# Thermo-Electro-Mechanical Modeling and Experimental Validation of Thickness Change of a Lithium-Ion Pouch Cell with Blend Positive Electrode

^{*}

*Batteries*)

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Investigated Cell

#### 2.2. Experimental Methodology

#### 2.3. Mechanical Model

#### 2.4. Simulation Methodology

#### 2.5. Tabulated Molar Volumes in Cantera

## 3. Model Parameterization

#### 3.1. Volume Changes of Active Materials

_{2}) of LCO by XRD experiments. Thus, data for the complete lithium stoichiometric range of LCO, is available as displayed in Figure 2a.

Material | Source | Primary Source | Comment |
---|---|---|---|

Graphite | |||

1 | Ref. [26] Figure 2a, Curve 1 | [62,63] | Calculated in [26] from average d-spacing |

2 | Ref. [26] Figure 2a, Curve 2 | [64,65] | Calculated in [26] from average d-spacing |

3 | Ref. [26] Figure 2a, Curve 3 | [65,66] | Calculated in [26] from average d-spacing |

4 | Ref. [26] Figure 2a, Curve 4 | [67] | Calculated in [26] from average d-spacing |

5 | Ref. [26] Figure 2a, Curve 5 | [65,68] | Calculated in [26] from average d-spacing |

6 | Ref. [40] Table A1 | [69,70] | |

7 | Ref. [32] Figure 3a | [71] | Average of charge and discharge |

8 | Ref. [55] Table S1, 2nd cycle | - | $\mathrm{Average}\mathrm{of}\mathrm{charge}\mathrm{and}\mathrm{discharge},\mathrm{linearly}\mathrm{extrapolated}\mathrm{to}X=1$ |

9 | Ref. [55] Table S1, 3rd cycle | - | $\mathrm{Average}\mathrm{of}\mathrm{charge}\mathrm{and}\mathrm{discharge},\mathrm{linearly}\mathrm{extrapolated}\mathrm{to}X=1$ |

NCA | |||

Ref. [72] Figure 3c | - | Average of charge and discharge | |

LCO | |||

Ref. [73] Table 1 | [56] | $\mathrm{Outliers}\mathrm{at}X=0.9$$\mathrm{and}X=0.51$ are not considered |

#### 3.2. Thermal Parameters

^{2}K) with a standard deviation of 0.58 W/(m

^{2}K) (again averaged for all five cooling phases). Data are shown exemplarily for the first cooling phase in Figure 5b. Note that the linear fit is only applied for times $t<450\mathrm{s}$, because the noise in the signal becomes more significant after this and in turn the signal deviates from its exponential characteristic. With the derived heat transfer coefficient and the thermal conductivity taken from the reference model, we furthermore performed a Biot number check to verify the assumption of a thermally thin cell [48]. The Biot number $Bi$ was calculated as

#### 3.3. Model Adjustment for Calendaric Ageing

## 4. Results and Discussion

#### 4.1. Thermo-Electrochemical Behavior

_{V}: 0.01 V (0.05 C), 0.03 V (1 C), 0.04 V (2 C), 0.07 V (5 C), 0.09 V (10 C); and for MAE

_{T}: 0.03 K (0.05 C), 0.11 K (1 C), 0.16 K (2 C), 0.49 K (5 C), 1.29 K (10 C). The MAE confirms the trend that is visible in Figure 8: With higher C-rate, the error is increasing. It is worthwhile noting that all conditions were simulated with the identical model, that is, there were no parameter adaptions for different C-rates. The model accuracy observed here is typical for P2D and P3D models [33,34,46]. It is owed to the simplifications in the underlying model assumption, in particular, the homogenization on all three pseudo scales that ignores the significant structural complexities of particle shape, electrode microstructure, and cell design. In this light, the observed magnitude of the MAE can be considered rather small.

#### 4.2. Mechanical Behavior

#### 4.3. Displacement Components

#### 4.4. Spatial Profiles

## 5. Conclusions and Outlook

- Cell expansion at low C-rates is dominated by intercalation-induced swelling.
- At high C-rates, thermal expansion resulting from electrochemical heating significantly contributes to the cell thickness change, leading to peaks in the displacement vs. time and an increased hysteresis in the displacement vs. charge throughput.
- In the investigated cell, displacement is dominated by the graphite NE. The LCO-NCA blend PE shows almost no displacement due to the opposite expansion behavior of LCO and NCA. These results show that electrode blends can be tailored with respect to their mechanical properties.
- At high C-rates, the expansion shows a significant spatial gradient in the direction along the electrode thickness. This can lead to spatially inhomogeneous electrode aging.

## Supplementary Materials

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Modeling Domain and Main Assumptions

^{+}and PF

_{6}

^{−}ions in the liquid electrolyte and electrons in the solid electrode components is modeled in one dimension along the thickness of the electrode pair. Again, this is perpendicular to the electrode sheet area. For ion transport, we describe species fluxes due to migration and diffusion with a Nernst-Planck approach with concentration- and temperature-dependent diffusion coefficients. Electronic conductivity within the electrodes is assumed high and not rate-limiting. On the particle level (z scale in Figure A1), the diffusion of intercalated lithium atoms in the bulk of the AM particles is modeled using a simple Fickian diffusion approach. The diffusion coefficients are assumed concentration and temperature dependent.

#### Appendix A.2. Model Equations and Parameters

Macroscale (x Direction): Heat Transport in Cell | |

Energy conservation | $\rho {c}_{p}\frac{\partial T}{\partial t}=\frac{\partial}{\partial x}\left(\lambda \frac{\partial T}{\partial x}\right)+{\dot{q}}^{\mathrm{V}}$ |

Heat flux at cell surface | ${J}_{q}={\alpha}_{\mathrm{surf}}\left(T-{T}_{\mathrm{amb}}\right)+\u03f5{\sigma}_{\mathrm{SB}}({T}^{4}-{T}_{\mathrm{amb}}^{4})$ |

Total heat sources | ${\dot{q}}^{\mathrm{V}}=\frac{{A}_{\mathrm{e}}}{{V}_{\mathrm{cell}}}\left(\underset{0}{\overset{{L}_{\mathrm{EP}}}{\int}}\left({\dot{q}}_{\mathrm{chem}}\left(y\right)+{\dot{q}}_{\mathrm{ohm}}\left(y\right)\right)\mathrm{d}y+{R}_{\mathrm{cc}}{i}^{2}\right)$ |

Chemistry heat source | ${\dot{q}}_{\mathrm{chem}}={\sum}_{n=1}^{{N}_{\mathrm{r}}}\left({r}_{n}{A}_{n}^{\mathrm{V}}\right(-\Delta {H}_{n}+F{\nu}_{\mathrm{e},n}\Delta {\varphi}_{n}\left)\right)$ |

Ohmic heating | ${\dot{q}}_{\mathrm{ohm}}={\sigma}_{\mathrm{elyt}}\xb7{\left(\frac{\partial {\varphi}_{\mathrm{elyt}}}{\partial y}\right)}^{2}$ |

Mechanical displacement of complete cell | $\Delta {L}_{\mathrm{cell}}={N}_{\mathrm{EP}}\xb7\Delta {L}_{\mathrm{EP}}$ |

Mesoscale (y direction): Mass and charge transport and mechanics in electrode pair | |

Mass conservation of species i | $\frac{\partial \left({\epsilon}_{\mathrm{elyt}}{c}_{i}\right)}{\partial t}=-\frac{\partial {J}_{i}}{\partial y}+{\dot{s}}_{i}^{\mathrm{V}}+{\dot{s}}_{\u0307i,\mathrm{DL}}^{\mathrm{V}}$ |

Charge conservation | ${C}_{\mathrm{DL}}^{V}\frac{\partial \left(\Delta \varphi \right)}{\partial t}=\sum _{i}{z}_{i}F\frac{\partial {J}_{i}}{\partial y}-{i}_{F}^{\mathrm{V}}$ |

Species fluxes: Nernst-Planck | ${J}_{i}=-{D}_{i}^{\mathrm{eff}}\frac{\partial {c}_{i}}{\partial y}-\frac{{z}_{i}F}{RT}{c}_{i}{D}_{i}^{\mathrm{eff}}\frac{\partial {\varphi}_{\mathrm{elyt}}}{\partial y}$ |

Mechanical displacement of individual finite volume k | $\Delta {L}_{k}=\left(\frac{{\sigma}_{\mathrm{n}}}{{E}_{k}}+{\alpha}_{\mathrm{th},k}\Delta T+\sum _{j=1}^{{N}_{\mathrm{A}\mathrm{M}}}{\epsilon}_{j,k}^{0}\xb7\frac{\Delta {V}_{\mathrm{m},j,k}}{{V}_{\mathrm{m},j,k}^{0}}\right)\xb7{L}_{k}^{0}$ |

Mechanical displacement of electrode pair | $\Delta {L}_{\mathrm{EP}}={\displaystyle \sum _{k=1}^{{N}_{\mathrm{FV}}}}\Delta {L}_{k}$ |

Microscale (z direction): Mass transport in active material particles | |

Mass conservation (Fick’s 2nd law) | $\frac{\partial {c}_{\mathrm{Li},\mathrm{AM}}}{\partial t}=\frac{1}{{z}^{2}}\frac{\partial}{\partial z}\left({z}^{2}{D}_{\mathrm{Li},\mathrm{AM}}\frac{\partial {c}_{\mathrm{Li},\mathrm{AM}}}{\partial z}\right)$ |

Chemical kinetics and thermodynamics * | |

Interfacial rate of electrochemical reaction n (Butler-Volmer) | ${r}_{n}=\frac{{{i}_{n}}^{0}}{F}\left[\mathrm{exp}\left(\frac{{\alpha}_{\mathrm{c}}zF}{RT}{\eta}_{\mathrm{act},n}\right)-\mathrm{exp}\left(-\frac{\left({1-\alpha}_{\mathrm{c}}\right)zF}{RT}{\eta}_{\mathrm{act},n}\right)\right]$ |

Exchange current density | ${i}_{n}^{0}={i}_{n}^{00}\xb7\mathrm{exp}\left(-\frac{{E}_{\mathrm{act},n}}{RT}\right)\xb7{\displaystyle \prod _{i=1}^{{N}_{\mathrm{R}}}}{\left(\frac{{c}_{i}}{{c}_{i}^{0}}\right)}^{\left(1-{\alpha}_{\mathrm{c},n}\right)}\xb7{\displaystyle \prod _{i=1}^{{N}_{\mathrm{P}}}}{\left(\frac{{c}_{i}}{{c}_{i}^{0}}\right)}^{{\alpha}_{\mathrm{c},n}}$ |

Overpotential | ${\eta}_{\mathrm{act},n}=\Delta {\varphi}^{\mathrm{eff}}-\Delta {\varphi}_{n}^{\mathrm{eq}}=\Delta \varphi -{R}_{\mathrm{SEI}}^{\mathrm{V}}{i}_{\mathrm{F}}^{\mathrm{V}}-\Delta {\varphi}_{n}^{\mathrm{eq}}$ |

Species source terms | ${\dot{s}}_{i}^{\mathrm{V}}={\sum}_{n=1}^{{N}_{\mathrm{r}}}\left({\nu}_{i}{r}_{n}{A}_{n}^{\mathrm{V}}\right)$ |

Equilibrium potential (Nernst equation) | $\Delta {\varphi}_{n}^{\mathrm{eq}}=-\frac{\Delta {G}_{n}^{0}}{zF}-\frac{\mathit{RT}}{\mathit{zF}}\mathrm{ln}\left({\displaystyle \prod _{i=1}^{{N}_{\mathrm{R}},{N}_{\mathrm{P}}}}{\left(\frac{{c}_{i}}{{c}_{i}^{0}}\right)}^{{\nu}_{i}}\right)$ |

Gibbs reaction energy | $\Delta {G}_{n}^{0}=\sum _{i=1}^{{N}_{\mathrm{R}},{N}_{\mathrm{P}}}{\nu}_{i}{\mu}_{\mathrm{i}}^{0}$ |

Standard-state chemical potential | ${\mu}_{i}^{0}={h}_{i}^{0}-T{s}_{i}^{0}+\left(p-{p}_{\mathrm{ref}}\right){\overline{V}}_{i}$ |

Current, voltage, potentials | |

Cell voltage | $E={\varphi}_{\mathrm{elde},\mathrm{ca}}-{\varphi}_{\mathrm{elde},\mathrm{an}}-i\xb7{R}_{\mathrm{cc}}$ |

Temperature dependence of current collection resistance | ${R}_{\mathrm{cc}}={R}_{\mathrm{cc}}^{0}\left[1+{\alpha}_{\mathrm{c}\mathrm{c}}\left(T-293\mathrm{K}\right)\right]$ |

Cell current | ${I}_{\mathrm{cell}}=\frac{{A}_{\mathrm{e}}}{{V}_{\mathrm{cell}}}\xb7\underset{y=0}{\overset{{L}_{\mathrm{electrode}}}{\int}}\left({i}_{\mathrm{F}}^{\mathrm{V}}+{i}_{\mathrm{DL}}^{\mathrm{V}}\right)\mathrm{d}y$ |

Faradaic current density | ${i}_{\mathrm{F}}^{\mathrm{V}}=F{\dot{s}}_{\mathrm{e}}^{\mathrm{V}}={\sum}_{n=1}^{{N}_{\mathrm{r}}}F\left({\nu}_{\mathrm{e},n}{r}_{n}{A}_{n}^{\mathrm{V}}\right)$ |

Double layer current density | ${i}_{\mathrm{DL}}^{\mathrm{V}}={C}_{\mathrm{DL}}^{\mathrm{V}}\frac{\mathrm{d}\left(\Delta \Phi \right)}{\mathrm{d}t}$ |

Species source term from double layer | ${\dot{s}}_{i,\mathrm{DL}}^{\mathrm{V}}=\frac{{z}_{i}}{F}{i}_{\mathrm{DL}}^{\mathrm{V}}$ $\mathrm{with}i={\mathrm{Li}}^{+}$ |

Potential step (positive and negative electrode) | $\Delta \varphi ={\varphi}_{\mathrm{e}\mathrm{l}\mathrm{d}\mathrm{e}}-{\varphi}_{\mathrm{e}\mathrm{l}\mathrm{y}\mathrm{t}}$ |

Multi-phase management | |

Volume fraction of phases | $\frac{\partial \left({\rho}_{j}{\epsilon}_{j}\right)}{\partial t}={\displaystyle \sum _{i=1}^{{N}_{\mathrm{R},j},{N}_{\mathrm{P},j}}}{\dot{s}}_{i}^{V}{M}_{i}$ |

Feedback on transport coefficients (porous electrode theory) | ${D}_{i}^{\mathrm{eff}}=\frac{{\epsilon}_{\mathrm{elyt}}}{{{\tau}_{\mathrm{elyt}}}^{2}}{D}_{i}$ |

Symbol | Unit | Meaning |
---|---|---|

$A$ | m² | Surface area |

${A}_{\mathrm{e}}$ | m² | Active electrode area |

${A}_{n}^{\mathrm{V}}$ | m²·m^{−3} | Volume-specific surface area of reaction n |

$Bi$ | 1 | Biot number |

${C}_{\mathrm{DL}}^{\mathrm{V}}$ | F·m^{−3} | Volume-specific double-layer capacity |

${c}_{i}$ | mol·m^{−3} | Concentration of species i in a bulk phase |

${c}_{\mathrm{max},i}$ | mol·m^{−3} | Maximum concentration of species i in a bulk phase |

${c}_{i}^{0}$ | mol·m^{−3} | Standard concentration of species i |

${c}_{p}$ | J·kg^{−1}·K^{−1} | Specific heat capacity |

${D}_{i}$ | m^{2}·s^{−1} | Diffusion coefficient of species i |

${D}_{i}^{\mathrm{eff}}$ | m^{2}·s^{−1} | Effective diffusion coefficient of species i |

${E}_{\mathrm{act}}$ | J·mol^{−1} | Activation energy of forward reaction |

$E$ | Pa | Young’s modulus |

$E$ | V | Cell voltage |

$F$ | C·mol^{−1} | Faraday’s constant |

$\Delta {H}_{n}$ | J·mol^{−1} | Enthalpy of reaction n |

${h}_{i}^{0}$ | kJ·mol^{−1} | Molar enthalpy of species i |

$\Delta {{G}_{n}^{0}}_{}$ | J·mol^{−1} | Gibbs energy of reaction n |

${I}_{\mathrm{cell}}$ | A | Cell current |

$i$ | 1 | Index of species |

$i$ | A·m^{−2} | Area-specific current (with respect to ${A}_{\mathrm{e}}$) |

${i}^{0}$ | A·m^{−2} | Exchange current density |

${i}^{00}$ | A·m^{−2} | Exchange current density factor |

${i}_{\mathrm{DL}}^{\mathrm{V}}$ | A·m^{−3} | Volume-specific double-layer current |

${i}_{\mathrm{F}}^{\mathrm{V}}$ | A·m^{−3} | Volume-specific faradaic current |

$j$ | 1 | Index of bulk phases |

${J}_{\mathrm{q}}$ | W·m^{−2} | Heat flux from cell surface |

${J}_{i}$ | mol·m^{−2}·s^{−1} | Molar flux of species i |

$k$ | 1 | Index of finite volume on $y$ scale |

$\Delta {L}_{k}$ | m | Displacement of finite volume $k$ |

$\Delta {L}_{\mathrm{cell}}$ | m | Total displacement of complete cell |

$\Delta {L}_{\mathrm{EP}}$ | m | Total displacement of single electrode pair |

$\Delta {L}_{\mathrm{int}}$ | m | Intercalation-induced displacement |

$\Delta {L}_{\mathrm{mech}}$ | m | Mechanical displacement |

$\Delta {L}_{\mathrm{th}}$ | m | Thermal displacement |

${L}^{0}$ | m | Initial length |

${L}_{k}^{0}$ | m | Initial length of finite volume $k$ |

${L}_{\mathrm{EP}}$ | m | Thickness of electrode pair |

$L$ | m | Length of thermal conduction |

${M}_{i}$ | kg·mol^{−1} | Molar mass of species i |

$m$ | kg | Cell mass |

$n$ | 1 | Index of reactions |

${N}_{\mathrm{AM}}$ | 1 | Number of active materials |

${N}_{\mathrm{FV}}$ | 1 | Number of compartments on mesoscale |

${N}_{\mathrm{EP}}$ | 1 | Number of electrode pairs |

${N}_{\mathrm{R}},{N}_{\mathrm{P}}$ | 1 | Number of reactants and products in reaction |

${N}_{\mathrm{r}}$ | 1 | Number of reactions |

$p$ | Pa | Pressure |

${p}_{\mathrm{ref}}$ | Pa | Reference pressure |

${\dot{q}}_{\mathrm{chem}}$ | W·m^{−2} | Heat source due to chemical reactions |

${\dot{q}}_{\mathrm{ohm}}$ | W·m^{−2} | Heat source due to ohmic losses |

${\dot{q}}^{\mathrm{V}}$ | W·m^{−3} | Volume-specific heat source |

$R$ | J·K^{−1}·mol^{−1} | Ideal gas constant |

${R}_{\mathrm{cc}}$ | Ω·m^{2} | Area-specific ohmic resistance of current collection system |

${R}_{\mathrm{cc}}^{0}$ | Ω·m^{2} | Area-specific ohmic resistance of current collection system at reference temperature |

${R}_{\mathrm{SEI}}^{\mathrm{V}}$ | Ω·m^{3} | Volume-specific ohmic resistance of SEI film |

${r}_{n}$ | mol·m^{−2}·s^{−1} | Interfacial reaction rate of reaction n |

${r}_{j}$ | m | Particle radius of bulk phase j |

${s}_{i}^{0}$ | J·mol^{−1}·K^{−1} | Molar entropy of species i |

${\dot{s}}_{i}^{\mathrm{V}}$ | mol·m^{−3}·s^{−1} | Volumetric species source term |

${\dot{s}}_{\u0307i,\mathrm{DL}}^{\mathrm{V}}$ | mol·m^{−3}·s^{−1} | Volumetric species source term due to double-layer charge/discharge |

$t$ | s | Time |

$\Delta T$ | K | Temperature difference |

$\Delta {T}_{0}$ | K | Initial temperature difference |

$T$ | K | Temperature |

${T}_{\mathrm{amb}}$ | K | Ambient temperature (cell surrounding) |

${T}_{\mathrm{surf}}$ | K | Cell surface temperature |

${V}_{\mathrm{cell}}$ | m³ | Volume of cell |

${\overline{V}}_{i}$ | m^{3}·mol^{−1} | Partial molar volume of species i |

${V}_{\mathrm{m},\mathrm{AM}}$ | m^{3}·mol^{−1} | Molar volume of active material |

${V}_{\mathrm{m},i}^{*}$ | m^{3}·mol^{−1} | Pure species molar volume |

${V}_{\mathrm{m},\mathrm{AM}}^{0}$ | m^{3}·mol^{−1} | Initial molar volume of active material |

${\Delta V}_{\mathrm{m},\mathrm{AM}}$ | m^{3}·mol^{−1} | Molar volume change of active material |

$x$ | m | Spatial position in dimension of battery thickness |

$X$ | 1 | Stoichiometry of lithium in the active material |

${X}_{i}$ | 1 | Mole fraction of species i |

$\mathrm{SOC}$ | 1 | State-Of-Charge |

$y$ | m | Spatial position in dimension of electrode pair thickness |

$z$ | m | Spatial position in dimension of particle thickness |

$z$ | 1 | Number of electrons transferred in charge-transfer reaction |

${\alpha}_{\mathrm{th}}$ | K^{−1} | Thermal expansion coefficient |

${\alpha}_{\mathrm{surf}}$ | W·m^{−2}·K^{−1} | Heat transfer coefficient |

${\alpha}_{\mathrm{c}}$ | 1 | Cathodic transfer coefficient of electrochemical reaction |

${\alpha}_{\mathrm{cc}}$ | 1 | Slope of temperature dependent expression of ohmic resistance of current collection system |

${\varphi}_{\mathrm{elde}},{\varphi}_{\mathrm{elyt}}$ | V | Electric potential in the solid phase and in the electrolyte |

$\Delta \varphi $ | V | Electric potential difference between electrode and electrolyte |

$\Delta {\varphi}^{\mathrm{eff}}$ | V | Effective electric potential difference |

$\Delta {\varphi}^{\mathrm{eq}}$ | V | Equilibrium potential difference |

${\Delta \varphi}_{n}$ | V | Electric potential difference of reaction n |

$\u03f5$ | 1 | Emissivity of the cell surface |

${\epsilon}_{\mathrm{AM}}^{0}$ | 1 | Initial volume fraction of active material |

${\epsilon}_{\mathrm{elyt}}$ | 1 | Volume fraction of the electrolyte |

${\epsilon}_{\mathrm{int}}$ | 1 | Intercalation-induced strain |

${\epsilon}_{j}$ | 1 | Volume fraction of bulk phase j |

${\epsilon}_{\mathrm{mech}}$ | 1 | Mechanical strain |

${\epsilon}_{\mathrm{th}}$ | 1 | Thermal strain |

${\eta}_{\mathrm{act}}$ | V | Activation overpotential |

$\lambda $ | W·m^{−1}·K^{−1} | Thermal conductivity |

${\mu}_{i}^{0}$ | J·mol^{−1} | Standard-state chemical potential |

${\nu}_{\mathrm{e},n}$ | 1 | Stoichiometric coefficient of electron in electrochemical reaction n |

$\rho $ | kg·m^{−3} | Density |

${\sigma}_{\mathrm{elyt}}$ | S·m^{−1} | Electrolyte conductivity |

${\sigma}_{\mathrm{n}}$ | Pa | Normal stress |

${\sigma}_{\mathrm{SB}}$ | W·m^{−2}·K^{−4} | Stefan-Boltzmann constant |

${\tau}_{\mathrm{elyt}}$ | 1 | Geometric tortuosity of the electrolyte |

**Figure A1.**Schematic representation of 1D+1D+1D (pseudo-3D, P3D) modeling domain [36].

**Figure A2.**Material data for the three AM (

**a**,

**d**,

**g**) LCO, (

**b**,

**e**,

**h**) NCA and (

**c**,

**f**,

**i**) graphite. The first row (

**a**–

**c**) shows the molar enthalpies and entropies of intercalated lithium within the three AM. The molar enthalpies and entropies of the vacancies are set to 0 (reference species), see Ref. [47]. The second row (

**d**–

**f**) shows the solid-state diffusion coefficients of lithium within the three AM at 20 °C. The diffusion is assumed thermally activated with activation energies of 28.95 kJ·mol

^{−1}, 115.78 kJ·mol

^{−1}and 44.0 kJ·mol

^{−1}for LCO, NCA and graphite, respectively. The third row (

**g**–

**i**) shows the molar volumes as function of lithium stoichiometry of the three AM. The vertical dashed lines indicate the stoichiometry ranges for every AM used in the studied cell within the cut-off voltage limits, as obtained through optimization. See Ref. [34] for details.

No. | Electrode | Reaction | Rate Coefficient | Activation Energy | Symmetry Factor |
---|---|---|---|---|---|

(1) | Negative | Li^{+}[elyt] + e^{−} + V[C_{6}] ⇄ Li[C_{6}] | ${i}^{00}$ = 8.84·10^{14} A/m^{2} [34] | 77.1 kJ/mol [34] | 0.5 [34] |

(2) | Positive | Li^{+}[elyt] + e^{−} + V[LCO] ⇄ Li[LCO] | ${i}^{00}$ = 8.20·10^{12} A/m^{2} [34] | 72.3 kJ/mol [34] | 0.5 [34] |

(3) | Positive | Li^{+}[elyt] + e^{−} + V[NCA] ⇄ Li[NCA] | ${i}^{00}$ = 2.63·10^{10} A/m^{2} [34] | 61.0 kJ/mol [34] | 0.5 [34] |

Layer | Phase | $\mathbf{Initial}\mathbf{Volume}\mathbf{Fraction}\mathit{\epsilon}$ |
$\mathbf{Density}$ $\mathit{\rho}$/kg·m ^{−3} | Species (Initial Mole Fraction X) |
---|---|---|---|---|

PE | LCO | 0.2856 | 4790 | Li[LCO], V[LCO] (depends on SOC) |

NCA | 0.2368 | 3900 | Li[NCA], V[NCA] (depends on SOC) | |

Electrolyte | 0.2976 | 1270 | ${\mathrm{C}}_{3}{\mathrm{H}}_{4}{\mathrm{O}}_{3}\left(0.52\right),{\mathrm{C}}_{4}{\mathrm{H}}_{8}{\mathrm{O}}_{3}\left(0.34\right),\mathrm{L}{\mathrm{i}}^{+}$$\left(0.07\right),\mathrm{P}{\mathrm{F}}_{6}^{-}$(0.07) | |

Gas phase | 0.030 | From ideal gas law | N_{2} (1) | |

Electron conductor | 0.150 | 2000 | No chemically active species | |

Separator | Separator | 0.5 | 777 | No chemically active species |

Electrolyte | 0.470 | 1270 | same as at PE | |

Gas phase | 0.030 | From ideal gas law | N_{2} (1) | |

NE | C_{6} | 0.5073 | 2270 | Li[C_{6}], V[C_{6}] (depends on SOC) |

Electrolyte | 0.4527 | 1270 | same as at PE | |

LEDC | 0.0008 | 1300 | (CH_{2}OCO_{2}Li)_{2} | |

Lithium carbonate | 0.0092 | 2100 | Li_{2}CO_{3} | |

Gas phase | 0.030 | From ideal gas law | N_{2} (1) |

Parameter | Value |
---|---|

$\mathrm{Cell}\mathrm{thickness}$ $\mathrm{Active}\mathrm{electrode}\mathrm{area}{A}_{\mathrm{e}}$ $\mathrm{Cell}\mathrm{thermal}\mathrm{conductivity}\lambda $ Cell heat capacity ${c}_{p}$ | 3 mm 0.02883 m² 0.9 W·m ^{−1}·K^{−1}0.95 J·g ^{−1}·K^{−1} |

Thickness of PE | 32.9 µm |

Thickness of separator | 15.7 µm |

Thickness of NE | 49.1 µm |

$\mathrm{Tortuosity}\mathrm{of}\mathrm{PE}\tau $ | 1.35 |

$\mathrm{Tortuosity}\mathrm{of}\mathrm{separator}\tau $ | 1.21 |

$\mathrm{Tortuosity}\mathrm{of}\mathrm{NE}\tau $ | 1.22 |

$\mathrm{Diffusion}\mathrm{coefficients}{D}_{{\mathrm{L}\mathrm{i}}^{+}}$ $,{D}_{{\mathrm{P}\mathrm{F}}_{6}^{-}}$ | See Equations (A1) and (A2) |

$\mathrm{Specific}\mathrm{surface}\mathrm{area}\mathrm{LCO}/\mathrm{electrolyte}{A}^{\mathrm{V}}$ | 6.67·10^{5} m^{2}/m^{3} |

$\mathrm{Specific}\mathrm{surface}\mathrm{area}\mathrm{NCA}/\mathrm{electrolyte}{A}^{\mathrm{V}}$ | 4.28·10^{6} m^{2}/m^{3} |

$\mathrm{Specific}\mathrm{surface}\mathrm{area}\mathrm{graphite}/\mathrm{electrolyte}{A}^{\mathrm{V}}$ | 2.79 00B7 10^{5} m^{2}/m^{3} |

$\mathrm{PE}\mathrm{double}\mathrm{layer}\mathrm{capacitance}{C}_{\mathrm{DL}}^{\mathrm{V}}$ | 1.5·10^{4} F·m^{−3} |

$\mathrm{NE}\mathrm{double}\mathrm{layer}\mathrm{capacitance}{C}_{\mathrm{DL}}^{\mathrm{V}}$ | 2.8·10^{5} F·m^{−3} |

$\mathrm{Ohmic}\mathrm{resistance}\mathrm{of}\mathrm{current}\mathrm{collection}\mathrm{system}{R}_{\mathrm{cc}}^{0}$ | 7·10^{−1} mΩ·m^{2} |

$\mathrm{Slope}{\alpha}_{\mathrm{cc}}$
$(\mathrm{ref}.T=293\mathrm{K}$) Electrical conductivity of the SEI layer ${\sigma}_{\mathrm{SEI}}$ | $-0.01$ 1.0·10 ^{−5} S/m |

$\mathrm{Graphite}\mathrm{stoichiometry}\mathrm{range}{X}_{\mathrm{Li}\left[{\mathrm{C}}_{6}\right]}$ (0…100% SOC) | 0.012…0.601 |

$\mathrm{LCO}\mathrm{stoichiometry}\mathrm{range}{X}_{\mathrm{Li}\left[\mathrm{LCO}\right]}$ (0…100% SOC) | 0.9922…0.448 |

$\mathrm{NCA}\mathrm{stoichiometry}\mathrm{range}{X}_{\mathrm{Li}\left[\mathrm{NCA}\right]}$ (0…100% SOC) | 0.790…0.186 |

$\mathrm{Radius}\mathrm{of}\mathrm{LCO}\mathrm{particles}{r}_{\mathrm{LCO}}$ | $4.5\mathrm{\xb7}{10}^{-6}\mathrm{m}$ |

$\mathrm{Diffusion}\mathrm{coefficient}\mathrm{of}\mathrm{Li}\mathrm{in}\mathrm{LCO}{D}_{\mathrm{Li},\mathrm{LCO}}$ | See Figure A2d |

$\mathrm{Radius}\mathrm{of}\mathrm{NCA}\mathrm{particles}{r}_{\mathrm{NCA}}$ | $0.7\mathrm{\xb7}{10}^{-6}\mathrm{m}$ |

$\mathrm{Diffusion}\mathrm{coefficient}\mathrm{of}\mathrm{Li}\mathrm{in}\mathrm{NCA}{D}_{\mathrm{Li},\mathrm{NCA}}$ | See Figure A2e |

$\mathrm{Radius}\mathrm{of}\mathrm{graphite}\mathrm{particles}{r}_{{\mathrm{C}}_{6}}$ | $1.075\mathrm{\xb7}{10}^{-5}\mathrm{m}$ |

$\mathrm{Diffusion}\mathrm{coefficient}\mathrm{of}\mathrm{Li}\mathrm{and}\mathrm{graphite}{D}_{\mathrm{Li},{\mathrm{C}}_{6}}$ | See Figure A2f |

## References

- Ziegler, M.S.; Trancik, J.E. Re-examining rates of lithium-ion battery technology improvement and cost decline. Energy Environ. Sci.
**2021**, 14, 1635–1651. [Google Scholar] [CrossRef] - Winter, M.; Barnett, B.; Xu, K. Before Li Ion Batteries. Chem. Rev.
**2018**, 118, 11433–11456. [Google Scholar] [CrossRef] - Armand, M.; Axmann, P.; Bresser, D.; Copley, M.; Edström, K.; Ekberg, C.; Guyomard, D.; Lestriez, B.; Novák, P.; Petranikova, M.; et al. Lithium-ion batteries—Current state of the art and anticipated developments. J. Power Sources
**2020**, 479, 228708. [Google Scholar] [CrossRef] - Birkl, C.R.; Roberts, M.R.; McTurk, E.; Bruce, P.G.; Howey, D.A. Degradation diagnostics for lithium ion cells. J. Power Sources
**2017**, 341, 373–386. [Google Scholar] [CrossRef] - Zhao, Y.; Stein, P.; Bai, Y.; Al-Siraj, M.; Yang, Y.; Xu, B.-X. A review on modeling of electro-chemo-mechanics in lithium-ion batteries. J. Power Sources
**2019**, 413, 259–283. [Google Scholar] [CrossRef] - Xu, R.; Zhao, K. Electrochemomechanics of Electrodes in Li-Ion Batteries: A Review. J. Electrochem. Energy Convers. Storage
**2016**, 13, 652. [Google Scholar] [CrossRef] [Green Version] - Escher, I.; Hahn, M.; Ferrero, G.A.; Adelhelm, P. A Practical Guide for Using Electrochemical Dilatometry as Operando Tool in Battery and Supercapacitor Research. Energy Technol.
**2022**, 10, 2101120. [Google Scholar] [CrossRef] - Popp, H.; Koller, M.; Jahn, M.; Bergmann, A. Mechanical methods for state determination of Lithium-Ion secondary batteries: A review. J. Energy Storage
**2020**, 32, 101859. [Google Scholar] [CrossRef] - Mendoza, H.; Roberts, S.A.; Brunini, V.E.; Grillet, A.M. Mechanical and Electrochemical Response of a LiCoO
_{2}Cathode using Reconstructed Microstructures. Electrochim. Acta**2016**, 190, 1–15. [Google Scholar] [CrossRef] [Green Version] - Miranda, D.; Costa, C.M.; Lanceros-Mendez, S. Lithium ion rechargeable batteries: State of the art and future needs of microscopic theoretical models and simulations. J. Electroanal. Chem.
**2015**, 739, 97–110. [Google Scholar] [CrossRef] - Franco, A.A. Multiscale Modelling and Numerical Simulation of Rechargeable Lithium Ion Batteries: Concepts, Methods and Challenges. RSC Adv.
**2013**, 3, 13027–13058. [Google Scholar] [CrossRef] - Krewer, U.; Röder, F.; Harinath, E.; Braatz, R.D.; Bedürftig, B.; Findeisen, R. Review—Dynamic Models of Li-Ion Batteries for Diagnosis and Operation: A Review and Perspective. J. Electrochem. Soc.
**2018**, 165, A3656–A3673. [Google Scholar] [CrossRef] - DeCaluwe, S.C.; Weddle, P.J.; Zhu, H.; Colclasure, A.M.; Bessler, W.G.; Jackson, G.S.; Kee, R.J. On the Fundamental and Practical Aspects of Modeling Complex Electrochemical Kinetics and Transport. J. Electrochem. Soc.
**2018**, 165, E637–E658. [Google Scholar] [CrossRef] - Kupper, C.; Weißhar, B.; Rißmann, S.; Bessler, W.G. End-of-Life Prediction of a Lithium-Ion Battery Cell Based on Mechanistic Aging Models of the Graphite Electrode. J. Electrochem. Soc.
**2018**, 165, A3468–A3480. [Google Scholar] [CrossRef] - Brosa Planella, F.; Ai, W.; Boyce, A.M.; Ghosh, A.; Korotkin, I.; Sahu, S.; Sulzer, V.; Timms, R.; Tranter, T.G.; Zyskin, M.; et al. A continuum of physics-based lithium-ion battery models reviewed. Prog. Energy
**2022**, 4, 42003. [Google Scholar] [CrossRef] - Zhang, X.; Shyy, W.; Marie Sastry, A. Numerical Simulation of Intercalation-Induced Stress in Li-Ion Battery Electrode Particles. J. Electrochem. Soc.
**2007**, 154, A910–A916. [Google Scholar] [CrossRef] - Garcı́a, R.E.; Chiang, Y.-M.; Craig Carter, W.; Limthongkul, P.; Bishop, C.M. Microstructural Modeling and Design of Rechargeable Lithium-Ion Batteries. J. Electrochem. Soc.
**2005**, 152, A255. [Google Scholar] [CrossRef] - Cheng, Y.-T.; Verbrugge, M.W. Evolution of stress within a spherical insertion electrode particle under potentiostatic and galvanostatic operation. J. Power Sources
**2009**, 190, 453–460. [Google Scholar] [CrossRef] - Renganathan, S.; Sikha, G.; Santhanagopalan, S.; White, R.E. Theoretical Analysis of Stresses in a Lithium Ion Cell. J. Electrochem. Soc.
**2010**, 157, A155. [Google Scholar] [CrossRef] - Bohn, E.; Eckl, T.; Kamlah, M.; McMeeking, R. A Model for Lithium Diffusion and Stress Generation in an Intercalation Storage Particle with Phase Change. J. Electrochem. Soc.
**2013**, 160, A1638–A1652. [Google Scholar] [CrossRef] - Fu, R.; Xiao, M.; Choe, S.-Y. Modeling, validation and analysis of mechanical stress generation and dimension changes of a pouch type high power Li-ion battery. J. Power Sources
**2013**, 224, 211–224. [Google Scholar] [CrossRef] - Roberts, S.A.; Brunini, V.E.; Long, K.N.; Grillet, A.M. A Framework for Three-Dimensional Mesoscale Modeling of Anisotropic Swelling and Mechanical Deformation in Lithium-Ion Electrodes. J. Electrochem. Soc.
**2014**, 161, F3052–F3059. [Google Scholar] [CrossRef] [Green Version] - Garrick, T.R.; Kanneganti, K.; Huang, X.; Weidner, J.W. Modeling Volume Change due to Intercalation into Porous Electrodes. J. Electrochem. Soc.
**2014**, 161, E3297–E3301. [Google Scholar] [CrossRef] [Green Version] - Gomadam, P.M.; Weidner, J.W. Modeling Volume Changes in Porous Electrodes. J. Electrochem. Soc.
**2006**, 153, A179. [Google Scholar] [CrossRef] [Green Version] - Rieger, B.; Schlueter, S.; Erhard, S.V.; Jossen, A. Strain Propagation in Lithium-Ion Batteries from the Crystal Structure to the Electrode Level. J. Electrochem. Soc.
**2016**, 163, A1595–A1606. [Google Scholar] [CrossRef] - Rieger, B.; Erhard, S.V.; Rumpf, K.; Jossen, A. A New Method to Model the Thickness Change of a Commercial Pouch Cell during Discharge. J. Electrochem. Soc.
**2016**, 163, A1566–A1575. [Google Scholar] [CrossRef] - Sauerteig, D.; Hanselmann, N.; Arzberger, A.; Reinshagen, H.; Ivanov, S.; Bund, A. Electrochemical-mechanical coupled modeling and parameterization of swelling and ionic transport in lithium-ion batteries. J. Power Sources
**2018**, 378, 235–247. [Google Scholar] [CrossRef] - Ai, W.; Kraft, L.; Sturm, J.; Jossen, A.; Wu, B. Electrochemical Thermal-Mechanical Modelling of Stress Inhomogeneity in Lithium-Ion Pouch Cells. J. Electrochem. Soc.
**2020**, 167, 13512. [Google Scholar] [CrossRef] - Pereira, D.J.; Fernandez, M.A.; Streng, K.C.; Hou, X.X.; Gao, X.; Weidner, J.W.; Garrick, T.R. Accounting for Non-Ideal, Lithiation-Based Active Material Volume Change in Mechano-Electrochemical Pouch Cell Simulation. J. Electrochem. Soc.
**2020**, 167, 80515. [Google Scholar] [CrossRef] - Zhang, X.; Klinsmann, M.; Chumakov, S.; Li, X.; Kim, S.U.; Metzger, M.; Besli, M.M.; Klein, R.; Linder, C.; Christensen, J. A Modified Electrochemical Model to Account for Mechanical Effects Due to Lithium Intercalation and External Pressure. J. Electrochem. Soc.
**2021**, 168, 20533. [Google Scholar] [CrossRef] - Pegel, H.; von Kessel, O.; Heugel, P.; Deich, T.; Tübke, J.; Birke, K.P.; Sauer, D.U. Volume and thickness change of NMC811 | SiO
_{x}-graphite large-format lithium-ion cells: From pouch cell to active material level. J. Power Sources**2022**, 537, 231443. [Google Scholar] [CrossRef] - Clerici, D.; Mocera, F.; Somà, A. Experimental Characterization of Lithium-Ion Cell Strain Using Laser Sensors. Energies
**2021**, 14, 6281. [Google Scholar] [CrossRef] - Kupper, C.; Bessler, W.G. Multi-Scale Thermo-Electrochemical Modeling of Performance and Aging of a LiFePO
_{4}/Graphite Lithium-Ion Cell. J. Electrochem. Soc.**2017**, 164, A304–A320. [Google Scholar] [CrossRef] - Carelli, S.; Quarti, M.; Yagci, M.C.; Bessler, W.G. Modeling and Experimental Validation of a High-Power Lithium-Ion Pouch Cell with LCO/NCA Blend Cathode. J. Electrochem. Soc.
**2019**, 166, A2990–A3003. [Google Scholar] [CrossRef] - Quarti, M.; Bessler, W.G. Model-Based Overpotential Deconvolution, Partial Impedance Spectroscopy, and Sensitivity Analysis of a Lithium-Ion Cell with Blend Cathode. Energy Technol.
**2021**, 9, 2001122. [Google Scholar] [CrossRef] - Carelli, S.; Bessler, W.G. Prediction of reversible lithium plating with a pseudo-3D lithium-ion battery model. J. Electrochem. Soc.
**2020**, 167, 100515. [Google Scholar] [CrossRef] - Goodwin, D.G.; Moffat, H.K.; Schoegl, I.; Speth, R.L.; Weber, B.W. Cantera: An Object-oriented Software Toolkit for Chemical Kinetics, Thermodynamics, and Transport Processes; Zenodo: Genève, Switzerland, 2022. [Google Scholar] [CrossRef]
- Mayur, M.; Yagci, M.C.; Carelli, S.; Margulies, P.; Velten, D.; Bessler, W.G. Identification of stoichiometric and microstructural parameters of a lithium-ion cell with blend electrode. Phys. Chem. Chem. Phys.
**2019**, 21, 23672–23684. [Google Scholar] [CrossRef] - Quarti, M.; Bayer, A.; Bessler, W.G. Trade-off between energy density and fast-charge capability of lithium-ion batteries: A model-based design study of cells with thick electrodes. Electrochem. Sci. Adv.
**2023**, 3, e2100161. [Google Scholar] [CrossRef] - Mohtat, P.; Lee, S.; Siegel, J.B.; Stefanopoulou, A.G. Towards better estimability of electrode-specific state of health: Decoding the cell expansion. J. Power Sources
**2019**, 427, 101–111. [Google Scholar] [CrossRef] - Schiffer, Z.J.; Cannarella, J.; Arnold, C.B. Strain Derivatives for Practical Charge Rate Characterization of Lithium Ion Electrodes. J. Electrochem. Soc.
**2016**, 163, A427–A433. [Google Scholar] [CrossRef] [Green Version] - Zhang, X.; He, J.; Zhou, J.; Chen, H.; Song, W.; Fang, D. Thickness evolution of commercial Li-ion pouch cells with silicon-based composite anodes and NCA cathodes. Sci. China Technol. Sci.
**2021**, 64, 83–90. [Google Scholar] [CrossRef] - Müller, V.; Scurtu, R.-G.; Richter, K.; Waldmann, T.; Memm, M.; Danzer, M.A.; Wohlfahrt-Mehrens, M. Effects of Mechanical Compression on the Aging and the Expansion Behavior of Si/C-Composite|NMC811 in Different Lithium-Ion Battery Cell Formats. J. Electrochem. Soc.
**2019**, 166, A3796–A3805. [Google Scholar] [CrossRef] - Oh, K.-Y.; Siegel, J.B.; Secondo, L.; Kim, S.U.; Samad, N.A.; Qin, J.; Anderson, D.; Garikipati, K.; Knobloch, A.; Epureanu, B.I.; et al. Rate dependence of swelling in lithium-ion cells. J. Power Sources
**2014**, 267, 197–202. [Google Scholar] [CrossRef] - Rieger, B.; Schlueter, S.; Erhard, S.V.; Schmalz, J.; Reinhart, G.; Jossen, A. Multi-scale investigation of thickness changes in a commercial pouch type lithium-ion battery. J. Energy Storage
**2016**, 6, 213–221. [Google Scholar] [CrossRef] - Doyle, M.; Fuller, T.F.; Newman, J. Modeling of Galvanostatic Charge and Discharge of the Lithium/Polymer/Insertion Cell. J. Electrochem. Soc.
**1993**, 140, 1526–1533. [Google Scholar] [CrossRef] - Mayur, M.; DeCaluwe, S.C.; Kee, B.L.; Bessler, W.G. Modeling and simulation of the thermodynamics of lithium-ion battery intercalation materials in the open-source software Cantera. Electrochim. Acta
**2019**, 323, 134797. [Google Scholar] [CrossRef] - Hetnarski, R.B. Encyclopedia of Thermal Stresses: With 371 Tables; Springer: Dordrecht, The Netherlands, 2014; ISBN 978-94-007-2738-0. [Google Scholar]
- Rieger, B.; Erhard, S.V.; Kosch, S.; Venator, M.; Rheinfeld, A.; Jossen, A. Multi-Dimensional Modeling of the Influence of Cell Design on Temperature, Displacement and Stress Inhomogeneity in Large-Format Lithium-Ion Cells. J. Electrochem. Soc.
**2016**, 163, A3099–A3110. [Google Scholar] [CrossRef] - Koerver, R.; Zhang, W.; de Biasi, L.; Schweidler, S.; Kondrakov, A.O.; Kolling, S.; Brezesinski, T.; Hartmann, P.; Zeier, W.G.; Janek, J. Chemo-mechanical expansion of lithium electrode materials–on the route to mechanically optimized all-solid-state batteries. Energy Environ. Sci.
**2018**, 11, 2142–2158. [Google Scholar] [CrossRef] - Deuflhard, P.; Hairer, E.; Zugck, J. One-step and extrapolation methods for differential-algebraic systems. Numer. Math.
**1987**, 51, 501–516. [Google Scholar] [CrossRef] - Ehrig, R.; Nowak, U.; Oeverdieck, L.; Deuflhard, P. Advanced Extrapolation Methods for Large Scale Differential Algebraic Problems. In High Performance Scientific and Engineering Computing; Lecture Notes in Computational Science and Engineering; Bungartz, H.-J., Durst, F., Zenger, C., Eds.; Springer: Berlin/Heidelberg, Germany, 1999; pp. 233–244. [Google Scholar]
- Atkins, P.W.; de Paula, J. Atkins' Physical Chemistry, 8th ed.; W.H. Freeman: New York, NY, USA, 2006; ISBN 0-7167-8759-8. [Google Scholar]
- Smith, J.M.; van Ness, H.C.; Abbott, M.M.; Swihart, M.T. Introduction to Chemical Engineering Thermodynamics, 8th ed.; McGraw-Hill Education: New York, NY, USA, 2018; ISBN 978-1-259-69652-7. [Google Scholar]
- Schweidler, S.; de Biasi, L.; Schiele, A.; Hartmann, P.; Brezesinski, T.; Janek, J. Volume Changes of Graphite Anodes Revisited: A Combined Operando X-ray Diffraction and In Situ Pressure Analysis Study. J. Phys. Chem. C
**2018**, 122, 8829–8835. [Google Scholar] [CrossRef] - Amatucci, G.G.; Tarascon, J.M.; Klein, L.C. CoO
_{2}, The End Member of the Li_{x}CoO_{2}Solid Solution. J. Electrochem. Soc.**1996**, 143, 1114–1123. [Google Scholar] [CrossRef] - Reimers, J.N.; Dahn, J.R. Electrochemical and In Situ X-Ray Diffraction Studies of Lithium Intercalation in Li
_{x}CoO_{2}. J. Electrochem. Soc.**1992**, 139, 2091. [Google Scholar] [CrossRef] - Van der Ven, A.; Aydinol, M.K.; Ceder, G.; Kresse, G.; Hafner, J. First-principles investigation of phase stability in Li
_{x}CoO_{2}. Phys. Rev. B Condens. Matter**1998**, 58, 2975–2987. [Google Scholar] [CrossRef] - Laubach, S.; Laubach, S.; Schmidt, P.C.; Ensling, D.; Schmid, S.; Jaegermann, W.; Thissen, A.; Nikolowski, K.; Ehrenberg, H. Changes in the crystal and electronic structure of LiCoO(2) and LiNiO(2) upon Li intercalation and de-intercalation. Phys. Chem. Chem. Phys.
**2009**, 11, 3278–3289. [Google Scholar] [CrossRef] [PubMed] - Winter, M.; Besenhard, J.O.; Spahr, M.E.; Novák, P. Insertion Electrode Materials for Rechargeable Lithium Batteries. Adv. Mater.
**1998**, 10, 725–763. [Google Scholar] [CrossRef] - Heng, Y.-L.; Gu, Z.-Y.; Guo, J.-Z.; Wang, X.-T.; Zhao, X.-X.; Wu, X.-L. Research progress on the surface/interface modification of high-voltage lithium oxide cathode materials. Energy Mater.
**2022**, 2, 200017. [Google Scholar] [CrossRef] - He, H.; Huang, C.; Luo, C.-W.; Liu, J.-J.; Chao, Z.-S. Dynamic study of Li intercalation into graphite by in situ high energy synchrotron XRD. Electrochim. Acta
**2013**, 92, 148–152. [Google Scholar] [CrossRef] - Billaud, D.; Henry, F.X.; Lelaurain, M.; Willmann, P. Revisited structures of dense and dilute stage II lithium-graphite intercalation compounds. J. Phys. Chem. Solids
**1996**, 57, 775–781. [Google Scholar] [CrossRef] - Dahn, J.R. Phase diagram of LixC
_{6}. Phys. Rev. B Condens. Matter**1991**, 44, 9170–9177. [Google Scholar] [CrossRef] - Dahn, J.R.; Fong, R.; Spoon, M.J. Suppression of staging in lithium-intercalated carbon by disorder in the host. Phys. Rev. B Condens. Matter
**1990**, 42, 6424–6432. [Google Scholar] [CrossRef] [Green Version] - Senyshyn, A.; Dolotko, O.; Mühlbauer, M.J.; Nikolowski, K.; Fuess, H.; Ehrenberg, H. Lithium Intercalation into Graphitic Carbons Revisited: Experimental Evidence for Twisted Bilayer Behavior. J. Electrochem. Soc.
**2013**, 160, A3198–A3205. [Google Scholar] [CrossRef] - Ohzuku, T.; Iwakoshi, Y.; Sawai, K. Formation of Lithium-Graphite Intercalation Compounds in Nonaqueous Electrolytes and Their Application as a Negative Electrode for a Lithium Ion (Shuttlecock) Cell. J. Electrochem. Soc.
**1993**, 140, 2490–2498. [Google Scholar] [CrossRef] - Yazami, R.; Reynier, Y. Thermodynamics and crystal structure anomalies in lithium-intercalated graphite. J. Power Sources
**2006**, 153, 312–318. [Google Scholar] [CrossRef] - Takami, N.; Satoh, A.; Hara, M.; Ohsaki, T. Structural and Kinetic Characterization of Lithium Intercalation into Carbon Anodes for Secondary Lithium Batteries. J. Electrochem. Soc.
**1995**, 142, 371–379. [Google Scholar] [CrossRef] - Qi, Y.; Guo, H.; Hector, L.G.; Timmons, A. Threefold Increase in the Young’s Modulus of Graphite Negative Electrode during Lithium Intercalation. J. Electrochem. Soc.
**2010**, 157, A558. [Google Scholar] [CrossRef] - Didier, C.; Pang, W.K.; Guo, Z.; Schmid, S.; Peterson, V.K. Phase Evolution and Intermittent Disorder in Electrochemically Lithiated Graphite Determined Using in Operando Neutron Diffraction. Chem. Mater.
**2020**, 32, 2518–2531. [Google Scholar] [CrossRef] - Louli, A.J.; Li, J.; Trussler, S.; Fell, C.R.; Dahn, J.R. Volume, Pressure and Thickness Evolution of Li-Ion Pouch Cells with Silicon-Composite Negative Electrodes. J. Electrochem. Soc.
**2017**, 164, A2689–A2696. [Google Scholar] [CrossRef] - Mukhopadhyay, A.; Sheldon, B.W. Deformation and stress in electrode materials for Li-ion batteries. Prog. Mater. Sci.
**2014**, 63, 58–116. [Google Scholar] [CrossRef] - Bauer, M.; Wachtler, M.; Stöwe, H.; Persson, J.V.; Danzer, M.A. Understanding the dilation and dilation relaxation behavior of graphite-based lithium-ion cells. J. Power Sources
**2016**, 317, 93–102. [Google Scholar] [CrossRef] - Grimsmann, F.; Brauchle, F.; Gerbert, T.; Gruhle, A.; Knipper, M.; Parisi, J. Hysteresis and current dependence of the thickness change of lithium-ion cells with graphite anode. J. Energy Storage
**2017**, 12, 132–137. [Google Scholar] [CrossRef] - Waldmann, T.; Kasper, M.; Wohlfahrt-Mehrens, M. Optimization of Charging Strategy by Prevention of Lithium Deposition on Anodes in high-energy Lithium-ion Batteries–Electrochemical Experiments. Electrochim. Acta
**2015**, 178, 525–532. [Google Scholar] [CrossRef] - Bitzer, B.; Gruhle, A. A new method for detecting lithium plating by measuring the cell thickness. J. Power Sources
**2014**, 262, 297–302. [Google Scholar] [CrossRef] - Rieger, B.; Schuster, S.F.; Erhard, S.V.; Osswald, P.J.; Rheinfeld, A.; Willmann, C.; Jossen, A. Multi-directional laser scanning as innovative method to detect local cell damage during fast charging of lithium-ion cells. J. Energy Storage
**2016**, 8, 1–5. [Google Scholar] [CrossRef] - Sturm, J.; Spingler, F.B.; Rieger, B.; Rheinfeld, A.; Jossen, A. Non-Destructive Detection of Local Aging in Lithium-Ion Pouch Cells by Multi-Directional Laser Scanning. J. Electrochem. Soc.
**2017**, 164, A1342–A1351. [Google Scholar] [CrossRef] - Spingler, F.B.; Wittmann, W.; Sturm, J.; Rieger, B.; Jossen, A. Optimum fast charging of lithium-ion pouch cells based on local volume expansion criteria. J. Power Sources
**2018**, 393, 152–160. [Google Scholar] [CrossRef] - Li, Z.; Fang, R.; Ge, H.; Liu, Z.; Spingler, F.B.; Jossen, A.; Zhang, J.; Liaw, B. Multiphysics Footprint of Li Plating for Li-Ion Battery and Challenges for High-Accuracy Detection. J. Electrochem. Soc.
**2022**, 169, 80530. [Google Scholar] [CrossRef] - Chikkannanavar, S.B.; Bernardi, D.M.; Liu, L. A review of blended cathode materials for use in Li-ion batteries. J. Power Sources
**2014**, 248, 91–100. [Google Scholar] [CrossRef] - Heubner, C.; Liebmann, T.; Schneider, M.; Michaelis, A. Recent insights into the electrochemical behavior of blended lithium insertion cathodes: A review. Electrochim. Acta
**2018**, 269, 745–760. [Google Scholar] [CrossRef] - Laresgoiti, I.; Käbitz, S.; Ecker, M.; Sauer, D.U. Modeling mechanical degradation in lithium ion batteries during cycling: Solid electrolyte interphase fracture. J. Power Sources
**2015**, 300, 112–122. [Google Scholar] [CrossRef] - Carelli, S.; Bessler, W.G. Coupling Lithium Plating with SEI Formation in a Pseudo-3D Model: A Comprehensive Approach to Describe Aging in Lithium-Ion Cells. J. Electrochem. Soc.
**2022**, 169, 50539. [Google Scholar] [CrossRef]

**Figure 1.**Schematic sketch (

**left**) and photograph (

**right**) of the experimental setup. LVDT, linear variable differential transformer.

**Figure 2.**Relative molar volume ${V}_{\mathrm{m},\mathrm{AM}}/{V}_{\mathrm{m},\mathrm{AM}}^{0}$ as function of lithium stoichiometry $X$ derived from published XRD experimental data for the three AM (

**a**) LCO, (

**b**) NCA and (

**c**) graphite. For graphite, different data sources are compared. See Table 1 for references and text for details.

**Figure 3.**Simulated displacement of the lithium-ion pouch cell as function of time for a 0.05 C cycle (first charge, then discharge) for different literature parameters of the graphite relative molar volume (Figure 2c), compared to the experimental measurement. The numbers indicated in the legend correspond to the references listed in Table 1.

**Figure 4.**Rapid cycling experiment for thermal characterization. (

**a**) Current (positive for charge) and voltage, (

**b**) thickness and surface temperature as function of time.

**Figure 5.**Identification of thermal parameters. (

**a**) Displacement as function of temperature difference of the five consecutive cooling phases. For clarity, only 30 data points of each data set are shown. (

**b**) Logarithmic temperature difference as function of time (exemplarily for the first of five cooling phases). The solid curves each show a linear fit. In panel (

**a**) this linear fit shows the averaged slope of all five cooling phases.

**Figure 6.**Rapid cycling experiment for thermal characterization: Comparison of simulation and experiment for (

**a**) displacement and (

**b**) temperature difference. Note that the time axis is set to zero at the beginning of the first heating phase.

**Figure 7.**Overview of the experimental data. (

**a**) Measured current (positive for charge), (

**b**) voltage, (

**c**) thickness, (

**d**) temperature as function of time. The vertical dashed lines separate the conditioning protocol from the beginning of the actual cycling experiment.

**Figure 8.**Experimental and simulated CCCV charge and discharge cycles at different C-rates. (

**a**) Voltage as function of charge throughput and (

**b**) cell surface temperature as function of normalized time. In panel (

**a**), the lower branches represent discharge (time progressing from left to right) and the upper branches charge (time progressing from right to left), where the charge throughput of both, charge and discharge curves are normalized to a fully-charged cell. In panel (

**b**), normalized time is defined as time after beginning-of-charge over time of end-of-discharge. The thick and thin parts of the solid lines represent the CC and CV phases, respectively.

**Figure 9.**Experimental and simulated CCCV charge and discharge cycles at different C-rates. (

**a**) Experimental displacement, (

**b**) simulated displacement. Normalized time is defined as time after beginning-of-charge over time of end-of-discharge. The thick and thin parts of the lines represent the CC and phases, respectively. Same experiment as in Figure 8.

**Figure 10.**Direct comparison of experimental and simulated displacement for 0.05 C and 10 C, (

**a**) as function of charge throughput and (

**b**) as function of normalized time. Same data as in Figure 9. In panel (

**a**), right-facing arrows indicate the charging phases and left-facing arrows the discharging. In panel (

**b**), normalized time is defined as time after beginning-of-charge over time of end-of-discharge. The data sets “Sim. ideal” are simulations with the assumption of ideal solid solution behavior.

**Figure 11.**Simulated displacement and its contributions during a charge-discharge cycle. Left panels (

**a**), (

**c**) at 0.05 C, right panels (

**b**), (

**d**) at 10 C. The upper panels (

**a**), (

**b**) show a stacked area plot of the intercalation-induced ${\Delta L}_{\mathrm{int}}$ and thermal-induced ${\Delta L}_{\mathrm{th}}$ displacement components. The lower panels (

**c**), (

**d**) show individual contributions of the three AM to the intercalation-induced displacement ${\Delta L}_{\mathrm{int}}$.

**Figure 12.**Simulated spatial profiles of the simulated strain distribution along the dimension of the electrode pair (y scale, cf. Figure A1 in the Appendix A) during charging at a rate of (

**a**,

**c**) 0.05 C and (

**b**,

**d**) 10 C. The upper panels (

**a**,

**b**) consider thermal expansion while at the lower panels (

**c**,

**d**) thermal expansion was switched off. Also note that for 10 C (

**b**,

**d**) the time steps are displayed in minutes and logarithmically spaced to better resolve the fast progression. Note that only the charge (CCCV) is shown here. SEP, separator.

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**MDPI and ACS Style**

Schmider, D.; Bessler, W.G.
Thermo-Electro-Mechanical Modeling and Experimental Validation of Thickness Change of a Lithium-Ion Pouch Cell with Blend Positive Electrode. *Batteries* **2023**, *9*, 354.
https://doi.org/10.3390/batteries9070354

**AMA Style**

Schmider D, Bessler WG.
Thermo-Electro-Mechanical Modeling and Experimental Validation of Thickness Change of a Lithium-Ion Pouch Cell with Blend Positive Electrode. *Batteries*. 2023; 9(7):354.
https://doi.org/10.3390/batteries9070354

**Chicago/Turabian Style**

Schmider, David, and Wolfgang G. Bessler.
2023. "Thermo-Electro-Mechanical Modeling and Experimental Validation of Thickness Change of a Lithium-Ion Pouch Cell with Blend Positive Electrode" *Batteries* 9, no. 7: 354.
https://doi.org/10.3390/batteries9070354