Modeling the Effect of Chemical Membrane Degradation on PEMFC Performance

The steady state, MEA-level performance is based on the conservation of mass, charge and momentum wherever applicable.²⁹ Figure 2 shows the geometry of the modeled domain which includes gas diffusion layer (GDL), micro-porous layer (MPL), and CL on each side of the polymer membrane. The left half of the domain is under the land region and experiences a compressive stress of 1.5 MPa. The cell is assumed to operate under high stoichiometry so that the gas concentration along the channel remains approximately the same as at the inlet. In the porous regions encompassing the electrodes, MPLs and the GDLs, the Maxwell-Stefan gas diffusion equations—binary diffusion (H₂-H₂O) in the anode-side and ternary (O₂-N₂-H₂O) in the cathode-side—were solved. The governing equations in various cell components are listed in Table I. The binary diffusivities were calculated using the expression in Table II and corrected in the porous regions using the Bruggeman relation.

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Table I. Transport equations modeled in various computational domains.

	Equation	Note
MPL / GDL
	∇ · ( − 1.5sσs∇φs) = 0	s, σs and φs  are respectively the phase fraction, conductivity and the electric potential of the electron-conducting phase in the respective domain.
		i = H2 and H2O (anode-side MPL / GDL) i = O2, H2O and N2 (cathode-side MPL / GDL) is the gas phase porosity in the respective domain
CL (anode)
	∇ · ( − 1.5−, sσs∇φs) = Sφ, 1	η = φs − φe − Eref−
	∇ · ( − 1.5−, eσe∇φe) = −Sφ, 1
		i = H2 or H2O
		η = φs − φe − Eref+, 2e
CL (cathode)
	∇ · ( − 1.5+, sσs∇φs) = Sφ, 3	η = φs − φe − Eref+
	∇ · ( − 1.5+, eσe∇φe) = −Sφ, 3
		i = O2, and H2O
		η = φs − φe − Eref+, 2e
Membrane
	∇.( − σe∇φe) = 0
		η = φs, theoretical − φe − Eref+, 2e φs, theoretical = 0.14l V; l = relative distance from anode − membrane interface(0 at anode − membrane interface; 1 at anode − membrane interface)
		i = HF and H2O2
		i = SO3, COOH, CF2, OH, calpha and cbeta

Table II. Expression of binary gas diffusivity and parameters associated with its calculation.

Variable/Parameter	Symbol	Expression/Value
Empirical expression of binary diffusivities50	Di, j
Molar mass of:
Hydrogen		0.002 (kg mol− 1)
Oxygen		0.032 (kg mol− 1)
Nitrogen		0.028 (kg mol− 1)
Water vapor		0.018 (kg mol− 1)
Diffusion volume of (Fuller et al.50):
Hydrogen		7.07
Oxygen		16.6
Nitrogen		17.9
Water vapor		12.7

The electrochemical reactions considered are (75°C):

Anode and Cathode:

Anode-only:

Cathode-only:

The kinetic expressions for Reactions 1–4 are given in Table I. The parasitic reactions, electrochemical reduction at anode and oxidation at the cathode, require consideration of the crossover of the reactant gases through the membrane, H₂ from anode to cathode and O₂ from cathode to anode which is discussed next.

The transport of H₂ across the membrane occurs under the gradient of the hydrogen partial pressure () and the gradient of total pressure across the membrane (P).³⁰ Under pure partial pressure gradient (i.e., ∇P = 0), the permeability () seems to be different than the permeability under total pressure gradient (), indicating either a separate mechanism of H₂ gas crossover operating under each condition or, alternatively, dominance of one of the two mechanisms proposed by Mittelsteadt and Liu.³¹ The permeation rate constant of each mechanism is governed by a unique Arrhenius Equation. The two types of permeability based on experimental observation^30,32 are further assumed to be independent of relative humidity above 75°C. Thus, the permeation rate of H₂ across the membrane,(A.m⁻²) can be described as follows:

The H₂ crossover fluxes induced by the two driving forces are assumed not to influence each other. The local concentration of the dissolved H₂, in the membrane is given by Henry's law and is equal to .

Likewise the permeation kinetics of oxygen is governed by:

where, (m²V⁻¹s⁻¹) is the permeability coefficient given by the expression 1.54 × 10⁻⁵ × e^−2755/T.³³ The oxygen permeation under a gradient of total pressure across the membrane is assumed zero (). An RH-dependent model for the oxygen permeation was used by Sethuraman et al.³⁴ However, like the case for the hydrogen permeation, we assume that the effect of humidity on the oxygen permeability is minimum and we shall use the recent data by Zhang et al.³³

The presence of H₂O₂ in the fuel cell membrane has been experimentally confirmed.³⁵ H₂O₂ can be generated by the chemical reaction of the crossover species or by the electro-chemical 2e⁻ reduction of cross-over oxygen at the anode. The anode potential, as discussed in the preceding section, favors the electrochemical route of H₂O₂ production which especially dominates under open circuit voltage conditions due to the high crossover rate. The H₂O₂ produced at the anode has a boiling point (150.2°C) higher than water and joins the dissolved water in the polymer and diffuses through the membrane. The diffusing H₂O₂ decomposes at Fenton reagents such as iron,³⁶ which are apparently present in the membrane in the range of [mol/m³], and generate hydroxyl radicals as:

Iron in the oxidation state three does not catalyze peroxide decomposition. Under a reducing environment Fe²⁺ is regenerated from Fe³⁺ as follows:

A dynamic equilibrium between Fe²⁺ and Fe³⁺ exists. The highly oxidative hydroxyl (·OH) radical is generally believed to attack the oxygen containing weak links such as the oxygen linking the side group to the main chain or the ―COOH end group on the main chain. Chemically stabilized grade of the PFSA-based membranes such as Nafion 211 possesses negligibly small quantity of oxygen containing main chain terminal group/s. Thus the chemical degradation of such membranes is initiated at the weakest ether link on the side chain, as evidenced in a recent ex-situ Fenton's test²⁴ and is assumed as such in this paper. The study by Ghassemzadeh & Holdcroft²⁴ on Nafion 211 further indicates that the α-OCF₂ is the first chemical group under attack by the hydroxyl radical. This is likely followed by the attack on the β-OCF₂ which was reported to occur after a time lag of over 24 h, while the concentrations of either of the main chain groups, CF-m and CF₂, remained unchanged even after 48 h of test signifying little or no degradation of the main chain.

Based on the above, the pathway for the chemical degradation of the membrane in an operating fuel cell is defined by the four sequential irreversible chemical reactions,³⁷ shown in Fig. 3. The rate constants and reaction rates for reactions (i) to (iv) in Fig. 3 and for the reaction in Eq. 7 are listed in Table III. The longer the ―(CF₂)_pCOOH molecule (i.e., higher value of p) in reaction (iv) in Fig. 3, the more susceptible it could be to attack by ^•OH radical.³⁸ In this work the rate constant for all the sequential reactions in Fig. 3 will be assumed constant and independent of the chain length. The chemical species SCF₂, α-OCF₂, β-OCF₂, CF₂, and COOH are always part of the polymer chain which is modeled as rigid. The species COOH refer to the ―COOH group attached to the main chain, cf. Eq. (iii) and (iv) in Fig. 3, and does not represent the (―(CF₂)_x―) group. Relative position of various functional groups with respect to the repeating unit of Nafion is depicted in Fig 1.

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Table III. Reaction rates and reaction constants for reactions (i) to (iv) in Fig. 3 and Fenton reactions.

Reaction	Rate constant (identifier)	Reaction rate (identifier)
(i) in Fig. 3	2.35 × 103 m3/mol/s (k1)51
(ii) in Fig. 3	9.04 × 103 m3/mol/s (k2)
(iii) in Fig. 3	8.5 × 104 m3/mol/s (k3)
(iv) in Fig. 3	1.0 × 103 m3/mol/s (k4)52 1.02 × 105 m3/mol/s38	k4.cCOOH.cOH (r4)
Eq. 7	1.05 × 105exp(−35397/R/T) m3/mol/s (k5)
Eq. 8	0.3 s−1 (k6)	k6.; is the concentration of iron (III) in the membrane

The dry density, ρ_dry, and the EW of Nafion are related as:³⁷

where c_i and M_i are respectively the concentration and molecular weight of species i (i = SCF₂, α-OCF₂, β-OCF₂, CF₂, and COOH).

Incorporation of the preceding elements in the degradation model requires the following assumptions:

• (1)  
  There is no oxygen containing end groups on the main chain at beginning of life (BOL). Hydrogen ion concentration or the pH of the membrane does not change with the degradation.
• (2)  
  The reaction order with respect to each reacting chemical species is one except for Eq. 8 where the reaction order with respect to ^•H is assumed zero. All of the degradation reactions in Fig. 3 are irreversible.
• (3)  
  The chemical species SCF₂, α-OCF₂, β-OCF₂, CF₂, and COOH are always part of the polymer chain. These species participate in degradation reactions but do not diffuse as they are chemically bonded to the main chain. Hydroxyl radical is another of the species for which diffusion is not considered due to its highly reactive and unstable nature. Its composition is determined by its production via Fenton reaction and its reaction in degradation reactions (Fig 3). All of the fluorine leaves the system as HF species. The remaining chemical species (containing C and S) are washed out and their effect is accounted for in characteristics such as EW, of degraded Nafion.
• (4)  
  The loss of species COOH (reaction (iv) in Fig. 3) is not allowed when is greater than c_COOH.
• (5)  
  The water uptake and associated swelling characteristics of the membrane is unchanged by the degradation.
• (6)  
  There is no degradation in PEMFC components other than the membrane.

The process of chemical degradation alters the material characteristics of Nafion slowly but continuously. A relation between these material characteristics and the polymer composition is needed to study the effect of chemical degradation of the membrane on cell performance.

The porosity of the membrane, ϕ_w, is calculated as volume fraction of the membrane occupied by the liquid water, given by:

where λ is the membrane water content defined as the number of water molecules per mole of charge site concentration and V_w and V_m are the molar volume of the water and the membrane respectively. The material loss originating from the degradation effects a dimensional change modeled as an equivalent volumetric strain, _d:

where V⁰_m and are the molar volume and the charge site concentration of the membrane at BOL. In addition, the membrane experiences a volumetric strain, 0.0256λ, induced by the liquid water uptake.

The time-dependent mass conservation of non-diffusing species, i, (i = SCF₂, α-OCF₂, β-OCF₂, CF₂, OH and COOH) is governed by:

where R_i is the source term of species i, expressed in terms of reaction rates and stoichiometry of reactions in Table IV. The second term on the RHS of Eq. 12 takes into account the diluting/concentrating effects arising from membrane swelling and material loss.

Table IV. Source terms for the chemical species SCF₂, α-OCF₂, β-OCF₂, CF₂, OH and COOH in the membrane.

Chemical species (i)	SCF2	α-OCF2	β-OCF2	CF2	OH	COOH	H2O2
Source term ()	-r1	r1 − r2	r2 − r3	14r3 − r4	− r1 − 3r2 − r3 − 2r4 + r5	*	− r5

^*H(x) is Heaviside function taking a value of 0 for x < 0 and 1 for x > 0.

The performance characteristics of a PEMFC under normal operating conditions are fairly stable and, within limits, independent of cell degradation processes at a short time scale of an hour. Thus, the performance of the cell is modeled assuming steady state operation of the cell. However, we note that transient effects arising from chemical degradation of the membrane can influence the performance characteristics by altering the degradation dependent parameters such as membrane porosity and conductivity which are inputs to the performance model. In Ref. 21, a feedback between the degradation mechanisms and the performance was taken into account, allowing the model to predict the cell voltage decay, including the OCV one. The present approach is consistent with concepts presented in Refs. 19,39.

The ageing processes linked to chemical degradation of the membrane are considered to be a function of time and field variables relevant to steady-state performance. The coupling of the performance and the chemical degradation models is relatively straightforward wherein the transient physico-chemical processes governing chemical degradation of the membrane are solved together with the steady state processes. The models and their coupling is validated against recently published experimental accelerated stress test data (chemical degradation and performance).²⁵

The boundary conditions are summarized in Table VI. The inlet hydrogen, oxygen, and water vapor concentrations are calculated according to the operating conditions presented in Lim et al.²⁵ Hydrogen peroxide and hydrogen fluoride concentrations are assumed to be zero at the CL and MPL interfaces. For the remaining ionomer species that participate in the membrane degradation, no boundary conditions but initial conditions are required, given these species are fixed inside the membrane. The initial concentrations are:

Table VI. Summary of the boundary conditions applied in the simulations.

	Cathode	Cathode	Cathode	Cathode CL|Membrane	Anode	Anode	Anode
Variables	GC|GDL	GDL|MPL	MPL|CL	& Membrane|Anode CL	MPL|CL	GDL|MPL	GC|GDL
cO2, g	continuous	continuous	continuous	No Flux	continuous	continuous	continuous
cH2, g	continuous	continuous	continuous	No Flux	continuous	continuous	continuous
cw, g	continuous	continuous	continuous	No Flux	continuous	continuous	continuous
λ			No flux	continuous	No flux
			No flux	continuous	No flux
cHF			No flux	continuous	No flux
Vs	No flux	continuous	continuous		continuous	continuous	0

Mesh sensitivity simulations performed with both an extremely fine mesh (24,346 elements) and a fine mesh (6,438 elements) yielded virtually identical results. The results presented here were obtained with the extremely fine mesh. A Massively Parallel Sparse direct Solver (MUMPS) was employed to solve the set of fully coupled equations. Initial conditions for the domains were set as shown in Table Vt6, and initial concentrations were prescribed for all gas species. Typical computational time on a workstation with AMD FX 8-core processors ranged from 15 to 60 min depending on the operating conditions.

The beginning of life (BOL) average polarization behavior over a five-cell stack is compared with the simulated polarization curve in Fig. 4. The simulated open circuit voltage (OCV) of 0.98 V was lower than the theoretical OCV of 1.16 V at a cell temperature of 348.15K. The drop in the OCV was fully accounted for by the hydrogen crossover from anode to cathode which results in parasitic hydrogen oxidation and oxygen reduction reaction at the cathode. Overall, the BOL polarization behavior of the cell is captured quite well by the steady state performance model which contains only two adjustable parameter (exchange current density and transfer coefficient).

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Apart from the steady-state performance the transient degradation model was validated against the membrane degradation measured by the fluoride emission rate under an OCV hold test described in Lim et al.²⁵ The cell operating parameters for the 10h OCV hold test as well as the simulation are listed in Table V. The simulated FER reproduces the experimental trend well, but as shown in Figure 5 the experimental FER lags behind the simulation; this is attributed to the extra diffusion path from the membrane-CL interface, where the simulated FER were calculated, to the fluoride detector located downstream along the exhaust stream.

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The ratio of the cathode to anode FER was 1.075 in the simulation vs. 1.15 in the experiment. The simulated ratio of the FER increases with a decrease in the gradient of the theoretical electrode potential in the membrane which was increased linearly from 5 mV at anode-membrane interface to 20 mV at the cathode-membrane interface. Although a further reduction of the gradient in the theoretical electrode potential in the membrane would have resulted in a larger FER ratio, the resulting potential profile would become inconsistent with the experimental profiles observed by Liu & Zuckerbrod.³⁵

The theoretical electrode potential in the membrane is influenced by the size of Pt particles which gets dissolved at the cathode, transported and re-deposited in the membrane.⁴⁰ The measured and model potential profile varied approximately linearly from anode up to about 80% of the membrane thickness. To keep the model complexity in check and ensure tractable simulations, the potential was assumed to vary linearly inside the membrane. The potential profile and the gas composition in the membrane determine the rate of the two-electron reduction of dissolved oxygen to hydrogen peroxide by controlling the overpotential for the electrochemical reactions. The selectivity of peroxide formation over water generation is taken as unity, although it has been shown to generally vary as a function of the electrode potential and humidity.³⁴

Formation of radicals from H₂O₂ decomposition is not a dominant membrane degradation mechanism in PEMFCs, rather membrane-degrading species appear to form in a reaction between molecular H₂ and O₂ on a catalytic site.⁴¹ Although a potential dependent H₂O₂-mediated indirect route of radical formation at Fenton's regents in the membrane was assumed here, the direct formation of membrane-attacking species cannot be ruled out. Thus, there is a need to delineate the role and selectivity of a potential-independent direct and a potential-dependent indirect route of radical formation.

Two of the consequences of membrane degradation are membrane thinning and OCV decay (Fig. 6). The OCV decrease resulting from the chemical membrane degradation induced by the OCV-hold test, is more rapidly in the experiments than in the simulation. Although not shown here, most of the decrease in the OCV was recoverable. A similar behavior was observed in another study where a voltage decay during 100 h of OCV hold was fully recoverable⁴² after treatment to remove contaminating species at electrodes. Hence the rapid and reversible decay in the experimentally-measured OCV can be traced to electrode contamination. The small OCV decay in the simulation is irreversible and due to high gas crossover associated with membrane thinning. A further source of voltage decay is loss of electrochemical surface area which is outside the scope of this study. The high frequency impedance yields an almost constant conductivity value (∼0.2 Ω⋅cm²) in the steady OCV phase from BOL to EOL, likely due to counteracting effects of reduced proton conductivity and membrane thinning.

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Without the addition of radical scavengers or additives such as ceria,^43,44 PFSA based membrane does not seem to stabilize chemically and structurally. When subjected to harsh conditions such as high temperature OCV-hold, these membranes start to degrade chemically. Figure 7 shows the evolution of the simulated concentration of various functional groups of Nafion during the 10 h OCV-hold test. The most notable observation is the fact that 15.66% of sulfonic acid head groups are lost over the 10 h period of OCV-hold test with only 1.4% reduction in thickness; whereas as noted above protonic conductivity remains essentially constant. Lim et al. surmised this is due to a counteracting effect of membrane thinning and a decrease in the membrane conductivity due to the loss of the sulfonic acid head groups. Conversely, considering negligible membrane thinning, it could be argued that some side chain fragments are not actually washed away and stay inside the membrane with intact sulfonic acid head groups. This is especially plausible given that the C-S bond in the sulfonic acid group is not the weakest group to be attacked by the radicals, whereas the two C-O-C ether bonds on the side chain are more susceptible to radical attack.

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After OCV-hold for 10 h, the measured cell polarization behavior shows a marked decrease especially in the activation regime (Fig. 8). The trend is captured in the simulated polarization behavior; however, the small loss in cell performance in the activation regime is not reproduced by the simulation. The activation loss in the measured cell polarization behavior could be due to loss of cathode catalytic activity. The present model does not account for any CL degradation such as possible CL poisoning by the product of the degradation (e.g., the (bi-) sulfate ion) or for loss in catalytic activity due to Pt coarsening, dissolution and re-deposition in the membrane. Ongoing work is aimed at incorporating CL degradation in the model.

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Experimental observations point to the complex interplay and equilibrium among various radicals (^•H, ^•OH, ^•OOH etc.) and this should also be considered in further model developments. The chemical degradation mechanism appears to be a function of the dominant radical in the membrane. Recent observations of ex-situ exposure of Nafion to selective ^•H and ^•OH radicals created by β–irradiation indicate that the radical ^•H attacks both main and side chain CF bonds, whereas ^•OH radical only attacks the oxygen containing ether bond in the side chain of Nafion.²³ The former mechanism (attack of ^•H on the CF group) results in lower fluoride emission, however, attack by the ^•OH radical results in higher fluoride emission rate. Irrespective of the above mechanism the decrease in conductivity and ion exchange capacity are commensurate with the loss of sulfonic acid group. The deterioration in mechanical properties such as a decrease in strain-to-failure, increase in Young's modulus and a decrease in the ultimate tensile strength are linked to structure of Nafion. In order to predict the performance degradation under combined chemical and mechanical stressors, the gaps in understanding of the interplay between chemical composition, Nafion morphology and mechanical properties need to be bridged.^45,46