Hyoung Jun Lim,Geonwoo Kim,Gun Jin Yun
Durability and Performance Analysis of Polymer Electrolyte Membranes for Hydrogen Fuel Cells by a Coupled Chemo-mechanical Constitutive Model and Experimental Validation
polymer electrolyte membrane fuel cell,perfluorosulfonic acids,continuum mechanics,chemical degradation,pinhole
In this paper, a chemo-mechanically coupled behavior of Nafion 212 is investigated through predictive multiphysics modeling and experimental validation. Fuel cell performance and durability are critically determined by the mechanical and chemical degradation of a perfluorosulfonic acid (PFSA) membrane. However, how the degree of chemical decomposition affects the material constitutive behavior has not been clearly defined. To estimate the degradation level quantitatively, fluoride release is measured. The PFSA membrane in tensile testing shows nonlinear behavior, which is modeled by J2 plasticity-based material modeling. The material parameters, which contain hardening parameters and Young’s modulus, are characterized in terms of fluoride release levels by inverse analysis. In the sequel, membrane modeling is performed to investigate the life prediction due to humidity cycling. A continuum-based pinhole growth model is adopted in response to mechanical stress. As a result, validation is conducted in comparison with the accelerated stress test (AST) by correlating the size of the pinhole with the gas crossover generated in the membrane. This work provides a dataset of degraded membranes for performance and suggests the quantitative understanding and prediction of fuel cell durability with computational simulation.
Durability and Performance Analysis of Polymer Electrolyte Membranes for Hydrogen Fuel Cells by a Coupled Chemo-mechanical Constitutive Model and Experimental Validation

In this paper, a chemo-mechanically coupled behavior of Nafion 212 is investigated through predictive multiphysics modeling and experimental validation. Fuel cell performance and durability are critically determined by the mechanical and chemical degradation of a perfluorosulfonic acid (PFSA) membrane. However, how the degree of chemical decomposition affects the material constitutive behavior has not been clearly defined. To estimate the degradation level quantitatively, fluoride release is measured. The PFSA membrane in tensile testing shows nonlinear behavior, which is modeled by J2 plasticity-based material modeling. The material parameters, which contain hardening parameters and Young’s modulus, are characterized in terms of fluoride release levels by inverse analysis. In the sequel, membrane modeling is performed to investigate the life prediction due to humidity cycling. A continuum-based pinhole growth model is adopted in response to mechanical stress. As a result, validation is conducted in comparison with the accelerated stress test (AST) by correlating the size of the pinhole with the gas crossover generated in the membrane. This work provides a dataset of degraded membranes for performance and suggests the quantitative understanding and prediction of fuel cell durability with computational simulation.

Renewable energy solutions are gaining worldwide attention by solving the increase in environmental pollution and global energy consumption due to fossil fuels. The renewable hydrogen economy, which includes hydrogen production, storage, and the conversion of hydrogen to electricity, is widely recognized as a innovative solution for future energy sources. Especially, polymer electrolyte (or proton exchange) membrane fuel cells (PEMFCs) have received much attraction as a promising technology due to their simple structure, manufacturing method, and weight/space efficiency. The potential of PEMFCs for use in a wide range of applications as efficient power systems depends on overcoming technical challenges. For now, two typical substances affecting fuel cells’ efficiency are the catalyst coating the electrodes and the electrolyte membrane for selective permeability. Issues of their durability and cost associated with producing PEMFCs make them challenging to commercialize. It is known that PEMFCs are exposed to mechanical and chemical stresses during operation, degrading performance as voltage loss increases. Among the cell components, the membrane itself has been reported to be vulnerable to significant sources of failure, such as mechanical damage and chemical decomposition. Therefore, the membrane should be durable enough to withstand mechanical stress and chemical attack to maintain its function in operating the cell. Perfluorosulfonic acid (PFSA) membranes are used as polymer electrolytes in the PEMFC system due to their high hydrogen ion permeability and durability in chemical degradation. During cell operation, the polymer electrolyte membrane (PEM) provides a pathway for proton ion transfer from the anode to the cathode, a barrier to prevent the mixing of oxygen and hydrogen, and an insulator for electron-proofing. The multifunctional roles of the PFSA membrane require high proton conductivity to guarantee the fuel cell performance and structural stability to ensure durability. Many attempts have been made to fulfill two prominent roles. One way is to increase the number of sulfonic acid groups (−SO3H) by lowering the equivalent weight (EW), making proton transport easier through water molecules. However, it is reported to be more vulnerable to radical attacks. It means that the membrane with a low EW has low chemical resistance. Another way is to shorten the movement distance of protons by thinning the membrane to minimize the ohmic loss. Fuel cell performance can be improved with increasing proton transfer efficiency, but chemical degradation is inevitably accelerated when damage is caused by mechanical stresses and crossover of reactant gas through a thinner membrane. It causes the creation and growth of pinholes, which promote the crossover of gases and consequently lead to a decrease in cell performance. Thus, mechanical and chemical interactions further accelerate membrane degradation. So far, the PFSA membranes’ chemical degradation mechanisms have not yet been fully elucidated. One believes that the chemical decomposition mechanism originates from the formation of radical species such as hydroxyl (·OH) and hydroperoxyl (·OOH) radicals. The formation of radicals or radical precursors results from the byproduct of an electrochemical reaction. Radical species attack the sulfonic acid group of the side chain and the carboxylic acid (−COOH) end group site of the backbone chain, resulting in the cleavage of chemical structure and defects. The enlarged defect greatly accelerates the membrane degradation rate. The effect of mechanical and chemical degradation on the membrane has been thoroughly examined through experiments. The accelerated stress testing (AST) developed by the U.S. Department of Energy comprises a humidity cycling test for mechanical durability and an open-circuit voltage (OCV) for chemical durability. The OCV hold test is generally conducted at a low relative humidity (RH, 30%) at a high temperature (90°C) where the reactant gas permeability from the anode to the cathode is higher than when drawing net cell current. For mechanical durability, the humidity test cycles between 0 and 100% supersaturation by the monitoring gas crossover and short circuit resistance. Although the AST has the advantage of enabling accurate evaluation in that it realizes the same environment as the real-world fuel cell operation, it is costly and time-consuming. In contrast, the ex situ Fenton test comprising hydrogen peroxide and ferrous ions is widely used for the chemical decomposition of the membrane within a short period. Although many studies have been conducted on experimental characterization, studies on PEM material modeling for simulation are relatively less advanced. The development of a simulation technique must be accompanied to reduce the cost and time incurred in manufacturing and designing a fuel cell. In particular, it is essential to predict the performance and lifespan of a polymer membrane during fuel cell operation. The strains of hydration (swelling) in the clamped constraint induce mechanical stresses on the membrane. Hence, the mechanical durability of the PEM is determined by the swelling strain, which is commonly evaluated by RH cycles. As a result, various constitutive relations have been introduced to demonstrate the mechanical behavior of the PFSA ionomer membrane. Considering the PEM as an elastomer, one represented the PEM as a rheological model. It has the advantage of well expressing the microcracks and crazing characteristics that occur in the behavior of polymers by expressing a spring and dashpot. For example, a physics-based model can effectively express a reputational plastic flow, chain entanglement, and entropic effect. However, it has critical limitations due to many calibration parameters that change according to the experimental input and poor convergence for implicit analysis, which is adequate for accurate prediction. Nevertheless, the rheological model is being actively applied in many works as it can demonstrate creep and relaxation mechanisms due to time-dependent behavior. Indeed, Solasi et al. stated that the creep mechanism had occurred consistently from hydration loading during fuel cell operation. On the other hand, the PEM is also represented by linear elasticity with isotropic hardening plasticity based on the von Mises yield criterion. This methodology shows a straightforward implementation for implicit analysis because of its excellent convergence, and the material parameters are directly related to the macroscopic stress–strain curve. Furthermore, modeling consideration is required since the PEM is exposed to a hygrothermal environment and shows remarkable changes according to temperature and humidity. Above all, what should be focused on in the PEM modeling is the performance prediction over time. Khorasany et al. developed a uniaxial cyclic tension test and fatigue model under various applied stress profiles and environmental conditions (RH and temperature). They revealed that the membrane fatigue lifetime decreases exponentially as the applied maximum stress is increased, and the effect of temperature was greater than that of RH. Moreover, these results were compared with the model of the assembled cell. Despite the good agreement, this work cannot represent the real PEM environment because only mechanical loading is taken into account in both outcomes. Next, Kusoglu and Weber suggest a mechanistic approach for defeat growth in the PEM by determining a defeat size enlarged with increasing plastic strain. Under swelling-driven cyclic stresses, they correlated the pinhole to the level of gas crossover, which is directly related to the cell performance. However, a chemical-induced defect size was roughly assumed based on the mechanical-only-induced growth. At the current stage, an attempt must be made to consider both mechanical and chemical degradations in the PEM modeling. Of course, several reports experimentally expressed the degradation of elastic modulus and yield stress according to the extent of chemical degradation through experiments, but more mechanics-based development is needed for simulation. In this study, we focus on the chemo-mechanically coupled degradation modeling for the material behavior of the PEM. The objective of this approach is to predict its performance and durability under repetitive stress cycle conditions. The outline of this work is as follows: First, chemical degradation is introduced and evaluated with the ex situ Fenton test. Characterizing the chemically decomposed PEM, especially defects, is performed in terms of different degradation levels. Next, the material constitutive model is realized with the J2 plasticity theory to describe a nonlinear behavior. Here, the material parameters are expressed as a function of chemical decomposition level, temperature, and humidity. The defect from chemical and/or mechanical degradation is idealized as a circular void. The void evolution under cyclic loading is modeled based on the continuum plasticity theory in the material model and employed to correlate with the cell performance. Collectively, a chemo-mechanically coupled material model is developed and examines the membrane behavior under the RH cycle. Finally, a comparison is carried out against the AST results for lifetime prediction.

The degradation of the PFSA ionomer membrane is divided into a mechanical and chemical-induced mechanism, reducing the PEMFCs’ durability. Mechanical degradation is related to membrane deformation due to local changes in relative humidity and cyclic stress applied to the membrane. During RH cycling in cell operation, the membrane repeats expansion and contraction, gradually reducing the membrane’s mechanical strength, leading to defects such as poor interfacial contact between the membrane and the electrode and pinhole formation. The gas crossover has deteriorated through the formed pinhole, and the membrane degradation is accelerated. Chemical degradation of the PFSA membrane involves a radical-induced chemically decomposed phenomenon where ·OH or ·OOH radicals generated during PEMFC operation attack the backbone and side chains of the membrane or ionomer binder. Radicals are generated from the decomposition of hydrogen peroxide in an oxygen reduction reaction or a chemical reaction between crossover hydrogen and oxygen on the cathode catalyst surface. Oxygen reacts in the reduction reaction with a small number of hydrogen ions at the cathode to produce hydrogen peroxide. After that, hydrogen peroxide is decomposed by heat or reacts with metal ions that resulted from corrosion to generate ·OH radicals. The generated ·OH radical or metal ion reacts with hydrogen peroxide to generate ·OOH radicals. This radical mechanism is accelerated in the gas crossover state. The radical-induced chemically decomposed mechanisms of the PFSA membrane can be divided into four major categories. The first mechanism is known as the unzipping reaction and starts with ·OH radicals by attacking the terminal −COOH group of the PFSA backbone chain, generating CF2 radicals, CO2, and H2O. The generated CF2 radical reacts with ·OH radicals again to form HF and then reforms COOH at the end. If the unzipping reaction occurs repeatedly, it leads to the decomposition of the CF2 backbone chain of the membrane. Second, the ·OH radical attacks the C–S bond of the PFSA structure to generate CF2 radicals, SO3, and H2O. The generated CF2 radical causes an unzipping reaction with the ·OH radical and leads to side-chain decomposition of the membrane. The third is a reaction where the ·OH radical attacks the ether group of the side chain. In the structure of Nafion, the decomposition of the ether group (α-OCF2) close to the sulfonic acid group begins first, and the ether group (β-OCF2) close to the backbone chain is then decomposed. Lastly, the ·H radical attacks the tertiary carbon near the ether group, resulting in CF chain scission of the side and backbone chains. The regions subjected to radical attack in the Nafion structure are summarized in Figure 1. Meanwhile, many attempts have been made to alleviate the Nafion degradation. For example, composite membranes with expanded polytetrafluoroethylene (ePTFE) reinforcement have been developed to improve operational durability by creating a more mechanically robust. An ionomer is incorporated into a central microporous PTFE-rich support layer, which enhances water transport pathways. Because of its sandwich structure and outer Nafion layers, it provides excellent mechanical strength and proton conductivity. By applying this structural modification, there has been a reduction in in-plane swelling when compared with thickness direction. It is therefore possible to increase the lifetime and durability of the reinforced membrane during RH cycling in fuel cells. In terms of chemical degradation, one of the ways to mitigate the degradation is to employ free radical scavengers (FRSs), which prevent Nafion from free radical attack. The reason FRSs can protect membranes is that hydroxyl radicals prefer to react with FRSs rather than Nafion membranes. Further, FRSs have the characteristic of being able to regenerate after reactions, which makes them suitable for long-term use. It is important that FRSs do not dissolve in an acidic atmosphere in order to be resistant to fuel cell environments. There are three representative types of FRSs including derivatives of ceria, noncerium inorganic, and organic FRSs. More detail about FRSs is investigated in the literature.

The Fenton test is utilized as an accelerated ex situ chemical degradation method for the PFSA membrane to achieve a sufficient degradation effect quickly. The Nafion 212 membranes manufactured by Dupont are cut into a square shape with a length of 55mm. The Fenton reagent is made by mixing 15 ppm ferrous ions, FeSO4 (Sigma-Aldrich, USA), and 30% hydrogen peroxide (Samcheon, South Korea) to generate free radicals. More details about the activation mechanism of radicals in the PEMFC can be found in the literature. The procedure of the Fenton test is composed of three steps. First, the membrane is washed with de-ionized water to remove impurities and then dried in a vacuum oven. Second, the Fenton reagent is heated on a hot plate to the desired reaction temperature (80 °C) and stirred (400 rpm) using a ϕ 60 × 20mm cross-shape magnetic bar. Then, the pretreated membrane is immersed in the Fenton reagent for targeted hours. After treatment, the aged membrane is washed and dried repeatedly. To obtain chemically degraded membranes with different levels, membranes are placed in the Fenton solutions for 24, 36, 48, 60, and 72 H. During the Fenton test, the solutions are replaced every 24 H. Here, the experiment is performed by fixing the membrane to the PTFE frame for uniform degradation, as shown in Figure 2.

The extent of membranes’ chemical degradation depends on many factors, such as the density of ferrous ion, hydrogen peroxide concentration, stirring speed, temperature, and degradation time. A total fluoride release is generally used to estimate the chemical degradation level. The concentration of fluoride ions in the drained solution is measured using an ion-selective electrode (ISE, Thermo Scientific Orion A212, USA). The electrode is calibrated using 1, 10, and 100 ppm fluoride (F–) standard solution with an equal volume ratio of total ionic strength adjustment buffer. It is used to adjust the pH of the solution, re-ionize the complexed fluoride, and keep the ionic strength constant. The density of fluoride in the solution is then measured using the ISE. Here, the drained solutions are filtered using a platinum wire before measurement to remove the metal precipitate and the residual hydrogen peroxide, which may influence fluoride ion determination. After that, normalization is conducted because the degree of release of fluoride ions varies depending on the membrane area. A total fluoride release can be expressed as a function of degradation time (Figure 3). The result in the graph is the average value of 10 samples per time, and the error bar indicates the standard deviation. It shows that the total fluoride release increases linearly in time, which means it has a constant fluoride release rate (FRR). To better indicate the degree of membrane degradation, the FRRs under other conditions in the literature are compared in Table 1 and the subset of Figure 3. As a result, the FRR and decomposition levels performed in this study are similar to those obtained by the AST. Indeed, the fluoride release solely is not always a good indicator to assess the degradation status of a fuel cell, given that the degradation mechanism has not yet been ascertained explicitly. However, fluoride release is still used as a quantitative indicator in many studies due to its ease of measurement.

Morphological changes in the surface of the Nafion 212 membrane after chemical decomposition are observed using a field-emission microscope (FE-SEM, Sigma, Carl Zeiss, Germany). The sample is attached to a copper conductive tape, and platinum is sputtered under a vacuum before observation. The observations of the membrane surface through FE-SEM according to decomposition time are summarized in Figure 4a. Before Fenton’s test, the as-is membrane shows a smooth surface without defects. However, defeats (e.g., bubbles and pinholes) are observed after 24 H of chemical degradation. Bubble formation is caused by membrane unit decomposition. As the duration increases, the distribution of the bubble increases. Finally, the surface of the membrane is torn due to the bubble expansion, called a pinhole. As a result of pinhole creation, the torn bubbles become smaller, which induces small and densely formed bubbles on the membrane surface. Because of the structural distortion caused by the bubble, the membranes become transparent (0 H) to opaque (72 H), as shown in Figure 5. The bubble and pinhole ratios are evaluated over time through image processing to characterize morphological changes by the extent of chemical degradation. In the case of pinholes, segmentation is relatively easy due to the apparent difference in light density. Meanwhile, bubbles are difficult to distinguish with threshold criteria since they have a stereoscopic shape. For this reason, multistep image processing is conducted for bubble segmentation. Preprocessing is performed by removing the nonuniform noise of the background through a Gaussian blurring operation. After that, the Otsu algorithm makes images more distinguishable, and the density of the bubble is measured (Figure 4b). Image processing is accomplished using a MATLAB image processing toolbox. Twenty image samples are assessed each time for adequate evaluation. Consequently, the area fractions of pinholes and bubbles on the membrane surface are characterized according to the degradation time in Figure 4c. The graph results consist of the average value and the standard deviation at each time. While the density of bubbles linearly increases in time, the one for pinholes shows exponential increase. Moreover, pinholes start to appear starting at 36 H. This phenomenon is because pinhole generation follows after bubble formation, and it can be understood by showing an increase in pinholes as the higher the number of bubbles. The size of the pinhole is retained, while the size of the bubble is reduced after the formation of the pinhole. Hence, the bubble’s linear increase can explain the pinhole’s exponential increase. Furthermore, it is noted that the distribution of pinholes can be utilized as an index to evaluate the degree of reactant gas crossover of the membrane, which is highly related to cell performance.

Uniaxial tensile tests are conducted to determine the mechanical behavior of chemically degraded membranes. An SEM in situ tensile testing machine (Emcraft, Cube II and Deben, Microtest) is utilized under room conditions (23 °C, 50% RH), as shown in Figure 6a. Rectangular specimens are prepared using fresh razor blades with a length and width of 25 and 10 mm, respectively. The strain rate for the cross-head is set to be 1 mm/min, which is 4% of the specimen length per minute. Ten samples are utilized for each time. The stress–strain curves of chemically degraded membranes with different hours are summarized in Figure 6b. The pristine membrane has high tensile yield stress and exhibits ductile behavior. Therefore, it exhibits a linear elastic behavior before the yield stress and a hardening behavior after that. It is observed that the membrane’s tensile modulus and yield stress diminish as the chemical decomposition continues. Interestingly, a considerable decline in ductility of membranes is shown as increasing decomposition time, which is described as the gradual increase in the hardening slope. Decomposition times between 48 and 60 H where pinholes are actively generated represent a sharp decrease in ductility by showing a high hardening slope. Notably, embrittlement is also observed during AST. There are several studies to demonstrate the reason why Nafion becomes brittle. However, there are no approaches to logically elucidate the embrittle mechanisms of the chemically decomposed Nafion. In this study, we postulate the embrittlement mechanism of Nafion through a chemi-crystallization phenomenon. As a polymeric material, Nafion after chemical decomposition loses its ductility as well as performance. It is supposed that embrittlement is caused by the random scission of the polymer chain. Polymeric materials show brittle properties when the molar mass is smaller than the specific molar mass (e.g., critical molar mass, Mc). When polymeric materials have a molar mass larger than Mc (M > Mc), they show ductility. Otherwise, they possess brittle properties. Generally, it is known that Mc is usually about 15 kg mol–1 and is 5–10 times greater than the entanglement molar mass (Me). This specific molar mass has something to do with the embrittlement mechanism. It is possible to explain the embrittlement mechanism of polymers in the following manner. There is chain entanglement in the polymers, which is implied that polymeric materials show ductile deformation under external loads. In other words, chain drawing in the entanglement of the amorphous region leads to ductility. The chemically degraded polymers, however, become brittle because they lose chain entanglement due to the chain scission. The cause of the chain entanglement loss is explained as a chemi-crystallization phenomenon. In other words, the molar mass of the polymers decreases because of the chain scission. Finally, it makes the thickness of the lamellar thicker. Then, residual entanglement is pushed into the interlamellar region. Meanwhile, the thicker the lamellar, the thinner the interlamellar thickness, resulting in entanglement loss and brittleness of the polymer. Consequently, Nafion undergoes chemi-crystallization as a result of the Fenton reaction breaking down the polymer chain, resulting in a thinner interlamellar region. It leads to entanglement loss of the polymer and makes it brittle. Property degradation and embrittlement of the membrane increase the risk of mechanical damage because the membrane cannot sustain structural deformation. Therefore, external loadings are more likely to form cracks and pinholes, leading to performance degradation. However, the embrittlement alone is insufficient to understand the membrane’s mechanical damage mechanism, thus the damage process and its mechanism are also considered. The surface morphologies of the 60 H chemically degraded membrane are observed using SEM under tensile loading (Figure 7). Bubbles and pinholes are formed on the membrane surface after chemical decomposition. The bubble and pinhole enlarge when the membrane is subjected to external loading. As summarized in Figure 7a, the stress concentration has occurred at the edge of the pinhole in the bubble, and the edge becomes the crack initiation point by starting to tear apart. Crack propagation is extended by deforming the material before the crack tip dissipates energy. Thus, crack growth along the pinholes is an effective way of enhancing the crack propagation resistance. This phenomenon is easily observed in the SEM image, as shown in Figure 7b.

In this section, we present a constitutive model that can represent the membrane’s hygro-thermo-mechanical response. We assume that the total strain tensor, εij, is given as the sum,where εijM, εijS, and εijT are the mechanical, swelling, and thermal strain, respectively. δij is Kronecker’s delta, which indicates that swelling and thermal strain comprise only of the normal component. More details for each of these strains are discussed as follows. As the Nafion 212 membrane shows elastoplastic behavior in the experiment, the mechanical strains, εijM, can be defined within the context of the J2 flow rule. The plasticity model defined by the plastic flow rule realized irreversible deformations of materials. The total mechanical strain components are divided into the elastic and plastic strains as follows:where the superscripts “el” and “pl” denote the elastic and plastic strains, respectively. Then, the stress–strain relationship is expressed with Hooke’s law as follows:where Cijkl represents the material’s fourth-order elasticity tensor. The behavior of the membrane in this work is assumed to be isotropic. Therefore, the elastic stiffness tensor Cijkl is expressed by eq 4In this equation, E and ν are the Young’s modulus and Poisson ratio, respectively. The material parameters of the PEM are determined as functions of relative humidity and temperature through experiments. The J2 flow rule is adopted to investigate the plastic deformation of the membrane. Plastic deformation arises when the von Mises stress reaches the membrane’s yield stress. Therefore, the plastic yield function of the membrane is expressed as the following condition:In this equation, σef is the effective stress, σy0 is the initial yield stress, r(p) is the hardening function, and p is the effective plastic strain. σef and r(p) are defined as follows: Here σ′ is the deviatoric stress and b and Q are material constants associated with isotropic hardening, which give an exponential hardening shape in the stress–strain curve. Since integrating eq 6 with an initial condition, the hardening function goes Q(1 – e–bp). It implies that Q is the saturated value, which means that the ultimate stress can be achieved with σy0 + Q. The constant b determines the rate of saturation. Figure 8 shows the uniaxial stress–strain behavior using the isotropic hardening function. In this work, isotropic hardening is only considered because the significant interest is the nonlinear hardening expression using plastic deformation. When the effective stress does not reach yield stress (ϕ < 0), the material shows elastic behavior. Otherwise, the material undergoes plastic deformation following the isotropic hardening rule, where the size of the yield surface expands in all directions. The direction of the plastic strain after yielding is determined by the normality hypothesis and consistency condition as follows:where dλ is the plastic multiplier. The partial derivative of the yield function with respect to the stress (∂ϕ/∂σ) is formulated as After plastic deformation is initiated, the stress states at the material points are always located on the expanded yield surface, which satisfies the consistency condition. This condition is described as follows: Following the consistency condition and radial return mapping method, numerical incremental forms of the isotropic hardening stress and the effective plastic strain can be described as follows: Following the incremental formations in eq 10, Newton Raphson iteration is conducted until these internal variables satisfy the consistency condition. σtr is the trial stress for elastic predictor. Details about the subsequent computations can be found in the literature, and the radial return mapping implicit algorithm for the elastoplastic model is summarized in a pseudo-code format in Table 2. This proposed model is implemented through the ABAQUS user material subroutine (UMAT). The thermal strain for a temperature change can be expressed aswhere α denotes the coefficient of thermal expansion and T and Tref are the current and reference temperatures, respectively. The swelling deformation caused by membrane water absorption can be expressed by an empirical relationship based on volume swellingwhere ξi is the anisotropy ratio in the i direction, satisfying the condition ∑i=1nξi = 1. For isotropic swelling, the swelling strain is identical in all directions, which becomes ξi = 1/3. ϕp is a volume fraction of the polymer, expressed aswhere λ is the water content (the number of water molecules attached to each sulfonic acid group) and EW and ρp are the equivalent weight and density of the PEM, respectively (EW = 1100, ρp=1970 kg/m3). Here, the water content, λ, can be expressed through water activity (relative humidity). There are experimentally determined and/or validated formulations in many previous studies. In this work, the empirical model from Springer et al. is utilized to match experimental environments in the validation work.

To better explain how mechanical properties are affected by chemical decomposition time, the membrane properties, such as Young’s modulus, yield stress, and strain hardening parameters, are characterized in terms of fluoride release. A finite element (FE) specimen model is created where a mesh is assigned for a square specimen with a square length of 0.01 mm. A 3D cubic element (C3D8) is used for the element. Figure 9 shows the stress and displacement distribution. Considering that the stress is uniformly distributed in the middle of the specimen, it is assumed that the specimen behavior can represent the material behavior. The material constitutive model is defined as described in Section 3.1 in the specimen model. In the FE simulation, the Poisson ratio is assumed to be 0.35 because of its low influence on the uniaxial stress response. Finally, inverse analysis is conducted to characterize material parameters concerning chemical degradation, which is summarized in Table 3. Our previous study describes the inverse analysis method in detail. Seeing that the material parameters obtained through inverse characterization, the elastic modulus and yield stress significantly drop as the chemical decomposition time is increased. In the case of Young’s modulus, 60 and 72 H chemically degraded membranes show a half reduction in performance compared to the pristine membrane. As Q and b are parameters that determine the hardening slope in the stress–strain curve, an increase in Q and a decrease in b indicate that the hardening slope becomes steeper, and the material shows a brittle behavior. Figure 10 depicts the change of material parameters according to the chemical decomposition as the value of fluoride release (correlated with the results in Figure 3). Figure 10a indicates the variations of Young’s modulus and yield stress. Although the change of both values at the beginning and the end is not noticeable, they decrease with an increase in fluoride release and change rapidly between 12 and 21 μmol cm–2. This trend is observed identically in the yield stress ratio (Q/σy0) and the hardening stress rate (EbQ). Since it is difficult to compare only with the independent values of b and Q, they are replaced with intuitive values (e.g., the yield stress ratio and the hardening stress rate) expressed in the stress–strain curve. The range between 48 and 60 H where pinholes occur suddenly shows a drastic change in mechanical properties. Based on these results, the chemi-crystallization causes entanglement loss of the polymer and can be predicted by the pinhole distribution observed on the membrane surface. A mathematical expression is conducted based on the polynomial regression model. Young’s modulus and yield stress are sufficiently expressed by a third-order polynomial, whereas a fourth-order polynomial is utilized for significant changes in the yield stress ratio and hardening stress rate. The mathematical expressions are summarized in eq 14:The defined material parameters are used for simulation in Section 4 to examine the membrane performance.

A pinhole is a defect caused by chemical and mechanical degradation. The pinhole acts as a passageway for the reactant gas and accelerates the degradation of fuel cell performance. Therefore, the density of the pinhole and the extent of gas crossovers are closely related. This pinhole size can be characterized by the formation from chemical decomposition and mechanically induced growth. Therefore, a pinhole growth model of the membrane subjected to cyclic loading is required to estimate the cell performance. The concept of the pinhole growth model is summarized in Figure 11. In this study, the shape of the pinhole is idealized as a circular shape. Interestingly, mechanical loading has no noticeable degradation alone. Therefore, it is assumed that pinhole formation is determined by chemical degradation. Since the pinhole shape is circular, the ratio that the pinhole occupies in the membrane can be expressed as a dimensionless number as expressed in eq 15.where Amem and Rpin are the surface area of the membrane and the pinhole radius, respectively. In this work, the pinhole with an ideal cavity has the potential to grow when the membrane is plastically deformed. The pinhole growth model under cyclic stress is inspired by the conventional theory of pore growth in solids based on continuum plasticity. The change in the pinhole radius is expressed aswhere σm and σef are the hydrostatic and equivalent (effective) stresses, respectively, dp denotes the equivalent plastic strain increment, and c is the scaling parameter relating the pinhole radius to strains. The use of far-field stresses means that local variations occurring in the pinhole are negligible, for example, the stress concentration at the pinhole interface.

Changes in the mechanical properties of PFSA membranes by temperature and humidity have been reported in many studies. A case study was conducted by independently changing various temperatures and humidity, and the mechanical property changes of Young’s modulus and yield stress were expressed mathematically as follows:Here, the superscript “std” indicates the properties of the membrane in a standard environment (23 °C, 50% RH), and n and m are fitting parameters. The corresponding fitting parameter is determined between 2 and 4 depending on the type of PEM, temperature, and humidity range. In the case of Nafion 212 of PEMFCs, the n and m are determined as 3.6 and 2.4, respectively, following the literature. Since the ability to transport hydrogen for the PEM varies according to external environment conditions (temperature and humidity), it dramatically impacts cell performance. Excessive humidity prevents the discharge of water from the cathode, blocking fuel gas passage and reducing cell performance. On the contrary, too low humidity is not suitable for fuel cell efficiency. Therefore, the proper level of humidity on the PEM is controlled through a humidifier in the PEMFC system. While the humidity change has happened via the amount of water generated at the cathode, the fuel cell operating temperature is maintained between 60 and 80 °C. Thus, the temperature consideration in the modeling can be negligible compared with humidity. PEMFC performance measurement through AST is generally performed at the same temperature. Computational simulation for fuel cell operation is required to predict PEM behavior accurately. The PEMFC structure based on the unit cell may be expressed as shown in Figure 12. The PEM is placed in the middle, and the gas diffusion layer and bipolar plates are symmetrically stacked. A cell stack for the PEMFC can be implemented when such unit cells are repeatedly stacked, and the number of cells varies depending on the target power of the application. Looking in more detail around the PEM, there are thin layers, such as the catalyst layer (CL) and microporous layer (MPL), which play a role in a chemical reaction and transportation rather than a mechanical support. Therefore, the force to compress the cell is transmitted from the bipolar plate to the membrane. At the same time, as the PEM is subjected to clamping constraints, compressive stress is applied to the PEM according to the humidity change. Since the channel is a passage of reactant gas (e.g., air and hydrogen), the compressive force by clamping pressure and hydration is not uniformly applied to the membrane. However, from focusing on the severe part observation, the model geometry in this work simplifies with uniform loading. In addition, the shape of the repeated cell is captured through the symmetric boundary condition. Moreover, this study assumes that the tensile and compressive behaviors of the membrane are identical. The mechanical response to the humidity change of PEM is investigated according to the five times of RH cycle. In order to observe only the effect of humidity, there is no clamping pressure. The hydration cycles applied to the membrane are set to three different amplitudes (i.e., Δλ = 1, 6, and 12 for H2O/SO3–). Changes in membrane water content have a sinusoidal shape. Figure 13a,b shows corresponding changes in swelling strain and properties. Hydration of the membrane increases the swelling strain and decreases material properties. The magnitudes of the induced stresses on the membrane due to clamping conditions show differently depending on the membrane’s properties and swelling strain. In this context, the increase in water content leads to an increase in properties and a decrease in swelling strain. Thus, the magnitude of the stress may not be relatively different because the properties and swelling strain work complementary. However, since the increase in the swelling strain is more pronounced than the decrease in properties, different behaviors according to each condition can be observed. In the case of Δλ = 6 and 12, pinhole growth can be observed since plastic deformation occurs within the cycle. When the compressive stress acting on the membrane during the hydration process causes plastic deformation, the membrane receives tensile stress during dehydration. Alternating stress modes between compression and tension during RH cycling results in periodic fatigue stress states. To demonstrate pinhole growth under plastic deformation, we investigate the growth of pinholes in plastically deformed membranes. Figure 13c shows the change in normalized pinhole radius R/R0 obtained in the mechanical response for Δλ with various expansion amplitudes. It can be inferred that under positive hydrostatic stress (σm > 0), the membrane is plastically deformed, and the radius of the pinhole increases. As a result, it is explained that the pinhole area grows significantly as the hydration change (Δλ) increases. The most significant increase in pinhole growth occurs at Δλ = 12, where the membrane is maximally expanded. In contrast, only elastic behavior is observed in Δλ = 1 without causing pinhole growth. Other aspects show that reducing the RH amplitude or decreasing the compressive stress state through expansion anisotropy is beneficial for mechanical durability. However, it is essential to recognize that the actual PEM lifetime is controlled by several factors and other phenomena related to transport, chemical degradation, and interactions with other cell components. This model is further extended to predict the membrane’s lifespan with the pinhole size. Because gas crossover determines the PEM performance, pinhole distribution in simulation results can be utilized for durability evaluation.

We compare the proposed model with the gas crossover data from AST. Pinhole characterization from chemical decomposition and mechanical deformation are considered concurrently by accumulating pinhole growth by the RH cycle based on the pinhole density determined through the Fenton test. This scheme can rigorously evaluate pinhole growth by considering mechanical and chemical factors. The flowchart of this methodology is summarized in Figure 14. The membrane undergoes RH cycling with reactant gas in AST. The AST was performed at 80 °C in H2/air with anode and cathode flow rates of 2 SLPM and outlet pressures of 101 kPa. Here, the RH cycle consists of 2 min at 148% RH (90 °C dewpoint) and 2 min at 0% RH (i.e., each cycle takes 4 min). The details of the experimental setup for mechanically/chemically combined in situ AST can be found in the literature. To measure the pinhole distribution, (i) define an initial pinhole density based on the amount of fluorine release and (ii) expand in pinhole growth under cyclic loading. A pinhole creates a pathway for reactant gas crossover, which can be an indicator of membrane failure and is therefore used to measure the life span and durability of the membrane. However, since it is difficult to compare the exact values of pinhole and gas crossover, we examine it as normalized. Figure 15 shows that the model predictions for normalized pinhole sizes in chemical/mechanical degradation agree reasonably well with the experimental data. Pinhole growth due to mechanical deformation during RH cycling shows a relatively small but significant difference compared to the only chemically degraded result. Since a small amount of crossover also occurs in the pristine membrane, the result of adding the initial crossover to the chemo-mechanically coupled pinhole density shows more similarity to the experiment. Consideration of mechanical and chemical effects in the modeling approach can demonstrate a synergistic effect accelerating crossover rates. Several simulations can be further performed according to the proposed methodology to account for membranes exposed to different conditions. As a result, it shows promising results by validating pinhole growth according to chemical/mechanical degradation with in situ AST crossover results. However, the thinning of the membrane results from chemical decomposition is neglected in this model. It plays an important role in fuel cell performance and durability in terms of proton conductivity and mechanical resistance. Despite not being covered in this study, it should be taken into account in the electrochemical modeling process in the future.

In this study, the mechanical properties of the PEM after chemical decomposition were investigated and applied to the material constitutive modeling. Total fluoride release was expressed as a function of decomposition time and showed a linear increase to characterize the degree of chemical degradation and correlate mechanical properties with degradation levels. It was identical to the outcomes obtained in the in situ accelerated stress test. This verification means that the mechanical behavior of the PEM obtained through the ex situ Fenton test can represent the state of the PEM in the in situ fuel cell operating. Based on the developed mechano-chemical material constitutive model, knowledge of fatigue stress response and lifetime prediction of the PEM against RH cycles can be examined by introducing the concepts of defect growth. In particular, we adopted the concept of continuum plasticity, where pinhole growth is promoted by two factors: hydrostatic stress and plastic strain. The effects of mechanical response under RH cyclic loading (hydration) on pinhole growth were investigated, and the possibility of retarding pinhole growth was revealed by reducing expansion and increasing the strength and modulus without increasing the equivalent weight. To explain the chemo-mechanical synergistic effect on the defects, we study the interacting degradation by integrating the mechanically driven pinhole growth and the semi-empirically expressed pinhole generation by chemical degradation. Despite the simplicity of the synergistic relationship of chemo-mechanical degradation, the trend predicted by the proposed model agrees well with that of the crossover data, representing the PEM performance. Although the mechanical influence caused by the humidity cycle alone does not cause rapid failure, this work understood the complex synergetic degradation mechanism by the mechanical/chemical concurrent degradation simulation. Nevertheless, a thorough investigation of PEM durability analysis needs to be revealed through a more solid interaction between mechanical and chemical degradation. In conclusion, this study presented a methodology for predicting the durability of membranes through a quantitative understanding of the mechanical and chemical coupling degradation of properties.