Modeling of PEM Fuel Cell Catalyst Layers: Status and Outlook

Abstract

Computational modeling has played a key role in advancing the performance and durability of polymer electrolyte membrane fuel cells (PEMFCs). In recent years there has been a significant focus on PEMFC catalyst layers because of their determining impact on cost and and durability. Further progress in the design of better performance, cheaper and more durable catalyst layers is required to pave the way for large scale deployment of PEMFCs. The catalyst layer poses many challenges from a modeling standpoint: it consists of a complex, multi-phase, nanostructured porous material that is difficult to characterize; and it hosts an array of coupled transport phenomena including flow of gases, liquid water, heat and charged occurring in conjunction with electrochemical reactions. This review paper examines several aspects of state-of-the-art modeling and simulation of PEMFC catalyst layers, with a view of synthesizing the theoretical foundations of various approaches, identifying gaps and outlining critical needs for further research. The review starts with a rigorous revisiting of the mathematical framework based on the volume averaging method. Various macroscopic models reported in the literature that describe the salient transport phenomena are then introduced, and their links with the volume averaged method are elucidated. Other classes of modeling and simulation methods with different levels of resolution of the catalyst layer structure, e.g. the pore scale model which treats materials as continuum, and various meso- and microscopic methods, which take into consideration the dynamics at the sub-grid level, are reviewed. Strategies for multiscale simulations that can bridge the gap between macroscopic and microscopic models are discussed. An important aspect pertaining to transport properties of catalyst layers is the modeling and simulation of the fabrication processes which is also reviewed. Last but not least, the review examines modeling of liquid water transport in the catalyst layer and its implications on the overall transport properties. The review concludes with an outlook on future research directions.

Graphical Abstract

Appendices

Appendix 1: Definition of Volume Averaging

The volume average of a variable ψ over a control volume, i.e. $\left\langle {{\varvec{\psi}}_{k} } \right\rangle$, is defined as

$$ {\left\langle {{\varvec{\psi}}_{k}} \right\rangle} = \frac{1}{V}\mathop \int \nolimits_{{V_{k} }}^{a} {\varvec{\psi}}_{k} {\text{d}}V $$

(60)

Where V_k is the volume of phase k and V is the total control volume for averaging, i.e., $V = \sum\limits_{k} {V_{k} }$.

The intrinsic volume average of variable ψ in phase k is

$$ {\left\langle {{\varvec{\psi}} _{k} } \right\rangle ^{k} } = \frac{1}{{V_{k} }}\mathop \int \nolimits_{{V_{k} }}^{a} {\varvec{\psi}}_{{k}} {\text{d}}V $$

(61)

The volume fraction of phase k is then $\varepsilon_{k} = \frac{{V_{k} }}{V}$ and naturally, $\sum\nolimits_{k} {\varepsilon_{k} }$.From (60) and (61), we have

$$ \left\langle {{\varvec{\psi}}_{k} } \right\rangle = \varepsilon_{k} \left\langle {{\varvec{\psi}}_{k} } \right\rangle^{k} $$

(62)

The fluctuating component, i.e., deviation due to the presence of other phases, is defined as

$$ {\varvec{\tilde{\psi }}}_{k} = {\varvec{\psi}}_{k} - \left\langle {{\varvec{\psi}}_{k} } \right\rangle^{k} $$

(63)

Following (61), the intrinsic volume average of the product for two variables becomes

$$ \left\langle {{\varvec{\psi}}_{k} \varphi_{k} } \right\rangle^{k} = \left\langle {{\varvec{\psi}}_{k} } \right\rangle^{k} \left\langle {\varphi_{k} } \right\rangle^{k} + \left\langle {\varvec{\tilde{\psi }}_{k} \tilde{\varphi }_{k} } \right\rangle^{k} $$

(64)

The volume average of a time derivative is

$$ {\left\langle {\frac{{\partial {\varvec{\psi}} _{k} }}{{\partial t}}} \right\rangle } = \frac{\partial }{\partial t}\left( {{\varvec{\varPsi}}_{k} } \right) - \frac{1}{V}\mathop \int \nolimits_{{A_{k} }}^{a} {\varvec{\varPsi}}_{k} \varvec{w}_{k} \varvec{n}_{k} {\text{d}}A $$

(65)

Where w_k is the velocity of interface and n_k is the normal vector of the interface.

The volume average of a gradient or divergent:

$$ \left\langle {\nabla {\varvec{\psi}} _{k} } \right\rangle = \nabla \left\langle {{\varvec{\psi}} _{k} } \right\rangle - \frac{1}{V}\mathop \int \nolimits_{{A_{k} }}^{a} {\varvec{\varPsi}}_{k} \varvec{n}_{\varvec{k}} {\text{d}}A $$

(66)

Appendix 2: Derivation of the Agglomerate Model and the PDL Model with the Volume Averaging Method

The following discussion is continued from Chapter 2.1.

The agglomerate model and the PDL model are similar in that there exists a difference of oxygen concentration between the bulk pore phase and the catalyst surface. For both models, we can derive from the VAM equations by making further assumptions.

Assumption 7

Zero net flux of oxygen through the ionomer phase in the REV. This eliminates the first term on the LHS of (6), i.e. $\nabla \cdot \left( {\left\langle {\varvec{j}_{{{\text{O}}_{2} }} } \right\rangle_{\text{m}} } \right) = 0$. This implies that oxygen consumed on the CPt surface is solely from the pore phase. This may not be valid under circumstances, e.g. when the pore phase is flooded, in which case oxygen could still diffuse through the ionomer pathway.

Adding (5) and (6), which cancels out the fluxes across interface A_pm, we have:

$$ \nabla \cdot \left( {\left\langle {\varvec{j}_{{{\text{O}}_{2} }} } \right\rangle_{\text{p}} } \right) + \frac{1}{V}\mathop \int \nolimits_{{A_{\text{mc}} }}^{a} \varvec{j}_{{{\text{O}}_{2} }} \cdot \varvec{n}_{\text{ms}} {\text{d}}A = 0 $$

(67)

Equation (67) simply states that the net flux of oxygen gas through the REV equals the consumption of oxygen (in ionomer) at the CPt surface.

2.1 Model closure: Mass Transfer in Pore Phase

Next we can provide model closure based on (67) with oxygen as an example. The mass diffusion in porous media is often modeled with an effective diffusivity:

$$ \langle \varvec{j}_{{{\text{O}}_{2} }} \rangle_{\text{p}} = - \rho D_{{{\text{O}}_{2} }}^{\text{eff}} \nabla \varvec{Y}_{{{\text{O}}_{2} }} $$

(68)

The effective diffusivity can be measured by experiment or computed by microscale models, e.g. [84, 97], with zero oxygen source, i.e.

$$ \nabla \cdot \left( {\varvec{j}_{k} } \right) = - \nabla \cdot \left( {\rho D_{{{\text{O}}_{2} }}^{\text{eff}} \nabla \varvec{Y}_{{{\text{O}}_{2} }} } \right) $$

(69)

Both Fickian diffusion and Knudsen diffusion can be formulated. It is noted that Assumption 3 can be relaxed and the diffusive mass flux can then be modeled according to the Stefan–Maxwell formulation [296].

2.2 Model Closure: Reactions and Phase Equilibrium

The model closure for the diffusion-reaction processes in the CL is the major difference among the aforementioned macroscopic models. From (67) and (68), the transport of gas species can be expressed in this form:

$$ - \nabla \cdot \left( {D_{i}^{\text{eff}} \nabla \varvec{c}_{i} } \right) = \dot{m}_{i} $$

(70)

Assumption #8

Phase equilibrium at the pore-ionomer interface.

The concentration of gas species on the ionomer surface can be related to that in the pore phase by the Henry’s law, which means the chemical potential of oxygen and that in the ionomer phase are the same and phase equilibrium is established. Another assumption of chemical equilibrium on the reaction site can be made to describe the balance of surface coverage, e.g. [73].

Assumption #9

First-order reaction for ORR

A first-order reaction for the oxygen reduction reaction can be assumed:

$$ \dot{m} = - k \cdot c(X) $$

(71)

For the ORR the parameter k takes the form of either the Butler–Volmer equation or the Tafel equation. Assuming a constant parameter k in (71) and a 1D configuration, Fig. 16, with boundary conditions of

$$ \begin{aligned} & c = c_{0} \;{\text{at}}\;X = 0,{\text{and}} \\ c = c_{1} \;{\text{at}}\;X = L \\ \end{aligned} $$

the solution of (70) is

$$ c\left( x \right) = \frac{{c_{1} - c_{0} {\text{e}}^{{ - \lambda {\text{L}}}} }}{{{\text{e}}^{{\lambda {\text{L}}}} - {\text{e}}^{{ - \lambda {\text{L}}}} }}{\text{e}}^{\lambda x} + \frac{{c_{0} {\text{e}}^{{\lambda {\text{L}}}} - c_{1} }}{{{\text{e}}^{{\lambda {\text{L}}}} - {\text{e}}^{{ - \lambda {\text{L}}}} }}{\text{e}}^{ - \lambda x} $$

(72)

where

$$ \lambda = \sqrt {\frac{k}{D}} $$

(73)

Fig. 16

Illustration of the coordinate and dimensions of the CL and the GDL

For the macro-homogeneous model, the effective diffusivity is the mass diffusivity of gas species in the ionomer phase. For the agglomerate model and the pore-diffusion limited model, the effective diffusivity of gas species in the pore is used. For the latter two models, which assume that there exists certain mass diffusion obstacles between the pore and the reaction site, further assumptions can be made about the interface condition and the reaction term in (70).

Consider a configuration that the ionomer phase in the REV is characterized by an equivalent thickness, or the inversed specific surface area (V/S), where V is the volume of the ionomer in the REV and S is the surface area of carbon/ionomer interfaces. This surface area can be later extended to the area of the catalyst if discrete distribution of catalysts on the carbon surface is assumed. The concentration to be used for (71) is then the concentration on the surface of catalyst, c*(x) as opposed to that in the “bulk”, c(x), i.e.,

$$ \dot{m} = - k \cdot c^{*} (X) $$

(74)

The local reaction can then be related to diffusion in ionomer by assuming a local flux as

$$ \dot{m} = - A\frac{{D_{m} (c - c^{*} )}}{\delta } $$

(75)

where A is the surface area of the catalyst, δ is the characteristic thickness of the ionomer.

Equating (74) & (75) and rewriting for c*, we have

$$ c^{*} = \frac{c}{{1 - \frac{k\delta }{{AD_{m} }}}} $$

(76)

Substituting (76) into (74) and (75), we have

$$ \lambda = \sqrt {\frac{k}{{D\left( {1 - \frac{k\delta }{{AD_{m} }}} \right)}}} $$

(77)

The agglomerate model in essence considers two mass transfer resistances along the transport pathway, i.e., mass diffusion through the ionomer coating and through the agglomerate. The final flux (oxygen consumption) is then related to oxygen concentration at the agglomerate surface by two resistors in series, i.e.

$$ R_{\text{diff}} = \frac{\delta }{{D_{m} }}, $$

(78)

and

$$ R_{\text{reac}} = k $$

(79)

The effective resistance (diffusivity) through the ionomer is in general a function of ionomer thickness, oxygen diffusivity, etc. The effective resistance for the transport inside the agglomerate is a function of the geometrical form of the agglomerate and the relative strength of diffusion versus the reaction rate, i.e. the Damköhler number.

As the reaction takes place in the reaction zone, concentration varies along the penetration direction and a gradient of concentration, therefore a deviation from the highest reaction rate, is formed. An effectiveness factor η can be introduced to relate the actual flux (oxygen consumption) into the agglomerate and the flux in the absence of mass transfer through it. In the catalyst industry the effectiveness factor is often expressed as a function of the Thiele modulus, which is a function of kinetics, diffusivity, catalyst pellet geometry, etc. For instance, effectiveness factor for a sphere can be expressed as

$$ \eta = \frac{\delta }{{\sqrt {An_{0} } }} \cdot \left[ {\frac{1}{{\tan (3\sqrt {An}_{0} )}} - \frac{1}{{3\sqrt {An}_{0} }}} \right], $$

(80)

where An₀ is the zeroth order of the Aris number, or a modified Thiele modulus. The idea of a structure-based model is then to establish a rational approach to determine the Thiele modulus (or equivalent formula) that represents the reconstructed microstructure and transport/kinetic parameters for a given catalyst layer.