Oxygen transport in the low–Pt catalyst layer of a PEM fuel cell: Impedance spectroscopy study

A model for PEM fuel cell impedance taking into account the pore size distribution (PSD) in the cathode catalyst layer is developed. Experimental PSD is approximated by pores of three sizes (small, medium and large) and in each kind of pores, the oxygen diffusion coefficient is allowed to have a separate value. The model is fitted to experimental impedance spectra of a low–Pt PEM fuel cell. The oxygen diffusivities of small and medium pores exhibit rapid growth with the cell current density, while in large pores, this parameter remains nearly constant. We show that oxygen reduction occurs mainly in the small and medium pores, leaving the large pores for mass transport only. This effect explains the discrepancy between small effective oxygen diffusivity of PEMFC catalyst layer measured in situ in operating cells by limiting current method, and much larger value of this parameter determined from ex situ experiments using Loschmidt cell.

Greek: η ORR overpotential, positive by convention, V λ Air flow stoichiometry σ N Nafion film proton conductivity, S cm −1 ω Angular frequency of the AC signal, s −1

Introduction
One of the key components of polymer electrolyte membrane (PEM) fuel cells is cathode catalyst layer (CCL) converting oxygen, protons and electrons into water. The electron conductivity of modern CCLs is usually high, while transport of oxygen and protons to Pt sites, where the oxygen reduction reaction (ORR) occurs requires quite a substantial overpotential. Success of PEMFC technology on the mass market depends on PEMFC cost, which could be reduced by lowering of Pt loading in the CCL. However, some ten years ago it has been found that low-Pt cell cathodes exhibit anomalous oxygen transport losses [1,2]. In CCL, Pt/carbon particles form agglomerates covered by a thin ionomer (typically Nafion) film. Reducing of Pt loading increases oxygen flux per agglomerate, making Nafion film a significant barrier for oxygen transport to the Pt surface. A simple model reveals a limiting current density associated with the Nafion film [3], much like the limiting current density due to oxygen transport through the gas diffusion layer (GDL). At high currents, the ionomer film covering agglomerate works as a limiting nano-scale barrier for oxygen.
The findings [1,2] triggered an avalanche of experimental and modeling studies on oxygen transport in low-Pt CCLs. On the theory side, the problem of O 2 transport in CCL has been considered on atomistic scale using molecular dynamics and DFT calculations [4,5], and on macroscale by developing mean-field models [6][7][8][9] and numerical structural models employing either mass conservation equations, or Lattice-Boltzmann approach [10][11][12]. Experimental studies have been focused on in situ measuring of film transport resistivity using limiting current method [13][14][15][16][17]. The references above indicate just several exemplary works; more papers could be found in reviews [18][19][20].
Fan et al [20] provided an excellent review of challenges on a way toward development of efficient catalyst layers for PEMFCs. Among other problems, careful engineering of a pore size distribution (PSD) with appreciable amount of mesopores (2-20 nm) to achieve the desired catalyst/ionomer distribution and ensure efficient supply of O 2 and protons to catalyst sites has been noted. Salari et al reported experimental measurements of ex situ and in situ effective oxygen diffusivities in the CCL [15]. They reported at least an orderof-magnitude lower in situ oxygen diffusivity (measured in operating cell) as compared to ex situ value (measured in Loschmidt diffusion cell). The difference has been attributed to inclusion of water film and Nafion film into the oxygen pathway in operating fuel cell, while in ex situ experiments, oxygen diffuses predominantly through the large pores between catalyst agglomerates, the so-called secondary pores. Below, the model forelectrochemical impedance of a low-Pt PEMFC [21,22] is extended to take into account variation of oxygen diffusion coefficient between pores of different size. This allows us to rationalize the effect of PSD on oxygen transport. In addition, the model separates oxygen transport in void pores from the transport in Nafion film. The experimental pore size distribution is approximated by the three-pore PSD. Fitting of experimental spectra returns the oxygen diffusion coefficient in each type of pores as a function of cell current.
We show that the difference between in situ and ex situ effective CCL oxygen diffusivities can be explained by different size of pores transporting oxygen in the two cases. In ex situ experiments, large pores are the main oxygen transport media. However, in the in situ measurements, small pores are the primary pathway for oxygen transport due to their much higher cumulative side surface film/Pt area, where the ORR runs. To the best of our knowledge this work reports estimates of oxygen diffusivities in pores of various size for the first time.

Model
The model includes oxygen transport in the channel, GDL, CCL pores, and in the Nafion film covering Pt/C agglomerates. The CCL is modeled as a set of straight cylindrical pores extending through the whole CCL depth. The pore volume is separated from the coaxial Pt surface by a thin ionomer film ( figure 1(a)). In real CLs, the pores of different size form a random network. Modeling of such a system requires much more complicated logic of calculations (continuity of oxygen concentration and fluxes) and much more computational resources. Our model is a first step on a way toward pore-network modeling of CCL impedance.
Transient mass and charge conservation equations in a single pore include oxygen diffusion equation along the void pore coupled in a 1d+1d manner to the oxygen diffusion equation in the radial direction through ionomer film ( figure 1(b)). The boundary conditions to radial equations contain Henry's law for oxygen dissolution at the pore/film interface and the Tafel rate of the oxygen reduction reaction at the ionomer/Pt interface. The system is completed by the proton charge conservation equation having a form of diffusion-like equation for the ORR overpotential η ( figure 1(b)). In the GDL, oxygen transport is described by the transient Fick's diffusion equation with zero right side ( figure 1(b)). For the list of symbols see Nomenclature section.
In this work, measured PSD of a similar standard Gore electrode [23] has been approximated by a three-pore spectrum (figure 2). The spectrum is represented by the small, medium and large pores (16.5, 75.1 and 230 nm, respectively), with the relative sizes illustrated in figure 3. An important feature of this model is that in each type of pores, the oxygen diffusion coefficient D p is allowed to have a separate value.
At the macro-scale, the cell is separated into N virtual segments (figure 4). In each segment, oxygen concentration at the channel/GDL interface is determined by the plug flow equation shown in figure 4. The right side of this equation describes the diffusive oxygen flux at the channel/GDL interface, which 'leaks' to the GDL along the through-plane axis x. The boundary and interface conditions for the system of equations in figures 1, 4 express continuity of the oxygen concentration and flux, and zero proton current at the membrane/GDL interface.
The system of transient equations has been linearized and Fourier-transformed to get a system of linear equations for the small perturbation amplitudes of the oxygen concentration c 1 and overpotential η 1 (see [21] for details). In every segment, solution of this system for the pore size k = 1, 2, 3 gives impedance of the kth type pores Z k = − η 1 /(σ N ∂η 1 /∂x)| x=0 . Total impedance Z seg of the nth segment is calculated according to Finally, taking into account cable inductance L cab and membrane (high-frequency) resistance R HFR , for the total system impedance we have

Experimental
The electrochemical evaluation was performed using Gore PRIMEA (A510.1/M715.18/C510.1) catalyst coated membrane (CCM) with 0.1 mg Pt cm −2 loading for anode and cathode. The CCM was evaluated by SEM and thicknesses of electrodes was found to be 3-4 microns, membrane thickness varied from 15 to 18 microns. SEM pictures of the MEA used can be found in [24]. Gas diffusion layer from Sigracet (25BC) was applied for both electrodes. Based on the specifications from the manufacture, 25 BC carbon paper substrate has 5 wt% of PTFE treatment, while microporous layer (MPL) typically contains 20%-25% PTFE. Teflon gaskets with thickness of 125 μm were employed to fabricate a membrane electrode assembly (MEA) and provide 25%-30% of compression ratio. The cell active area was 76 cm 2 . The anode/cathode operating conditions were H2/air, 2/4 stoichiometry, 100/50% RH, 150 kPa back pressure for both electrodes and cell temperature of 80°C. The operating conditions were chosen based on the US DOE Hydrogen and Fuel Cell Technology Office protocols for evaluation of the single PEMFCs. Fuel and oxidant humidification levels were selected to be 100% and 50%, respectively, to ensure proper humidification at low current operation and prevent cathode flooding at high current conditions. The EIS curves of the overall cell were recorded under galvanostatic mode with 11 steps/ decade and frequency sweep from 10 kHz to 0.1 Hz. The current perturbation signal was set to ensure the cell voltage response of 10-20 mV. Details of the experimental work can be found in our previous publications [21].

Results and discussion
The oxygen diffusivities in the GDL D b , in void pores D p,k , in the ionomer film D N , the film proton conductivity σ N , the ORR Tafel slope b and the double layer capacitance C dl has been declared as fitting parameters (indicated in red in figures 1, 4) and impedance equation (3) has been fitted to experimental spectra. The cell geometrical and operation parameters are listed in table 1. With L cab , R HFR and three diffusion coefficients D p,k corresponding to the three types of pore, the total number of fitting parameters is eleven. Calculations have been performed using a custom parallel Python code. Spectra fitting was done using the least_squares subroutine with trf method from the Python SciPy library. Typical calculation requires about 10 to 20 min on 24 cores of Intel Xeon processors. The examples of measured and fitted spectra are shown in figures 5, 6. Interestingly, a better fitting of the low-frequency (LF) part of the spectra has been achieved at higher cell current density (figures 5, 6). The second (LF) loop is attributed to oxygen transport in the GDL and channel. Our model ignores liquid water transport in the cell and better quality of LF domain fitting could be attributed to higher air flow rate at high currents, which facilitates liquid water removal from the cell. The parameters resulted from fitting are shown and discussed in appendix ( figure 9). Here, we focus attention on the pore oxygen diffusivities D p,k ( figure 7).
Qualitatively, the ratio between oxygen diffusivities follows the trend prescribed by Knudsen diffusion: D p ∼ R p . The ratio of diffusion coefficients in medium and small pores first increases with the cell current density, and then remains nearly constant at currents above 400 mA cm −2 ( figure 7). The decrease of D ox in small pores in the range of current between 100 and 300 mA cm −2 is, perhaps, due to preferential accumulation of liquid water in pores of that size.
An interesting, yet unclear feature is rapid growth of D p in small and medium pores with the cell current density J, while in large pores, this parameter remains fairly insensitive to J ( figure 7). In fine pores, oxygen transport may proceed through surface diffusion, which provides a parallel, current-dependent transport pathway [25]. A more detailed model with the account of liquid water transport in the pores and surface diffusion effects is necessary to fully understand the curves in figure 7.
The constancy of D p (J) in large pores can be explained using the following arguments. In our model, conversion of proton current into water occurs at the side surface of Pt cylinder (figure 1). Consider a single large pore and a number of small pores of the same total volume. For example, the volume of large pore with the radius R = 3 units is equivalent to the total volume of 9 small pores with the radius r = 1 unit From equation (5) it follows that if small and large pores occupy equal fractions of the total CCL volume, the side surface of small pores would convert R/r times larger current. The current conversion efficiency is thus Table 1. Cell geometrical and operating parameters. A/C stands for anode/cathode.
Catalyst loading A/C, mg cm −2 0.1/0.1 CCL thickness l t , cm 3 · 10 −4 (3 μm) Nafion film thickness l N , cm (assumed) 10 −6 (10 nm) GDL thickness l b , cm 230 · 10 −4 (230 μm) ORR exchange current density i * , A cm −3 (assumed) 10 −3 Henry's constant for oxygen solubility in water at 80°C, mol/mol 6.76 · 10 −3 Flow stoichiometry A/C 2 /4 Relative humidity A/C 100%/50% Cathode absolute pressure, kPa 150 Cell temperature, K 273+80 The PSAD calculated from the experimental PSD [23] is shown in figure 2. As can be seen, a dominant fraction of proton current coming to the CCL is converted in small and medium pores, leaving no job for large pores. Nearly zero current converted in large pores explains insensitiveness of large pore oxygen diffusivity on cell current. The discussed PSAD effect can explain the discrepancy between the low value of CCL effective oxygen diffusivity (10 −5 -10 −4 cm 2 s −1 ) reported from in situ impedance measurements in operating PEM fuel cells  ( figure 9(b), see also [26]) and one to two orders of magnitude larger values of D p ; 10 −3 cm 2 s −1 reported from ex situ experiments with Loschmidt cell [27]. Indeed, the Loschmidt cell measurements return D p in large pores, as in the absence of current these pores provide the main oxygen pathway. However, in an operating fuel cell, the  impedance 'feels' D p in small and medium pores, as these pores transport the largest oxygen flux needed for conversion of proton current.

Conclusions
The experimental pore size distribution of the cathode catalyst layer in PEM fuel cell is approximated by pores of three size: small, medium and large. A model for PEM fuel cell cathode impedance is extended to take into account different oxygen diffusivities in the pores of different size. The model is fitted to experimental impedance spectra of a low-Pt PEM fuel cell. The oxygen diffusion coefficients in small and medium pores exhibit rapid and yet unexplained growth with the cell current density J, while the oxygen diffusion coefficient in large pores does not change with J. Simple geometrical arguments show that most part of the proton current is converted on the side surface of small and medium pores, which is much larger than the side surface of large pores. This effect explains the difference between large value of CCL effective oxygen diffusivity measured ex situ and small value of this parameter returned from in situ experiments. In ex situ situation, large pores are the main oxygen pathway, while under in situ conditions, proton current is converted mainly in small pores whose oxygen diffusivity is much smaller.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).
Appendix A. Transport parameters resulted from fitting Figure 9 summarizes the system parameters resulted from fitting. Ionomer film oxygen permeability  and transport resistivity  are closely related as ∼1/ (figures 9(a), (d)). In this calculation, the average resistivity  is about 0.8 s cm −1 , which is about two times higher than the value obtained for the same sample from the model with 9 pores [28]. Note, however, that the 9-pore model [28] did not take into account variation of the parameter D p between the pores. In section Results and Discussion it has been shown that this variation is quite large. Overall, the less accurate approximation of PSD and variation of D p between the pores could lead to higher  in the present calculation.
GDL oxygen diffusivity D b varies between 0.02 cm 2 s −1 and 0.04 cm 2 s −1 ; a single value of 0.06 cm 2 s −1 obtained at the cell current of 400 mA cm −2 seems to be an overestimate ( figure 9(b)). The ionomer proton conductivity σ N is fairly low, about 1 mS cm −1 (figure 9(c)). A study of Gasteiger's group [29][30][31] have shown strong dependence of this parameter on ionomer-to-carbon ratio and relative humidity. The value of 1 mS cm −1 falls into the range reported in the aforementioned works. The double layer capacitance C dl is about 25 F cm −3 (figure 9(e)), which agrees well with our previous calculations for similar Pt/C electrodes. The ORR Tafel slope varies between 70 and 80 mV/decade with the single 'escape' of 90 mv/decade at 300 mA cm −2 (figure 9(f)). 70-80 mV/decade agrees well with the value reported in literature [29].