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# A system-level modeling framework for predicting Pt catalyst degradation in proton exchange membrane fuel cells

Walid Touil, Zhongliang Li, Rachid Outbib, Daniel Hissel, Samir Jemeï

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Walid Touil, Zhongliang Li, Rachid Outbib, Daniel Hissel, Samir Jemeï. A system-level modeling framework for predicting Pt catalyst degradation in proton exchange membrane fuel cells. Journal of Power Sources, 2025, 625, pp.235628. 10.1016/j.jpowsour.2024.235628. hal-05057229

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# A system-level modeling framework for predicting Pt catalyst degradation in proton exchange membrane fuel cells

Walid Touil<sup>a,b</sup>, Zhongliang Li<sup>b,*</sup>, Rachid Outbib<sup>a</sup>, Daniel Hissel<sup>b,c</sup>, Samir Jemei<sup>b</sup>

## HIGHLIGHTS

* Improving platinum catalyst durability in PEM fuel cells for transport applications.

* Innovative dual-scale model combines multi-physics and degradation model for platinum catalyst.

* Framework enables optimal control strategies to mitigate degradation effects.

* Model reliability supported by validation against stress tests and real-world data, including NEDC.

## ARTICLE INFO

**Keywords:**

PEM fuel cell

Catalyst degradation

Dual-scale modeling

Micro-scale modeling

Durability enhancement control

## ABSTRACT

Enhancing the durability of fuel cell components, especially the platinum catalyst, remains a significant challenge for proton exchange membrane fuel cells in transportation applications. The long-term optimal performance and high durability of fuel cells rely on appropriate system-level operations and controls, while an accuracy model able to predict the fuel cell degradation evolution is essential to support such a control design. In this study, an innovative dual-scale modeling framework comprising a multi-physics model and a Pt catalyst degradation model designed for PEM fuel cells is presented. Specifically, we propose a physics-based degradation model for the Pt catalyst, which takes into account various factors influencing platinum nanoparticle distribution, the multi-physics model interacts with the degradation model to provide the real-time fuel cell's internal state, and interpret the catalyst degradation in the fuel cell performance. The developed dual-scale model is tested against the experimental data obtained from a range of accelerated stress tests and vehicle driving cycle tests. The test results demonstrate interesting agreements between the model predictions and experimental data. The potential of the proposed model framework in fuel cell degradation prediction and optimal control design is therefore justified.

## 1. Introduction

Fuel cell vehicles (FCVs) offer a promising solution to reduce carbon emissions from consumer vehicles and mitigate global warming [1]. In transportation applications, the fuel cell stack is subjected to high load changes, presenting a significant challenge to the durability of fuel cell components, particularly in Proton Exchange Membrane (PEM) fuel cell technology [2].

The primary challenges hindering the commercialization of FCVs are cost and lifetime concerns, primarily attributed to platinum usage as a catalyst. Platinum (Pt) catalyst degradation can result in various detrimental effects on fuel cell performance, including a decrease in catalyst activity [3], and deposition of platinum in the membrane.

These effects may lead to a decline in power output and efficiency, as well as an overall reduction in fuel cells' lifetime. To enhance the affordability of FCVs, it is imperative to increase the stability of deposited Pt to improve the fuel cell's lifetime [4].

The catalyst of fuel cells consists of carbon loaded with nano-sized catalytic Pt particles, which are mixed with an ionomer recast dispersion to create a thin catalytic layer. This layer is positioned between the membrane and a gas diffusion layer (GDL) [5]. Pt is often used as the catalyst for the hydrogen oxidation reaction (HOR) and oxygen reduction reaction (ORR) in PEM fuel cells. In addition, carbon provides the necessary electrical conductivity to transport electrons from/to the active sites of the catalyst particles.

<footnote><sup>a</sup> Université de Aix-Marseille, LIS-LAB, Faculty of Sciences, CNRS, Marseille, France</footnote>

<footnote><sup>b</sup> Université de Franche-Comté, FEMTO-ST, FCLAB, CNRS, Belfort, France</footnote>

<footnote><sup>c</sup> Institut Universitaire de France (IUF), France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: zhongliang.li@univ-fcomte.fr (Z. Li).</footnote>
In high-volume manufacturing, catalysts make up about 10% of fuel cell system costs [6]. For PEM fuel cell stacks in heavy-duty vehicle production in 2022 (50,000 systems annually), catalyst expenses comprise 59% of the stack cost, €112/kW for stacks and €66/kW for catalysts [7]. Platinum plays a significant role in the cost of PEM fuel cell systems. As the Pt catalyst is an essential and relatively expensive component in a fuel cell, improving its durability and reducing the amount of Pt loaded in fuel cells are attracting significant research focuses aimed at making PEM fuel cells more sustainable and economical [8].

To mitigate Pt catalyst degradation, researchers have been investigating various approaches. A first approach involves catalyst design through the development of more robust catalyst materials that are resistant to dissolution, making it a key research area. Researchers are exploring different alloy compositions, surface treatments, and protective coatings to enhance the stability of the Pt catalyst [9–11]. The second approach is based on optimal control, optimizing the operating conditions of the fuel cell [12]. This includes managing load dynamics, reactant relative humidities, temperatures, and reactant supplies in a manner that mitigates Pt catalyst degradation.

Optimizing the operating conditions to mitigate the degradation of the Pt catalyst involves the development of strategies and addressing specific challenges in control design. Such an optimal control can be designed efficiently in model predictive control paradigm [13]. A reliable predictive model is thus necessary to make control decisions aimed at optimizing a set of inputs. Consequently, there is a growing need to establish robust degradation models capable of accurately predicting fuel cell durability under various operations. These degradation models play a pivotal role in the development of advanced monitoring and control strategies, ultimately resulting in enhanced performance and extended lifetimes of fuel cells [14].

Research in this field is ongoing, and new models and methods are continuously being developed to improve the understanding and prediction of Pt catalyst degradation in PEM fuel cells [15]. Darling and Meyers [16,17] put forward the initial degradation model for Pt catalysts. This model describes the dissolution of Pt and its subsequent electrochemical deposition on Pt nanoparticles located at the cathode, without accounting for carbon corrosion, Pt migration, and coalescence. Building upon their work, Wu Bi and Thomas F. Fuller [18] proposed a more comprehensive model, which includes novel features such as the mass loss of cathode Pt into the membrane, leading to the formation of a Pt band. However, both models rely on a simplified bimodal approach, considering only a two-particle size profile to simulate the loss of cathode active surface area due to Pt dissolution and deposition.

Rinaldo et al. [19] introduced a novel nanoparticle population dynamics model for a variety of catalysts. This advanced model not only predicts the changes in the distribution of particle radii over time but also estimates the loss of electrochemically active surface area (ECSA). It takes into consideration the initial distribution of particle radii and incorporates various kinetic processes such as Pt dissolution, platinum oxide (PtO) formation, and PtO chemical dissolution. Furthermore, the authors extended their research by developing a dynamic model that encompasses the phenomena of surface area loss [20]. They utilized population balance equations in particle space to account for nanoparticle dissolution, redeposition, coagulation, and detachment, providing a comprehensive understanding of the complex dynamics involved [20].

To apply the developed catalyst degradation models in practical situations, Mayur et al. [21] developed a lifetime prediction model that separates the fuel cell's performance and degradation mechanism using a first-order approximation. This multi-scale model links a fuel cell car model with a cell level catalyst degradation model that can be customized by applying real-time drive data. Nonetheless, there are some limitations to the degradation model. For instance, the model does not account for the direct impacts of voltage cycling on platinum degradation. In addition, the model employs a degradation factor that

appears empirical which makes the model lack of generalization consideration. Similarly, Kneer et al. [22] developed a semi empirical model to predict how platinum surface area will decrease after being subjected to voltage cycling based on the accelerated stress test (AST) data, then coupled to an analytical polarization curve model to estimate the MEA performance. However, this model remains effective only when the primary causes of degradation are Pt dissolution and Pt particle growth. Furthermore, different-sized particles within an electrode may undergo specific aging or degradation mechanisms. Therefore, establishing the correlation between particle size distribution and ECSA enables precise predictions and the seamless incorporation of various degradation mechanisms within the Pt catalyst.

The majority of current degradation model, exhibits a limitation in its scope by concentrating on just one degradation process. Even the dynamic population model, although it attempts to describe various degradation mechanisms, fails to provide a clear understanding the effects of different internal physical states (temperatures, relative humidities, etc.). Moreover, these initial models have not been seamlessly integrated into dynamic or multiphysics fuel cell models which limits the interpretation of the model output at stack/system level and the model prediction accuracy since the dynamic internal states can hardly be estimated.

On the other side, empirical degradation models offer the advantage of often being examined in conjunction with multiphysics models. However, they fall short in terms of explaining the impact of a specific degradation on the catalyst functionality. The use of aggregated degradation models, where a simplified empirical model replaces the entire sequence of degradation mechanisms, severely limits their effectiveness in widely varied operating conditions.

This study presents a novel dual-scale modeling framework that combines a multi-physics model and a physical degradation model of the Pt catalyst for PEM fuel cells. The multi-physics model is employed to ascertain the internal state of the fuel cell, including variables such as cell voltage, relative humidity, and temperature, at each time step. The degradation model of Pt catalyst is proposed, considering various factors affecting platinum particle size distribution, including platinum oxidation, dissolution, redeposition, detachment, and agglomeration. By coupling these two models, a connection between microscopic degradation processes and macroscopic performance, as well as operating parameters is established. Consequently, it becomes feasible to estimate the extent of platinum loss and predict the degradation performance of the catalyst under arbitrary time-varying operating conditions.

Finally, to enhance usability and reliability, the model undergoes validation through a diverse array of ASTs and is compared against experimental data. Moreover, given the primary need for predictions in real-world environments, the model is simulated and validated using real-world data, specifically the New European Driving Cycle (NEDC) dataset.

The paper is structured as follows: it begins by examining the degradation mechanisms of Pt catalysts and explores the influence of different operating conditions in Section 2. Section 3 introduces a micro-scale modeling approach aimed at addressing Pt catalyst degradation, encompassing both kinetic and physical degradation models. Subsequently, a macro-scale model of PEMFC is introduced, along with a dual-scale framework that illustrates the interaction between the two model levels. Section 4 details the simulation methodology and presents results, while Section 5 concludes the paper by summarizing the main findings and discussing their implications.

## 2. Description of Pt catalyst degradation

### 2.1. Pt catalyst degradation mechanisms

Pt catalysts are susceptible to a range of degradation mechanisms that can significantly affect both their performance and lifespan. In this context, understanding these diverse degradation phenomena is
<figure><img src="image_4.png" /><figcaption>Fig. 1. Pt catalyst degradation mechanism.</figcaption></figure>

crucial for accurate modeling and effective management of Pt catalysts degradation within PEM fuel cells.

Experimental studies have been conducted to examine the stability of Pt at the cathode in order to gain insights into the various degradation mechanisms and develop strategies for mitigation [23–26]. Several recent reviews have highlighted the four primary degradation modes of a Pt catalyst as shown in Fig. 1, which include [27,28]: (1) platinum dissolution are the primary degradation mechanisms, which can initiate secondary degradation processes such as (2) Ostwald ripening, (3) particle migration and agglomeration, and (4) detachment.

Platinum dissolution and redeposition (Ostwald Ripening) refer to a phenomenon involving dissolved platinum particles. Smaller platinum particles preferentially dissolve due to their higher chemical potential compared to larger particles. The dissolved platinum species then migrate through the electrolyte or ionomer, eventually redepositing onto the surfaces of larger platinum particles. This process leads to a loss of active platinum surface area, diminishing the effectiveness of the catalyst [23].

Platinum dissolution and precipitation in the fuel cell membrane are critical processes. Pt ions migrate from the cathode to the anode by diffusion [29], where they interact chemically with hydrogen permeating from the anode, forming a platinum band in the membrane [18,28,30]. The impact of the platinum band on degradation is debated due to conflicting findings [31,32].

Coalescence is the merging of platinum nanoparticles on carbon supports into larger particles without platinum dissolution. Driven by the reduction of overall surface energy, this process results in a distinct particle size distribution as the surface energy decreases with particle growth. Differentiating between coalescence and Ostwald ripening is challenging, particularly regarding their roles in the loss of electrochemical active area [33–35].

Detachment of the platinum catalyst is a key consequence of carbon corrosion in the catalyst layer, as depicted in Fig. 1. Electrochemical oxidation of the carbon support not only causes electrical insulation but also leads to catalyst detachment, loss of porosity, and increased performance losses due to mass transport limitations [36]. Significant carbon corrosion can occur during rapid jumps to higher potentials, such as in vehicle start-stop operations, and prolonged operation at normal potentials can also damage the carbon support [36].

These degradation mechanisms are considered irreversible losses within PEM fuel cells during operation. The material structure of the Pt catalyst, fabrication process, and operating conditions all significantly influence these mechanisms, which will be further discussed in the next section.

## 2.2. The effect of operating conditions

Apart from the cell's operating potential, factors like temperature and gas humidity also play a crucial role in affecting the rate at which the catalyst layer degrades [37]. Researchers have extensively studied

the impacts of these operational parameters in various experimental studies documented in the literature [38].

**Cycling profile:** Wang et al. [39] found that platinum dissolves significantly faster under potential cycling conditions compared to potentiostatic conditions, with rates increasing by three to four orders of magnitude. This underscores the need to include cycling conditions in catalyst degradation studies. The upper potential limit (UPL) influences degradation rates, with higher UPLs generally leading to increased degradation, while the impact of the lower potential limit (LPL) remains debated [40,41].

Mitsushima et al. [42] studied platinum consumption in sulfuric acid during potential cycling in PEM fuel cells, noting that rectangular waveforms result in a steady consumption rate of about 10 ng cm⁻² cycle⁻¹. Triangular waveforms showed increased consumption with frequency, peaking at 40 Hz. Specifically, a slow cathodic triangular waveform with an anodic scan rate of 20 V s⁻¹ and a cathodic scan rate of 0.5 V s⁻¹ led to a consumption rate ten times higher than symmetric triangular waves, highlighting the influence of potential and transient conditions on platinum dissolution.

In fuel cell stacks, start and stop may cause the electrode potential in the cathode region to exceed 1.4 V, potentially leading to carbon corrosion [29].

**Relative humidity:** Wu Bi et al. [43] explored how the changes of relative humidity affect catalyst deterioration. They noted that when the humidity in the air is lowered from 100% to 50%, the catalyst exhibits a threefold reduction in the rate of Pt mass loss and a twofold decrease in the loss of active surface area. However, there were no notable disparities of Pt surface oxide between the two humidity levels. As a result, the researchers concluded that the slower degradation observed at lower relative humidity levels was due to a decrease in the mobility of dissolved Pt ions under these conditions.

**Temperature:** The degradation of fuel cell components accelerates at elevated temperatures, which cause faster Pt particle breakdown, reducing the electrochemically active surface area [44]. In [45], valuable insights are provided into how temperature influences Pt dissolution and its subsequent impact on the degradation of Pt catalyst. High temperatures and voltages during potential cycling also promote carbon corrosion and Pt dissolution redeposition, increasing particle size [46]. Experiments at 90 °C and 0.9 V show that Pt catalyst degradation, driven by dissolution and re-deposition, is amplified by both temperature and potential increases [47].

Conversely, operating PEM fuel cells at low temperatures reduces Pt catalyst activity, electrochemical kinetics, and proton conductance, potentially causing reversible performance drops. This work aims to clarify the factors influencing Pt catalyst degradation, with a focus on temperature.

# 3. Methodology

## 3.1. Micro-scale model: Pt catalyst degradation model

### 3.1.1. Pt degradation processes and kinetics

Although platinum is generally considered a noble metal with higher resistance to oxidation in the presence of air compared to other materials, thermodynamic analysis based on the Pourbaix diagram suggests that platinum can still undergo oxidation and even dissolve under relevant conditions of potential and pH in fuel cells [48]. This behavior, known as Pt dissolution, arises from the electrochemical instability exhibited by platinum particles when subjected to an applied potential. Two primary mechanisms contribute to the loss of Pt particles. The first mechanism involves the electrochemical dissolution of Pt, resulting in the formation of platinum ion (Pt²⁺) as described in Reaction (1). The second mechanism entails the chemical dissolution of the PtO film, which forms due to the oxidation of platinum, as indicated in Reactions (2) and (3) [18,49].

$$Pt \leftrightarrow Pt^{2+} + 2e^{-} \qquad (1)$$
$$Pt + H_2O \leftrightarrow Pt-O + 2H^+ + 2e^- \quad (2)$$

$$Pt-O + 2H^+ \leftrightarrow Pt^{2+} + H_2O \quad (3)$$

The rates of the electrochemical reactions, namely the dissolution and oxidation rates of Pt, are described by the modified Butler-Volmer equations. These equations, initially developed by Darling and Meyers [16], consider the electrochemical potentials of the Pt particles in relation to their radius. The Pt dissolution rate, relating to (Reaction. (1)), where the first term describes the dissolution of Pt and the second term defines the re-deposition of dissolved Pt²⁺ ions, is given by:

$$R_1(r,t) = k_1(T, RH)(1 - \theta(r,t)) \left[ \exp\left(\frac{\alpha_1 F}{RT}(E(t) - E_1^{eq}(r))\right) - \frac{c_{Pt}(t)}{c_{Pt}^{ref}}\exp\left(\frac{-(1-\alpha_1)F}{RT}(E(t) - E_1^{eq}(r))\right) \right] \quad (4)$$

The rate of platinum oxidation (Reaction. (2)) employed in this study is adapted from the models originally developed by Harrington et al. [50–52], with further modifications and adjustments made by Darling et al. [16].

$$R_2(r,t) = k_2(T, RH) \left[ \exp\left(-\frac{\omega\theta(r,t)}{RT}\right) \exp\left(\frac{\alpha_2 F}{RT}(E(t) - E_2^{eq}(r))\right) - \theta(r,t) \left(\frac{c_{H^+}^{ref}}{c_{H^+}^{eq}}\right)^2 \exp\left(-\frac{(1-\alpha_2)F}{RT}(E(t) - E_2^{eq}(r))\right) \right] \quad (5)$$

The reaction rates, denoted as $k_i$, are $k_1$ for the Pt dissolution reaction and $k_2$ for the Pt oxidation reaction. The applied potential, $E(t)$, refers to the external voltage applied to the fuel cell (FC). The concentrations of dissolved Pt and protons (H⁺) are denoted as $c_{Pt}$ and $c_{H^+}$, with corresponding reference concentrations $c_{Pt}^{ref}$ and $c_{H^+}^{ref}$. The surface coverage of Pt particles covered with oxides is described by $\theta(r,t)$, which ranges between 0 and 1. The equilibrium potential for the electrochemical reactions ($i$ = 1,2), denoted as $E_i^{eq}(r)$, depends on the particle radius. The parameters $a_i$, $R$, $F$, $T$, and $\omega$ represent the electron transfer coefficient in reaction ($i$ = 1,2), the universal gas constant, Faraday's constant, temperature, and PtO–PtO interaction energy, respectively.

It should be remarked that the above two equations, even not fully validated experimentally, show high potential to be used in the proposed modeling framework. The two models are theoretically justified as they are based on first-principle laws. In addition, the model is also in accordance with the experimental findings in [24].

Since the catalyst exists in the form of nanoparticles, it is necessary to modify the standard equilibrium potentials to account for the nanospherical structure [16]. This adaptation involves applying the Gibbs–Thomson correction. Consequently, the adjusted thermodynamic dissolution potential $E_1^{eq}(r)$, and thermodynamic oxidation potential $E_2^{eq}(r)$ are as follows:

$$E_1^{eq}(r) = E_1^0 - \frac{\Delta\mu_{Pt}(r)}{2F} \quad (6)$$

$$E_2^{eq}(r) = E_2^0 - \frac{\Delta\mu_{Pt}(r)}{2F} + \frac{\Delta\mu_{PtO}(r)}{2F} \quad (7)$$

where the Gibbs–Thomson correction terms $\Delta\mu_{Pt}(r)$ and $\Delta\mu_{PtO}(r)$ are given by

$$\Delta\mu_{Pt}(r) = \frac{\sigma_{Pt} M_{Pt}}{\rho_{Pt}} \frac{1}{r} \quad (8)$$

$$\Delta\mu_{PtO}(r) = \Delta\mu_{PtO}^{ref} + \frac{\sigma_{PtO} M_{PtO}}{\rho_{PtO}} \frac{1}{r} \quad (9)$$

These terms describe the correlation between the particle radius $r$ and the surface tensions of Pt ($\sigma_{Pt}$) and PtO ($\sigma_{PtO}$). Here, $\rho_{Pt}$ and $\rho_{PtO}$ denote the densities of platinum and platinum oxide, respectively, while $E_1^0$ and $E_2^0$ represent the standard equilibrium potentials for dissolution and oxidation.

Finally, the rate of the chemical dissolution of PtO (Reaction. (3)) is given by [41]

$$R_3(r,t) = k_3 \theta(r,t) c_{H^+}^2 \quad (10)$$

The chemical dissolution rate, denoted as $k_3$, represents the rate at which the chemical dissolution process occurs. $c_{H^+}$ denotes the concentration of protons.

In addition to the change in radius of each particle, particles can be permanently detached from carbon support due to carbon corrosion, causing a change in the amount of particles. At high voltage, the catalyst support material undergoes degradation due to Reaction (11) [53], diminishing the adhesion of Pt and resulting in particle loss. This corrosion process unfolds in aqueous solutions as outlined by Schneider et al. [38,54].

$$C + 2H_2O \leftrightarrow CO_2 + 4H^+ + 4e^- \quad (11)$$

This mechanism is implemented in the model by the following equation:

$$R_4(r,t) = k_4(T, RH) \frac{M_c}{\rho_c} \exp\left(\frac{\alpha_4 F}{RT}(E(t) - E_4^{eq})\right) \frac{1}{r} \quad (12)$$

Here, $k_4$ represents the corrosion reaction rate, $\alpha_4$ describes the charge transfer coefficient, and $E_4^{eq}$ signifies the equilibrium constant of the reaction. $M_c$ and $\rho_c$ are the carbon molar mass and density, respectively. The ratio $1/r$ accounts for the effect that smaller Pt particles have a reduced contact surface, which leads to decreased adhesion.

Adapting this model to account for operating conditions such as relative humidity and temperature is required [45,53]. Given that increased humidification and temperature of the electrode facilitate the electrochemical reactions ($i$ = 1,2,4) [54,55], the relative humidity (RH) and temperature are taken to reflect their effect on the reaction rate as follows:

$$k_i(T, RH) = k_i^{ref} \left( \frac{RH}{RH_0} \right)^{n_i} T \cdot \exp\left( - \frac{G_i}{k_B \cdot T} \right) \quad (13)$$

where $k_b$ denotes the Boltzmann constant, $T$ is the temperature. $k_i^{ref}$ represents the rate constants for dissolution ($i$ = 1), oxidation ($i$ = 2) and carbon corrosion ($i$ = 4), respectively. Gibbs free energy $G_i$ defines activation energy barrier influence [54].

The kinetic model provides the reaction rates of platinum losses process ($R_1$, $R_2$, $R_3$ and $R_4$) as functions of the applied potential, particle radius, and operation-dependent variables such as relative humidity, temperature, and proton concentration. Integrating the kinetic models of platinum dissolution into the physical catalyst degradation model is crucial. This integration allows us to assess the effect of platinum dissolution on particle distribution and predict the electrochemical surface area over time and under time varying operations.

### 3.1.2. Physical Pt catalyst degradation model

As part of our model input, we employ an initial particle radius distribution (PRD) which is conventionally quantified using a log-normal distribution $f_N(r,t)$. This initial distribution serves to characterize the pristine state of the catalyst layer and can be characterized through Transmission Electron Microscopy (TEM) image analysis of a Pt catalyst or X-ray Diffraction (XRD) measurements [20].

During the aging and under potential load, the PRD undergoes a noticeable change that indicates the level of degradation [56]. The increase in particle size is thought to result from the coagulation and redeposition mechanisms. The loss of platinum mass from the support can be attributed to detachment or dissolution. The dissolved platinum in the electrolyte may be redeposited or migrate into the membrane to form the Platinum-in-the-Membrane (PITM) band [57].

A continuity equation was developed based on the theories of Wagner, Lifshitz, and Slyozov [58], to incorporate the changes in the particle radius distribution, $f_N(r,t)$. This equation takes into account the contributions from each mechanism and allows for the evolution of
the particle radius distribution over time. The PRD evolution over time is given by:

$$ \frac{\partial f_N(r, t)}{\partial t} = -\frac{\partial}{\partial r} \left[ f_N(r, t) \frac{dr}{dt} \right] + J^+ - J^- - R_4(r, t) f_N(r, t) \quad (14) $$

The first term presented in Eq. (14) represents the process of dissolution and redeposition. To incorporate the influence of dissolution and redeposition, the rate of change in particle radius r for individual particles was derived from Darling and Meyers work [16] and can be expressed as follows:

$$ \frac{dr}{dt} = -\frac{M_{Pt}}{\rho_{Pt}} (R_1(r,t) + R_3(r,t)) \quad (15) $$

The phenomenon of coagulation, which involves the physical merging of particles, is characterized by the radius-scaled Smoluchowski creation and extinction terms, as described in [58,59]

$$ J^+ = \frac{1}{2} k_5 \int_{r_{min}}^{r_{max}} \frac{r^2}{(r^3 - \hat{r}^3)^{\frac{1}{3}}} f_N \left( (r^3 - \hat{r}^3)^{\frac{1}{3}}, t \right) f_N(\hat{r}, t) d\hat{r} \quad (16) $$

$$ J^- = k_5 \int_{r_{min}}^{r_{max}} f_N(r,t) f_N(\hat{r}, t) d\hat{r} \quad (17) $$

where $k_5$ is the coagulation rate constant.

The platinum that is present in a dissolved form is not limited to being redeposited, as it could also potentially be lost to the membrane. The mass balance equation relates to the concentration of Pt²⁺ ions, which we denote as $c_{Pt}$ in the electrolyte. It can be expressed by the following equation

$$ \epsilon \frac{dc_{Pt}}{dt} = L_v \int_{r_{min}}^{r_{max}} 4\pi r^2 f_N(r,t) (R_1(r,t) + R_3(r,t)) dr + q_{Pt}(t) \quad (18) $$

where $L_v$ is the platinum mass loading per unit volume. The loss of platinum ions, indicated as $q_{Pt}$, within the catalyst layer with a thickness of $h_{CL}$, can be expressed by

$$ q_{Pt}(t) = -D_{Pt} \frac{c_{Pt}(t)}{h_{Cl} h_{Pt}} \quad (19) $$

where $D_{Pt}$ is Pt ions diffusion coefficient in membrane and $h_{Pt}$ is the diffusion length to Pt band [30]. The platinum oxide coverage is defined by

$$ \frac{d\theta}{dt} = \left( \frac{R_2(r,t) - R_3(r,t)}{\Gamma_{max}} \right) - \left( \frac{2\theta(r,t)}{r} \right) \frac{dr}{dt} \quad (20) $$

The right side explains oxide formation/reduction via reaction paths and how varying particle size affects oxide coverage. Here, $\Gamma_{max}$ represents the active site quantity in moles per platinum area.

The microscale model quantifies the degradation on platinum nano particles and PRD change, focusing on sizes about nanometers without the need for complex atomic simulations, enhancing efficiency for real-time degradation mitigation.

## 3.2. Macro-scale model: Multiphysics dynamic model of PEMFC

The degradation model is combined with a control oriented model, which allows for the control of the factors that impact the catalyst degradation. A control-oriented fuel cell system model, integrating both multiphysical principles and stack polarization data [60], has been utilized. More details on the applied model and its implementation can be found in [60]. This model has been modified to integrate the microscale catalyst degradation model and thus the ensemble form a dual-scale model.

The ECSA is determined through the micro-scale model which is employed for the catalyst. This calculation takes into account the temporal progression of PRD while assuming a consistent spatial development across all areas of the catalyst surface. This approach simplifies the integration between the two models and significantly reduces computational time. Inversely, this macro-scale model considers the dynamic influences, allowing us to estimate the internal state of the fuel cell,

<figure><img src="image_5.png" /><figcaption>Fig. 2. Flowchart for simulating the dual-scale model of PEM fuel cell degradation.</figcaption></figure>

including parameters like humidity, partial pressure, membrane water content. As shown in Fig. 2, the flowchart details the interactions between the two submodels. Furthermore, the fuel cell model readily encompasses the entire system, facilitating the seamless integration of all essential auxiliary components to guarantee the proper operation of the fuel cell.

The micro-scale degradation model impacts the macroscale performance model by altering the ECSA. The change in PRD caused by the different degradation processes allows for the assessment of the time evolution of ECSA.

$$ S(t) = 4\pi \int_{r_{min}}^{r_{max}} r^2 f_N(r, t) dr, \quad (21) $$

$$ L_{pt}(t) = \frac{4\pi}{3} \rho_{Pt} \int_{r_{min}}^{r_{max}} r^3 f_N(r, t) dr, \quad (22) $$

$$ A_{Pt}(t) = S(t) L_{pt}(t) \quad (23) $$

where $S(t)$ and $L_{pt}(t)$ are the electrochemical surface area and platinum mass loading, respectively. $A_{Pt}(t)$ is the effective surface area of Pt.

The voltage generated by the fuel cell is calculated by subtracting various losses from the open circuit voltage. These losses, namely activation loss, mass transfer loss, and ohmic loss, are comprehensively explained in the Ref. [61]. The activation loss is determined using the Tafel equation:

$$ E_{act} = \frac{RT}{\alpha F} \ln \left( \frac{i}{i_0} \right) \quad (24) $$

The activation loss resulting from the decay of ECSA can be described by the direct relationship between ECSA and the value of the exchange current density $i_0$ as mentioned in the Ref. [61]. Additionally, the coverage of Pt sites by reversible Pt oxide can be incorporated into the kinetics of the oxygen reduction reaction (ORR) [62]:

$$ i_0 = i_0^{\text{ref}} A_{Pt}(t) (1 - \theta) \left( \frac{P_{O_2}}{P_{\text{ref}}} \right)^{\gamma} \exp \left[ -\frac{G_{\text{red}}}{RT} \left( 1 - \frac{T}{T_{\text{ref}}} \right) \right] \exp \left( -\frac{\omega\theta}{RT} \right) \quad (25) $$
<table><caption>**Table 1**
Pt catalyst specification.</caption><thead><tr><th>Parameter</th><th>Meaning</th><th>Value/unit</th></tr></thead><tbody><tr><td>S(0)</td><td>Initial ECSA</td><td>[20–80] m<sup>2</sup> g<sub>Pt</sub><sup>-1</sup></td></tr><tr><td>L<sub>pt</sub>(0)</td><td>Initial mass loading of Pt</td><td>[0.2–0.4] mg<sub>Pt</sub> cm<sup>-2</sup></td></tr><tr><td>r<sub>min</sub></td><td>Min particle radius</td><td>0.5 nm</td></tr><tr><td>r<sub>max</sub></td><td>Max particle radius</td><td>10 nm</td></tr><tr><td>r<sub>avg</sub></td><td>Initial radius mean</td><td>[2–5.5] nm</td></tr></tbody></table>

The concentration loss results from the drop in concentration of the reactants. These losses are the reason for rapid voltage drop at high current density.

$$E_{\text{conc}} = \frac{RT}{\beta F} \ln \left( 1 - \left( \frac{i}{i_{\text{max}}} \right) \right) \quad (26)$$

where $i_{max}$ is the limiting current density, which is related to the total oxygen transfer resistance $R_{O_2}$. The oxygen transfer resistance of the catalyst layer is directly influenced by the catalyst degradation [63].

$$i_{\text{max}} = R_{O_2} i_{\text{max}}^{\text{ref}}, \quad R_{O_2} = \frac{a}{A_{\text{Pt}}(t)} + b \quad (27)$$

The ohmic losses, denoted as $E_{ohm}$, in PEM fuel cells, refer to the electrical resistance faced by the flow of ions through the proton exchange membrane and the current passing through various components of the cell

$$E_{\text{ohm}} = i(R_{\text{ohm}} + r_{\text{ohm}}t) \quad (28)$$

The ohmic resistance $R_{ohm}$ also increases with time by a fitted parameter $r_{ohm}$.

Finally the fuel cell voltage expression given by:

$$E_{\text{cell}} = E_{\text{ocv}} - E_{\text{act}} - E_{\text{ohm}} - E_{\text{conc}} \quad (29)$$

The proton concentration represents the density of protons within the membrane and is related with the water content of the membrane material through a specific equation.

$$c_{\text{H}^+} = \frac{1}{\frac{EW}{\rho_{\text{naf}}} + \lambda_m \frac{M_{\text{H2O}}}{\rho_{\text{H2O}}}} \quad (30)$$

where EW represents the equivalent weight of the ionomer, while $\rho_{naf}$ ion refers to the density of the ionomer material Nafion. $\lambda$ represents the number of water molecules per acid group, which is directly related to the water activity $a_w$.

The membrane water content can be calculated using the activities of the gas in the anode and the cathode:

$$\lambda(a_w) = \begin{cases} 0.043 + 17.81a_w - 39.85a_w^2 + 36a_w^3, & \text{if } 0 < a_w < 1 \\ 14 + 1.4(a_w - 1), & \text{if } 1 < a_w < 3 \end{cases}$$

where $a_w$ is calculated and controlled based on dynamic macro-scale model.

## 4. Results and discussion

We utilize an initial PRD, typically generated using a log-normal distribution $f_N(r, t)$ given by

$$f_N(r, 0) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left(-\frac{(\ln(r) - \mu)^2}{2\sigma^2}\right) \quad (31)$$

Our choice of utilizing a log-normal distribution is informed by its good alignment with experimental PRD data derived from various catalyst layers [19]. Where the standard deviation $\sigma$ and the mean $\mu$ are fitted based on the available experimental PRD data. This distribution comprises a finite number of particle groups, each categorized by their particle radius and the number of particles contained within. Fig. 3 visually represents the assumed PRD, divided into N distinct groups, with fresh catalyst specifications detailed in Table 1.

<figure><img src="image_6.png" /><figcaption>**Fig. 3.** Initial particle radius distribution.</figcaption></figure>

### 4.1. Results and discussion I: validation under AST

The objective of this stage is to confirm the reliability of the degradation model across various experimental conditions and assess its accuracy using AST experimental data. Following the original design of experiments, the U.S. Department of Energy (DOE) AST protocol design, two voltage cycles were performed: a square wave (SW) and a triangle wave (TW), both with a 6-second period. The LPL was set at 0.6 V, while the UPL was varied.

During this stage, degradation tests were conducted with variations in temperature, relative humidity, and potential range. This step is crucial for understanding the impact of different operating conditions on degradation and for validating the model's ability to predict degradation under diverse scenarios, thereby extending its applicability beyond focused Accelerated Stress Testing (AST) to real-world driving conditions. The experimental data from the literature concerning ECSA losses is summarized in Table 2, which is utilized to validate catalyst degradation in PEM fuel cells under AST by comparing ECSA evolution across various operating conditions.

The initial model simulation compared ECSA loss over 30k load cycles under the conditions outlined in dataset (1), considering the experimental setup and the initial characteristics of the Pt catalyst, with full details provided in [64]. The simulation was performed on a standard desktop PC with an Intel Core i9 processor clocked at 2.6 GHz. It took approximately 15 min to complete for 50 particle radii groups. However, the computation time depends on the number of particle groups, with a minimum of 50 radii groups required to ensure reliable simulation results.

At the nanoscale, we can examine how Pt dissolution and redeposition rates change in response to cyclic potential and assess how well the model aligns with prior knowledge from both experiences and the existing data regarding the degradation mechanisms. The results in nano, microscale in SW 0.98 are shown in Fig. 4. The applied potential cycle is illustrated in Fig. 4(a). In Fig. 4(b), we can observe the simulated evolution of Pt oxide coverage for three distinct particle sizes (2, 4, and 6 nm). Notably, smaller particles exhibit a greater overall coverage, primarily due to the shift in equilibrium potential $E_2^{eq}(r)$. The dissolution rates of two particle sizes (0.5 and 2 nm) and the concentration of Pt$^{2+}$ ions dissolved in the ionomer, illustrated in Figs. 4(b) and 4(d), are influenced by variations in the applied potential, oscillating between maximum and minimum values.

Furthermore, As the potential increases from 0.6 to 0.98 V, we observe a rapid onset of Pt dissolution. It is noteworthy that smaller particles exhibit significantly higher dissolution rates when compared
<table><caption>**Table 2**
Summary of experimental conditions used for validation purposes.</caption><thead><tr><th>Purpose</th><th>Dataset</th><th>Signal</th><th>UPL/LPL (V)</th><th>Holds time (s)</th><th>Cycles</th><th>Temp (°C)</th><th>RH (%)</th><th>Ref</th></tr></thead><tbody><tr><td rowspan="4">Potential</td><td>(1)</td><td>SW 0.98</td><td>0.98/0.6</td><td>3/3</td><td>30,000</td><td>80</td><td>85</td><td>[64]</td></tr><tr><td>(2)</td><td>TW 0.9 V</td><td>0.9/0.6</td><td>-</td><td>300,000</td><td>80</td><td>100</td><td>[38]</td></tr><tr><td>(3)</td><td>SW 0.9 V</td><td>0.9/0.6</td><td>3/3</td><td>300,000</td><td>80</td><td>100</td><td>[38]</td></tr><tr><td>(4)</td><td>SW 0.95 V</td><td>0.95/0.6</td><td>3/3</td><td>300,000</td><td>80</td><td>100</td><td>[38]</td></tr><tr><td rowspan="2">Relative humidity</td><td>(5)</td><td>SW 0.95 V</td><td>0.95/0.6</td><td>3/3</td><td>300,000</td><td>80</td><td>60</td><td>[38]</td></tr><tr><td>(6)</td><td>SW 0.95 V</td><td>0.95/0.6</td><td>3/3</td><td>300,000</td><td>80</td><td>30</td><td>[38]</td></tr><tr><td rowspan="2">Temperature</td><td>(7)</td><td>SW 0.95 V</td><td>0.95/0.6</td><td>3/3</td><td>300,000</td><td>90</td><td>100</td><td>[38]</td></tr><tr><td>(8)</td><td>SW 0.95 V</td><td>0.95/0.6</td><td>3/3</td><td>300,000</td><td>60</td><td>100</td><td>[38]</td></tr></tbody></table>

<figure><img src="image_7.png" /><figcaption>(a) Applied cyclic potential, SW 0.98v</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>(b) Evolution of PtO for three radius r = 2,4,6 nm</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>(c) Dissolution/redeposition rates for two radius</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>(d) Concentration of Pt²⁺ ions in the ionomer</figcaption></figure>

<caption>**Fig. 4.** Nanoscale evolution.</caption>

to their larger counterparts, primarily due to their lower equilibrium potential $E_1^{eq}(r)$. Concurrently, as the dissolution process evolves, the concentration of Pt²⁺ ions within the catalyst layer's ionomer increases. Conversely, when the potential decreases once again, the redeposition of dissolved Pt ions commences immediately, causing the ion concentration to decrease within short time.

Fig. 4(c) illustrates that larger particles have the ability to accept redeposited ions from the electrolyte at a much faster rate than smaller particles. This phenomenon promotes the preferential growth of larger particles. When coupled with the higher dissolution rates of smaller particles, it leads to an overall shift in the PRD towards larger particles. Consequently, this shift results in a loss of ECSA.

Fig. 5 provides a visual representation of how ECSA and the evolution of the PRD model. In Fig. 5(b), it is clear that as time progresses,

particle size increases, causing a concurrent shift in the PRD towards larger particles. This phenomenon can account for both redeposition and agglomeration mechanisms, as well as the increased susceptibility of smaller particles to corrosion. Furthermore, the overall reduction in the distribution function accounts for the dissolution occurring across all particle radii. Upon comparing the ECSA generated by the developed degradation model with real-world experimental data, a significant and acceptable level of agreement becomes evident, as shown in Fig. 5(a). Nevertheless, achieving a more precise tracking of the degradation rate continues to present a challenge, especially as we contend with intricate interactions among various degradation processes involving different fuel cell components. This challenge becomes even more pronounced as we are close to the end of the cell's operational lifespan, where such interactions may become increasingly conspicuous.
<figure><img src="image_11.png" /><figcaption>(a) ECSA evolution over 30 k cycles</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>(b) Particle radius evolution</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>(a) Evolution of ECSA for datasets (2) & (3) & (4)</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>(b) Evolution of ECSA for datasets (4) & (5) & (6)</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>(c) Evolution of ECSA for datasets (4) & (7) & (8)</figcaption></figure>

<caption>Fig. 6. ECSA evolution under different operating conditions : (a) Applied potential, (b) Relative humidity, and (c) Temperature.</caption>

We conducted additional calculations to evaluate the ECSA's re-
sponse to various load profiles, contrasting our results with experi-
mental data from datasets (2), (3), and (4) as shown in Fig. 6(a). Our
analysis reveals notable ECSA losses, prominently seen with the SW
profile, in contrast to lower losses associated with the TW profile. These
discrepancies can be attributed to extended dwell times at elevated
voltages, which accelerate Pt catalyst degradation. Specifically, pro-
longed periods of heightened Pt²⁺ concentration during square load

profiles increase ion flux into the membrane, thereby maximizing
platinum loss. Moreover, when compared with dataset (4), the pivotal
role of UPL in accelerating catalyst layer degradation becomes evident.
Operating the fuel cell at high UPL significantly deteriorates perfor-
mance and exponentially amplifies degradation effects. The model's
alignment with the data underscores its sensitivity to voltage profiles
and highlights the crucial role they play in Pt catalyst degradation.
<figure><img src="image_16.png" /><figcaption>Fig. 7. Model simulation compared with long-term dynamic durability test data.</figcaption></figure>

The results depicted in Fig. 6(b) illustrates how relative humidity impacts catalyst degradation. Higher levels of humidification correlate with increased ECSA loss, as electrochemical reactions are favored under higher RH conditions. Additionally, higher humidity exposure corresponds to greater ion diffusion, resulting in elevated platinum losses.

Under various temperatures, our simulated model demonstrates strong agreement with the corresponding experimental data, as illustrated in Fig. 6(c). These results affirm the reliability of our predictive model across a range of temperature conditions. This enhancement can be attributed to our method of incorporating the temperature-dependent reaction rate, as outlined in Eq. (13).

Finally, our model improves accuracy under varying operating conditions such as voltage, relative humidity, and temperature. This capability enables us to optimize these parameters beyond simply managing applied loads through power management [65]. Moreover, with this model, we can optimize all factors that influence the catalyst layer, thereby significantly contributing to an enhanced lifespan. However, a slight deviation from the experimental data can be observed due to not fully accounting for the intricate effects of membrane deterioration, especially concerning the penetration of hydrogen into the catalyst layer. This phenomenon represents an important area for potential model refinement and further investigation to enhance the accuracy of the model under varying operating conditions.

## 4.2. Results and discussion II: validation under NEDC

After analyzing the model's ability to capture the different operating conditions and observing that the simulated data aligns well with the experimental data, this section highlights the primary goal of this dual-scale model, which is to predict catalyst degradation during real-world driving cycles, specifically the NEDC. The long-term dynamic durability test experiment aims to measure ECSA over a duration of 500 h. All relevant details regarding the experiment, including specific tests, can be found in Ref. [66].

The predicted ECSA loss and relative performance degradation under driving cycle conditions are presented in Fig. 7(b). Incorporating the detailed degradation model into a comprehensive performance model will enable the direct conversion of driving speed into voltage cycle data, as depicted in Fig. 7(a). The results appear to be reasonable and suggest that with adequate validation data, this tool could prove to be valuable. However, fitting the model under AST conditions may explain the slight overestimation of degradation during the last 250 operational hours, as shown in Fig. 7(b). To enhance precision, additional experimental data, particularly from fuel cell vehicles field data, is required.

In Fig. 7(c), we examine the voltage drop rate calculated using the dual-scale model. It is therefore verified that this dual-scale model enables us to refine and update the voltage drop rate and performance losses. However, it should be noted again that the overall performance loss results from superimposed degradation processes which deserves
<table><caption>**Table 3**
Comparison with different modeling approaches.</caption><thead><tr><th>Model</th><th>Focus area</th><th>Key parameters</th><th>Description</th></tr></thead><tbody><tr><td>Darling-Meyer [16,17]</td><td>- Pt dissolution/redeposition<br>- Pt oxidation</td><td>- Voltage cycling</td><td>- Bimodal catalyst model: limited to one or two particle sizes<br>- Humidity and temperature effects need evaluation<br>- Validated only under AST</td></tr><tr><td>Wu Bi [18]</td><td>- Pt dissolution/redeposition<br>- Pt oxidation<br>- Pt ion band formation</td><td>- Voltage cycling</td><td>- Bimodal catalyst model: limited to one or two particle sizes<br>- Humidity and temperature effects need evaluation<br>- Needs refinement for accuracy<br>- Validated only under AST</td></tr><tr><td>Mayur et al. [21]</td><td>- Pt dissolution</td><td>- Voltage cycling<br>- Temperature<br>- Humidity</td><td>- Computationally expensive multiphysics models<br>- Lack of detailed degradation mechanisms<br>- Multi-scale modeling approach<br>- Indirect coupling based on a lookup table</td></tr><tr><td>Kneer et al. [22]</td><td>- Pt dissolution/redeposition</td><td>- Voltage cycling<br>- Temperature<br>- Humidity</td><td>- First-order kinetic rates model of degradation.<br>- Underestimation of kinetic losses due to unaccounted particle shifts<br>- Evaluated under AST conditions.</td></tr><tr><td>Our model framework</td><td>- Pt dissolution/redeposition<br>- Pt oxidation<br>- Carbon corrosion<br>- Agglomeration</td><td>- Voltage cycling<br>- Temperature<br>- Humidity</td><td>- Consider particle radius distribution<br>- Calibrated parameters for improved kinetics model<br>- Detailed degradation modeling<br>- Direct integrated into performance model<br>- Online prediction and degradation mitigation<br>- Evaluated under AST and real-world conditions</td></tr></tbody></table>

further deeper investigations. Further investigation into the heterogeneity within the cell is needed. Dhanushkodi et al. [67] revealed that variations in local operating conditions, such as relative humidity, temperature, and pressure, significantly impact platinum dissolution rates and kinetic losses across the segmented cell. The use of spatially resolved catalyst layer performance and degradation, rather than assuming uniform temporal degradation on the catalyst surface, is crucial for improving the accuracy of our model.

Finally, to evaluate the effectiveness of our predictive modeling framework, we present a comparative analysis of different modeling approaches in Table 3. Our framework is not limited to individual degradation processes but integrates multiple mechanisms, including Pt dissolution and redeposition, agglomeration, and carbon corrosion, across a range of operating conditions. It also models the particle radius distribution to derive the ECSA, accounting for temperature variations and the effects of relative humidity. This enables accurate prediction of performance losses under various operating conditions. Additionally, the framework allows for the integration of detailed degradation mechanisms into dynamic system performance updates, making it suitable for real-time degradation mitigation during real-world driving cycles.

## 5. Conclusion

This study addresses the challenge of improving the durability of fuel cell components, specifically focusing on the platinum catalyst of PEM fuel cells for transportation applications. An innovative dual-scale modeling framework is proposed, in which a multi-physics model and a degradation model of the platinum catalyst are combined.

The integration of these models enables accurate predictions of catalyst degradation under various operating conditions, facilitating the implementation of optimal control strategies. Rigorous validation against accelerated stress tests and real-world data, including the New European Driving Cycle, supports the model's reasonable assumptions and predictive capabilities.

However, achieving a more precise tracking of the degradation rate remains challenging, particularly in dealing with intricate interactions among various components.

Future work is therefore oriented to investigate the modeling of superimposed degradation processes by enhancing the proposed models in this dual-scale modeling framework. The optimal control design based on the developed model will also be the continuous part of this work.

## Nomenclature table

<table><thead><tr><th>Symbol</th><th>Description</th><th>Value</th><th>Unit</th></tr></thead><tbody><tr><td>$k_1^{ref}$</td><td>Pt dissolution reaction constant</td><td>2.5 × 10<sup>-10</sup></td><td>[mol/(cm<sup>2</sup> s K)]</td></tr><tr><td>$k_2^{ref}$</td><td>Pt oxidation reaction constant</td><td>1.5 × 10<sup>-12</sup></td><td>[mol/(cm<sup>2</sup> s K)]</td></tr><tr><td>$k_3$</td><td>Pt chemical reaction constant</td><td>1 × 10<sup>-15</sup></td><td>[mol/(cm<sup>2</sup> s)]</td></tr><tr><td>$k_4^{ref}$</td><td>Corrosion reaction constant</td><td>1 × 10<sup>-18</sup></td><td>[mol/(cm<sup>2</sup> s K)]</td></tr><tr><td>$k_5$</td><td>Coagulation rate constant</td><td>1 × 10<sup>-15</sup></td><td>[s<sup>-1</sup>]</td></tr><tr><td>$\theta$</td><td>Fraction of platinum surface covered by PtO</td><td></td><td>[-]</td></tr><tr><td>$\alpha_i$</td><td>Charge transfer coefficient for reaction i = 1,2,4</td><td>0.5</td><td>[-]</td></tr><tr><td>$E_1^0$</td><td>Standard equilibrium potential for Pt dissolution</td><td>1.188</td><td>[V]</td></tr><tr><td>$E_2^0$</td><td>Standard equilibrium potential for Pt oxidation</td><td>0.98</td><td>[V]</td></tr><tr><td>$E_4^{eq}$</td><td>Equilibrium constant of the corrosion reaction</td><td>0.2</td><td>[V]</td></tr><tr><td>$F$</td><td>Faraday's constant</td><td>96 485</td><td>[C/mol]</td></tr><tr><td>$R$</td><td>Universal gas constant</td><td>8.314</td><td>[J/(mol K)]</td></tr><tr><td>$\sigma_{Pt}$</td><td>Surface tension of Pt nanoparticles</td><td>2.37 × 10<sup>-4</sup></td><td>[J/cm<sup>2</sup>]</td></tr><tr><td>$\sigma_{PtO}$</td><td>Surface tension of PtO nanoparticles</td><td>1 × 10<sup>-4</sup></td><td>[J/cm<sup>2</sup>]</td></tr><tr><td>$\omega$</td><td>PtO interaction coefficient</td><td>27 × 10<sup>3</sup></td><td>[J/mol]</td></tr><tr><td>$\Gamma_{max}$</td><td>Maximum oxide at Pt surface</td><td>2.8 × 10<sup>-9</sup></td><td>[mol/cm<sup>2</sup>]</td></tr><tr><td>$\rho_{Pt}$</td><td>Pt metal density</td><td>21.45</td><td>[g/cm<sup>3</sup>]</td></tr><tr><td>$\rho_{PtO}$</td><td>PtO metal density</td><td>14.1</td><td>[g/cm<sup>3</sup>]</td></tr><tr><td>$M_{Pt}$</td><td>Pt molecular weight</td><td>195.1</td><td>[g/mol]</td></tr></tbody></table>
<table><tbody><tr><td>M<sub>PtO</sub></td><td>PtO molecular weight</td><td>211</td><td>[g/mol]</td></tr><tr><td>M<sub>H2O</sub></td><td>Water molecular weight</td><td>18</td><td>[g/mol]</td></tr><tr><td>c<sup>ref</sup><sub>Pt</sub></td><td>Reference Pt ion concentration</td><td>1 × 10<sup>-3</sup></td><td>[mol/cm<sup>3</sup>]</td></tr><tr><td>c<sup>ref</sup><sub>H+</sub></td><td>Reference proton concentration</td><td>1 × 10<sup>-3</sup></td><td>[mol/cm<sup>3</sup>]</td></tr><tr><td>RH<sub>0</sub></td><td>Reference relative humidity</td><td>100</td><td>[%]</td></tr><tr><td>ρ<sub>naf</sub></td><td>Density of the ionomer material</td><td>1.95</td><td>[g/cm<sup>3</sup>]</td></tr><tr><td>ρ<sub>EW</sub></td><td>Equivalent weight of the ionomer</td><td>1100</td><td>[g/mol]</td></tr><tr><td>h<sub>Cl</sub></td><td>Catalyst layer thickness</td><td>6 × 10<sup>-4</sup></td><td>[cm]</td></tr><tr><td>ε</td><td>Electrode porosity</td><td>[-]</td><td></td></tr><tr><td>D<sub>Pt</sub></td><td>Pt ions diffusion coefficient in membrane</td><td>[-]</td><td>[cm<sup>2</sup>/s]</td></tr><tr><td>c<sub>Pt</sub></td><td>Pt ion concentration</td><td>[-]</td><td>[mol/cm<sup>3</sup>]</td></tr><tr><td>c<sub>H+</sub></td><td>Proton concentration</td><td>[-]</td><td>[mol/cm<sup>3</sup>]</td></tr><tr><td>f<sub>N</sub></td><td>Normalized particle radius distribution function</td><td>[-]</td><td></td></tr><tr><td>J<sup>+</sup></td><td>Creation term in the Smoluchowski coagulation equation</td><td>[-]</td><td></td></tr><tr><td>J<sup>-</sup></td><td>Extinction term in the Smoluchowski coagulation equation</td><td>[-]</td><td></td></tr><tr><td>q<sub>Pt</sub></td><td>Platinum loss flux</td><td>[-]</td><td>[mol/(cm<sup>3</sup> s)]</td></tr><tr><td>D<sub>Pt</sub></td><td>Pt ions diffusion coefficient in membrane</td><td>[-]</td><td>[cm<sup>2</sup> s]</td></tr><tr><td>h<sub>Cl</sub></td><td>Catalyst layer thickness</td><td>[-]</td><td>[cm]</td></tr><tr><td>S</td><td>Electrochemical Surface Area</td><td>[-]</td><td>[cm<sup>2</sup>/g<sub>Pt</sub>]</td></tr><tr><td>A<sub>Pt</sub></td><td>Pt surface area per unit area of MEA</td><td>[-]</td><td>[cm<sup>2</sup>/cm<sup>2</sup>]</td></tr><tr><td>L<sub>pt</sub></td><td>Platinum mass loading</td><td>[-]</td><td>[g/cm<sup>2</sup>]</td></tr><tr><td>L<sub>U</sub></td><td>platinum mass loading per unit volume</td><td>[-]</td><td>[g/cm<sup>3</sup>]</td></tr><tr><td>i</td><td>Current density</td><td>[-]</td><td>[A/cm<sup>2</sup>]</td></tr><tr><td>i<sub>0</sub></td><td>Exchange current density</td><td>[-]</td><td>[A/cm<sup>2</sup>]</td></tr><tr><td>i<sup>ref</sup><sub>0</sub></td><td>Reference exchange current density</td><td>[-]</td><td>[A/cm<sup>2</sup>]</td></tr><tr><td>P<sub>O2</sub></td><td>Oxygen partial pressure</td><td>[-]</td><td>[bar]</td></tr><tr><td>P<sub>ref</sub></td><td>Reference pressure</td><td>[-]</td><td>[bar]</td></tr><tr><td>G<sub>red</sub></td><td>The activation energy for O<sub>2</sub> reduction on Pt</td><td>[-]</td><td>[J/mol]</td></tr><tr><td>a<sub>1</sub>, b<sub>1</sub></td><td>Fitted parameters</td><td>[-]</td><td></td></tr><tr><td>i<sub>ref</sub></td><td>Maximum current density</td><td>[-]</td><td>[A/cm<sup>2</sup>]</td></tr><tr><td>a<sub>w</sub></td><td>Water activity in membrane</td><td>[-]</td><td></td></tr><tr><td>λ</td><td>Number of water molecules per acid group</td><td>[-]</td><td></td></tr></tbody></table>

CRediT authorship contribution statement

**Walid Touil:** Writing – original draft, Visualization, Validation, Software, Methodology, Formal analysis, Conceptualization. **Zhongliang Li:** Writing – review & editing, Funding acquisition, Conceptualization, Supervision, Project administration, Methodology. **Rachid Outbib:** Writing – review & editing, Supervision. **Daniel Hissel:** Writing – review & editing, Supervision. **Samir Jemei:** Writing – review & editing, Supervision.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work has been supported by French National Research Agency via project DEAL (Grant no. ANR-20-CE05-0016-01) and the EIPHI Graduate School (contract ANR-17-EURE-0002) and the Bourgogne Franche-Comté Region.

Data availability

Data will be made available on request.

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