# Digitalization Platform for Mechanistic Modeling of Battery Cell Production

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## Abstract

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## 1. Introduction

#### 1.1. Motivation for a Model-Based Digitalization Platform

#### 1.2. Existing Approaches to Make Cause–Effect Relations Transparent

## 2. Methodology

#### 2.1. Concept of the Modeling Framework

#### 2.2. Process Chain Model (I)

#### 2.2.1. Meta-Modeling of Bottom-Up Models

#### 2.2.2. Combining Process Models to Formulate the Process Chain Model

#### 2.3. Battery Cell Model (II)

_{l}). Various models can be found in the literature to calculate the additional parameters based on the structural parameters. While conventional approaches such as the Bruggeman correlation for calculating the tortuosity only provide a rough estimate of the physical property of the electrode [34], more recent approaches were presented by [32,33] that reflect the three-dimensional structure of the electrode, thus providing a more accurate description of the additional parameters. The battery model parameters can then be used in a DFN-type model to compute the performance of the battery cells.

#### 2.4. Analysis (III)

## 3. Use Case

#### 3.1. Exemplary Implementation

^{−2}and a coating density of 2.95 g cm

^{−3}is created virtually. A normal distribution with a standard deviation of 0.5% was assumed for the formulation parameters and process parameters for all processes, resulting in subcell deviation. Material-intrinsic parameters such as the particle density of the individual materials (active material, conductive additive, and polymer binder) were set constant. Cell-to-cell deviations due to temporal changes in the production processes were not regarded. A product-oriented workflow of the combined process chain and battery cell models was selected where the mass loading and the energy density were predefined by the reference cell. Subsequently, the influence of propagating uncertainties along the process chain and into the battery cell were investigated, and a sensitivity analysis was conducted as part of the analysis module. Since the process chain model exclusively focuses on the production of cathodes, only a cathode half-cell P2D model was implemented to evaluate the performance properties. The sensitivity analysis is an extension of the previously published work in [9] and focuses on the composition of the dry mixture and slurry as well as the processing of the coating on the substrate.

#### 3.2. Process Chain Model

^{−2}were computed to provide a sufficient data basis for the meta-model. The DEM simulation at each of the seven different input wet mass loadings was repeated five times for stochastically varying particle configurations of the cathode. The resulting porosity data were fitted with a linear, quadratic, and exponential regression function. The exponential function shows the highest agreement with the DEM data, i.e., a higher coefficient of determination R

^{2}or a lower root mean square error RMSE and a corrected Akaike information criterion AIC

_{c}, which is why it was selected as the meta-model for the drying process.

#### 3.3. Battery Cell Model

#### 3.4. Analysis

^{−2}and 2.95 g cm

^{−3}, respectively. A standard deviation of 1% was assumed for the formulation and process parameters. Material-intrinsic parameters such as the densities were considered constant. A comprehensive overview of the process and structural parameter values can be found in Table A2 in Appendix A. The structural parameters were transferred to the structure surrogate model to compute the battery model parameters. Finally, the battery model parameters were utilized to determine the performance properties of the battery cell. The volumetric energy density and the discharge capacity for the given battery cell were 1892 ± 8 Wh ${\mathrm{L}}_{\mathrm{electrode}}^{-1}$ and 2.41 ± 0.07 mAh cm

^{−2}, respectively, for a current density of 1 mA cm

^{−2}(approx. 0.4 C). The volumetric energy density is solely related to the volume of the cathode coating and not the volume of the battery cell.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Table A1.**Overview of included formulation, process, and structural parameters for the process chain model.

Parameter | Unit | Results | |
---|---|---|---|

Formulation | ${\omega}_{AM}$ | - | 0.94 ± 1.0% |

${\omega}_{CA}$ | - | 0.03 ± 1.0% | |

${\omega}_{PB}$ | - | 0.03 ± 1.0% | |

${x}_{solid}$ | - | 0.7 ± 1.0% | |

Raw material | ${\rho}_{AM}$ | g cm^{−3} | 4.75 |

${\rho}_{CA}$ | g cm^{−3} | 1.80 | |

${\rho}_{PB}$ | g cm^{−3} | 1.70 | |

${\rho}_{solvent}$ | g cm^{−3} | 1.03 | |

Mixing | ${\rho}_{solids}$ | g cm^{−3} | 4.31 ± 0.9% |

Dispersion | ${\rho}_{slurry}$ | g cm^{−3} | 2.20 ± 1.2% |

Coating | $\dot{V}$ | cm^{3} h^{−1} | 696 ± 1.0% |

v | m min^{−1} | 1.0 ± 1.0% | |

w | mm | 125 ± 1.0% | |

${h}_{wet}$ | µm | 92.8 ± 1.7% | |

${M}_{wet}$ | mg cm^{−2} | 20.5 ± 2.1% | |

Drying | ${\u03f5}_{dry}$ | % | 43.9 ± 0.3% |

${m}_{dry}$ | mg cm^{−2} | 14.3 ± 2.8% | |

${h}_{dry}$ | µm | 59.2 ± 2.6% | |

${\rho}_{coa}$ | g cm^{−3} | 2.42 ± 0.9% | |

Calendering | ${q}_{L}$ | N mm^{−1} | 84 ± 1.0% |

${\u03f5}_{min}$ | % | 17.6 ± 0.3% | |

${\gamma}_{c}$ | N mm^{−1} | 129 ± 2.4% | |

${\u03f5}_{cal}$ | % | 31.5 ± 0.4% | |

${h}_{cal}$ | µm | 48.5 ± 3.0% | |

${\rho}_{coa}$ | g cm^{−3} | 2.95 ± 0.9% | |

${X}_{AM}$ | - | 0.58 ± 0.3% | |

${X}_{CA}$ | - | 0.05 ± 1.3% | |

${X}_{PB}$ | - | 0.05 ± 1.2% |

**Table A2.**Battery model parameters used in the applied model previously presented in [9]. The diffusion coefficient in the electrolyte, ionic conductivity, and transference number are a function of the electrolyte concentration. (* CBM—carbon binder matrix).

Parameter | Symbol | Unit | Separator | Cathode |
---|---|---|---|---|

Coating thickness | $\delta $ | m | - | 4.85 ×
10^{−5} |

Porosity | $\u03f5$ | - | 0.5 | 0.315 |

Particle size | ${R}_{p}$ | m | - | 5.5 × 10^{−6} |

Tortuosity | $\tau $ | - | 1 | 5.9449 |

Maximum capacity solid | ${c}_{max}$ | mol m^{−3} | - | 4.3221 × 10^{4} |

Initial capacity solid | ${c}_{0}$ | mol m^{−3} | - | 1.5467 × 10^{4} |

Initial capacity electrolyte | ${c}_{e}$ | mol m^{−3} | - | 1 × 10^{3} |

Diffusion coefficient solid | ${D}_{s}$ | m^{2} s^{−1} | - | 9.5594 × 10^{−15} |

Diffusion coefficient electrolyte | ${D}_{e}$ | m^{2} s^{−1} | - | $f\left({c}_{Li}\right)$ [48] |

Electronic conductivity AM | ${\kappa}_{AM}$ | S m^{−1} | - | 0.0309 |

Electronic conductivity CBM * | ${\kappa}_{CBM}$ | S m^{−1} | - | 760 |

Ionic conductivity | ${\kappa}_{e}$ | S m^{−1} | - | $f\left({c}_{Li}\right)$ [48] |

Transference number | ${t}_{p}$ | - | - | $f\left({c}_{Li}\right)$ [48] |

Charge transfer coefficient | $\alpha $ | - | - | 0.5 |

Reaction rate constant | k | s^{−1} | - | 1.1717 × 10^{−9} |

Double layer capacitance | ${C}_{DL}$ | F m^{−2} | - | 0.2 |

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**Figure 1.**Schematic concept of the modeling framework consisting of three modules: (

**I**) process chain model, (

**II**) battery cell model, and (

**III**) analysis.

**Figure 2.**Simulation workflow for a production-oriented and product-oriented utilization of the modeling framework.

**Figure 3.**Process chain model implemented by a concatenation of model containers, which are connected via structural parameters. Exemplary models are presented for the process steps of mixing, coating, and drying.

**Figure 4.**Meta-modeling approach consisting of (

**I**) virtual design of experiment, (

**II**) fit of meta-models, and (

**III**) validation and meta-model selection.

**Figure 5.**Cell-to-cell and subcell deviations within different segments (S1–S10) throughout the process.

**Figure 10.**Sankey diagram of the sensitivities between the different parameter domains focusing on the structural parameters that are required for the battery cell model.

**Table 1.**Sensitivity indices (total, uncorrelated, and correlated) of the battery cell model for the input parameter weight fractions of the active material ${\omega}_{AM}$ and conductive additive ${\omega}_{CA}$, dry mass loading ${m}_{dry}$, and coating density ${\rho}_{cal}$ for a discharge rate of 1 C.

Vol. Energy Density | Discharge Capacity | |||||
---|---|---|---|---|---|---|

${\mathit{S}}_{\mathit{i}}$ | ${\mathit{S}}_{\mathit{i}}^{\mathit{U}}$ | ${\mathit{S}}_{\mathit{i}}^{\mathit{C}}$ | ${\mathit{S}}_{\mathit{i}}$ | ${\mathit{S}}_{\mathit{i}}^{\mathit{U}}$ | ${\mathit{S}}_{\mathit{i}}^{\mathit{C}}$ | |

${\omega}_{AM}$ | 0.76 | 0.77 | −0.01 | 0.25 | 0.23 | 0.02 |

${\omega}_{CA}$ | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

${m}_{dry}$ | 0.00 | 0.00 | 0.00 | 0.75 | 0.72 | 0.03 |

${\rho}_{cal}$ | 0.24 | 0.24 | 0.00 | 0.00 | 0.00 | 0.00 |

∑ | 1.00 | 1.01 | −0.01 | 1.00 | 0.95 | 0.05 |

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## Share and Cite

**MDPI and ACS Style**

Thomitzek, M.; Schmidt, O.; Ventura Silva, G.; Karaki, H.; Lippke, M.; Krewer, U.; Schröder, D.; Kwade, A.; Herrmann, C.
Digitalization Platform for Mechanistic Modeling of Battery Cell Production. *Sustainability* **2022**, *14*, 1530.
https://doi.org/10.3390/su14031530

**AMA Style**

Thomitzek M, Schmidt O, Ventura Silva G, Karaki H, Lippke M, Krewer U, Schröder D, Kwade A, Herrmann C.
Digitalization Platform for Mechanistic Modeling of Battery Cell Production. *Sustainability*. 2022; 14(3):1530.
https://doi.org/10.3390/su14031530

**Chicago/Turabian Style**

Thomitzek, Matthias, Oke Schmidt, Gabriela Ventura Silva, Hassan Karaki, Mark Lippke, Ulrike Krewer, Daniel Schröder, Arno Kwade, and Christoph Herrmann.
2022. "Digitalization Platform for Mechanistic Modeling of Battery Cell Production" *Sustainability* 14, no. 3: 1530.
https://doi.org/10.3390/su14031530