# A Thermodynamic Model for Lithium-Ion Battery Degradation: Application of the Degradation-Entropy Generation Theorem

^{*}

## Abstract

**:**

## 1. Introduction

#### Background/Literature Review

_{i}are numbers of moles, and G is Gibbs free energy. The dot above quantities represents differentiation with time t. The entropy produced S’ from the energy dissipated via the Ohmic or chemical reaction, was evaluated from Prigogine’s minimum entropy generation theorem [19] as.

## 2. Degradation-Entropy Generation Theorem Review

#### 2.1. Statement

#### 2.2. Generalized Degradation Analysis Procedure

- It identifies the degradation measure w, dissipative process energies ${p}_{i},$ and phenomenological variables ${\zeta}_{ij}$;
- It finds the entropy generation ${S}_{i}\prime $ = ${S}_{i}\prime $(${p}_{i}$) caused by the dissipative processes ${p}_{i}$;
- It evaluates the coefficients ${B}_{i}$ by measuring increments, accumulation or rates of degradation versus increments, accumulation or rates of entropy generation;
- It relates degradation measure to entropy generation, via Equations (7) or (9).

## 3. Formulations

#### 3.1. Fundamental Thermodynamic Formulations

#### 3.1.1. First Law—Energy Conservation

#### 3.1.2. Second Law and Entropy Balance—Irreversible Entropy Generation

#### 3.1.3. Combining First and Second Laws

#### 3.2. Li-Ion Battery Analysis

#### 3.2.1. Combining First and Second Laws with Gibbs Potential

_{e}or entropy flow dS

_{e}into or out of the battery, neither dG nor dS measures the permanent changes in the battery. On the other hand, entropy generation, Equation (24) or (26), evolves monotonically as per the second law. With $\delta S\prime =0$ indicating an ideal (reversible) battery-process interaction, Equation (26) also indicates that a portion of any real battery’s energy is never available for external work, $\delta S\prime >0$. In Equation (26), the entropy generated by the battery’s internal irreversibilities alone, is in accordance with experience, similar to the Gouy-Stodola theorem used in availability (exergy) analysis [43,45,48,49]. Note since a discharged battery cannot ‘revert’ to a charged state without external recharging, all batteries are thermodynamically irreversible.

#### 3.2.2. Relaxation/Settling and Self Discharge

_{C}, Boltzmann constant k

_{B}, distance x and dynamic friction coefficient $\eta $ (inverse of electrical mobility). Here, $-SdT$, positive for decreasing temperature dT ≤ 0, represents both voltage and thermal relaxation, and the last right-hand side term represents diffusion during settling, all of which proceed spontaneously and significantly slower than the active Ohmic processes [18,19]. Equation (24) also applies to self-discharge during storage, with the external Ohmic term Vdq representing spontaneous charge leakage [2].

_{0}to t gives the total Gibbs energy and entropy generation

#### 3.3. Entropy Content S and Internal Free Energy Dissipation −SdT

^{+}). The molar Gibbs energy at constant temperature and pressure gives the following, via Equation (17),

#### 3.4. Degradation-Entropy Generation (DEG) and Capacity Fade in Batteries

- Let available battery capacity or charge content $\mathcal{C}$ be a DEG transformation measure and capacity fade (lost discharge/charge capacity) $\Delta \mathcal{C}$ be the observed/measured degradation, the DEG Equation (9), with $\Delta \mathcal{C}$ replacing $\Delta $w becomes,$$\Delta \mathcal{C}={\displaystyle \sum}_{i}{B}_{i}S{\prime}_{i}$$
- From Equation (41), entropy generation S’ = S’$\left\{V\left(t\right),I\left(t\right),T\left(t\right)\right\}$, suggesting via Equation (43) that $\mathcal{C}$ = $\mathcal{C}\left\{V\left(t\right),I\left(t\right),T\left(t\right)\right\}$. Substituting the entropy generation terms of Equation (41) into Equation (43) gives$$\Delta \mathcal{C}={B}_{VT}\underset{{t}_{0}}{\overset{t}{{\displaystyle \int}}}\frac{\mathcal{C}\dot{V}}{T}dt+{B}_{\mathsf{\Omega}}\underset{{t}_{0}}{\overset{t}{{\displaystyle \int}}}\frac{VI}{T}dt-{B}_{G}\underset{{t}_{0}}{\overset{t}{{\displaystyle \int}}}\frac{{V}_{OC}{I}_{rev}}{T}dt$$
- Via Equation (8), with $\mathcal{C}$ replacing w, DEG coefficients$${B}_{VT}=\frac{\partial \mathcal{C}}{\partial S{\prime}_{VT}},{B}_{\mathsf{\Omega}}=\frac{\partial \mathcal{C}}{\partial S{\prime}_{\mathsf{\Omega}}},{B}_{G}=\frac{\partial \mathcal{C}}{\partial {S}_{rev}}$$

## 4. Experiments

#### 4.1. Apparatus

#### 4.2. Setup and Procedure

#### 4.2.1. Setup and Initial Measurements

#### 4.2.2. Cycling

- the battery’s capacity fell to less than two-thirds the initial capacity or
- the battery began to inflate in geometric volume (close monitoring of Li-ion batteries was required during cycling).

## 5. Results, Analysis, and Discussion

#### 5.1. Gibbs Energy and Entropy Components

#### 5.2. DEG—Capacity Versus Entropy

#### 5.2.1. Phenomenological Charge, Measured Charge, and Reversible Charge

#### 5.2.2. Evaluating Capacity Fade—Battery Cycle Life Model

- Cyclic values of DEG capacity fade $\Delta {\mathcal{C}}_{DEG}$ and Coulomb-Counted capacity fade $\Delta {\mathcal{C}}_{CC}$ presented in Table 2 for cycles 1 to 32, were obtained as follows:

- Evaluate phenomenological charge ${\mathcal{C}}_{phen}$ in Table 2, columns 2 (discharge) and 7 (charge), from Equation (47), by combining reference DEG coefficients with each cycle’s phenomenological entropy components $S{\prime}_{\Omega}$ and $S{\prime}_{VT}$ given in Table 1. For example, the discharge step of cycle 6, row 6 of Table 1, has ${S\prime}_{\mathsf{\Omega}}=$ −0.08 Wh/K, and ${S\prime}_{\mathrm{VT}}=$ –0.005 Wh/K. When combined with ${B}_{\mathsf{\Omega}}=76.6$ Ah K/Wh and ${B}_{VT}=113$ Ah K/Wh from cycle 1, Equation (47) gives ${\mathcal{C}}_{\mathrm{phen}}$ = (76.6 × −0.08) + (113 × −0.005) = −6.7 Ah.
- Evaluate reversible charge ${\mathcal{C}}_{rev}$ (Table 2, columns 3 and 8) from Equation (49). For example, the cycle 1 starting current ${I}_{1}\left({t}_{1}\right)$ = −5.2 A gives the cycle 6 discharge ${I}_{rev}=$ −5.2 A which, with the cycle 6 discharge duration $\Delta t$ = 1.53 h (Table A1 in Appendix B), gives ${\mathcal{C}}_{\mathrm{rev}}$ = −5.2 × 1.53= −8.0 Ah.
- Evaluate DEG capacity fade (Table 2, columns 4 and 9) from Equation (51), $\Delta {\mathcal{C}}_{DEG}=$ ${\mathcal{C}}_{phen}-{\mathcal{C}}_{rev}$. For cycle 6 discharge, $\Delta {\mathcal{C}}_{\mathrm{DEG}}=$ −6.7 − (−8.0) = 1.3 Ah. Similarly, for cycle 6 charge, $\Delta {\mathcal{C}}_{\mathrm{DEG}}=$ 0.3 Ah.
- Evaluate cyclic Coulomb-Counted charge transfer (Table 2, columns 5 and 10), ${\mathcal{C}}_{\mathrm{CC}}={{\displaystyle \int}}_{{\mathrm{t}}_{0}}^{\mathrm{t}}I\left(t\right)\mathrm{dt}$ where I(t) is instantaneously measured unsteady current. For cycle 6 discharge and charge, row 6 of Table 2, ${\mathcal{C}}_{\mathrm{CC}}$= −7.2 Ah and 4.4 Ah respectively.
- Evaluate Coulomb-Counted capacity fade (Table 2, column 6) from Equation (42), $\Delta {\mathcal{C}}_{CC}=$ $\left|{\mathcal{C}}_{1}\right|-\left|{\mathcal{C}}_{CC}\right|$, where ${\mathcal{C}}_{1}=$ −6.1 Ah is Coulomb-Counted charge transfer during cycle 1 discharge. For cycle 6, $\Delta {\mathcal{C}}_{\mathrm{CC}}$ = |−6.1| − |−7.2| = −1.1 Ah.

## 6. Discussion

- phenomenological entropy generation $S{\prime}_{phen}$ is the sum of Ohmic entropy $S{\prime}_{\Omega}$ and electro-chemico-thermal ECT entropy S′
_{VT}; - entropy generation is the difference between phenomenological $S{\prime}_{phen}$ and reversible $S{\prime}_{rev}$ Gibbs entropies, at every instant;
- entropy generation is always non-negative, in accordance with the second law, whereas components $S{\prime}_{phen}$ and $S{\prime}_{rev}$ are directional—positive during charge and negative during discharge. This implies $\left|S{\prime}_{phen}\right|\ge \left|S{\prime}_{rev}\right|$ during charge and $\left|S{\prime}_{phen}\right|\le \left|S{\prime}_{rev}\right|$ during discharge, in accordance with experience and thermodynamic laws. This article demonstrated the significance of the previously neglected reversible $S{\prime}_{rev}$ and ECT $S{\prime}_{VT}$ entropies in evaluating entropy generation in batteries.

#### 6.1. Features of the DEG Theorem and Coefficients

#### 6.1.1. DEG Trajectories, Surfaces, and Domains

## 7. Summary and Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Nomenclature | Name | Unit |

$A$ | chemical affinity | J/mol |

B | DEG coefficient | Ah K/Wh |

$\mathcal{C}$ | charge, charge transfer or capacity | Ah |

$\Delta \mathcal{C}$ | charge or capacity fade | Ah |

F | Faraday’s constant | C/mol |

G | Gibbs energy | Wh |

I | discharge/charge current or rate | A |

k_{B} | Boltzmann constant | J/K |

m | mass | kg |

n’ | number of charge species | |

N | cycle number | Kg/mol |

N, N_{k} | number of moles of substance | mol |

p | dissipative process energy | J |

P | pressure | Pa |

q | charge | Ah |

Q | heat | J |

R | gas constant | J/mol·K |

S | entropy or entropy content | Wh/K |

S’ | entropy generation or production | Wh/K |

t | time | sec |

T | temperature | degC or K |

U | internal energy | J |

V | voltage | V |

$\mathcal{V}$ | volume | m^{3} |

w | degradation measure | |

W | work | J |

Symbols | ||

μ | chemical potential | |

ζ | phenomenological variable | |

Subscripts & acronyms | ||

Ω | Ohmic | |

0 | initial | |

c | charge | |

d | discharge | |

ECT, VT | Electro-Chemico-Thermal | |

t | time | |

rev | reversible | |

irr | irreversible | |

phen | phenomenological | |

CC | Coulomb-Counted | |

DEG | Degradation-Entropy Generation |

## Appendix A

#### Appendix A.1. Electrochemical Kinetics

#### Appendix A.1.1. Charge Intercalation (Absorption) and Deintercalation (Release)

#### Appendix A.1.2. Reaction Rates

#### Appendix A.1.3. Charge Transport

^{+}in both regions. Activity ${a}_{k}^{low}$ could also be used for convenience.

^{2}·s) in terms of Li

^{+}concentration M

_{c}(${x}_{k}$,t) (mol/cm

^{3}) and the diffusion coefficient D (cm

^{2}/s) can be written as

_{B}gives the Stokes-Einstein equation for electrical mobility

#### Appendix A.2. Coupling Reaction and Transport Kinetics with the Electrochemical Potential

## Appendix B

#### Experimental Results

**Table A1.**Monitored and processed parameters for the Li-ion battery #2 (initial discharge current: ~5 A, charge current: 3 A).

Discharge | Charge | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

N | $\mathbf{\Delta}\mathbf{t}$ h | ${\mathcal{C}}_{\mathbf{t}}$ Ah | $\mathbf{W}$ Wh | $\mathbf{\Delta}\mathbf{T}$ degC | ${\mathbf{T}}_{\mathbf{\infty}}$ degC | ${\mathbf{V}}_{\mathbf{e}\mathbf{n}\mathbf{d}}$ V | ${\mathbf{Z}}_{0.5\mathbf{h}\mathbf{r}}$ $\mathsf{\Omega}$ | SoH | $\mathbf{\Delta}\mathbf{t}$ hr | ${\mathcal{C}}_{\mathbf{t}}$ Ah | $\mathbf{W}$ Wh | $\mathbf{\Delta}\mathbf{T}$ degC | ${\mathbf{T}}_{\mathbf{\infty}}$ degC |

1 | 1.47 | −6.10 | −19.9 | 6.5 | 25.9 | 0.77 | 0.042 | 1.00 | 3.49 | 10.47 | 41.4 | 0.2 | 27.6 |

2 | 1.79 | −8.37 | −28.6 | 10.2 | 24.8 | 2.43 | 0.025 | 0.98 | 1.72 | 5.14 | 20.2 | 0.8 | 26.7 |

3 | Half cycle | 1.33 | 3.97 | 16.1 | 8.1 | 25.7 | |||||||

4 | 2.00 | −9.52 | −32.9 | 6.2 | 27.8 | 2.66 | 0.031 | 0.98 | 1.28 | 3.83 | 15.1 | −2.7 | 27.7 |

5 | 0.64 | −3.00 | −10.2 | 8.6 | 24.6 | 2.69 | 0.081 | 0.95 | 1.81 | 5.42 | 21.8 | −0.3 | 26.5 |

6 | 1.53 | −7.23 | −24.6 | 7.2 | 26.1 | 2.61 | 0.026 | 0.97 | 1.47 | 4.41 | 17.4 | 0.4 | 26.7 |

7 | 1.40 | −6.71 | −22.9 | 6.8 | 24.8 | 2.67 | 0.032 | 0.99 | 2.73 | 8.17 | 32.8 | 0.0 | 25.3 |

8 | 2.03 | −9.74 | −33.4 | 8.8 | 24.6 | 2.64 | 0.028 | 1.00 | Missing data | ||||

9 | 1.36 | −6.39 | −21.5 | 6.3 | 24.9 | 2.58 | 0.026 | 0.98 | 3.60 | 10.80 | 43.3 | 2.6 | 27.1 |

10 | 1.77 | −8.70 | −30.3 | 7.1 | 26.8 | 2.87 | 0.024 | 0.99 | 0.61 | 1.81 | 7.0 | −1.7 | 27.7 |

11 | 1.09 | −5.18 | −17.5 | 9.2 | 23.5 | 2.42 | 0.027 | 0.99 | 1.62 | 4.86 | 19.2 | 0.9 | 26.2 |

12 | 1.82 | −8.52 | −29.0 | 6.9 | 26.6 | 1.88 | 0.032 | 0.99 | 1.80 | 5.40 | 21.5 | 2.8 | 26.3 |

13 | Half cycle | 2.13 | 6.37 | 25.3 | −0.7 | 28.5 | |||||||

14 | 2.12 | −10.54 | −37.0 | 6.3 | 29.7 | 2.32 | 0.028 | 1.03 | 0.99 | 2.96 | 11.5 | −0.7 | 29.5 |

15 | 1.33 | −6.24 | −21.0 | 8.8 | 24.7 | 2.42 | 0.040 | 0.98 | 1.53 | 4.58 | 18.1 | −0.1 | 26.7 |

16 | 1.73 | −8.27 | −28.1 | 7.9 | 24.8 | 2.14 | 0.025 | 1.00 | 1.46 | 4.38 | 17.3 | −0.7 | 25.8 |

17 | 1.67 | −7.97 | −27.0 | 8.9 | 25.4 | 2.31 | 0.029 | 1.00 | 1.81 | 5.42 | 21.5 | −0.5 | 27.5 |

18 | 2.13 | −10.01 | −33.6 | 7.3 | 26.2 | 1.46 | 0.027 | 1.02 | 1.28 | 3.84 | 15.0 | 0.7 | 27.5 |

19 | 1.36 | −6.42 | −21.4 | 5.6 | 25.6 | 2.18 | 0.026 | 0.99 | 1.47 | 4.40 | 17.4 | −0.6 | 26.1 |

20 | 2.00 | −8.95 | −29.2 | 8.0 | 25.8 | 0.95 | 0.027 | 1.01 | 1.69 | 5.06 | 20.1 | 0.1 | 28.2 |

21 | 2.16 | −9.45 | −30.4 | 9.0 | 27.2 | 1.10 | 0.028 | 1.02 | 1.27 | 3.79 | 14.9 | −0.1 | 26.7 |

22 | 1.50 | −6.78 | −21.8 | 7.4 | 25.6 | 1.58 | 0.031 | 0.98 | 1.41 | 4.22 | 16.7 | 0.9 | 25.8 |

23 | 1.74 | −7.85 | −25.8 | 5.9 | 25.3 | 1.93 | 0.021 | 0.95 | 1.50 | 4.50 | 17.8 | 2.0 | 26.0 |

24 | 1.80 | −8.19 | −26.7 | 11.1 | 24.0 | 1.68 | 0.025 | 0.97 | 1.73 | 5.17 | 20.5 | −0.2 | 26.5 |

25 | 0.68 | −3.35 | −11.7 | 3.4 | 25.6 | 3.44 | 0.025 | 0.97 | 4.16 | 12.47 | 50.9 | 2.1 | 26.3 |

26 | 2.69 | −12.37 | −40.6 | 8.0 | 25.0 | 1.56 | 0.036 | 1.00 | 1.48 | 4.43 | 17.5 | 2.4 | 26.8 |

27 | 2.44 | −8.84 | −25.7 | 6.9 | 24.9 | 0.57 | 0.034 | 0.95 | 2.76 | 8.27 | 33.1 | 3.1 | 26.7 |

28 | Half cycle | 1.77 | 5.30 | 21.1 | 3.1 | 26.3 | |||||||

29 | 1.91 | −6.41 | −18.1 | 8.4 | 25.5 | 0.58 | 0.045 | 0.93 | 3.08 | 9.22 | 37.2 | 2.3 | 25.9 |

30 | 0.91 | −3.75 | −11.4 | 13.0 | 25.2 | 1.15 | 0.074 | 0.96 | 3.33 | 9.97 | 40.4 | 2.7 | 26.3 |

31 | 1.31 | −5.76 | −17.7 | 8.3 | 25.8 | 1.80 | 0.051 | 0.95 | 3.21 | 9.63 | 39.2 | 7.2 | 26.1 |

32 | 0.75 | −3.19 | −9.7 | 12.3 | 25.4 | 1.51 | 0.129 | 0.94 | 1.69 | 5.05 | 20.0 | 0.9 | 25.7 |

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**Figure 1.**Photo of a portion of actual experimental set-up used in Li-ion battery measurements, showing four Li-ion batteries, a commercial charger, high-power discharge resistors, connector cables, and thermocouple wires.

**Figure 3.**Monitored parameters during cycling (cycle 6 data) showing 1.5 h discharge starting at ~5 A, followed by 0.4 h settling and 1.5 h constant-current charge at 3 A.

**Figure 4.**Phenomenological entropy components (Ohmic and Electro-Chemico-Thermal (ECT)) vs. measured charge transfer during discharge (solid curves) and charge (dashed curves).

**Figure 5.**Entropy generation component rates (Ohmic, ECT, and reversible Gibbs) over time, during discharge and charge.

**Figure 6.**Entropy generation $S\prime $ and components during cycle 6′s discharge and charge steps. The region between reversible entropy and phenomenological entropy curves represents entropy generation.

**Figure 7.**(

**a**) 3D plots and linear surface fits of Charge (vertical axes) vs. Ohmic and ECT entropies (horizontal axes) during discharge (red plane, curve ‘D’ starts from upper right corner) and charge (green plane, curve ‘C’ starts from lower left corner) steps of cycle 6, indicating a linear dependence on two active processes. (

**b**) The end projection of the planes and curves of (

**a**), and a visual of the goodness of fit. Axes are not to scale.

**Figure 8.**(

**a**) 3D scatter of cyclic phenomenological ${\mathcal{C}}_{phen}$ (vertical axis) vs. Ohmic and ECT entropies (horizontal axes) for all discharge steps (purple dots on red plane) and charge steps (white dots on green plane) shows that all 32 cycles lay on cycle 1’s DEG planes, a visual of Equations (52) and (53); (

**b**) end projection of (

**a**) shows a linear relationship. Axes are not to scale.

**Figure 9.**Capacity fade and components for cycle 6′s discharge and charge. The region between reversible and phenomenological charge transfer curves is capacity fade $\Delta \mathcal{C}$.

**Table 1.**Processed Gibbs energy and entropy parameters for Li-ion battery #2 (Discharge rate: ~5 A, Charge rate: 3 A). Cycle 6 (in bold) is used in the breakdown in this section.

Discharge | Charge | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

N | ${\mathbf{G}}_{\mathbf{\Omega}}$ Wh | ${\mathbf{G}}_{\mathbf{V}\mathbf{T}}$ Wh | ${{\mathbf{S}}^{\prime}}_{\mathbf{\Omega}}$ Wh/K | ${{\mathbf{S}}^{\prime}}_{\mathbf{V}\mathbf{T}}$ Wh/K | ${{\mathbf{S}}^{\prime}}_{\mathbf{r}\mathbf{e}\mathbf{v}}$ Wh/K | ${\mathbf{G}}_{\mathbf{\Omega}}$ Wh | ${\mathbf{G}}_{\mathbf{V}\mathbf{T}}$ Wh | ${{\mathbf{S}}^{\prime}}_{\mathbf{\Omega}}$ Wh/K | ${{\mathbf{S}}^{\prime}}_{\mathbf{V}\mathbf{T}}$ Wh/K | ${{\mathbf{S}}^{\prime}}_{\mathbf{r}\mathbf{e}\mathbf{v}}$ Wh/K |

1 | −19.93 | −3.00 | −0.07 | −0.010 | −0.11 | 41.40 | 1.00 | 0.14 | 0.003 | 0.13 |

2 | −28.62 | −2.22 | −0.09 | −0.007 | −0.13 | 20.15 | 0.55 | 0.07 | 0.002 | 0.07 |

3 | Half cycle | 16.14 | 0.08 | 0.05 | 0.000 | 0.05 | ||||

4 | −32.85 | −2.72 | −0.11 | −0.009 | −0.15 | 15.08 | 0.43 | 0.05 | 0.001 | 0.05 |

5 | −10.20 | −0.73 | −0.03 | −0.002 | −0.05 | 21.75 | 0.38 | 0.07 | 0.001 | 0.07 |

6 | −24.57 | −1.64 | −0.08 | −0.005 | −0.11 | 17.44 | 0.50 | 0.06 | 0.002 | 0.06 |

7 | −22.92 | −1.73 | −0.08 | −0.006 | −0.10 | 32.75 | 0.80 | 0.11 | 0.003 | 0.11 |

8 | −33.43 | −2.63 | −0.11 | −0.009 | −0.16 | Missing data | ||||

9 | −21.52 | −1.39 | −0.07 | −0.005 | −0.10 | 43.26 | 0.71 | 0.14 | 0.002 | 0.14 |

10 | −30.25 | −1.55 | −0.10 | −0.005 | −0.13 | 6.99 | 0.16 | 0.02 | 0.001 | 0.02 |

11 | −17.48 | −1.13 | −0.06 | −0.004 | −0.07 | 19.23 | 0.56 | 0.06 | 0.002 | 0.06 |

12 | −29.00 | −2.89 | −0.10 | −0.010 | −0.13 | 21.49 | 0.62 | 0.07 | 0.002 | 0.07 |

13 | Half cycle | 25.34 | 0.87 | 0.08 | 0.003 | 0.08 | ||||

14 | −37.01 | −3.11 | −0.12 | −0.010 | −0.16 | 11.49 | 0.36 | 0.04 | 0.001 | 0.04 |

15 | −20.97 | −2.00 | −0.07 | −0.007 | −0.10 | 18.07 | 0.57 | 0.06 | 0.002 | 0.06 |

16 | −28.13 | −1.95 | −0.09 | −0.006 | −0.12 | 17.29 | 0.51 | 0.06 | 0.002 | 0.06 |

17 | −26.99 | −2.28 | −0.09 | −0.008 | −0.12 | 21.45 | 0.65 | 0.07 | 0.002 | 0.07 |

18 | −33.60 | −3.65 | −0.11 | −0.012 | −0.16 | 15.00 | 0.46 | 0.05 | 0.002 | 0.05 |

19 | −21.43 | −1.62 | −0.07 | −0.005 | −0.10 | 17.43 | 0.42 | 0.06 | 0.001 | 0.06 |

20 | −29.16 | −3.74 | −0.10 | −0.012 | −0.15 | 20.05 | 0.65 | 0.07 | 0.002 | 0.06 |

21 | −30.39 | −4.53 | −0.10 | −0.015 | −0.16 | 14.90 | 0.24 | 0.05 | 0.001 | 0.05 |

22 | −21.75 | −2.56 | −0.07 | −0.008 | −0.11 | 16.70 | 0.27 | 0.06 | 0.001 | 0.05 |

23 | −25.83 | −1.72 | −0.09 | −0.006 | −0.13 | 17.79 | 0.51 | 0.06 | 0.002 | 0.06 |

24 | −26.72 | −2.70 | −0.09 | −0.009 | −0.13 | 20.47 | 0.37 | 0.07 | 0.001 | 0.07 |

25 | −11.73 | −0.31 | −0.04 | −0.001 | −0.05 | 50.85 | 0.56 | 0.17 | 0.002 | 0.16 |

26 | −40.64 | −5.32 | −0.13 | −0.017 | −0.21 | 17.52 | 0.50 | 0.06 | 0.002 | 0.06 |

27 | −25.68 | −5.24 | −0.08 | −0.017 | −0.18 | 33.06 | 0.51 | 0.11 | 0.002 | 0.11 |

28 | Half cycle | 21.13 | 0.26 | 0.07 | 0.001 | 0.07 | ||||

29 | −18.09 | −3.55 | −0.06 | −0.012 | −0.14 | 37.18 | 0.49 | 0.12 | 0.002 | 0.12 |

30 | −11.39 | −1.53 | −0.04 | −0.005 | −0.07 | 40.44 | 0.44 | 0.13 | 0.001 | 0.13 |

31 | −17.74 | −2.32 | −0.06 | −0.008 | −0.09 | 39.20 | 0.54 | 0.13 | 0.002 | 0.13 |

32 | −9.67 | −1.21 | −0.03 | −0.004 | −0.06 | 19.96 | 0.74 | 0.07 | 0.002 | 0.07 |

SUMMARY/TOTAL | ||||||||||

−2.34 | −0.234 | −3.47 | 2.43 | 0.053 | 2.39 |

**Table 2.**Processed Degradation-Entropy Generation (DEG) and Coulomb-Counted capacity fade parameters for Li-ion battery #2 (Discharge rate: ~5 A, Charge rate: 3 A). Cycle 6 (in bold) is used in the Capacity Fade discussion.

Discharge | Charge | ||||||||
---|---|---|---|---|---|---|---|---|---|

N | ${\mathcal{C}}_{\mathbf{p}\mathbf{h}\mathbf{e}\mathbf{n}}$ Ah | ${\mathcal{C}}_{\mathbf{r}\mathbf{e}\mathbf{v}}$ Ah | $\mathbf{\Delta}{\mathcal{C}}_{\mathbf{D}\mathbf{E}\mathbf{G}}$ Ah | ${\mathcal{C}}_{\mathbf{C}\mathbf{C}}$ Ah | $\mathbf{\Delta}{\mathcal{C}}_{\mathbf{C}\mathbf{C}}$ Ah | ${\mathcal{C}}_{\mathbf{p}\mathbf{h}\mathbf{e}\mathbf{n}}$ Ah | ${\mathcal{C}}_{\mathbf{r}\mathbf{e}\mathbf{v}}$ Ah | $\mathbf{\Delta}{\mathcal{C}}_{\mathbf{D}\mathbf{E}\mathbf{G}}$ Ah | ${\mathcal{C}}_{\mathbf{C}\mathbf{C}}$ Ah |

1 | −6.5 | −7.6 | 1.1 | −6.1 | 0.0 | 10.7 | 10.1 | 0.6 | 10.5 |

2 | −7.7 | −9.3 | 1.6 | −8.4 | −2.3 | 5.3 | 5.0 | 0.3 | 5.2 |

3 | Half cycle | 3.8 | 3.8 | 0.0 | 4.0 | ||||

4 | −9.4 | −10.4 | 1.0 | −9.5 | −3.4 | 3.8 | 3.7 | 0.1 | 3.8 |

5 | −2.5 | −3.3 | 0.8 | −3.0 | 3.1 | 5.3 | 5.2 | 0.1 | 5.4 |

6 | −6.7 | −8.0 | 1.3 | −7.2 | −1.1 | 4.6 | 4.3 | 0.3 | 4.4 |

7 | −6.8 | −7.3 | 0.5 | −6.7 | −0.6 | 8.4 | 7.9 | 0.5 | 8.2 |

8 | −9.4 | −10.6 | 1.2 | −9.7 | −3.6 | Missing data | |||

9 | −5.9 | −7.1 | 1.2 | −6.4 | −0.3 | 10.6 | 10.4 | 0.2 | 10.8 |

10 | −8.2 | −9.2 | 1.0 | −8.7 | −2.6 | 1.5 | 1.5 | 0.0 | 1.6 |

11 | −5.0 | −5.7 | 0.7 | −5.2 | 0.9 | 4.6 | 4.6 | 0.0 | 4.8 |

12 | −8.8 | −9.5 | 0.7 | −8.5 | −2.4 | 5.3 | 5.2 | 0.1 | 5.4 |

13 | Half cycle | 6.1 | 6.1 | 0.0 | 6.3 | ||||

14 | −10.3 | −11.0 | 0.7 | −10.5 | −4.4 | 3.0 | 2.9 | 0.1 | 3.0 |

15 | −6.2 | −6.9 | 0.7 | −6.2 | −0.1 | 4.6 | 4.4 | 0.2 | 4.6 |

16 | −7.6 | −9.0 | 1.4 | −8.3 | −2.2 | 4.6 | 4.2 | 0.4 | 4.4 |

17 | −7.8 | −8.7 | 0.9 | −8.0 | −1.9 | 5.3 | 5.2 | 0.1 | 5.4 |

18 | −9.8 | −11.1 | 1.3 | −10.0 | −3.9 | 3.8 | 3.7 | 0.1 | 3.8 |

19 | −5.9 | −7.1 | 1.2 | −6.4 | −0.3 | 4.6 | 4.3 | 0.3 | 4.4 |

20 | −9.0 | −10.4 | 1.4 | −9.0 | −2.9 | 5.3 | 4.9 | 0.4 | 5.1 |

21 | −9.4 | −11.2 | 1.8 | −9.5 | −3.4 | 3.8 | 3.7 | 0.1 | 3.8 |

22 | −6.3 | −7.8 | 1.5 | −6.8 | −0.7 | 4.6 | 4.1 | 0.5 | 4.2 |

23 | −7.6 | −9.0 | 1.4 | −7.9 | −1.8 | 4.6 | 4.3 | 0.3 | 4.5 |

24 | −7.9 | −9.4 | 1.5 | −8.2 | −2.1 | 5.3 | 5.0 | 0.3 | 5.2 |

25 | −3.2 | −3.5 | 0.3 | −3.4 | 2.8 | 12.9 | 12.0 | 0.9 | 12.5 |

26 | −11.9 | −14.0 | 2.1 | −12.4 | −6.3 | 4.6 | 4.3 | 0.3 | 4.4 |

27 | −8.0 | −12.7 | 4.7 | −8.8 | −2.7 | 8.4 | 8.0 | 0.4 | 8.3 |

28 | Half cycle | 5.3 | 5.1 | 0.2 | 5.3 | ||||

29 | −6.0 | −9.9 | 3.9 | −6.4 | −0.3 | 9.1 | 8.9 | 0.2 | 9.2 |

30 | −3.6 | −4.7 | 1.1 | −3.8 | 2.4 | 9.8 | 9.6 | 0.2 | 10.0 |

31 | −5.5 | −6.8 | 1.3 | −5.8 | 0.3 | 9.9 | 9.3 | 0.6 | 9.6 |

32 | −2.8 | −3.9 | 1.1 | −3.2 | 2.9 | 5.3 | 4.9 | 0.4 | 5.1 |

SUMMARY/TOTAL | |||||||||

−205.7 | −245.0 | 39.3 | −213.8 | 185.0 | 176.8 | 8.2 | 183.2 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Osara, J.A.; Bryant, M.D.
A Thermodynamic Model for Lithium-Ion Battery Degradation: Application of the Degradation-Entropy Generation Theorem. *Inventions* **2019**, *4*, 23.
https://doi.org/10.3390/inventions4020023

**AMA Style**

Osara JA, Bryant MD.
A Thermodynamic Model for Lithium-Ion Battery Degradation: Application of the Degradation-Entropy Generation Theorem. *Inventions*. 2019; 4(2):23.
https://doi.org/10.3390/inventions4020023

**Chicago/Turabian Style**

Osara, Jude A., and Michael D. Bryant.
2019. "A Thermodynamic Model for Lithium-Ion Battery Degradation: Application of the Degradation-Entropy Generation Theorem" *Inventions* 4, no. 2: 23.
https://doi.org/10.3390/inventions4020023