# Predicting Capacity Fade in Silicon Anode-Based Li-Ion Batteries

^{*}

## Abstract

**:**

_{0.8}Co

_{0.15}Al

_{0.05}O

_{2}(NCA) and LiNi

_{1/3}Mn

_{1/3}Co

_{1/3}O

_{2}(NMC 111), were used in combination with silicon to study capacity fade effects using simulations in COMSOL version 5.5. The results of these studies provide insight into the effects of anode particle size and electrolyte volume fraction on the behavior of silicon anode-based batteries with different positive electrodes. It was observed that the performance of a porous matrix of solid active particles of silicon anode could be improved when the active particles were 150 nm or smaller. The range of optimized values of volume fraction of the electrolyte in the silicon anode were determined to be between 0.55 and 0.40. The silicon anode behaved differently in terms of cell time with NCA and NMC. However, NMC111 gave a high relative capacity in comparison to NCA and proved to be a better working electrode for the proposed silicon anode structure.

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Model Development

_{6}) dissolved in a 3:7 liquid mixture of ethylene carbonate (EC) and dimethyl carbonate (DMC).

- ${\mathsf{\epsilon}}_{1,\mathrm{s}}$ = Solid phase volume fraction of the negative electrode
- ${\mathsf{\epsilon}}_{2,\mathrm{s}}$ = Solid phase volume fraction of the positive electrode

- ${\mathrm{i}}_{\mathrm{loc},\mathrm{SEI}}$ = local current density,
- HK = dimensionless silicon expansion factor function, defined as zero during de–intercalation and the value depends on the state of charge of the negative electrode,
- J = exchange current density for the parasitic reaction (dimensionless),
- α = transfer coefficient of the electrochemical reduction reaction,
- ${\mathsf{\eta}}_{\mathrm{SEI}}$ = over-potential (assumes an equilibrium potential of 0 vs. lithium),
- ${\mathrm{q}}_{\mathrm{SEI}}$ = local accumulated charge from the SEI formation,
- f = parameter based on the SEI film properties (dimensionless)

_{SEI}(moles/m

^{3}),

- ${\mathsf{\upsilon}}_{\mathrm{SEI}}$ = reaction coefficient of the SEI species,
- n = number of electrons involved in the reaction,

- ${\mathrm{A}}_{\mathsf{\upsilon}}$ = electrode surface area (1/m)
- F = Faraday’s constant

- $\mathsf{\kappa}$ = SEI film conductivity (S/m)

#### 2.2. Model Assumptions

_{6}as the electrolyte. Since is there is no available experimental data for silicon, the following parameters in Table 1 and Table 2 are taken from COMSOL materials library validated by comparing the experimental data of graphite and EC in LiPF

_{6}[20,21]. The Electrochemical model parameters in Table 3, and governing equations, boundary conditions of this model in Appendix A and Appendix B are used from comsol library [19].

## 3. Results

## 4. Discussion

#### 4.1. Effect of Changing the Si Anode Particle Size

#### 4.2. The Effect of Variations of the Electrolyte Volume Fraction in the Si Electrode

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

${\mathrm{a}}_{\mathrm{s},\mathrm{i}}$ | Specific surface area, m^{2}/m^{3} |

${\mathrm{i}}_{\mathrm{l}}$ | Electronic current density in the solid phase (A m^{−2}) |

${\mathrm{Q}}_{\mathrm{l}}$ | Electrolyte current source |

${\mathrm{i}}_{\mathrm{total}}$ | The sum of all electrochemical current sources |

${\mathrm{t}}_{+}$ | Li^{+} Transference number |

${\mathrm{c}}_{\mathrm{l}}$ | Electrolyte salt concentration (mol m^{−3}) |

$\mathrm{f}$ | Average molar activity coefficient |

${\mathrm{Q}}_{\mathrm{s}}$ | Current source term |

${\mathrm{N}}_{\mathrm{l}}$ | Flux of ions |

${\mathrm{R}}_{\mathrm{l}}$ | Total Li^{+} source term in the electrolyte |

r | Radius distance variable of the solid particles (m) |

${\mathrm{i}}_{\mathrm{loc}}$ | Local current density (A m^{−2}) |

${\mathrm{i}}_{\mathrm{o}}$ | Exchange current density (A m^{−2}) |

T | Battery Temperature (K) |

R | Gas constant, 8.314 (J mol^{−1} K^{−1}) |

${\mathsf{\sigma}}_{\mathrm{l}}$ | Electronic conductivity of solid phase (S m^{−1}) |

${\mathsf{\sigma}}_{\mathrm{s}}$ | Ionic conductivity of the electrolyte (S m^{−1}) |

${\mathsf{\epsilon}}_{1,\mathrm{s}}$ | Solid phase volume fraction of the negative electrode |

${\mathsf{\epsilon}}_{2,\mathrm{s}}$ | Solid phase volume fraction of the positive electrode |

${\mathsf{\epsilon}}_{1,\mathrm{e}}$ | Electrolyte phase volume fraction negative electrode |

${\mathsf{\epsilon}}_{2,\mathrm{e}}$ | Electrolyte phase volume fraction positive electrode |

${\mathsf{\epsilon}}_{\mathrm{l}}$ | Electrolyte volume fraction |

${\mathsf{\epsilon}}_{\mathrm{s}}$ | Electrode volume fraction |

${\mathrm{t}}_{+}$ | Li^{+} Transference number |

H | Overpotential, V |

αa | αc Anodic and cathodic transfer coefficients |

${\varnothing}_{\mathrm{l}}$ | Electrolyte Potential |

${\varnothing}_{\mathrm{s}}$ | Electric Potential |

T | Battery Temperature (K) |

R | Gas constant, 8.314 (J mol^{−1} K^{−1}) |

Subscripts: | |

l | Solution Phase |

s | Solid Phase |

eff | Effective value of transport property in porous medium |

## Appendix A

## Appendix B

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**Figure 2.**Simulated discharge curves comparing NCA cathodes for the first and 2000th cycle when the Si anode particle sizes are (

**a**) 4$\mathsf{\mu}$m, (

**b**) 2$\mathsf{\mu}$m, (

**c**) 1$\mathsf{\mu}$m, (

**d**) 150 nm, (

**e**) 100 nm.

**Figure 4.**Simulated Electrolyte volume fraction at the two boundaries of the negative electrode at the interfaces with the current collector and the separator for porosity values of (

**a**) 0.40; (

**b**) 0.45; (

**c**) 0.50; (

**d**) 0.55.

**Figure 5.**Simulated Comparisons of the discharge curves of 2000 cycles at four rates of discharge (0.5C, 1C, 1.5C, 2C) for NCA cathodes.

**Figure 6.**Simulated Discharge curves comparing NMC cathodes for the first and 2000th cycle when the negative electrode particle sizes are (

**a**) 4$\mathsf{\mu}$m, (

**b**) 3$\mathsf{\mu}$m, (

**c**) 1$\mathsf{\mu}$m, (

**d**) 150 nm, (

**e**) 100 nm.

**Figure 8.**Simulated Electrolyte volume fraction at the two boundaries of the negative electrode at the interfaces with the current collector and the separator for porosity values of (

**a**) 0.40 (

**b**) 0.45 (

**c**) 0.50 (

**d**) 0.55.

**Figure 9.**Simulated Comparisons of the discharge curves after 2000 cycles at four rates of discharge (0.5C, 1C, 1.5C,2C) for NMC cathodes.

Name | Value | Units |
---|---|---|

α | 0.67 | 1 |

J | 8.4 × 10^{−4} | 1 |

f | 2 × 10^{2} | 1/s |

H | 6.7 | 1 |

Parameter | Symbol | Value | Units |
---|---|---|---|

SEI layer conductivity | $\mathsf{\kappa}$ | 5 × 10^{−6} | S/m |

Molar mass of product of side reaction | M_{P} | 0.16 | Kg/mol |

Density of product of side reaction | ${\mathsf{\rho}}_{\mathrm{P}}$ | 1.6 × 10^{3} | Kg/m^{3} |

Parameter | Symbol | NMC | NCA | Separator | Silicon |
---|---|---|---|---|---|

Thickness ($\mathsf{\mu}\mathrm{m}$) | L | 40 | 40 | 30 | 55 |

Particle size (nm) | r_{P} | 100^{assumed} | 100^{assumed} | Varied | |

Volume fraction of the active material | ${\mathsf{\epsilon}}_{\mathrm{s},1},{\mathsf{\epsilon}}_{\mathrm{s},2}$ | Varied | Varied | ||

Volume fraction of the electrolyte | ${\mathsf{\epsilon}}_{1,\mathrm{l}},{\mathsf{\epsilon}}_{2,\mathrm{l}}$ | Varied | Varied | ||

Electrolyte phase volume fraction separator | ${\mathsf{\epsilon}}_{\mathrm{c}}$ | 0.370 | |||

Maximum Lithium concentration in the solid phase (mol/m^{3}) | c_{s, max} | 49,000 | 48,000 | 278,000 | |

Maximum electrode state of charge | ${\mathrm{SOC}}_{\mathrm{max}}$ | 0.975 | 1 | - | 0.98 |

Minimum electrode state of charge | ${\mathrm{SOC}}_{\mathrm{min}}$ | 0 | 0.25 | - | 0 |

Diffusion coefficient of electrodes (m^{2}/S) | ${\mathrm{D}}_{\mathrm{s}}$ | 5× 10^{−13} | 1.5× 10^{−15} | 1× 10^{−13} | |

Diffusion coefficient of electrolyte (m^{2}/S) | ${\mathrm{D}}_{\mathrm{l}}$ | Equation (A1) | |||

Transfer coefficient | $\mathsf{\alpha}$ | 0.5 | 0.5 | 0.5 | |

Transport number | ${\mathrm{t}}_{+}$ | 0.363 | |||

Electrolyte Lithium concentration (mol/m^{3}) | c_{e, max} | 1200 | |||

Bruggeman coefficient for tortuosity | $\mathsf{\gamma}$ | 1.5 | 1.5 | 1.5 | |

Electronic conductivity (S/m) | $\mathsf{\sigma}$ | 100 | 91 | Equation (A2) | 0.1 [22] |

Faraday’s Constant (C mole ^{−1}) | F | 96,487 | |||

Universal gas constant (J/mol/K) | R | 8.314 | |||

Temperature | T | 318.15 K |

**Table 4.**Summary of the effect of particle sizes of the Si electrode (with NCA cathode) on the shape of the discharge curve after 2000 cycles.

Particle Size | Effect on the Discharge Curve |
---|---|

4$\text{}\mathsf{\mu}$m | Polarization persists |

$2\text{}\mathsf{\mu}$m | Polarization persists |

$1\text{}\mathsf{\mu}$m | Polarization persists |

150 nm | No Polarization |

100 nm | No Polarization |

**Table 5.**Summary of the effect of particle sizes of the Si electrode (with NMC cathode) on the shape of the discharge curve after 2000 cycles.

Particle Size | Effect on the Discharge Curve |
---|---|

4$\text{}\mathsf{\mu}$m | Polarization persists |

$3\text{}\mathsf{\mu}$m | Polarization persists |

$1\text{}\mathsf{\mu}$m | Polarization persists |

150 nm | No Polarization |

100 nm | No Polarization |

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**MDPI and ACS Style**

Dasari, H.; Eisenbraun, E. Predicting Capacity Fade in Silicon Anode-Based Li-Ion Batteries. *Energies* **2021**, *14*, 1448.
https://doi.org/10.3390/en14051448

**AMA Style**

Dasari H, Eisenbraun E. Predicting Capacity Fade in Silicon Anode-Based Li-Ion Batteries. *Energies*. 2021; 14(5):1448.
https://doi.org/10.3390/en14051448

**Chicago/Turabian Style**

Dasari, Harika, and Eric Eisenbraun. 2021. "Predicting Capacity Fade in Silicon Anode-Based Li-Ion Batteries" *Energies* 14, no. 5: 1448.
https://doi.org/10.3390/en14051448