Relative-Humidity Dependence of Electrochemically Active Surface Area in Porous Carbon Catalyst Layers

Surface adsorbed water on hydrophilic and oxidized surfaces plays a critical role in many fields such as biology, tribology, etc. ⁴⁸ Within HSC supports, surface oxygen groups render the surface hydrophilic resulting in water adsorption. Focusing on the more critical cathode PEFC, where air is fed to the cell, the gas molecules are simplified to water vapor and N₂ and their interactions with the carbon pore walls and subsequent adsorption modeled using Lennard-Jones (LJ) and dipole-dipole interaction potentials as detailed below. Extensions to other gases (e.g, H₂) is straightforward although the assumption of only N₂ and water vapor does not induce significant error due to the ideal nature of the gas at typical PEFC operating conditions (see SI).

Both nitrogen and water molecules in the gas phase exhibit intermolecular interactions with the carbon wall that can be described using LJ potentials. Because we consider only pores with pore sizes greater than 0.5 nm (water molecular size ∼0.2 nm ^49,50 ), we approximate the carbon pore walls as planar infinite and use the additive 9–3 LJ interaction potential for the gas-wall interaction potential, ^51,52

$$\begin{eqnarray}&&{U}_{{LJ},{wall}}(x)=4\pi {\varepsilon }_{G-C}{\sigma }_{G-C}^{3}{\rho }_{C}\left[\frac{1}{45}{\left(\frac{{\sigma }_{G-C}}{x}\right)}^{9}-\frac{1}{6}{\left(\frac{{\sigma }_{G-C}}{x}\right)}^{3}\right]\end{eqnarray} \tag{ 2 }$$

where ${\boldsymbol{x}}$ is the normal distance from the wall surface, ${{\boldsymbol{\varepsilon }}}_{{\boldsymbol{G}}-{\boldsymbol{C}}}$ is the 12–6 LJ potential well depth for gas (N₂ or water)/carbon interaction, ${{\boldsymbol{\sigma }}}_{{\boldsymbol{G}}-{\boldsymbol{C}}}$ is related to the corresponding 12–6 LJ potential collision diameter, and ${{\boldsymbol{\rho }}}_{{\boldsymbol{C}}}$ is the solid carbon density. The first term on the right in the above equation represents repulsive forces, whereas the second term represents attractive forces. Oxygen groups are only present on the surface of pore walls and present a planar 2-D surface for intermolecular interactions, thus the interaction potential due to surface oxygen groups is modeled with a 10–4 potential, ⁵³

$$\begin{eqnarray}&&{U}_{{LJ},{surface}}(x)=4\pi {\varepsilon }_{G-O}{\sigma }_{G-O}^{2}{\rho }_{A}\left[\frac{1}{5}{\left(\frac{{\sigma }_{G-O}}{x}\right)}^{10}-\frac{1}{2}{\left(\frac{{\sigma }_{G-O}}{x}\right)}^{4}\right]\end{eqnarray} \tag{ 3 }$$

where ${{\boldsymbol{\varepsilon }}}_{{\boldsymbol{G}}-{\boldsymbol{O}}}$ is the 12–6 LJ potential well-depth for gas (N₂ or water)/oxygen group interaction, ${{\boldsymbol{\sigma }}}_{{\boldsymbol{G}}-{\boldsymbol{O}}}$ is the corresponding 12–6 LJ potential collision diameter, and ${{\boldsymbol{\rho }}}_{{\boldsymbol{A}}}$ is the surface density of oxygen groups. The micropore walls exhibit varying ${{\boldsymbol{\rho }}}_{{\boldsymbol{A}}}$ depending on exposure to ${{\bf{O}}}_{{\bf{2}}},$ processing conditions, etc ${{\boldsymbol{\rho }}}_{{\boldsymbol{A}}}$ on each pore wall is thus assigned a random value from a gaussian distribution around a mean value instead of a constant fixed value. Pores with higher ${{\boldsymbol{\rho }}}_{{\boldsymbol{A}}}$ have greater adsorbed water thickness due to stronger interactions.

Here, we treat the interacting molecules such as H₂O, N₂ and surface oxygen groups as LJ spheres. ^54–56 Properties of a hydroxyl group are used to model the surface oxygen groups since OH is the most abundant type of oxygen group in carbonaceous materials. ⁵⁵ Values of various LJ parameters are listed in Table S2. 12–6 LJ potential parameters for N₂/carbon interactions were obtained from the study by Kim et al. ⁵⁴ Lorentz-Berthelot mixing rules were employed to calculate other cross-species 12–6 LJ potential parameters from pure species 12–6 LJ potential parameters, ⁵⁷

$$\begin{eqnarray}&&{\varepsilon }_{{ab}}=\sqrt{{\varepsilon }_{{aa}}{\varepsilon }_{{bb}}},\,{\rm{and}}\,{\sigma }_{{ab}}=({\sigma }_{{aa}}+{\sigma }_{{bb}})/2\end{eqnarray} \tag{ 4 }$$

In addition to intermolecular interactions, water molecules and surface oxygen groups exert strong dipole-dipole interactions due to their polar nature, which play a critical role in enhancing surface adsorption of water. Dipole-dipole interactions depend strongly on the orientation of the interacting molecules and have been extensively studied by Molecular Dynamics and Monte Carlo samplings. ^56,58–60 By assuming that surface dipoles are distributed uniformly and oriented perpendicular to the surface and by approximating the pore walls as planar surfaces, the potential energy of water (${{\boldsymbol{U}}}_{{\boldsymbol{Dipole}}},$ defined per mole) due to dipole-dipole interactions with the surface oxygen group is obtained from the expression (see SI for derivation)

$$\begin{eqnarray}&&{U}_{{Dipole}}\left(x\right)=\frac{\underset{0}{\overset{2\pi }{\int }}\underset{0}{\overset{2\pi }{\int }}-{\rho }_{A}{N}_{A}\frac{{\mu }_{S}{\mu }_{W}}{{\varepsilon }_{0}x}\cos \alpha \cos \beta \exp \left(-{\rho }_{A}\frac{{\mu }_{S}{\mu }_{W}}{{\varepsilon }_{0}x}\frac{\cos \alpha \cos \beta }{{k}_{B}T}\right)\,d\alpha \,d\beta }{\underset{0}{\overset{2\pi }{\int }}{\int }_{0}^{2\pi }\exp \left(-{\rho }_{A}\frac{{\mu }_{S}{\mu }_{W}}{{\varepsilon }_{0}x}\frac{\cos \alpha \cos \beta }{{k}_{B}T}\right)\,d\alpha \,d\beta }\end{eqnarray} \tag{ 5 }$$

where ${{\boldsymbol{\mu }}}_{{\boldsymbol{S}}}$ and ${{\boldsymbol{\mu }}}_{{\boldsymbol{W}}}$ are the dipole moments of water and surface oxygen group, respectively, ${\boldsymbol{\varepsilon }}$ is the dielectric permittivity of the gas phase, and ${{\boldsymbol{k}}}_{{\boldsymbol{B}}}$ and ${{\boldsymbol{N}}}_{{\boldsymbol{A}}}$ are Boltzmann's constant and Avogadro's number, respectively. In 3-D space, the water dipole vector is formed by three component vectors along the three principal axis directions. Only the component dipole vector aligned parallel to the surface-oxygen-group dipoles contributes to ${{\boldsymbol{U}}}_{{\boldsymbol{Dipole}}},$ while the other component vectors perpendicular to the surface-oxygen-group dipoles have net zero interaction potential (see SI). Equation 5 is derived by summing the interaction potential over the various orientations of the gaseous water dipole. At each ${\boldsymbol{x}},$ Eq. 5 is evaluated by converting the integral to a Riemann sum (See SI for details).

Note that the above formulation of ${{\boldsymbol{U}}}_{{\boldsymbol{Dipole}}}$ assumes that the distance ${\boldsymbol{x}}$ is greater than the size of the molecule (i.e., the distance over which the dipole charges are separated). At small ${\boldsymbol{x}},$ the overall dipole interaction potential is a result of Coulombic interactions between the individual poles. However, the model employs Eq. 5 for all ${\boldsymbol{x}}$ as an approximation, thus over-predicting adsorbed-film thickness. To the best of our knowledge, dipole-dipole interactions for molecules separated by small distances cannot be expressed analytically and are typically estimated from molecular dynamics. ^54,56

As noted above, the gas-molecule wall-interaction potentials result in adsorption of water molecules on the carbon surface. To estimate the thickness of the adsorbed water layer, we examine the local chemical potential of H₂O and ${{\bf{N}}}_{{\boldsymbol{2}}}$ molecules close to the wall. The LJ and dipole interaction potentials with the carbon surface, in addition to size and LJ interactions between the adsorbate molecules, allinfluence the local chemical potential of water and nitrogen molecules close to the carbon surface. A local density approximation along with Amagat's ideal mixing approximation (i.e., the Lewis fugacity rule) ^61,62 and the Peng-Robinson equation of state ^61–63 yields ⁶⁴ (see SI for details)

$$\begin{eqnarray*}&&{\mu }_{i}\left(x,T\right)={\mu }_{o,i}\left(T\right)+{RT}\mathrm{ln}\left({y}_{i}\right)+{RT}\mathrm{ln}\left(\frac{{\rho }_{i}\left(x\right){b}_{i}}{1-{\rho }_{i}\left(x\right)}\right)\end{eqnarray*}$$

$$\begin{eqnarray*}&&+\,\frac{{\rho }_{i}\left(x\right){b}_{i}{RT}}{1-{\rho }_{i}\left(x\right){b}_{i}}-\frac{{a}_{i}{\alpha }_{i}\left(T\right)}{{2}^{1.5}{b}_{i}}\mathrm{ln}\left[\frac{1+{\rho }_{i}\left(x\right){b}_{i}\left(1+\sqrt{2}\right)}{1+{\rho }_{i}\left(x\right){b}_{i}\left(1-\sqrt{2}\right)}\right]\end{eqnarray*}$$

$$\begin{eqnarray*}&&-\,\frac{{a}_{i}{\alpha }_{i}\left(T\right){\rho }_{i}\left(x\right)}{1+2{\rho }_{i}\left(x\right){b}_{i}-{\left({\rho }_{i}\left(x\right){b}_{i}\right)}^{2}}+{U}_{{LJ},\frac{{wall}}{i}}\left(x\right)\end{eqnarray*}$$

$$\begin{eqnarray}&&+\,{U}_{{LJ},{surface}/i}\left(x\right)+{U}_{{Dipole}/i}(x)\end{eqnarray} \tag{ 6 }$$

where ${{\boldsymbol{\mu }}}_{{\boldsymbol{o}},{\boldsymbol{i}}}$ is a reference temperature-dependent standard-state chemical potential of component i, ${{\boldsymbol{\rho }}}_{{\boldsymbol{i}}}$ is the local molar density of component i, and ${{\boldsymbol{y}}}_{{\boldsymbol{i}}}$ is the mole fraction of component i expressed in terms of molar densities

$$\begin{eqnarray}&&{y}_{i}=\frac{{\rho }_{i}}{\displaystyle {\sum }_{i}{\rho }_{i}}\end{eqnarray} \tag{ 7 }$$

Characteristic constants ${{\boldsymbol{a}}}_{{\boldsymbol{i}}}$ and ${{\boldsymbol{b}}}_{{\boldsymbol{i}}}$ of component ${\boldsymbol{i}}$ in Eq. 6 are calculated using the corresponding critical-state parameters, ⁶³ as outlined in the SI. ${{\boldsymbol{\alpha }}}_{{\boldsymbol{i}}}({\boldsymbol{T}})$ modulates the temperature dependence of attractive forces between molecules, ⁶³

$$\begin{eqnarray}&&\alpha \left(T\right)={\left[{c}_{1}+{c}_{2}\left(1-\sqrt{{T}_{r}}\right)\right]}^{2}\end{eqnarray} \tag{ 8 }$$

where ${{\boldsymbol{T}}}_{{\boldsymbol{r}}}$ is the reduced temperature (gas temperature divided by the its critical temperature). Peng and Robinson initially defined the constants ${{\boldsymbol{c}}}_{{\boldsymbol{1}}}$ and ${{\boldsymbol{c}}}_{{\boldsymbol{2}}}$ using the acentric factor for nonpolar hydrocarbon compounds. ⁶³ Since then, over 100 modifications for ${\boldsymbol{\alpha }}({\boldsymbol{T}})$ have been proposed to accommodate different types of compounds, operating conditions, and mixtures. ⁶⁵ We utilize the initial definitions of ${{\boldsymbol{c}}}_{{\boldsymbol{1}}}$ and ${{\boldsymbol{c}}}_{{\boldsymbol{2}}}$ in terms of the acentric factor for N₂. For water, we use the values ${{\boldsymbol{c}}}_{{\boldsymbol{1}}}$ and ${{\boldsymbol{c}}}_{{\boldsymbol{2}}}$ reported by Peng and Robinson in a later publication ⁶⁶ (see Table S2).

Upon equating the local chemical potential at distance ${\boldsymbol{x}}$ from the wall to the bulk chemical potential for each component gas (i.e., ${{\bf{N}}}_{{\bf{2}}}$ and ${{\bf{H}}}_{{\bf{2}}}{\bf{O}}$), Eqs. 6–8 yield two coupled nonlinear algebraic relations to describe ${{\boldsymbol{\rho }}}_{{{\boldsymbol{N}}}_{{\boldsymbol{2}}}}\left({\boldsymbol{x}}\right),{\bf{and}}\,{{\boldsymbol{\rho }}}_{{{\boldsymbol{H}}}_{{\boldsymbol{2}}}{\boldsymbol{O}}}({\boldsymbol{x}}).$ Once solved, the adsorbed water film thickness ${\boldsymbol{h}}$ from the calculated density profile is given by

$$\begin{eqnarray}&&h=\frac{1}{{\rho }_{{H}_{2}O\,{Liq}}}{\int }_{0}^{{\rm{\infty }}}({\rho }_{{H}_{2}O}(x)-{\rho }_{{H}_{2}O,{Bulk}}){dx}\end{eqnarray} \tag{ 9 }$$

where ${{\boldsymbol{\rho }}}_{{{\boldsymbol{H}}}_{{\boldsymbol{2}}}{\boldsymbol{O}},{\boldsymbol{Bulk}}}$ is the molar density of gas-phase water far from the wall (where LJ and dipole surface interaction potentials are non-existent), and ${{\boldsymbol{\rho }}}_{{{\boldsymbol{H}}}_{{\boldsymbol{2}}}{\boldsymbol{O}},{\boldsymbol{Liq}}}$ is the liquid-phase pure-water molar density. Due to the strong dipole-dipole interactions between water molecules and the surface oxygen groups, the adsorbed film is primarily composed of water. Hence, water bulk density is used as the normalizing factor in Eq. 9. Density profiles at various RHs are shown in Fig. 5 for x ≫ 0.15 nm. Below this distance, the potential energy increases sharply due to strong repulsive intermolecular forces. The steep slope of the potential function in this region coupled with system accuracy limits causes convergence issues of the iteration algorithm. Additionally, 0.15 nm is less than the distance of closest approach ${{\boldsymbol{\sigma }}}_{{\boldsymbol{G}}-{\boldsymbol{C}}}.$ At this distance, the molar density calculated is ∼${10}^{-6};$ hence, the adsorbed film thickness at distance ${\boldsymbol{x}}$ < 0.15 nm is neglected. Even at 99% RH, the thickness of the adsorbed film is ∼0.15 nm, which is somewhat less than a continuous water monolayer thickness. The thickness of adsorbed layer is highly sensitive to the value of ${\rho }_{A}$ and multilayer adsorption is observed just by doubling the value of ${\rho }_{A}$. Note that ${\rho }_{A}$ can vary widely between carbon types based on processing condition, air exposure during storage, etc.

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In the model, only micropores adsorb water. The mesopores and macropores are modeled with ionomer deposition on the pore walls, preventing water from adsorbing directly on the bare carbon surface. In an actual CL system with heterogeneous ionomer distributions, bare carbon surfaces in meso and macropores can also adsorb water. This contribution is, however, minimal given the small surface area of meso and macropores compared to that of the micropores, and thus is neglected. Water adsorption on Pt surfaces is similarly neglected due to their relatively small SA compared to carbon SA in the micropores. ⁶⁷

At typical PEFC operating temperatures (60 to 80 °C), there are multiple roots of the Peng-Robinson EOS. ⁶⁴ The smallest and largest roots correspond to gas and liquid phase respectively, while the middle root is considered aphysical. Hence, different adsorbed film thicknesses can be calculated depending on the initial guess. Since we are modeling the adsorption curve of the isotherm here, the bulk gas-phase density of water and nitrogen are used as initial guesses for the densities of various components for solving Eq.6 via Newton-Raphson iterative method.

Capillary condensation describes liquid filling of small capillary pores (2 to 50 nm) below the planar saturation pressure due to surface tension and concave meniscus curvature lowering the chemical potential. ^68,69 Kelvin's equation for gas mixtures governs the condensation conditions. ⁶⁹ By assuming that the condensed liquid consists of only water, and the gas phase includes both carrier gas (inert N₂) and water vapor, the Kelvin equation is written as

$$\begin{eqnarray}&&\mathrm{ln}\left(\frac{{y}_{{H}_{2}O}P}{{P}_{{H}_{2}O}^{{sat}}}\right)=\frac{1}{{\rho }_{{H}_{2}O,{Liq}}{RT}}\left((,-2\gamma H\cos \theta +(P-{P}_{{H}_{2}O}^{{sat}})\right)\end{eqnarray} \tag{ 10 }$$

where ${\boldsymbol{\gamma }}$ is the water/air surface tension, ${\boldsymbol{H}}$ is the total mean curvature of the meniscus and depends on pore radius/size and shape, ${{\boldsymbol{P}}}_{{{\bf{H}}}_{{\bf{2}}}{\bf{O}}}^{{\boldsymbol{sat}}}$ is the saturation water vapor pressure at temperature T, and ${{\boldsymbol{y}}}_{{{\bf{H}}}_{{\bf{2}}}{\bf{O}}}$ is the mole fraction of water vapor in the bulk gas phase, and ${\boldsymbol{\theta }}$ is the contact angle of water with the solid surface measured in the water phase (assumed to be 0 due to hydrophilic oxygen-groups and adsorbed water on the pore walls). Contact angle is not an independent parameter and can be calculated from adsorption isotherm. ⁷⁰ Here we use an assumed value of $\theta$ to limit the scope of the paper. The constricted cylindrical pores in CLs have a mean curvature equal to the inverse of pore radius ${\boldsymbol{r}},$ different from that for straight cylindrical capillaries with mean curvature ${\boldsymbol{1}}/{\boldsymbol{2}}{\boldsymbol{r}}$ to account mainly for presence of narrow constrictions in the CL pores. Water uptake due to capillary condensation at a given RH thus equals the total volume of pores with pore radius less than the critical radius for capillary condensation ${{\boldsymbol{r}}}_{{\boldsymbol{c}}},$ which equals ${\boldsymbol{1}}/{\boldsymbol{H}}.$

Ionomer water uptake is estimated from CL water content, λ, defined as the moles of water absorbed per mole of sulfonic acid groups. Based on experiments, Kusoglu and Weber report a polynomial fit for i as a function of RH, ^19,71

$$\begin{eqnarray}&&\lambda =0.316+4.833{RH}-10.07R{H}^{2}+11.56R{H}^{3}\end{eqnarray} \tag{ 11 }$$

Equation 11 is obtained from bulk ionomer data that should be representative of the response of the heterogeneous CL due to the presence of ionomer aggregates that behave similar to bulk ionomer and dominate water uptake (compared to ionomer thin films that are reported to exhibit reduced water uptake). ¹⁹ The water volume absorbed by ionomer is equal to the calculated ${\boldsymbol{\lambda }}$ value multiplied by water molar volume ${{\boldsymbol{v}}}_{{\boldsymbol{liq}},{{\boldsymbol{H}}}_{{\boldsymbol{2}}}{\boldsymbol{O}}}$ and the total moles of sulfonic groups, given by the ionomer mass divided by ionomer EW (1000 g mol⁻¹).

The above sections quantify water uptake via different modes in the CL. The total water uptake at a given RH, hence, is the sum of water uptake from the individual modes i.e., surface adsorption, capillary condensation, and ionomer absorption, as represented by each term on the right side of the following expression, respectively.

$$\begin{eqnarray}&&\begin{array}{l}{Total}\,{Water}\,{Uptake}({RH})\left({{cm}}^{3}/{g}_{{carbon}}\right)=h{\int }_{{r}_{c}}^{6{nm}}{\left(\frac{d\left({SA}\right)}{{dr}}\right)}_{{Micro}}{dr}+\displaystyle \sum _{\begin{array}{c}i={micro},\\ {meso}\,{and}\,{macro}\end{array}}{\int }_{{r}_{l}}^{{r}_{c}}{\left(\frac{{dV}}{{dr}}\right)}_{i}{dr}+\left(\frac{I:C\,{Ratio}}{{EW}}\right)\lambda {v}_{{liq},{H}_{2}O}\end{array}\end{eqnarray} \tag{ 12 }$$

where ${r}_{c}$ is the critical radius for capillary condensation, and ${r}_{l}$ is the smallest pore radius in respective pore type. SA represents pore surface area and $h$ is the adsorbed film thickness calculated from Eq. 9. The first term on the right in Eq. 12 represents water adsorbed on carbon walls and includes the larger micropores above ${r}_{c}$ because only non-liquid-filled micropores exhibit bare carbon walls. The second term represents capillary condensation and includes all pore-types. The last term representing ionomer water uptake is calculated using $\lambda$ from Eq. 11 using CL parameters i.e., I:C ratio, and ionomer EW. The relative contribution by each mode depends on several factors such as CL PSD, carbon surface properties, and I:C ratio. These findings are examined below.

In PEFCs, Pt particles in the interior micropores of HSC are accessed by ${{\rm{H}}}^{+}$ through liquid-water pathways, i.e., adsorbed and/or capillary-condensed water. ^31,38,39 In one of the early studies examining ${{\rm{H}}}^{+}$ conduction via adsorbed water layers, Khanna et al. suggested the dominance of a ${{\rm{H}}}^{+}$ tunneling mechanism at low humidity and vehicular transport at high humidity. ⁷² Furthermore, hydroxyl terminated surface defects are known to facilitate ${{\rm{H}}}^{+}$ transport. ^73–75 Therefore we define a threshold adsorbed film thickness ${h}_{o}$ at which the adsorbed water film can conduct water, thus rendering the Pt particles in electrochemical contact and leading to the electrochemically active surface area (ECSA). At adsorbed film thickness lower than ${h}_{o},$ the adsorbed films are not appreciably conductive and hence the Pt particles in contact do not contribute to ECSA. The conductivity of the adsorbed films is expressed as a step-function with the films exhibiting bulk liquid conductive at thickness ${h}_{o}$ and above. The total ECSA at any RH is determined by summing the Pt area in (i) meso and macropores pores since they are in contact with ionomer, and (ii) micropores that are either flooded due to capillary condensation or have adsorbed film thickness greater than ${h}_{o}.$ It is assumed that all micropores are in contact with meso/macropores and can thus conduct ${{\rm{H}}}^{+}$ when they have sufficient adsorbed film thickness or capillary-condensed water. In the model, ${h}_{o}$ is fit within reason (fractions of a monolayer) to reproduce experimental data.