Can We Quantify Oxygen Transport in the Nafion Film Covering an Agglomerate of Pt/C Particles?

An elementary unit of the structure in Figure 1 providing electrochemical conversion is a single agglomerate. Below, we will consider a spherical agglomerate of Pt/C particles, covered by a thin Nafion film. Let the agglomerate radius, including Nafion film be R_a, and the thickness of the Nafion film be d_N (Figure 2).

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To derive a model for the CCL impedance, we need transient conservation equations for oxygen in agglomerate and a transient performance model for the whole CCL. The oxygen transport equations in the Nafion film and in the agglomerate read

where t is time, r is the radial coordinate, D_N and D_a are the oxygen diffusion coefficients in the Nafion film and in the agglomerate, respectively, c_N, c_a are the oxygen concentrations in the Nafion and agglomerate, respectively, c_ref is the reference oxygen concentration, i_* is the electrode volumetric exchange current density (A cm^{− 3}), η is the ORR overpotential, positive by convention, and b is the ORR Tafel slope. Note that Eq. 1 contains the ORR rate on the right side, which means that the reaction runs also in the Nafion film (see discussion below).

To link the CCL and agglomerate models, consider a cylindrical pore of the radius R_p in the catalyst layer, with the pore walls formed by agglomerates (Figure 3). The oxygen mass transport equation along the pore reads

where x is directed along the pore axis, and ρ is the radial position in cylindrical coordinates (ρ, x). The right side of Eq. 3 is the divergence of the oxygen flux through the unit surface of the pore cylinder.

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The meaning of the divergence operator in Eq. 3 needs to be specified. This operator represents a divergence of the oxygen flux through the side wall of the cylindrical pore formed by agglomerates partly penetrating into the pore volume (Figure 3). To calculate this divergence we note, that in the limit of D_a = D_N = ∞, the right side of Eq. 3 must transform to the standard macro–homogeneous ORR rate.²⁷ This requirement leads to the following relation

where

is the oxygen flux through the unit agglomerate surface. With this, Eq. 3 transforms to

Below, we will see that the right side of Eq. 6 depends on the local overpotential η, which varies with x. The proton charge balance equation, including Ohm's law for the proton transport is

Here, the right side describes the "sink" of overpotential consistent with the sink of oxygen in Eq. 6. Physically, the rates of oxygen consumption and proton current conversion along the pore axis must be coupled by the Faraday law. The chain of Equations 1, 2, 6 and 7 describe oxygen transport along the void pore in the CCL, with the sink through the Nafion film to the water–filled agglomerates of Pt/C particles.

It is convenient to introduce dimensionless variables

where the characteristic time t_* is given by

With the dimensionless variables 8, the system 1, 2, 6, 7 transforms to the agglomerate model

and the pore (CCL) model

where the parameters δ, β and ε are given by

and the flux is nondimensionalized according to

The boundary conditions to the system 10, 11 express continuity of the oxygen concentration and flux at the Nafion/agglomerate interior interface (), and symmetry of the oxygen concentration profile at :

Here,

is the dimensionless thickness of the Nafion film, and is the dimensionless concentration of oxygen dissolved in Nafion at the film outer surface. This concentration is related to the gaseous oxygen concentration in the void pores of the CCL by the Henry's law with the constant K_ox (mol/mol):

The boundary conditions to Eqs. 12 and 13 are

The first of Eq. 20 means zero oxygen flux in the membrane, while the second fixes the oxygen concentration at the CCL/GDL interface. The first of Eq. 21 fixes the total ORR overpotential at the membrane interface, and the second expresses zero proton current to the GDL.

To obtain the CCL impedance, we apply a small–amplitude harmonic perturbation of potential to the CCL. Due to smallness of the AC signal, the response of the system is linear and harmonic, and we can write

where is the dimensionless angular frequency of the applied signal, and the superscripts 0 and 1 indicate the steady–state profile and the small–amplitude perturbation, respectively.

The equations for static shapes are obtained from Eqs. 10–13 by chalking out the time derivatives. Substituting 22 into the system 10–13, subtracting the static equations and neglecting the terms with the perturbation products, we get a system of linear equations for the perturbation amplitudes of the agglomerate model

and for these amplitudes of the CCL model

Here,

is the flux of dissolved oxygen perturbation through the outer surface of Nafion film. The boundary conditions to the system 23–26 are

and

As can be seen, the problem 23–26 is split into the local problem for and in a single agglomerate and the global problem for the through–plane shapes of and . The local problem is given by Eqs. 23, 24 with being a parameter. The global CCL problem is given by Eqs. 25, 26, with resulting from solution of the local problem.

The static shapes of the oxygen concentration in the agglomerate and in Nafion film are

Due to agglomerate smallness, the overpotential is assumed to be independent of . The expressions for coefficients in Eqs. 32, 33 are very cumbersome and not displayed here. Exact solution to the problem 23, 24 for the perturbation amplitudes and can be obtained with the aid of mathematical software. For the CCL problem 25, 26 we need only an expression for the perturbation of the oxygen concentration flux trough the Nafion film outer surface, Eq. 27. This flux can be represented as

Unfortunately, the exact expressions for the coefficients and are very cumbersome. However, in this problem, and are large, while is small.Performing asymptotic expansion of the exact over , Taylor series expansion over , and neglecting the terms on the order of , and O(²), we come to a compact result

Note that at leading order, the terms with vanish, as . Note also that in the limit of , Eqs. 35, 36 simplify to , , and in Eqs. 25, 26 we get the standard macro–homogeneous expressions for the Tafel ORR rate.

Before we proceed to numerical solution of the CCL system 25, 26, it is advisable to consider the limit of fast oxygen transport through the CCL depth. Note that considering this limit only makes sense if the cell current density is small; the respective criterium is given in the next section. In this case, in the system 25, 26 we can set (fast oxygen transport), and , (small cell current density). With these changes, , is independent of parameter and Eq. 26 can easily be solved. By definition, the CCL impedance is

Using here the solution to Eq. 26 we get

The spectrum of impedance Eq. 38 is shown in Figure 4 together with the charge–transfer and proton transport impedance corresponding to zero oxygen transport losses in the agglomerate. The spectrum of is obtained from Eq. 38 by passing to the limit . Noting that at small cell currents, the polarization curve of the CCL is given by the Tafel equation

for the coupled charge–transfer and proton transport impedance we find¹²

In dimension form this impedance reads

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The charge–transfer and the total Z_ccl spectra are very close to each other (Figure 4). Note that the difference is visible due to the upper estimate of the Nafion film thickness of 30 nm taken for the calculations; for a more realistic value of d_N = 10 nm, the spectra in Figure 4 would be indistinguishable. Nonetheless, Figure 4 gives us a hope that for higher currents and lower oxygen concentrations, the effect of Nafion film could be measurable.

From Eq. 38 we can obtain the formula for the CCL static resistivity . In PEMFCs, the parameter is large, typically ≃ 10²–10³. Setting in Eq. 38 , expanding the result over and calculating asymptotic expansion over , at leading order we get

In dimension form, Eq. 42 reads

The first term on the right side is the CCL resistivity to proton transport. The second term is the Faraday charge–transfer resistivity. The third term represents the oxygen transport resistivity of the Nafion film.

Remarkably, the Nafion film resistivity is inversely proportional to the CCL thickness l_t, Eq. 43. Figure 3 helps to understand this effect. Indeed, individual agglomerates are connected to the pore volume as parallel resistivities; thus, the longer the representative pore, the more agglomerates we have along the pore, and the smaller the total resistivity of the agglomerates ensemble.

Subtracting from , we get an explicit expression for the Nafion film impedance

where ϕ and ϕ₀ are given in Eqs. 38 and 40. Eq. 44 can be simplified: expanding the right side over small , and calculating asymptotic expansion of the result over large , at leading order we find

In dimension form, Eq. 45 reads

Figure 5 shows that Eq. 46 provides a good approximation of the exact expression 44.

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From Eq. 45 we can get the characteristic (summit) frequency ω_N of the Nafion film impedance (Figure 5). Clearly, at this frequency the following relation holds: . Calculating this derivative with Eq. 45, we come to

where . With the parameters in Table I and j₀ = 40 mA cm^{− 2}, for the regular frequency f_N = ω_N/(2π) we get f_N ≃ 6 Hz.

Small cell current density

The cell current density is small, if it obeys to¹⁹

Physically, j₀ must be much less than the characteristic current densities for the proton transport j_p and the oxygen transport j_ox in the CCL. If Eq. 48 holds, the CCL impedance is well approximated by the sum of the three impedances:

where Z_{ct + p} and Z_N are given by Eqs. 41 and 46, respectively. The low–current oxygen transport impedance Z_ox is given by²⁸

where

are the characteristic frequencies, and

is the Warburg–like impedance. In Eq. 49, Z_N is obtained as a solution of the CCL problem neglecting Z_ox, and Z_ox is obtained in Ref. 28 as a solution of the CCL problem neglecting Z_N. In the limit of small j₀, the total impedance is a sum of all the three impedances, Eq. 49. Figure 6 illustrates this statement.

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Further, if j₀ is small, we can simply add to Eq. 43 the term describing the CCL resistivity due to oxygen transport through the CCL depth²⁸, which yields

This equation can also be obtained by passing to the limit ω = 0 in Eq. 49.

It is advisable to compare the third and fourth terms in Eq. 53. With the data from Table I, for these terms we get R_N ≃ 8 mOhm cm² and R_ox ≃ 35 mOhm cm², respectively. Thus, the Nafion film resistivity is more than four times lower, than the through–plane oxygen transport resistivity. Note that the through–plane oxygen difffusivity D_ox taken for the estimate is an upper value; standard MEA may exhibit three to four times lower D_ox.¹⁹ In this case, R_N would be more than an order of magnitude lower than R_ox.

It is useful to compare the characteristic summit frequencies of the oxygen transport in Nafion film, Eq. 47 and of the through–plane oxygen transport. The latter frequency is given by²⁸

With the data from Table I and the current density of 50 mA cm^{− 2}, we get f_ox = ω_ox/(2π) ≃ 40 Hz. The value of f_N estimated above is 6 Hz, i.e., the Nafion film and through–plane transport impedances are well separated in the frequency domain. However, the frequency gap between f_N and f_ox would be zero for the oxygen diffusion coefficient D_ox ≃ 1.5 · 10^{− 5} cm² s^{− 1}, which is quite a realistic value for standard Pt/C electrodes.¹⁹ In that case both the transport losses are indistinguishable by impedance methods.

Medium cell current density

Eqs. 35, 36 are derived assuming that is large. In 32, appears in the combination . Thus, the approximation of large works as long as ; this limits the cell current density by the value of 250 mA cm^{− 2}. For the currents just below this value, we have to solve the Equations 25, 26, taking into account that the expression for the flux , Eqs. 34 contains the –dependent static shapes of and . These shapes obey to the static version of Eqs. 12, 13 with the right sides being the local ORR rate in the agglomerate:

Here, the right side is calculated as with given by Eq. 33.

The CCL impedance spectra resulting from solution of the general problem 25, 26 with the static shapes from Eqs. 55, 56 are shown in Figure 7. As can be seen, the Nafion film contribution to the CCL impedance is small but visible already at the cell current density on the order of 100 mA cm^{− 2}, and it increases with the current (Figure 7).

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Figure 8 shows the components of the total CCL impedance at the current density of 235 mA cm^{− 2}. The largest contribution gives the oxygen transport through the CCL depth; the contribution of the Nafion film impedance is about 10% (Figure 8). The impedances in Figure 8 are represented as separate arcs by shifting Z_ox and Z_N to the right along the real axis by the resistivities R_{ct + p} and R_{ct + p} + R_ox, respectively. This can be done using the model equations; however, separation of the impedances from the experimental spectrum is a much more difficult task. We note again, that the separation of Z_N and Z_ox is only possible if the characteristic frequencies ω_N and ω_ox are different.

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It should also be noted that the spectra in Figure 8 are plotted for the upper estimates of D_ox and d_N (Table I). For more realistic values of three to four times lower D_ox and three times lower d_N, the contribution of Z_N to the total impedance would be less than 1%. The experimental impedance spectra becomes progressively noisy with the increase in the cell current density, and reliable determination of Z_N with the standard 10 μm–thick CCL seems to be rather problematic. However, Eq. 53 shows that with the decrease in the CCL thickness l_t, the contribution of Z_ox to the total CCL impedance decreases, while the contribution of Z_N increases. This gives us a chance to determine Z_N by fitting the impedance models above to experimental spectra from MEA with the thin, low–loaded catalyst layers.

Finally, the following numerical experiment has been performed. An impedance spectrum for the current density j₀ = 100 mA cm^{− 2} has been generated using Eq. 49 and the parameters indicated in brackets in Table II. The total spectrum and its components are shown in Figure 9a. Then, the imaginary part of the spectrum has been perturbed by adding a 3% random noise and the model of Eq. 49 has been fitted to the perturbed spectrum. The fitting has been performed using the Maple least–squares procedure NonlinearFit.

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The perturbed and fitted spectra, and the components of the fitted spectrum are shown in Figure 9b. As can be seen, the fitted spectrum and its components are quite close to the "exact" spectra in Figure 9a. The fitting and prescribed parameters are listed in Table II. The Nafion film impedance, Eq. 46, depends on the ratio of the parameters d_N/D_N; hence the parameter K_N = d_N/D_N has been introduced and claimed as a fitting parameter. Fitting nicely captures the Tafel slope, the through–plane oxygen diffisivity and the double layer capacitance (Table II). The fitted values of σ_p and K_N are less accurate; nonetheless, these values are determined with 10% accuracy. Taking into account a very small value of Z_N (Figure 9a), this "experiment" gives us a hope that even small Z_N can be found from fitting real experimental spectra.

To conclude discussions, a following note should be made. The non–zero ORR rate in the Nafion film is justified by the following arguments. Formally, the "gap" in the ORR rate due to Nafion film would make it difficult to link the local agglomerate and the through–plane problems. In the present formulation, a simple equation for the divergence of the oxygen flux in Eq. 3 results from condition that in the limit of infinite D_a and D_N, this divergence should reduce to the macro–homogeneous expression i_*(c/c_ref)exp (η/b). In the absence of ORR rate in the Nafion film, this condition does not hold, as the "gap" in the oxygen conversion inside the Nafion film contradicts to the macro–homogeneous paradigm. Second, in real fuel cells, Pt is dissolved during operation and it migrates to membrane, where it is re–deposited, sometimes forming quite a dense band.²⁹ This process may lead to deposition of Pt in the Nafion film covering agglomerates. As this film is thin, Pt particles may have an electric contact with the main cluster of Pt/C particles, and hence ORR would run also in the film.

List of Symbols

	Marks dimensionless variables
	Marks dimensionless variables
b	ORR Tafel slope b = RT/αF, V
Cdl	Double layer volumetric capacitance, F cm− 3
cox	Oxygen molar concentration in the CCL pores, mol cm− 3
ca	Oxygen molar concentration in the agglomerate, mol cm− 3
cN	Oxygen molar concentration in Nafion film, mol cm− 3
cref	Reference oxygen molar concentration, mol cm− 3
Dox	Effective oxygen diffusion coefficient in the CCL pores, cm2 s− 1
Da	Oxygen diffusion coefficient in the agglomerate, cm2 s− 1
DN	Oxygen diffusion coefficient in the Nafion film, cm2 s− 1
dN	Nafion film thickness, cm
F	Faraday constant, C mol− 1
f	Regular frequency, Hz
j	Local proton current density in the CCL, A cm− 2
j0	Cell current density, A cm− 2
i	Imaginary unit
i*	Volumetric exchange current density, A cm− 3
lt	Catalyst layer thickness, cm
NN	Oxygen flux through the outer side of Nafion film, mol cm− 2 s− 1
N1η, Nox1	Coefficients in Eq. 34
Ra	Agglomerate radius, cm
Rccl	Static differential resistivity of the CCL, Ω cm2
r	Radial coordinate in the agglomerate, cm
t	Time, s
t*	Characteristic time, s, Eq. 9
x	Coordinate through the CCL, cm
Z	Total impedance of the cathode side, Ω cm2
Zccl	CCL impedance, Ω cm2
Zct + p	Charge–transfer and proton transport impedance, Ω cm2
ZN	Nafion film impedance, Ω cm2

Greek

αN, βN	Dimensionless coefficients in Eq. 33
δ	Dimensionless ratio δ = Ra/lt, Eq. 14
	Dimensionless ratio = lN/Ra
	Newman's dimensionless reaction penetration depth, Eq. 14
η	Local ORR overpotential (positive by convention), V
β	Dimensionless parameter, Eq. 14
σp	CCL ionic conductivity, Ω− 1 cm− 1
ϕ	Dimensionless parameter, Eq. 38
ω	Angular frequency (ω = 2πf), s− 1
ωox	Characteristic frequency of oxygen transport in the CCL, s− 1, Eq. 54
ωN	Characteristic frequency of oxygen transport in the Nafion film, s− 1, Eq. 47

Subscripts

Superscripts