PEM Single Cells under Differential Conditions: Full Factorial Parameterization of the ORR and HOR Kinetics and Loss Analysis

In this work we used 12 cm² Gore® PRIMEA® MEAs (W.L. Gore & Associates, Elkton, MD, USA) with a membrane thickness of 18 μm and platinum loadings of 0.4 mg_Pt cm⁻² in the cathode and 0.05 mg_Pt cm⁻² in the anode. Figure 2 superposes typical cathode and anode voltammograms for such an MEA. The roughness factors rf were calculated by integrating the colored H_UPD areas and averaging over the adsorption and desorption processes (see also Fig. S1 of the supplementary material, available at stacks.iop.org/JES/169/014503/mmedia). The results were rf_cathode = 166 ± 11 ${{\rm{m}}}_{\mathrm{Pt}}^{2}$ m${}_{\mathrm{geo}}^{-2}$ and rf_anode = 30 ± 3 ${{\rm{m}}}_{\mathrm{Pt}}^{2}$ m${}_{\mathrm{geo}}^{-2}$, determined over at least 10 CVs of each electrode.

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The 12 cm² MEAs were laminated into polyethylene naphthalate frames at about 100 °C. The MEAs were sandwiched between Sigracet® 22BB gas diffusion layers (GDL) from SGL® Carbon and mounted in a differential cell setup from Baltic FuelCells (quickConnect®) with straight graphite flow fields. The equivalent clamping pressure on the active area was about 1.3 MPa.

For all our tests we employ a fully automated FuelCon Evaluator C50 test station. The cell temperature is measured by a type K thermocouple and regulated by a Huber Ministat 125 thermostat. For the EIS measurements, a Gamry reference 3000 plus Booster 30 K device in combination with the test station load were used and high precision was ensured by using a four-point measurement setup. More information about our setup is given in our previous publication. ⁷

Our full factorial test run was carried out in four stages:

• 1.  
  Cell conditioning.
• 2.  
  Proton pump (H₂/H₂) and blocking cathode (H₂/N₂) experiments including CV and EIS measurements.
• 3.  
  Fuel cell (H₂/O₂) experiments including polarization curves and EIS. Second separate run with heliox (O₂ and He mixtures in the cathode).
• 4.  
  Separate test run in H₂/O₂ mode for limiting current measurements.

The conditioning was done in the H₂/air mode at p_total = 1.5 bar_a, rH = 100%, T = 50 °C, and with H₂/air flows of 1300/2000 sccm, according to the procedure described by Harzer et al.: ⁴¹ 0.6 V during 45 min, OCV during 5 min, and 0.85 V during 10 min. The cell performance was stable after repeating this ten times ( ≈ 10 h) and then operating the cell at 0.3 V for 2 h.

The proton pump and blocking cathode measurements allowed us to characterize the proton conductivities, the hydrogen permeation coefficient and also the anode resistance in a full factorial manner under system relevant conditions. We varied the operating conditions from rH = 30% to rH = 100% and T = 50 °C to T = 80 °C at p_total = 1.5 bar_a, leading to 32 operating conditions. For each of these conditions, EIS and CVs were recorded in H₂/N₂ mode and EIS and polarization curves in H₂/H₂ mode. The sequence was as follows for each humidity/temperature condition:

• 1.  
  Setting the temperature and the dew point (from low to high dew points) in H₂/N₂ mode and letting the cell equilibrate for 2 h before starting the measurements in order to ensure a steady state.
• 2.  
  Carrying out the polarization curves and the potentiostatic EIS at OCV in H₂/H₂ mode (proton pump) for the anode impedance from f = 100 kHz to f = 0.1 Hz with 10 points per decade and an amplitude of 4 mV.
• 3.  
  Switching back to H₂/N₂ mode, waiting for 10 min and carrying out potentiostatic EIS for R_Ω and R_p. Then, recording CVs to determine the hydrogen permeation properties, the roughness factor rf (or electrochemical surface area, ECSA) and the double layer capacity. This step was explained in detail in our previous publication. ⁷ Hereafter, going back to the first step with the next humidity and temperature condition.

In a second measurement campaign in H₂/H₂ mode we also implemented recovery steps to avoid CO-poisoning of the catalyst layers (see below for further details).

The main test run in fuel cell mode was done with 7 humidity conditions, from rH = 30% to rH = 100%, 4 temperatures (T = 50 °C to T = 80 °C), 4 concentrations on the cathode side (${x}_{{{\rm{O}}}_{2}}^{\mathrm{dry}}=1$, ${x}_{{{\rm{O}}}_{2}}=0.5$, ${x}_{{{\rm{O}}}_{2}}=0.25$, ${x}_{{{\rm{O}}}_{2}}=0.16$), and 16 current densities (j = 2 A cm⁻² to j = 4 mA cm⁻²), leading to 1792 operating points. For our EIS measurements we chose 10 currents out of the 16 (j = 2 A cm⁻² to j = 20 mA cm⁻²), leading to 1120 spectra. The constant dry flow rates were 1300 sccm on the anode side and 2000 sccm on the cathode side and the system pressure was p_total = 1.5 bar_a, leading to pressure drops Δp < 100 mbar for all the conditions. To vary ${p}_{{{\rm{O}}}_{2}}$ we mixed O₂ with N₂ ahead of the humidification bubbler, and we also conducted heliox measurements in a separate experiment for rH = 30% and rH = 80%. The run was carried out in the order of increasing dew points and from high currents to low currents. Whenever adjusting the temperature or the dew point, the cell was stabilized in H₂/air at 0.7 V for 2 h. The currents at which we performed EIS were stabilized for 25 min prior to the measurement to ensure high measurement quality, and the other currents were held for 5 min. The impedance measurements were carried out in hybrid mode with 11 mV AC amplitude from f = 100 kHz to f = 0.1 Hz and with 10 points per decade. We also calculated the distribution of relaxation times to deconvolute the physicochemical processes, given by

$$\begin{eqnarray}&&{Z}_{\mathrm{DRT}}(f)={R}_{{\rm{\Omega }}}+{\int }_{-\infty }^{+\infty }\displaystyle \frac{\gamma \mathrm{ln}(\tau )}{1+i2\pi f\tau }{\rm{d}}(\mathrm{ln}\tau )\end{eqnarray} \tag{ 2 }$$

For this, we used the tool by Wang et al. ⁴² which we have adapted to allow an automated evaluation of multiple spectra at the same time (See Figs. S2a to S2d).

Finally, we performed steady-state limiting current (SLC) measurements according to the method proposed by Baker et al. ³⁸ and transient limiting current (TLC) measurements as proposed by Göbel et al.. ⁴⁰ We chose to carry out the SLC measurements for 4 pressure levels (p_total = 1.5, 2, 3, 4 bar_a), 3 humidity levels (rH = 30, 80, 100%), 4 temperature levels (T = 50, 60, 70, 80 °C) and 4 concentrations (${x}_{{{\rm{O}}}_{2}}=0.02,\,0.05,\,0.1,\,0.15$). For stabilizing the operating conditions, the same procedures and times as mentioned above were used. Before going to the limiting current, the cell was held at 0.7 V for 8 min at given pressure, humidity, temperature and concentration. Then, the voltage was set to 0.3 V for 5 min, 0.2 V for 5 min and 0.15 V for 5 min and the limiting current was determined from these three operating points. The TLC measurements were carried out for ${x}_{{{\rm{O}}}_{2}}=0.16$ and ${x}_{{{\rm{O}}}_{2}}=0.25$ at p_total = 1.5 bar_a by conditioning the cell current for 10 min and going to the limiting current for 10 s in the potentiostatic mode. Since in limiting current operation the conditions were not differential anymore, we considered a log-averaged O₂ concentration to calculate the transport resistance R_MT.

An overview on how we ensured good measurement quality (state-of-health checks and Kramers-Kronig test) is given in the supplementary material of this work (see Fig. S3). Furthermore, we show the methodology used to extract the kinetic parameters for the anode and cathode kinetics based on a flowchart in Fig. 11 in the appendix.

In order to carry out loss analyses, the Ohmic resistance R_Ω containing the electronic losses and the protonic losses of the membrane, as well as the effective protonic resistance of the CCL ${R}_{{\rm{p}}}^{\mathrm{eff}}$, have to be known for each condition. These two resistances cause a voltage loss that is proportional to the cell current density as ${\rm{\Delta }}U=j\cdot ({R}_{{\rm{\Omega }}}+{R}_{{\rm{p}}}^{\mathrm{eff}})$. Further, it is essential to correct the cell current for the hydrogen crossover current ${j}_{{{\rm{H}}}_{2},\mathrm{crossover}}$ for accurate Tafel analyses of the ORR at small current densities. We characterized these ionomer properties extensively in our previous publication and provide a brief summary of the parameterized models in the supplementary material (see Fig. S4). ⁷ Also, the EIS spectra of an rH-variation under load for two currents and their DRTs are shown exemplarily in Fig. S5 to illustrate the importance of the ionomer humidification on the performance .

Since the Pt-loading in our anode is eight times lower than the loading in the cathode, we did not expect the anode overpotential to be completely negligible during normal fuel cell operation, even though the Pt-area specific exchange current density of the HOR is several orders of magnitude higher than the one of the ORR. Thus, we carried out measurements in the H₂/H₂ configuration to investigate the anode contribution to the overall cell performance. Due to the fast diffusion of H₂ from the channel to the ACL and the thinness ( ≈ 2 μm) of the electrode in our system, we neglect mass transport and assume that the anode impedance is limited by proton transport within the ACL and by the electrochemical oxidation of hydrogen according to

$$\begin{eqnarray}&&{{\rm{H}}}_{2}\rightleftharpoons 2{{\rm{H}}}^{+}+2{{\rm{e}}}^{-}\end{eqnarray} \tag{ 3 }$$

In previous works, the H₂ kinetics was generally assumed to be describable by a sequence of two out of the three following reaction pathways:

$$\begin{eqnarray}&&\mathrm{Tafel}:\quad {{\rm{H}}}_{2}\rightleftharpoons 2{{\rm{H}}}_{\mathrm{ad}}\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&\mathrm{Heyrovsky}:\quad {{\rm{H}}}_{2}\rightleftharpoons {{\rm{H}}}_{\mathrm{ad}}+{{\rm{e}}}^{-}+{{\rm{H}}}^{+}\end{eqnarray} \tag{ 5 }$$

$$\begin{eqnarray}&&\mathrm{Volmer}:\quad {{\rm{H}}}_{\mathrm{ad}}\rightleftharpoons {{\rm{H}}}^{+}+{{\rm{e}}}^{-}\end{eqnarray} \tag{ 6 }$$

There have been many studies focusing on nailing down the exact reaction mechanism and trying to find out whether the hydrogen adsorption on the metal or the electron transfer step is dominant (see references in Table I). The latest research of Stühmeier et al. showed that a Tafel-Volmer mechanism is most likely to occur within the anode of PEM fuel cells at small overpotentials. ¹⁴ However, the HOR is often described by a single Butler-Volmer (BV) law in the modeling literature for the sake of simplicity. Moreover, since the kinetics are typically operating in the linear regime over a large current density range, the Butler-Volmer expression can even be linearized. Thus, the overall voltage loss contribution from the anode reaction (in fact including the effect of proton transport limitations in the ACL) is written as

$$\begin{eqnarray}&&{\eta }_{\mathrm{anode}}=\displaystyle \frac{{RT}}{{{Fj}}_{{\rm{a}},\mathrm{eff}}^{0}({\alpha }_{c}+{\alpha }_{a})}\cdot j\end{eqnarray} \tag{ 7 }$$

where both the current density j and the effective exchange current density ${j}_{{\rm{a}},\mathrm{eff}}^{0}$ are referenced to the geometrical surface area. The sum of anodic and cathodic transfer coefficients (α_c + α_a ) cannot be determined from the linear regime and is set to unity in this case. ¹⁴ Table I summarizes various parameterizations of the HOR/HER from the literature that were referenced to the platinum surface area for the sake of comparableness (rf = 30 ${{\rm{m}}}_{\mathrm{Pt}}^{2}$/m${}_{\mathrm{geo}}^{2}$ for our measurements). However, due to the likely inhomogeneous through-plane-distribution of the reaction rate, this approach to reference the exchange current density presumably introduces errors and makes the comparison between values from different sources difficult. The above expression of the overpotential leads to the impedance of the charge transfer process

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\eta }_{\mathrm{anode}}}{\partial j}={R}_{\mathrm{anode}}=\displaystyle \frac{{RT}}{{{Fj}}_{{\rm{a}},\mathrm{eff}}^{0}}\,\to \,{j}_{{\rm{a}},\mathrm{eff}}^{0}=\displaystyle \frac{1}{{R}_{\mathrm{anode}}}\cdot \displaystyle \frac{{RT}}{F}\end{eqnarray} \tag{ 8 }$$

which does not depend on the current density. This means that the exchange current density can be parameterized by EIS and that measurements at OCV are sufficient. Further, it means that the Tafel slope (symmetry factor) is not accessible from these measurements and that we get no information regarding the rate limiting step.

Table I. HOR and/or HER data from the literature. Data obtained from measurements in acidic and in alkaline media as well as in full PEM fuel cells. In PEMFCs, the j₀ values are given for ${p}_{{{\rm{H}}}_{2}}\approx 1\,\mathrm{bar}$ and the information on the catalyst are only given for the working electrode if not mentioned otherwise (A stands for anode and C stands for cathode).

Catalyst	Electrolyte	T	${j}_{{\rm{a}},\mathrm{eff}}^{0}$	Eact	Tafel slope	Year / References
		(°C)	(mA cm${}_{\mathrm{Pt}}^{-2}$ )	(kJ mol−1)	(mV/dec)
5wt% Pt/C	96% H3PO4	160	144	17	43 a)	1975 / 43
Mono- & Polycrystalline Pt	HCLO4	25	1.7–3		30	1987 / 44
Polycrystalline Pt	0.01 N HCLO4	25	2.48		118 b)	1989 / 45
	0.01 N—0.1 N H2SO4		1.75–1.16
	0.1 N H3BO3		≪ 0.01
	0.08 N NaOH		0.55
Monocrystalline Pt	0.5 M H2SO4	25	≈1–1.5		32-35	1992 / 46
Monocrystalline Pt	0.05 M H2SO4	60	0.83–1.35c)	9.5–18 c)	28–112 c)	1997 / 47
1.7 mg cm−2 symmetrical	PEMFC (N112)	50	7 d)			1999 / 15
20wt% Pt/C, 0.5 mg cm−2	1 M H2SO4	25	1			2001 / 48
47wt% Pt/C, 0.05–0.4 mg cm−2	PEMFC (900EW)	60	≈27 e)	9.5	33	2004 / 49
10wt% Ptnano/C	0.10 M H2SO4	25	≈ 20		30	2004 / 50
0.7 mg cm−2 symmetrical	PEMFC (N117)	40–70	2.6 mA/mgPt f)	50–55		2005 / 17
Polycrystalline Pt	0.15 M HCLO4	25	1.08			2006 / 8
50wt% Pt/C, 0.35/0.5 mg cm−2 (A/C)	PEMFC (1100EW)	80	7–36 A/mgPt g)			2007 / 51
5wt% Pt/C, 0.003 mg cm−2	PEMFC	80	235–600		70–140	2007 / 52
Symmetrical, Pt/C	HT-PEM (PBI/H3PO4)	80–140	550–1450			2009 / 10
46wt% Pt/C, 0.007 mg cm−2	0.1 M KOH	21	0.57	30	120	2010 / 11
4.7wt% Pt/C, 0.002 mg cm−2	PEMFC	40-80	120–260	16	124–149	2014 / 12
0.2/0.4 mg cm−2 (A/C)	PEMFC	80	≈ 19 A/mgPt h)			2018 / 18
4.8wt% Pt/C, 1.2–1.6 μg cm−2	PEMFC	30–90	88–580	25	135–144	2021 / 14
0.05/0.4 mg cm−2 (A/C)	PEMFC	40–80	170–408 i)	17.8 i)		This work

^a)Determined with 2.3RT/(2F) for 160 °C. ^b)Determined with 2.3RT/(0.5F) for 25 °C. ^c)Dependent on the different Pt facets. ^d)Determined by j₀ = RT/(zFR_a), with R_a being the resistance of the HF loop. ^e)Data taken from ⁴⁷ for Pt(110) (j₀ = 1.35 mA cm${}_{\mathrm{Pt}}^{-2}$) and adjusted to match the observed polarization curves. ^f)Calculated based on the HF loop at 60 °C. Since rf is not given, data is mass specific. ^g)Based on the polarization curves slopes for 60% rH and 100% rH (12 mΩcm² and 2.5 mΩcm²). Values obtained by considering the ratio of loadings of the electrodes. Since rf is not given, data is mass specific. ^h)Estimated for ≈ (0.4/(0.4 + 0.2)) · 12 mΩcm² at OCV in H₂/H₂, T = 80 °C, ${p}_{{{\rm{H}}}_{2}}=0.67\,\mathrm{bar}$ and rH = 70%. ⁱ⁾Values obtained for our final test run containing micropolarization curves, EIS and recovery steps to keep the catalyst layers clean. Calculated for rf=30 ${{\rm{m}}}_{\mathrm{Pt}}^{2}$/m${}_{\mathrm{geo}}^{2}$ obtained by CV measurements.

Figure 3a shows the EIS at OCV and Fig. 3b the corresponding DRT of a relative humidity variation at 80 °C with pure H₂ on both sides. As can be seen from these typical H₂/H₂ spectra, two main processes are identified, which confirms previous literature findings. ^9,15–17 Apparently, both processes exhibit a strong humidity dependency. For the high-frequency loop, this dependency is believed to come from the proton transport resistance of the catalyst layer, ¹⁸ or from a water dependency in the hydrogen kinetics itself. ⁵³ However, EIS and DRT results from a humidity variation with a low ${p}_{{{\rm{H}}}_{2},\mathrm{anode}}$ (see Figs. S6a to S6b) show that the high frequency loop in fact contains two processes with separate time constants and that only the one occurring at the highest frequencies seems to be humidity-dependent. This hints at combined rH-dependent proton transport and rH-independent electrode kinetics, thus confirming the former hypothesis. In Fig. 3c we fitted the HF loop (R_P1A = R_HF) to the low-frequency real part of the transmission line model

$$\begin{eqnarray}&&Z=\sqrt{{R}_{{\rm{p}}}{R}_{\mathrm{ct}}}\coth \left(\displaystyle \frac{{R}_{{\rm{p}}}}{{R}_{\mathrm{ct}}}\right)\end{eqnarray} \tag{ 9 }$$

and assumed that the cathode impedance (counter electrode) is negligible (due to its eight times higher ECSA), which was previously discussed in the literature for comparable counter electrodes. ⁵² Here, R_ct represents the kinetics charge transfer resistance and R_p the proton resistance of the catalyst layer. The Fig. shows the resistance over rH at 80 °C for three different H₂ concentrations on the anode side (${x}_{{{\rm{H}}}_{2},{\rm{A}}}=0.75$, ${x}_{{{\rm{H}}}_{2},{\rm{A}}}=0.5$ and ${x}_{{{\rm{H}}}_{2},{\rm{A}}}=0.25$, obtained by diluting with N₂). We further assumed that the proton resistance is a fraction of the cathode proton resistance we characterized in blocking cathode configuration ⁷ and found R_p,anode = 0.18 · R_p,cathode, reflecting approximately the ratio of anode to cathode thickness (2/13 ≈ 0.15). The HF loop is therefore well explained by a combination of proton transport and charge transfer limitations as given by Eq. 9. It is worth noting that we generally found R_ct to be in the same order of magnitude or smaller (1 mΩ cm² < R_ct < 30 mΩ cm²) than R_p (7 mΩ cm² < R_p < 60 mΩ cm²). In this limit R_p > R_ct, Eq. 9 simplifies to $Z=\sqrt{{R}_{{\rm{p}}}\cdot {R}_{\mathrm{ct}}}$, such that the true exchange current density ${j}_{{\rm{a}},\mathrm{ct}}^{0}={RT}/(F\cdot {R}_{\mathrm{ct}})$ is related to the apparent exchange current density by ${j}_{{\rm{a}},\mathrm{ct}}^{0}={j}_{{\rm{a}},\mathrm{eff}}^{0}\cdot \sqrt{{R}_{{\rm{p}}}/{R}_{\mathrm{ct}}}$.

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Even though Neyerlin et al. ⁵² explained that with pure hydrogen the gas-phase related transport resistances are expected to be very small in the proton pump measurements with state-of-the-art diffusion media (high diffusion coefficient ${D}_{{{\rm{H}}}_{2}}\approx 0.5$ cm²/s), ⁵² Heinzmann et al. recently showed that low-frequency EIS loop depends on the partial pressure and assigned it to H₂ mass transport from the channel to the reaction sites. ¹⁸ We also see this ${p}_{{{\rm{H}}}_{2}}$-dependency of the LF loop (see below); however, the influence of rH is just as strong in our case. In the literature, this dependency on rH was attributed to ionomer dry out effects caused by the electroosmotic drag. ^54,55 Based on these analyses, we believe that the interplay of R_p and R_ct in the transmission line causes the combined rH and ${p}_{{{\rm{H}}}_{2}}$ influence on the low-frequency loop. The other hypotheses of the low frequency process being linked to H₂ adsorption or desorption processes, ^15,16 or to the HER, ¹⁰ could not be validated yet. Further parameter variations in the H₂/H₂ configuration can be seen in the supplementary material (Figs. S6 and S7).

When trying to characterize the temperature dependency of the anode kinetics, we were initially puzzled by a far too high apparent activation energy of E_act > 70 kJ mol⁻¹ in our first two H₂/H₂ measurement campaigns. We already observed this high T-dependency during the H₂/N₂ characterization in our previous publication ⁷ . After identifying an undesired and highly T-dependent anodic contribution via EIS, we chose to account for it through an additional R-CPE element in order to achieve a clean parameterization of the ionomer resistance R_p. We also found that this spurious process was correlated to long stabilization times ( > 2 h) prior to the measurements and that it was in fact due to CO poisoning of the catalyst layer. This poisoning effect is absent in the normal fuel cell configuration since O₂ crossover from the cathode to the anode helps keeping the anode catalyst clean. It also perfectly explains the observed high T-dependency. ⁵⁶ We refer to the supplementary material for a more detailed discussion.

To get a clean dataset we reproduced H₂/H₂ EIS and micropolarization measurements and introduced recovery steps after each operating condition to keep the catalyst layers clean. To do so, we carried out CV measurements from −0.9 V to 0.9 V to oxidize the CO layer in both the anode and cathode electrode by diluting H₂ in N₂ and stopping the flow before starting the measurements. Such a CV measurement is shown in Fig. 4a, where the CO peaks are highlighted by the gray areas and disappear completely after the second cycle. We chose rH = 100% in this test run to keep the ionomer resistances as low as possible and therefore make the evaluation of the kinetics as precise as possible. In Fig. 4b we show the micropolarization curves of a hydrogen partial pressure variation at rH = 100% and T = 40 °C and Fig. 4c displays corresponding EIS at OCV. The polarization curves show a perfectly linear behavior in the range that we used for the parameter fitting; however, the HOR direction (negative voltage and current in this case) shows non-linearities at higher overpotentials for the low H₂ concentrations. In this special case, this behavior is not linked to the anodic overpotential-dependent limiting current observed in the literature (most probably potential-independent rate limiting Tafel step), ^12,14 but rather to mass transport contributions because of the dropping stoichiometry (λ < 4). In the Nyquist plot of the EIS measurements at OCV, we can only observe one loop at high partial pressures. With decreasing partial pressures, several capacitive processes and a low-frequency inductive feature appear. According to detailed analyses of these EIS features (see Fig. S8), the high-frequency loop has an activation energy of around 20 kJ mol⁻¹ which is much higher than for the low-frequency capacitive loop (≈2 kJ mol⁻¹) and for the low-frequency inductive loop (≈4 kJ mol⁻¹). Moreover, the size of the inductive loop is inversely proportional to ${p}_{{{\rm{H}}}_{2}}$ (exponent ≈−1.0) and therefore stronger pressure-dependent than the capacitive loops (exponent ≈−0.8).

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However, for fitting our simple kinetics model we chose to evaluate the polarization curves. The according procedure for extracting the parameters is shown in the appendix in Fig. 11. Figure 5a shows an Arrhenius plot of the HOR/HER exchange current density for three different concentrations and the corresponding Ohmic resistance used for the performance corrections. The resulting activation energies are virtually independent of ${p}_{{{\rm{H}}}_{2}}$ and lie in the range 15 kJ mol⁻¹ to 18 kJ mol⁻¹, which is close to the value from Durst et al.. ¹² Fig. 5b shows the partial pressure dependency of the exchange current density (apparent reaction order with respect to ${p}_{{{\rm{H}}}_{2}}\approx 0.4$) at T = 40 °C. This is in line with the results of Stühmeier et al.. ¹⁴ At lower partial pressures the apparent exchange current density deviates from the expected behavior and gets too small. This effect partially comes from the presence of the other effects that are also visible in the EIS response as mentioned above. If we analyze only the high-frequency loops from the EIS (triangles in Fig. 5b), the exchange current densities are closer to the regression curve but still too low. This effect may come from a change in the influence of the ionomer contribution on the effective anode resistance when the charge transfer resistance rises due to lower ${p}_{{{\rm{H}}}_{2}}$, such that the approximation R_p > R_ct of Eq. 9 is not justified anymore. Another possible explanation is a change in the mechanisms of the kinetics at very low partial pressures. However, the current parameterization of the hydrogen kinetics is already sufficiently accurate in the range of operating conditions that are meaningful for typical PEM operation. Hence, we chose to exclude all the data points recorded at ${p}_{{{\rm{H}}}_{2}}\lt 0.1\,\mathrm{bar}$ in our final fitting and obtained the expression

$$\begin{eqnarray}\begin{array}{rcl}{j}_{{\rm{a}},\mathrm{eff}}^{0} & = & (1.57\cdot {10}^{2}\,{\rm{A}}{/\mathrm{cm}}_{\mathrm{Pt}}^{2})\cdot {\left(\displaystyle \frac{{p}_{{{\rm{H}}}_{2}}}{{p}_{\mathrm{ref}}}\right)}^{0.36}\exp \left(-\displaystyle \frac{17.8\,\mathrm{kJ}\,{\mathrm{mol}}^{-1}}{{RT}}\right)\end{array}\end{eqnarray} \tag{ 10 }$$

which was normalized by the roughness factor 30 m${}_{\mathrm{Pt}}^{2}$ m${}_{\mathrm{geo}}^{-2}$ (see above). The correlation plot in Fig. 5c demonstrates the model and the measured values match quite well apart from the points that were excluded from the fitting. Further, the resulting fit parameters are close to the values from the literature for proton pumping experiments (see Table I). ^12,14,52 Nevertheless, we emphasize that if we consider the true exchange current density ${j}_{{\rm{a}},\mathrm{ct}}^{0}={({j}_{{\rm{a}},\mathrm{eff}}^{0})}^{2}\cdot {R}_{{\rm{p}}}\cdot F/({RT})$ instead of the apparent exchange current density ${j}_{{\rm{a}},\mathrm{eff}}^{0}$, the activation energy becomes 23.4 kJ mol⁻¹ and the reaction order with respect to hydrogen doubles from 0.36 to 0.72. Further, the true exchange current density is two to three times higher (${j}_{{\rm{a}},\mathrm{ct}}^{0}\approx \,1\,{\rm{A}}\,{\mathrm{cm}}_{\mathrm{Pt}}^{-2}$) than the apparent exchange current density (${j}_{{\rm{a}},\mathrm{eff}}^{0}\approx \,0.4\,{\rm{A}}\,{\mathrm{cm}}_{\mathrm{Pt}}^{-2}$) at the same conditions.

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With this evaluation we obtain a ready-to-use model for the anode kinetics that can directly be implemented into simulation models or used for loss contribution analyses. Furthermore, we emphasize the importance of our CO recovery procedure for getting accurate data and that caution is required even when working at high cell temperatures since surface poisoning effects can easily be overlooked.

The ORR is the major loss process under the most relevant operating conditions. Thus, well-parameterized models are needed for the ORR to accurately predict the performance in fuel cell operation. As mentioned in the introduction, this is still subject to research as there is no consensus regarding its mechanistic description. However, simple Tafel kinetics is typically employed since the overpotential η_ORR is high and can be expressed as a logarithmic function of the current density according to

$$\begin{eqnarray}&&{\eta }_{\mathrm{ORR}}=\displaystyle \frac{{RT}}{{\alpha }_{{\rm{c}}}F}\cdot \mathrm{ln}\displaystyle \frac{j}{{j}_{{\rm{c}},\mathrm{eff}}^{0}}\end{eqnarray} \tag{ 11 }$$

where the cathodic transfer coefficient α_c is the product of the symmetry factor β and the number of transferred electrons z, and ${j}_{{\rm{c}},\mathrm{eff}}^{0}$ is the effective exchange current density. This leads to the impedance expression of the charge transfer process

$$\begin{eqnarray}&&\displaystyle \frac{\partial {\eta }_{\mathrm{ORR}}}{\partial j}={R}_{\mathrm{ct}}=\displaystyle \frac{{RT}}{{\alpha }_{{\rm{c}}}{Fj}}=\displaystyle \frac{b}{j}\end{eqnarray} \tag{ 12 }$$

that does not depend on the exchange current density ${j}_{{\rm{c}},\mathrm{eff}}^{0}$ and thus not on ${p}_{{{\rm{O}}}_{2}}$. This expression directly links the intrinsic Tafel slope b (slope of the curves U_jR−free versus ${\rm{ln}}(j+{j}_{{{\rm{H}}}_{2},{\rm{crossover}}})$) to the size of the charge transfer loop in the Nyquist spectra. In contrast to the anode kinetics (see Eqs. 7 and 8), EIS cannot be used to determine the exchange current density, yet would enable direct characterization of b. However, this is complicated by the impact of slow Pt oxidation kinetics on the apparent Tafel slope in steady-state measurements. ⁵⁷ These oxidation mechanisms lead to low-frequency inductive features in the EIS spectra and discrepancies are therefore observed between an EIS-based evaluation of b through the high-frequency charge transfer resistance R_ct by b = j · R_ct and a steady-state-based evaluation of b trough the slope of the polarization curve in the Tafel plot. The first approach yields Tafel slopes b ≥ 100 mV/dec (see Fig. S5), and the second approach yields slopes b ≈ 70 mV/dec (see below). We will analyze this effect in a forthcoming paper. In the present work, we aim for a simple representation of the ORR kinetics and understanding of its limits by parameterizing a Tafel law like shown in Eq. 11 using exclusively polarization data at low currents.

We evaluated the kinetics overpotential η_ORR by subtracting the jR-free voltage from the equilibrium voltage U₀ according to

$$\begin{eqnarray}\begin{array}{rcl}{\eta }_{\mathrm{ORR}} & = & {U}_{0}-{U}_{\mathrm{measurement}}+j\cdot ({R}_{{\rm{p}}}^{\mathrm{eff}}+{R}_{{\rm{\Omega }}})\\ & = & {U}_{0}-{U}_{{jR}-\mathrm{free}}\end{array}\end{eqnarray} \tag{ 13 }$$

where the jR-correction was made based on the evaluation of EIS spectra under load as described in our previous publication. ⁷ In the literature, the ORR parameters are most often obtained from measurements gathered under fully saturated conditions ¹⁹ and a full factorial parameterization containing data at varying rH does not exist to our knowledge; however, some authors discussed the influence of rH on the ORR performance. Xu et al. found out that their CCL performs badly at elevated temperature and dry conditions even after correcting the data for the effective protonic resistance. ^58,59 In contrast to these findings, Neyerlin et al. and Liu et al. saw a way smaller influence of rH on the jR-free performance. ^60,61 Even though the humidity influence on the ORR initially appeared to be small in our dataset (see Fig. S9a), we investigated different modeling approaches. Firstly, we used an approach were the water activity was set to unity as it is usually the case in literature studies which we call here the 'water-independent' approach. There, the equilibrium voltage U₀ can be calculated by the Nernst equation according to

$$\begin{eqnarray}\begin{array}{rcl}{U}_{0} & = & 1.23\,{\rm{V}}-(0.9\cdot {10}^{-3}\,{\mathrm{VK}}^{-1})(T-298\,{\rm{K}})\\ & & +\displaystyle \frac{{RT}}{2F}\mathrm{ln}\left[\left(\displaystyle \frac{{p}_{{{\rm{H}}}_{2}}}{{p}_{\mathrm{ref}}}\right){\left(\displaystyle \frac{{p}_{{{\rm{O}}}_{2}}}{{p}_{\mathrm{ref}}}\right)}^{\tfrac{1}{2}}\right]\end{array}\end{eqnarray} \tag{ 14 }$$

The exchange current density contains the dependencies on temperature and on oxygen partial pressure through the activation energy E_act and the reaction order γ, respectively, and was expressed as

$$\begin{eqnarray}&&{j}_{{\rm{c}},\mathrm{eff}}^{0}={rf}\cdot {j}_{{\rm{c}},\mathrm{ref}}^{0}\cdot {\left(\displaystyle \frac{{p}_{{{\rm{O}}}_{2}}}{{p}_{\mathrm{ref}}}\right)}^{\gamma }\exp \left[\displaystyle \frac{-{E}_{\mathrm{act}}}{{RT}}\left(1-\displaystyle \frac{T}{{T}_{\mathrm{ref}}}\right)\right]\end{eqnarray} \tag{ 15 }$$

with rf the roughness factor of the cathode electrode in ${{\rm{m}}}_{\mathrm{Pt}}^{2}$ m${}_{\mathrm{geo}}^{-2}$ and ${j}_{{\rm{c}},\mathrm{ref}}^{0}$ the reference exchange current density in A cm${}_{\mathrm{Pt}}^{-2}$. In our second approach that we call in the following 'water-dependent' we accounted for the actual partial pressure of water in the gas phase ${p}_{{{\rm{H}}}_{2}{\rm{O}}}$ in the Nernst equation instead of setting the activity to unity and thus calculated the cell equilibrium voltage by

$$\begin{eqnarray}\begin{array}{rcl}{\tilde{U}}_{0} & = & 1.23\,{\rm{V}}-(0.9\cdot {10}^{-3}\,{\mathrm{VK}}^{-1})(T-298\,{\rm{K}})\\ & & +\displaystyle \frac{{RT}}{2F}\mathrm{ln}\left[{\left(\displaystyle \frac{{p}_{{{\rm{H}}}_{2}{\rm{O}}}}{{p}_{\mathrm{ref}}}\right)}^{-1}\left(\displaystyle \frac{{p}_{{{\rm{H}}}_{2}}}{{p}_{\mathrm{ref}}}\right){\left(\displaystyle \frac{{p}_{{{\rm{O}}}_{2}}}{{p}_{\mathrm{ref}}}\right)}^{\tfrac{1}{2}}\right]\end{array}\end{eqnarray} \tag{ 16 }$$

We further added a possible water dependency in the exchange current density through a power law for ${p}_{{{\rm{H}}}_{2}{\rm{O}}}$ therein with the supplementary parameter n. This change yields

$$\begin{eqnarray}\begin{array}{rcl}{\tilde{j}}_{{\rm{c}},\mathrm{eff}}^{0} & = & {rf}\cdot {j}_{{\rm{c}},\mathrm{ref}}^{0}\cdot {\left(\displaystyle \frac{{p}_{{{\rm{H}}}_{2}{\rm{O}}}}{{p}_{\mathrm{ref}}}\right)}^{n}\cdot {\left(\displaystyle \frac{{p}_{{{\rm{O}}}_{2}}}{{p}_{\mathrm{ref}}}\right)}^{\gamma }\\ & & \times \exp \left[\displaystyle \frac{-{E}_{\mathrm{act}}}{{RT}}\left(1-\displaystyle \frac{T}{{T}_{\mathrm{ref}}}\right)\right]\end{array}\end{eqnarray} \tag{ 17 }$$

Therefore, the basic model for saturated conditions contains the four fitting parameters α_c, E_act, γ, and ${j}_{{\rm{c}},\mathrm{ref}}^{0}$, and the modified ${p}_{{{\rm{H}}}_{2}{\rm{O}}}$-dependent model adds a fifth parameter, n. It can easily be shown algebraically however that for n = 0.5 both models are in fact identical. Thus, a noticeable water dependency is only observed if n ≠ 0.5.

We depict the method to get the kinetic parameters schematically in Fig. 11 of the appendix. If not mentioned otherwise, the reference conditions for all fitted models were T_ref = 80 °C and p_ref = 1 bar. We took $j={j}_{\mathrm{measured}}+{j}_{{{\rm{H}}}_{2},\mathrm{crossover}}$ for the global fitting and excluded j > 0.1 A cm${}_{\mathrm{geo}}^{-2}$ in most cases. To verify the meaningfulness of the latter condition and see the influence of one-at-a-time parameter variations, we performed local fits at given humidity, temperature or current density (see Table S1 and Table S2 of the supplementary material) prior to the global fit over all the conditions simultaneously. These local fits confirmed that the parameters are constant and the Tafel slope was captured properly for j ≤ 0.1 A cm⁻². The model for the H₂ permeation coefficient ${{\rm{\Psi }}}_{{{\rm{H}}}_{2}}$ and the corresponding measurements are summarized in Fig. S4c over T and rH and the resulting crossover current taken for the current correction is shown in Fig. S4d for our standard condition p_total = 1.5 bar_a. Since the membranes of our samples showed very high electrical resistances ( ≫ 600 Ω cm²) we neglected electrical membrane shorts.

Optimized parameters from the global fit simultaneously over all the operating conditions are shown in Table II. It can be seen that the activation energy is around 70 kJ mol⁻¹, the reference exchange current density is around 2 · 10⁻⁸ A cm${}_{\mathrm{Pt}}^{-2}$, the reaction order is around 0.5 and the exponent of the power law for the water partial pressure dependency is between 0.4 and 0.5. Moreover, the fit results from cases in which α_c was not fixed to 1 (not shown in the table) yielded results that were very close to those of the fitting procedures in which α_c was fixed to 1, matching our expectations of Tafel slopes around 70 mV/dec (see above). Figures 6a and 6b show the good agreement between the fitted local exchange current densities (symbols, α_c = 1) and the global model (black lines) for an rH variation with the ${p}_{{{\rm{H}}}_{2}{\rm{O}}}$-dependent model at T = 50 °C and T = 80 °C (Fig. 6a) and with the ${p}_{{{\rm{H}}}_{2}{\rm{O}}}$-independent model (Fig. 6b) at the same conditions. The rH = 90% conditions are outliers and thus were excluded from the fitting (probably partial flooding of the electrode). In Fig. 6c and Fig. 6d we depict a ${p}_{{{\rm{O}}}_{2}}$ variation for four different temperature and humidity combinations for the ${p}_{{{\rm{H}}}_{2}{\rm{O}}}$-dependent model and the ${p}_{{{\rm{H}}}_{2}{\rm{O}}}$-independent model, respectively (see Fig. S9b for a similar plot in the heliox case). The slopes in the double-logarithmic plots of this Fig. represent the reaction order γ (for the ${p}_{{{\rm{O}}}_{2}}$ variation) and the exponent of the ${p}_{{{\rm{H}}}_{2}{\rm{O}}}$ influence n (rH variation). Both models seem to fit the data well. This is not surprising, as n ≈ 0.5 means that both models are essentially equivalent and there is effectively no influence of rH on the ORR. Thus, the exchange current density can be parameterized as

$$\begin{eqnarray}\begin{array}{rcl}{\tilde{j}}_{{\rm{c}},\mathrm{eff}}^{0} & = & {rf}\cdot (2.13\cdot {10}^{-8}\,{\rm{A}}{/\mathrm{cm}}_{\mathrm{Pt}}^{2}){\left(\displaystyle \frac{{p}_{{{\rm{H}}}_{2}{\rm{O}}}}{{p}_{\mathrm{ref}}}\right)}^{0.51}{\left(\displaystyle \frac{{p}_{{{\rm{O}}}_{2}}}{{p}_{\mathrm{ref}}}\right)}^{0.49}\\ & & \times \exp \left[\displaystyle \frac{-68.7\,\mathrm{kJ}\,{\mathrm{mol}}^{-1}}{{RT}}\left(1-\displaystyle \frac{T}{{T}_{\mathrm{ref}}}\right)\right]\end{array}\end{eqnarray} \tag{ 18 }$$

if the Nernst equation contains the effective water partial pressure, or as

$$\begin{eqnarray}\begin{array}{rcl}{j}_{{\rm{c}},\mathrm{eff}}^{0} & = & {rf}\cdot (2.15\cdot {10}^{-8}\,{\rm{A}}{/\mathrm{cm}}_{\mathrm{Pt}}^{2}){\left(\displaystyle \frac{{p}_{{{\rm{O}}}_{2}}}{{p}_{\mathrm{ref}}}\right)}^{0.51}\\ & & \times \exp \left[\displaystyle \frac{-71.3\,\mathrm{kJ}\,{\mathrm{mol}}^{-1}}{{RT}}\left(1-\displaystyle \frac{T}{{T}_{\mathrm{ref}}}\right)\right]\end{array}\end{eqnarray} \tag{ 19 }$$

if the water activity is set to unity in the Nernst equation.

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Table II. Exchange current density parameters extracted from globally fitting all the polarization points U_jR−free for j ≤ 0.1 A cm⁻². The conditions where no n is given were fitted with the ${p}_{{{\rm{O}}}_{2}}$-independent model (water activity set to unity in U₀ and ${j}_{{\rm{c}},\mathrm{eff}}^{0}$). The symmetry factor was fixed at α_c = 1.

αc	${j}_{{\rm{c}},\mathrm{ref}}^{0}\cdot {10}^{8}$	γ	n	Eact
(—)	(A/cm${}_{\mathrm{Pt}}^{2}$)	(—)	(—)	(kJ mol−1)
${p}_{{{\rm{H}}}_{2}{\rm{O}}}$-dependent model without the rH = 90% conditions
1.0	2.13	0.49	0.51	68.74
${p}_{{{\rm{H}}}_{2}{\rm{O}}}$-dependent model with all conditions
1.0	1.82	0.50	0.41	73.41
${p}_{{{\rm{H}}}_{2}{\rm{O}}}$-independent model with all the rH conditions
1.0	2.05	0.50	—-	69.52
${p}_{{{\rm{H}}}_{2}{\rm{O}}}$-independent model only with rH = 80% conditions
1.0	2.15	0.51	—-	71.27
${p}_{{{\rm{H}}}_{2}{\rm{O}}}$-independent model only with rH = 90% conditions
1.0	1.81	0.54	—-	70.88

Figure 7a depicts the measured exchange current density versus the modeled exchange current density for the ${p}_{{{\rm{H}}}_{2}{\rm{O}}}$-dependent model and Fig. 7b for the ${p}_{{{\rm{H}}}_{2}{\rm{O}}}$-independent model. The measured U_jR−free cell voltage versus the modeled cell voltage is also shown for both model variants in Figs. 7c and 7d. Even though both models yield very good agreement with the experimental data, the water-dependent model seems to slightly outperform the water-independent model, which is not surprising considering the additional degree of freedom in the fitting procedure. Figure S9c to S9f additionally depict the good match between measured U_jR−free polarization curves and the according model curves at four different rH and T combinations (exemplarily for the water-dependent model). Furthermore, we checked our specific values against the data published by Neyerlin et al. ¹⁹ that still represent a reliable reference to date for Pt/C catalysts. This comparison is given in Table III: first, we compared the exchange current density parameters referenced to 80 °C and 1 bar; second, we compared the jR-free cell voltage referenced to 80 °C 1 bar and 0.9 V. The equation for the latter was given as in Eq. 20: ¹⁹

$$\begin{eqnarray}&&\begin{array}{l}{U}_{{jR}-\mathrm{free}}=0.9\,{\rm{V}}-\displaystyle \frac{{RT}}{{\alpha }_{{\rm{c}}}F}\\ \quad \mathrm{ln}\left[\displaystyle \frac{j+{j}_{\mathrm{crossover}}}{{rf}\cdot {j}_{0,\mathrm{ref}}^{0.9{\rm{V}}}\cdot {\left(\tfrac{{p}_{{{\rm{H}}}_{2}}}{{p}_{\mathrm{ref}}}\right)}^{{\alpha }_{{\rm{c}}}/2}{\left(\tfrac{{p}_{{{\rm{O}}}_{2}}}{{p}_{\mathrm{ref}}}\right)}^{m}\exp \left[\tfrac{-{E}_{\mathrm{act}}^{0.9{\rm{V}}}}{{RT}}\left(1-\tfrac{T}{{T}_{\mathrm{ref}}}\right)\right]}\right]\end{array}\end{eqnarray} \tag{ 20 }$$

Table III shows that there is a good agreement between the reference from the literature and our measurements. Hence, we have an accurate ready-to-use model for the ORR mechanism in our full cell described by a simple Tafel law with an intrinsic slope determined by α_c = 1. This model can reasonably be applied over a wide range of current densities (U_jR−free > 0.8 V) and, according to the literature, more complex models with ORR intermediates only gain importance with low-loaded cathodes or at lower potentials. ²⁷ Subramanian et al. parameterized oxide-coverage-dependent ORR kinetics to model their jR-corrected performance curves measured with a cathode loading of 0.06 mg cm${}_{\mathrm{geo}}^{-2}$. ²⁷ This model contains a change in the apparent Tafel slope at around 0.75 V to 0.8 V and they ruled out mass transport effects in their measurement based on limiting current measurements. In our data, we also see deviations from the ORR model below 0.8 V. These effects could also result from more complex ORR kinetics, from inaccurate jR corrections at high current densities (decreasing fitting quality with decreasing CCL utilization), or from mass transport contributions. A more detailed discussion regarding these observations is given below.

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Table III. Comparison of our reference data (rH = 80%) with the data of Neyerlin et al.. ¹⁹ The parameters were acquired by fitting Eq. 15 and Eq. 20 to our data.

Parameter	Neyerlin 19	This work
		{O2,N2}	{O2,He}
Exchange current density parameters
${j}_{{\rm{c}},\mathrm{ref}}^{0}\cdot {10}^{8}$ (A/cm${}_{\mathrm{Pt}}^{2}$)	2.47	2.15	2.45
γ (—)	0.54	0.51	0.62
Eact (kJ mol−1)	67	71.27	68.62
UjR−free parameters
${j}_{0,\mathrm{ref}}^{0.9{\rm{V}}}\cdot {10}^{4}$ (A/cm${}_{\mathrm{Pt}}^{2}$)	2.47	2.11	2.49
m (—)	0.79	0.75	0.87
${E}_{\mathrm{act}}^{0.9{\rm{V}}}$ (kJ mol−1)	10	10.85	10.89

Investigating the contribution to the cell resistance of O₂ mass transport from the gas channels to the catalytic surface of the Pt particles is essential for modeling purposes since it usually contributes to the overpotential at operating conditions that are relevant from a system point of view (medium to high current densities). Furthermore, the importance of this contribution grows as the roughness factor decreases, ⁶² which is a main target to pursue for cost-reductions. In this work, we want to clarify whether O₂ transport contributions are relevant in the ORR parameterization to ensure that the parameters are free of unwanted effects. For this purpose, both the steady-state limiting current (SLC) technique and the transient limiting current (TLC) technique are used and discussed. The principle of the first method lies in measuring the limiting current density with strongly diluted oxygen to avoid high currents and therefore much water production (dry transport resistance), whereas the principle of the second method lies in recording a limiting current density for an arbitrarily conditioned cell state by going to a low cell voltage only for a few seconds ( < 10 s) in order not to significantly change the cell state. In our case, the TLC was measured along polarization curves for different operating parameters to investigate the impact of current density on the transport resistance.

We measured steady-state limiting currents for strongly diluted O₂ in N₂ gas streams and different temperature and humidity levels according to the method described by Baker et al.. ³⁸ During this procedure, the cell was operated up to the O₂ mass transport limitation (vertical part of the polarization curve at high currents) by jumping to cell voltages below 0.3 V in the potentiostatic mode. Low oxygen concentrations are used to avoid high limiting currents and therefore much liquid water production that would alter the measured transport resistance in the gas phase. From this limiting current density the oxygen transport resistance (in s/m) from the channel to the electrode can be calculated by

$$\begin{eqnarray}&&{R}_{{{\rm{O}}}_{2},\mathrm{MT}}^{}=\displaystyle \frac{{p}_{{{\rm{O}}}_{2}}^{\mathrm{channel}}\cdot 4\cdot F}{R\cdot T\cdot {j}_{\mathrm{lim}}}\end{eqnarray} \tag{ 21 }$$

Since the conditions were not perfectly differential during the limiting current measurements, we chose to use a log-mean O₂ partial pressure in the gas channel which was calculated based on the inlet and the outlet mole fraction. The outlet mole fraction was estimated by

$$\begin{eqnarray}&&{x}_{{{\rm{O}}}_{2}}^{\mathrm{outlet}}=\displaystyle \frac{{\dot{n}}_{{{\rm{O}}}_{2}}^{\mathrm{inlet}}-{j}_{\mathrm{lim}}/(4\cdot F)}{{\dot{n}}_{{{\rm{O}}}_{2}}^{\mathrm{inlet}}+{\dot{n}}_{{{\rm{N}}}_{2}}+{\dot{n}}_{{{\rm{H}}}_{2}{\rm{O}}}^{\mathrm{inlet}}+{j}_{\mathrm{lim}}/(4\cdot F)}\end{eqnarray} \tag{ 22 }$$

with $\dot{n}$ being the molar fluxes and under the assumption that the product water is completely in the gas phase on the cathode side, ${\dot{n}}_{{{\rm{H}}}_{2}{\rm{O}}}^{\mathrm{outlet}}={\dot{n}}_{{{\rm{H}}}_{2}{\rm{O}}}^{\mathrm{inlet}}+{j}_{\mathrm{lim}}/(2\cdot F)$. Based on such an O₂ transport resistance from limiting current experiments, it is possible to estimate the voltage drop during a polarization curve by considering both the effect on the Nernst voltage and the drop of the exchange current density of the ORR:

$$\begin{eqnarray}&&{\rm{\Delta }}{U}_{{{\rm{O}}}_{2},{\rm{MT}}}(j)=\displaystyle \frac{{RT}}{F}\left(\displaystyle \frac{1}{4}+\displaystyle \frac{{{\gamma }}_{{\rm{ORR}}}}{{{\alpha }}_{{\rm{ORR}}}}\right){\rm{ln}}\left(\displaystyle \frac{{p}_{{{\rm{O}}}_{2}}^{{\rm{in}}}-{\textstyle \tfrac{{RT}}{4F}}\cdot {R}_{{{\rm{O}}}_{2}}\cdot j}{{p}_{{{\rm{O}}}_{2}}^{{\rm{in}}}}\right)\end{eqnarray} \tag{ 23 }$$

Here, γ_ORR = 0.5 is the ORR reaction order and α_ORR = 1 the ORR transfer coefficient. In Fig. 8a we depict the measured transport resistance ${R}_{{{\rm{O}}}_{2},\mathrm{MT}}$ over the limiting current density ${j}_{\mathrm{lim}}\propto {x}_{{{\rm{O}}}_{2}}$ for 80% and 100% rH and four temperatures (T = 50 °C to T = 80 °C). As expected, the resistance grows with decreasing temperature, which can be attributed to the decreasing diffusion coefficients. Also, the 80%-rH resistances seem to be located in an intermediate range between dry conditions and fully saturated conditions as they converge to very low values at low limiting currents and to similar values as for 100% rH at high limiting currents. We also observed such a transition behavior between a dry and a wet plateau in our transient limiting current measurements (see Fig. 8c). Such behavior is known in the literature for some GDLs. ^39,40,63,64 As the resistances are below 1 s cm⁻¹ even at fully satured gas and low temperature conditions, the loss contributions will be very small at typical operating conditions.

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In Fig. 8b we show the voltage drop occasioned by O₂ transport over the current density for four different temperature and humidity combinations, which reflects the higher transport resistances of the fully saturated conditions. Owing to the fact that the loss contribution initially increases linearly, it is very small in the low current density range (j < 0.5 A cm⁻²) no matter the temperature and partial pressure. In the range that is interesting for our ORR investigations, the voltage drop does not exceed 3 mV and is therefore not significant for the ORR parameterization. Subramanian et al. used such SLC measurements as an argument to exclude the possibility of O₂-transport-related effects in the change of the apparent Tafel slope they observed experimentally (mainly with their low-loaded cathode). ²⁷ However, according to us there are two uncertainties coming with such considerations. First, the resistance from the SLC experiment does not correspond to the resistance along the polarization curve since it is a function of saturation and temperature of the porous media and therefore depends on current density. This is indicated in Fig. 8c, where the transient limiting current measurements show the significant influence of the pre-polarization current on the mass transport resistance. Secondly, we might underestimate the resistance and mainly characterize the GDL resistance because the oxygen concentration may drop toward zero at the CL/GDL boundary during limiting current operation such that the catalytic reaction concentrates near that interface. Thus, the measured resistance may not contain the CCL resistance that is effective during a normal polarization curve in the non-transport-limited region. Moreover, even though ${R}_{{{\rm{O}}}_{2},\mathrm{MT}}$ in the SLC case (Fig. 8a) seems to converge to similar values as the wet plateaus in the TLC case (Fig. 8c) at high limiting currents (and thus high O₂ concentrations), the TLC resistances are higher than the SLC resistances at low currents, indicating a different water management in the cell. This effect was also observed by Göbel et al. ⁴⁰ but is not yet well enough understood. Therefore, the polarization curve corrections based on limiting current techniques have to be interpreted with caution. However, mass transport corrections of the performance based on both the SLC and the TLC technique are discussed below. For the SLC method we parameterized a model of the transport resistance ${R}_{{{\rm{O}}}_{2},\mathrm{MT}}$ by fitting it to the values measured for the highest O₂ concentration (highest limiting current and thus highest resistance) ${x}_{{{\rm{O}}}_{2}}=0.15$, over all the humidity, temperature, and total pressure levels (see above for the values). For this model we got

$$\begin{eqnarray}\begin{array}{rcl}{R}_{{{\rm{O}}}_{2},\mathrm{SLC}} & = & {(8.74\cdot {10}^{-1}\,{\rm{s}}/\mathrm{cm})({rH})}^{0.07}{\left(\displaystyle \frac{{p}_{\mathrm{total}}}{{p}_{\mathrm{ref}}}\right)}^{0.87}\\ & & \times \exp \left(\displaystyle \frac{10.7\,\mathrm{kJ}\,{\mathrm{mol}}^{-1}}{{RT}}\right)\end{array}\end{eqnarray} \tag{ 24 }$$

with p_ref = 1 bar_a. The good match between this model and the measured data is shown in Fig. 9.

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Finally, we show typical jR-corrected polarization curves for the N₂ and the He cases in Fig. 10a (T = 50 °C, rH = 80%) and Fig. 10b (T = 80 °C, rH = 80%) for all our O₂-concentration levels in the Tafel representation. These polarization curves show that the kinetics part at low currents can be accurately described by a Tafel slope of around 70 mV/dec, which meets our expectations. Moreover, the use of helium instead of nitrogen does not seem to influence the kinetics part (j ≤ 0.2 A cm⁻² at T = 50 °C and j ≤ 0.5 A cm⁻² at T = 80 °C) and starts to have an influence at current densities above 0.3 A cm⁻². While the heliox curves clearly show less voltage loss at high current densities, deviations from ideal Tafel behavior still remain in this region. Plausible explanations would be either pressure-independent transport effects (platinum-near effects that are unaffected by the use of heliox) or intrinsic changes of the ORR Tafel slope. Interestingly, the deviations are more important at cold conditions, where the diffusion coefficients are the lowest and two-phase issues (liquid water) the most pronounced. Also, the ${x}_{{{\rm{O}}}_{2}}^{\mathrm{dry}}=1$ curves are parallel to the ${x}_{{{\rm{O}}}_{2}}=0.5$ curves with nitrogen but the distance between the ${x}_{{{\rm{O}}}_{2}}^{\mathrm{dry}}=1$ curves and the ${x}_{{{\rm{O}}}_{2}}=0.5$ curves with helium decreases over the current density. One explanation for this effect could be a different water management with the helium mixture since the diffusion coefficient of water in helium is roughly ten times higher than the one of water in nitrogen (3.1 · 10⁻⁴ m²s⁻¹ for He against 3.9 · 10⁻⁵ m²s⁻¹ for N₂ at 300 K) and the thermal conductivity of helium is roughly six times higher than the one of nitrogen(157 mWm⁻¹K⁻¹ for He against 26 mWm⁻¹K⁻¹ for N₂ at 300 K and 2 bar). ⁶⁵ Fig. 10c exhibits a typical polarization curve for ${x}_{{{\rm{O}}}_{2}}=0.16$ with its according jR correction, anode correction based on our parameterization of the HOR, and mass transport correction based either on the SLC method (curve i) or on the TLC method (curve ii). One can see that even for this humid condition (rH = 80%), the major loss mechanism apart from the ORR still comes from the protonic losses in the membrane and cathode catalyst layer over a wide range of current densities. Figure 10c further confirms that voltage losses from anode kinetics and mass transport are negligible at the lowest current densities used for the parameterization of the ORR kinetics. Across our full factorial DOE, the voltage losses from these effects at j = 1 A cm⁻² do not exceed 5 mV ( ≈ 4.6 mV at T = 50 °C and ≈ 2.5 mV at T = 80 °C, anode kinetics) and 7 mV (mass transport, based on SLC measurements), respectively. At higher current densities however, these losses gain in importance and thus meaningful characterization is needed in order not to draw wrong conclusions. Even though the Ohmic contributions and the HOR kinetics are properly accounted for, the mass transport loss correction brings uncertainties in the analysis: while the SLC-corrected curve predicts a deviation from the Tafel line beyond 0.5 A cm⁻², the TLC-corrected curve lies on the Tafel line over the whole current density range. Further measurement techniques such as EIS could be brought into play to strengthen the investigation of mass transport contributions. Nevertheless, the model for ${R}_{{{\rm{O}}}_{2}}$ parameterized by SLC measurements (see Eq. 24) can be employed to delimit the performance range in which the ORR obeys a simple Tafel law for sure, which corresponds to steady-state cathodic half-cell potentials above 0.8 V in our case as shown in Fig. 10d.

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