# The Holby–Morgan Model of Platinum Catalyst Degradation in PEM Fuel Cells: Range of Feasible Parameters Achieved Using Voltage Cycling

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## Abstract

**:**

## 1. Introduction

- Platinum is fully dissolved from catalyst;
- Platinum surface is fully covered by oxide;
- Platinum nano-particles, which were initially uniformly distributed, agglomerate in a single band in the middle of the catalyst layer.

## 2. Materials

## 3. Methods

## 4. Results

## 5. Discussion

- For basis values of the model parameters, the critical value of maximum feasible temperatures is about 110 °C.
- ECSA ratio loss rate $\dot{E}$ in (14) increases monotonically with the increase in the temperature in the feasible range.

- For the temperature fixed at $T=80$ °C, the maximum value for feasible pH is around 1.4.
- Active area RLR $\dot{E}$ decreases monotonically with the increase in pH within the feasible range 0–1.4.

- For pH = 0 fixed, tested diameters of platinum particles are feasible when larger than the minimum of 2 nm.
- ECSA RLR decreases monotonically when increasing the Pt diameter in the range of 2–10 nm.
- $\dot{E}$ drops significantly for ${d}_{\mathrm{Pt}}$ less than 2.5 nm, and shows few changes for ${d}_{\mathrm{Pt}}$ larger than 5 nm.

- For ${d}_{\mathrm{Pt}}=3$ nm fixed, ECSA RLR decreases monotonically when increasing the Pt particle loading within all tested ranges of 0–10 mg/cm${}^{2}$.
- $\dot{E}$ shows a larger drop for ${p}_{\mathrm{Pt}}=$ 0–1 mg/cm${}^{2}$ than for ${p}_{\mathrm{Pt}}=$ 1–10 mg/cm${}^{2}$, where it is close to a linear relationship.

- For ${p}_{\mathrm{Pt}}=0.4$ mg/cm${}^{2}$ fixed, the platinum to carbon volume fraction has the minimum of 0.002 for feasible values.
- ECSA RLR raises monotonically with the increase in the Pt/C volume fraction within the feasible range of 0.002–1.
- $\dot{E}$ lifts significantly for $\epsilon \le 0.05$, and changes very little for $\epsilon \ge 0.4$.

## 6. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

2D/3D | Two/three dimensions |

AST | Accelerated stress test |

BoL/EoL | Beginning/end of life |

C | Carbon |

CL | Catalyst layer |

CV | Cyclic voltammetry |

ECSA | Electrochemical surface area |

FC | Fuel cell |

FCH JU | Fuel cell and hydrogen joint undertaking |

GDL | Gas diffusion layer |

HOR | Hydrogen oxidation reaction |

LPL/UPL | Lower/upper potential level |

MEA | Membrane electrode assembly |

OAT | One parameter at a time |

ORR | Oxygen reduction reaction |

PEM | Polymer electrolyte membrane/proton exchange membrane |

PEMFC | Polymer electrolyte fuel cell |

pH | Potential of hydrogen |

PSD | Particle size distribution |

Pt/PtO/Pt${}^{2+}$ | Platinum/platinum oxide/platinum ion |

Pt/C | Platinum on carbon |

RLR | Ratio loss rate |

RH | Relative humidity |

SW/TW | Square/triangle wave |

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**Figure 2.**Feasible solution: (

**a**) Pt ion concentration c [mol/cm${}^{3}$]. (

**b**) Pt particle diameter d [nm]. (

**c**) PtO coverage ratio $\theta $ [1] versus $t\in (0,1.\overline{1})$ [h] and $x\in (0,10)$ [$\mathsf{\mu}$m].

**Figure 4.**Unfeasible solution at $T=120$ °C: (

**a**) Pt ion concentration c [mol/cm${}^{3}$]. (

**b**) Pt particle diameter d [nm]. (

**c**) PtO coverage ratio $\theta $ [1] versus $t\in (0,1.\overline{1})$ [h] and $x\in (0,10)$ [$\mathsf{\mu}$m].

**Figure 5.**Unfeasible solution at pH = 1.5: (

**a**) Pt ion concentration c [mol/cm${}^{3}$]. (

**b**) Pt particle diameter d [nm]. (

**c**) PtO coverage ratio $\theta $ [1] versus $t\in (0,1.\overline{1})$ [h] and $x\in (0,10)$ [$\mathsf{\mu}$m].

**Figure 6.**RLR parameter $\dot{E}$ [${10}^{-3}$/h] in dependence on: Pt particle diameter ${d}_{\mathrm{Pt}}$ [nm] (

**a**). Pt particle loading ${p}_{\mathrm{Pt}}$ [mg/cm${}^{2}$] (

**b**). Pt/C volume fraction $\epsilon $ [1] (

**c**).

**Figure 7.**Unfeasible solution at ${d}_{\mathrm{Pt}}=1$ nm: (

**a**) Pt ion concentration c [mol/cm${}^{3}$]. (

**b**) Pt particle diameter d [nm]. (

**c**) PtO coverage ratio $\theta $ [1] versus $t\in (0,1.\overline{1})$ [h] and $x\in (0,10)$ [$\mathsf{\mu}$m].

**Figure 8.**Unfeasible solution at $\epsilon =0.001$: (

**a**) Pt ion concentration c [mol/cm${}^{3}$]. (

**b**) Pt particle diameter d [nm]. (

**c**) PtO coverage ratio $\theta $ [1] versus $t\in (0,1.\overline{1})$ [h] and $x\in (0,10)$ [$\mathsf{\mu}$m].

Symbol | Value | Units | Description |
---|---|---|---|

${\nu}_{1}$ | $1\times {10}^{4}$ | Hz | Dissolution attempt frequency |

${\nu}_{2}$ | $8\times {10}^{5}$ | Hz | Backward dissolution rate factor |

${\beta}_{1}$ | 0.5 | Butler transfer coefficient for Pt dissolution | |

n | 2 | Electrons transferred during Pt dissolution | |

${U}_{\mathrm{eq}}$ | 1.118 | V | Pt dissolution bulk equilibrium voltage |

$\Omega $ | 9.09 | cm${}^{3}$/mol | Molar volume of Pt |

$\gamma $ | $2.4\times {10}^{-4}$ | J/cm${}^{2}$ | Pt [1 1 1] surface tension |

${c}_{\mathrm{ref}}$ | 1 | mol/cm${}^{3}$ | Reference Pt ion concentration |

${H}_{1,\mathrm{fit}}$ | $4.4\times {10}^{4}$ | J/mol | partial molar Pt dissolution activation enthalpy |

${D}_{\mathrm{Pt}}$ | $1\times {10}^{-6}$ | cm${}^{2}$/s | Diffusion coefficient of Pt ion in the membrane |

${\nu}_{1}^{\u2605}$ | $1\times {10}^{4}$ | Hz | Forward Pt oxide formation rate constant |

${\nu}_{2}^{\u2605}$ | $2\times {10}^{-2}$ | Hz | Backward Pt oxide formation rate constant |

$\Gamma $ | $2.2\times {10}^{-9}$ | mol/cm${}^{2}$ | Pt surface site density |

${\beta}_{2}$ | 0.5 | Butler transfer coefficient for PtO formation | |

${n}_{2}$ | 2 | Electrons transferred during Pt oxide formation | |

${U}_{\mathrm{fit}}$ | 0.8 | V | Pt oxide formation bulk equilibrium voltage |

$\lambda $ | $2\times {10}^{4}$ | J/mol | Pt oxide dependent kinetic barrier constant |

$\omega $ | $5\times {10}^{4}$ | J/mol | Pt oxide-oxide interaction energy |

${H}_{2,\mathrm{fit}}$ | $1.2\times {10}^{4}$ | J/mol | Partial molar oxide formation activation enthalpy |

Symbol | Value | Units | Description |
---|---|---|---|

T | 353.15 | K | Temperature |

$pH$ | 0 | Potential of hydrogen | |

${d}_{\mathrm{Pt}}$ | $3\times {10}^{-7}$ | cm | Pt particle diameter |

${p}_{\mathrm{Pt}}$ | $4\times {10}^{-4}$ | g/cm${}^{2}$ | Pt particles loading |

$\epsilon $ | 0.02 | Pt/C volume fraction | |

${\rho}_{\mathrm{Pt}}$ | 21.45 | g/cm${}^{3}$ | Pt particles density |

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**MDPI and ACS Style**

Kovtunenko, V.A.
The Holby–Morgan Model of Platinum Catalyst Degradation in PEM Fuel Cells: Range of Feasible Parameters Achieved Using Voltage Cycling. *Technologies* **2023**, *11*, 184.
https://doi.org/10.3390/technologies11060184

**AMA Style**

Kovtunenko VA.
The Holby–Morgan Model of Platinum Catalyst Degradation in PEM Fuel Cells: Range of Feasible Parameters Achieved Using Voltage Cycling. *Technologies*. 2023; 11(6):184.
https://doi.org/10.3390/technologies11060184

**Chicago/Turabian Style**

Kovtunenko, Victor A.
2023. "The Holby–Morgan Model of Platinum Catalyst Degradation in PEM Fuel Cells: Range of Feasible Parameters Achieved Using Voltage Cycling" *Technologies* 11, no. 6: 184.
https://doi.org/10.3390/technologies11060184