Transmission Line Impedance Models Considering Oxygen Transport Limitations in Polymer Electrolyte Membrane Fuel Cells

The experimental data used for this work were obtained using two distinct cells showing significantly different behavior in terms of high frequency impedance.

EIS with a blocked electrode is used as a complement to in operando characterization to investigate PEMFC cathode structure and transport properties. For instance, Makharia et al. studied the effect of the ionomer concentration in the catalyst layer on electrolyte resistance³ and Gaumont et al. measured the proton resistance of catalyst layers as a possible marker of their degradation.²⁴ Figure 2 shows a Nyquist plot of a PEMFC impedance spectrum obtained with a blocked electrode characterized by a straight line at high frequencies, generally associated to proton conduction losses within the CCL.^14–18 The slope higher than 45° is in agreement with inhomogeneous ion transport properties through the CCL expected for thick electrodes which is the case for cell B (MEA self-assembled by hot pressing). At low frequencies, the imaginary part of the impedance increases steeply so that it approaches a vertical line associated with a purely capacitive behavior. However, significant deviation from the vertical line can still be observed, which is generally explained by the inhomogeneous distribution of pore radii within the electrode.²⁶ Each pore has a different penetration depth and therefore a different transport impedance response depending on the applied signal.

Zoom In Zoom Out Reset image size

Impedance spectra of blocked electrodes are very well represented using transmission line models. TLM were first introduced by de Levie in 1963 to analyze the electrochemical behavior of porous electrodes.^14,27 Figure 3 shows a discretized transmission line EEC, which is based on the assumption that the electrode consists of an assembly of horizontal cylindrical pores of identical diameter and homogeneous electrical properties (i.e. C^δ_dl and σ_ion). Each pore has a length equal to the thickness of the electrode and contains an ion conducting phase (electrolyte) in the upper rail of the circuit and an electron conducting phase (carbon and Pt agglomerates) in the lower rail.

Zoom In Zoom Out Reset image size

In the case of carbon supported catalyst layers, the resistance of the carbon/Pt agglomerates is assumed negligible so that the EEC consists only of a parallel network of double layer capacitances C^δ_dl connected via ionomer linear resistances 1/σ_ion. In the MEA equivalent circuit, the parameters of the transmission line are usually replaced by the average macroscopic parameters of the catalyst layer: C_dl = C^δ_dlδ, with δ the electrode thickness, and (Figure 3, bottom).¹³ In addition, the MEA equivalent circuit includes an ionic resistance R_mem representing the membrane and an electronic resistance R_{GDL + FF} accounting for the Flow Field (FF) plates, GDL and possibly other contact resistances. Note that the impedance of the anode is generally neglected due to the fast kinetics of the H₂ oxidation reaction compared to the O₂ reduction reaction.

Ultimately, the global impedance of the cell can be written as:³

Where the high frequency resistance is defined as R_HF =  R_mem + R_{GDL + FF}. ω is the angular frequency and i the imaginary unit. Equation 1 was obtained under the assumption of homogeneous electrical properties, as mentioned above.

Equation 1 can be simplified in the high frequency domain, considering that:

And in the low frequency domain, knowing that :

Therefore, R_HF and R_ion can be easily estimated from the profile of the Nyquist plots. The high-frequency resistance R_HF is given by the intersection of the impedance spectrum with the x-axis and the intersection between the low-frequency line and the x-axis is R_ion/3 + R_HF, as shown in Figure 2.

The ionic resistance R_ion  plays an important role in the electrochemical behavior of PEMFC cathodes, although the main contribution to performance losses remains the slow ORR kinetics. R_ct and the double layer capacitance C_dl, are the two main parameters characterizing the ORR kinetics. To study their impact on the FC impedance, we performed in operando EIS (under H₂/air) with both cells. An example of corresponding experimental spectra is shown in Figure 4.

Zoom In Zoom Out Reset image size

Figure 4 (top) shows the Nyquist plot of impedance data measured with cell A. This impedance spectrum is composed of two intricated loops: the high frequency loop is governed mostly by electrochemical reaction kinetics and the low frequency loop by oxygen transfer limitations. The EEC typically associated with such FC spectra is the Randles circuit,^28,29 shown in Figure 5. In a more general way, this EEC is commonly used to interpret the impedance of thin electrodes. It is composed of a charge transfer resistance R_ct in parallel with a double layer capacitance C_dl characterizing the electrochemical reaction, in series with a high-frequency resistance R_HF. Finally, a Warburg impedance Z_W connected in series to R_ct accounts for oxygen transport limitations within the cathode catalyst layer. This is compatible with an electrode description based on agglomerate models but this is intrinsically in contradiction with the thin film assumption of the Randles EEC.

Zoom In Zoom Out Reset image size

The impedance of this EEC is given by:

With,

In Equation 5, R_d is the diffusion resistance and τ the time constant characterizing the processes of oxygen diffusion through the catalyst layer.

The Randles circuit cannot be used to fit and analyze spectra such as that of Figure 4 (bottom), obtained with cell B, because it does not consider the high frequency straight line associated with the ionic resistance through the porous structure of the electrode. This must be done using a TLM-like model. Eikerling and Kornyshev¹⁸ applied a TLM model (similar to that developed by De Levie for blocked electrodes) to operating FC and calculated the impedance analytically for low currents in the absence of oxygen supply limitations. Later, Makharia et al.³ proposed a TLM model neglecting oxygen transport limitations and showed that this model and the physical model of Eikerling and Kornyshev were equivalent for low currents. More recently, Cruz-Manzo and Chen²⁵ derived a TLM like model for intermediate and high currents including a Warburg element to account for oxygen diffusion and a charge transfer resistance in parallel with a constant phase element to account for the ORR kinetics.

In order to analyze our impedance spectra, we modified the TLM model of Figure 3 regarding charge transfer during the ORR -introducing R_ct - and mass transfer through the CCL -introducing Warburg elements in series with R_ct- as shown in Figure 6. Note that although TLM are more detailed than Randles EEC and can take some aspect of the electrode 3D structure into account, such impedance models must not be mistaken for agglomerate models, which are among the most detailed physical description of charge and mass transport through Gas Diffusion Electrode (GDE).

Zoom In Zoom Out Reset image size

Ultimately, the global impedance of the cell can be described as:

Where Z_CCL(ω) stands for the modified expression of the TLM impedance accounting for charge transfer and mass transport through the cathode catalyst layer. Similar expressions were already used in the literature²¹ but to the best of our knowledge, none of them took mass transfer into account using a Warburg element or proposed simple analytical expressions. However, Eikerling and Kornyshev considered Knudsen oxygen diffusion through the electrode.¹⁸ Note that Equation 6 was obtained under the hypothesis of a homogeneous current density j through the thickness of the catalyst layer, so that the charge transfer resistance can also be assumed homogeneous (R_ct = b/j, with b the Tafel slope^30,31). In a more general manner, all parameters appearing in a TLM-like model are considered homogeneous when analytic solutions are proposed. This is clearly a strong limitation of these approaches but on the other hand, their ability to predict the appearance of the 45° high frequency line shows that the homogeneous parameter hypothesis may not be far from reality, at least in some operating conditions. Finally, one can assume that deviation from ideal behavior, i.e. high frequency lines with slopes different than 45° are due to non-homogenous conditions through the catalyst layer.¹³ In our case (Figure 4 and in the following), the slope was close to 48°.

Occurrence of the 45° high frequency line

In this section we consider the conditions for which the 45° straight line appears in EIS Nyquist plots. These conditions depend on the reaction kinetics parameters, R_ct and C_dl, as well as on the ionic resistance R_ion and the angular frequency ω. To simplify the equations, the oxygen transport limitations (i.e. the Warburg impedances) are not taken into account in the following, which can be easily justified by their negligible impact at high frequencies. Figure 7 shows impedance spectra computed for different values of R_ion, the other parameters remaining unchanged and close to values typically measured with our cells (R_ct = 0.13 Ω.cm² and C_dl = 0.0562 Ω/cm²). We can see from this figure that there is a threshold value of the angular frequency above which the 45° straight line is always present. Furthermore, the value of this angular frequency threshold increases as R_ion decreases. That is the reason why the 45° straight line may not be observed for FC with a high ionomer content in the cathode catalyst layer and/or thin electrodes. On top of that, a review of existing literature shows that this straight line is observed in most of, if not all cases, with home-made MEA assembled by hot pressing^12,13 which is consistent with this numerical analysis.

Zoom In Zoom Out Reset image size

To understand the relationship between the ion transport resistance through the CCL and the extent of the 45° line in the Nyquist plot, we consider the dimensionless ratio K between the ionic resistance R_ion and the impedance of the other elements of the branch Z(R_ct||C_dl) characterizing the reaction kinetics. In Equation 7, Z(R_ct||C_dl) is function of the angular frequency ω:

And,

If ||K|| < 1 at the highest scanning frequency, ion transport has a low impact on the cell impedance compared to reaction kinetics and the 45° straight line will not appear in the Nyquist plot and the shape of the impedance spectra approaches that of usual Randles circuits. If ||K|| is close to one or higher, the influence of ion transport on the impedance spectra will be visible above a threshold value of the angular frequency ω_c. In the example shown in Figure 7, ω_c is reached for ||K|| ≈ 7 independently of the values of the impedance parameters, such as:

Therefore, if ion transport is negligible compared to the reaction kinetics limitations, the threshold frequency ω_c could be sufficiently high so that the 45° line is not visible in the Nyquist plot. An important consequence of this result is that the modified TLM model presented in Figure 6 corresponds to a general EEC of FC impedance that can be used as well with volumetric, as with thin CCL, since it tends toward the Randles circuit when R_ion is sufficiently low.

This can be demonstrated starting from Equation 6 and considering the first order Taylor expansion of . Thus when, R_ion → 0, and the modified TLM impedance becomes:

Which corresponds to the cathode catalyst layer resistance Z_Randles of the Randles EEC given in Equation 4. To illustrate this convergence between TLM-like and Randles EEC, Figure 8 shows experimental data that were obtained with Cell A. We can see that both models allow to fit the spectrum similarly with identical residuals (2.13 10⁻⁴ for the both models), although there is no 45° straight line at high frequencies. As a consequence, the value of the ionic resistance is close to zero, but the other parameters are identical with both models (Table I).

Zoom In Zoom Out Reset image size

Table I. Estimated parameters obtained with the experimental data and models in Figure 8.

	RHF	Rct	Cdl	Rion	Rd	τ
	(Ω.cm2)	(Ω.cm2)	(F/cm2)	(Ω.cm2)	(Ω.cm2)	(s)
Randles EEC with ZW in CCL σ = 2.13 × 10−4	0.0893	0.1240	0.0302	-	0.0754	0.0671
TLM with ZW in CCL σ = 2.13 × 10−4	0.0893	0.1240	0.0302	1.09 × 10−8	0.0754	0.0671

Location of the main mass transfer resistance

We have just shown that the modified TLM in Figure 6 is equivalent to the usual Randles EEC when the ionic resistance through the catalyst layer is negligible, which is the case for instance with a surface electrode. This leads us to the conclusion that the Warburg impedance in the Randles circuit corresponds to an oxygen transport resistance that is physically located in the catalyst layer, which is inherently in contradiction with the hypothesis of a surface electrode, often put forward with Randles EEC. Significant mass transfer resistance may appear through the gas diffusion layer and the most appropriate way to take them into account consists in using a modified Randles EEC with the Warburg impedance connected in series to the reaction kinetics parameters R_ct and C_dl, such as presented in Figure 9 (top). Of course, the same approach can be followed with TLM-like EEC such as shown in Figure 9 (bottom). It must be noted that both EEC in Figure 9 consider oxygen transport resistance in the GDL or CCL only, although it would still be possible, at least from a mathematical point of view, to keep Warburg elements in series with the charge transfer resistance R_ct (and in parallel with C_dl) to account also for oxygen transport resistances through the CCL. Nevertheless, these two mass transfer resistances -through the CCL and in the GDL- would reveal highly correlated. This is the reason why, in this work, we limited ourselves to four options:

• Usual Randles EEC (Figure 5) with Warburg impedance in the CCL vs. modified Randles EEC with Warburg impedance in series -Figure 9 (top)-, applied to impedance spectra measured with cell A (see the experimental section), i.e. without a straight line at high frequencies.
• TLM-like EEC (Figure 6) with Warburg impedance in the CCL vs. TLM-like EEC with Warburg impedance in series -Figure 9 (bottom)-, applied to impedance spectra measured with cell B (see the experimental section), i.e. with a straight line at high frequencies.

Zoom In Zoom Out Reset image size

The data and graphs in Figure 9 show that both approaches -Warburg elements in the CCL or in series (in the GDL)- allow to fit rather well the experimental spectra. This is the case with cell A (Randles EEC) as well as with cell B (TLM-like EEC). The standard deviations are similar, although consistently lower when the Warburg impedance is in the CCL. The corresponding values of the impedance parameters are given in Table II. In both cases, their order of magnitude is consistent with those usually encountered in PEMFC.^3,28,31,32 However, despite of their ability to fit equally well the experimental data, both approaches cannot be considered as equivalent since the corresponding values of the kinetic parameters differ significantly (i.e. up to 40%). The fit quality alone does not allow a clear conclusion about the most appropriate EEC and thus about the location of the main mass transfer impedance: a more detailed analysis of the diffusion impedance parameters (R_d and τ) is necessary. Since they are related to the physical parameters governing charge and mass transfer through the MEA³¹ it is possible to estimate the thickness of an equivalent diffusion layer starting from R_ct, R_d and τ:

Equation 11 can be derived considering that R_ct = b/j_f, R_d = bδ/4FD_effc*_O2 and τ = δ²/D_eff,^25,31 with j_f the current density (0.5 A/cm² in our case), F the Faraday constant, D_eff the effective oxygen diffusion coefficient and c_O2 the oxygen concentration at the GDL/channel interface (air at 1 atm and 333 K in cell A or 343 K in cell B). c*_O2 in the expression of R_d is the oxygen concentration at the reaction sites. Solving Fick equation in steady-state, gives c*_O2 = c_O2 − j_fδ/4FD_eff.²⁹

Table II. Estimated parameters starting from the experimental data and models in Figure 9.

	RHF	Rct	Cdl	Rion	Rd	τ	δ
	(Ω.cm2)	(Ω.cm2)	(F/cm2)	(Ω.cm2)	(Ω.cm2)	(s)	(μm)
Cell A: Randles EEC
ZW in CCL σ = 3.55 × 10−4	0.086	0.132	0.034	-	0.097	0.066	265
ZW in GDL σ = 4.65 × 10−4	0.078	0.102	0.053	-	0.135	0.077	230
Cell B: TLM EEC
ZW in CCL σ = 2.35 × 10−4	0.069	0.108	0.034	0.234	0.122	0.066	218
ZW in GDL σ = 2.79 × 10−4	0.071	0.074	0.057	0.191	0.172	0.073	183

Note that this expression of the thickness δ 11 of the equivalent diffusion layer is independent of the effective diffusion coefficient so that no hypothesis has to be made about the nature of oxygen diffusion: wet air through the pores of the GDL and/or CCL or ionomer layer covering the reaction sites. The values of δ estimated with Equation 11 for both approaches (i.e. Warburg elements in the CCL or in the GDL) and both cells (A with Randles EEC and B with TLM) are given in Table II. It can be clearly seen that in all cases, the estimated values are of the order of magnitude of the GDL thickness (200 μm and 150 μm for cells A and B, respectively) rather than the CCL thickness (about 10 μm) or the very thin ionomer film covering the catalyst sites (about 5–10 nm). These results thus are in favor of a Warburg element in series with the reaction kinetic parameters, i.e. accounting for oxygen diffusion through the GDL.

At that point, it is worth mentioning that locating the main mass transfer resistance(s) is one of the many complications linked with the low frequency part of fuel cell impedance spectra. Firstly, because the low frequency -apparent- impedance is also strongly impacted by oxygen concentration oscillations induced by the measuring signal.^30,33,34 Then, because the derivation of the Warburg impedance supposes Fickian oxygen diffusion only, while, at least in a fuel cell fed with air, Stefan-Maxwell equations should be used as in most stationary models.^35,36 In this regard, oxygen diffusion is strongly dependent on the overall convective flux, and thus on water management.³⁷ Such issues are however out of the scope of this work. That is the reason why only the order of magnitude of the diffusion thickness δ was considered in this discussion.

The main conclusion of this work is that TLM-like EEC modified to consider charge transfer resistance as well as oxygen transport resistance can be used instead of Randles EEC to model PEMFC CCL impedance since they correspond to a general representation: the impedance of such TLM-like EEC tends toward that of a Randles EEC when the ion transport resistance through the catalyst layer becomes negligible compared to the ORR kinetics parameters, i.e. the charge transfer resistance and the double layer capacitance. Similarly, the occurrence of the straight line at high frequencies, depends on the ratio between the ionic resistance and the elements characterizing the reaction kinetics. This is the reason why the straight line is always observed with blocked electrodes, but appears only sometimes during fuel cell operation, i.e. with thick electrodes or electrodes with low ionomer content.

Finally, these considerations lead us to the conclusion that the Warburg impedance in the Randles circuit corresponds to an oxygen transport resistance that is physically located in the cathode catalyst layer, which is in contradiction with the hypothesis of a surface electrode that governs the derivation of this EEC. Assuming that the main oxygen transport limitation is in the GDL would be more consistent with the Randles EEC, but this hypothesis means that the Warburg impedance should be connected in series with the charge transfer resistance and double layer capacitance. We have seen that such a modification has a significant impact on the values of the kinetics parameters that can be identified from experimental impedance spectra, with Randles EEC, as well as with TLM-like EEC.

J. Mainka 0000-0003-1270-0023

O. Lottin 0000-0002-4607-203X