_{e}+{\kappa}_{e}^{elyte,eff}\frac{1-{t}_{+}}{F}\left(\frac{\partial {\mu}_{e}}{\partial {c}_{e}}\right)\overrightarrow{\nabla}{c}_{e}$ | |

AM | s |
$$0=-\overrightarrow{\nabla}{\overrightarrow{J}}_{s}^{AM}-{a}_{v}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{i}_{inter}$$
| ${\overrightarrow{J}}_{s}^{AM,eff}=-{\kappa}_{s}^{AM,eff}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\overrightarrow{\nabla}{\Phi}_{s}$ |

s |
$$\frac{\partial {c}_{s}}{\partial t}=\frac{1}{{r}^{2}}\frac{\partial}{\partial r}\left({r}^{2}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\overrightarrow{N}}_{s}\right)$$
| ${\overrightarrow{N}}_{s}=-{D}_{s}\frac{\partial {c}_{s}}{\partial r}$ | |

Sep | e |
$$\frac{\left({\epsilon}_{e}\partial {c}_{e}\right)}{\partial t}=-\overrightarrow{\nabla}{\overrightarrow{N}}_{e}^{sep}$$
| ${\overrightarrow{N}}_{e}^{sep}=-{D}_{e}^{sep,eff}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\overrightarrow{\nabla}{c}_{e}+\frac{{t}_{+}}{F}{\overrightarrow{J}}_{e}^{sep}$ |

e |
$$0=-\overrightarrow{\nabla}{\overrightarrow{J}}_{e}^{sep}$$
| ${\overrightarrow{J}}_{e}^{sep}=-{\kappa}_{e}^{sep,eff}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\overrightarrow{\nabla}{\phi}_{e}+{\kappa}_{e}^{sep,eff}\frac{1-{t}_{+}}{F}\left(\frac{\partial {\mu}_{e}}{\partial {c}_{e}}\right)\overrightarrow{\nabla}{c}_{e}$ |

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**MDPI and ACS Style**

Knorr, T.; Hein, S.; Prifling, B.; Neumann, M.; Danner, T.; Schmidt, V.; Latz, A.
Simulation-Based and Data-Driven Techniques for Quantifying the Influence of the Carbon Binder Domain on Electrochemical Properties of Li-Ion Batteries. *Energies* **2022**, *15*, 7821.
https://doi.org/10.3390/en15217821

**AMA Style**

Knorr T, Hein S, Prifling B, Neumann M, Danner T, Schmidt V, Latz A.
Simulation-Based and Data-Driven Techniques for Quantifying the Influence of the Carbon Binder Domain on Electrochemical Properties of Li-Ion Batteries. *Energies*. 2022; 15(21):7821.
https://doi.org/10.3390/en15217821

**Chicago/Turabian Style**

Knorr, Tobias, Simon Hein, Benedikt Prifling, Matthias Neumann, Timo Danner, Volker Schmidt, and Arnulf Latz.
2022. "Simulation-Based and Data-Driven Techniques for Quantifying the Influence of the Carbon Binder Domain on Electrochemical Properties of Li-Ion Batteries" *Energies* 15, no. 21: 7821.
https://doi.org/10.3390/en15217821