Performance of a PEM fuel cell cathode catalyst layer under oscillating potential and oxygen supply

A model for impedance of a PEM fuel cell cathode catalyst layer under simultaneous application of potential and oxygen concentration perturbations is developed and solved. The resulting expression demonstrates dramatic lowering of the layer impedance under increase in the amplitude of the oxygen concentration perturbation. In--phase oscillations of the overpotential and oxygen concentration lead to formation of a fully transparent to oxygen sub--layer. This sub--layer works as an ideal non polarizable electrode, which strongly reduces the system impedance.


I. INTRODUCTION
Electrochemical impedance spectroscopy (EIS) has proven to be a unique non-destructive and non-invasive tool for fuel cells characterization 1 .In its classic variant, EIS implies application of a small-amplitude harmonic perturbation of the cell current or potential and measuring the response of the cell potential or current, respectively.In recent years, there has been interest in alternative techniques based on application of pressure (Engebretsen et al. 2 , Shirsath et al. 3 , Schiffer et al. 4 , Zhang et al. 5 ) or oxygen concentration (Sorrentino et al. [6][7][8] ) perturbation to the cell and measuring the response of electric variable (potential or current), keeping the second electric variable constant.
Application of pressure oscillations at the cathode channel inlet or outlet inevitably leads to flow velocity oscillations (FVO).Kim et al. 9 and Hwang et al. 10 reported experiments showing dramatic improvement of PEM fuel cell performance under applied FVO.The effect of FVO on the cell performance was more pronounced with lower static flow rates and with increasing the FVO amplitude 9 .In 9,10 , the effect has been attributed to improvement of diffusive oxygen transport through the cell due to FVO.Kulikovsky 11,12 developed a simplified analytical model for impedance of the cell subjected to simultaneous oscillations of potential and air flow velocity.The model have shown reduction of the cell static resistivity upon increase of the FVO amplitude.Yet, however, due to system complexity, the mechanism of cell performance improvement is not clear.
Below, a much simpler system (PEM fuel cell cathode catalyst layer) subjected to oscillating in-phase potential and oxygen supply is considered.Analytical model for the CCL impedance under these conditions is solved.The result demonstrates the effect of impedance reduction due to oscillating oxygen supply.In-phase oxygen concentration and a) Electronic mail: A.Kulikovsky@fz-juelich.de  overpotential oscillations make part of the catalyst layer at the cathode catalyst layer (CCL)/gas diffusion layer (GDL) interface fully transparent to oxygen, which leads to dramatic decrease of the system impedance.

II. MODEL
Consider a problem for impedance of the cathode catalyst layer (CCL) under oscillating potential and oxygen supply (Figure 1).For simplicity, we will assume that the proton transport is fast.The model is based on two equations: the proton charge conservation and the oxygen mass transport equation Here, x is the distance through the CCL, C dl is the double layer capacitance, η is the positive by convention ORR over-arXiv:2310.08945v1[physics.chem-ph]13 Oct 2023 potential, t is time, j is the local proton current density, x is the coordinate through the CCL, i * is the ORR volumetric exchange current density, c ref is the reference oxygen concentration, and b is the ORR Tafel slope.Introducing dimensionless variables where ω is the angular frequency of applied AC signal, Z is the CCL impedance, Eqs.(1), (2) take the form where µ is the constant parameter Substituting Fourier-transforms of the form into Eqs.(4), (5), neglecting terms with the perturbation products and subtracting the respective static equations, we get a system of equations relating the perturbation amplitudes η1 1 (ω), j1 (x, ω) and c1 (x, ω) 13 : where the superscripts 0 and 1 mark the static variables and the small perturbation amplitudes, respectively, c0 1 is the static oxygen concentration at the CCL/GDL interface, k ≥ 0 is the constant model parameter, η1 1 is the amplitude of applied potential perturbation.
The boundary condition for Eq.( 8) means zero proton current at the CCL/GDL interface.The left boundary condition for Eq.( 9) expresses zero oxygen flux through the membrane.The feature of this problem is the right boundary condition for Eq.( 9), meaning external control of oxygen concentration perturbation c1 at the CCL/GDL interface: c1 (1) varies in-phase with the overpotential perturbation η1 1 .Note that the perturbations of applied cell potential and overpotential have different signs, assuming that the electron conductivity of the cell components is much larger than the CCVL proton conductivity.
Due to fast proton transport, η1 1 is nearly independent of x.For simplicity we will also assume that the variation of static oxygen concentration along x is also small and we set c0 ≃ c0 1 .Introducing electric Y and concentration G admittances and taking into account the static polarization curve j0 = c0 1 exp η0 (11)   we can rewrite the system (8), ( 9) in terms of Y and G: Eq.( 13) can be directly solved: Using this solution and solving Eq.( 12), for the CCL impedance Z = 1/Y | x=0 we get where ZRC is the parallel RC-circuit impedance, and ZW is the Warburg-like impedance: CCL impedance Z, Eq.( 15), for the current density of 100 mA cm −2 and the indicated values of parameter k is shown in Figure 2. The other parameters are listed in Table I.Note that to emphasize the effect, the CCL oxygen diffusivity is taken to be very low, D ox = 10 −5 cm 2 s −1 .
The Nyquist spectrum corresponding to k = 0 (no concentration perturbation at the CCL/GDL interface) consists of two partly overlapping arcs, of which the left one is due to charge transfer and the right one is due to oxygen transport (Figure 2a).In Figure 2b, the left peak of − Im (Z) (red curve) corresponds to oxygen transport and the right shoulder is due to faradaic reaction.
The perturbation of oxygen concentration at the CCL/GDL interface dramatically changes the Nyquist spectra.With the growth of k, the arc due to oxygen transport strongly decreases (Figure 2).Calculation of Eq.( 15) limit as ω → 0 gives the CCL static resistivity which decreases as k increases (Figure 2a).
Figure 3a shows what happens to the x-shape of the phase shift between the oxygen concentration c1 and overpotential η1 1 (the phase angle of the function G(x) = c1 /η1 1 ) as k increases.At k = 0, there is a large phase shift between c1 and η1 1 , excluding a single point at x = 1, where c1 = 0.With k = 0.5, a finite-thickness domain where c1 and η1 1 oscillate almost in-phase forms, and with k = 0.95 this domain of zero phase shift increases (Figure 3).Zero phase angle between η1 1 and c1 means no oxygen transport losses in this domain.
Figure 3b shows the modulus of the right side of Eq.( 12), which is the modulus of the ORR rate perturbation.As can be seen, in the domain transparent for oxygen, the ORR rate strongly increases.Thus, with k > 0, a relatively thin sub-layer at the CCL/GDL interface forms, which works as an ideal non polarizable catalyst layer with fast oxygen transport and large ORR rate.The concerted action of both effects leads to dramatic lowering of the CCL transport impedance.
Further increase in k changes the sign of the phase angle at the membrane interface, meaning formation of inductive loop in the Nyquist spectrum.In this state, the static resistivity drops below the charge-transfer value under constant oxygen supply R ct = b/j; this regime deserves special consideration and it is not discussed here.The limiting value k lim at which R = Rct = 1/ j0 is The states with k > k lim could be unstable, which needs further studies.From practical perspective, in a real fuel cell it is not feasible to directly perturb the oxygen concentration at the CCL/GDL interface in-phase with the overpotential perturbation.However, the experiments of Kim et al. 9 and Hwang et al. 10 suggest that the effect could be achieved by perturbing air flow velocity in the channel.Such a perturbation could cause in-phase oscillations of the oxygen concentration and overpotential, which greatly improves the transient cell performance.η ORR overpotential, positive by convention, V η 1 1 Amplitude of applied overpotential perturbation, V µ Dimensionless parameter, Eq.( 6) ϕ Dimensionless parameter, Eq.( 17) ψ Dimensionless parameter, Eq.( 14) ω Angular frequency of the AC signal, s −1

FIG. 1 .
FIG.1.Schematic of the cathode catalyst layer and typical shapes of the oxygen concentration, proton current and overpotential through the layer.In-phase harmonic perturbations of the overpotential η 1 1 and oxygen concentration c 1 1 are applied at the CCL/GDL interface.

FIG. 3 . 3
FIG. 2. (a) Nyquist spectra of the CCL, Eq.(15), for the indicated values of parameter k.The other parameters are listed in Table I.The point Rct indicates the static chargetransfer resistivity b/j0.(b) The frequency dependence of the impedances in (a).

TABLE I .
The cell parameters used in calculations.Parameter Dox is taken to be small to emphasize the effect of oxygen transport in the CCL.C dl Double layer volumetric capacitance, F cm −3 c Oxygen molar concentration, mol cm −3 c ref Reference oxygen concentration, mol cm −3 Dox Oxygen diffusion coefficient in the CCL, cm 2 s −1