Modeling the Morphological Effects of Catalyst and Ionomer Loading on Porous Carbon Supports of PEMFC

We consider a CCL of constant thickness d_ccl spread over the cell active area A_ccl and hence with the volume

$$\begin{eqnarray}&&{V}_{{ccl}}={A}_{{ccl}}{d}_{{ccl}}\end{eqnarray} \tag{ 1 }$$

From the given platinum loading L_Pt , we can calculate the total Pt mass within the CCL

$$\begin{eqnarray}&&{m}_{{Pt},{total}}={L}_{{Pt}}{A}_{{ccl}}\end{eqnarray} \tag{ 2 }$$

and subsequently the total mass of solid carbon

$$\begin{eqnarray}&&{m}_{C,{total}}=\displaystyle \frac{{m}_{{Pt},{total}}}{{{wt}}_{{Pt}/C}}\end{eqnarray} \tag{ 3 }$$

where wt_Pt/C is the weight percentage of the catalyst compared to the support for the given CCL. Note many sources provide the weight percentage

$$\begin{eqnarray}&&{{wt}}_{{Pt}/{cat}}=\displaystyle \frac{{m}_{{Pt},{total}}}{{m}_{{Pt},{total}}+{m}_{C,{total}}}\end{eqnarray} \tag{ 4 }$$

instead of wt_Pt/C , in which case the latter can be obtained as

$$\begin{eqnarray}&&{{wt}}_{{Pt}/C}=\displaystyle \frac{{{wt}}_{{Pt},{cat}}}{1-{{wt}}_{{Pt},{cat}}}\end{eqnarray} \tag{ 5 }$$

We assume that the catalyst support structure consists of spherical carbon primary particles of radius r_C which feature multiple nanopores. Assuming that, on average, a certain partition x_C,p of the sphere volume is devoted to nanopores, the total number of primary particles within the CCL for a given carbon mass is

$$\begin{eqnarray}&&{N}_{C}=\displaystyle \frac{{m}_{C,{total}}}{\tfrac{4}{3}\pi {r}_{C}^{3}{\rho }_{C}\left(1-{x}_{C,p}\right)}\end{eqnarray} \tag{ 6 }$$

where ρ_C is the density of the solid carbon phase. The overall volume taken up by solid carbon and nanopores is hence

$$\begin{eqnarray}&&{V}_{C,{total}}=\displaystyle \frac{4}{3}\pi {r}_{C}^{3}{N}_{C}\end{eqnarray} \tag{ 7 }$$

and the overall nanopore volume is

$$\begin{eqnarray}&&{V}_{p,{total}}={x}_{C,p}{V}_{C,{total}}\end{eqnarray} \tag{ 8 }$$

In most cases, the specific pore volume V_p,total /m_C,total is given by experimental data and can be used to calculate x_C,p by combining Eqs. 6 to 8

$$\begin{eqnarray}&&{x}_{C,p}=\displaystyle \frac{1}{1+{m}_{C,{total}}{\left({\rho }_{C}{V}_{p,{total}}\right)}^{-1}}\end{eqnarray} \tag{ 9 }$$

If there is no information available on the CCL thickness d_ccl , we propose to estimate this value from the obtained number of carbon primary particles in an approach similar to that of Darling. ²⁶ Assuming a certain packing density x_C,pack < 0.74 where the latter is the maximum packing density of spheres, d_ccl is calculated as

$$\begin{eqnarray}&&{d}_{{ccl}}=\displaystyle \frac{1}{{x}_{C,{pack}}}\displaystyle \frac{{V}_{C,{total}}}{{A}_{{ccl}}}\end{eqnarray} \tag{ 10 }$$

In this work, we assume nanopores of cylindrical shape and uniform length l_p . The volume of a single pore for a given pore opening radius r_p is hence

$$\begin{eqnarray}&&{V}_{p}({r}_{p})={l}_{p}\pi {r}_{p}^{2}\end{eqnarray} \tag{ 11 }$$

The surface of the pore walls, excluding the bottom of the pore is

$$\begin{eqnarray}&&{A}_{p,{wall}}({r}_{p})=2\pi {r}_{p}{l}_{p}\end{eqnarray} \tag{ 12 }$$

and the surface of the pore opening, which equals that of the pore bottom, is

$$\begin{eqnarray}&&{A}_{p,{opening}}({r}_{p})=\pi {r}_{p}^{2}\end{eqnarray} \tag{ 13 }$$

We assume a distribution dn_p,C (r_p )/dr_p which describes the number of nanopores of a certain size class r_p in the unloaded catalyst support. For simpler notation, we will use the abbreviation

$$\begin{eqnarray}&&n^{\prime} (r)=\displaystyle \frac{{\rm{d}}n(r)}{{\rm{d}}r}\end{eqnarray} \tag{ 14 }$$

for distributions throughout this work. In the model, all distributions can either be an analytical relation or discrete values for each size class, e.g. obtained from TEM data. The Eqs. stated in the following refer to a continous distribution, but analogous relations hold for the discrete case with the integrals replaced by an appropriate summation rule. The pore size distribution can either be adapted directly from measured data or be any relation $n^{\prime} ({r}_{p})$ which is then normalized to

$$\begin{eqnarray}&&n{{\prime} }_{p,C}({r}_{p})=\displaystyle \frac{{V}_{p,{total}}}{{\int }_{0}^{\infty }{V}_{p}({\hat{r}}_{p})n^{\prime} ({\hat{r}}_{p}){\rm{d}}{\hat{r}}_{p}}n^{\prime} ({r}_{p})\end{eqnarray} \tag{ 15 }$$

to ensure that the distribution yields the correct overall pore volume for the given pore morphology. Using this distribution, we can estimate the overall surface area of the catalyst layer. We assume that the carbon surface area of a primary particle which is lost due to a pore opening approximately equals the area gained at the pore bottom. Any surface area increase due to nanopores in otherwise spherical carbon primary particles is then due to the pore walls such that the overall support surface area is calculated as

$$\begin{eqnarray}&&{A}_{C,{total}}=4\pi {r}_{C}^{2}{N}_{C}+{\int }_{0}^{\infty }{A}_{p,{wall}}({\hat{r}}_{p})n{{\prime} }_{p,C}({\hat{r}}_{p}){\rm{d}}{\hat{r}}_{p}\end{eqnarray} \tag{ 16 }$$

The pore size distribution of the carbon support before Pt and ionomer loading can be transferred into a pore volume distribution (PVD)

$$\begin{eqnarray}&&v{{\prime} }_{p,C}({r}_{p})=V({r}_{p})n{{\prime} }_{p,C}({r}_{p})\end{eqnarray} \tag{ 17 }$$

which will be used frequently throughout this work. Analogous to Eq. 14, we define $v^{\prime} (r)={\rm{d}}v(r)/{\rm{d}}r$.

Similar to the pore size distribution, we introduce continous or discrete distributions $n{{\prime} }_{{Pt},{inner}}({r}_{{Pt}})$ and $n{{\prime} }_{{Pt},{outer}}({r}_{{Pt}})$ which denote the catalyst particle size distributions inside the nanopores and on the support surface, respectively. The particles are assumed to be spherical with a volume

$$\begin{eqnarray}&&{V}_{{Pt}}({r}_{{Pt}})=\displaystyle \frac{4}{3}\pi {r}_{{Pt}}^{3}\end{eqnarray} \tag{ 18 }$$

and surface area

$$\begin{eqnarray}&&{A}_{{Pt}}({r}_{{Pt}})=4\pi {r}_{{Pt}}^{2}\end{eqnarray} \tag{ 19 }$$

and density ρ_Pt . For a given catalyst layer, we assume that a mass fraction wt_Pt,inner of the overall Pt loading resides within the support nanopores and obtain appropriately normalized distributions from initially unnormalized relations $n{{\prime} }_{{inner}}({r}_{{Pt}})$ and $n{{\prime} }_{{outer}}({r}_{{Pt}})$ as

$$\begin{eqnarray}&&n{{\prime} }_{{Pt},{inner}}({r}_{{Pt}})=\displaystyle \frac{{{wt}}_{{Pt},{inner}}{m}_{{Pt},{total}}}{{\rho }_{{Pt}}{\int }_{0}^{\infty }{V}_{{Pt}}({\hat{r}}_{{Pt}})n{{\prime} }_{{inner}}({\hat{r}}_{{Pt}}){\rm{d}}{\hat{r}}_{{Pt}}}n{{\prime} }_{{inner}}({r}_{{Pt}})\end{eqnarray} \tag{ 20 }$$

and

$$\begin{eqnarray}&&n{{\prime} }_{{Pt},{outer}}({r}_{{Pt}})=\displaystyle \frac{(1-{{wt}}_{{Pt},{inner}}){m}_{{Pt},{total}}}{{\rho }_{{Pt}}{\int }_{0}^{\infty }{V}_{{Pt}}({\hat{r}}_{{Pt}})n{{\prime} }_{{outer}}({\hat{r}}_{{Pt}}){\rm{d}}{\hat{r}}_{{Pt}}}n{{\prime} }_{{outer}}({r}_{{Pt}})\end{eqnarray} \tag{ 21 }$$

(compare to the approach in Ref. 27). Alternatively, the respective partitions of inner and outer particles may be defined by particle counts (e.g. from STEM data ² ), rather than weight percentages. In this case, wt_Pt,inner must be calculated such that, next to normalizing to the overall Pt loading (Eqs. 20 and 21), the measured relation between inner and outer particle numbers is also satisfied

$$\begin{eqnarray}&&\displaystyle \frac{{N}_{{Pt},{inner}}}{{N}_{{Pt},{outer}}}=\displaystyle \frac{{\int }_{0}^{\infty }n{{\prime} }_{{Pt},{inner}}({\hat{r}}_{{Pt}}){\rm{d}}{\hat{r}}_{{Pt}}}{{\int }_{0}^{\infty }n{{\prime} }_{{Pt},{outer}}({\hat{r}}_{{pt}}){\rm{d}}{\hat{r}}_{{Pt}}}\end{eqnarray} \tag{ 22 }$$

From the normalized distributions, the ECSA can be calculated as

$$\begin{eqnarray}&&{A}_{{Pt},{inner}}={\int }_{0}^{\infty }{A}_{{Pt}}({\hat{r}}_{{Pt}})n{{\prime} }_{{Pt},{inner}}({\hat{r}}_{{Pt}}){\rm{d}}{\hat{r}}_{{Pt}}\end{eqnarray} \tag{ 23 }$$

and analogously for the outer Pt. ²⁸ Next, we link the inner Pt particle size distribution to the pore size distribution in order to define how Pt is distributed within the nanopores. A Pt particle of radius r_Pt can only be situated in a nanopore of size r_p ≥ r_Pt . We assume that inner Pt is distributed according to pore volume, that is, the larger the pore, the more particles can be found there. Among pores of the same size, each pore is attributed the same number of Pt particles. The total number of Pt particles of size r_Pt that can be found in the pores of size r_p can hence be described by the two-dimensional distribution

$$\begin{eqnarray}&&\begin{array}{l}n{{\prime\prime} }_{{Pt},p}({r}_{{Pt}},{r}_{p})\\ =\,\left\{\begin{array}{ll}n{{\prime} }_{{Pt},{inner}}({r}_{{Pt}})\displaystyle \frac{{V}_{p}({r}_{p})n{{\prime} }_{p,C}({r}_{p})}{{\displaystyle \int }_{{r}_{{Pt}}}^{\infty }{V}_{p}({\hat{r}}_{p})n{{\prime} }_{p,C}({\hat{r}}_{p}){\rm{d}}{\hat{r}}_{p}} & {{for}\,{r}}_{{Pt}}\leqslant {r}_{p}\\ 0 & {otherwise}\end{array}\right.\end{array}\end{eqnarray} \tag{ 24 }$$

with n''_Pt,p (r_Pt , r_p ) = dn_Pt,p (r_Pt , r_p )/dr_Pt dr_p . In the above expression, the factor ${V}_{p}({r}_{p})n{{\prime} }_{p,C}({r}_{p})/{\int }_{{r}_{{Pt}}}^{\infty }{V}_{p}({\hat{r}}_{p})n{{\prime} }_{p,C}({\hat{r}}_{p}){\rm{d}}{\hat{r}}_{p}$ compares the differential volume contributed by the observed pore size class r_p with the total volume of pores which are large enough to host a particle of size r_Pt . With $n{{\prime} }_{{Pt},{inner}}({r}_{{Pt}})$ corresponding to the total number of particles of size class r_Pt slated for distribution into pores with r_p ≥ r_Pt , the amount of particles which is attributed to the pore size class r_p then corresponds to $n{{\prime} }_{{Pt},{inner}}({r}_{{Pt}})$ multiplied by the aforementioned factor. The expression n''_Pt,p will allow us to derive several quantities of interest throughout this work, one example being the ionomer coverage on Pt particles of a given size within support nanopores.

To conclude this Section, it should be noted that loading Pt into the pores causes the PVD $v{{\prime} }_{p,C}({r}_{p})$ of the support to change to a different distribution $v{{\prime} }_{p,{Pt}/C}({r}_{p})$. ^1,2,8 We will illustrate how $v{{\prime} }_{p,{Pt}/C}$ was obtained from n''_Pt,p for different qualitative scenarios in the results part, and assume that $v{{\prime} }_{p,{Pt}/C}$ is given for the following considerations on ionomer loading.

A central contribution of our model is the calculation of the distribution of available ionomer on the support surface and in the nanopores, depending on the given Pt/C morphology and ionomer loading. The total mass of ionomer within the CCL can be obtained using the weight percentage wt_io which links the ionomer mass to that of the Pt/C catalyst

$$\begin{eqnarray}&&{m}_{{io},{total}}=\displaystyle \frac{{{wt}}_{{io}}}{1-{{wt}}_{{io}}}\left({m}_{C,{total}}+{m}_{{Pt},{total}}\right)\end{eqnarray} \tag{ 25 }$$

If the ionomer to carbon mass ratio I/C is given instead of wt_io , the weight percentage can be derived as

$$\begin{eqnarray}&&{{wt}}_{{io}}=\displaystyle \frac{I/C}{{{wt}}_{{Pt}/C}+I/C+1}\end{eqnarray} \tag{ 26 }$$

The total ionomer volume for a given ionomer density ρ_io is hence

$$\begin{eqnarray}&&{V}_{{io},{total}}=\displaystyle \frac{{m}_{{io},{total}}}{{\rho }_{{io}}}\end{eqnarray} \tag{ 27 }$$

We assume that the ionomer forms a film of constant thickness d_io . In order to calculate how the available ionomer volume is distributed on the support and inside the nanopores, we formulate the schematic proposed by Kobayashi et al. ² into three qualitative cases:

• 1.  
  At low ionomer contents, ionomer impregnation sets out on the outer surface of the support primary particles. Ionomer may not penetrate the nanopores until all available outer support surface is covered by an ionomer film with a set minimum thickness ${d}_{{io}}={d}_{{io},\min }$.
• 2.  
  If full outer coverage with ${d}_{{io}}={d}_{{io},\min }$ is reached, ionomer starts to penetrate nanopores which are sufficiently large, that is, which have pore openings larger than a cutoff radius r_p,cut which is considered to equal the film thickness d_io . With increasing ionomer content, a growing part of these pores is filled until 100% impregnation of the valid pores is reached.
• 3.  
  In case the ionomer content exceeds the volume that would suffice for complete impregnation of the outer surface and pores larger than ${d}_{{io},\min }$ with a film of that same thickness, the model solves for the value ${r}_{p,{cut}}={d}_{{io}}\gt {d}_{{io},\min }$ that satisfies the condition that the support surface and all pores larger than r_p,cut be impregnated with a film of constant thickness d_io .

Figure 1 illustrates the possible cases of ionomer filling schematically. In case 3, the increased film thickness and, subsequently, the larger cutoff radius have the effect that more pores are excluded from ionomer impregnation as compared to lower ionomer contents. Consequently, the overall amount of carbon and Pt surface which is covered by ionomer increases with increasing I/C ratio in cases 1 and 2, but the inner ionomer coverage is reduced again in CCLs with sufficiently high ionomer content to trigger case 3. Also, the accessible nanopore volume decreases much more dramatically when pores are cut off by ionomer (case 3) as opposed to merely being slowly filled with ionomer (case 2). It should be noted that the total ionomer loading determines in which qualitative case a CCL is, and the different configurations correspond to different CCL entities. An individual CCL cannot e.g. transit from case 2 to 3 with a sudden reversion of pore impregnation. Besides being adapted from reference, ² the proposed qualitative behavior concurs with the observations of Andersen et al. ¹³ who suggested a distinct change in ionomer structure toward aggregation at high ionomer contents. The overall ionomer coverage on Pt consequently increased with ionomer content when the weight percentage was low, but decreased at higher-than-optimum ionomer contents. The proposed increase in ionomer film thickness, either proportional to the I/C ratio ^10,14 or only at high ionomer contents ¹¹ is likewise supported by experimental findings. We will show below that further aspects observed in literature, such as no change in ionomer coverage with (low) I/C ratio ^11,12 or constant film thickness in certain conditions do not necessarily contradict these assumptions, but can actually be predicted by the proposed model depending on the combination of support morphology, catalyst particle distribution and ionomer content.

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In the remainder of this Section, we illustrate how parameters which quantify the ionomer distribution are calculated for the different impregnation cases. At full coverage, the ionomer volume on the outer support surface can be estimated as

$$\begin{eqnarray}&&\begin{array}{l}{V}_{{io},{outer},\max }({d}_{{io}},{r}_{p,{cut}})=\displaystyle \frac{4}{3}\pi {N}_{C}\left({\left({r}_{C}+{d}_{{io}}\right)}^{3}-{r}_{C}^{3}\right)\\ \quad -\,{d}_{{io}}{\displaystyle \int }_{{r}_{p,{cut}}}^{\infty }{A}_{p,{opening}}({\hat{r}}_{p}-{d}_{{io}})n{{\prime} }_{p,{Pt}/C}({\hat{r}}_{p}){\rm{d}}{\hat{r}}_{p}\end{array}\end{eqnarray} \tag{ 28 }$$

where

$$\begin{eqnarray}&&n{{\prime} }_{p,{Pt}/C}({r}_{p})=v{{\prime} }_{p,{Pt}/C}({r}_{p})/{V}_{p}({r}_{p})\end{eqnarray} \tag{ 29 }$$

is the number of pores remaining after Pt loading. The first term in Eq. 28 describes a homogenous ionomer film of thickness d_io around a carbon primary particle. ²⁹ As illustrated in Fig. 1a, this film is interrupted above all nanopores that have an opening radius larger than r_p,cut = d_io , i.e. which can potentially be impregnated by ionomer if sufficient ionomer loading is available. For simplification, we assume that the gaps in the otherwise continous ionomer film above these pores can be approximated as disc-like shapes. The overall contribution of these discs is described by the second term in Eq. 28. We distinguish between d_io and r_p,cut in the given Eqs. since these quantities can generally be set to different values, although we consider them to be equal in our model setup to reduce experimentally unaccessible degrees of freedom.

Note that calculating $n{{\prime} }_{p,{Pt}/C}$ from $v{{\prime} }_{p,{Pt}/C}$ as in Eq. 29 implies that we consider the nanopores to still be of cylindrical shape after Pt loading and to approximately have retained their radius-specific volume V_p (r_p ). Any losses in $v{{\prime} }_{p,{Pt}/C}({r}_{p})$ compared to $v{{\prime} }_{p,C}({r}_{p})$ are taken to be caused by a decreased number of pores of size class r_p , rather than a changed pore volume. It is likely that both these processes have some influence on experimental PVDs, but they cannot be distinguished without further measurements. However, particle size distributions measured in literature ^1,2 suggest that the inner Pt volume is small compared to the total nanopore volume and thus insufficient to account for the losses in pore volume usually observed. It is more probable that inner or outer Pt blocks the pore entrances, ^1,8 thereby reducing the number of accessible pores, but not significantly changing the shape and volume of pores that remain accessible which is why we deem Eq. 29 valid.

To determine the appropriate impregnation case, next to ${V}_{{io},{outer},\max }$, we need the maximum ionomer volume that can theoretically be stored within the nanopores of a given Pt/C for a certain film thickness. We consider a fully impregnated pore to be covered in a film of thickness d_io along the pore walls and bottom, see Fig. 1c. The corresponding ionomer volume in an individual pore is then composed of a cylinder mantle and a disc at the pore bottom, both with thickness d_io , which is expressed as

$$\begin{eqnarray}\begin{array}{rcl}{V}_{{io},p,\max }({r}_{p},{d}_{{io}}) & = & \left(\pi {r}_{p}^{2}{d}_{{io}}+\left({l}_{p}-{d}_{{io}}\right)\right.\\ & & \left.\times \left(\pi {r}_{p}^{2}-\pi {\left({r}_{p}-{d}_{{io}}\right)}^{2}\right)\right)\end{array}\end{eqnarray} \tag{ 30 }$$

In most cases, this is the maximum ionomer content that can be located in a pore of size class r_p > r_p,cut under the assumption of cylindrical pores and uniform film thickness. However, the deposition of Pt in the pores can in principle cause the remaining available pore volume after Pt loading to be smaller than ${V}_{{io},p,\max }$. We take this into account by comparing the differential volumes $v{{\prime} }_{p,{Pt}/C}({r}_{p})$ and $n{{\prime} }_{p,{Pt}/C}({r}_{p}){V}_{{io},p,\max }({r}_{p},{d}_{{io}})$ and choosing the smaller value for each pore size class. Expressed as a distribution, the maximum ionomer volume that can be stored in the pores in this way is

$$\begin{eqnarray}&&\begin{array}{l}v{{\prime} }_{{io},p,\max }({r}_{p},{d}_{{io}},{r}_{p,{cut}})\\ =\,\left\{\begin{array}{ll}0 & {{if}\,{r}}_{p}\lt {r}_{p,{cut}}\\ \min \left(v{{\prime} }_{p,{Pt}/C}({r}_{p});n{{\prime} }_{p,{Pt}/C}({r}_{p}){V}_{{io},p,\max }({r}_{p},{d}_{{io}})\right) & {otherwise}\end{array}\right.\end{array}\end{eqnarray} \tag{ 31 }$$

and the maximum total ionomer volume within the support nanopores is

$$\begin{eqnarray}&&\begin{array}{l}{V}_{{io},{inner},\max }({d}_{{io}},{r}_{p,{cut}})\\ \quad ={\displaystyle \int }_{0}^{\infty }v{{\prime} }_{{io},p,\max }({\hat{r}}_{p},{d}_{{io}},{r}_{p,{cut}}){\rm{d}}{\hat{r}}_{p}\end{array}\end{eqnarray} \tag{ 32 }$$

For a given Pt/C combination and minimum film thickness, we abbreviate two characteristic volumes ${V}_{{io},1}={V}_{{io},{outer},\max }({d}_{{io},\min },{d}_{{io},\min })$ and ${V}_{{io},2}={V}_{{io},{inner},\max }({d}_{{io},\min },{d}_{{io},\min })$. The impregnation case is determined by comparing the available ionomer volume V_io,total with these quantities:

• 1.  
  For V_io,total ≤ V_io,1, all ionomer volume is located on the support surface such that V_io,outer = V_io,total and V_io,inner = 0. We assume that the ionomer film cannot become arbitrarily thin but, due to surface tension, rather forms islands which exhibit the set minimum thickness ${d}_{{io}}={d}_{{io},\min }$. The overall coverage of the available support surface can hence be smaller than unity (Fig. 1a).
• 2.  
  For V_io,1 < V_io,total ≤ V_io,1 + V_io,2, there is full coverage with the minimum film thickness on the outer support surface such that V_io,outer = V_io,1. Pores with openings larger than ${r}_{p,{cut}}={d}_{{io},\min }$ can now become partially impregnated with ionomer, the degree of coverage depending on the available inner ionomer volume V_io,inner = V_io,total − V_io,outer (Fig. 1b).
• 3.  
  For V_io,total > V_io,1 + V_io,2, all pores larger than the cutoff radius are 100% impregnated. In order to use all available ionomer and still fulfill the condition that the cutoff radius be the same value as the film thickness, the film thickness is adapted to the value ${d}_{{io}}={r}_{p,{cut}}\gt {d}_{{io},\min }$ which satisfies the condition

  $$\begin{eqnarray}&&{V}_{{io},{outer},\max }({d}_{{io}},{d}_{{io}})+{V}_{{io},{inner},\max }({d}_{{io}},{d}_{{io}})={V}_{{io},{total}}\end{eqnarray} \tag{ 33 }$$

  with ionomer volumes ${V}_{{io},{inner}}={V}_{{io},{inner},\max }({d}_{{io}},{d}_{{io}})$ and ${V}_{{io},{outer}}={V}_{{io},{outer},\max }({d}_{{io}},{d}_{{io}})$ (Fig. 1c).

The distribution behavior of ionomer within the porous CCL is thus fully determined by the given ionomer content, the minimum thickness of the formed ionomer film, the carbon primary particle size, the nanopore distribution and the distribution of Pt within the pores. Lastly, we use the derived quantities to give a mathematical description of the Pt/ionomer interface. For Pt located on the outer support surface, the coverage with ionomer is independent of the particle size. An individual particle can either be covered or not covered by ionomer, but the statistical coverage of all particles in a size class equals the coverage found on the outer surface

$$\begin{eqnarray}&&{{\rm{\Theta }}}_{{io}/{Pt},{outer}}=\displaystyle \frac{{V}_{{io},{outer}}}{{V}_{{io},{outer},\max }({d}_{{io}},{r}_{p,{cut}})}\end{eqnarray} \tag{ 34 }$$

For Pt particles located in nanopores, the case is different. Particles situated in pores below the cutoff radius are never contacted by ionomer while the statistical coverage on particles in pores above the cutoff radius corresponds to the percentage to which the pore is impregnated (Fig. 1b). In order to find a general expression for the ionomer coverage on inner particles of a size class r_Pt , we therefore need information on how many particles of that size class are located in pores that are contacted by ionomer. This information is provided by the previously introduced distribution n''_Pt,p (r_Pt , r_p ) and we can derive

$$\begin{eqnarray}\begin{array}{rcl} & & {{\rm{\Theta }}}_{{io}/{Pt},{inner}}({r}_{{Pt}},{d}_{{io}},{r}_{p,{cut}})\\ & = & \displaystyle \frac{{V}_{{io},{inner}}}{{V}_{{io},{inner},\max }({d}_{{io}},{r}_{p,{cut}})}\\ & & \times \,\displaystyle \frac{{\int }_{{r}_{p,{cut}}}^{\infty }n{{\prime\prime} }_{{Pt},p}({r}_{{Pt}},{\hat{r}}_{p}){\rm{d}}{\hat{r}}_{p}}{n{{\prime} }_{{Pt},{inner}}({r}_{{Pt}})}\end{array}\end{eqnarray} \tag{ 35 }$$

where the second fraction expresses the partition of inner Pt particles of size r_Pt sitting in impregnated pores. In summary, for a morphological description of the CCL, the model requires the material composition, pore and particle size distributions, partition of particles inside pores and assumptions on carbon primary particle size, basic pore morphology and minimum ionomer film thickness. We will show that this information, in our case obtained from literature, is sufficient to reproduce various measurement methods with good qualitative and quantitative correspondence.

First, the log-differential PVDs extracted from the reference publications were transferred to the corresponding differential distributions $v{{\prime} }_{p}$. ³⁰ Under the assumption of cylindrical pores, the pore size distributions $n{{\prime} }_{p}$ were then obtained. The inner and outer Pt particle size distributions were assumed to be Gaussian distributions with mean particle radius and standard deviation corresponding to the values obtained from STEM data in the reference publications. By normalizing the particle size distributions as described above, it was ensured that they corresponded to the Pt loading as well as the percentage of Pt particles in the nanopores and on the support surface that were reported in the reference publications. From the particle and pore size distributions, the distribution of Pt in the nanopores n''_Pt,p was calculated. Concurrent with the reference publications, we consider nanopores to denote pores with r_p ≤ 5 nm.

With the basic distributions given, we sought to calculate the PVD after Pt loading $v{{\prime} }_{p,{Pt}/C}$ which is a quantity that can be directly compared to literature. As an example for our considerations, we use Pt/CB which has an intermediate surface area and exhibits nanopores in the whole observed size range from 0.25 to 5 nm pore radius. In the experiment, Pt loading decreased the accessible nanopore volume by about 40% for this support. As stated before, this cannot be explained purely by the comparatively small additional Pt volume, but is rather attributed to Pt blocking the nanopore entrances ^1,8 with subsequent loss of the full nanopore volume below and hence a reduction in the number of pores. We used the model to explore how this process could work in a qualitative way.

The first case we investigated was the assumption that a Pt particle whose size is close to that of the pore in which it is lodged is sufficient to block the pore. With the previously obtained distribution n''_Pt,p , the number of Pt particles whose radius is 60% or more of the radius of their pore can be calculated. Assuming that each of these particles blocks an entire pore, the fraction of blocked pores is

$$\begin{eqnarray}&&{x}_{p,{Pt}/C,{blocked}}({r}_{p})=\displaystyle \frac{{\int }_{0.6{r}_{p}}^{{r}_{p}}n{{\prime\prime} }_{{Pt},p}({\hat{r}}_{{Pt}},{r}_{p}){\rm{d}}{\hat{r}}_{{Pt}}}{n{{\prime} }_{p,C}({r}_{p})}\end{eqnarray} \tag{ 36 }$$

and the remaining nanopore volume distribution after Pt loading is

$$\begin{eqnarray}&&v{{\prime} }_{p,{Pt}/C}({r}_{p})=\left(1-{x}_{p,{Pt}/C,{blocked}}({r}_{p})\right)v{{\prime} }_{p,C}({r}_{p})\end{eqnarray} \tag{ 37 }$$

Figure 2a shows the resultant PVD compared to the data measured in Ref. 1. For better comparison to literature, all PVDs in this work are plotted in the log-differential form. ³⁰ Clearly, the simulated distribution fails to capture the qualitative behavior of the experimental one. The nanopore volume is reduced between 0.5 and 2.5 nm which is the size range corresponding to that of the inner Pt particles (r_Pt = 1.15 ± 0.25 nm, see Table I). Larger pores are unaffected because the radii of the Pt particles stored in the pores do not amount to the 60% of the pore radius we assumed as a criterion for blockage. Smaller pores are likewise unaffected because Pt particles are generally too large to be stored in these pores. However, the experimental data suggests nanopore volume loss in all pore size ranges. Therefore, we considered a different scenario where the cross-section of all particles sitting inside the pores of a size class was compared to the pore opening surface area. We assumed that nanoparticles can effectively block an area twice that of their collective cross-sections, hence, in this scenario, the fraction of blocked pores is approximated as

$$\begin{eqnarray}&&{x}_{p,{Pt}/C,{blocked}}({r}_{p})=\displaystyle \frac{2{\int }_{0}^{\infty }\pi {\hat{r}}_{{Pt}}^{2}n^{\prime\prime} ({\hat{r}}_{{Pt}},{r}_{p}){\rm{d}}{\hat{r}}_{{Pt}}}{{A}_{p,{opening}}({r}_{p})n{{\prime} }_{p,C}({r}_{p})}\end{eqnarray} \tag{ 38 }$$

In this scenario, it is possible for several small particles to block a pore as opposed to necessitating one large particle with a radius close to the pore size. Note that the model does not distiguish where in the nanopore or in which particular nanopores of a given size class the particles sit. It has been suggested that Pt deposition preferentially occurs near the entrance of a pore due to the higher reactivity of edge sites, ⁸ hence, we consider the statistically expressed partition of blocked pores in Eq. 38 reasonable. The resultant PVD is again calculated according to Eq. 37 and is shown in Fig. 2b. For pores larger than 1.5 nm, the model now captures the experimental data very well, but the predicted volume loss for smaller pores is still insufficient. It is likely that additional factors contributed to the measured volume loss, but deriving their nature from the data was not straightforward. An obvious possibility is the blockage of pores due to outer Pt. However, in the case of Pt/CB, the accumulated cross-sections of all outer Pt particles only amount to about 14% of the surface area constituted by the openings of pores smaller than r_p = 1.5 nm. Even if every outer Pt particle sat on the entrance of one or several of these pores, losses would not be sufficiently high due to the vast number of small pores in the given sample. Since we have no information on the Pt deposition method of this commercial catalyst, we can only speculate that the smallest pores are potentially more vulnerable to blockage during the process, possibly by small Pt particles which may have been unresolved in STEM measurements or by carbon fragments or neighbouring primary particles. In addition, calculating pore volumes in adsorbents that feature both pores below a radius of 1 nm and larger pores is challenging due to the concurrent processes of micropore filling and monolayer-multilayer adsorption in such materials. ^8,31 This was addressed in reference publication ¹ by using quenched solid density functional theory analysis to calculate the PVDs, but it is possible that small pore sizes are associated with a higher measurement uncertainty. For the purposes of modeling Pt deposition, we remain with the second scenario in the following. The proposed mechanism showed the best match for the five other supports and reasonable accordance with the larger pores of Pt/CB. The experimental and simulated PVDs after Pt deposition for all six supports can be found in the next Section in Fig. 4, together with the distributions of the unloaded support and after ionomer addition.

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We conclude our considerations on loading porous supports with Pt by comparing characteristics of the simulated CCLs, calculated from the described pore and particle size and volume distributions, to experimental data. Figure 3 shows the support surface area, the total nanopore volume after Pt addition and the ECSA of the inner and outer Pt. For calculating the support surface area, we only took into account the contribution of pores with r_p ≥ 0.5 nm to approximate the limit ³¹ of the nitrogen adsorption based BET surface area measurements performed in the reference publications. This only affects Pt/CB and Pt/GCB as the characterized pore size range for the other supports starts at 1 nm. Generally, the depicted integral quantities match the experimental data well, suggesting that our assumptions on basic pore morphology and catalyst distribution on the support are reasonable when large numbers are involved, even though representing an individual pore by a cylinder may seem a strong simplification. A notable fact in Fig. 3 is that the relative deviation of the calculated support surface area is highest for the least porous supports, GCB and V1. It is likely that these are not represented well by cylindrical pores of uniform length l_p . Presumably, nanopore morphology is closer to a semi-sphere in these support types ¹ . At the same opening radius and volume, the specific surface area of a semi-spherical pore is less than that of a cylindrical pore which would explain why the model overestimates the surface area in these cases. By replacing Eqs. 11 to 13, a different representation of basic pore morphology can easily be formulated into the model in case corresponding information on the support is available. The data shown in Fig. 3 is also summarized in Table I.

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With correspondence of the simulated and experimental Pt/C combinations established, we next focused on the effects of adding ionomer to the catalyst and support. An I/C ratio of 0.7 was set for correspondence with the reference work ¹ where pore volume distributions after both Pt and ionomer loading are provided. A minimum ionomer film thickness of ${d}_{{io},\min }=1\,\mathrm{nm}$ was chosen for all support types. This value is in accordance with the findings of Park et al. ¹ who reported a uniform and continous ionomer film of 1–2 nm thickness on Pt/CB. However, they also observed that the film tended to be less uniform and more concentrated near Pt particles with increased degree of graphitization of the support. This was not taken into account in the model, but is discussed in the context of the Pt/ionomer interface in the next Section.

With the total ionomer volume V_io,total and minimum film thickness ${d}_{{io},\min }$ given, the inner and outer ionomer volumes, the cutoff radius and the actual film thickness were calculated as described above. Using these quantities, the remaining PVD after ionomer impregnation for each support was calculated as

$$\begin{eqnarray}&&\begin{array}{l}v{{\prime} }_{p,{Pt}/C/{io}}({r}_{p},{d}_{{io}},{r}_{p,{cut}})\\ =\,\left\{\begin{array}{ll}v{{\prime} }_{p,{Pt}/C}({r}_{p})-\displaystyle \frac{{V}_{{io},{outer}}}{{V}_{{io},{outer},\max }({d}_{{io}},{r}_{p,{cut}})}v{{\prime} }_{p,{Pt}/C}({r}_{p}) & {{for}\,{r}}_{p}\leqslant {r}_{p,{cut}}\\ v{{\prime} }_{p,{Pt}/C}({r}_{p})-\displaystyle \frac{{V}_{{io},{inner}}}{{V}_{{io},{inner},\max }({d}_{{io}},{r}_{p,{cut}})}v{{\prime} }_{{io},p,\max }({r}_{p},{d}_{{io}},{r}_{p,{cut}}) & {otherwise}\end{array}\right.\end{array}\end{eqnarray} \tag{ 39 }$$

The first case accounts for the fact that, at very low ionomer contents, the available ionomer volume may not be sufficient to cover all support surface and pore openings smaller than r_p,cut in a film of thickness ${d}_{{io},\min }$. In this case, on average, only a percentage equal to ${V}_{{io},{outer}}/{V}_{{io},{outer},\max }({d}_{{io}},{r}_{p,{cut}})$ of these pores is cut off with their respective pore volumes being subtracted from the overall remaining pore volume. If the ionomer content is high enough to achieve full outer coverage, there is no contribution to the pore volume from the pores smaller than r_p,cut anymore. In the second case of pores larger than r_p,cut , the volume of a pore with size r_p is diminished by the ionomer volume stored in that pore.

Figure 4 shows the simulated PVDs of the carbon support before and after loading with Pt and ionomer in comparison with the experimental data. Note that measured PVDs after ionomer addition are only available for Pt/CB and Pt/GCB since the reference publication ² did not feature this data. After the addition of ionomer, the remaining nanopore volume is significantly reduced. For all supports except Pt/K13, the calculated ionomer film thickness and cutoff radius are larger than the set minimum film thickness, meaning that ionomer impregnation case 3 prevails at I/C = 0.7. The available ionomer volume is more than the volume theoretically needed to cover the outer support surface and all pores with ${r}_{p}\gt {d}_{{io},\min }=1\,\mathrm{nm}$ in a film of that same thickness. Consequently, the model solves for a larger value d_io = r_p,cut at which all ionomer can be stored in the CCL, with any pores above the cutoff radius being fully impregnated (compare to Fig. 1c). The calculated cutoff radius depends on the support morphology and tends to be smaller for supports with high nanopore volume. For these, more ionomer can be stored in the pores and case 3 is only triggered at comparatively high ionomer contents. However, for instance, Pt/K8 has a smaller cutoff radius than Pt/CB despite the pore volume of the latter being larger. This is due to the fact that a large contribution to nanopore volume comes from small pores <1 nm in Pt/CB. For a set minimum film thickness of 1 nm, these pores are always cut off and no ionomer can be stored inside. Contrary to the other supports, Pt/K13, with the highest specific surface area and nanopore volume, is in case 2, with the cutoff radius still at ${r}_{p,{cut}}={d}_{{io},\min }$. The ionomer volume is sufficient to reach full outer coverage and enter the nanopores, reducing the nanopore volume, but not enough for 100% impregnation of pores >1 nm. While this is plausible, it must be noted that the simulated PVD after Pt deposition, which in turn influences the PVD after ionomer loading, deviates significantly from the measured data for Pt/K13 because pore volume gain in certain size classes cannot be emulated by the model. For CB and GCB, we exemplify in the last Section of the results part how the model can be adapted to capture the available validation data better.

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In order to compare our model to further data, we performed simulations corresponding to measurements in the reference publication of Kobayashi et al. ² The authors did not provide pore volume distributions after ionomer loading, however, they ran N₂ adsorption measurements for varying I/C ratios to obtain the adsorption/desorption hysteresis volume from which they derived the status of pore impregnation. In order to generate comparable data, we used our model to calculate the accessible volume within the CCL at different I/C ratios. As accessible volume, we consider the secondary pore space between carbon primary particles plus the nanopore volume which is not taken up or blocked by ionomer. With a given ionomer volume and corresponding impregnation case, the secondary pore volume is calculated as

$$\begin{eqnarray}&&{V}_{{ccl},{secondary}}={V}_{{ccl}}-{V}_{{io},{outer}}-{V}_{{Pt},{outer}}-{V}_{C,{total}}\end{eqnarray} \tag{ 40 }$$

The accessible volume is then calculated by adding the cumulative contribution of the nanopores after ionomer loading

$$\begin{eqnarray}&&{V}_{{ccl},{accessible}}={V}_{{ccl},{secondary}}+{\int }_{0}^{\infty }v{{\prime} }_{p,{Pt}/C/{io}}({\hat{r}}_{p},{d}_{{io}},{r}_{p,{cut}}){\rm{d}}{\hat{r}}_{p}\end{eqnarray} \tag{ 41 }$$

Figure 5 shows the reference data and simulated accessible volume in the CCL for I/C ratios between 0.2 and 1.7. The curves of the more porous supports (Pt/CB, Pt/K4, Pt/K8, Pt/K13) exhibit inflection points where the slope of the curve changes to a steeper decrease and then, for Pt/CB, Pt/K4 and Pt/K8, back to the initial slope. These points mark the transition between different ionomer impregnation cases. At low ionomer contents, in cases 1 and 2, any additional ionomer volume reduces the accessible pore volume by an equal amount (Figs. 1a and 1b). However, at higher ionomer contents in case 3, d_io and r_p,cut increase, cutting off more nanopores than at lower I/C ratios and resulting in more drastic loss of accessible volume (Fig. 1c). Once r_p,cut is in size ranges above which there exist only few pores whose blockage hardly impacts the PVD, adding yet more ionomer mainly increases the outer film thickness and the curve reassumes its initial slope. For supports with small overall nanopore volume such as Pt/GCB and Pt/V1, pore impregnation takes place in a very narrow range of I/C values and changes in the curve slope are hardly noticeable.

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The qualitative behavior corresponds well to that observed by Kobayashi and co-workers for Pt/K8 and Pt/K13 (Fig. 5a), which is to be expected since their considerations on pore impregnation formed the basis for our model. From a quantitative point of view, the I/C ratios at which the inflection points taken to be related to pore blockage occur also match the experimental data well. The ordinate values cannot be compared directly due to the conversion factor between pore volume and liquid nitrogen volume ^7,8 and also because we considered the entire contribution of the secondary pore volume in the model. In the measurement, N₂ adsorption in primary pores is to some extent accompanied by adsorption in larger pores and on the external surface of the primary particles. ⁸ Since the contributions of these processes cannot be easily distinguished, we considered the simulations to correspond best to the experiment when the offset due to secondary pores was taken into account.

Our considerations regarding ionomer in porous supports so far focused on illustrating in which way nanopores can be blocked or impregnated by ionomer. In this Section, we return to the Pt particles located within the nanopores and on the support surface and the consequences for the protonic accessibility of these particles. Equation 35 was used to calculate the ionomer coverage on the inner Pt particles Θ_io/Pt,inner for various I/C ratios. Note that in this equation, the decision whether a Pt particle can be contacted by ionomer depends on the size of the pore in which the particle sits, but not on whether the pore is considered to be blocked by Pt. One could argue that the statistical portion of Pt which blocks pores should be removed from the pool of particles potentially contacted by ionomer. However, we stated above that Pt blocks the entrance of a pore such that the entire pore is removed from the PVD and does not appear in $n{{\prime} }_{p,{Pt}/C}$ (compare to inset in Fig. 2b). The consequence is that the outer ionomer film evolves to be continous (i.e. not interrupted) above this pore according to Eq. 28. The particles in turn sitting at the pore entrance, we consider that coverage on these particles is guaranteed by the outer film in this particular case.

Figure 6 shows the ionomer coverage on particles inside pores against the particle size class. At low I/C ratios, in case 1, the coverage on inner particles is zero until full outer coverage has been reached. In case 2, the maximum coverage that can be reached on particles larger than the cutoff radius (and hence in pores larger than the cutoff radius) is ${V}_{{io},{inner}}/{V}_{{io},{inner},\max }({d}_{{io},\min },{d}_{{io},\min })$. Smaller particle size classes exhibit coverages smaller than this value due to the fact that some of the particles in the size class can be located in non-impregnated pores below the cutoff radius (Fig. 1a). At the I/C ratio where the maximum Θ_io/Pt,inner first reaches 100%, case 3 commences. The cutoff radius becomes larger, and is discernable in the plots as the radius at which the curve assumes a value of 1. Particles with r_Pt > r_p,cut are always fully covered with ionomer in case 3 whereas the overall coverage of particles below the cutoff radius actually becomes less since more of the pores they are sitting in are blocked at higher ionomer contents.

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The consequence of the predicted behavior is that for each support, there is an I/C ratio where a maximum overall ionomer coverage on Pt is reached before it is reduced again. In many publications, ^17,32–35 the existance of such an optimum ionomer content was shown and is usually considered as the balance point between sufficient protonic binding and sufficient remaining accessible volume for gas and water transport. Our findings add a further dimension to this by suggesting that, regardless of reactant transport, also the protonic accessibility of Pt might not grow indefinitely with adding more ionomer but in fact exhibit a maximum at intermediate ionomer contents, a fact that is supported by recent findings of Andersen et al. ¹³ For a better visualization of this, we used the model to calculate the actual protonically accessible ECSA on each support for different I/C ratios. The contribution of each particle size class was weighted with its respective ionomer coverage, both for inner Pt and outer Pt particles. The coverage on the latter is independent of particle size such that

$$\begin{eqnarray}&&\begin{array}{l}{A}_{{Pt},{total},{protonic}}={{\rm{\Theta }}}_{{io}/{Pt},{outer}}{A}_{{Pt},{outer}}\\ \,+{\displaystyle \int }_{0}^{\infty }{{\rm{\Theta }}}_{{io}/{Pt},{inner}}({\hat{r}}_{{Pt}},{d}_{{io}},{r}_{p,{cut}})\\ \,\times \,{A}_{{Pt}}({\hat{r}}_{{Pt}})n{{\prime} }_{{Pt},{inner}}({\hat{r}}_{{Pt}}){\rm{d}}{\hat{r}}_{{Pt}}\end{array}\end{eqnarray} \tag{ 42 }$$

The obtained ECSA, normalized to the total Pt loading, is shown in Fig. 7 in comparison to ECSA measurements from the reference publication of Kobayashi et al. ² The modeled results match the experimental data very well. The more porous supports Pt/CB, Pt/K8 and Pt/K13 exhibit distinct maxima around I/C values of 0.8 and 1.2, respectively. For the less porous supports Pt/GCB, Pt/V1 and Pt/K4, the ECSA increases up to an I/C value around 0.5 and then remains mostly constant. The reason is that on these supports, the largest contribution to the ECSA comes from Pt located on the outer support surface. In contrast to the ionomer coverage on inner particles, the coverage on the outer particles always increases with I/C or remains constant which is why there is little change in the ECSA after full outer coverage has been reached. From a quantitative point of view, the model slightly overestimates the ECSA values, particularly for Pt/V1 and Pt/K4. As mentioned in the previous Section, Park et al. ¹ noticed in their work that the outer ionomer film tended to be less continous with less porous supports and increased degree of support graphitization. The largest part of the ECSA being located on the surface of Pt/GCB, Pt/V1 and Pt/K4, a discontinous outer ionomer film would therefore significantly reduce the protonically connected ECSA which could explain the discrepancy between simulation and experiment. This could be represented in the model by reducing the aspired maximum outer ionomer coverage proportional to the degree of graphitization of the support.

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Finally, we recapitulate that there have been different findings in literature as to whether the ionomer coverage remains constant ¹² or changes with I/C ^10,11 and whether the ionomer film thickness remains constant ³⁶ or increases. ^10,14,15 In the presented simulation results, we can actually find all these aspects. The outer coverage increases in case 1, but stays constant in case 2. The film thickness stays constant in cases 1 and 2, but increases in case 3. Thus, all these processes appear, but when and how depends on the support morphology and material combination. It is likely that the situation is similar in real CCLs which encourages our distribution-based modeling approach.

This last Section is intended to exemplify how the presented model framework can be refined to obtain good quantitative correspondence to a given support when suitable experimental information is available. For Pt/CB and Pt/GCB, the work of Park et al. ¹ provides reference PVDs after ionomer loading. In Fig. 4a and 4b, we can observe that for the used parameters, the model underestimates the remaining pore volume after ionomer addition. In the experiment, ionomer addition caused pores smaller than approximately 0.6 nm to mostly vanish from the PVD. This generally agrees with our assumption of small pores being blocked entirely by ionomer while the volume of larger pores is reduced. Consequently, we would expect the calculated cutoff radius to be around 0.6 nm instead of 1.4 nm for Pt/CB and 2.0 nm for Pt/GCB. However, with the condition that d_io = r_p,cut should always be the case, these are the minimal values that can be attained, having to store all given ionomer volume in the CCL. We illustrate the effects of two qualitative modifications in the model which led to much better correspondence.

A uniform thickness d_io = d_io,inner = d_io,outer and the condition d_io = r_p,cut were so far assumed in order to limit experimentally unaccessible degrees of freedom in the model to a minimum. Nevertheless, we consider it quite plausible that the inner ionomer film should be of lesser thickness than the outer film, for instance if the ionomer film above a pore stretches and 'drips' into a pore. First, therefore, we de-coupled the outer and inner film thickness and assumed that the cutoff radius equals the inner film thickness, setting d_io,inner = r_p,cut = 0.6 nm. Consequently, in order to store all available ionomer in the CCL, the outer film thickness d_io,outer must be flexible such that all ionomer which does not clad pores with r_p > 0.6 nm in a film of that same thickness is accomodated. Figures 8a and 8b show the corresponding PVDs and calculated d_io,outer after ionomer loading for Pt/CB and Pt/GCB. The simulations now capture the jump in the curves around 0.6 nm well. However, for larger pores, the remaining pore volume is now overestimated. This is due to the fact that with d_io,inner = r_p,cut , all pores, no matter which size, are only impregnated with a rather thin film of thickness 0.6 nm with little loss of pore volume. Hence, as a further refinement, we considered it plausible that the ionomer film thickness within a pore can actually become larger when a pore is larger. If all nanopores were entirely filled with ionomer, for instance, the inner film thickness inside a pore would be equal to the respective pore radius. We therefore assumed the inner ionomer film thickness to be dependent on pore size and amount to 40% of the pore radius. This factor was motivated from the reference work of Park et al. ¹ As mentioned above, the authors reported a uniform and continous ionomer coverage with a thickness of 1–2 nm or less on Pt/CB. The coverage was found to be less uniform for less porous supports, but no further quantitative information was given. The observed primary pore size range being 0.25 to 5 nm, d_io,inner was hence set to 40% of r_p in order to obtain film thicknesses no larger than 2 nm in agreement with the observations of Park et al. The cutoff radius remained at 0.6 nm, reflecting the fact that surface tension will likely keep ionomer from dripping into the smallest pores below this size. Again, the outer ionomer film thickness was calculated such that it accounts for all ionomer not stored in the nanopores. The resultant pore volume distributions are depicted in Figs. 8c and 8d and match the experimental data very well. For both supports, the calculated outer film thickness is also within the size range of 1–2 nm.

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In summary, with these exemplary considerations, we intend to illustrate that our model description of a CCL, while designed to capture the effects of the different material combinations without necessitating hard to assess parameters like the inner ionomer film thickness, still offers to control these parameters in order to obtain solid quantitative correspondence with available experimental data.