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

We report a model of the cathode catalyst layer (CCL) impedance, which includes impedances due to oxygen transport in the Nafion film covering Pt/C agglomerates, and due to oxygen transport through the CCL depth. In the case of small cell current density, analytical solutions for the CCL impedance Zccl are derived; for larger currents, we analyze numerical solution for Zccl . The characteristic frequencies of the oxygen transport through the Nafion film and through the CCL depth are close to each other, and the contribution of the Nafion film impedance Z N to Zccl is small up to the current densities ≃ 100 mA cm−2. This makes it difficult reliable determination of Z N from experimental spectra of a standard 10 μm–thick CCL. However, with the decrease in the CCL thickness, the relative contribution of Z N to Zccl increases. It gives us a chance to determine Z N by fitting the models of this work to measured spectra of a low–loaded MEA with a thin CCL. An example of fitting the model to a synthetic numerical impedance is given. © The Author(s) 2017. Published by ECS. This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 License (CC BY, http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse of the work in any medium, provided the original work is properly cited. [DOI: 10.1149/2.1261704jes] All rights reserved.

Polymer electrolyte membrane fuel cells are ready to take their place in the spectrum of electrochemical sources powering the future society.Still, however, a lot needs to be done to improve stability and reduce the cost of these cells.Development of simple and robust ex situ characterization techniques for the fuel cell research and applications (cars, home appliances etc.) is an urgent task.
PEMFC impedance is determined mainly by the cathode catalyst layer (CCL).The kinetics of oxygen reduction reaction (ORR) are sluggish and the oxygen transport through the catalyst layer is slow; thus, the CCL gives the major contribution to the cell impedance.SEM pictures show that a typical CCL is a porous structure formed by numerous spherical agglomerates of Pt/C particles covered by Nafion film. 20,21Recent experiments of Singh et al. 22 have shown that the addition of Nafion significantly changes the catalyst layer impedance.At medium and large overpotentials, the presence of Nafion leads to formation of the second, low-frequency arc in the impedance spectrum of the electrode, which is associated by the authors with the oxygen transport in Nafion film covering Pt/C agglomerates. 22he nature of the oxygen transport loss in the CCL is still controversial.Basically, this loss can be associated with the oxygen transport in agglomerates, and with the oxygen transport through the CCL depth.What is the contribution of each mechanism into the total transport loss in the CCL, and how can we separate and quantify each mechanism?Answering these questions is a challenge for the fuel cell modeling community.
Numerous papers have been devoted to static polarization curve modeling, taking into account Nafion-covered Pt/C agglomerates; references can be found in a recent work. 23One of the first physicsbased models for PEMFC cathode impedance has been developed by Springer et al. 3 The underlying transient model for the CCL performance was based on standard macro-homogeneous equations for the charge and oxygen mass conservation; no explicit account of the oxygen transport in agglomerates was done.A first attempt to incorporate the effect of oxygen transport in Pt/C agglomerates has seemingly been done by Raistrick. 24He considered a planar agglomerate at the surface of a pore having a form of a slit.An equation for the perturbed oxygen concentration in the agglomerate has been derived; the solution for the pore impedance was obtained numerically.Recent SEM pictures show, however, that the agglomerates are spherical rather than the planar structures. 20,21Jaouen and Lindbergh 5 considered oxygen transport in spherical agglomerates in their impedance model of the CCL; however, they neglected oxygen transport through the electrode depth.The electrode impedance was calculated numerically in their work.Similar numerical impedance model with neglect of the oxygen transport through the CCL depth has been developed by Gerteisen et al. 8 It is worth noting that Gerteisen et al. assumed that the agglomerates are fully filled with Nafion.In a recent paper, Gerteisen developed a numerical impedance model of the CCL, which includes oxygen transport through the CCL depth and through the Nafion film covering agglomerate. 25However, the transport in the Nafion film/agglomerate was assumed to be infinitely fast, and hence this transport contributes to the real component of impedance only, i.e., it merely shifts the impedance spectrum as a whole along the real axis.
In this work, we report a model of the CCL impedance, which includes transient oxygen transport equations in the spherical Pt/C agglomerates and through the CCL depth.The agglomerate is assumed to be covered by a thin spherical Nafion film, while the agglomerate interior volume is filled with water.First, we construct a system of transient equations for the CCL performance; this system is then linearized and Fourier-transformed to yield a system of equations for the AC perturbation amplitudes.In the case of small cell current density, this system is solved and the analytical expression for the CCL impedance is derived.For larger currents (see below), we analyze numerical solution for the CCL impedance.

Impedance Model
A conceptual picture of the CCL structure suggested by Liu et al. 26 is depicted in Figure 1.Spherical agglomerates of Pt/C particles covered by Nafion film form a contiguous cluster, which provides proton transport to the catalyst sites.The void space between agglomerates serves as a pathway for oxygen transport through the CCL depth.To reach Pt particles, oxygen must be dissolved and transported through the Nafion film.Nafion film be R a , and the thickness of the Nafion film be d N (Figure 2).
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.
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. 27This 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 [11]   and the pore (CCL) model where the parameters δ, β and ε are given by and the flux ÑN 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 (r = 1 − ǫ), and symmetry of the oxygen concentration profile at r = 0: Here, is the dimensionless thickness of the Nafion film, and c * 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 cox 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.
Linearization and Fourier transform.-Toobtain 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 ω = ωt * 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 1 [31]   As can be seen, the problem 23 Due to agglomerate smallness, the overpotential η0 is assumed to be independent of r .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 c1 a and c1 N 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 Ñ 1 N trough the Nafion film outer surface, Eq. 27.This flux can be represented as Unfortunately, the exact expressions for the coefficients Ñ 1 η and Ñ 1 ox are very cumbersome.However, in this problem, Da and DN are large, while ǫ is small.Performing asymptotic expansion of the exact Ñ 1 N (1) over Da , Taylor series expansion over ǫ, and neglecting the terms on the order of Note that at leading order, the terms with D−1 a vanish, as Da ≫ DN .Note also that in the limit of DN →∞ ,E q s .35,36 simplify to Ñ 1 η ≃ K ox c0 ox exp( η0 )/3, Ñ 1 ox ≃ K ox exp( η0 ) + iω /3 ,a n di nE q s .25, 26 we get the standard macro-homogeneous expressions for the Tafel ORR rate.

Results and Discussion
Fast oxygen transport through the CCL depth.-Beforewe 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 c1 ox = 0 (fast oxygen transport), and c0 = c1 = const, η0 = η0 = const (small cell current density).With these changes, Ñ 1 ox = 0, Ñ 1 η is independent of x 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 i ss h o w ni nF i g u r e4 together with the charge-transfer and proton transport impedance Zct+p corresponding to zero oxygen transport losses in the agglomerate.The spectrum of Zct+p is obtained from Eq. 38 by passing to the limit DN →∞ .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 Zct+p we find 12 Zct+p =− In dimension form this impedance reads 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 Rccl .In PEMFCs, the parameter ε is large, typically ε ≃ 10 2 -10 3 .Setting in Eq. 38 ω = 0, expanding the result over ǫ and calculating asymptotic expansion over ε, 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 Zct+p from Zccl , we get an explicit expression for the Nafion film impedance where φ and φ 0 are given in Eqs.38 and 40.Eq.44 can be simplified: expanding the right side over small ǫ, and calculating asymptotic  I.
expansion of the result over large ε, 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.
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: ∂Im Z N /∂ ω ω= ωN = 0. Calculating this derivative with Eq. 45, we come to ωN = ˜j0 where c * K ox c1 .With the parameters in Table I and j 0 = 40 mA cm −2 , for the regular frequency f N = ω N /(2π)w eg e t f N ≃ 6 Hz.

Finite rate of oxygen transport through the CCL depth.-Small cell current density.-The cell current density is small, if it obeys
Physically, j 0 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 by 28 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 ,andZ ox is obtained in Ref. 28 as a solution of the CCL problem neglecting Z N .In the limit of small j 0 , the total impedance is a sum of all the three impedances, Eq. 49. Figure 6 illustrates this statement.Further, if j 0 is small, we can simply add to Eq. 43 the term describing the CCL resistivity due to oxygen transport through the CCL depth 28 , which yields [53] This equation can also be obtained by passing to the limit ω = 0in 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 ≃ 8mOhmcm 2 and R ox ≃ 35 mOhm cm 2 , 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 . 19In 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 by 28 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 cm 2 s −1 , which is quite a realistic value for standard Pt/C electrodes. 19In that case both the transport losses are indistinguishable by impedance methods.
Medium cell current density.-Eqs.35, 36 are derived assuming that Da is large.In 32, Da appears in the combination exp( η0 )/ Da .Thus, the approximation of large Da works as long as exp( η0 ) ≪ Da ; 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 Ñ 1 N ,E q s .34contains the x-dependent static shapes of η0 and c0 .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 3 DN ∂ c0 N /∂ r | r =1 with c0 N 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).The parameters for calculations are listed in Table I.I.
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.
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.
Fitting.-Finally, the following numerical experiment has been performed.An impedance spectrum for the current density j 0 = 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.
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    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 throughplane 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. 29This 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.

Conclusions
A physics-based model for impedance of the cathode catalyst layer is developed.The model takes into account oxygen transport through the CCL depth, and through the Nafion film covering agglomerates of Pt/C particles.Transient mass and charge conservation equations are linearized and Fourier-transformed to get a system of linear equations for small perturbation amplitudes.In the case of small cell current density and fast oxygen transport through the CCL, the system is solved and analytical expressions for the CCL impedance and static resistivity are derived.With typical CCL parameters, at small currents, the contribution of the Nafion film to the CCL impedance appears to be small.Furthermore, for the Nafion film thickness of 10 nm and through-plane oxygen diffusivity on the order of 10 −5 cm 2 s −1 , the characteristic frequencies of the oxygen transport in the Nafion film and in the CCL pores are nearly the same, and the respective impedances cannot be separated.For larger current densities, the system of equations for perturbation amplitudes is solved numerically.With the growth of the cell current, the contribution of Nafion film impedance increases up to 10% of the total CCL impedance.The Nafion film impedance can best be measured by fitting the model equations of this work to impedance of a low-loaded, thin CCL, in which the effect of the though-plane oxygen transport is small.

Figure 1 .
Figure 1.Conceptual sketch of a PEMFC electrode composed of catalyzed high-surface area high-structure carbon-support and proton-conducting ionomer.Grayand black-shaded structures: primary carbon-support agglomerates with high electronic conductivity; blue dots: catalytically active platinum or platinum-alloy nanoparticles; red lines and red block: proton-conducting ionomer films on the carbon-support and ionomeric membrane; gray-shaded blocks: electrona n d gas-conducting gas diffusion medium and micro-porous layer (MPL).Reprinted with permission from Ref. 21.

Figure 2 .
Figure 2. Cartoon of the spherical agglomerate of Pt/C particles covered by a Nafion film of the thickness d N .

Figure 3 .
Figure 3. Schematic of a CCL pore; the pore walls are formed by Pt/C agglomerates.

Figure 4 .
Figure 4. Line -the charge-transfer plus proton transport spectrum corresponding to zero transport losses in the agglomerate.Points -the spectrum including the Nafion film thickness of 30 nm.The ORR overpotential is 420 mV, which corresponds to the cell current density of 40 mA cm −2 The other parameters are listed in TableI.Note the coordinates ˜j0 Z .

Figure 5 .
Figure5.Points -the exact spectrum of Nafion film impedance, Eq. 44, line -the asymptotic expression for this spectrum, Eq. 46.Parameters for the calculation are listed in TableI.

Figure 7 .
Figure7.Lines -the spectra corresponding to zero transport losses in the Nafion film.Points -the spectra, which include the Nafion film impedance.The parameters for calculations are listed in TableI.

Figure 8 .
Figure 8. Solid line -the total CCL impedance, short-dashed line -the charge-transfer impedance, including proton transport impedance, longdashed line -the through-plane oxygen transport impedance, dotted linethe Nafion film impedance.The cell current density is 240 mA cm −2 ,t h e parameters for calculations are listed in TableI.
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

Solution for the local oxygen flux.-The static
shapes of the oxygen concentration in the agglomerate and in Nafion film are c0 a -26 is split into the local problem for c1 a and c1 N in a single agglomerate and the global problem for the through-plane shapes of c1 ox and η1 .The local problem is given by Eqs. 23, 24 with η1 ( x) being a parameter.The global CCL problem is given by Eqs. 25, 26, with Ñ 1 N resulting from solution of the local problem.

Table I . Parameters used in calculations. To emphasize the effects of oxygen transport in the Nafion film, the film thickness is taken to be 30 nm.
D a ,cm 2 s −1 (Ref.30)4 • 10 −5 Oxygen diffusion coefficient in Nafion film D N ,cm 2 s −1 (Ref.31)0 .85• 10 −6 Oxygen diffusion coefficient through the CCL depth D ox ,cm 2 s −1 (upper estimate) 10 −4 Exchange current density i * ,Acm −3

Table II . Fitted and prescribed (indicated in brackets) parameters. The other parameters are listed in Table I.
Symbols ˜Marks dimensionless variables ˆMarks dimensionless variables b ORR Tafel slope b = RT/αF,V C dl Double layer volumetric capacitance, F cm −3 c ox Oxygen molar concentration in the CCL pores, mol cm −3 c a Oxygen molar concentration in the agglomerate, mol cm −3 c N Oxygen molar concentration in Nafion film, mol cm −3 c ref Reference oxygen molar concentration, mol cm −3 D ox Effective oxygen diffusion coefficient in the CCL pores, cm 2 s −1 D a Oxygen diffusion coefficient in the agglomerate, cm 2 s −1 D N Oxygen diffusion coefficient in the Nafion film, cm 2 s −1 Greek α N , β N Dimensionless coefficients in Eq. 33 δDimensionless ratio δ = R a /l t ,Eq.14 ǫDimensionless ratio ǫ = l N /R a Angular frequency (ω = 2πf ), s −1 ω ox Characteristic frequency of oxygen transport in the CCL, s −1 ,Eq.54 ω N Characteristic frequency of oxygen transport in the Nafion film, s −1 ,Eq.47