Multiscale Model of Carbon Corrosion in a PEFC: Coupling with Electrocatalysis and Impact on Performance Degradation

Water transfer phenomenon is a mandatory issue that has been addressed in several low-temperature PEFC models.⁴¹ As a first approximation in our model, we consider the case of an isothermal fuel cell fed with pure oxygen and hydrogen fully saturated with water vapor (to be in agreement with experiments now occurring in our group). In particular, we assume that the polymer electrolyte membrane (PEM) and electrodes are already fully humidified at the simulated operation time (PEFC starting operation time). In these conditions, it is not necessary to take into account the transient pore phase flooding by liquid water in the electrode:^{36, 37} in fact, chemical potential of water becomes the same in pore and ionomer phases, inducing equilibrium conditions and thus no transients. The additional water production in the cathode is assumed to be evacuated through the liquid phase with zero thermodynamic resistance (no chemical potential gradient). These comments are extensive to water description in the gas diffusion layer (GDL). Pressure drop is supposed to be negligible in the GDL and the pore phase, and the additional water production is assumed to be evacuated through the liquid phase.

As we consider pure feeding gases (no nitrogen), hydrogen and oxygen pressure drops in the pore phases are neglected.³⁴ Produced pressure is regarded to be negligible [there is no accumulation of in the cathode; the diffuse through the GDL, and is evacuated by the channels (cf. COR section)].

In this first step model, oxygen crossover transients are not taken into account. The anodic carbon oxidation and the oxidation-dissolution-electrochemical ripening of Pt nanoparticles are not considered here.

Our global model organization is presented in Fig. 2. The model consists of four interacting parts: the anode region operating with hydrogen (anode 1), the anode region exposed to oxygen (anode 2), the cathode Pt/ionomer interface (cathode 1), and the cathode C/interface (cathode 2). The couple anode 1/cathode 1 constitutes the direct fuel cell, and the couple anode 2/cathode 2, the reverse one.

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In cathode 1, only the ORR is assumed to occur. In cathode 2, the COR is the only electrochemical reaction taking place. These two electrode models interact, e.g., as in the description of the platinum coarsening in cathode 1 driven by the carbon degradation occurring in cathode 2 (see COR section).

In the anode, we account for two different electrochemical reactions: the HOR and the ORR. As shown in Fig. 2, electrons produced by the COR (reverse current ) in cathode 2, are consumed by the ORR in cathode 1; the protons produced are consumed by the ORR occurring in anode 2 (this favors the COR in cathode 2). Then, the total current entering in the cathodic platinum phase is the addition of the external load current and the reverse current.

The cathodic electrostatic potential is assumed to be the same for platinum and carbon phases (i.e., the same for both cathodes). The anodic electrostatic potential is assumed to be the same for the two anodes (excellent electronic conductivity is assumed between the two anode regions).

The electrode morphology adopted here is similar to that previously introduced in Ref. ³⁷: carbon agglomerate networks (and supported Pt) are assumed to be embedded in a hydrated ionomer medium (the so-called impregnated Nafion phase in our previous papers) (Fig. 3). These structures also include nano/micropores, allowing the reactant gas transport. Obviously, this is a simplistic representation of the high-level complexity of the electrode morphology. However, it allows one to capture the elementary processes taking place in the MEA and to describe, on a physical basis, the coupling between them.

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Within a continuum framework, each electrode submodel is designed to describe coupled physicochemical phenomena taking place at different geometrical scales: a microscale transport phenomena description of reactants and charges (protons and electrons) through the electrodes and the membrane thickness, and a spatially distributed microscale model of the reactant diffusion through the hydrated Nafion ionomer layer covering the networks, are coupled to a 1D nanoscale mechanistic description of the catalyst-electrolyte interface (anodes 1 and 2, cathode 1) and of the carbon-electrolyte interface (cathode 2) (Fig. 4).

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The nanoscale models in anodes 1 and 2 and cathode 1 are similar to those introduced in our previous papers:^{5, 36–38} they consist of a compact layer submodel describing the competitive adsorption of the intermediate reaction species and the parasite water molecules on the catalyst surface, and of a diffuse layer submodel in the electrolyte, constituted of protons and spatially fixed counterions (sulfonate sites). The nanoscale model of the C/ionomer interface (cathode 2) is also shown in Fig. 4. Because of the highly complex nature of the C interface, we account here only for the inner layer describing the competitive adsorption of C oxides and water molecules (see COR section).

We note as the microscale coordinate in the thickness of the impregnated Nafion layer. The interfacial nanoscale model is located at the microscale point , and the nanoscale coordinate inside the diffuse layers of some nanometers thickness is noted by . The Nafion/Pt and Nafion/C interfaces are supposed to be flat and located at .

Adopted cathodic carbon structure

In order to simulate the temporal evolution of the cathode thickness (cf. COR section), we account for high carbon surface area/volume ratio by representing the carbon phase as a two-dimensional Sierpinski's carpet projected in the cathode thickness direction (Fig. 5).

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The total volume occupied by carbon is given by

where is the Sierpinski's fractal order, the electrode geometrical area, the (time-dependent) cathode thickness, the carbon density, and the instantaneous carbon mass constituting the cathode.

The internal total carbon surface area is

By assuming an initial cathodic carbon mass , , [an initial cathode thickness (cf. Tables I and II)] and using Eq. 3, the fractal order is ; this value is adopted for the calculations presented in this paper. By using Eq. 4 with these parameter values, the specific carbon surface area in the cathode is , in quantitative agreement with experimental measurements.⁹ Equation 3 also implies that if we increase the initial carbon mass, the initial cathode thickness is increased.

In the anode pore phase, the hydrogen and oxygen partial pressures (, ) are assumed to obey

where is the water saturation pressure (cf. Table I). In a first approximation, we assume a linear relation between anode hydrogen and oxygen partial pressures and total anode pressure, according to

where is a dimensionless constant comprised between 0 and 1 (cf. Table II).

Table I. Parameters values used in the numerical simulations.

	Geometrical or physicochemical parameter value	Unit	References
Anode			Estimated
Cathode 1			Estimated
Cathode 2			Estimated
Cathode 2			Estimated and 9
Cathode 1			Estimated
		m	Measured
		m	Estimated 5–7, 36–40
		m	Measured
		m	Measured
		m	Measured
			Measured
		m	5–7, 36–40
		m	Calculated 5
		—	Calculated (Eq. 4)
			36–38
Anode 1			Fitted
Anode 2		—	Fitted
Cathode 1		—	Fitted
Cathode 2		—	Fitted
Anode 1			This work and 5–7, 36–40
			This work and 5–7, 36–40
Anode 2			Fitted
			Fitted
			Fitted
Cathode 1			This work and 5–7, 36–40
			Fitted
			Fitted
Cathode 2			Fitted
			Fitted
			Fitted
			Fitted
			Fitted
			Fitted
			Fitted
			5
			36, 37
	(the same value in both electrodes)		36, 37
	(the same value in both electrodes)		5–7, 36, 37
		—	53
			5, 37
			5, 37
			5, 37
			50
			Assumed
			Assumed
			36
			36
			36
		Pa	5–7, 36–40
			38
			5
			53
			Assumed
		Ω	Assumed

Table II. Default parameters and operating conditions used for the simulations.

Default operating parameter value

The anodic platinum surface area is assumed to be partially exposed to oxygen. The platinum surface area exposed to hydrogen is given by

where is a coefficient comprised between 0 and 1 (cf. Table II). The platinum surface area exposed to oxygen is given by

In the cathode pore phase, the oxygen partial pressure is assumed to obey

In the impregnated Nafion layers hydrogen and oxygen transport is assumed to be governed by diffusion^{36, 37}

where and are, respectively, the concentration and the diffusion coefficient of the hydrogen (anode 1) and oxygen (anode 2 and cathode 1).

By assuming equilibrated absorption of and in hydrated Nafion, we have at (Henry's laws)^{36, 37}

where and are the Henry's coefficients given in Table I.

The fluxes are given by the kinetics at the compact layer level, as described in the following section.

The nanoscale models describe the interfacial electrochemical phenomena taking place in anodes 1 and 2 as well as in cathode 2 (cf. Fig. 5). According to our approach,^{5–7, 36–40} in the compact layer we have

where is the catalyst surface coverage by the free sites, is the surface coverage by the reaction intermediates, and and θ⃖ the surface coverage by parasite adsorbed water molecules with dipolar moment opposed (up) and directed (down) to the platinum layer. Note that all the coverages are defined at the compact layer level: a simple physically based description of the electrochemical interface is provided, taking into account the coupling between the double layer structure and the electrochemical reactions.

In anode 1, we describe the HOR according to the Tafel–Heyrovsky–Volmer reaction model³⁸

In anode 2 and cathode 1, for the ORR description, we assume the mechanism^{5, 36, 42}

The rates of these elementary steps can be written for the HOR as³⁸

and for the ORR in cathode 1^{5, 37}

and, respectively, , , and for the ORR in anode 2 with kinetic parameter values being not necessarily the same as in Eq. 24, 25, 26.

In the previous kinetic equations, is the hydrogen concentration at , the oxygen concentration in cathode 1 or in anode 2, the proton concentration, and , and are the surface coverages by adsorbed intermediates. , denotes the electronic transfer factors, are the standard rate constants, and is the electrostatic potential drop through the compact layer, defined as^{36, 38}

where is the electrostatic potential in the catalyst phase and is the electrostatic potential in the Nafion phase just outside the water layer adsorbed on the catalyst with charge surface density . is given by

where is the electrostatic potential drop related to the thickness of the adsorbed water layer, is the potential drop related to the dipolar nature of these molecules, where is the "effective thickness" of the compact layer (e.g., this layer could consist of several water layers), , is the electric permittivity of the compact layer, and is the electric permittivity of the diffuse layer. We have demonstrated that in Eq. 28 is given by the solution of the transcendental equation³⁸

The value of (anode 1) and the values of and (anode 2 and cathode 2) are given by the solution of the balance equations

For the surface coverage by dipoles directed toward the electrode, we have^{5, 37, 38}

and for the opposed ones, we have

is given by^{5, 37, 38}

The diffuse layer models describe the proton diffusion/electromigration transport (in the nanometric neighborhood of the particles), coupled with the electric field generated by the resulting charge distribution. All species are considered as punctual, and the interparticle electrical interaction and water solvatation are neglected. The proton-driving equations are given by

and the electrostatic potential in the diffuse layer by the Poisson's equation as

where is the electrical potential in the electrolyte phase, and the concentration of the spatially immobile anions modelling the Nafion sulfonate sites.³⁸

For the anode 1 diffuse layer model, the boundary conditions are given by

where is obtained as explained in the following section. This diffuse layer model allows calculating and . Following the procedure described in Ref. ^36–40, the combination of Eq. 27, 28, 29 gives , the electrostatic potential in the anode phase.

For the cathode 1 ORR, we have

Again, in cathode 1, this diffuse layer model allows calculating and . The combination of Eq. 27, 28, 29 gives , the electrostatic potential in the cathode Pt and C phases.

For the anode 2 ORR, we write

The anode 2 diffuse layer model allows calculating for kinetics Eq. 24, 26, as well as as described in the section on membrane models.

The charge density is calculated by the conservation law at the catalyst/electrolyte interface surface^{5, 36–38} as

for anode 1 where

where is given by Eq. 8 and by

where is the external load current (the input of the model), and is the reverse current produced by the carbon oxidation (cf. next section).

For cathode 1, we have

where

where is the surface of the platinum interface in cathode 1 (cf. next section).

Finally, for the anode 2 ORR, we have

where

with given by Eq. 9.

The boundary conditions to coupling with the microscale transport description of the reactants (Eq. 11) in the impregnated Nafion are

for anode 1 and

for cathode 1 (anode 2).

On the basis of Eq. 27, 28, the anode 1 and cathode 1 nanoscale models calculate the electrostatic potential in the Pt phase for both electrodes; thus, the cell potential is given by .

In an analogous way to anode 1 and cathode 1, here we introduce a compact layer model describing the carbon oxidation mechanism (cf. Fig. 4). In a first approximation, the carbon oxidation is modeled by the two-step mechanism in Eq. 1, 2 with an O intermediate specie. In this compact layer, we have the global equation

The reaction rates are, respectively, for and

where are the standard rate parameters (cf. Table I), is the proton concentration [supposed to be the same as in the bulk ], is the concentration in the cathode, and is the surface coverage by the absorbed monoatomic oxygen on the carbon interface surface given by the solution of the balance equation

In Eq. 63, is the carbon pure phase activity,and is given by⁴³

with the initial carbon mass in the cathode and the cumulated lost carbon mass given by

where is the carbon molar mass.

The concentration of in the cathode is given by the balance equation

where is the molar flux from the cathode to the GDL, given by

is the molar flux from the GDL to the cathode chamber (channel), given by

where is the diffusion coefficient of into the GDL. As it shows in Fig. 6, expression 70 can be written as

where is the concentration in the cathodic chamber given by

where and are, respectively, the inlet and outlet gas flow in the cathode channel, is the concentration in the oxygen inlet flow gas, the concentration in the channel, and the cathode channel thickness.

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The C/electrolyte interfacial potential in Eq. 63, 64 is given by

where the electrostatic potential in the carbon phase is known [ is assumed], and the electrostatic potential at the compact layer level is assumed to be the same as in the bulk [, calculated as explained later, in the section on membrane models]. The current produced by the COR is

which constitutes the input for Eq. 53, 59.

The different coverages are

with the same transcendental equation calculating as in the other electrodes (cf. Eq. 29), with provided by the solution of

In our model, we assume three phenomena to be directly linked with the COR: the decrease of the cathode electrode thickness , the decrease of the platinum surface area of the cathode , and the increase of the cathode/GDL contact resistance .

Cathode thickness temporal evolution

From Eq. 3, we derive a linear relation between the cathode thickness and the lost carbon mass

The total cathode carbon surface area decreases following Eq. 4.

COR-driven platinum coarsening

Under operating PEFC conditions, the platinum oxidation/dissolution,^5–7 the electrochemical Ostwald's ripening,^5–7 and the platinum coarsening driven by the COR cathode degradation,³ contribute to decrease the platinum surface area.

In the model in this paper, we take into account only the platinum coarsening (the two other mechanisms being already implemented in our previous model^5–7). This mechanism is assumed to be driven by the decrease of the cathode thickness: when the carbon support degrades, the platinum nanoparticles coarsening is favored and they can detach from the carbon support becoming useless.^44–46 Both processes contribute to the active surface area decrease (Fig. 7).

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In a first approximation, we assume the mathematical relation for the platinum surface area of cathode 1

where β is a dimensionless positive constant, the initial thickness of the cathode electrode, and is the initial surface area of platinum in cathode 1 given by

where is the platinum total mass in the cathode, is the radius of the platinum nanoparticles (spherical representation),and is the platinum density.

Time-dependent contact resistance

denotes the time-dependent contact resistance between the cathode electrode and the GDL (Fig. 8). When the cathode is not damaged (no carbon degradation), we assume to have , and the electrostatic potential in the electrode carbon phase and the one on the current collector are the same. However, when the carbon layer is damaged, the electric connectivity among the electrode, the GDL, and the current collector is degraded and then increases.^{15, 47, 48}

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In a first approximation, we assume that is a function of according to

where is a positive constant (cf. Table I).

The measured cathode potential becomes

Remark. In a first approximation, the impregnated Nafion thickness is assumed to be constant. Actually, because the carbon thickness decreases, the Nafion thickness may increase (if Nafion chemical degradation is not significant), increasing even more the reactant transport resistances and the impact on MEA performance degradation rate.

We have solved our model numerically by means of an in-house code developed within a Simulink/Matlab environment. Discretization of the equations describing the reactants diffusion through the Nafion phase has been directly performed by using Simulink; five grid points have been used for the impregnated layer models. Solutions of the diffuse layer transport equations are analytically approximated as described in Appendix 1.

For the simulations shown in this paper, we have chosen the set of physicochemical and geometrical parameters given in Table I. The orders of magnitude of some of the involved parameters in our model are known (e.g., electrical permittivity ), others are estimated [e.g., ] or measured (e.g., ). Electrokinetics parameters are obtained following the procedure in Ref. ³⁶: calculated observables (e.g., potential) are fitted with experimental values under well-known operating conditions [e.g., predicted open-circuit voltage (OCV) value obeys Nernstian behavior under equilibrium conditions (i.e., at ]. Results shown in all figures in this paper are obtained in the conditions detailed in Table II, unless stated otherwise. The numerical code allows the analysis of the dynamic behavior of the different state variables for different simulated operation times , including the electrode electronic potentials and and the cell potential , in response to a static or dynamic current demand at given temperature , pressures and , and initial platinum, Nafion, and carbon loadings. Also, the model allows analyzing the temporal evolution of the reverse current , the concentration, the cathode electrode thickness , the cumulated carbon mass loss, and the platinum surface area.

Electrochemical impedance spectroscopy diagrams are also calculated for different and operating conditions. These transient simulations are carried out by following the numerical method previously introduced by us in the context of PEFC modeling:^{5–7, 36–40} the calculation is performed by controlling the current in the time domain with subsequent fast Fourier transformation. For the determination of the impedances, as in real experiments, a small sinusoidal signal (1% of the nominal current) is superimposed on the fixed (dc) current level in order to obtain a linear response of the potentials. This is performed for frequencies ranging from in the anticlockwise direction. As in our previous models, prior to starting the simulation of the periodic signal response, the pseudo-steady-state is calculated.

We underline that the full coupled model (i.e., including anode 1, cathode 1, anode 2, and cathode 2) is solved here. Simulations have been carried out on a computer with an Intel Pentium 4 processor , of RAM, using the commercial Simulink solver ode15s with variable integration time step. The approximate calculation running time for a simulated operation time of is . Calculation time (including the calculation of the pseudostationary state) of an EIS with 40 frequency points is .

Impact of the COR-driven degradation mechanisms on the performance temporal evolution

In Fig. 9, we compare the polarization curves obtained without and with cathode carbon oxidation for three different simulated operation times (the calculation without the coupling with the COR corresponds to the case ).

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For the shorter simulated operation times ( and ), the cell potential shape agrees with typical experimental cell polarization curves. At , the OCV potential appears to be lower because of the simultaneous presence of hydrogen and oxygen in the anode chamber.

For longer simulated operation times ( and ), the aging impact on performance reveals higher potential at OCV and low-current potential (no considerable impact is observed for intermediate and high operating currents). This behavior agrees with experimental observations⁹ and theoretical results discussed later in this paper (cf. Fig. 16).

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On the basis of these results, we underline here that carbon oxidation could be another explanation of the experimentally observed PEFC OCV degradation under power cycling conditions, which some authors attribute to the presence of in the cathode (resulting from membrane crossover from the anode).¹⁰

Figure 10 shows the simulated long-term behavior of the anode, cathode, and cell potential, for two static nominal currents. First, we note that and decrease with time because of the COR-driven degradation mechanisms [by not taking into account the COR description, the model calculates time-invariant potentials (not shown here)]. When the cathode carbon phase is degraded, the cathode thickness and the platinum surface area decrease with time (Fig. 11a and 11b), the latter following the platinum coarsening mechanism (cf. Eq. 79). Thus, the current density at the platinum level increases, despite the corrosion current decline shown in Fig. 12 (this current decreases because carbon mass decreases). In fact, there exists a competition between the COR rate, inducing the decrease, and the COR-driven platinum coarsening, inducing the increase. Prediction of cathode thickness and corrosion current decays has been already reported in experimental literature.^{11, 31}

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For both nominal currents in Fig. 10, at a given simulated operation time , the simulated cathodic potential decreases dramatically. This potential failure occurs when the cathode platinum surface area (driven by the temporal evolution of the cathode thickness) becomes too small to ensure the demanded current .

The predicted "potential collapse" operation times probably overestimate the durability of typical PEFC because, in the model discussed here, we do not include coupling with other degradation mechanisms. Our previous model in Ref. ^5–7 can be coupled to the model here in order to explore the competition between the different aging mechanisms, as we propose for future work⁴⁹ (cf. Conclusion section).

The temporal evolution of the cumulative carbon mass loss and the concentration in the cathode layer at are provided in Fig. 13. The predicted behavior is governed by the carbon corrosion reaction rates (cf. Eq. 63, 64). The cumulative carbon mass loss increases with time, but the slope decreases because of the platinum surface area decrease (cf. discussion of Fig. 28). The transient concentration behavior is in qualitative agreement with some in-line measurements reported in literature.¹⁸

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The calculated effect of the external nominal current on the MEA durability [cell potential "failure time" (cf. Fig. 10)] is plotted in Fig. 14, which shows a bell-shaped curve suggesting that an "optimal" external current inducing a maximal durability exists. The predicted shape is explained as follows. Lower currents favor COR-driven Pt coarsening (COR is favored). Then, decreases faster as the current decreases: durability decreases as the nominal current decreases. At higher currents, COR-driven Pt coarsening is slower; however, because the demanded current is higher, failure potential is reached faster as the current increases, and durability decreases again as the current increases.

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As shown in Fig. 15, the model captures the effect of the simulated operation time on the EIS pattern. At , for short simulated operation times, EIS consists of a high-frequency capacitive and of a low-frequency inductive loop, in agreement with experimentally observed patterns.⁵⁰ The low-frequency inductive loop is related to the competition between intermediates species on the catalytic surface,⁵ and its size decreases with simulated operation time (because increases). Furthermore, the high-frequency capacitive arc diameter, also linked to interfacial electrochemical phenomena, looks less sensitive to short operating times.

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The degradation impact is more significant when the simulated operation time is longer. We can see that the inductive loop becomes a capacitive loop at , as already reported,¹⁸ and the high-frequency loop diameter decreases considerably.

The increase of the amplitude of the low-frequency capacitive loop is associated with the increase of the oxygen transport limitation through the microscale Nafion layer because of the COR-driven rise of .

Moreover, we can see that the high-frequency resistance increases with time. Because in our simulations the membrane resistance remains unchanged (no ionomer degradation mechanisms included), this behavior is due to the increase originated by the COR.

Impact of the operating conditions on the degradation

In Fig. 16, we show the impact of the external current on the cumulative carbon mass loss {defined as }, for two simulated operation times. It can be observed that when the current increases, the cathodic potential decreases and then the COR becomes slower (cf. Fig. 9); lost carbon mass is higher at lower currents, in agreement with ex situ experimental behaviors.⁹ This suggests that, to mitigate cathode carbon oxidation during start-up/shutdown flow experiments, it seems better to operate the MEA at intermediate/high current values.

Figures 17 and 18 illustrate the effect of the external current on the EIS shape for two simulated operation times. At , the impact of the carbon corrosion on performance is not yet significant and the EIS sensitivity to the nominal current is analogous to the tendency already reported by us by simulating a MEA without describing the aging phenomena^{36, 37, 39, 40} (Fig. 17a, 17b and 17c). At low currents, the EIS high-frequency capacitive arc diameter decreases with the demanded current and the inductive loop decreases. At a sufficiently high current, a low-frequency capacitive loop related to the oxygen transport limitation through the microscale hydrated Nafion layer appears. As expected, the magnitude of this low-capacitive arc increases as the nominal current increases.

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At , the COR-driven degradation is quite significant at , which translates as the presence of a low-frequency capacitive loop (not observed for the same current at ) (Fig. 18a). However, for intermediate currents ( and ) the EIS shapes are qualitative and quantitative similar to the ones calculated at because COR impact on performance decreases with current. At higher currents (Fig. 18b), despite that COR process and the driven platinum coarsening are slower, the available platinum surface area becomes too low to ensure the demanded external current. Then, transport limitation low-frequency capacitive arcs appear at lower currents than in the case.

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Figure 19 shows the impact of the temperature on the cumulated carbon mass loss and on the performance, for two simulated operation times. As expected from the experimental knowledge,³¹ the cathode damage increases with the temperature, up to ; the temperature favors the COR (cf. Table I). For higher temperatures, however, cumulated carbon mass loss slightly decreases; this interesting feature occurs because the remaining at a given simulated operation time decreases with temperature (COR-driven Pt coarsening is faster as temperature increases), slowing down the carbon corrosion process (cf. discussion of Fig. 28).

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Figures 20 and 21 show the simulated temperature impact on EIS at , for two different operation simulated times ( and ). In both cases, because the hydrated Nafion proton conductivity increases with the temperature (cf. Table I), the high-frequency resistance shifts toward the Nyquist's plan origin.

The predicted sensitivity of the capacitive and inductive loop sizes is related to the temperature-dependent mathematical expressions used to describe the electrochemical reactions, the carbon oxidation mechanism, and the reactant transports. It can be observed that at , the amplitude of the high-frequency capacitive arc is slightly bigger at the highest temperature (Fig. 20).

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Otherwise, at , the impact of the COR-driven degradation is significant (cf. Fig. 11a): the amplitude of the high-frequency arc is smaller at the highest temperature (Fig. 21). As shown in Fig. 21, the COR also contributes to decrease the amplitude of these low-frequency inductive arcs. These loops are masked, for a sufficiently high temperature, by a low-frequency capacitive loop, with an amplitude increasing with temperature. In fact, temperature favors the COR-driven platinum coarsening and again, this leads to the decreasing of and to the increasing of . The increase of this current density leads to early hydrogen and oxygen limitation and manifests as a bigger low-frequency arc amplitude at the highest temperature.

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In Fig. 22, we focus on the reactant pressures impact on the cumulated lost carbon mass at two simulated operation times (we assume ). The lost carbon mass increases when the anode and cathode pressures are increased together; this is because the increase of the oxygen concentration in the anode side, which improves the cathode COR mechanism, as shown in Fig. 23a. As expected, the cell potential decreases when the oxygen concentration in the anode increases, as shown in Fig. 23b.

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Impact of initial loadings and cathode structural properties on COR degradation

As shown in Fig. 24, the model captures the impact of the initial cathodic platinum mass ratio on the cumulated carbon mass loss, for two simulated operation times. Increasing the platinum mass in the cathode considerably increases the carbon support damage, in agreement with some experimental results.³⁰ Hence, it is clear that the cathodic ORR (cathode 1) plays a major role on the COR (cathode 2), by directly consuming the electrons produced by the last reaction (the cathode platinum "catalyzes" the carbon corrosion).

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The slopes of these curves decrease as the initial increases; this is because decreases during the simulated PEFC operation, and then, COR is slowed down (cf. Fig. 28).

The impact of on EIS is provided at two simulated operation times in Fig. 25a and 25b. At , the impact of platinum loading reveals small (EIS shapes are similar for the three platinum loadings). For , the increase of the initial platinum loading accelerates the carbon damage (low-frequency capacitive arcs increase with the initial platinum loading). In fact, the decrease of the platinum loading delays the COR impact on the PEFC MEA performances.

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In Fig. 26, we show the influence of the initial carbon mass on the cumulated carbon mass loss at , for three cathodic platinum mass ratios. It is not surprising to observe that the increase of the initial carbon mass leads to the increase of the cathode damage, because of the increase of the platinum loading (platinum mass ratio kept constant).

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In contrast to this, in Fig. 27 we show that if we keep constant the initial platinum loading when the carbon mass is increased, then the cathode carbon damage appears to be decreased.

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Figure 28 illustrates the calculated impact of the platinum nanoparticle radius on the cumulative carbon mass loss for two simulated operation times (in contrast to our previous model in Ref. ^5–7, this radius is time invariant here). It can be observed that increasing the radius (decreasing ) reduces the cumulative carbon mass loss. This is because decreases as the radius increases, and then the COR appears to be less catalyzed by the platinum. This theoretical behavior is in agreement with the experimental tendencies provided by Stevens et al.²⁹ and by Adjemian.⁵¹

The above analysis means that a mechanism such as the electrochemical Ostwald's ripening [contributing to the mean radius increase with time (cf. Fig. 3 and 10 in Ref. ⁵)] will contribute to slowing down the COR process.

Figure 29 shows the simulated EIS sensitivity to the platinum nanoparticle radius for two simulated operation times. We emphasize that for the shorter operation time , the radius does not considerably impact the EIS pattern. At , a low-frequency capacitive arc associated with the increase of and the oxygen transport limitation appears for the smaller radius considered in the calculations. This also illustrates that performance degradation is faster for small catalyst nanoparticle radius, in agreement with results discussed in Fig. 28.

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