Electrochemical impedance spectroscopy provides unique opportunity for testing and characterization of PEM fuel cells without interruption of current production mode (Lasia, 2014). A classic approach to interpretation of EIS data is construction of equivalent electric circuit having impedance spectrum close to the measured one. However, a more attractive option provides the distribution of relaxation times (DRT) technique. Note that DRT is sometimes called distribution function of relaxation times.

In the context of PEM fuel cell studies, the idea of DRT can be explained as follows. To a first approximation, PEM fuel cell impedance Z can be modeled as impedance of a parallel RC circuit Z = R 1 + i ω R C , (1) where ω is the angular frequency of applied AC signal. This approximation corresponds to the cell with ideal (fast) transport of reactants in all transport media (Kulikovsky, 2017). In that case, R describes Tafel resistivity of the oxygen reduction reaction (ORR), and C represents the superficial double-layer capacitance of the electrode.

In general, to calculate transport contributions to cell impedance, one has to develop a transport model for the ORR reactants. However, if information on transport resistivities and characteristic frequencies suffices, DRT provides a simpler option. Denoting RC = τ, multiplying the right side of Eq. 1 by nonnegative function γ(τ) and integrating over τ, we get Z ω = R p o l ∫ 0 ∞ γ τ d τ 1 + i ω τ (2) where R _pol is the total polarization resistivity of the cell, and the function γ is the DRT of impedance Z. Mathematically, Eq. 2 means expansion of Z(ω) into infinite sum of RC impedances, with the resistivity of each elementary RC circuit being R _pol γ dτ. The function K R C ω , τ = 1 1 + i ω τ (3) under integral in Eq. 2 is usually called “Debye model” (Barsoukov and Macdonald, 2018). Eq. 2 can also be considered as integral transform of γ(τ), which justifies the term “RC kernel” for Eq. 3.

Quite evidently, DRT of a single parallel RC circuit is Dirac delta function γ = δ(τ − τ _*) positioned at τ _* = RC. This example illustrates the main feature of DRT: it converts any RC-like impedance into a single, more or less smeared on the τ scale δ-like peak.

All transport processes in a fuel cell eventually are linked to the double-layer capacitance in the catalyst layer; thus, it is usually assumed that impedance of every process is not far from impedance of a parallel RC circuit. That means that the DRT of a PEMFC is expected to consist of several delta-like peaks. As the regular frequency f = 1/(2πτ), it is convenient to plot γ(f) instead of γ(τ). Position of each γ(f) peak marks a characteristic frequency of the respective transport process and R n = R p o l ∫ τ n τ n + 1 γ τ d τ (4) gives the contribution of process resistivity to the total cell polarization resistivity R _pol . Here, τ _n and τ _n+1 are the peak boundaries on the τ scale.

Fuel cell impedance is usually measured on equidistant in log-scale frequency mesh {f _n , n = 1, … , N}, with ln (f _n+1) − ln (f _n ) being independent of n. From numerical perspective, it is beneficial to deal with the function G(τ) satisfying to Z ω = R ∞ + R p o l ∫ − ∞ ∞ G τ d ⁡ ln τ 1 + i ω τ , (5) where the term R _∞ is added to describe pure ohmic (high-frequency) fuel cell resistivity. Clearly, γ = G/τ, and Eq. 4 in terms of G takes the form R n = R p o l ∫ τ n τ n + 1 G τ d ⁡ ln ⁡ τ = 2 π R p o l ∫ f n f n + 1 G f d ⁡ ln f , f n < f n + 1 (6) where the frequencies f _n = 1/(2πτ _n+1) and f _n+1 = 1/(2πτ _n ) mark the peak boundaries. Note that Eqs 4, 6 return a single peak resistivity if the DRT peaks are not overlapped; otherwise, R _n would represent the sum of overlapped peak resistivities. The problem of overlapped peak resistivity is alleviated if the DRT in analytical form is found, as in, for example, the ISGP method (Avioz Cohen et al., 2021).

Setting in Eq. 5 ω = 0 and taking into account that Z (0) − R _∞ = R _pol , we see that G obeys to normalization condition ∫ − ∞ ∞ G τ d ⁡ ln ⁡ τ = 2 π ∫ − ∞ ∞ G f d ⁡ ln ⁡ f = 1 (7)

In the following, Eq. 5 will be discussed, as G is usually used instead of γ in practical calculations.

DRT technique, Eq. 2, was invented by Fuoss and Kirkwood (1941) in the context of polymer materials impedance and brought to the fuel cell community seemingly by Schichlein et al. (2002). Since 2002, a lot of works from the group of Ivers–Tiffée have been devoted to deciphering of solid oxide fuel cell spectra by means of DRT [see a review (Ivers-Tiffee and Weber, 2017)]. Analysis of PEMFC impedance spectra using DRT is a relatively new field (Heinzmann et al., 2018; Avioz Cohen et al., 2021; Reshetenko and Kulikovsky, 2021; Wang et al., 2021). Heinzmann et al. (2018) measured impedance spectra of a small (1 cm²) laboratory PEMFC and studied DRT peak behavior depending on cell temperature, relative humidity (RH), oxygen concentration, and current density. They obtained DRT with up to five peaks; the leftmost on the frequency scale peak P1 was attributed to oxygen diffusion in the gas-diffusion and cathode catalyst layers (CCLs). Avioz Cohen et al. (2021) performed impedance measurements of a 5-cm² cell varying temperature, RH, and current density. The calculated DRT consisted of four peaks, attributed (in ascending frequencies) to (1) oxygen transport in the GDL/CCL, (2) ORR, (3) proton transport in the CCL, and (4) proton transport in membrane. Note that Heinzmann et al. and Cohen et al. used different codes for DRT calculation. Wang et al. (2021) measured impedance spectra of application-relevant 25-cm² PEMFC and obtained a three–peak DRT; the lowest frequency peak was attributed to oxygen diffusion processes in the cell. In our recent work (Reshetenko and Kulikovsky, 2021), DRT spectra of a low-Pt PEMFC have been reported; we attributed the low-frequency peak to oxygen transport in the GDL and, possibly, in the channel.

In PEMFCs, the supplied oxygen (air) is transported through the four quite different media: channel, GDL, open pores of the CCL, and finally through Nafion film covering Pt/C agglomerates. One therefore could expect four corresponding peaks in the DRT spectra. However, in Heinzmann et al. (2018), Avioz Cohen et al. (2021), and Wang et al. (2021), a single oxygen transport peak has been reported. There are two options to explain this result: either some of the oxygen transport peaks overlap with each other (or with the ORR peak) and DRT is not able to separate them, or the codes used were unable to resolve all the transport processes. It is important to note that the code for DRT calculation of Wan et al. (2015) used in Heinzmann et al. (2018) and Wang et al. (2021), the ISGP code (Hershkovitz et al., 2011) used in Avioz Cohen et al. (2021), and our code using Tikhonov regularization (Tikhonov, 1995) in combination with NNLS solver (Kulikovsky, 2020a; Kulikovsky, 2021a) are based on the RC kernel, Eq. 5.

The real part of RC-circuit impedance, Eq. 1, is positive, and the imaginary part is negative. This imposes limits on functions Z, which could be represented by Eq. 5. For example, impedance of an inductive loop cannot be expanded in infinite series of RC impedances, as the imaginary part of inductive impedance is positive. Quite similarly, the impedance Z having negative real part in some frequency domain also cannot be represented by Eq. 5.

Below, we show that the DRT calculated with RC kernel could completely miss some of the transport peaks in PEMFC spectra. An alternative K ₂ kernel better capturing oxygen transport processes in the cell cathode is suggested. The kernel is illustrated by calculation of DRT of the recent analytical PEM fuel cell impedance spectrum (Kulikovsky, 2021b). Finally, we show that the new kernel well separates the channel, GDL, and ORR peaks in the DRT spectra of a standard Pt/C-based PEM fuel cell.

Below, the following dimensionless variables will be used c ̃ = c c h i n , J ̃ = J i * l t , η ̃ = η b , D ̃ b = 4 F D b c h i n i * l t 2 , l ̃ b = l b l t , ω ̃ = ω t * , Z ̃ = Z i * l t b . (8)

Here, t _* is the characteristic time of double-layer charging t * = C d l b i * , (9) c is the oxygen concentration, c h i n is the reference oxygen concentration, C _dl is the volumetric double-layer capacitance, J is the mean current density in the cell, η is the ORR overpotential, positive by convention, b is the ORR Tafel slope, i _* is the volumetric ORR exchange current density, l _t is the CCL thickness, D _b is the oxygen diffusion coefficient in the GDL, and l _b is the GDL thickness.

Analytical GDL impedance Z ̃ g d l has been derived in Kulikovsky and Shamardina (2015). For the cell current densities well below the limiting current density due to oxygen transport in the GDL, Z ̃ g d l has the form Z ̃ g d l = tanh μ l ̃ b i ω ̃ / D ̃ b μ i ω ̃ D ̃ b 1 + i ω ̃ / J ̃ (10) where μ is the constant parameter μ = 4 F c h i n C d l b , (11)

Eq. 10 has been obtained from the general expression for the cathode side impedance in the limit of infinite air flow stoichiometry (Kulikovsky and Shamardina, 2015). Eq. 10 is a Warburg finite-length impedance divided by the factor ( 1 + i ω ̃ / J ̃ ) . The factor describes the effect of double-layer charging by the cell current density J ̃ transported in the form of oxygen flux through the GDL to the attached CCL (Kulikovsky and Shamardina, 2015). In his classic work (Warburg, 1899), Warburg used static polarization curve to derive the boundary condition for calculation of transport impedance of a semi-infinite electrode [Eq. 4 of Ref. (Warburg, 1899)]. Later, Warburg model was extended for the case of finite-length transport layer, using the same boundary condition on the electrode side [see Lasia (2014), page 104]. However, account of capacitive term in the electrode charge conservation equation changes the boundary condition for the oxygen transport equation (Kulikovsky and Shamardina, 2015), leading to the additional factor ( 1 + i ω ̃ / J ̃ ) in the denominator of Eq. 10. Nyquist plot of impedance (10) is shown in Figure 1A; frequency dependence of Re Z ̃ g d l is depicted in Figure 1B. As can be seen, between 20 and 200 Hz, the real part of Z _gdl is essentially negative, and at higher frequencies Re Z ̃ g d l tends to zero. As the real part of Warburg finite-length impedance is positive, the negative real part of impedance (10) is due to the term ( 1 + i ω ̃ / J ̃ ) in the denominator.

Impedance of oxygen transport in channel, Appendix Eq. A1, and in the CCL (Kulikovsky, 2017) also exhibits negative real part (Figure 2). Last but not least, in low-Pt cells, an important role plays oxygen transport through a thin Nafion film covering Pt/C agglomerates in the CCL (Greszler et al., 2012; Weber and Kusoglu, 2014; Kongkanand and Mathias, 2016). The spectrum of this transport layer is quite similar in shape to the spectrum in Figure 1 (Kulikovsky, 2021a). Thus, all the oxygen transport processes in a PEMFC cannot be described by the standard RC kernel, and another kernel suitable for description of impedance elements with the negative real part is needed.

In a standard PEMFC, unless the cell current density is small, the DRT peaks of oxygen transport in the GDL, CCL, and channel are expected to locate at frequencies below the frequency f _ct of faradaic (charge transfer) processes. Thus, to capture the oxygen transport peaks, correction for the negative real part of impedance is needed in the range of frequencies f < f _ct . The kernel K ₂ suggested in his work consists, thus, of two parts: K 2 = tanh i ω τ i ω τ 1 + i ω τ , f ≤ f ∗ , TL kernel 1 1 + i ω τ , f > f ∗ , R C k e r n e l (12) where f _* ≃ f _ct is the threshold frequency. Selection of optimal f _* is discussed below. A function similar to impedance of a transport layer (TL) (10) forms the low-frequency part of K ₂; the real part of this function is negative at ωτ > 1.81052. The high-frequency (RC) part of K ₂ is the standard RC-circuit kernel. Switching between TL and RC kernels is necessary, as the TL part itself does not describe well RC-circuit impedance (see below). The real part of GDL impedance becomes negative at ≃20 Hz (Figure 1B), while the imaginary part at this frequency is quite large (Figure 1A). This shape of impedance around and below 20 Hz is thus far from the RC-circuit impedance and the TL kernel is needed to “recognize” this shape. At higher frequencies, transport impedances tend to zero, and the Debye kernel works better. The idea behind Eq. 12 is thus to expand the low-frequency components of cell impedance using the TL kernel and the high-frequency components using the standard RC kernel.

It is convenient to combine Eq. 12 into one function K 2 = tanh α i ω τ α i ω τ 1 + i ω τ (13) where α is a step function of the frequency f α = 1 − H f − f ∗ + ϵ , (14) H(x) is the Heaviside step function, and ϵ = 10^–10 is a small parameter to avoid zero division error. Parameter α therefore changes from 1 to 0 at the threshold frequency f = f _*. With α → 0, the Warburg factor in Eq. 13 tends to unity: tanh α i ω τ α i ω τ → 1 , as α → 0 (15) and hence the α function serves as a switch between the two kernels in Eq. 12.

With Eq. 13, 5 takes the form Z = R ∞ + R p o l ∫ − ∞ ∞ tanh α i ω τ G τ d ⁡ ln τ α i ω τ 1 + i ω τ . (16)

Setting in Eq. 16 ω → 0, it is easy to show that G still obeys to normalization conditions, Eq. 7.

Impedance of all oxygen transport processes in a PEM fuel cell exhibits negative real part in some frequency range. This makes it difficult to accurately calculate the respective DRT peaks using the standard RC kernel 1/(1 + iωτ). A novel kernel K ₂, Eq. 13, is suggested. K ₂ combines the low-frequency transport layer kernel having a domain with negative real part and the standard RC kernel for description of faradaic and high-frequency processes in the cell. Calculation of DRT for analytical PEMFC impedance shows that K ₂ kernel captures the peak due to oxygen transport in the gas-diffusion layer, whereas the RC kernel can miss this peak. Comparison of Pt/C PEMFC DRT calculated using RC and K ₂ kernel shows that the K ₂ kernel resolves the GDL oxygen transport peak, which otherwise is merged to the ORR peak when using the standard RC kernel. Overall, the K ₂ spectra of a standard Pt/C PEMFC operating at the air flow stoichiometry λ = 2 consist of five peaks. In the frequency ascending order, these peaks are due to (1) oxygen transport in channel, (2) oxygen transport in the GDL, (3) faradaic reactions, (4) oxygen transport in the CCL, and (5) proton transport in the CCL. If the CCL proton conductivity is high, the peak (6) shifts to the frequencies well above 1 kHz, and it may not be resolved due to inductance of measuring system.