A Comprehensive Physical Impedance Model of Polymer Electrolyte Fuel Cell Cathodes in Oxygen-free Atmosphere

Electrochemical impedance spectroscopy (EIS) is an indispensable tool for non-destructive operando characterization of Polymer Electrolyte Fuel Cells (PEFCs). However, in order to interpret the PEFC’s impedance response and understand the phenomena revealed by EIS, numerous semi-empirical or purely empirical models are used. In this work, a relatively simple model for PEFC cathode catalyst layers in absence of oxygen has been developed, where all the equivalent circuit parameters have an entire physical meaning. It is based on: (i) experimental quantification of the catalyst layer pore radii, (ii) application of De Levie’s analytical formula to calculate the response of a single pore, (iii) approximating the ionomer distribution within every pore, (iv) accounting for the specific adsorption of sulfonate groups and (v) accounting for a small H2 crossover through ~15 μm ionomer membranes. The derived model has effectively only 6 independent fitting parameters and each of them has clear physical meaning. It was used to investigate the cathode catalyst layer and the double layer capacitance at the interface between the ionomer/membrane and Pt-electrocatalyst. The model has demonstrated excellent results in fitting and interpretation of the impedance data under different relative humidities. A simple script enabling fitting of impedance data is provided as supporting information.

In this work, a fully physical impedance model for PEFC CCLs in absence of O 2 has been developed and proposed to use in PEFC impedance analysis. The modeling is based on five pillars, which account for the independent experimental data on the pore properties, de Levie's analytical formula to calculate the impedance response of a single cylindrical pore, the ionomer distribution in a single pore, the specific adsorption of sulfonate groups (from Nafion polymer) at the surface of the Pt-catalyst and possible side reactions (e.g. caused by hydrogen crossover through the membrane). The developed model has no CPE elements and has demonstrated excellent results in fitting and interpretation of the impedance responses characterizing the interface between the ionomer (membrane thickness ~15 μm) and Pt-electrocatalyst supported on carbon at the cathode side under different humidities.

Experimental
A FuelCon Evaluator C-1000-LT test station was used to operate the fuel cells. AC-impedance measurements were conducted using a Zahner electrochemical workstation IM6 together with a potentiostat PP241. A state-of-the-art high-performance membrane electrode assembly (MEA) provided by Johnson Matthey plc with an active area of 43 cm 2 was placed between two gold-plated flow field plates in cross-flow arrangement. During the measurement using H 2 /N 2 mode, the anode was fed with H 2 serving as the analog of the reversible hydrogen electrode (RHE). The cathode was fed with nitrogen. The cathode potential was set at 0.5 V vs the anode potential (0 V). The probing AC signal amplitude was set to 20 mV, and a frequency range from 20 kHz to 20 mHz was chosen. The choice of the amplitude value was based on a compromise between a good signal-to-noise ratio and linearity of the system. Using standard Krammers-Krönig tests and Fourier transform analysis it has been found that 20 mV amplitude was the optimal to satisfy both criteria. Ten frequency points were recorded per decade and each measurement point was measured four times and averaged. For each condition, two spectra were recorded in order to ensure quasi-stationarity. The gas and cell temperatures were set to 80 °C. The setup was flushed with nitrogen for about 2 h trying to remove oxygen from the fuel cell and the test station before starting the measurement. In order to investigate the impedance's dependence on the relative humidity (RH), the bubblers of the anode and cathode side were set to dew point temperatures of T d = 44.8 °C 59.0 °C, 67.9 °C 74.6 °C and 80 °C corresponding to RH = 20%, 40%, 60%, 80% and 100% respectively. In order to keep the partial pressures of the supplied gases constant, the gas outlet pressures were adjusted according to the RH to 1.59 bar, 1.69 bar,1.78 bar, 1.88 bar, and 1.97 bar at RH of 20%, 40%, 60%, 80%, and 100% respectively. This way the reference potential of the anode was maintained. Scanning electron microscopy (Zeiss SUPRA 55VP, 10-20 kV) was used to determine membrane, cathode CL and anode CL thickness of the used CCM (see Fig. 1). Cryo-fracturing of the sample using liquid nitrogen was applied in order to access the sample's cross-sectional area. The investigated structure was sputtered with gold in order to increase its electronic conductivity. Since the developed model is based on the description of pores being covered with ionomer (pore radius > 10 nm) 34 , mercury porosimetry (QUANTACHROME POREMASTER 60-GT), being able to provide information about the whole relevant pore radius range, was used to quantify pore radius and pore number. Three samples (~6 × 25 cm²) were measured and averaged for further considerations. Table 1 summarizes further experimental information.

Results and Discussion
Model Description. Figure 2A shows the relation between PEFC's components and the developed model.
Electric resistance of flow field (FF) plates and gas diffusion layer (GDL) as well as the ionic resistance of the membrane are modeled by resistive elements R el and R mem . Anode's and cathode's catalyst layers (ACL and CCL) are modeled by sub-models represented by Z ACL and Z CCL , respectively. With the assumption of the absence of any reactive gases, the impedance response of the CCL is given only by the double layer charging of the porous structure. The impedance response of the ACL under the two-electrode configuration is dominated by the hydrogen oxidation reduction. As a result of the fast hydrogen reaction kinetics, the impedance response of the anode can be neglected. The remaining equivalent circuit (EC) is given by a series connection of a resistance, accounting for all ohmic resistances and the sub-model of the CCL. In order to simplify the CCL model's description, its explanation is started with five main conceptual constituents.
(1) The real porous structure of the fuel cell is taken into account. Therefore, in this model the pore radius distribution experimentally determined by mercury porosimetry was implemented as the input parameter. Without including pore size information, it was not possible to fit the impedance response at lower frequencies.
(2) De Levie's analytical formula is used to calculate the impedance response of a single cylindrical pore.
(3) The ionomer distribution within the CL is taken into consideration. (4) The specific adsorption of functional sulfonate groups on Pt electrocatalysts 35 is taken into account. (5) Side reactions are assumed to occur e.g. of hydrogen being crossed through the membrane or traces of oxygen in the nitrogen gas. Figure 2B shows the MEA architecture used for the experiments and the modeling. Initially, it is assumed that the catalyst layer can be appropriately modeled as an assembly of cylindrical pores of different radii but equal lengths. Subsequently, Fig. 2A schematically explains the situation, where no background Faradaic reactions are expected. The cathode catalyst layer is approximated by a well-known transmission line model (TLM) where the EC of a single pore consists of the parallel connection of N double layer capacitances C dl connected via an ionomer resistance R ion . Figure 2C shows the corresponding schematics within a pore, which is completely filled with the ionomer. However, in contrast to the common arbitrary assumption about the pore size distribution, it is proposed that experimental data should be used. Figure 2D shows an example of such experimental assessment based on Hg-porosimetry. The corresponding EC of the CL in the absence of any Faradaic reactions as a parallel connection of the individual pores is quantified by this assessment.
It should be noted that the modified transmission line model by means of the experimental pore radius distribution (PTLM) introduced in this work can be considered as consisting of two TLMs accounting for two dispersions in the impedance spectra occurring in the high-and mid-frequency range (see supplementary information). According to the frequency dependent penetration depth λ of the AC-potential perturbation in a pore 36,37 , the spectra in the Nyquist representation should show a straight line in the high frequency range, where real and imaginary parts are equal to each other, thus creating a phase angle of −45° (see an example in Fig. 3A).
The ordinary TLM describes the 45°-region and is given by de Levie's formula 38 (see the corresponding fitting in Fig. 3A): where κ CL,ion is the ionomer conductivity in the CL, c dl the specific double layer capacitance, l pore the pore length, r pore the pore radius, ω the angular frequency of the impedance signal and i the imaginary unit. The formula is valid under following general assumptions: Electronic resistance of the CL is neglected • Pore lengths l pore are equal to the thickness of the CCL t ccl multiplied by τ (tortuosity factor) of the CCL τ = ⋅ l t pore ccl (see supplementary information) • The ionomer/catalyst interface is purely capacitive resulting in an ideal penetration depth • One-dimensional treatment of pores (due to deep pores) • No occurrence of Faradaic reactions at the cathode • Cylindrical pore shape De Levie's TLM approach is based on dividing one pore in N slices. Each slice is assumed to have an ideal double layer capacitance at the I/C interface. The ideal capacitances of the single slices are hereby connected in parallel via additional ionomer resistances. Therefore, the capacitances at the electrode side of the pore have a decisive larger ionomer resistance connected in front. When decreasing the frequency, the probing signal penetrates deeper into the pore. Accordingly, this results in an equal increase in imaginary and real part in the Nyquist plot (45° phase shift) followed by a straight 90° line (purely capacitive behavior), as shown in Fig. 3A.
However, taking a further look at the experimental spectrum in Fig. 3A in the mid-frequency range, an obvious non-ideal behavior is depicted as a significant deviation from the 90° degree inclination in the Nyquist representation of the impedance data. This can mainly be attributed to the pore radius distribution in the real  36 . Pores of different sizes possess different penetration depth for the applied potential perturbation and, therefore, possess different impedance responses. The parallel connection of these different impedance responses leads to the observed non-ideal behavior. In this work, this is taken into account by using experimental pore radius distribution (Fig. 2C) and using the parallel connection of single pores of different sizes. This significantly improves the fitting, as shown in Fig. 3A.
Considering liquid electrolytes, Robert de Levie assumed pores being completely filled with an electrolyte. This however is not valid for solid electrolytes (ionomer) applied in PEFCs. For PEFCs, a three phase boundary is present in the CCL. According to scanning electron microscopy imaging, the ionomer is assumed to form a homogenous thin film at the pore surface 34 . In addition, ionomer is distributed heterogeneously within the CL forming films of varying thickness 39 . In order to maintain the distribution of single pore's ionic resistance and to account for the ionomer film thickness distribution within the CL, a pore radius dependent ionomer film thickness (t ion = x*r pore ) is introduced. According to the ionomer to carbon ratio (I/C) and the density of the carbon structure in the electrode and of the ionomer, the corresponding volume of ionomer in the CCL, V ionomer , is calculated and is used together with the determined pore surface area to quantify the ionomer film thickness proportionality factor x (for further details see supplementary information). With respect to the reduction of ionomer cross section area, A CS , compared to completely filled pores, the CCL's ionomer conductivity determined by fitting is reduced according to the definition of conductivity (κ = 1/R*l eff /A CS ). Here R represents the measured impedance and l eff the effective length. Figure 3C displays the EC accounting for the fuel cell's ohmic resistance R ohmic and the impedance of the catalyst layer Z CCL , which is determined by the use of de Levies analytical formula and the use of the experimental CL's pore number and ionomer distribution. A detailed derivation of the pore number distribution is given in the supplementary information. In the following, the model being described by this EC will be referred to by pore . The PND-model shows good fitting performance in the high-and mid-frequency region, outperforming other models used in the literature (see Fig. 3A). However, it must be noted, that the fitting in the low frequency region is rather far from the experimental data (see Fig. 3B). Therefore, further possible processes at the cathode side should be taken into account.
At the applied working DC-potential close to U = 0.5 V specific adsorption of functional sulfonate groups of the ionomer on Pt-electrocatalysts and specific adsorption of *OH/*O adsorbates coming from water should be taken into account 15,35 . Accounting for the adsorption process, a series connection of adsorption capacitance C ads and adsorption resistance R ads is introduced (see Fig. 3D). Namely this combination of elements represents physical equations describing so-called surface limited and quasi-reversible specific adsorption of anionic adsorbates 33,40,41 . Here, R ads = −1/(∂i ads /∂E) and C ads = −q a (∂θ ads /∂E); q a is the charge necessary to form an adsorbate layer, i ads is the Faradaic current due to e.g. sulfonate group adsorption/desorption, θ is the fractional adsorbate surface coverage which "oscillates" around quasi-steady-state values at a given potential under AC-probing. In addition, the occurrence of other ("continuous" in contrast to the surface limited, e.g. hydrogen oxidation) Faradaic side reactions has to be taken into consideration. Therefore, a resistance accounting for possible side reactions, R SR = −1/(∂i SR /∂E), is introduced (i SR is the Faradaic current due to side reactions) in parallel, which could be caused by hydrogen cross over through the membrane or traces of oxygen in the nitrogen gas. R SR and series connection of C ads and R ads (classical physical model of reversible surface limited adsorption 42 ) are connected in parallel to the catalyst layer model Z CCL as shown in Fig. 3D. The parallel connection is justified by double layer charging, sulfonate adsorption and possible side reactions occurring simultaneously with different time constants. The model being described by the EC accounting for possible side reactions and sulfonate adsorption is in the following referred to by PASR-model (parallel adsorption and side reaction model). Figure 4 shows the improved fitting performance of the PASR-model at the whole frequency range for different relative humidities, as further discussed in the next sections. One should note that effectively, the applied model has only 6 independent fitting parameters (R mem , κ CL,ion , c dl, R SR , C ads and R ads ) and each of them has clear physical meaning, i.e. no constant phase elements or additional semi-empirical diffusion elements. This is achieved by the use of the experimental pore size distribution and careful accounting for the most probable Faradaic reactions. Table 2 gives an example of values of the fitting parameters obtained by fitting an impedance spectrum. The values of the fitting parameters are in good accordance with literature 43,44 . Please, see the supporting information for a relatively simple MATLAB code used to implement the model.

Model verification under different experimental conditions.
Fitting of impedance spectra. Properties of the CCL (i.e. ionic conductivity κ CL,ion and specific double layer capacitance c dl obtained from the fitting) were   Eikerling 19 , the developed models perform very good fitting of measured data in the high frequency region (see Fig. 3). Figure 3A shows the superior fitting in the mid-frequency region of the PND-model compared to de Levie's and Eikerling's model. With the introduction of C ads , R ads and R SR (see Fig. 3D), accounting for sulfonate adsorption and possible side reactions, the PASR-model also performs excellent fitting in all frequency regions (see Fig. 3B). Figure 4 shows impedance data measured at different relative humidities. The PASR-model demonstrates excellent fitting for humidities up to RH = 60%. At higher RHs the modeled impedance deviates slightly from the measured one, however within few percent of root-mean-square deviation. It is most likely due to the fact that at high RH, ionomer structure can change, resulting in a deviating double layer formation and deviating conductive ionomer channels in the porous structure. Moreover, water formed by means of capillary condensation can create additional double layer surfaces in pores not being covered with ionomer. In addition, water droplets formed in the cell hardware can influence the measured impedance data. In the following sections, a closer look is taken towards the dependencies of the double layer capacitance and the ionic resistance of single pores. In the supplementary information, examples of the fitting related the applicability of the model in the case of structurally different MEAs are also given. Double layer capacitance. The single pore's double layer capacitance C pore,dl at the I/C-interface is given by the product of pore's surface area A S and CCL's specific double layer capacitance c dl , which was determined by fitting the impedance data. According to the assumption of cylindrical pores of constant length, the pore's surface area (A S = 2π*r pore *l pore ) and C pore,dl possess linear dependences on the pore radius (see Fig. 5A). With regard to the increasing surface area larger pores possess a higher capacitance. Figure 5B displays the sum of double layer capacitances of all pores of a given size as a function of pore radius. Although the capacitance of the single pores increases with increasing radius, pores in the range of 30 nm have the highest impact towards the overall double layer capacitance of the CCL. This can be explained by means of the CCL's PRD having its peak pore number in the range of 30 nm (see Fig. 2D). Note however, that the shapes of the curves shown in Figs 2D and 5B are not the same: significant "non-trivial" differences are clearly seen. Figure 5A,B also shows an increase of specific double layer capacitance with increasing relative humidity. Possible explanations for this effect can be the appearance of additional capacitance from capillary condensation 45 , non-uniform change in the dielectric constant of the double layer 46,47 , or geometric expansion of the ionomer film 48 .
Pore resistance. The ionic resistance of a single pore is given by R pore,ion = 1/ κ CL,ion *l pore /A CS , where the cross sectional area of the ionomer is given by A CS = π•r 2 •(2x-x 2 ). According to this definition, the ionic resistance of a single pore is an inverse quadratic function of pore radius (see Fig. 5C).
The single pore's ionic resistance increases with decreasing radius. Figure 5D displays the resistance of a parallel connection of all pores of a given size. Pores in the radius range of around 30 nm possess the lowest proton resistance, thus dominating the CCL's proton resistance. This can once again be explained by means of the CCL's PRD.

Conclusions
An all-physical-element impedance model for PEFCs at the presence of hydrogen at the anode and nitrogen at the cathode has been developed. The catalyst layer impedance modeling is based on five pillars, which include application of the de Levie's formula to describe a single pore's impedance, experimental quantification of pore number and radii, considering ionomer distribution and taking into account specific adsorption of sulfonate groups at the surface of the Pt-catalyst as well as the occurrence of side reactions such as hydrogen oxidation. It should be emphasized that all parameters used in the model possess physical meaning. Thus, the model is predestinated for PEFC's CCL analysis, by means of fitting CCL's characteristic parameters like ionomer conductivity or specific double layer capacitance. Notably, the model has only 6 independent parameters for the fitting. Possible fields of application range from benchmarking and receiving inspection to development-, aging-and failure-analysis. Initial results using one of the state-of-the-art fuel cells at BMW show very promising results. The model performs good fitting in the whole frequency spectrum of impedance data of a PEFC operated in H 2 / N 2 mode. Considering the assembly of all pores of a given radius (parallel connection), in the particular example used in this work, pores in the radius range of 30 nm have the highest impact towards the CL's overall double layer capacitance and proton conductivity. Data availability. All data including MATLAB code for the fitting are available from the Authors upon request.