Predictive system-level modeling framework for transient ...

The short version of the paper was presented at the 10th International Conference on Applied Energy (ICAE2018) on August

22–25, 2018, Hong Kong. This paper is a substantial extension of the short version of the conference paper.

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Predictive system-level modeling framework for transient

operation and cathode platinum degradation of high temperature

proton exchange membrane fuel cells

Ambrož Kregara,b,*, Gregor Tavčara, Andraž Kravosa, Tomaž Katrašnika

aUniversity of Ljubljana, Faculty of Mechanical Engineering, Aškerčeva 6, SI-1000 Ljubljana, Slovenia

bUniversity of Ljubljana, Faculty of Education, Kardeljeva ploščad 16, SI-1000 Ljubljana, Slovenia

*Corresponding author. E-mail address: [email protected]

Abstract

High temperature proton exchange membrane fuel cells (HT-PEMFCs) are a promising and emerging

technology, which enable highly efficient, low-emission, small-scale electricity and heat generation.

The simultaneous reduction in production costs and prolongation of service life are considered as major

challenges toward their wider market adoption, which calls for the application of predictive virtual tools

during their development process. To present significant progress in the addressed area, this paper

introduces an innovative real-time capable system-level modeling framework based on the following:

a) a mechanistic spatially and temporally resolved model of HT-PEMFC operation, and b) a degradation

modeling framework based on interacting individual cathode platinum degradation mechanisms.

Additional innovative contributions arise from a consistent consideration of the varying particle size

distribution in the transient fuel cell operating regime. The degradation modeling framework

interactively considers the carbon and platinum oxidation phenomena, and platinum dissolution,

redeposition, detachment, and agglomeration; hence, covering the entire causal chain of these

phenomena. Presented results confirm capability of the modeling framework to accurately simulate the

platinum particle size redistribution. Results clearly indicate more pronounced platinum particle growth

towards the end of the channel since humidity is the main precursor of oxidation reactions. In addition,

innovative modeling framework elucidate contributions of agglomeration, which is more pronounced at

voltage cycling, and Ostwald ripening, which is more pronounced at higher voltages, to the platinum

particles growth. These functionalities position the proposed modeling framework as a beyond state-of-

the-art tool for model-supported development of the advanced clean energy conversion technologies.

Keywords: Fuel cell; Proton-exchange membrane; High temperature; Modeling; Mechanistically

based; Platinum degradation

1. Introduction

Global concerns on sustainable energy use and environmental protection call for innovative energy

conversion technologies. Among the alternatives, high temperature proton exchange membrane fuel

cells (HT-PEMFCs) are a viable solution for highly efficient, low-emission, small-scale electricity and

heat generation. The advantages of HT-PEMFCs in small-scale stationary power generation arise mainly

from their ability to use reformed hydrocarbons directly (in particular the widely available natural gas

or bio-methane), as well as in their relatively simple design compared to non-PEM based technologies.



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However, to achieve a higher market share and large market penetration (even without subsidies),

significant cost reduction needs to be achieved while simultaneously ensuring a long service life [1]. To

attain this objective, further optimization is required on the component and system levels. This study

focuses on the system level, because the optimization of an HT-PEMFC system—comprising

component sizing and devising control logic—crucially influences its performance and service life.

To address the requirements of shorter product development cycles and reduced development cost

efficiently, while striving to approach engineering limits in power density and service life, it is necessary

to rely on modeling and simulation tools in the development process of HT-PEMFC systems. While

three-dimensional (3D) computational fluid dynamics (CFD) models of PEMFCs, e.g., AVL FIRE [2],

Ansys Fluent [3], CFX [4], CFD-ACE+ [5], OpenFOAM [6], COMSOL Multiphysics [7], and STAR-

CD [8], have already reached a relatively high level of maturity, there are still major challenges to be

solved in the area of system-level PEMFC models. These models are specifically applied to designing

the entire system. For HT-PEMFCs, this includes the stack, and the balance-of-plant (BoP) components

Nomenclature

𝛼 electron transfer coefficient

𝜎 surface tension

Γ surface density of sites available for oxidation

𝜀 volume fraction

Θ fraction of surface covered by oxide species

𝜌 density

𝑎 activity

𝑏 Tafel slope reduced for transfer coefficient

𝑐 concentration of species

�� average conc. in computational domain

𝑑 width

𝐷 diffusion coefficient matrix

𝐸 equilibrium potential

𝑓𝑁 continuous particle size distribution

𝐹 Faraday constant

𝐻 Heaviside theta function

𝑘 reaction rate constant

𝑘𝑟𝑒𝑣 reverse reaction deceleration

𝑙 depth of computational slice

𝑚 mass loading

𝑀 molar mass

𝑁𝑖 particle size class population

𝑟 reaction rate

𝑅 universal gas constant

𝑅𝑖 particle radius

𝑆 surface

𝑣 velocity

𝑉 scalar velocity potential

𝑉 volume

𝑈𝐹𝐶 fuel cell voltage

𝑈𝑐𝑎𝑡 local voltage in cathode catalyst layer

𝑋𝑆 renormalization factor

𝑧𝑒𝑛𝑡𝑒𝑟 coordinates of channel entrance

𝑧𝑒𝑥𝑖𝑡 coordinates of channel exit

Abbreviations

𝐶 carbon

𝐶𝑂 stable 𝐶 = 𝑂 carbon surface group

𝐶𝑂2 carbon dioxide

𝐶𝑂𝐻 𝐶 − 𝑂𝐻 carbon surface group

𝑃𝑡 platinum

𝑃𝑡2+ platinum ion (2+)

𝑃𝑡𝑂 stable 𝑃𝑡 = 𝑂 platinum surface group

𝑃𝑡𝑂𝐻 𝑃𝑡 − 𝑂𝐻 platinum surface group

𝐻+ proton

𝐻2𝑂 water

𝑎𝑔𝑔 agglomeration

𝑎𝑛 anode

𝑎𝑡𝑡 attachment

𝑐𝑎𝑡 cathode

𝑐𝑜𝑟𝑟 corrosion

𝑑𝑒𝑡 detachment

𝑑𝑖𝑠𝑠 dissolution

𝑖𝑜𝑛 ionomer

𝑚𝑎𝑥 maximum

𝑚𝑒𝑎𝑛 mean

𝑚𝑒𝑟 merging

𝑜𝑥 oxidation

𝑡𝑜𝑡𝑎𝑙 total



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(the thermal management component and the control unit). The ultimate aim is to design a system that

meets the required performance and service life. Furthermore, in the common development process, the

system-level models are applied in the early development stages for exploration of the design space,

designing the system, and subsequently in the validation and calibration phases, where they serve as

hardware-in-the-loop (HiL) applications and are used for the development of control functionalities and

strategies.

To support tasks in the early development stages, a system-level model must feature a high level of

predictiveness and generality, in addition to short computational times that enables the assessment of

numerous system configurations. In the validation and calibration phases (and especially in HiL

applications), the real-time capability of the model and its realistic response to variations in actuator

parameters are required. A high level of predictiveness of the models applied in the early development

stages is clearly motivated by the fact that in those development phases, hardware components are not

available. In addition, in both early and late development stages, the high level of predictiveness (and

thus realistic response to variations in actuator parameters) are required when approaching engineering

limits to support efficient exploration of the very large design space including selection and control of

the BoP components, which greatly influence the optimal parameters of the fuel cell (FC) stack.

Therefore, applicability of the data driven models, e.g., [9–13], which are commonly applied in the

system level, is limited to studies aimed at combined performance and service life optimizations. This

is because the accuracy of the data-driven model does not reach beyond the trained variation space of

parameters, and that it is generally not practically possible to train the model over a variation space of

parameters that features a very large dimensionality.

A further challenge motivating high level of predictiveness of the models arises from the objective of

ensuring prolonged service life, which requires coupling of the model of FC operation with FC

degradation models. For the case of the FC catalyst layer degradation, it is well known [14–16] that the

local rates of platinum (Pt) dissolution, redeposition, migration, and agglomeration, which will

subsequently be denoted as individual degradation mechanisms (Fig. 2 a)), strongly depend on the local

values and temporal dynamics of temperature, reactant concentrations, and electric potential field

(subsequently denoted as degradation stimuli). This in turn means that the degradation processes are

only modeled in a sufficient level of detail if spatial and temporal variations in degradation stimuli,

provided by the FC operation model, are considered. Such an approach is not possible with the use of

lumped degradation models, in which the entire chain of individual degradation mechanisms is

substituted with a simple empirical model, for example, in [17–20]. Because the causal chain of

degradation processes is poorly modeled in such an approach, its usefulness in supporting the FC

performance and degradation optimization in the early development stages and under a wide range of

parameter variations is very limited.

Models addressing individual degradation processes in the FC catalyst layer have already been

established. The carbon corrosion was modelled either by simple electrochemical model [21], by

coupling between carbon and Pt surface groups [14,22] or by more complex microscopic models [23].

A wide variety of approaches was proposed for modelling Pt surface oxidation, describing it by single

[16,24] or multi-step [14,25] electrochemical process, or coupled with other electrochemical reactions

[25,26], leading to its dissolution [16,26–28], redeposition [15] and migration in the membrane [15,27].

The effects of Pt detachment and agglomeration [29], and Ostwald ripening [28,30] on changes in the

Pt particle size distribution in the catalyst layer and the consequent loss of active surface have also been

studied using simplified models of these processes [31,32]. The degradation of alloyed Pt catalysts was

also studied by Franco et al. in [33]. The models of these individual processes have also been combined



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into a more extensive multi-scale models, taking into account the interplay between degradation

processes. This was done by modelling the detailed dynamics of individual degradation processes on

different scales using methods such as coarse-grained molecular dynamics and kinetic Monte Carlo

methods, and combining the results into a fully coupled degradation model [23,31,34,35] The feedback

effects of degradation of individual fuel cell components on the degradation stimuli were also modeled

[36,37]. The reader is referred to the review article, recently published by Jahnke et al., for an extended

overview of the state-of-the-art in catalyst degradation modelling of PEM-FC [38].

In addition to the degradation of the Pt catalyst, membrane degradation is another important factor in

determining the FC performance and lifetime [38]. Several membrane degradation models have been

proposed in the literature, mostly focused on degradation of Nafion or similar membranes in low-

temperature PEMFCs (LT-PEMFCs) [39–42], but some also address HT-PEMFC membrane

degradation [43]. It is well proven that catalyst dissolution, diffusion, and redistribution in the membrane

play an important role as a precursor of membrane degradation processes [15,38,44]. Therefore, proper

modeling of catalyst degradation plays a crucial role in determining the degradation stimuli used in the

membrane degradation model and in fully developed system-level degradation model, the degradation

of the membrane should be included. However, the degradation stimuli for the catalyst degradation

model are mostly independent of the direct back-influences of membrane degradation, and can therefore,

for the testing purposes of catalyst degradation model, be considered on the level of FC operation model

via changing the macroscopic membrane parameters, as will be explained in Section 4.2 [42].

Despite a wide variety of approaches to the degradation modelling, the existing models, covering the

entire chain of degradation mechanism, are too complex to be applied as the system-level tools for FC

simulations. To the best of the authors’ knowledge, the models of individual degradation mechanism,

have not yet: a) been coupled to a system-level model of HT-PEMFC operation providing spatially and

temporally resolved traces of degradation stimuli; and b) achieved sufficient computational efficiency,

i.e., real-time capability, which enables direct modeling of HT-PEMFCs over longer time scales. To

achieve significant progress in the addressed area, this paper presents an innovative real-time capable

system-level modeling framework of the HT-PEMFC stack comprising a) a mechanistic spatially and

temporally resolved model of HT-PEMFC operation serving as the provider of degradation stimuli and

b) a degradation modeling framework based on interacting individual degradation mechanisms of Pt in

the cathode catalyst layer.

The paper is structured as follows. In Section 2, the key features of proposed modeling framework are

outlined, explaining the requirements the model need to fulfill to serve as a system-level tool, useful in

process of FC design. In Section 3, the framework components are discussed in more detail. In case of

already developed FC operation model, the basic ideas are outlined in Section 3.1 and the references are

provided where additional explanation of model features can be found. The novel physically based and

computationally efficient degradation model is presented in detail in Section 3.2 with the explanation of

all relevant degradation mechanisms and the equations used to model their effects. The coupling

between both components is explained in Section 3.3. The validation of the modeling framework is

presented in Section 4, with validation of operation model being presented in Section 4.1. Section 4.2

provides the analysis of experimental data, used in degradation model validation, with the simulation

results being presented in Section 4.3 and additional capabilities of the model, not used in validation,

being presented in Section 4.4. Concluding remarks are given is Section 5.



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2. Model requirements and key features of the modeling framework

The main objective of the model is to ensure high level of predictiveness of the entire causal chain from

1) the input in the form of FC operation and control parameters over to the 2) FC operational conditions

determining the rates of individual degradation mechanisms and 3) of the Pt degradation at the cathode

through modeling the Pt size redistribution in a transient operating regime. This requires innovatively

combining different nanoscale phenomena driven by transient degradation stimuli being determined by

the FC design and operating conditions.

The requirements of the FC operation model can thus be summarized as mechanistically based, spatially

resolved, transient, and computationally fast. On the other hand, the requirements of the degradation

modeling framework can be summarized as based on (the most influential) individual degradation

mechanisms, capable of considering (the most influential) interactions between different individual

degradation mechanisms, transient, and computationally fast.

The core principle of the physically based spatially and temporally resolved model of HT-PEMFC

operation is a novel computationally efficient approach that combines a one-dimensional (1D) numerical

and a two-dimensional (2D) analytic solution, denoted as hybrid 3D analytic-numerical (HAN) [45–48].

A real-time capable HAN modeling approach thus, on one hand, allows for achieving high level of

predictiveness in the FC performance modeling crucial for an adequate virtual integration of FC in the

plant model, and on the other hand, provides spatially and temporally resolved data of degradation

stimuli, which are prerequisite input parameters for the degradation modeling framework.

This ensures a plausible response of the degradation model to varying design and control parameters or

functionalities and thus allow it to reflect the degradation dynamics in FCs consistently. To attain high

computation speeds, the presented model relies on a discreet particle size distribution, which is

consistently applied to all relevant individual degradation mechanisms. This treatment of particle size

distribution is crucial for the correct treatment of two mechanisms:

- platinum and carbon oxidation reaction models that feature the Kelvin term addressing the

surface tension, which depends on the particle size, and

- calculation of detachment rate of Pt particles considering both the electrochemical kinetics of

carbon corrosion and the particle size.

Thereby, the proposed modeling framework enables full modeling of the coupled catalyst degradation

phenomena during the FC’s entire operating lifespan by continuously simulating the varying particle

size distribution through a consistent application of the conservation laws. Note that since the listed

degradation processes are not unique to pure Pt catalyst, very similar framework can be used also for

binary catalyst, provided that appropriately modified degradation rates are used. The degradation

mechanisms, specific for binary catalysts, such as component-dependent dissolution and leeching from

particles ([33]), however, obviously cannot be explained by the proposed model. The membrane

degradation is not modeled directly in the proposed framework; however, its influence on the catalyst

degradation stimuli is considered as a slow temporal change of membrane thickness and ionic

conductivity in the FC operation model.

It is important to note that the proposed innovative modeling framework is intended for a system-level

simulation over longer time scales. To comply with the listed model requirements, care was devoted to

meticulously balancing between the level of detail and the computational expense to model and replicate

the most significant performance and degradation phenomena and their interactions adequately. The

significant contribution of the study, therefore, originates from the fact that it presents a system-level



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mechanistic modeling framework for performance and degradation modeling bringing the FC system-

level modeling to a new level. The proposed innovative modeling framework enables—compared to the

current state of the art—an exploration of the design space not only in terms of the FC performance, but

also its longevity. Additionally, it offers a higher fidelity model-supported design of FC control

functionalities tailored to both performance and longevity objectives.

3. Modeling framework

To preserve the focus of the study, the modeling framework of the HT-PEMFC stack, being the

innovative contribution of this study, is presented in this section. A description of the models of BoP

components is omitted, as they are already widely reported in the literature [49–52].

3.1. Model of FC operation

The physically based spatially and temporally resolved model of HT-PEMFC operation (HAN) is based

on a modeling approach of the nonlinear electrochemical phenomena fully coupled to the species

transport framework that was presented in [45,46]. A computationally optimized HAN modeling

approach that complies with the real-time constraints denoted as HAN-RT (real-time) was presented in

[47] and the extensions of the HAN-RT modeling approach to high temperature FC was presented in

[48].

For completeness and brevity of this paper, the main principles of the model of FC operation are outlined

subsequently (please refer to [45–48] for more information). The detailed description of hydrogen

oxidation and oxygen reduction electrochemical reaction kinetics, membrane water transport, and proton

conductivity are omitted, as they follow the conventional expressions found in the literature (the exact

formulations used can be found in [45–48]). However, the electrochemical reaction kinetics relevant to

the degradation processes are given in detail in Section 3.2. Thus, the focus of this HAN model principles

summary is on the innovative modeling of species transport to and from the catalyst surface.

In the presented approach, the modeled FC is of a straight parallel channel geometry, which is

numerically discretized in the direction along the channels shown in Fig. 1 a) and b). This 1D numerical

discretization yields shallow 2D slices (Fig. 1 c)), which are subjected to the analytical solution. A 2D

slice consists of three parts: a cathode feed part, a membrane electrode assembly (MEA) part, and an

anode feed part, where each feed part is divided into one channel and two gas diffusion layer (GDL)

domains (under the channel and under the rib) yielding seven domains (Fig. 1 d)).

The complete solution is obtained by coupling the individual 2D analytic slice solutions to one another

via the perpendicular numerically resolved 1D gas flow. This gives the approach its name “hybrid 3D

analytic-numerical.” The HAN model thus yields a full 3D result on species concentration distribution.



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The basis of the HAN modeling approach is solving a steady-state species transport equation [45–47]:

𝜕𝑐

𝜕𝑡= −𝛁 ⋅ 𝑱 = 𝐷𝛁2𝑐 − 𝛁 ⋅ (𝒖 𝑐) = 0, (1)

with 𝑱 representing the 3D species fluxes, 𝛁 the 3D nabla operator, 𝐷 the diffusion coefficient matrix, 𝑐

the species concentrations vector, and 𝒖 the 3D gas velocity. This hybrid 3D approach introduces a

distinction that splits the general 3D notation into a 2D+1D notation reflecting a different treatment of

physical phenomena in the dimensions perpendicular to the channel gas flow (𝑥 − 𝑦 plane) and in the

dimension along the channel gas flow (𝑧 axis) yielding

𝜕𝑐

𝜕𝑡= ∇2(𝐷 𝑐 − �� 𝑈)⏟

2D

−1

𝑙((𝑣 𝑐)|𝑧=𝑧𝑒𝑥𝑖𝑡 − (𝑣 𝑐)|𝑧=𝑧𝑒𝑛𝑡𝑒𝑟)⏟

1𝐷

= 0, (2)

with 𝑈 being the scalar velocity potential as defined in [25,26], 𝑙 = 𝑧𝑒𝑥𝑖𝑡 − 𝑧𝑒𝑛𝑡𝑒𝑟, which is the depth

of the slice with 𝑧𝑒𝑛𝑡𝑒𝑟 and 𝑧𝑒𝑥𝑖𝑡 denoting the domain entry and exit coordinates of species flow,

respectively. Eq. (2) can be interpreted as a simple diffusion equation on a 2D plane with a source term

(1

𝑙(𝑣 𝑐)|𝑧=𝑧𝑒𝑛𝑡𝑒𝑟) and a sink term (−

1

𝑙(𝑣 𝑐)|𝑧=𝑧𝑒𝑥𝑖𝑡).

The analytic part of the HAN modeling approach is obtained by the following: a) first, devising a general

analytic solution of the 2D diffusion problem in each of the seven computational domain types (Fig. 1

d)) by finding the domain-specific particular integral and eigenfunctions (also called harmonics) of the

Laplace (∇2) operator; and b) second, expressing this specific solution in the form of a linear expansion

comprising the particular integral and a Fourier series of eigenfunctions in such a way that

- the continuity of the solution and its derivative is ensured at the interfaces between

computational domains, and

- the boundary conditions are met.

The smallest number of terms considered in such an expansion, which still leads to satisfactory results,

is used in the real-time capable HAN model (HAN-RT):

Fig. 1: Schematic fuel cell geometry broken down into elementary units used in the HAN modeling approach. Blue regions represent the membrane and spotted translucent the GDLs; the green surfaces represent the rib symmetry plane and the yellow surfaces the symmetry plane between two ribs. a) A parallel channel co- or counter-flow fuel cell geometry. b) Representative unit with a highlighted slice (red). c) A sliced-out section of the representative unit. d) Computational domains of a slice (from top down: cathode channel domain, two cathode GDL domains, MEA domain, two anode GDL domains, and anode channel domain).



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𝑐(𝑥, 𝑦) = 𝐴0𝑥2

ℎ2+ 𝐵0,0 + 𝐵0,1 𝑐𝑜𝑠 (

𝜋

ℎ𝑥) . (3)

The physical significance of the three terms in (3) is explained in [47].

With this number of harmonics taken into consideration, the solution distinguishes two distinct species

concentration values pertaining to the two regions of catalyst per feed part per slice: the one under the

channel and the one under the rib (Fig. 1 d)).

In addition to distinct concentration values (reactant and product concentration), the solution of the

diffusion problem coupled to the electrochemical kinetics yields also distinct values of temperature,

electrode potential, reaction overvoltage, and electric current density. This offers not only spatially

resolved performance results, but also spatially resolved physical conditions that represent the

degradation stimuli.

3.2. Degradation modeling framework

The degradation modeling framework comprises interacting models of six individual degradation

mechanisms relevant for Pt degradation in the cathode catalyst layer:

1. carbon support surface oxidation,

2. carbon corrosion,

3. platinum particle detachment and agglomeration,

4. platinum particle surface oxidation,

5. platinum dissolution and redeposition/Ostwald ripening, and

6. platinum diffusion into membrane.

These are presented in the schematic of interacting individual degradation mechanisms in Fig. 2 a). To

provide a more intuitive insight, Fig. 2 b) shows a schematic of the intertwined individual degradation

mechanisms.

All individual degradation mechanisms are modeled as 0D reactors, which obtain spatially and

temporally resolved data of degradation stimuli as input parameters from the HAN-RT model of the FC

operation. The interaction between the model of FC operation and the degradation model is presented

in Section 3.3.

Fig. 2: Schematics of a) interacting individual degradation mechanisms and b) intertwined degradation mechanisms. The schematics aim at highlighting the importance of coupling the models of several degradation mechanisms due to their entanglement.

a) b)



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The oxidation dynamics on the surface of platinum particles depend strongly on the particle size due to

the effects of surface tension. The modeling approach is, therefore, based on the particle size

distribution, as explained in Section 3.2.1, where the particle size is considered in all relevant

electrochemical equations (see Sections 3.2.2 and 3.2.3), resulting in different oxidation dynamics for

each size class and a consequent change in the particle population of each class, which is explained in

Sections 3.2.4 and 3.2.5.

The platinum diffusion into the ionomer was modeled by assuming a linear concentration profile

between the catalyst and the position of the platinum band in the membrane [44] considering the Fickian

diffusion [27]. The diffusion of platinum into the membrane causes a net loss of platinum mass in the

catalyst, resulting in the loss of an active area; however, using the parameters taken from [27] and [44],

this contribution turned out to be very small compared to other degradation mechanisms. Nevertheless,

the modeling of this process is an important step in providing a physically plausible coupling to

membrane degradation models, where the platinum diffusion is a crucial initiator of further degradation

processes [44].

Particle size distribution model

The particles are sorted into 𝑀 size classes, indexed by integer 𝑖, with each class being populated with

𝑁𝑖 Pt particles featuring radii between 𝑅𝑖 −1

2Δ𝑅 and 𝑅𝑖 +

1

2Δ𝑅, where 𝑅𝑖 is the mean particle radius in

the 𝑖-th size class. The main goal of the degradation model is to calculate the rate of change of class

populations 𝑁𝑖(𝑡) over time caused by Ostwald ripening and particle agglomeration. This in turn allows

to track the changes of several important catalyst parameters, such as catalyst platinum mass loading

and catalyst surface area. Note, however, that the catalyst surface area, calculated from the particle size

distribution, does not directly relate to the electrochemically active surface area (see [53]), which should

be considered when the degradation model is coupled back to the FC operation model.

This approach is similar to the approach of Rinaldo et al. [54–56], where the particle size distribution is

expressed as a continuous function. However, the use of discreet particle size distribution with relatively

small number of size classes is essential for obtaining the high computation speeds required to simulate

long-term degradation effects on a system level. However, the use of such an approach requires various

adaptations in existing models of particle agglomeration and Ostwald ripening, as will be explained in

Sections 3.2.4 and 3.2.5.

Platinum and carbon oxidation and corrosion model

The precursors of catalyst degradation processes, addressed in our model, are oxidation processes on

the surface of Pt particles and carbon support in the catalyst layer, which are initiated by a combination

of high electric potential, temperature, and the presence of water vapor. On the one hand, oxidation leads

to carbon corrosion and consequent particle detachment and agglomeration; on the other hand, the part

of the surface of platinum particles, which is free of oxides, is prone to dissolution into the ionomer [16].

The carbon and Pt oxidation model is based on the oxidation mechanisms proposed by Pandy [14].

However, to improve the performance of the proposed model and enable its use in long-term degradation

simulations, the oxidation model was reduced to a minimal number of reactions required to describe the

transient phenomena such as the 𝐶𝑂2 emission peaks after voltage change [57], which will be discussed

in Section 4.4.2. The mechanism was originally developed for LT-PEMFCs; however, because the

carbon support and catalyst materials used in HT-PEMFCs are similar [58], a similar mechanism



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comprising the most significant reaction pathways was applied in the proposed model. However, the

difference in operating temperature of LT- and HT-PEMFCs results in different values of equilibrium

potentials and reaction constants of the proposed electrochemical reactions, as will be further discussed

in this section.

The core of the reduced mechanism consists of a set of six electrochemical reactions (4), describing the

oxidation processes of the carbon support surface and platinum particles in the catalyst layer. The

equilibrium potential 𝐸 of each equation is given in reference to the reversible hydrogen electrode

(RHE).

𝐶1: 𝐶∗ +𝐻2𝑂 ⇌ 𝐶 − 𝑂𝐻 + 𝐻+ + 𝑒−; 𝐸𝐶1,0 = 0.29 V RHE

𝐶2: 𝐶 − 𝑂𝐻 ⇌ 𝐶 = 𝑂 + 𝐻+ + 𝑒−; 𝐸𝐶2,0 = 0.80 V RHE

𝐶3: 𝐶 − 𝑂𝐻 + 𝐻2𝑂 → 𝐶∗ + 𝐶𝑂2 + 3𝐻+ + 3𝑒−; 𝐸𝐶2,0 = 0.96 V RHE

𝑃𝑡1: 𝑃𝑡∗ +𝐻2𝑂 ⇌ 𝑃𝑡 − 𝑂𝐻 +𝐻+ + 𝑒−, 𝐸𝑃𝑡1,0 = 0.79 V RHE

𝑃𝑡2: 𝑃𝑡 − 𝑂𝐻 ⇌ 𝑃𝑡 = 𝑂 + 𝐻+ + 𝑒−; 𝐸𝑃𝑡2,0 = 0.80 V RHE

𝑃𝑡𝐶: 𝑃𝑡 − 𝑂𝐻 + 𝐶 − 𝑂𝐻 → 𝐶∗ + 𝑃𝑡∗ + 𝐶𝑂2 + 2𝐻+ + 2𝑒−; 𝐸𝑃𝑡𝐶,0 = 0.62 V RHE

(4)

Oxidation of both the carbon support surface (reactions 𝐶1 and 𝐶2 in (4)) and the platinum particles

(reactions 𝑃𝑡1 and 𝑃𝑡2 in (4)) occurs in two stages. Hydroxide groups (𝐶 − 𝑂𝐻 and 𝑃𝑡 − 𝑂𝐻) are

formed on suitable defect sites on the carbon and Pt surface (𝐶∗ and 𝑃𝑡∗) in the presence of water

(reactions 𝐶1 and 𝑃𝑡1) at sufficiently high electric potential and are further oxidized into stable oxide

groups (𝐶 = 𝑂 and 𝑃𝑡 = 𝑂) at even higher potentials (reactions 𝐶2 and 𝑃𝑡2).

The surface groups induce the formation of carbon dioxide and consequential carbon corrosion via two

reactions: 𝐶3 and 𝑃𝑡𝐶 in (4). The carbon hydroxide groups 𝐶 − 𝑂𝐻 , on one hand, result in 𝐶𝑂2

formation in the presence of water at a high potential of 0.96 V (reaction 𝐶3), leaving behind new defect

site 𝐶∗ , and on the other hand, they also interact with 𝑃𝑡 − 𝑂𝐻 groups on the platinum surface at

potentials above 0.62 V (reaction 𝑃𝑡𝐶), again resulting in 𝐶𝑂2 formation and leaving behind both free

platinum and carbon defect sites 𝐶∗ and 𝑃𝑡∗ [14].

The temperature dependence of equilibrium potentials in reactions (4) is based on

𝑛 𝑒 𝐸0 = Δ𝐺 = Δ𝐻 − 𝑇 Δ𝑆, (5)

relating the equilibrium potential to the difference in Gibbs potential between reactants and products

[59]. To calculate the equilibrium potentials at HT-PEMFC conditions, as indicated in (4), from the

values given in [14] for LT-PEMFC, the entropy changes Δ𝑆 of the given reactions need to be evaluated.

Because the concentrations of reactants at equilibrium potential are equal to their reference values, only

the entropy changes due to the phase transitions of reactants were considered. Due to high temperature,

both water and 𝐶𝑂2 in reactions (4) can be assumed to be in gaseous state, and therefore, the entropy

change of condensation of both substances [60,61] was used to approximate the shifts of equilibrium

potentials in reactions 𝐶1, 𝐶3, 𝑃𝑡1, and 𝑃𝑡𝐶 in (4), which were calculated to be 0.09, 0.01, 0.09, and

−0.03 V, respectively, compared to the values given in [14].

In addition to the equilibrium potential change, an important difference between LT- and HT-PEMFCs

also lies in the origin of water and protons participating in oxidation reactions. In LT-PEMFCs, the

concentration of both water and protons depends strongly on the membrane humidity. By contrast, in

HT-PEMFCs, water is present in the gaseous phase; thus, its concentration is more directly related to



11

feed gas properties, while the protons participating in the reactions originate from phosphoric acid,

which is used as the membrane ionic conductor. Hence, their concentration is directly related to pH

values of the acid. In the proposed degradation model, a constant value of pH = 1.25 (related to 85%

phosphoric acid used for doping of polybenzimidazole membrane) is assumed [62,63]. Time-dependent

proton concentration could be obtained from a dedicated membrane degradation model describing the

acid leakage out of the membrane, but this effect was assumed to be negligible in the proposed model.

The state of surface oxidation of platinum and carbon is defined by the fraction of the surface covered

by each oxide species, i.e., Θ𝑃𝑡𝑂𝐻 , Θ𝑃𝑡𝑂 , Θ𝐶𝑂𝐻 , and Θ𝐶𝑂 , and the fraction of the surface free of

oxides, Θ𝑃𝑡 and Θ𝐶. It is also assumed that not more than one monolayer of oxides can cover the carbon

or platinum surface, Θ𝑋 < 1, (𝑋 = 𝑃𝑡, 𝑃𝑡𝑂𝐻, 𝑃𝑡𝑂, 𝐶, 𝐶𝑂𝐻, 𝐶𝑂), and that the sum of coverage fractions

on either carbon or platinum is constant over time:

Θ𝑃𝑡 + Θ𝑃𝑡𝑂𝐻 + Θ𝑃𝑡𝑂 = 1,

Θ𝐶 + Θ𝐶𝑂𝐻 + Θ𝐶𝑂 = 1. (6)

To obtain realistic results, it is important to consider how the particle size affects the equilibrium electric

potential of reactions 𝑃𝑡1, 𝑃𝑡2, and 𝑃𝑡𝐶 due to the surface tension. This is addressed by the introduction

of the Kelvin term, shifting the equilibrium potential from that of a flat Pt surface (denoted 𝐸0) by a

value that is inversely proportional to the particle radius 𝑅𝑖 [16,27]:

𝐸𝑃𝑡1(𝑅𝑖) = 𝐸𝑃𝑡1,0 +1

2𝐹𝑅𝑖(𝜎𝑃𝑡𝑂𝐻𝑀𝑃𝑡𝑂𝐻

𝜌𝑃𝑡𝑂𝐻 −𝜎𝑃𝑡𝑀𝑃𝑡

𝜌𝑃𝑡 ),

𝐸𝑃𝑡2(𝑅𝑖) = 𝐸𝑃𝑡2,0 +1

2𝐹𝑅𝑖(𝜎𝑃𝑡𝑂𝑀𝑃𝑡𝑂

𝜌𝑃𝑡𝑂 −𝜎𝑃𝑡𝑂𝐻𝑀𝑃𝑡𝑂𝐻

𝜌𝑃𝑡𝑂𝐻 ),

𝐸𝑃𝑡𝐶(𝑅𝑖) = 𝐸𝑃𝑡𝐶,0 −1

2𝐹𝑅𝑖

𝜎𝑃𝑡𝑂𝐻𝑀𝑃𝑡𝑂𝐻

𝜌𝑃𝑡𝑂𝐻 .

(7)

The magnitude of the Kelvin shift depends on the surface tensions 𝜎, molar mass 𝑀, and density 𝜌 of

the surface groups involved in the reaction (𝑃𝑡, 𝑃𝑡𝑂𝐻, and 𝑃𝑡𝑂).

The inclusion of the Kelvin term into the rates of reactions (4) is an important improvement over the

oxidation dynamics found in [14]. The rates of reactions (4) are [14,16,27]

a) 𝑟𝐶1,𝑖 = 𝛾𝐻𝑇𝑘𝐶1(𝑎𝐻2𝑂Θ𝐶,𝑖𝑒𝛼𝐶1(𝑈𝑐𝑎𝑡−𝐸𝐶1)/𝑏 − 𝑎𝐻+Θ𝐶𝑂𝐻,𝑖𝑒

−(1−𝛼𝐶1)(𝑈𝑐𝑎𝑡−𝐸𝐶1)/𝑏),

b) 𝑟𝐶2,𝑖 = 𝛾𝐻𝑇𝑘𝐶2(Θ𝐶𝑂𝐻,𝑖𝑒𝛼𝐶2(𝑈𝑐𝑎𝑡−𝐸𝐶2)/𝑏 − 𝑎𝐻+Θ𝐶𝑂,𝑖𝑒

−(1−𝛼𝐶2)(𝑈𝑐𝑎𝑡−𝐸𝐶2)/𝑏),

c) 𝑟𝐶3,𝑖 = 𝛾𝐻𝑇𝑘𝐶3𝑎𝐻2𝑂Θ𝐶𝑂𝐻,𝑖𝑒3𝛼𝐶3(𝑈𝑐𝑎𝑡−𝐸𝐶3)/𝑏,

d) 𝑟𝑃𝑡1,𝑖 = 𝛾𝐻𝑇𝑘𝑃𝑡1(𝑎𝐻2𝑂Θ𝑃𝑡,𝑖𝑒𝛼𝑃𝑡1(𝑈𝑐𝑎𝑡−𝐸𝑃𝑡1(𝑅𝑖)−𝑟𝑂𝑥ΘOx)/𝑏 −

𝑘𝑟𝑒𝑣𝑎𝐻+Θ𝑃𝑡𝑂𝐻,𝑖𝑒−(1−𝛼𝑃𝑡1)(𝑈𝑐𝑎𝑡−𝐸𝑃𝑡1(𝑅𝑖))/𝑏),

e) 𝑟𝑃𝑡2,𝑖 = 𝛾𝐻𝑇𝑘𝑃𝑡2(Θ𝑃𝑡𝑂𝐻,𝑖𝑒𝛼𝑃𝑡2(𝑈𝑐𝑎𝑡−𝐸𝑃𝑡2(𝑅𝑖)−𝑟𝑂𝑥ΘOx)/𝑏 − 𝑘𝑟𝑒𝑣𝑎𝐻+Θ𝑃𝑡𝑂,𝑖𝑒

−(1−𝛼𝑃𝑡2)(𝑈𝑐𝑎𝑡−𝐸𝑃𝑡2(𝑅𝑖))/𝑏),

f) 𝑟𝑃𝑡𝐶,𝑖 = 𝛾𝐻𝑇𝑘𝑃𝑡𝐶Θ𝐶𝑂𝐻,𝑖Θ𝑃𝑡𝑂𝐻,𝑖𝑒2𝛼𝑃𝑡𝐶(𝑈𝑐𝑎𝑡−𝐸𝑃𝑡𝐶(𝑅𝑖))/𝑏,

(8)

where 𝑟𝑗,𝑖 denotes the rate of the 𝑗-th reaction in (4) (𝑗 = 𝐶1, 𝐶2, 𝐶3, 𝑃𝑡1, 𝑃𝑡2, 𝑃𝑡𝐶) for particles in the

𝑖-th size class with mean radius 𝑅𝑖.

General scaling of the reaction rates is given by the reaction rate constants 𝑘𝑗, given for LT-PEMFC in

[14,15]. The reaction rates are additionally affected by temperature via the change in Gibbs free energy

of the transition states of each electrochemical reaction. As a detailed analysis of transition states

required to completely describe these effects requires insights on the atomistic scale [59], which is not

considered in the scope of this study, the reaction rate constants for the case of HT-PEMFCs were

modeled by multiplying all LT-PEMFC reaction rates, reported in [14], by the same temperature

dependent factor 𝛾𝐻𝑇 in (8), determined by the calibration of the model using experimental data.



12

Because the reaction rates change exponentially with the enthalpy of transition states Δ𝐻++, the change

in temperature from 80℃, typical for LT-PEMFCs, to 160℃ for HT-PEMFCs, can result in a relatively

large reaction rate acceleration 𝛾𝐻𝑇 of several thousands, assuming that the enthalpies of transition state

are of the order of 1 eV [59]. Another effect of temperature 𝑇 on the reaction rates (8) is captured in the

Tafel slope, 𝑏 = 𝑅𝑇/𝐹, with the gas constant 𝑅 and the Faraday constant 𝐹.

The balance between forward and backward rates of reactions is affected by two factors: first, the rate

and direction are determined by the concentration of reactants and products, namely the surface

coverages of different oxide species (Θ𝑋), water activity (𝑎𝐻2𝑂 = 𝑐𝐻2𝑂/𝑐𝐻2𝑂,𝑟𝑒𝑓), and proton activity

(𝑎𝐻+ = 𝑐𝐻+/𝑐𝐻+,𝑟𝑒𝑓). Second, the rate changes exponentially with the difference between the local

electric potential in the catalyst layer 𝑈𝑐𝑎𝑡, which is defined as the potential difference between catalyst

particles and the adjacent ionomer, and the equilibrium potential of each reaction (𝐸𝑗), which is affected

by the particle size via the Kelvin term (7) for the reactions featuring platinum surface oxides (𝐸𝑗(𝑅𝑖)

in (8) d)–f)). Note that the potential 𝑈𝑐𝑎𝑡 is in general different from the FC potential 𝑈𝐹𝐶 (see Section

4.2). The reaction rates are further shifted due to the presence of oxide layers [64,65], expressed as an

additional exponential term 𝑟𝑂𝑥ΘOx = 𝑟𝑂𝑥(ΘPtOH + ΘPtO) (Eq. (8) d)–e)). The slower rates of

reduction, in comparison to oxidation, are addressed by an additional parameter 𝑘𝑟𝑒𝑣 in (8) d)–e)

introduced in [16,27].

The reaction rates (8) determine the temporal dynamics of surface groups’ production and

consequentially their surface coverage Θ𝑋,𝑖 for each particle size class 𝑅𝑖:

a) 𝑑ΘPt,i

𝑑𝑡=

𝐹

ΓPt(−𝑟𝑃𝑡1,𝑖 +

𝑑𝑃𝑡𝐶

𝑅𝑖𝑟𝑃𝑡𝐶,𝑖),

b) 𝑑ΘPtOH,i

𝑑𝑡=

𝐹

ΓPt(𝑟𝑃𝑡1,𝑖 − 𝑟𝑃𝑡2,𝑖 −

𝑑𝑃𝑡𝐶

𝑅𝑖𝑟𝑃𝑡𝐶,𝑖),

c) 𝑑ΘPtO,i

𝑑𝑡=

𝐹

ΓPt𝑟𝑃𝑡2,𝑖,

d) 𝑑ΘC,i

𝑑𝑡=

𝐹

ΓC(−𝑟𝐶1,𝑖 + 𝑟𝐶3,𝑖 + 𝑟𝑃𝑡𝐶,𝑖),

e) 𝑑ΘCOH,i

𝑑𝑡=

𝐹

ΓC(𝑟𝐶1,𝑖 − 𝑟𝐶2,𝑖 − 𝑟𝐶3,𝑖 − 𝑟𝑃𝑡𝐶,𝑖),

f) 𝑑ΘCO,i

𝑑𝑡=

𝐹

ΓC𝑟𝐶2,𝑖.

(9)

The set of differential equations (9) by definition conserves the sum of surface coverages on the platinum

and carbon support (6). Because the reaction rates are defined per unit area ([𝑚𝑜𝑙 ∙ 𝑚−2 ∙ 𝑠−1]), the rate

of change of oxide coverages is calculated via the surface density of sites available for oxidation (ΓPt, ΓC)

[15,66,67]. Note that the reaction rates, given by Pandy [14] already consider the factor 𝐹/Γ and are,

therefore, given as dimensionless numbers. Additionally, it is important to consider that the reaction

between Pt and carbon OH groups (reaction 𝑃𝑡𝐶 in Eq. (5)) only takes place on a narrow band of width

𝑑𝑃𝑡𝐶 at the junction between the Pt and carbon surface (see Fig. 3) and the oxide coverage of carbon

therefore only relates to this region.



13

Electrochemical processes, described in this section, affect the particle size distribution 𝑁𝑖 in two ways:

First, the part of platinum surface not covered by oxides, is prone to dissolution and redeposition of

𝑃𝑡2+ions, as explained in Section 3.2.3, causing some particles to grow and others to shrink, leading to

changes in particle size distribution, as explained in Section 3.2.4. Second, the carbon corrosion,

addressed by reactions 𝐶3 and 𝑃𝑡𝐶, results in the detachment of Pt particles from the carbon support

and their agglomeration into larger particles, as discussed in Section 3.2.5. Thus, a relatively small set

of reactions, describing basic oxidation phenomena on carbon and Pt surface, provides a physically

grounded and computationally efficient method to distinguish between underlying causes of different Pt

particles growth mechanism.

Platinum dissolution and redeposition model

The dissolution of platinum is described by the electrochemical reaction [16,27]

𝑃𝑡 ⇌ 𝑃𝑡2+ + 2𝑒−; 𝐸𝑑𝑖𝑠𝑠,0 = 1.155 V RHE. (10)

Similar to the case of carbon and Pt oxidation, the equilibrium potential of dissolution, obtained from

[16,27], is shifted by the temperature due to the entropy of hydration of 𝑃𝑡2+ ions [68], causing a

decrease of −0.033 V (Eq. (5)) compared to that of LT-PEMFC, resulting in a value 𝐸𝑑𝑖𝑠𝑠,0 = 1.155 V.

In the proposed model of platinum dissolution, it is assumed that the dissolution and redeposition of

𝑃𝑡2+ ions can only take place on the parts of the platinum surface, which are not covered by oxides

[16]. Similar to the rates of oxidation (8), the dynamics of platinum dissolution are given by the reaction

rate, consisting of forward and backward terms:

𝑎) 𝑟𝑑𝑖𝑠𝑠,𝑖 = 𝛾𝐻𝑇𝑘𝑑𝑖𝑠𝑠Θ𝑃𝑡 (𝑒𝛼𝑑𝑖𝑠𝑠(𝑈𝑐𝑎𝑡−𝐸𝑑𝑖𝑠𝑠(𝑅𝑖))

𝑏𝑑𝑖𝑠𝑠 −𝑐𝑃𝑡2+

𝑐𝑃𝑡2+,𝑟𝑒𝑓𝑒−(1−𝛼𝑑𝑖𝑠𝑠)(𝑈𝑐𝑎𝑡−𝐸𝑑𝑖𝑠𝑠(𝑅𝑖))

𝑏𝑑𝑖𝑠𝑠 ),

𝑏) 𝐸𝑑𝑖𝑠𝑠(𝑅𝑖) = 𝐸𝑑𝑖𝑠𝑠,0 −1

2𝐹

𝜎𝑃𝑡𝑀𝑃𝑡𝜌𝑃𝑡𝑅𝑖

.

(11)

The dissolution rate is proportional to the reaction rate constant 𝑘𝑑𝑖𝑠𝑠 , multiplied by the oxide-free

surface fraction ΘPt. The increase in reaction rate due to temperature is described by the same factor

𝛾𝐻𝑇 as for oxidation reactions (8). The balance between the forward and backward reactions is again

determined by the difference between the local catalyst electric potential 𝑈𝑐𝑎𝑡 and the equilibrium

Fig. 3: Schematic of microscopic geometric parameters of degradation model. Carbon corrosion, relevant for particle detachment, only takes place in a narrow region of width 𝑑𝑃𝑡𝐶 on the junction of platinum and carbon; therefore, the oxide coverage of carbon is tracked only in this region.



14

dissolution potential 𝐸𝑑𝑖𝑠𝑠(𝑅𝑖), affected by the Kelvin term (11) b), as well as the concentration of Pt

ions dissolved in the ionomer 𝑐𝑃𝑡2+. It is assumed in the proposed model that platinum particles of

various sizes are randomly spread throughout the catalyst and, therefore, all particles, regardless of size,

interact with a single reservoir of 𝑃𝑡2+ ions with a uniform concentration. Therefore, the time dynamics

of 𝑃𝑡2+ ion concentration 𝑐𝑃𝑡2+ is expressed as a sum of contributions of particles from all size classes:

𝑑𝑐𝑃𝑡2+

𝑑𝑡=

1

𝑉𝑖𝑜𝑛∑ 4𝜋𝑅𝑖

2𝑀𝑖=1 𝑁𝑖𝑟𝑑𝑖𝑠𝑠,𝑖. (12)

The model also considers the total volume of the ionomer in the catalyst layer, 𝑉𝑖𝑜𝑛 = 𝑆𝐹𝐶𝑑𝑐𝜀𝑖𝑜𝑛 ,

calculated as a product of the FC surface 𝑆𝐹𝐶, the catalyst layer thickness 𝑑𝑐, and the volume fraction

of ionomer 𝜀𝑖𝑜𝑛.

The coupling of particles of all sizes to a shared reservoir of 𝑃𝑡2+ ions is crucial for modeling the so-

called Ostwald ripening [16]. The equilibrium concentration 𝑐𝑃𝑡2+, at which the dissolution rate (11) a)

is zero, depends on the particle size due to the Kelvin term: it is larger for small particles and smaller

for large particles. The global stationary concentration 𝑐𝑃𝑡2+ can, therefore, only be achieved when

small particles are dissolving and the platinum is redepositing on large particles [26]. This is an

important factor in the degradation of the FC catalyst layer, which leads to an increase in mean particle

size as will be explained in Section 3.2.4.

Particle size redistribution: Ostwald ripening

The dissolution or redeposition of platinum on a particle of typical size 𝑅𝑖 in the 𝑖-th size distribution

class leads to its growth or shrinkage depending on dissolution rate (11):

𝑑𝑅𝑖

𝑑𝑡= −

𝑀𝑃𝑡

𝜌𝑃𝑡𝑟𝑑𝑖𝑠𝑠,𝑖. (13)

The growth or shrinkage of particles in a particular size class leads to a decrease in its population unless

it is outweighed by new entries to the class coming in as shrinking particles from a larger size class, or

growing particles from a smaller size class. The number of particles transferred between classes depends

also on the width Δ𝑅𝑖 of each size class and the number of particles 𝑁𝑖 it contains. The governing

equations written below are given for an equidistant distribution with a uniform class width Δ𝑅:

𝑑𝑁𝑑𝑖𝑠𝑠,1

𝑑𝑡=

1

Δ𝑅[−𝑁1|��1| + 𝑁2|��2|𝐻(−��2)],

𝑑𝑁𝑑𝑖𝑠𝑠,𝑖

𝑑𝑡=

1

Δ𝑅[𝑁𝑖−1|��𝑖−1|𝐻(��𝑖−1) − 𝑁𝑖|��𝑖| + 𝑁𝑖+1|��𝑖+1|𝐻(−��𝑖+1)], 2 ≤ 𝑖 ≤ 𝑀 − 1,

𝑑𝑁𝑑𝑖𝑠𝑠,𝑀

𝑑𝑡=

1

Δ𝑅[𝑁𝑀−1��𝑀−1𝐻(��𝑀−1) + 𝑁𝑀��𝑀𝐻(−��𝑀)],

(14)

where the Heaviside theta function 𝐻(. ) is used to properly distinguish the contributions of neighboring

size classes in case of growth or shrinkage of particles populating them. Note that (14) do not conserve

the total number of particles because the shrinking particles from size class 𝑖 = 1 are completely

dissolved and vanished from the distribution. The growing particles in largest size class 𝑖 = 𝑀, are on

the other hand considered as staying in the same size class.

In the limit of infinite number of classes (𝑀 → ∞,Δ𝑅 → 0), equations (14) transform into the expression

for continuous particle size distribution used in [54–56]:



15

𝑑𝑓𝑁(𝑟, 𝑡)

𝑑𝑡=𝜕

𝜕𝑟[𝑓𝑁(𝑟, 𝑡)

𝑑𝑟

𝑑𝑡]. (15)

For long-term degradation modeling, however, when a small number of particle size classes are desired,

as is the case in computationally efficient system-level models, aiming at short simulation times, the

expression (14) is more physically plausible than one that would be obtained from (15) with the use of

the finite difference method.

As explained in Section 3.2.3, dissolution and redeposition lead to shrinkage of smaller particles and

growth of larger ones. Applied to (14), this leads to a decrease in population of classes with small

particles, and an increase in population of classes with large particles. Even in the case of negligible

platinum diffusion into the membrane and consequent mass conservation, this leads to a reduction in

catalyst surface area. The effect of this process will be demonstrated in Section 4.4.1, where the results

of degradation modeling are presented.

Particle size redistribution: particle detachment and agglomeration

The platinum and carbon surface oxidation processes in the presence of water at elevated voltage and

temperature lead to carbon corrosion, as explained in Section 3.2.2. Because corrosion is most prominent

at the junction of the Pt particle and its carbon support, it inevitably leads to a decrease in contact surface

between the particle and its carbon support and therefore to the eventual detachment of the particle. In

this section, a model of platinum particle detachment from its carbon support, caused by corrosion, and

the consequent agglomeration is presented.

The detailed mechanisms of the agglomeration processes are not yet completely understood [69];

therefore, some assumptions have been used to model these effects on a system level. First, it is assumed

that at each moment, the number of particles attached to the carbon support is much larger than the

number of detached particles. This means that the detached particles will always merge with particles

attached to the support. This is closely related to the second assumption that a detached particle instantly

merges with another particle. This eliminates the need for tracking the number of detached particles. It

is also assumed that the probability of two particles merging is independent of their size: a detached

particle will merge with an attached particle of a random size.

To calculate the change in the particle size distribution, three variables need to be calculated for each

size class 𝑖: the rate of particle detachment ��𝑑𝑒𝑡,𝑖, the rate at which the detached particles are attached

to the particles in the respective size class ��𝑎𝑡𝑡,𝑖, and the rate of new particles appearing in the respective

class due to merging ��𝑚𝑒𝑟,𝑖. The net dynamics of particle population in a class due to agglomeration

depend on all three contributions:

𝑑𝑁𝑎𝑔𝑔,𝑖

𝑑𝑡= ��𝑑𝑒𝑡,𝑖 + ��𝑎𝑡𝑡,𝑖 + ��𝑚𝑒𝑟,𝑖, (16)

where ��𝑑𝑒𝑡,𝑖 and ��𝑎𝑡𝑡,𝑖 are defined as negative quantities. A similar model was proposed in [56] for the

case of a continuous particle size distribution.

The rate of particle detachment is directly linked to the carbon corrosion rate ((8) c) and f)): a particle

will detach when a sufficient amount of carbon underneath it is corroded. The mass of corroded carbon,

required for detachment, depends on the size of the Pt particle and on the strength of its attachment to

the support. In the proposed model, it is assumed that even for particles of similar size, the strength of

attachment to the carbon support can vary, and thus, even for a small amount of corroded carbon, some

particles will already detach. As another limit, the amount of corroded carbon required to detach all



16

particles of size 𝑅𝑖 will be calculated by assuming that the most strongly attached particles have half the

surface submerged into the carbon support. The mass of corroded carbon required to detach such a

particle is

𝑚𝐶,𝑚𝑎𝑥,𝑖 = 2𝜋𝑅𝑖2𝑑𝑃𝑡𝐶𝜌𝐶 , (17)

assuming that the corrosion takes place mostly at the Pt–carbon junction of width 𝑑𝑃𝑡𝐶 (Fig. 3). The

assumption that the number of detached particles increases linearly with the amount of corroded carbon,

yields the result

𝑑𝑁𝑑𝑒𝑡,𝑖𝑑𝑡

= −𝑘𝑑𝑒𝑡��𝐶,𝑐𝑜𝑟𝑟,𝑖𝑚𝐶,𝑚𝑎𝑥,𝑖

𝑁𝑖 = −𝑘𝑑𝑒𝑡𝑀𝐶𝜌𝐶(𝑟𝑃𝑡𝐶,𝑖 + 𝑟𝐶3,𝑖)

𝑁𝑖2𝜋𝑅𝑖

, (18)

which indicates that smaller particles are more likely to detach than larger ones. The model includes one

physically motivated calibration parameter 𝑘𝑑𝑒𝑡, which covers all other influences not accounted for

specifically in the simple model, as for example, the effects of possible surface defects in carbon, which

may decrease the strength of particle attachment and result in higher detachment rates, or exact topology

of interaction between the Pt particle and its support. Eq. (18) is an important improvement over the

existing models of particle detachment [31,32,56] because it considers the effects of particle size on the

detachment rate as well as the detailed electrochemical background of carbon corrosion via reaction

rates 𝑟𝑃𝑡𝐶 and 𝑟𝐶3 (8).

Following the assumption that the detached particles merge with the attached particles independently of

their size, the rate of attachment to particles of size 𝑅𝑖 is proportional to the number of such particles 𝑁𝑖

and the total rate of particle detachment, summed over all classes, ��𝑑𝑒𝑡,𝑡𝑜𝑡 = ∑ ��𝑑𝑒𝑡,𝑖𝑀𝑖=1 :

𝑑𝑁𝑎𝑡𝑡,𝑖𝑑𝑡

=��𝑑𝑒𝑡,𝑡𝑜𝑡∑ 𝑁𝑖𝑀𝑖=1

𝑁𝑖 . (19)

The rate of particle production due to their merging, ��𝑚𝑒𝑟,𝑖, is based on the conservation of mass of

platinum particles during merging. When a detached particle of size 𝑅𝑗 attaches to a particle of size 𝑅𝑘,

the size of the newly formed particle is

𝑅𝑖 = √𝑅𝑗3 + 𝑅𝑘

33. (20)

The rate at which the merged particles appear in size class 𝑅𝑖 is therefore obtained by the summation of

the contributions of all detachment rates in class 𝑅𝑗 and attachment rates in class 𝑅𝑘, such that 𝑅𝑗 and

𝑅𝑘 fulfill condition (20):

𝑑𝑁𝑚𝑒𝑟,𝑖𝑑𝑡

=𝑅𝑖𝑋𝑆𝑛𝑖Δ𝑅

∑��𝑑𝑒𝑡,𝑗��𝑎𝑡𝑡,𝑘

��𝑑𝑒𝑡,𝑡𝑜𝑡

√𝑅𝑗3+𝑅𝑘

33≈𝑅𝑖

. (21)

Note that Eq. (21) is equivalent to the expression for continuous particle size distributions, given in [56],

but much more suitable for numerical calculations with a limited number of particle size classes.

However, to properly account for different numbers of possible pairs, 𝑅𝑗 and 𝑅𝑘, contributing to the new

particles in class 𝑅𝑖, the sum needs to be properly renormalized by comparing the number of terms in

the sum in (21), denoted as 𝑛𝑖, to the area of integration in the continuous case, expressed by 𝑋𝑆 =𝑑

𝑑𝑅∫ √𝑅3 − 𝑥3

3𝑅

0𝑑𝑥 ≈ 1.766. This approach ensures the conservation of Pt mass in agglomeration



17

model even at small number of particle size classes desired to ensure sufficient computational efficiency

of system-level degradation model.

Because the newly formed particles in the distribution (Eq. (21)) will always be larger than either of

both particles involved in the merging, the total change in distribution due to agglomeration (Eq. (16))

will lead to a decrease in the number of small particles and an increase in the number of large ones. The

net effect is similar to the Ostwald ripening (14), which will be further discussed in Section 4.4.

Interaction of individual degradation mechanisms

The full time dynamics of the particle size redistribution is calculated by combining the effects of the

Ostwald ripening (14) and the particle agglomeration (16):

𝑑𝑁𝑖𝑑𝑡

= 𝑑𝑁𝑑𝑖𝑠𝑠,𝑖𝑑𝑡

+𝑑𝑁𝑎𝑔𝑔,𝑖

𝑑𝑡. (22)

This set of 𝑀 differential equations for all size classes in the particle size distribution determines the

changes in surface area of the FC catalyst, which is a standard way of tracking the degradation processes

in FCs. However, as both terms in (22) depend on oxidation rates and surface oxide coverages, these

also need to be calculated using Eq. (9). Two equations for each size class can be eliminated based on

the constant sum of surface coverages (6), resulting in 5 ×𝑀 differential equations. One additional

equation is required to track the 𝑃𝑡2+ ion concentration (12), resulting in a set of 5 × 𝑀 + 1 differential

equations. What is important is that the number of size classes 𝑀 can be used to balance between the

level of details in the modeled particle size distribution and the performance efficiency of the model

based on the intended use.

3.3. Interaction of models

As depicted in Fig. 4, the spatial resolution of the degradation model consisting of one dedicated 0D

reactor for each respective computational domain of the FC operation model is defined by the resolution

of the FC operation model. This approach is justified by the fact that the model of FC operation has a

sufficiently coarse discretization to support the assumption that the short-range Pt phenomena of

detachment and agglomeration as well as the Pt redeposition due to dissolution occur well within one

computational domain. Since the degradation study in the anode catalyst layer are very similar, the

proposed model could in principle also be used to model the anode degradation by coupling it to the

degradation stimuli outputs of FC operation model, calculated for the anode catalyst layer. The

experimental data on the anode degradation in HT PEMFC are scarce and the proposed model was

therefore tested only for cathode catalyst layer.



18

The proposed modeling framework also allows for modeling the back influences of the degradation

phenomena (Fig. 4) on the parameters of the operational model via the reduced catalyst surface area

(Section 3.2.1), which in turn affects the catalyst’s exchange current density. However, to simplify the

analysis of the degradation mechanisms, this sort of back-coupling of the degradation to operation model

is a task deserving its own dedicated study and was therefore omitted.

4. Results and discussion

4.1. Validating the model of FC operation

The model of HT-PEMFC operation was extensively validated on three levels by benchmarking its

simulation results with the following:

1. spatially resolved results generated by a validated [2,70] 3D multiphase CFD tool [47],

2. experimental results of a steady-state FC operation at various boundary conditions [70], and

3. experimental results of a transient FC operation [48].

Therefore, its validation is only briefly addressed in this study as more emphasis is placed on the

validation and the results of modeling the degradation phenomena, which are being reported for the first

time.

The spatial variations in degradation stimuli lead to important differences in degradation rates at

different locations in the FC [71,72]; therefore, it is instructive to demonstrate that the model of HT-

PEMFC operation produces high-reliability results, which serve as the degradation stimuli used in the

degradation modeling.

The performance of the HAN-RT model in simulating the steady-state operation of the HT-PEMFC is

validated by comparing the results of the spatial distribution of important physical quantities obtained

with the HAN-RT simulation to those obtained by the 3D multiphase CFD tool.

Fig. 4: Schematic of the interaction between the performance model (Fig. 1) and the degradation modeling framework (Fig. 2).



19

The spatial distributions of four variables were compared as follows:

1) distribution of oxygen concentration at cathode catalyst surface,

2) distribution of water vapor concentration at cathode catalyst surface to validate gas flow,

diffusion, convection, and general species transport modeling,

3) distribution of electric current density, and

4) distribution of catalyst electric potential to validate the modeling of electrochemical reaction

kinetics and the thermodynamic equilibrium potential.

All plots in Fig. 5 show close agreement between the results obtained by the high-resolution high-

accuracy 3D CFD model and the results obtained by the computationally optimized HAN-RT model.

The HAN-RT model distinguishes average values of variables at the (cathode) catalyst surface under

the channel and values under the rib [47]. For easier comparison, the detailed 3D CFD results are also

gathered and averaged to provide corresponding “under channel” and “under rib” values and plotted

alongside the HAN-RT results in Fig. 5.

To avoid the effects of too many physical phenomena overshadowing one another, a case with a high

hydrogen concentration in the anode feed gas and a close to uniform temperature has been simulated.

Thus, predominantly, the effects of variation in oxygen and water vapor concentration are reflected in

Fig. 5: Plots of distribution of values of various variables over cathode catalyst surface and in the cathode channel of the representative unit at the output current level of the steady-state operating point. The surface under channel and surface under rib are distinguished. The x-axis runs along the channel gas flow direction with the anode inlet and the cathode outlet at x = 0 mm. Solid lines refer to 3D CFD results and dots to HAN results with blue color denoting values pertaining to “under channel” location and orange to “under rib” location.



20

the variation in the plotted variables. As the oxygen is being consumed on the way from the inlet (𝑥 =

0.018 m) to the outlet (𝑥 = 0.000 m), its concentration at the catalyst surface correspondingly decreases

as shown in Fig. 5 a). Additionally, the difference between the conditions at the catalyst surface under

the channel and under the rib is clearly visible with oxygen levels under the rib being lower due to the

longer pathway from the channel. Inversely, water is being produced at the catalyst surface; thus, on the

way from the inlet to the outlet, its concentration at the catalyst surface increases and also the

concentrations “under rib” are higher than those “under channel” with the same rationale (Fig. 5 b)). As

shown in Fig. 5 c), the variation in the distribution of current density closely follows that of the oxygen

catalyst surface concentration. This is an exemplary effect of the reactant concentration term in the

Butler–Volmer equation. The effect of oxygen concentration levels also dominates the cathode potential

as plotted in Fig. 5 d), overshadowing the effect of higher current density, which, on its own, would

assume the opposite trend, i.e., lower electrode potentials at higher current densities.

Within the scope of this study, it is important to present the capability of the HT-PEMFC operation

model to demonstrate the transient operation of an FC adequately. Thus, a challenging simulation case

of a start-up sequence, featuring temporal variations in volume flows of feed gases, FC temperature, and

electric current, is shown in Fig. 6. The FC is started up according to a specific electric current ramp-up

sequence and the accompanying reactant supply and cooling adjustment. The effects of these three

controlled processes are recorded as time variations in electric current (green), air (orange), and

hydrogen (purple) volume flows, and stack temperatures (black) are used as the time-varying boundary

conditions for the simulation.

The simulation objective and the main result are then the temporal variation in the stack voltage (Fig.

6). The voltage trace predicted by the simulation (red) and plotted alongside the experimentally

measured voltage data (blue) shown in Fig. 6 validates the model’s accuracy and predictive capabilities

in transient operation, which inherently depend on the accurate prediction of the reactant concentrations

being validated in [48].

The average real-time factor (simulation time divided by physical time) over the entire simulation was

less than 0.2 and it never exceeded 0.5, confirming the computational efficiency of the HAN-RT model.

Based on the presented results, it can be concluded that the model of FC operation fulfills all four

requirements listed in the introduction and thus represents an adequate basis for coupling to the

degradation model.

Fig. 6: Temporal variation in boundary conditions (normalized output current I, volume flows of reactant gases, and temperature) and comparative evaluation of the temporal variation in the simulated voltage output against the measured output stack voltage (output voltage 𝑈𝐹𝐶 plotted relative to the stack open circuit voltage OCV) during fuel cell start-up procedure. Close agreement between measured and modeled voltages during the entire simulation time validates the transient capabilities of the HAN-RT model.



21

4.2. Analysis of measured degradation data

The degradation model was calibrated and validated based on experimental data obtained by long-term

degradation tests performed on a real FC system. In [53], two HT-PEMFCs having an initial catalyst

particle size distribution with mean radius of 𝑅𝑚𝑒𝑎𝑛 = 2.06 nm and standard deviation of 𝜎 =

0.58 nm (MEA_t0) were operated at identical steady-state conditions at a temperature of 160℃ and

current density 0.2 A/m2, one for 2700 h (MEA_t1) and the other for 4800 h (MEA_t2). The input

variables of the degradation model were generated using the HAN-RT model, replicating these operating

conditions. As no information is provided in [53] on the position in the FC where the aged MEA was

analyzed, it was assumed that the samples used to determine the particle size distribution were taken

halfway between the inlet and outlet of the feed gas channels.

To distinguish the effects of catalyst and membrane degradation, allowing for a correct calculation of

local catalyst potential, the polarization curves of the fresh and the two aged MEA samples given in [53]

were reproduced using the HAN-RT model. The changes in electrochemical surface area and membrane

thickness during degradation, measured in [53], were used as input parameters of the HAN-RT model,

while the membrane ionic conductivity was used as a fitting parameter. The polarization curves for all

three samples, generated by the HAN-RT model, are compared to experimental data from [53] in Fig.

7. The fitted values of the membrane ionic conductivity show a decrease of 24% for MEA_t1 and 52%

for MEA_t2 compared to the fresh sample MEA_t0, indicating the importance of membrane

degradation.

However, although membrane degradation is not covered in the proposed degradation model, the

estimation of membrane degradation based on polarization curves enables the generation of correct

degradation stimuli for the catalyst degradation model, even in the absence of a dedicated membrane

degradation model. Most notably, the local catalyst potential, defined as the potential difference between

the Pt particles and the adjacent ionomer, can be estimated as

𝑈𝑐𝑎𝑡 = 𝑈𝐹𝐶 + 𝜂𝑎𝑛 + 𝜂𝑚𝑒𝑚, (23)

where 𝑈𝐹𝐶 is the operational voltage of the FC, 𝜂𝑎𝑛 is the anode activation voltage, and 𝜂𝑚𝑒𝑚 is the

membrane voltage drop, all of which are calculated using the HAN-RT model. Based on the polarization

curves for samples MEA_t0, MEA_t1, and MEA_t2, the degradation stimuli were calculated at 0, 2700,

and 4800 h of operation and the values at intermediate times were obtained by quadratic interpolation.

Fig. 7: Comparison between experimentally measured [53] and HAN-RT generated polarization curves for fresh and two

aged MEAs. Excellent agreement between modeled and measured values was achieved by using the measured values of

exchange current density and membrane thickness for all three samples and determining membrane conductivity as a

fitting parameter.



22

4.3. Degradation model calibration and validation

The calibration of the degradation model was aimed at achieving the best agreement of the modeled

particle size distribution 𝑁𝑖, calculated as explained in Section 3.2.6, for both aged MEA samples with

the ones given in the supplementary materials of [53]. To ensure a sufficiently smooth particle size

distribution and to avoid boundary effects of the largest particles in the distribution, M = 15 size classes

were used between 0 and 15 nm to cover the entire spectra of particle sizes reported in [53]. Numerical

integration of the set of 76 governing equations (see Section 3.2.6) was performed on a personal

computer using the Bulirsch–Stoer semi-implicit extrapolation method with an adaptive time step,

implemented in programming language C [73]. The computational time needed to simulate 4800 h of

FC degradation in steady state operation regime was ~10 s, indicating high computational efficiency of

the proposed model.

The model was calibrated by adapting only two parameters, 𝛾𝐻𝑇 and 𝑘𝑑𝑒𝑡 , which were then kept

constant during the entire degradation simulation. The first adaptive parameter was reaction rate

acceleration due to high temperature 𝛾𝐻𝑇 (introduced in Section 3.2.2). This factor also covers possible

differences in reaction rates originating from the fact that the catalyst material used in [53] is not pure

platinum but Pt/Ru nanoparticles, which might lead to different oxidation kinetics. The second adaptive

parameter was particle detachment factor 𝑘𝑑𝑒𝑡 (introduced in Section 3.2.5), addressing the detailed

mechanism of particle detachment not covered in the proposed model. Other parameters used in the

model were determined based on the established literature. To ensure the consistency between the rates

of electrochemical reactions, most values were taken from the paper by Pandy [14], covering most of

the relevant processes. The values of parameters used in the present study are given in Appendix 1.

Good agreement between experimental and modeled particle size distribution, as shown in Fig. 8, was

obtained at parameter values of 𝛾𝐻𝑇 ≈ 1300, consistent with the predictions made in Section 3.2.2, and

𝑘𝑑𝑒𝑡 ≈ 8900. Note that values of both calibration parameters directly depend on the values of other

degradation model parameters, obtained from the literature and listed in Appendix 1, which differ from

source to source [14,16,22,27]. Both values, as well as other model parameters, should therefore be

further verified by additional experiments, as proposed in Section 4.4. The deviations between both

distributions could (to some extent) also be attributed to the limited precision of the experimentally

Fig. 8: Comparison between experimentally measured particle size distribution (white-filled histogram) [53] and one produced by degradation model (colored histogram) for fresh and aged MEA. The measured particle distribution on a fresh sample MEA_t0 (black) was used as the model input. The modeled distribution of particles in both aged samples MEA_t1 (red) and MEA_t2 (blue) shows good agreement with experimentally measured data.



23

measured distribution, as measurement uncertainty is not reported in the source. The result clearly shows

that the proposed degradation model is capable of correctly reproducing the changes in particle size

distribution for two different stages of catalyst aging using the same calibration parameters, indicating

a valid approach to degradation modeling. The capability of the model to reproduce the degradation

phenomena mechanistically over an extended period of 2700 and 4800 h of FC operation can be

considered as a significant progress in the field of system-level FC degradation modeling.

4.4. Analysis of individual degradation mechanisms

This section presents a more detailed analysis of the degradation model results, elucidating the model

capabilities in distinguishing different degradation mechanisms. Furthermore, acknowledging that the

degradation tests, given in [53], were performed at a single FC operating point (i.e., at fixed operating

conditions) additional experiments supporting more precise model calibration are proposed.

Detailed analysis of Oswald ripening and particle agglomeration

To elucidate the details of the Oswald ripening modeling and particle agglomeration, as well as the

individual contributions of these effects to the overall particle growth, Fig. 9Fig. shows the rates of

change of particle size distribution due to the platinum particle dissolution (Oswald ripening, (14)) and

agglomeration (Eq. (16)) as a function of particle size for two FC operation modes and two particle size

distributions with M = 30 size classes. Fig. 9 a) shows the degradation rates after 48 h of long-term

degradation test, used in the model calibration (see Section 4.2), i.e., at a current density of 0.2 A/cm2

[53], resulting in a catalyst voltage (as calculated by HAN-RT) of 𝑈𝑐𝑎𝑡 ≈ 0.6 V. Fig. 9 b) shows the

same quantities at a higher FC voltage of 0.96 V on a partially aged sample (MEA_t1 in [53]) with mean

particle radius of 𝑅𝑚𝑒𝑎𝑛 = 5.49 nm and a standard deviation of 𝜎 = 2.28 nm. As for the case of model

calibration, the degradation stimuli were generated by the HAN-RT model.

The shape of blue curves in Fig. 9 shows the rates of change in particle size distribution due to the

Ostwald ripening (��𝑑𝑖𝑠𝑠, Eq. (14)). The Kelvin term initiates a rapid dissolution of smaller particles and

redeposition of platinum on larger particles, as explained in Section 3.2.4. The sharp peak in the

dissolution rate of 5.7 × 10−5 at 0.75 nm in Fig. 9 b), not completely shown on plot, results from the



24

competition between the rapid dissolution rate of small particles due to the Kelvin term (11) and the

small number of available small particles in the size distribution.

The detachment of particles from the carbon support (orange dashed line, ��𝑑𝑒𝑡 in Eq. (18)), and their

attachment to the remaining particles (green dashed line, ��𝑎𝑡𝑡 in Eq. (19)) are both proportional to the

total number of particles. The detachment is more prominent for smaller particles due to the faster carbon

corrosion (Kelvin term in reaction rate Eq. (8) f)) and smaller contact surface between the particle and

its carbon support (Eq. (18)). The merged particles (black dashed line, ��𝑚𝑒𝑟 in Eq. (21)) appear at larger

sizes (Eq. (20)), resulting in the combined effect of particle agglomeration (��𝑎𝑔𝑔, red line, Eq. (16))

being similar to the Ostwald ripening, initiating a net particle growth. Note that the left (dissolution rate)

and the right (agglomeration rate) axes in Fig. 9 are scaled differently. At a high voltage (Fig. 9 b)), the

effects of dissolution seem to be orders of magnitude more pronounced than the particle agglomeration,

while at a lower voltage during long-term operation (Fig. 9 a)), the contributions of both effects are

comparable, which can be explained by the decreased platinum dissolution at lower voltages (Eq. (11)).

The presented results indicate that both phenomena are modeled in a physically/chemically plausible

way. Because the proposed models of Ostwald ripening and particle agglomeration are based on

different mechanisms, they yield different size-dependence redistribution rates, as shown in Fig. 9, and

different relative rates depending on the FC voltage. These phenomena have already been reported as

experimental observation in some studies [74], which further confirms the validity of the proposed

degradation model and suggests that the relative contribution of Oswald ripening and particle

agglomeration could be determined from more precise measurements of the shape of particle size

distribution of aged samples obtained from degradation tests performed at different voltages.

Transient degradation processes

To highlight the importance of the impact of transient FC conditions on the relative rate of different

degradation phenomena, the model was tested in one of the standard DOE voltage cycling regimes

between 𝑈𝐹𝐶 = 0.6 V and 0.96 V with a cycle period of 6 s [75]. The model was tested on a partially

aged sample (MEA_t1 in [53]) with a mean particle radius of 𝑅𝑚𝑒𝑎𝑛 = 5.49 nm and a standard

deviation of 𝜎 = 2.28 nm, with the time traces of degradation stimuli being generated by the HAN-RT

model. The effects of this cycling were analyzed based on simulated 𝐶𝑂2 emissions, indicating carbon

corrosion, 𝑃𝑡2+ ion concentration in the ionomer, Pt particles dissolution, and surface oxide

concentrations, revealing detailed microscopic processes in the catalyst layer.

Fig. 9: Relative rates of particle redistribution ��/𝑁𝑡𝑜𝑡𝑎𝑙 due to Pt dissolution (blue) and agglomeration (red), split down to individual contributions due to detachment (orange dashed), attachment (green dashed), and merging (black dashed). a) Operation at current density of 0.2 A/cm2 (𝑈𝑐𝑎𝑡 ≈ 0.6 V) on 2.06 nm particles (MEA_t0). b) High voltage operation at 𝑈𝐹𝐶 =0.96 V on 5.49 nm particles (MEA_t1). Both contributions to particle redistribution show clear differences in particle size dependence as well as strong dependence on fuel cell potential.

b) a)



25

The black dotted line in Fig. 10 b) shows that the electric potential in the catalyst layer 𝑈𝑐𝑎𝑡, calculated

by the HAN-RT model, is slightly higher than the FC potential 𝑈𝐹𝐶, as expected based on Eq. (23). Fig.

10 a) shows the rate of carbon mass loss (orange) and the 𝑃𝑡2+ion concentration in the ionomer (blue),

resulting from voltage cycling as a function of time. The platinum ion concentration shows a sharp

increase when the voltage is increased and a rapid decline when it is decreased. This is in agreement

with the prediction that platinum dissolution is caused by high voltage, whereas this figure also shows

that the reaction rates, determined during model calibration, result in a very fast establishment of steady-

state concentrations. Such fast transitions suggest that the inclusion of diffusion effects to describe the

redistribution of 𝑃𝑡2+ions in the ionomer might improve the quality of the proposed degradation model.

The rate of carbon corrosion, expressed as a mass loss of carbon per second per catalyst geometric area,

shows typical peaks after the increase in voltage, similar to the ones measured by Borup [57] and

explained by Pandy [14] for LT-PEMFCs. The peaks are much narrower that the ones measured in [57],

which is the result of a much faster reaction kinetics in the HT-PEMFC compared to that in LT-PEMFC.

The appearance of carbon corrosion peaks is explained by the interaction of the surface oxide groups on

platinum and carbon. As shown in Fig. 10 b), after the change in voltage (black dotted line), the surface

oxide coverages are converging toward a steady-state value, which depends on both the voltage and the

particle size (Eq. (8)). The 𝐶 − 𝑂𝐻 surface groups accumulate on the carbon surface during the low

voltage part of the cycle [14], as shown in Fig. 10Fig. b). Afterward, when the voltage is increased far

above the equilibrium potential of the carbon corrosion reaction 𝑈𝑃𝑡𝐶 = 0.62 V, the 𝐶 − 𝑂𝐻 groups

rapidly react with the accumulated 𝑃𝑡 − 𝑂𝐻 groups via the 𝑃𝑡𝐶 reaction (8) f), resulting in intense

carbon corrosion. This can be observed in Fig. 10 b) as a rapid drop in Θ𝑃𝑡𝑂𝐻 and Θ𝐶𝑂𝐻 coverages after

voltage increase and as a peak in carbon corrosion rate in Fig. 10 a).

With the relative values of reaction rates taken from [14], the model predicts higher carbon corrosion

rates at low voltage compared to that at high voltage (Fig. 10Fig. a)). This is explained by the larger

availability of the 𝐶 − 𝑂𝐻 groups, initiating carbon corrosion at low voltage, as shown in Fig. 10 b). At

high voltage, although the exponential part of the reaction rate for the reaction 𝑃𝑡𝐶 (Eq. (8) f)) is being

increased, the concentration of the available 𝐶 − 𝑂𝐻 and 𝑃𝑡 − 𝑂𝐻 groups is decreased due to their

concurrent transformation to stable 𝐶 = 𝑂 and 𝑃𝑡 = 𝑂 groups, which leads to reduced rates of carbon

corrosion at these operating conditions.

These results highlight the capability of the proposed model to plausibly address the transient

degradation phenomena and suggest that the time resolved measurement of 𝐶𝑂2 emissions and 𝑃𝑡2+

Fig. 10: Results of degradation simulation during accelerated stress test of HT-PEMFC by fuel cell voltage cycling between𝑈𝐹𝐶 = 0.6 and 0.96 V every 3 s. a) Rate of corrosion of carbon support (orange) and concentration of dissolved platinum ions 𝑃𝑡2+ in catalyst ionomer (blue). The carbon corrosion rate shows typical peaks, reported by experiments in voltage cycling regime. b) Electric potential in catalyst layer (𝑈𝑐𝑎𝑡, black dotted) and surface oxide coverage of Pt particles and carbon support for two different particle sizes: 2 nm (solid lines) and 5 nm (dashed lines). The coverages change with time due to changes in voltage (Eq. (8)), but also depend on particle size due to the Kelvin term (7).

b) a)



26

ion concentration in the catalyst layer could be used to improve the model calibration and further

distinguish between the contribution of the Oswald ripening and the particle agglomeration to the overall

catalyst degradation.

Position dependence of catalyst degradation

Several experiments show that catalyst degradation is more pronounced toward the outlet side of cathode

gas channels compared to the inlet side [71,72]. Because the HAN-RT model is fully capable of

comprehending the differences in degradation stimuli along the gas paths, it was used for simulating

long-term degradation tests, described in [53], for regions of catalyst at the cathode inlet, in the middle

of the FC, and at the cathode outlet. The resulting changes in mean radius and standard deviations of

particle size distribution over time, shown in Fig. 11, indicate that the degradation model indeed

reproduces the trends observed in experiments. It is discernible from Fig. 11, that the particle growth is

more prominent at the cathode outlet (blue line), which can be explained by the increase in humidity of

cathode gas toward the outlet, as shown in Fig. 5 d). Because humidity is the main precursor of oxidation

reactions (4), the regions of the catalyst closer to the outlet will experience faster carbon corrosion and

consequently faster particle growth, which is in agreement with the results of experimental carbon

corrosion analysis on segmented FCs [71,72]. This result indicates the importance of spatially resolved

fields of degradation stimuli for accurate modeling of catalyst aging and the importance of knowledge

on sample positions for correct degradation model calibrations.

The results presented in this section show that all considered catalyst degradation processes are modeled

in a plausible way, and are compatible with experimental results given in the established literature. In

addition, the results highlight additional capabilities of the model to track individual degradation

processes, which are valuable for the efficient model calibration and crucial for the analysis of aging

processes in real FC systems and development of suitable mitigation strategies.

5. Conclusion

This paper presents an innovative predictive real-time capable system-level modeling framework for the

performance and cathode platinum degradation modeling of high temperature proton exchange

membrane fuel cells. The unique features of this computationally efficient model on the system level

Fig. 11: Increase in mean radius (solid line) and standard deviation (dashed line) of Pt particle size distribution during fuel cell operation for inlet, middle, and outlet of cathode channel, compared to experimental data (black dots with error bars) [53]. Particle agglomeration is more severe toward the end of the channel where water concentration is higher, resulting in faster carbon corrosion.



27

arise from the interaction of the mechanistic spatially and temporally resolved model of the high

temperature proton exchange membrane fuel cells operation and the degradation modeling framework,

based on interacting individual cathode platinum degradation mechanisms. The degradation modeling

framework considers various size distributions of platinum particles and comprises carbon and platinum

oxidation phenomena as well as platinum dissolution, redeposition, detachment, and agglomeration

phenomena, combined in physically consistent and computationally efficient model, enabling consistent

treatment of the entire causal chain while simulating the degradation phenomena over longer time scales

up to the fuel cell’s operating lifespan.

The presented results confirm the credibility of the proposed framework in modeling the FC

performance and platinum degradation at the cathode as well as its computational speed and efficiency.

The model is capable to reproduce and explain several phenomena, observed in fuel cell degradation

experiments. More pronounced cathode catalyst degradation towards the fuel cell gas channel outlet

compared to inlet is explained in the model by the increase in humidity along the channel. Furthermore,

the innovative modeling framework elucidates the differences in contributions to platinum particle

growth: the agglomeration, caused by carbon corrosion, is more pronounced at voltage cycling, while

the Ostwald ripening, caused by platinum dissolution, is more pronounced at higher voltages,

The proposed modeling framework, therefore, represents a significant progress in the area of model-

supported design of fuel cells and in-depth understanding of the cause and effect chain from the fuel cell

operation to its degradation. Hence, it efficiently contributes to the model-supported development of

advanced clean energy conversion technologies. Application areas of the proposed modeling framework

comprise the development and optimization of balance-of-plant design and control functionalities as

well as the optimization of fuel stack parameters to simultaneously prolong the service life and high fuel

cell performance and to contribute to the reduced production costs.

Acknowledgments

The research is partially funded by Slovenian Research Agency (research core funding No. P2-0401)

and by Austrian Research Promotion Agency (research project no. 848810: MEA Power and research

project no. 854867: SoH4PEM). The authors are also grateful to Freudenberg Sealing Technologies

GmbH & Co. KG for sharing the experimental data for model validation.

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33

Appendix 1

The input parameters of the degradation model were divided into three groups. The rates and equilibrium

potentials of degradation processes and various additional degradation parameters, introduced in Section

3.2, are listed in Table A1 as “Degradation model parameters.” These parameters were either extracted

directly from the literature or estimated based on the data available in the literature [14–16,53,66,67].

Table A1: Degradation model parameters

Parameter Value Units Description Source

𝐸𝑃𝑡1 0.79 𝑉 Equilibrium potential for reaction 𝑃𝑡1 Calculated from Ref. [14]

𝐸𝑃𝑡2 0.80 𝑉 Equilibrium potential for reaction 𝑃𝑡2 Calculated from Ref. [14]

𝐸𝑃𝑡𝐷𝑖𝑠 1.155 𝑉 Equilibrium potential for Pt dissolution Calculated from Ref. [16]

𝐸𝐶1 0.29 𝑉 Equilibrium potential for reaction 𝐶1 Calculated from Ref. [14]



𝐸𝑃𝑡𝐶 0.62 𝑉 Equilibrium potential for reaction 𝑃𝑡𝐶 Calculated from Ref. [14]

𝑘𝑃𝑡1,𝐿𝑇 2.2 × 10−5 𝑚𝑜𝑙/𝑚2𝑠 Reaction rate of reaction 𝑃𝑡1 Calculated from Ref. [14] and [15]

𝑘𝑃𝑡2,𝐿𝑇 2.2 × 10−5 𝑚𝑜𝑙/𝑚2𝑠 Reaction rate of reaction 𝑃𝑡2 Calculated from Ref. [14] and [15]

𝑘𝑃𝑡𝐷𝑖𝑠𝑠,𝐿𝑇 3.4 × 10−9 𝑚𝑜𝑙/𝑚2𝑠 Reaction rate of Pt dissolution Ref. [15]

𝑘𝐶1,𝐿𝑇 4.6 × 10−8 𝑚𝑜𝑙/𝑚2𝑠 Reaction rate of reaction 𝐶1 Calculated from Ref. [14] and [66]



𝑘𝑃𝑡𝐶,𝐿𝑇 1.1 × 10−5 𝑚𝑜𝑙/𝑚2𝑠 Reaction rate of reaction 𝑃𝑡𝐶 Calculated from Ref. [14] and [66]

𝑟𝑂𝑥 30 𝑘𝐽/𝑚𝑜𝑙 Pt oxide interaction Ref. [15]

𝑘𝑟𝑒𝑣 0.3 − Pt oxidation reversibility Ref. [16]

Γ𝑃𝑡 2.15 𝐴𝑠/𝑚2 Pt oxidation surface area Ref. [15]

Γ𝐶 4.6 𝐴𝑠/𝑚2 Carbon oxidation surface area Ref. [66], [67]

𝑑𝑃𝑡𝐶 1 𝑛𝑚 Carbon corrosion depth Assumed

𝑐𝑃𝑡2+,𝑟𝑒𝑓 1 𝑚𝑜𝑙/𝑙 Reference 𝑃𝑡2+concentration Ref. [15]

𝑘𝑑𝑒𝑡,𝐶𝑉 8900 − Particle detachment factor Fitted on [53]

γ𝐻𝑇 1300 − Particle detachment factor Fitted on [53]

The parameters, listed as “Material constants” in Table A2, define the physical properties of carbon,

platinum, and water and are independent of detailed FC and MEA properties.

Table A2: Material constants


𝜌𝑃𝑡 21090 𝑘𝑔/𝑚3 Pt density Ref. [15]

𝜌𝑃𝑡𝑂𝐻 14170 𝑘𝑔/𝑚3 Pt-OH density Calculated from Ref. [15]

𝜌𝑃𝑡𝑂 14100 𝑘𝑔/𝑚3 Pt=O density Ref. [15]

𝜌𝐶 2000 𝑘𝑔/𝑚3 Carbon density -

𝜎𝑃𝑡 2.73 𝐽/𝑚2 Pt surface tension Ref. [15]

𝜎𝑃𝑡𝑂𝐻 1.34 𝐽/𝑚2 Pt-OH surface tension Calculated from Ref. [15]

𝜎𝑃𝑡𝑂 1 𝐽/𝑚2 Pt=O surface tension Ref. [15]

𝑀𝑃𝑡 195 𝑔/𝑚𝑜𝑙 Pt molar mass Ref. [15]

𝑀𝑃𝑡𝑂𝐻 212 𝑔/𝑚𝑜𝑙 Pt-OH molar mass Calculated from Ref. [15]

𝑀𝑃𝑡𝑂 211 𝑔/𝑚𝑜𝑙 Pt=O molar mass Ref. [15]

𝑀𝐶 12 𝑔/𝑚𝑜𝑙 Carbon molar mass -

𝑀𝐻2𝑂 18 𝑔/𝑚𝑜𝑙 Water molar mass -

The parameters listed as “MEA properties” in Table A3 depend on the detailed structure and materials

used in the membrane and catalyst layer of the FC to be modeled. As it was not possible to extract all

parameters for the analysis from the source providing the experimental data used for validation [53],

some estimations were made based on other sources [76–78].



34

Table A3: MEA properties


𝑆0 50 𝑐𝑚2 Fuel cell surface Ref. [53]

𝑑𝑐 48 𝜇𝑚 Catalyst layer thickness Ref. [53]

𝑑𝑚𝑒𝑚

𝑁𝑡𝑜𝑡𝑎𝑙,0

85

1.0 × 1017

𝜇𝑚

−

Membrane thickness

Initial number of Pt particles

Ref. [53]

Calculated from Ref. [78]

𝜀𝑖𝑜𝑛 0.3 − Ionomer volume ratio in catalyst layer Ref. [76]

𝜀𝑃𝑡𝐶 0.2 − Carbon volume ratio in catalyst layer Ref. [76]

𝜀0 0.5 − Void volume ratio in catalyst layer Ref. [77]

Predictive system-level modeling framework for transient ...

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