Electrochemical Energy Reviews

, Volume 2, Issue 3, pp 428–466 | Cite as

Modeling of PEM Fuel Cell Catalyst Layers: Status and Outlook

  • Pang-Chieh Sui
  • Xun Zhu
  • Ned DjilaliEmail author
Review article


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


Simulation Transport phenomena Pore scale modeling Macroscopic modeling Fuel cell electrode Catalyst layer 



polymer electrolyte membrane fuel cell


gas diffusion layer


microporous Layer


catalyst layer


membrane electrode assembly


computational fluid dynamics


molecular dynamics


kinetic Monte Carlo




representative element volume


volume averaging method


direct numerical simulation


lattice Boltzmann model

1 Introduction

A polymer electrolyte membrane fuel cell (PEMFC) is comprised of a membrane electrode assembly (MEA) sandwiched by bipolar plates with embedded fluid channels. A typical MEA is made of an ion-conducting membrane coated on both sides subsequently with a catalyst layer (CL) and a microporous layer (MPL), and bonded by a gas diffusion layer (GDL). The CL is the heart of the MEA because most transfer processes and energy release take place within it. The CL also contributes to a significant portion of material cost because of the precious metal used for electrocatalysis. Historically it was the breakthrough in MEA fabrication in the late 1980s that drove the rapid development of PEMFCs during the 1990s, which then led to mature products in the early 2000s. The research and development on PEMFCs since the turn of this century has addressed many issues pointed out in [1, 2], however, a major quantum jump into large scale commercialization of PEMFCs has not happened after two decades of hard work and heavy investment. The delay of such advancement can be, arguably, attributed to the lack of understanding for the intimately coupled, complex transport phenomena taking place in the CL, and thus the lack of know-hows to optimize its performance/cost ratio and durability. Understanding through modeling of the transport phenomena in CLs, in conjunction with simulation and characterization tools may help to advance our knowledge in this aspect. While computational simulations at both cell and stack levels have played a key role in the development and design of PEMFCs [3, 4, 5, 6, 7], the predictive abilities of such simulations remain limited by the modeling and resolution of transport phenomena in the MEA [6, 7], and especially in the CL. Modeling of the CL is the most critical to further progress in performance, durability and cost, and yet the most challenging and least complete because of the length scales and the transport/reactions involved. The primary goal of this review is therefore to critically survey past literature on modeling of the CL, identify critical gaps and limitations, and outline future research needs to allow the next quantum jump for PEMFCs.

1.1 Salient Features of Transport Phenomena in PEMFC Catalyst Layers

A conventional PEMFC catalyst layer is made first by mixing ionomer and catalyzed carbon with solvent to become a colloid and then applying it to either the membrane (together termed as catalyst coated membrane, or CCM) or porous diffusion layers (together termed as the gas diffusion electrode or GDE), e.g. [8, 9, 10]. In an operating fuel cell, the catalyst layer primarily functions as a gateway for the transport of charged species (protons and electrons) and non-charged, gas species (O2, H2, H2O). The transport of charged species takes place in the solid phases (protons in the ionomer, electrons in carbon/Pt), whereas the transport of non-charged species and phase change of water take place in the gas pores and the ionomer phase. Electrochemical reactions occur in the vicinity of the catalyst where ionomer and carbon are present, and their rates are functions of concentration of the non-charged species as well as the potential difference between the ionomer and the carbon phases. Heat released from these electrochemical reactions and heat generated due to charge transfer is transferred throughout all phases. Product water from the cathodic reaction may exist in several phases simultaneously (gas, liquid, absorbed water in the ionomer, and ice under extreme conditions) depending on the thermodynamic state, which in turn depends on local heat transfer and mass transport.

The catalyst layer presents several challenges to modeling, such as (1) coupled transport and reactions, (2) wide range of length scale, and (3) multiphase flow. The transport processes in a CL occur over at least three orders of magnitude of length scales (nm to μm,) and a few of these processes are coupled, which necessitates spatial resolution down to the length scale of the catalyst, hence it is inherently a multi-scale problem. From a modeling standpoint, the catalyst layer of a PEMFC is analogous to the flame zone in a combustor in the sense of its small characteristic length scale over which chemical reactions occur versus that of mass transfer of the reactants. The catalyst layer has the smallest dimension (ca. 10 μm) among all components in a unit cell, yet it is there that a most complex array of transport phenomena take place. However, the PEMFC catalyst layer differs from a flame zone in that multiple species are transported into the reaction zone through different pathways: gas species diffuse through the pore space, positive ions migrate within the electrolyte, electrons move in the carbon-Pt solid phase, water moves through space other than the carbon/catalyst (CPt) in different forms, and heat transfer occurs among all phases in the catalyst layer, see Fig. 1. In the catalyst layer, each transport pathway has its morphology and network, yet all these pathways are closely intertwined down to the level of gas pores in the CL. Modeling works on the transport and chemical reactions in the PEMFC catalyst layer that were aimed to gain understanding of the coupled problem began around the same time when PEMFCs saw their breakthrough in the early 1990s, e.g., the pioneering work by [11]. The macro-homogeneous model proposed in that work stemmed from models developed for the transport in liquid electrolyte, therefore a catalyst layer flooded with water was implied. A different category of catalyst layer models, i.e. the agglomerate model and its variants, which assume that CPt particles form certain agglomerates, were developed to describe the transport and had gained popularity due to its closer representation of physical observations. However, both types of catalyst layer models are essentially phenomenological models and their predictive capabilities are rather limited. In order to utilize modeling and simulation tools to guide the design of the next generation of PEMFCs, a comprehensive catalyst model based on thorough understanding of the physics and microstructure is needed.
Fig. 1

Transport pathways in a cathodic catalyst layer

1.2 Scope and Objectives

1.2.1 Classification of Modeling and Simulation Methods

Figure 2 shows a diagram listing available computational methods relevant to the modeling and simulation of the PEMFC catalyst layer, from large scale on the left towards atomistic scale on the right. These models can be divided into two different categories by the frame of reference, i.e., continuum and particle-based models. The continuum approach describes the transport based on a continuum space and it is a natural choice for engineering applications. The continuum models are further divided into the volume-averaged type, where spatial homogenization within the control volume is performed, and the geometry-resolved type, where information of actual microstructure of porous media is included in the model. These particle-based models describe the transport processes for a single molecule or a cluster of molecules. The particle-based models are always time-dependent and some sort of time/volume averaging is needed in order to extract useful information. Since these particle-based models deal with movement of molecules or clusters of particles, the length scale the models can handle is thus limited to sub-μm and often smaller, and the time scale is typically sub-μs or shorter. The particle-based models can be divided into two groups according to the principles of motion employed, i.e. the first principles, or ab initio methods which follow established laws of physics and do not make ad hoc assumptions or involve fitting parameters. The non-first-principles methods are an emerging type of modeling that simulates the process of aggregation for particles suspending in colloid phase and the formation of microstructure of porous media by considering the interactions among forces using similar methodology as particle dynamics.
Fig. 2

Classification of models for the PEMFC catalyst layer

The catalyst layer considered in this review is limited to that fabricated with the conventional methods, i.e. by mixing the ingredients and applying directly to the membrane or diffusion media. In this regard, catalyst layers fabricated by nano-fabrication methods, such as the ultra-thin CL reported by [12, 13], are not considered in this review. Although catalyst layers are situated on both sides of the membrane, it is the cathodic catalyst layer that is considered throughout this review unless otherwise noted, because it is where transport and kinetics are likely limited under normal and high current operating conditions. Nonetheless, most of the modeling works and methodologies are readily applicable to the anodic catalyst layer as well.

1.2.2 Structure of this Review

In the literature of fuel cell research, there have been a few review articles that focus on different aspects of modeling, e.g. [4, 14, 15, 16, 17, 18, 19, 20]. These reviews are comprehensive overviews of the models for all the components of PEMFCs. Modeling of catalyst layer is mentioned, but because of the wide scope of these papers, fundamental issues related to CLs are not discussed in depth. A couple of recent review papers [21, 22] focus on some aspects of the CL. Various methods of catalyst layer fabrication are summarized in [8, 9, 10, 23, 24]. These reviews focus on technological aspects of catalyst layer but do not provide insights into modeling of the catalyst layer. Several reviews have been published recently on specific topics such as modeling water transport [25, 26, 27], and on degradation in catalyst layer [28]. The focus of the present review is on the theoretical framework of PEMFC catalyst layer modeling and major computational tools employed to implement the modeling. A systematic examination on fundamental transport principles pertaining to catalyst layer is attempted. In addition to classical theories, promising methodologies in progress are also surveyed. The objective of this review is two-fold: to first revisit the rigorous theoretical framework based on which catalyst models can be derived and built, and secondly by reviewing literature in different disciplines in order to identify gaps among existing CL models and to propose strategies to bridge them. The remaining discussion of this review is organized as follows.

In Chapter 2, classical theories based on the volume averaging method and the governing equations derived for PEMFC CL are discussed. In Chapter 3, macroscopic models for CL are reviewed and compared to the rigorous set of equations from Chapter 2. In Chapter 4, a new trend of pore-scale modeling and numerical simulation performed on resolved microstructural geometry of the PEMFC CL is introduced. In Chapter 5, mesoscopic and microscopic simulation methods for even smaller length scales than Chapter 4 are briefly explained and their strengths and limitations are assessed. The strategy of multiscale simulations involving methodologies discussed in Chapters 2–5 is summarized at the end of this chapter. An emerging research direction that demands further investigation and modeling is the fabrication procedure of the CL, which is discussed in Chapter 6. A special topic, i.e. modeling of liquid water transport, is discussed in Chapter 7. The present review concludes in Chapter 8 with recommendations for future study.

2 Modeling Framework

In Chapters 2–5, macroscopic and microscopic models developed for the transport in the CL over a wide range of length scale and time scale will be discussed. Among the models shown in Fig. 2, there still exist gaps of several orders of magnitude in the length scale within which each model is valid. Ideally one could develop a strategy of multi-scale modeling to bridge these gaps, however, this idea is currently at a stage far from completion for CL modeling due to at least two reasons: (1) for the macroscopic modeling, there lacks a rigorous, theoretical framework to fully account for the transport processes that are coupled within phase networks and the interfaces among these networks; (2) between the macroscopic and microscopic models, there lacks a common platform upon which simulations using both methodologies can be linked directly. In this chapter we revisit classical theories with conservation equations derived based on the volume averaging method, which will be employed to address the aforementioned missing elements. The complete set of thus derived conservation equations along with appropriate initial conditions, boundary conditions and the equation of state discussed in this chapter constitute the modeling framework to link macroscopic models and microscopic models.

In this chapter, a brief derivation of conservation equations based on the volume averaging method (VAM) is provided. The derivation of generalized conservation equations will be used later to demonstrate that existing CL models in the literature are in fact special cases of the generalized formulation after certain assumptions. Furthermore, the derivation also identifies places for model closure where mesoscopic or microscopic models can be employed.

2.1 Volume Averaging Method

The volume averaging method is a thorough method developed to model the transport phenomena in multi-component, multi-phase systems, e.g. see [29, 30, 31] for more details. In essence, this method describes the transport within the representative element volume (REV) for each individual phase in the system for all primary variables following a definitive set of volume-averaging principles. In each transport equation, both volumetric quantities and interfacial quantities are considered. More general formulation and derivation of the method can be found in [29]. In the fuel cell literature, formulation based on the VAM has been reported by [32, 33] and more recently by [34]. The major advantage of the VAM is its completeness in terms of accounting systematically every possible transport phenomenon encountered in common practice. However, the challenge is not in the derivation of these conservation equations but rather it is in the closure of terms in the equations that arise from volume averaging steps.

2.1.1 Field Equation for Each Individual Phase

The volume averaged transport equation is derived from a microscopic, homogeneous field equation, which describes the transport of a conserved scalar or vector in phase k:
$$ \frac{\partial }{\partial t}\left( {\rho_{k} \varPsi_{k} } \right) + \nabla \cdot \left( {\rho_{k} \varvec{v}_{k} \varPsi_{k} } \right) = - \nabla \cdot \left( {\varvec{j}_{k} } \right) + \rho_{k} \varphi_{k} $$
where ρ is the density of phase k (subscript), ψ is the primary variable, v is the fluid velocity, j is the diffusional flux, and φ is the volume source term. By employing the principles for volume averaging, the macroscopic volume-averaged equation is obtained by integrating over a control volume with a representative length scale lrep, where pore size ≪ lrep ≪ system length scale:
$$ \begin{aligned} & \frac{\partial }{{\partial t}}\left( {\rho _{k} \left\langle {{\varvec{\psi}} _{k} } \right\rangle } \right) + \nabla \cdot \left( {\rho _{k} \left\langle {\varvec{v}_{k} } \right\rangle \left\langle {{\varvec{\psi}} _{k} } \right\rangle ^{k} } \right) + \nabla \cdot \left( {\rho _{k} \left\langle {\tilde{\varvec{v}}_{k} \cdot {\varvec{\tilde{\psi }}}_{k} } \right\rangle } \right) + \mathop \int \nolimits_{{A_{i} }}^{a} \rho _{k} \left\langle {\left( {\varvec{v}_{k} - \varvec{v}_{i} } \right) \cdot \varvec{n}_{k} {\varvec{\psi}} _{k} } \right\rangle {\text{d}}A \\ & \quad = - \nabla \cdot \left( {\left\langle {\varvec{j}_{k} } \right\rangle } \right) - \frac{1}{V}\mathop \int \nolimits_{{A_{i} }}^{a} \varvec{j}_{k} \cdot \varvec{n}_{k} {\text{d}}A + \rho _{k} \left\langle {\varphi _{k} } \right\rangle \\ & \left[ {{\text{Advection}}\;{\text{by}}\;{\text{the}}\;{\text{mean}}\;{\text{flow}}} \right] + \left[ {{\text{Dispersion}}} \right] + \left[ {{\text{Advection}}\;{\text{across}}\;A_{i} } \right] \\ & [{\text{Flux}}\;{\text{due}}\;{\text{to}}\;{\text{driving}}\;{\text{forces]}} + \left[ {{\text{Flux}}\;{\text{across}}\;A_{i} } \right] + \left[ {{\text{Production}}} \right] \\ \end{aligned} $$

Definition of the terms in Eq. (2) can be found in Appendix A. Comparing (1) and (2), one can see that some terms in (2), namely the advection term, diffusion flux term and source term, are in the same form as (1), yet several additional terms associated with phase interface, i.e. the dispersion term as well as advection and diffusion flux across interface, also appear.

2.1.2 Volume-Averaged Transport Equations for the CL

The transport in a catalyst layer of PEMFC can be described with conservation equations in the form of Eq. (2). The phases involved in the catalyst layer model are solid ionomer (M), solid carbon-Pt (S), liquid (L), and gas (G). The primary variable ψ can be substituted by 1, v, h (enthalpy), Y (mass fraction), ϕ (electrical potential), and s (liquid saturation) for the conservation of mass, momentum, energy, and non-charged species and charged species, and liquid water, respectively. In a H2/air PEMFC, the gas species are H2, O2, H2O, and N2, whereas the charged species are H+ and e. For a broader range of applications, gas species such as CO, CO2, SO2etc., ions such as OH, Pt2+ (for platinum dissolution) and Fe3+ (for contamination from metallic bipolar plate), and solid such as carbon (for carbon corrosion problem) and electrolyte (membrane degradation) can be included in the model for the species equations. Water appears in several forms in the CL model, i.e., as vapor, liquid, absorbed water in electrolyte, and ice if sub-zero temperature conditions exist. An equation of state is needed for all non-charged species for the evaluation of density.

With the aforementioned variables, the governing equations for the CL then include (a) continuity equation for the fluid phases (2 equations); (b) momentum equation for the fluid phases (2 equations), (c) energy equation for all phases (4 equations), (d) species equation for all phases (4 × N equations), where N is the number of species involved in the model. For N = 6 (H2, O2, H2O, and N2, H+ and e), the maximum number of equations involved is 32. The number of equations for a complex system may be in excess of one hundred for the catalyst layer model, thus listing the equations is beyond the scope of this review. For the four phases of LGMS, there exist six two-phase interfaces, i.e. MS, ML, MG, SL, SG, and LG. The number of interfacial terms in the governing equations for all species is massive, which makes closing the model, rather than deriving the model, a major challenge. Although the number of equations and interfacial terms can be reduced by the nature of certain transport processes (e.g. charge species through a GS or a GM interface is not likely to occur), there remain substantial unknowns that require constitutive equations to be modeled for these interfacial terms. It should be pointed out that the actual number of equations can be reduced significantly with appropriate assumptions, e.g. phase equilibrium on the interfaces or thermal equilibrium established in the control volume.

2.1.3 Interfaces in the CL

The conventional method of fabrication of a CL by mixing ionomer and CPt particles yields a porous medium full of interfaces. Transport of different species take place simultaneously across these interfaces, therefore, modeling the interfacial transport is crucial to the understanding of these coupled transport processes. For the sake of simplicity, the CPt particle is considered as one single phase and the interface between carbon and Pt is ignored. For water in the ionomer, it is assumed that water appears as water molecules absorbed in the ionomer solid, although in theory when water content exceeds its saturation value, excess water may appear as another phase that moves freely. Figure 3 illustrates the four material phases considered in the VAM model. Description of the transport across the interfaces in a CL is given as follows.
Fig. 3

Schematic diagram of four phases in a CL


This is the catalyst surface where electrochemical reactions take place. Also dissolved species in the ionomer phase may diffuse in on this interface before they reach the reaction sites.


When liquid water exists in the pores, the liquid may cover the ionomer. The area of such interface depends strongly on the wettability of the ionomer surface. In some agglomerate-type of models, it is assumed that the ionomer surface is covered by a layer of liquid water. Such assumption may be plausible, though rigorous validation has never been attempted thus far.


The remaining ionomer in the pores that is not covered by liquid constitutes this interface. The water content in the ionomer phase is often assumed in equilibrium with the water vapor in the pore. Similarly the same equilibrium may exist between liquid water and the ionomer, which is different from that for water vapor. However, this would cause a conflict in terms of actual water content, or the so-called Schroeder’s paradox.


This is the interface where CPt particles are covered by liquid water. Ideally the only transport that occurs on this interface is heat transfer.


Similar to SL.


The liquid vapor interface is the only interface that both phases between the interface move due to ambient conditions such as pressure difference and phase change that may take place due to heat transfer or mass transport. Modeling the movement of such interface has been the center of the classical theory on multi-phase flows.

In addition to these two-phase interfaces, there are the so-called triple-phase boundaries that exist in the CL, i.e. interfaces of MSL, MSG and SLG. Electrochemical reactions may take place on MSL and MSG. It is also noted that among all the phases in a CL, the interfaces associated with liquid water are dynamic in nature due to movement of water in the pore phase. The ionomer interface with the gas pore may change according to the water content in the ionomer phase. It is anticipated that for a densely packed configuration for the CL, the swelling/shrinkage of ionomer may affect gas pore structure significantly. It is also noted that carbon particles are subject to the so-called “carbon corrosion” under high cell voltage conditions and may vanish from the REV.

2.1.4 Interface with Membrane and Porous Electrode

In addition to the interfaces within the CL, there are two more interfaces of interest, i.e., the interface the CL in contact with the membrane and that with the GDL or the MPL. The adhesion of conducting materials (ionomer on the CL-membrane interface, and carbon on the CL-GDL or CL-MPL interface) keeps the continuity of charged species to flow into the CL. The bonding of these interfaces is in general much weaker than those within the CL and disconnection of the CL from its neighboring interfaces, i.e. delamination, which often occurs due to uneven stress distribution or fatigue of materials. Strictly speaking, the volume averaging method is not applicable to this interface of a CL because the interface separates two dissimilar materials and the underlying assumption of representative volume is not valid if one chooses to include this interface within the REV. Nevertheless, pore-scale models which resolve the geometry of any interface can be employed to study the transport problems on such interfaces and the findings can be applied to assist the VAM model.

2.1.5 Simplification of the Governing Equations—the First Step

Some assumptions can be made to the complete set of governing equations for the PEMFC CL to simplify analysis. We begin with the following assumptions:

Assumption #1

The system reaches a steady state. This assumption may be valid for most processes with constant boundary conditions, except when the liquid water generation rate is sufficiently high and the motion of water becomes time-dependent.

Assumption #2

Electrochemical reaction occurs on the Pt surface. This eliminates the volumetric source term in Eq. (1) for gas species:
$$ \nabla \cdot \left( {\rho_{k} \varvec{v}_{k} \varPsi_{k} } \right) = - \nabla \cdot \left( {\varvec{j}_{k} } \right) $$

For the volume-averaged equation Eq. (2), additional assumptions can be made to further simplify the problem.

Assumption #3

Negligible convective velocity (versus diffusion). This eliminates all convective terms including the dispersion term and advection across interfaces.

Assumption #4

No liquid water. This is questionable, but is assumed as the baseline of which comparison will be made when liquid water is involved.

With Assumptions 1–4, we have the volume-average equation for gas species as
$$ \nabla \cdot \left( {\varvec{j}_{k} } \right) + \frac{1}{V}\mathop \int \nolimits_{{A_{i} }}^{a} \varvec{j}_{k} \cdot \varvec{n}_{k} {\text{d}}A = 0 $$
and for all other variables:
$$ \nabla \cdot \left( {\varvec{j}_{k} } \right) + \frac{1}{V}\mathop \int \nolimits_{{A_{i} }}^{a} \varvec{j}_{k} \cdot \varvec{n}_{k} {\text{d}}A - \rho_{k} \varphi_{k} = 0 $$

For the transport of gas species, we can make the following assumptions.

Assumption #5

No gas diffusion into CPt.

Assumption #6

Carbon and Pt have the same transport properties. CPt is then represented by C in the equation.

For gas species that participate in electrochemical reactions, e.g. oxygen and hydrogen, we considertheir transport in two phases, i.e. the pore phase (p) and the ionomer phase (m), respectively, and Eq. (4) for both phases can be written as:
$$ - \nabla \cdot \left( {\varvec{j}_{{{\text{O}}_{2} ,{\text{p}}}} } \right) + \frac{1}{V}\mathop \int \nolimits_{{A_{\text{pm}} }}^{a} \varvec{j}_{{{\text{O}}_{2} ,{\text{p}}}} \cdot \varvec{n}_{\text{p}} {\text{d}}A = 0 $$
$$ - \nabla \cdot \left( {\varvec{j}_{{{\text{O}}_{2} ,{\text{m}}}} } \right) + \frac{1}{V}\mathop \int \nolimits_{{A_{\text{pm}} }}^{a} \varvec{j}_{{{\text{O}}_{2} ,{\text{m}}}} \cdot \varvec{n}_{\text{m}} {\text{d}}A + \frac{1}{V}\mathop \int \nolimits_{{A_{\text{mc}} }}^{a} \varvec{j}_{{{\text{O}}_{2} ,{\text{m}}}} \cdot \varvec{n}_{\text{m}} {\text{d}}A = 0 $$

It is noted that the normal vectors on both sides of an interface are equal in magnitude but opposite sign. The remaining task for species transport is to model all the terms in (5) and (6).

2.2 Summary of the VAM Formulation for CL

The complete set of volume-averaged conservation equations in the form of Eq. (2) for all phases in the REV, along with all necessary initial and boundary conditions and the equations of state for all species, constitutes the modeling framework for the transport phenomena in the CL of the PEMFC. Constitutive equations for the terms in these conservation equations are needed to close the model. The development of a CL model is in practice equivalent to the determination of these constitutive equations based on the characteristics of the CL microstructure. For instance, for each phase, the microstructure information such as connectivity of the phase manifests in the effective volumetric properties such as the effective diffusivity. Between two phases, the microstructure information such as the specific active area manifests in the interface transfer.

From the discussion of the VAM for the CL model, we can see that there remain a few obstacles before this method can become a feasible and rigorous approach to deal with the complexity of the problem. In connecting the VAM with investigations on the microstructure and microscopic modeling, several challenges are identified: (1) Volumetric properties for evaluation of apparent conductivity of different species (non-charged and charged) and heat, (2) Surface properties including active catalyst areas and all two-phase and triple-phase interfaces, (3) Equilibrium conditions in the vicinity of the interfaces, including thermal equilibrium, phase equilibria, and dynamic equilibrium between the fluid phases, (4) Mechanisms and kinetics of the chemical reactions—the reactants and products of these reaction may go through several steps to reach or leave the reaction sites, i.e. adsorption/desorption and surface diffusion etc., therefore knowledge of the mechanisms and transport pathways is crucial to modeling of transport in the CL. The discussion of the following chapters is arranged surrounding the modeling framework.

3 Macroscopic Models for Transport Phenomena

As was stated previously, transport of many physical and chemical quantities take place in different phases of a catalyst layer. Methodologies for analyzing and modeling multi-component, multi-phase flow in porous media have been developed for some time, e.g. [29, 31], among many others. However, these methodologies developed thus far is only applicable to some aspects of the transport phenomena in the PEMFC catalyst layer for several reasons: (1) Multiple interfaces and interfacial transport exist in the porous media; (2) The length scales involved in some transport are in the nano-meter range, under which thermodynamic and transport properties deviate significantly from those at larger length scales [35]; (3) Heterogeneous reactions take place on the surfaces that are strongly affected the transport of other physical and chemical quantities. Questions such as “What is the state of water in the different phases of the catalyst layer?” or “Do phase equilibria establish in the catalyst layer?” cannot be answered unless we can confidently describe the microstructure, determine the thermodynamics in all phases, and characterize the chemical reactions taking place therein. Understanding the microstructure of the solid phase will aid dealing with the complications due to these factors. Understanding the thermodynamics and transport in the nano-scale, confined space will also shed some lights on the unknown phenomena on the interface of the catalyst.

Despite the lack of accurate knowledge of the coupled transport taking place in the PEMFC CL, numerous CL models have been reported since the early 1990s. In this and the next chapter we review the CL models in two major groups respectively, i.e. the macroscopic models which describe the transport based on the volume averaging method, and the models which attempt to resolve the microstructure of the CL. The macroscopic models view the catalyst layer from afar, e.g. the SEM image in Fig. 4a at low magnification, which does not differentiate the material phases. The models which resolve the microstructure, in general termed as “pore scale models” view the catalyst layer in details, e.g. Fig. 4b.
Fig. 4

SEM images at a 3 k magnification and b 250 k magnification

3.1 Macroscopic Models

The macroscopic catalyst layer models reported to date can be classified into three types, according to [15], i.e., (1) interface model (2) macro-homogeneous model and (3) agglomerate model. Description on these models is given in the following sections.

3.1.1 Interface Models

The interface models treat the CL as an interface between the gas diffusion layer (GDL) and the membrane. The CL is assumed to be infinitesimally thin, thus its microstructure is ignored. With this approach, the CL is the location where depletion of oxygen and hydrogen and production of water occur [36, 37, 38], and detailed information of the potential is not required. Some more sophisticated approaches of the interface model account for electrochemical kinetics at the interface, which considers Faraday’s law and kinetic expressions are used as boundary condition at this interface [36, 39, 40, 41]. Such models are adequate to provide approximated solutions for rapid computation of large scale simulation. This type of models cannot account for all relevant interactions in order to permit optimization of fabrication parameters such as catalyst loading and pore size distribution.

3.1.2 Macro-Homogeneous Models

The macro-homogeneous model assumes that void space, the solid conductive material and the electrolyte are uniformly distributed in the catalyst layer, thus the CL is treated as a homogeneous layer. Such models stemmed from early models for electrochemical systems that involved the liquid electrolyte that floods most of the porous electrode. In [42] and similar models, the primary variables solved are: concentration of oxygen (\( C_{{{\text{O}}_{2} }}^{\text{g}} \)), vapor water (\( C_{\text{v}}^{\text{g}} \)), ionic potential in the CL (Φ+) and solid potential (Φs). The governing equations for this kind of models are basically diffusion-type of equation for each of the primary variables, shown as follows.

The conservation equation of oxygen:
$$ - \nabla \varvec{J}_{{{\text{O}}_{2} }} - \varvec{R}_{{{\text{O}}_{2} }} = 0,\;{\text{with}} $$
$$ \varvec{J}_{{{\text{O}}_{2} }} = - D_{{{\text{O}}_{2} }}^{\text{eff}} \nabla \varvec{C}_{{{\text{O}}_{2} }} , $$
where \( \varvec{J}_{{{\text{O}}_{2 } }} \) is oxygen flux, \( \varvec{R}_{{{\text{O}}_{2 } }} \) is consumption rate of oxygen, \( D_{{{\text{O}}_{2} }}^{\text{eff}} \) is effective diffusion coefficient of oxygen, and \( \varvec{C}_{{{\text{O}}_{2} }} \) is oxygen concentration.
Conservation of water vapor:
$$ - \nabla \varvec{J}_{\text{v}} - \varvec{R}_{\text{w}} = 0,{\text{where}} $$
$$ \varvec{J}_{\text{v}} = - D_{\text{v}}^{\text{eff}} \nabla \varvec{C}_{\text{w}} , $$
where \( \varvec{J}_{\text{v}} \) is water flux, \( \varvec{R}_{\text{w}} \) is the interfacial transfer rate of water between liquid and vapor water, and \( \varvec{C}_{\text{w}} \) is water vapor concentration.
Conservation of proton:
$$ - \nabla \varvec{i}_{ + } - \varvec{R}_{ + } = 0,\;{\text{with}} $$
$$ \varvec{i}_{ + } = - \kappa_{\text{N}}^{\text{eff}} \nabla \phi_{ + } , $$
where \( \varvec{i}_{ + } \) is proton current, \( \varvec{R}_{ + } \) is the proton consumption rate, \( \kappa_{\text{N}}^{\text{eff}} \) is effective proton conductivity, and \( \phi_{ + } \) is potential in the ionomer phase.
Conservation of electron:
$$ - \nabla \varvec{i}_{\text{s}} + \varvec{R}_{\text{s}} = 0,\;{\text{with}} $$
$$ \varvec{i}_{\text{s}} = - \kappa_{\text{s}}^{\text{eff}} \nabla \phi_{\text{s}} , $$
where \( \varvec{i}_{\text{s}} \) is electronic current, \( \varvec{R}_{\text{s}} \) is the electron consumption rate, \( \kappa_{\text{s}}^{\text{eff}} \) is effective electric conductivity, and \( \phi_{\text{s}} \) is potential in the ionomer phase.
With proper equations for the electrochemical reactions and appropriate boundary conditions, solution of the primary variables can be obtained. There have been a few macro-homogeneous models in the literature [11, 43, 44, 45, 46].Some other macro-homogeneous models considered dissolution of reactant gas in the electrolyte or water (if a water layer is assumed to exist) [34, 47, 48]. In addition to the above transport equations, one may also include the transport for liquid water, e.g. Conservation of liquid water:
$$ - \nabla \varvec{J}_{\text{w}} + 2\varvec{R}_{{{\text{O}}_{2} }} + \varvec{R}_{\text{w}} = 0 $$
$$ \varvec{J}_{\text{w}} = - \frac{{\rho_{\text{w}} K_{{{\text{w}},0}} }}{{M_{\text{w}} \mu_{\text{w}} }}\left( { - \frac{{{\text{d}}p_{\text{c}} }}{{{\text{d}}s}}} \right)^{k} s\nabla s, $$
where \( \varvec{J}_{\text{w}} \) is water flux, \( \varvec{R}_{{{\text{o}}_{2} }} \) is oxygen consumption rate, \( \varvec{R}_{\text{w}} \) is the water production rate, \( \rho_{\text{w}} \) is density of liquid water, \( K_{{{\text{w}},0}} \) is relative permeability, \( M_{\text{w}} \) molecular weight of water, \( \mu_{\text{w}} \) is viscosity of liquid water, \( p_{\text{c}} \) is capillary pressure (difference of liquid and vapor pressure), s is liquid saturation.
The effects of liquid water on the transport phenomenon in the catalyst layer were investigated by [49] using the macro-homogeneous model. In their model, the site coverage and volume blockage effects due to liquid water in the CL are taken into account through an electrochemically active reduction model \( ECA^{\text{eff}} = ECA\left( {1 - s} \right)^{\text{c}} \) and a Bruggeman type of correction for the effective oxygen diffusivity and water diffusivity is written respectively as
$$ D_{{{\text{O}}_{2} }}^{\text{eff}} = D_{{{\text{O}}_{2} }}^{\text{g}} \left[ {\varepsilon^{\text{m}} \left( {1 - s} \right)^{\text{b}} } \right] $$
$$ D_{\text{w}}^{\text{eff}} = D_{\text{w}}^{\text{g}} \left[ {\varepsilon^{\text{m}} \left( {1 - s} \right)^{\text{b}} } \right], $$
where \( D_{{{\text{O}}_{2} }}^{\text{eff}} \) is effective diffusion coefficient of oxygen, \( D_{{{\text{O}}_{2} }}^{\text{g}} \) is the oxygen diffusion coefficient, ε is porosity, and s is liquid water saturation, \( D_{\text{w}}^{\text{eff}} \) is the diffusion coefficient of water vapor, \( D_{\text{w}}^{\text{g}} \) is the water vapor diffusion coefficient.

The CL is considered as a homogeneous porous medium in the macro-homogeneous model, as a result, information of the CL microstructure is lacking in the model. This is the major drawback of the macro-homogeneous models. Nevertheless, the assumptions of the homogeneous porous medium for the CL may be appropriate under certain conditions, e.g. when the pores of the CL are all flooded with liquid water.

3.1.3 Agglomerate Models

Recent studies in catalyst layer composition suggest that the conductive carbon support and the platinum particles group in small agglomerates bounded by electrolytes [50, 51, 52, 53, 54]. Iczkowski and Cutlip [55] were among one of the first to report such model for fuel cell catalyst layers. Their model was originally developed to simulate the performance of phosphoric acid fuel cells (PAFCs). For PEMFC catalyst layers, the agglomerates are assumed to be either spheres of electrolyte -usually Nafion filled with carbon and Pt particles and approximately one micron in radius [50, 51, 52, 53, 56, 57] or spheres of carbon particles and platinum around 50 nm in radius that are void and are filled with liquid water during cell operation [8, 10]. When the cathode transfer coefficient is one, the results for either type of agglomerate are similar if the size of the agglomerates is the same [58].

According to both agglomerate models, the reaction inside the agglomerate can be modeled in a similar fashion to the reaction in a porous catalyst [59]. These models assume that oxygen diffuses through the gas pores, dissolves into the electrolyte/water around the agglomerate, diffuses through the electrolyte/water into the agglomerate and thereby reaches the reaction site. The transport process described is similar to the one suggested in macro-homogeneous models; however, the macro-homogeneous models do not take into account the characteristics of the agglomerate or the diffusion of oxygen into the agglomerate. Therefore, the macro-homogeneous models are less likely to be accurate. Models that take into account the agglomerate structure are known as agglomerate models [51, 56, 60, 61]. Several studies have shown that agglomerate models give a better prediction of experimental results [52, 62]. However, these agglomerate models require more empirically determined parameters, and this could be a reason for the better fit to experimental data [62].

Based on the assumption of the agglomerate shape, the agglomerate model can be further classified as slab [63, 64], cylindrical [60] and spherical [52, 53] configurations. Based on the length scale of agglomerate models considered, the agglomerate model can be divided into two types:
  1. 1.

    Simple agglomerate model [15, 51, 56]

The simple agglomerate model considers only effects that occur on the agglomerate length scale. In essence, the model assumes a uniform reaction-rate distribution, which implies a uniform gas concentration and surface overpotential throughout the thickness of the CL. The governing equations for this model are similar to those of the macro-homogeneous model except that the conservation equations for charged species (electrons and protons) are modified with the combination of agglomerate characteristic parameters:
$$ \nabla \varvec{i}_{\text{s}} = 4F\frac{{p_{{{\text{O}}_{2} }} }}{H}\left( {\frac{1}{{E_{r} k_{\text{c}} \left( {1 - \varepsilon_{\text{cat}} } \right)}} + \frac{{\left( {r_{\text{agg}} + \delta } \right)\delta }}{{a_{\text{agg}} r_{\text{agg}} D}}} \right)^{ - 1} $$
where the kinetic constant can be written as:
$$ k_{\text{c}} = \left( {\frac{{\varepsilon_{\text{l}} m_{{{\text{p}}t}} S_{\text{ac}} }}{{4Ft_{\text{cl}} \left( {1 - \varepsilon_{\text{cat}} } \right)}}} \right)\left[ {\frac{{i_{0}^{\text{ref}} }}{{C_{{{\text{O}}_{2} }}^{\text{ref}} }}} \right]\left[ {\exp \left( { - \frac{{\alpha_{\text{c}} F}}{RT}\eta_{\text{local}} } \right) - { \exp }\left( {\frac{{\left( {1 - \alpha_{\text{c}} } \right)F}}{RT}\eta_{\text{local}} } \right)} \right] . $$
The reaction effectiveness factor for spherical agglomerate is given as:
$$ E_{r} = \frac{1}{{\varPhi_{\text{L}} }}\left( {\frac{1}{{\tanh \left( {3\varPhi_{\text{L}} } \right)}} - \frac{1}{{3\varPhi_{\text{L}} }}} \right) $$
where \( \varPhi_{\text{L}} \) is a dimensionless group, commonly known as Thiele’s modulus for chemical reactions:
$$ \varPhi_{\text{L}} = \xi \sqrt {\frac{{k_{\text{c}} }}{{D_{\text{eff}} }}} $$
It should be noted that the simple agglomerate model does not consider reaction distributions and proton migration across the CL, i.e., reaction distribution and overpotential distribution are not considered. Therefore, some researchers only looked into the transport on a single agglomerate region, e.g., [65], who investigated the dynamic behavior of a single spherical agglomerate and extended their single spherical agglomerate models to the optimization studies.
  1. 2.

    Agglomerate/macro-homogeneous models

The afore-mentioned macro-homogeneous models do not consider the agglomerate structure, whereas the simple agglomerate models do not take into account the reaction distribution and overpotential distribution. Therefore, it is natural to combine these two CL models, hence the name of agglomerate-macro-homogeneous models.
In this model, the transport in the agglomerate follows the same equation of the agglomerate model [50, 66, 67]
$$ \nabla \varvec{i}_{\text{s}} = 4F\frac{{P_{{{\text{O}}_{2} }} }}{H}\left( {\frac{1}{{E_{r} k_{\text{c}} \left( {1 - \varepsilon_{\text{cat}} } \right)}} + \frac{{\left( {r_{\text{agg}} + \delta } \right)\delta }}{{a_{\text{agg}} r_{\text{agg}} D}}} \right)^{ - 1} $$
The local transfer current is a dependent on the density of agglomerates [58, 61, 68, 69, 70]:
$$ \nabla \varvec{i}_{ + } = - \rho_{\text{a}} \varvec{l}_{\text{a}} $$
where the density of agglomerates is defined as:
$$ \rho_{\text{a}} = \frac{{1 - \varepsilon_{\text{cat}} }}{{(4/3)\uppi R_{\text{a}}^{3} }} $$
and the current produced in an agglomerate Ia is the analytical solution of diffusion–reaction equation of an agglomerate, which is given as
$$ I_{\text{a}} = nF\left( {4\uppi R_{\text{a}}^{ 2} } \right)N_{0} = - 4\uppi nFR_{\text{a}} D_{\text{a}}^{\text{eff}} c_{{{\text{O}}_{ 2} }}^{\text{s}} \left[ {\phi R_{\text{a}} \coth \left( {\phi R_{\text{a}} } \right) - 1} \right] $$
$$ \phi R_{\text{a}} = \sqrt {\frac{{A_{\text{a}} i_{0}^{\text{ref}} }}{{nFD_{\text{a}}^{\text{eff}} c_{0}^{\text{ref}} }}} R_{\text{a}} \exp \left( {\frac{{\alpha_{r} F}}{2RT}\eta } \right) $$

The gas composition and the overpotential change across the catalyst layer due to ohmic, mass transfer, and reaction effects.

The agglomerate/macro-homogeneous models include more physical considerations and relevant effects in the CL than the macro-homogeneous type of models. This type of models are used as an optimization tool for the CL as demonstrate by various researchers [61, 68]. However, these models do not reflect the complex microstructure (pore size distribution, agglomerate size distribution and really reaction interface) on the performance of catalyst layer. Similar models that take into account species adsorption and desorption on the catalyst surface can be found in [71, 72].

3.1.4 Models for Hypothetical Agglomerate’s Inner Structure

In the agglomerate models, the CL is considered to be composed of agglomerates formed by carbon/catalyst particles with porous inter-agglomerate pores formed among the agglomerates. The inter-agglomerate space may be occupied by electrolyte and/or mixture of reactant/products. Inside the agglomerates, the space may be filled with ionomer or water. The agglomerate models are thus further classified as ionomer-filled and water-filled models. Ionomer-Filled Agglomerate Model
The ionomer-filled agglomerate model assumes that the conductive carbon support and platinum particles are grouped in small spherical agglomerates bonded and surrounded by electrolyte [50, 51, 52, 53, 54, 56, 60]. A representation of the cathode electrode according to the agglomerate model can be seen in Fig. 5. The reaction inside the agglomerate is then modeled as a reaction in a porous catalyst [59]. Oxygen is assumed to (a) diffuse through the gas pores between agglomerates; (b) dissolve into the electrolyte phase; and, (c) diffuse, in the electrolyte inside the agglomerate, to the reaction site. The governing equations of the ionomer-filled agglomerate model basically follow those shown in (1927).
Fig. 5

Catalyst layer and gas diffusion layer microstructure. Reprinted with permission from Ref. [295]. Copyright © 2007, Elsevier

Although macro-homogeneous and agglomerate models are both in use today, several studies have shown that agglomerate models provide a better fit to experimental results [52, 62]. A comprehensive comparative study of the three catalyst layer models was recently presented by using three-dimensional numerical solutions [73]. This comparison highlighted the importance of a physically representative model for the catalyst layer, showing that, at low current densities, the thin film model results in different current density distributions compared to the macro-homogeneous and agglomerate models. Furthermore, only the agglomerate model was capable of predicting the performance drop at high currents due to the mass transport limitations that are observed in an actual fuel cell. In order to predict the voltage drop at high current density, the key assumption is that a 50–100 nm uniform ionomer film exists around agglomerates (see Fig. 5), which constitutes a major diffusion barrier for oxygen.

Various ionomer-filled agglomerate models all capture the mass transport limiting effects at high current densities [56, 73, 74, 75], with higher correlation to experimental results. The governing equations show that three additional parameters are included in the model: the agglomerate radius, the electrolyte fraction within the agglomerate, and the electrolyte thin film bonding the agglomerate. These parameters give extra flexibility to the model, allowing polarization curves to be obtained that show these characteristic mass transport losses. Specifically, it captures the effect of reduced effectiveness of the reaction at high overpotentials, as explored by [56]. At high overpotentials, the utilization of the platinum is very low as diffusion of oxygen through the electrolyte phase becomes limiting. This effect is compounded by lower oxygen concentrations under the current collector. The thin film thickness is frequently investigated, as it has a pronounced effect on the polarization curve [56, 73, 74]. It is also a parameter that is difficult to characterize and cannot be confirmed experimentally. However, selection of all the parameters is a critical step in the modeling procedure as each will affect the results of the simulation. Attempts to confirm the size of the agglomerate radius by visualization or molecular dynamics are not consistent; consequently, a wide range of these structural parameters have been used in numerical catalyst layer models. The effects of the agglomerate size are studied numerically.

Further complexity can be added to the model by considering not only agglomerate size, but also the structure and the layout of the agglomerates within the layer, i.e. if the agglomerates are distributed as a packed bed or one on top of another. In [75] the effects of packing were studied in three-dimensions and it was concluded that complex arrangement of the agglomerates has a significant effect on the oxygen transport and activation losses within the catalyst layer. Similarly, it has been noted that the size of the agglomerate may have an effect on the effective transport through the pores. Average pore sizes may decrease as the agglomerate radius is reduced, limiting oxygen transport [56]. Agglomerate models have been presented that propose a reduction in the effective diffusivity from a reduced agglomerate radius based on a tortuosity argument rather than a pore size distribution [76]. However, control over the agglomerate size and arrangement is not yet possible in catalyst layer fabrication, nor can the proposed structure be confirmed. This approach to catalyst layer modeling will continue to be criticized until the physical structure can be accurately observed and the effects characterized experimentally. Water Filled Agglomerate Model

The water-filled agglomerate model assumes that the conductive carbon support and platinum particles are grouped in small spherical agglomerates surrounded by electrolytes but filled with liquid water in the mesopores that exist within the agglomerates. Claims are often made that only Pt in contact with the ionomer phase is electrocatalytically active because electroneutrality should apply at all scales. However, charge separation in the double layer region of a metal electrolyte interface is a fundamental electrochemical phenomenon. This phenomenon forms the basis of a model of reactivity in water-filled agglomerates of the CL.

In [58], a model of ionomer-free, water-filled agglomerates was considered. Based on simple size considerations recently confirmed by coarse-grained MD simulations [77], it was assumed that the ionomer is not able to penetrate nm-sized primary pores inside of agglomerates. Protons are delivered through a thin film of the ionomer phase to the surface of agglomerates, from which they diffuse into water-filled primary pores. The Poisson-Nernst-Planck equation and the oxygen diffusion equation were solved, including a sink term to account for the oxygen reduction reaction.

3.1.5 Pore Diffusion Limited (PDL) Model

In modeling catalytic, electrochemical reactions in porous media, it is a common practice to assume that diffusion of gas species in the pores is relatively slow compared to the reaction taking place on the solid surface. Such model has been adopted in modeling the transport in the CL for some CFD framework, e.g. CFD-ACE+ [78] and Fluent [79]. In such models, there exists a mass transfer between the bulk flow in the pore and the catalyst surface, see Fig. 6. At the catalyst surface the diffusion flux is balanced by the reaction flux, yielding a balance equation, which may be written as:
$$ \sum\limits_{j = 1}^{{N_{\text{ steps }} }} {M_{i} } \frac{{{\varvec{j}}_{{{\text{T}},j}} }}{F} = \rho D_{i} \nabla Y_{i} $$
where \( {\varvec{j}}_{{{\text{T}},j}} \) is the transfer current. In discrete form, the reaction–diffusion balance equation (Eq. 28) may be expressed as:
$$ \sum\limits_{j = 1}^{{N_{\text{ steps }} }} {M_{i} } \left( {a_{ij}^{\prime \prime } - a_{ij}^{\prime } } \right)\frac{{{\varvec{j}}_{{{\text{T}},j}} }}{F} = \rho D_{i} \frac{{Y_{i} - Y_{{{\text{p}},i}} }}{\delta } $$
where Yp,i denotes the species mass-fraction in the pore fluid, while Yi denotes the mass-fraction at the pore-fluid/catalyst interface. The diffusion length scale is denoted by δ and it is assumed to be equal to the average pore size. The transfer current, jT is obtained from the Butler-Volmer equation, and may be written in its most general form as:
$$ {\varvec{j}}_{{{\text{T}},j}} = \frac{{{\varvec{j}}_{0,j} }}{{\prod\limits_{k = 1}^{N} {\left[ {\varLambda_{{k,{\text{ref}}}} } \right]^{{\alpha_{k,j} }} } }}\left[ {\exp \left( {\frac{{\alpha_{{{\text{a}},j}} F}}{RT}\eta } \right) - \exp \left( { - \frac{{\alpha_{{{\text{c, }}j}} F}}{RT}\eta } \right)} \right]\prod\limits_{k = 1}^{N} {\left[ {\varLambda_{k} } \right]^{{\alpha_{k,j} }} } $$
where \( {\varvec{j}}_{0,j} \) is the reference current for the jth reaction step. αa,j and αc, j are kinetic constants determined from experimentally generated Tafel plots. \( \left[ {\varLambda_{k} } \right] \) represents the average interfacial molar concentration of the k-th species (related to the interfacial mass-fractions by \( \left[ {\varLambda_{k} } \right] = \rho Y_{k} /M_{k} \), and \( \alpha_{k,j} \) are the concentration exponents of the k-th species for the jth step. The subscript “ref” refers to concentration values at the reference state at which the reference current density is prescribed. The electrode overpotential is equal to η = (ΦS − ΦF). Substitution of Eq. (30) into Eq. (29) yields a set of non-linear equations for the mass fractions at the pore-catalyst interface. This set of equations is solved numerically to obtain Yi for all species. Once this is known, the species sources in each cell of the catalyst layer can be computed by using:
$$ \dot{\omega }_{i} = \rho D_{i} \frac{{Y_{i} - Y_{p,i} }}{\delta }\left( {\frac{S}{V}} \right)_{\text{eff}} $$
Fig. 6

Schematic diagram showing the relationship of phases and current flows in the PDL model. Reprinted with permission from Ref. [78]. Copyright © 2003, Electrochemical Society, Inc.

The effective surface-to-volume ratio, (S/V)eff, is a direct representation of catalyst loading. In reality, it is usually a small fraction of the geometrically available surface-to-volume ratio.

The formulation of the local species source term in (31) stems from conventional formulation to model chemical reactions in porous media which is often limited by species diffusion in the pore level, hence the pore size is used as the characteristic length scale. However, in the catalyst layer of a PEMFC, the gas diffusion in the pores may not be rate-limiting but rather, diffusion of reactant gases through the electrolyte may be the slower transport process. In this case the diffusion of gas species through the electrolyte and the thickness of the electrolyte film covering the platinum should be used. In addition to the species diffusion through the electrolyte, the species transfer may be affected by the presence of liquid water along the transport pathway, which will change the characteristic length scale and Henry’s constant of gas in water will be needed. With the formulation of (31) these additional mass transfer resistance can be lumped into the model parameter of (S/V)eff.

3.2 Application of VAM in Explaining Classical CL Models

Generally speaking, the aforementioned CL models differ primarily in their treatment for mass transport, in particular, for oxygen into the cathodic CL. In this section, we attempt to compare these models using the VAM formulation. Figure 7a–c show the relationship of the involved phases for different CL models. For the transport equation for oxygen in the ionomer phase:
$$ \frac{1}{V}\int_{{A_{i} }} {\left( {\varvec{j}_{{{\text{O}}_{ 2} }} \cdot \varvec{n}_{\text{ms}} } \right)} {\text{d}}A = - \dot{m}_{{{\text{O}}_{ 2} }} $$
where the RHS of (32) represents surface reaction on the CPt surface. For the macro-homogeneous model, the transfer of oxygen is assumed to take place through the electrolyte phase. The concentration of oxygen on the GDL/CL boundary, cf. the right boundary in Fig. 7a, is assumed to be the equilibrium concentration of oxygen in the electrolyte. In the ionomer phase, oxygen diffuse through the CL and reaction takes place on the MS interface. For the agglomerate model, cf. Fig. 7b, on the interface of the electrolyte and the pore (mp interface), phase equilibrium of oxygen is assumed. In the ionomer phase, dissolved oxygen on this interface then diffuses further towards the catalyst sites on the surface of the agglomerate. At a steady state, the flux on the electrolyte surface equals that depleted on the catalyst sites. Inside the ionomer phase, conservation of oxygen can be written as
$$ \nabla \cdot (\left\langle {\varvec{j}_{{{\text{O}}_{2} }} } \right\rangle_{\text{m}} ) + \frac{1}{V}\int_{{A_{i} }} {\left( {\varvec{j}_{{{\text{O}}_{ 2} }} \cdot \varvec{n}_{\text{mp}} + \varvec{j}_{{{\text{O}}_{ 2} }} \cdot \varvec{n}_{\text{MS}} } \right)} {\text{d}}A = 0 $$
Fig. 7

Illustration of the relationship among phases assumed in different CL models: a the macro-homogeneous model b the agglomerate model c PDL model

Similarly, in the gas phase, conservation of oxygen can be written as
$$ \nabla \cdot \langle (\varvec{j}_{{{\text{O}}_{ 2} }} \rangle_{\text{p}} ) + \frac{1}{V}\int_{{A_{i} }} {\left( {\varvec{j}_{{{\text{O}}_{ 2} }} \cdot \varvec{n}_{\text{mp}} + \varvec{j}_{{{\text{O}}_{ 2} }} \cdot \varvec{n}_{\text{ps}} } \right)} {\text{d}}A = 0 $$
For the model with transport limited by diffusion in the gas pore, cf. Fig. 7c, conservation of oxygen can be written as
$$ \nabla \cdot \left( {\left\langle {\varvec{j}_{{{\text{O}}_{ 2} }} } \right\rangle_{\text{p}} } \right) + \frac{1}{V}\int_{{A_{i} }} {\left( {\varvec{j}_{{{\text{O}}_{ 2} }} \cdot \varvec{n}_{\text{ps}} } \right)} {\text{d}}A = 0 $$

One can see that these classical models for the CL are in some ways simplifications from the comprehensive set of equations by using the VAM.

3.3 Issues of Model Validation

In the literature most of the validation for CL models is performed by comparing polarization curves of model results versus experimental data obtained under similar operating conditions. Characterization of the porous microstructure reported thus far is rather limited. The assumption on the morphology of the catalyst/ionomer is solely based on 2D images of the catalyst layer specimen (TEM or SEM). Whether the catalyst/ionomer possesses certain structures or exhibits certain uniformity such that the assumptions for the homogeneous model or the agglomerate hold, is by far more a speculation than a fact. The practice of using polarization curves for validation is problematic for several reasons, e.g. (1) polarization curves are obtained for entire test cell, not only the catalyst layer; (2) the parameters in the catalyst layer models outnumber the characteristics in these curves, namely the kinetic, ohmic resistance, and voltage drop-off due to mass transfer limitation; (3) the variations in the polarization curves are a result of coupled transport and contribution due to the CL alone cannot be determined. The real problem above all is the lacking of validation and characterization performed down to the same length scale of the catalyst layer ingredient and understanding of the actual microstructure of the layer.

3.4 Summary

In this chapter we have reviewed the classical, macroscopic models for the CL of PEMFCs. The macrohomogeneous models represent one group of models that treat the CL as a homogeneous medium, which does not contain information of the microstructure in the CL. The agglomerate models, on the other hand, consider the transport and reaction in a separate, smaller length scale of the CL. The agglomerate models appear to yield model results that agree with experimental data in a global sense (polarization curves). These models are examined in this chapter by using the VAM formulation in order to elucidate the underlying physics and assumptions made for these models. It is found that this type of models, however, is built based on the assumption that mass transfer to the reaction sites is the rate-limiting process such that the transport of gas species through the ionomer phase in the CL thickness direction is negligible. From a VAM perspective, both types of models lack the rigor needed for the accurate description of the transport phenomena because of several reasons: (1) the microstructure and morphology of the CL are not properly characterized, (2) these models break down when other transport processes become rate-limiting, (3) the transport properties used in these models are often obtained measurement of the bulk material, while they may deviate from their bulk values when synthesized in the CL because of interactions among materials in the nanometer scale may occur during fabrication,(4) phase equilibrium conditions are subject to verification. In order to address these drawbacks, several approaches, including the pore scale modeling and the meso- and microscopic modeling, are currently being employed or under development. In Chapter 4 we review the pore scale models that solve a limited set of field equations of the transport upon resolved material geometry of the CL. In Chapter 5, mesoscopic models including the Lattice Boltzmann models, kinetic Monte Carlo method, and several levels of atomistic models are reviewed.

4 Numerical Simulation with Resolved Microstructural Geometry

In order to address the ambiguity issues encountered in CL model validation, new modeling approaches that are capable of resolving the microstructural geometry of the CL at the length scale down to near the nanometer level have emerged in the past few years. These models in essence attempt to solve the field equations, e.g., Eq. (1), within the entire or part of the CL. Several challenges arise immediately for implementing the solution procedure for this approach: (1) accurate reconstruction of the microstructures of the different phases in the computational domain; (2) validity of continuum equations at the nanometer scale; (3) high demand in computational resources for simulation over a large domain. The challenge of reconstruction of actual porous media can be helped by experimental observation with advanced microscopes (TEM, SEM, etc.,) and numerically by adopting robust algorithms to reconstruct the microstructure that matches the actual structure statistically. The second challenge almost necessitates certain level of atomistic simulation and some sort of multiscale modeling to reveal the phenomena in the nanometer and sub-nanometer range. The last challenge can be dealt with modern large-scale, parallel computing with computer codes with domain decomposition implementation.

The concept of solving coupled transport equations at the length scale of pores in a porous medium is not new; however, development of such approach did not take off until recent years when high-resolution microscopy and high performance computing technology became mature. Computational methods solving transport equations with resolved microstructural geometry have been termed differently in the literature, e.g., as explicit numerical simulation (ENS) [80], direct numerical simulation (DNS) [81]—which is confusing with the DNS in turbulent flow simulation, and most recently converged to the pore scale model (PSM) [82, 83, 84, 85]. The PSM has been developed to solve complex problems such as fluid flow in porous media [80], drying of porous media [82, 84, 86], conjugated heat and mass transfer with reactions [87, 88, 89, 90], and microstructure formation [91].

The first pore-scale modeling for the PEMFC CL can be dated back to [92], which investigated the concentration distribution of discrete catalyst particles by using a set of diffusion-reaction equations in a 2-D domain. A 3-D reconstruction of the porous CL was first attempted in [62], and later expanded in [93]. Comprehensive development of pore-scale modeling and simulations was reported by [81] and a series of papers following that [85, 94, 95, 96]. More recently, systematic investigation and development of the PSM for the PEMFC CL are reported by [83, 97, 98].

In this chapter we review the recent modeling works that solve the transport problem in the CL with spatially resolved microstructures. Due to the heterogeneous nature of the catalyst layer microstructure and the need to determine average transport properties in the catalyst layer for macroscopic models such as the agglomerate model, several groups have attempted to directly simulate transport and reactions in the catalyst layer microstructure. This approach is a two-step process. First, the catalyst layer microstructure must be computationally reconstructed in a manner that preserves its physical properties (e.g. volume fraction, two-point correlation function, pore size distribution). Second, appropriate numerical methods and governing equations must be chosen to simulate species transport, chemical reactions, and the effects of temperature and the water content in the catalyst layer microstructure. Advanced numerical models should take into account the effects of two-phase flow in the catalyst layer, due to the potential for liquid water to block pores or cover the electrochemical reaction sites. Finally, the mesoscale model must be coupled to an agglomerate model and a macroscale model, to truly take into account the multiscale nature of PEM fuel cells. A summary of the different existing approaches along with their limitations is given in the following subsections.

4.1 Reconstruction of the Catalyst Layer Microstructure

A summary on reconstruction of porous media is given by [99]. The construction of realistic three-dimensional representations of porous media has been the subject of several studies in the recent years [100, 101, 102]. Different methods have been introduced in the last decade, such as direct imaging of the pore space at a resolution of a few microns by using micro-CT scanning [103] and the process-based technique. The resolution of the state-of-the-art technique of micro-CT scanning is limited to a few microns, therefore still not sufficient in resolving the CL. The process-based technique needs detailed understanding and accurate modeling of the complicated fabrication process of the CL. With this technique the reconstructed microstructure of the CL is represented as the combination of pores and throats. This technique has been successfully applied in the construction of Berea sandstone [104, 105]. The process-based method is capable of reflecting the effects of manufacturing processes on the microstructure of the CL as well as the real pore size distribution. The capabilities of this technique make it a promising path to approach a good way to study the microstructure of the CL. Figure 8 shows representative results of process-based simulations by [106].
Fig. 8

a Void space of a sandstone produced by process-based simulation; b Pore and throat representation of sandstone. Reprinted with permission from Ref. [106]. Copyright © 2005, Cambridge Univ. Press

4.1.1 Statistical Catalyst Layer Reconstruction

Historically, the problem of reconstructing random porous media has been considered by those in the geoscience field. In particular, stochastic methods have been developed that use the experimentally observed volume fraction and two-point correlation function (taken from two-dimensional images) to create a statistically equivalent computational microstructure [107, 108, 109, 110]. Initially, a random Gaussian field of data is created. This field is passed through a linear and a nonlinear filter in order to produce discrete values that indicate the phase of the structure. The problem of reconstruction is converted into an optimization problem where coefficients are computed that allow one to specify the phase of each computational cell.

One of the first attempts to reconstruct the catalyst layer used two-dimensional TEM images to obtain the volume fraction and two-point correlation function for a PEM catalyst layer [96]. Stochastic methods taken from porous media reconstruction in the geoscience were applied to reconstruct the catalyst layer. Two phases were specified in the catalyst layer: the pore phase and a mixed electrolyte/carbon/platinum phase. Pore cells and mixed electrolyte/carbon/platinum cells are tagged as “dead” or “transport”, depending on whether or not there is an active pathway for species transport through the computational domain.

An alternative approach for reconstructing random porous media was developed by using the “simulated annealing” method [111, 112]. In the simulated annealing method, the random structure is initialized in such a way that the volume fraction of each phase is satisfied. The energy of the system is specified as the sum of the squares of the difference between the higher-order correlation functions of the computational and experimental data. Next, random points with different phases are interchanged in the structure. The interchange is accepted with a probability p according to the Metropolis method
$$ p(\Delta E) = \left\{ {\begin{array}{*{20}l} {1,} \hfill & {\Delta E \leqslant 0} \hfill \\ {\exp \left( { - \frac{\Delta E}{T}} \right)} \hfill & {\Delta E > 0} \hfill \\ \end{array} } \right. $$
where \( \Delta E \) is the change in energy of the system, and T is a fictitious temperature. The method is derived from the principle that a system at high temperature which cools down will tend to move towards the greatest entropy. Over time the fictitious temperature is reduced, allowing the system to reach a state of minimum energy.
A simulated annealing method that allowed for three different phases (the pore, the ionomer, carbon/platinum) to be specified was developed for two-dimensional catalyst layer reconstruction [113]. The reconstructed catalyst layer was then analyzed for mechanical stresses that are induced by load cycling. This method was recently applied to a three-dimensional reconstruction of the catalyst layer with a slightly different methodology [114]. Based on the widely accepted representation of the catalyst layer as agglomerated carbon black spheres covered with the ionomer, the initial mesh was constructed by placing random spheres in a three-dimensional domain. Instead of interchanging pixels, random movements of carbon spheres were considered. The Metropolis method was used to determine whether or not to accept a sphere movement. The spheres were allowed to overlap by a probability given by
$$ p(\Delta E) = \exp \left( { - \frac{v}{\varepsilon }} \right) $$
where v is the ratio of the overlapping volume to the volume of a sphere. The two-point correlation function was used as the target function. This procedure was first applied for the pore and solid phases. After this step was completed, the ionomer and carbon phases were distinguished based on the radii from the sphere centers. In order to validate the reconstructed catalyst layer, experimental measurements of the pore size distribution were compared with the numerical results. The computational results captured the qualitative behavior of the pore size distribution obtained by using mercury porosimetry. Another method that has been used for catalyst layer reconstruction of solid oxide fuel cells is through the use of non-destructive three-dimensional imaging techniques [115]. X-ray computed tomography was used to resolve the catalyst layer microstructure at a resolution of less than 50 nm. It is possible that non-destructive imaging techniques may be applicable for PEM fuel cells as well. One of the current difficulties in imaging PEM fuel cell catalyst layers is in distinguishing the ionomer phase from the carbon phase.

4.2 Numerical Methods Used to Simulate Transport and Chemical Reactions

4.2.1 Continuum Model Approaches

Some of the first work in catalyst layer simulation on the mesoscale was done by using so-called “direct numerical simulation” [81, 94, 95, 96]. In this particular approach, Ohm’s law is used for electron and proton transport, while Fick’s law is used for oxygen diffusion. In addition, the electrochemical reaction is assumed to be governed by Tafel kinetics. Taking into account the current generated by electrochemical reactions, the governing equations for electron, proton, and oxygen concentration are given respectively as:
$$ \nabla \cdot \left( {\sigma \nabla\varvec{\phi}_{\text{s}} } \right) + a\int {\varvec{j}} \delta \left( {x - x_{\text{ int }} } \right){\text{d}}s = 0 $$
$$ \nabla \cdot \left( {\kappa \nabla\varvec{\phi}_{\text{e}} } \right) + a\int {\varvec{j}} \delta \left( {x - x_{\text{ int }} } \right){\text{d}}s = 0 $$
$$ \nabla \cdot \left( {D_{{{\text{O}}_{ 2}^{{}} }} \nabla \varvec{C}_{{{\text{O}}_{ 2} }} } \right) + a\int {\frac{\varvec{j}}{4F}} \delta \left( {x - x_{\text{ int }} } \right){\text{d}}s = 0 $$
where σ is the electron conductivity, ϕs is the electron potential in the solid phase, a is the interfacial area, j is the current generated from the electrochemical reaction, κ is the proton conductivity, ϕe is the proton potential in the electrolyte phase, D is the oxygen diffusivity, \( C_{{{\text{O}}_{2} }} \) is the oxygen concentration and F is Faraday’s constant.

These equations were first modeled in two dimensions using an idealized microstructure [159]. For the two-dimensional simulation, the oxygen concentration was specified at one boundary along with the electron potential. At the opposite boundary, the proton potential was specified. Symmetry boundary conditions were used for the other boundaries. The computational domain spanned a length of 20 μm. The equations were discretized by using a finite difference method.

The equations were modified in three dimensions so that only proton transport and oxygen transport were considered [94]. A regular microstructure was used again with a domain size of 20 μm × 3 μm × 3 μm. The computational cells resolved the catalyst layer down to 250 nm. This is one order of magnitude larger than the size of the carbon particles. Several parametric studies were then performed to analyze the different types of polarization losses through the catalyst layer when the oxygen diffusivity and proton conductivity were changed. Three-dimensional simulations were also performed for random microstructures [95]. In this case, the Bruggeman correlation factor was found to be in the range of 3.5–4.5 instead of the often used value of 1.5.

This work was further extended when the transport of water was added to the numerical model [96]. Water was assumed to be in the gas phase, and the proton conductivity was computed from the water uptake value and the temperature. The diffusivity of the water vapor was determined from the temperature and pressure. They found the optimum relative humidity for the catalyst layer to be at fifty percent. The commercial CFD software package FLUENT was used to solve the equations for water vapor, oxygen, and proton transport.

4.2.2 Lattice Boltzmann Models

In addition to direct numerical simulations, Lattice Boltzmann methods have been used very recently to model transport in the catalyst layer [114]. Lattice Boltzmann methods solve for the particle distribution function by using a collection of pseudo-particles which resides on a lattice. Interparticle forces and collisions are taken into account. In this particular model, a higher-order Lattice Boltzmann method was used to account for the non-equilibrium effects for flows with finite Knudsen numbers. The simulations were performed on a grid with a resolution of 8 nm. The density and momentum of each species can be computed by taking successive moments of the particle distribution function. In this work, oxygen diffusion and ionic conduction were taken into account. Separate Bruggeman correlations for the oxygen diffusion and ionic conductivity were computed. An exponent of 3.2 was obtained for the oxygen diffusivity, while an exponent of 2.0 was obtained for the ionic conductivity. Lattice Boltzmann methods have also been used to simulate two-phase flow in the catalyst layer and the gas diffusion layer [96, 114]. Initially, work was done by coupling direct numerical simulation with a Lattice Boltzmann model to simulate pore blockage and reaction site coverage in the catalyst layer [114]. Later, a potential-interaction Lattice Boltzmann method was used to simulate two-phase flow [116] of water draining through the gas diffusion layer and the catalyst layer [96]. This method was used due to its simplicity in handling flows in complex geometry with multiple phases. Numerical experiments were performed to obtain the fluid/fluid interaction parameter and the fluid/solid interaction parameter, which relate to surface tension at both interfaces. Inertial forces, viscous forces, and gravitational forces were neglected in the model, while surface tension was assumed to be the predominant force in the system. The simulations indicated that at low capillary pressures, liquid water advanced through the catalyst layer in the capillary fingering regime. At high saturation levels, however, liquid water moved through the catalyst layer as an advancing front.

4.2.3 Outlook for Research

The published results from existing catalyst layer simulations have been limited to a very small number of catalyst layer microstructures, with resolutions that can at best only resolve the morphology of an agglomerate. The typical approach is to reconstruct a catalyst layer that has been used in experiments and then to perform simulations by using the reconstructed geometry. While this approach is useful for modeling a particular catalyst layer, there is an inherent randomness in catalyst layer structures that is not accounted for when only one microstructure is considered. A better approach would be to do a statistical study of the transport properties of catalyst layers by using a large number of different microstructures. In addition, a finer resolution should be used to more accurately represent the agglomerated carbon black particles. There is a need to validate the numerical methods that are used for catalyst layer simulations.

In addition, there is a need to compare solutions of the same catalyst layer microstructure that have been computed using different numerical methods. Furthermore, numerical solutions should be validated with experimental data from numerous different catalyst layer microstructures. It appears that using finite differences or the finite volume method would be the best approach for simulating the catalyst layer microstructure, due to the computational efficiency of the methods and the fact that the methods are capable of handling chemical reactions. Lattice Boltzmann methods are very computationally intensive and require coupling with another numerical method to handle chemical reactions. Any comprehensive simulation should account for electron transport, proton transport, oxygen transport, water transport, temperature effects, and electrochemical reactions. These effects should all be coupled together in a numerical model to accurately simulate catalyst layer behavior.

5 Mesoscopic and Microscopic Simulations

In contrast to engineering disciplines that mostly deal with phenomena as continuum, chemistry and disciplines related to chemistry usually work on the phenomena at the molecular level or below. The disparity in length scales and perspectives on analyzing the phenomena in these scales no doubt present a rather large gap between engineering and chemistry disciplines. However, scientific development in both disciplines has thrived in the past few centuries without the need to reconcile. Arguably, this is partly because that the forces in the molecular level do not affect directly those in the macro-scale, i.e. what take place in the molecular level can be ignored or treated as a global behavior, and vice versa for the effects of macro-scale on the phenomena in the molecular level. Length scale becomes an issue when interactions between the molecular scale and the macro-scale become significant, e.g. micro-scale devices and nano-scale materials. From the macro-scale perspective, many of the theories established for continuum break down when the length scale approaches nano-meters where collisions among the molecules and within confined space need to be taken into account. The impact of macro-scale forces at the molecular level is rather in the time scale than in the length scale, e.g. in addition to the common time scales in the fento-second level for atomic vibration and rotation, other processes of longer time scales such as adsorption and diffusion from the macro-scale need to be involved in the analysis.

Computer simulations at the molecular level have been developed for some time but only until the past decade saw explosive progress in their implementation and application in practical use, thanks to the available high performance computer architecture and efficient computational algorithms. In the literature, computational methods for micro-scale simulations can be divided into four classifications based on the methodology, i.e., by the order of length scale each method is suitable for: quantum mechanics (QM), molecular dynamics (MD), the lattice Boltzmann method (LBM) and kinetic Monte Carlo (KMC). Reviews and books on these methods are abundant, e.g. the book by [117] and the book by [118]. In the following sections, introduction of these methods and their application to modeling of catalyst layer of PEMFCs are given, followed by a summary on these methods and comparison.

5.1 Quantum Mechanics

Quantum chemistry has been an indispensable tool in homogeneous system studies. Computational modeling using QM in electrochemistry is delayed due to the complex nature of the interface problem. However, with recent advances in computer technology and electronic structure calculation algorithms, the QM calculation is fast becoming a powerful tool in the electrochemistry field [119]. Studies using QM for the reactions in the CL (especially in the cathode CL) have shown that QM is a powerful tool to calculate adsorption geometry, energy, dissociation energy barriers, reversible potential, activation energy, and potential dependent properties for elementary electron transfer steps. The information is not only useful in the screening of novel electro-catalysts but also potentially useful for determination of parameters in larger systems.

For modeling on the reactions in the cathode CL, application of QM has been mainly focused on two aspects, i.e., chemisorption and oxygen reduction reaction mechanism. Quantum calculations on the chemisorption can provide information about the molecular (or atomic) oxygen adsorption properties on catalysts (different transition metals or alloys); adsorption properties include adsorption energy, distance to the catalyst surface, distance between oxygen atoms and magnetic moment [119, 120, 121].

Quantum calculation on the oxygen reduction reaction mechanism can provide information on the thermodynamic and kinetics parameters of oxygen reduction reaction, including reversible potential [122], activation energy [123], effect of electronic fields [124], and electro-catalytic activity [125, 126] Recently, the ReaxFF method (Reactive force field) that describes complex reactions [127, 128, 129, 130, 131, 132, 133, 134, 135, 136] (including catalysis) provides nearly as accurately as QM but at cost comparable to force field (FF) based methods such as molecular dynamics [137, cf. Fig. 9]. These ReaxFF studies can be carried out for non-equilibrium systems in which flow of protons occurs. The critical component of their approach is the overlapping simulation methodologies, in which QM data are used to train the first-principles based ReaxFF reactive force field. The ReaxFF has the ability to describe the formation and dissociation of chemical bonds, which can be used for large-scale molecular dynamics simulation to characterize properties of fuel cell systems. The ReaxFF has been re-coded for a parallel environment and completely reactive simulations of about half a billion atoms have been reported, thus allows realistic simulations of complex fuel cell chemistry to be carried out.
Fig. 9

Comparison of predictions using QM and ReaxFF methods. Reprinted with permission from Ref. [137]. Copyright © 2005, the Am. Phys. Soc.

5.2 Molecular Dynamics

Molecular dynamics (MD) is a method of computer simulation that explicitly solves the motion of molecules and interactions among them. In essence, atoms in this method are considered as solid spheres and the motion of the molecules basically follows Newton’s second law:
$$ m_{i} \frac{{{\text{d}}^{2} \varvec{r}_{i} }}{{{\text{d}}t^{2} }} = \sum\limits_{j} {\varvec{F}_{ij} } + \sum {\varvec{F}_{k} } , $$
where mi is the mass of the ith molecule, ri is the molecule’s position vector, Fij and Fk are the forces exerted by other molecules and body force respectively.
Ideally most of the force fields can be expressed as gradients of potential energy function V:
$$ \varvec{F}_{k} = - \nabla_{{r_{i} }} \varvec{V} $$
Some common forces involved in the simulation include non-bonded interactions and bonded interactions. In general the non-bonded interactions include the interaction that has a Coulomb potential in the form of
$$ V = \sum\limits_{ij} {\frac{{q_{i} q_{j} }}{{4\uppi \varepsilon_{0} r_{ij} }}} , $$
where qi and qj are the signed magnitudes of two charges, rij is the distance between two particles, and ε0 is the permittivity of free space. And that has a Leonard–Jones potential in the form of
$$ V = 4\uppi \varepsilon \left[ {\left( {\frac{\sigma }{r}} \right)^{12} - \left( {\frac{\sigma }{r}} \right)^{6} } \right], $$
where ε is the depth of the potential well, σ is the finite distance at which the inter-particle potential is zero, r is the distance between the particles. The bonded interactions include bond stretching, bond angle and dihedral angle interactions, which have energy potential in the forms of
$$ V = \sum\limits_{\text{bonds}} {\frac{{k_{r} }}{2}} \left( {r_{ij} - r_{ij}^{0} } \right)^{2} , $$
$$ V = \sum\limits_{\text{ angles }} {\frac{{k_{\theta } }}{2}} \left( {\theta_{ijk} - \theta_{ijk}^{0} } \right)^{2} ,\;{\text{and}} $$
$$ V = \sum\limits_{\text{dihedrals}} {\left( {1 + \cos \left( {n\phi - \phi_{0} } \right)} \right)^{2} } ,\;{\text{respectively,}} $$
where rij, θijk, ϕ are the new distance, the angle between two bonds, and the angle between a molecule with respect to an angled molecule group (with their reference variable value with superscript 0), kr and kθ are the corresponding coefficients for these interactions.

The solution procedure for the MD in general requires computing the aforementioned force fields Eqs. (4347), and integrating the equations of motion, Eq. (42). There are several algorithms developed for integration, e.g. the Verlet algorithm, the velocity-Verlet algorithm and the predictor-corrector algorithm. The choice of the integration is a balance of accuracy and computation efficiency.

A major limitation of MD is its lacking of the capability for modeling processes that involve chemical reactions. Nevertheless MD has found many applications in dealing with transport of water and protons in the proton conducting membrane [138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150]. It is also noted that in [151], MD simulations were employed to study casting polymer films by evaporation, which can shed some lights in modeling the drying process in CL fabrication. Modeling using MD in this area provides valuable insights to the transport in the nanometer level that cannot be obtained by most other methods. However, for the coupled transport taking place in the CL, modeling using conventional MD techniques is not complete and incorporating other methodology that handles chemical reactions is needed, e.g. that proposed by Goddard et al. [137].

5.3 Lattice Boltzmann Method

The lattice Boltzmann model (LBM), which originated from the lattice gas automata (LGA), has received much attention since the late 1980s as an alternative computational method for fluid flow and multiphysics problems. The LBM has advantages over conventional CFD methodology in its mesoscopic features and computation efficiency. When the distribution functions are chosen correctly, the LBM formulation can fully recover continuum transport equations, such as the Navier–Stokes equation [152],and many others usually encountered in common engineering problems. Furthermore, the LBM considers flows on a lattice as a collection of pseudo-particles that are represented by a velocity distribution function. This approach is thus a mesoscopic computational method in nature; therefore, it is also an ideal approach for meso-scale and scale-bridging simulations. LBM has several advantages over the conventional CFD methods in terms of implementation and efficiency, e.g., in dealing with complex boundaries and its readiness for parallel computing. Methodologies and applications of the LBM have been well documented in review papers and books [118, 152, 153, 154, 155, 156].

5.3.1 LBM Formulation

The LBM in general can be expressed in the format of the lattice Boltzmann equation (LBE):
$$ f_{i} \left( {\varvec{x} + \varvec{c}_{i} \Delta t,t + \Delta t} \right) - f_{i} (\varvec{x},t) = \varOmega_{i} + \delta_{i} F_{i} $$
where fi is the particle velocity distribution function in the ith direction with velocity|ci| = Δxt, Ωi is the collision operator, and Fi is external force. DnQb lattice (n dimension with b discrete velocities) is often adopted for velocity ci. A widely accepted form of the collision term is the so-called lattice BGK approximation [157]:
$$ \varOmega_{i} = - \frac{1}{{\tau_{f} }}\left[ {f_{i} (\varvec{x},t) - f_{i}^{\text{eq}} (\varvec{x},t)} \right] $$
where \( {\tau_{f} }\) is the collision time and feq is the local equilibrium distribution function.
The local equilibrium distribution function can be expressed in the following form [158] in order to recover the Navier–Stokes equation:
$$ f_{i}^{\text{eq}} = \omega_{i} \rho \left[ {1 + \frac{{\varvec{e} \cdot \varvec{u}}}{{c_{\text{s}}^{ 2} }} + \frac{{\varvec{uu}:\left( {\varvec{e}_{i} \varvec{e}_{i} - c_{\text{s}}^{2} \varvec{l}} \right)}}{{2c_{\text{s}}^{ 4} }}} \right], $$
By employing the Chapman–Enskog expansion, i.e.,
$$ \frac{\partial }{\partial t} = \varepsilon \frac{\partial }{{\partial t_{1} }} + \varepsilon^{2} \frac{\partial }{{\partial t_{2} }},\frac{\partial }{\partial X} = \varepsilon \frac{\partial }{{\partial X_{1} }} $$
where e represents a small deviation of t and x, one can then derive the LBE to recover the Navier–Stokes equation (momentum):
$$ \frac{\partial \rho }{\partial t} + \nabla \cdot (\rho \varvec{u}) = 0 $$
$$ \frac{{\partial (\rho \varvec{u})}}{\partial t} + \nabla \left( {\frac{{\rho \varvec{uu}}}{\varepsilon }} \right) = - \nabla \varvec{p} + \nabla \cdot \left[ {\rho v_{\text{e}} (\nabla \varvec{u} + \varvec{u}\nabla )} \right] + F, $$
Fluid density and velocity are the first and second moments of the particle distribution functions:
$$ \rho = \sum\limits_{i} {f_{i} } $$
$$ \rho {\varvec{u}} = \sum\limits_{i} {f_{i} {\varvec{c}}_{i} } $$
It is noted that in the LBM formulation, material parameters do not appear explicitly in the distribution function, e.g., for the momentum equation, kinematic viscosity is related to relaxation time as
$$ \nu = c_{\text{s}}^{ 2} \left( {\tau_{f} - 0.5} \right)\Delta t $$

5.3.2 Development and Application of the LBM

The theoretical foundation of the LBM technique was laid during the late 1980s and mid-1990s [116, 159, 160, 161, 162, 163, 164, 165]. The development during that period and applications of the LBM for fluid flows were well summarized in the review paper by Chen and Doolen [152]. One of the major advantages over other computational methods for fluid dynamics is its easy implementation of boundary conditions; therefore, with the technique became mature, a great deal of research employing this technique to investigate single phase and multiphase flows in porous media with the solid structure was resolved in the computational domain, notably by [154, 158, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176], among many others. Algorithms to address instability issues that arise in flows with large density ratio were proposed by a few groups [167, 175, 177]. Formulation of the LBM method systems beyond the ordinary flow conditions has also been reported, e.g. for non-ideal gases [178, 179, 180], the equation of state [181, 182], reaction-diffusion equations [183, 184]. Recent research works performed with the LBM technique for complex flows are thoroughly documented in the excellent review paper by [155].

5.3.3 Multiphase and Multicomponent Formulation

For multiphase and multicomponent flows, four major models have been reported:

(1) The color model: Based on the original lattice gas model, Ref. [185, 186] used colored particles to distinguish between phases and applied a perturbation step so that Laplace’s law is approximately recovered at an interface. The color model is further developed by later studies [187]. The major drawback of this method is that it is difficult to incorporate microscopic physics into the model. (2) The SC model: the SC model was derived by Shan and Chen [116, 179] and later extended by others [169, 170]. In the SC model, a non-local interaction force between particles at neighboring lattice sites is introduced. The net momentum modified by inter-particle forces is not conserved by the collision at each local lattice node, yet the system’s global momentum conservation is satisfied when boundary effects are excluded. The SC model was used to obtain liquid water distribution within the CL microstructure for different saturation levels as a manifestation of the interplay between the two-phase dynamics and the underlying pore morphology [96].The SC model is widely used due to its simplicity in implementing boundary conditions in complex porous structures and its remarkable versatility in terms of handling fluid phases with different densities, viscosities and wettability, as well as the capability of incorporating different equations of state. (3) The FE model: this model is based on the free-energy (FE) approach, developed by Swift et al. [178, 188], who imposed an additional constraint on the equilibrium functions. The FE model is formulated to account for equilibrium thermodynamics of non-ideal fluids, allowing for the introduction of well-defined temperature and thermodynamics. Its major drawback is the unphysical non-Galilean invariance for the viscous terms in the macroscopic Navier–Stokes equation. (4) The fourth type of LB model has evolved from efforts to derive a thermodynamically consistent multiphase theory based upon the continuous Boltzmann equation [189, 190, 191]. This new class of model overcomes the unphysical features involved in the previous three types of multiphase LB models. It should be noted that this type of models is still in active development, and it has not yet been extended to porous medium systems.

5.3.4 Application of the LBM in Fuel Cell Modeling and Simulation

The major part of transport phenomena of fuel cells occur in their porous components. The LBM technique was naturally employed for numerical investigation for transport processes in fuel cells once the method became well established, e.g. for solid oxide fuel cells [192, 193] and for PEMFCs [114, 194, 195, 196, 197, 198]. Application of the LBM technique has also been extended to modeling water transport in the PEMFC gas channel, e.g., [199, 200]. The majority of these works basically employed the LBM technique to study the multiphase flow in the porous electrode, mainly on the GDL, and the solution of all coupled transport phenomena, namely diffusion, reaction and ion transport in catalyst layers, has not been attempted.

The LBM method has been developed as an alternative way to solve the fluid dynamics problem and the basic idea is to find the equilibrium distribution functions and collision terms which together will recover the continuum governing equation. One drawback of this method is that physical parameters such as transport properties and thermodynamic properties are not explicitly appearing in the formulation but often in an integral form, which makes modeling of the physics less straightforward. A promising methodology of combining the LBM with other computational methods, such as the finite volume methods [201, 202] and discrete element modeling [203], has emerged recently. These works have demonstrated the potential of such coupling to capture the pore-scale information by the LBM as well as large scale fluid motion with macroscopic flow solvers.

5.4 Kinetic Monte Carlo

The kinetic Monte Carlo method (KMC), or the dynamic Monte Carlo method, is a popular and powerful method being used in many areas to model and simulate transport processes at small length scales. The method was first proposed in the 1960s by Young and Elcock [204]. An excellent introduction to this method is written by Voter [205]. The paper by Fichthorn and Weinberg [206] provides theoretical explanations on the KMC methodology. It is concluded that the KMC can yield accurate results if the following criteria are met: (1) a dynamical hierarchy of transition probabilities created that satisfies the detailed-balance criterion; (2) appropriately calculated time increment; and (3) effective independence of various events comprising the system. In general the KMC method requires the knowledge of a set of rate constants connecting states of a system. In principle, an exact solution can be achieved if one can have a complete set of all rate constants, termed as “rate catalog”. A major advantage of the KMC is its ability to extend the time scale in atomistic simulation of materials [207], which makes it an attractive method to model processes that have long time scales (e.g. diffusion) with respective to that typically handled in MD or QM.

The solution procedure of KMC can be best demonstrated with the example used in [206] described as follows. The case in consideration is a simple Langmuir absorption problem, which has desorption and adsorption with rates of rD and rA respectively. The coverage of the species on a surface is described by
$$ \frac{{{\text{d}}\theta }}{{{\text{d}}t}} = r_{\text{A}} (1 - \theta ) + r_{\text{D}} \theta , $$
where θ is coverage, rA is absorption rate, and rD is desorption rate.
The analytical solution of the surface coverage at equilibrium is
$$ \theta_{\text{e}} = \frac{{r_{\text{A}} }}{{r_{\text{A}} + r_{\text{D}} }} $$
Figure 10 shows the flow chart of the solution procedure. The implementation for the Langmuir adsorption problem is straightforward and requires minimum of coding. Figure 11 shows a comparison of computed results versus the analytical solution. The rates used in the calculation are rA = 1 site s−1, rD = 2 site s−1 (WA = 1 and WD = rD/rA, both satisfy criterion #1). The real time for each step, which satisfies criterion #2, is
$$ t_{i} = \left[ {\left( {N - M_{i} } \right)r_{\text{A}} + M_{i} r_{\text{D}} } \right]^{ - 1} $$
Fig. 10

Flow chart of the solution procedure of the sample calculation used in [206]. Reproduced with permission from [206]. Copyright © 1991, AIP Publishing

Fig. 11

Comparison of surface coverage of Eq. (57) computed by using KMC and analytical solution Eq. (58)

With a large number of sites used in the simulation and an appropriate random number generator, criterion #1 can be met as well. The asymptotic value of the computed results approaches the analytical value of the state at equilibrium 0.333.

The KMC methodology has been applied in many areas. In the literature, KMC is a popular method to simulate transport processes on surfaces, e.g. for chemical vapour deposition [208, 209], phase formation such as nanostructuring [210, 211, 212] and crystal growth. The method has been used to simulate geometry based nucleation and phase change, in particular in modeling phenomena of the near critical state [213]. Studies that employed KMC on problems related to fuel cells are reported thus far mostly on solid oxide fuel cells [214, 215, 216, 217, 218]. KMC has also been applied in the investigation of the triple phase boundaries in a CL [219]. An interesting investigation that is related to water generation on Pt surface for catalytic combustion is by Hu et al. [220].

5.5 Multi-Scale, Hierarchical Simulations

The length scale and the time scale of the transport processes taking place in the catalyst layer span over a wide range. Microscopic simulations developed thus far for the atomic or molecular level are not practical, if possible at all with current state-of-the-art computer technology and computational algorithms, for simulating processes that occur at much larger time scale or length scales. The macroscopic methodology, on the other hand, lacks predictive capability to handle accurately the intricacy of coupled transport at micro-scale and nano-scale levels. In order to build an advanced simulation tool, strategies for multi-scale, hierarchical simulations are needed.

The literature on the strategy and development of multi-scale, hierarchical simulations is abundant. General discussion on computational methods for multi-scale simulation can be found in the review by Raabe [153], see Fig. 12.The past few years have seen rapid increase in the number of papers on this subject due to the emergence of nanotechnology, which calls for fundamental understanding for engineering materials in the nano-scale and micro-scale range. Overview on the methodology of different schemes and hybrid methods can be found in [221, 222, 223, 224]. Examples for multi-scale simulations can be found for micro-mechanics and nano-mechanics in [225, 226, 227], which focus on simulation of the materials and mechanical processes associated with them. Application and development of multi-scale simulations are also reported for different areas including catalyst design [228, 229, 230], thin film deposition [231, 232, 233, 234], heat transfer [235], polymer materials [236], and most relevant to the present review, works related to fuel cells [137, 140, 142, 237]. The methodology reported for multi-scale simulation all employ hybrid methods of macroscopic models and microscopic/atomistic models. The methods of choice depend on the largest scale and the smallest scale, e.g. for systems as large as a reactor, CFD-based methods are used [238, 239, 240, 241] and the finite element method for micro-scale materials [242, 243, 244]. KMC is ideal for most thin film deposition processes. The atomistic methods are QM, MD or KMC-based methods. It is noted that the aforementioned agglomerate and pore-level diffusion limited catalyst layer models in some sense are two-scale models with the transport in the agglomerates being the micro-scale.
Fig. 12

Computational methods versus the length scale. Reprinted with permission from Ref. [153]. Copyright © 2004, IOP science

In the multi-scale simulation approaches reported in the literature, it has been shown that the meso-scale model, which bridges the macroscopic and the microscopic models, plays a central role for the success of the simulations because it is responsible for the exchange of computational results between two different systems. For a multi-scale model that involves CFD, an appropriate meso-scale model can be the representative element volume (REV), based on which the complete set of volume averaged governing equations is derived. It should be noted that in the case of the catalyst layer of a PEMFC, such REV is still far larger than the typical domain considered for KMC or MD. For a catalyst layer of approximately 10 μm, the size of an REV suitable for CFD calculation is about 10−18 m3, which translates into 25 million of molecules if the REV is filled with gas (for solid it would be two or three orders of magnitude more). Therefore certain level of coarse-graining must be applied for microscopic simulation for this meso-scale domain. Furthermore, in the meso-scale model for a PEMFC catalyst layer, the meso-scale model should also reflect the microstructures, i.e. the topology of each continuous phase, and the coupled transport among all involved variables in the macroscopic models but inconveniently become less defined in the microscopic models.

Thus far, very little has been published which contains a mesoscale model of the catalyst layer coupled with the macroscale model of the fuel cell. Models of this type are necessary to accurately simulate PEM fuel cells and could plausibly be done by coupling together a mesoscale catalyst layer model, an agglomerate model, and a macroscopic model of the fuel cell. A major issue remained to be resolved is how the information should be transferred from a model at one scale to a different model at a larger or smaller scale. This can be done via computational homogenization [245, 246, 247], where the transfer of information is fully coupled to the different model solutions. Alternatively, the transfer of information can be decoupled from the model solutions. The computational benefits and costs of each approach should be investigated. In addition, one must consider what information needs to be transferred in the multiscale model. For instance, one could conceive of a multiscale fuel cell model where the effective diffusivity (computed from pore scale simulation) is provided as an input to an agglomerate model. If this is the only transfer of information in the multiscale model, it would be considered as a one-way coupling. However, if the agglomerate model was also providing information to the pore scale simulation model about boundary conditions, this would be two-way coupling, since information would be transferred from the pore scale simulation model to the agglomerate model, and vice versa. These issues must be clarified for an effective multiscale fuel cell simulation.

In conclusion, while the advances in meso-scale and micro-scale models have enhanced their practicality, multi-scale simulation for the CL of PEMFCs remains limited in the scope and is not yet practical for engineering applications. Nevertheless, meaningful and reasonable multi-scale simulation can be constructed with theoretical frameworks as the guideline for including micro-scale models in the existing, mature macroscopic models such as CFD methodology. For computational methods reviewed in this paper, appropriate length scale range and corresponding PEMFC components/materials for each method are shown in Fig. 13.
Fig. 13

Summary of length scale versus the computational method

6 Fabrication and Relevant Literature

Recent microscopic observations on the microstructure in the PEMFC CL have helped us gain striking insights to the morphology of the layer. The conventional perception of the microstructure of the layer as packed agglomerates seems to shift to intertwined networks of different conducting materials. These insights are in turn inspiring research works to model and simulate the fabrication processes in order to establish links between these processes and the final microstructure. Reports on industrial scale of processing and screening tools are scarce in the literature, e.g. [248, 249]. From a modeling standpoint, however, this is a rather complex realm where no single discipline excels. In general, typical fabrication for the PEMFC CL involves the following steps: (1) mixing of catalysts with dispersion solvents using sonication and homogenization apparatus to make catalyst ink; (2) application of the catalyst ink by printing or spraying on the membrane or the GDL; (3) crying or baking of the applied catalyst ink and final assemblage. The operating parameters of these processes all contribute to the resulting microstructure of the CL. During these processes, the CL components are subjected to long-range (hydrodynamic [250], electrostatic etc.) and short-range (molecular repulsion, van der Waal force [251] etc.) fluctuations, shear stress during atomization, and capillary force [252] during phase change of solvent and pore formation. Various aggregates of catalysts and ionomers form due to different carbon-catalyst, ionomers, and fabrication procedures [253, 254, 255]. In the past decade, a few modeling works focusing on part of the fabrication procedure and conditions have been performed, e.g., [256, 257, 258], it is felt that these works addressed only very limited aspects of the fabrication-microstructure problem and a holistic approach has not yet been attempted.

This chapter is aimed to review existing modeling work on the fabrication processes.

6.1 Dispersion and Aggregation of Particles in Colloids

When particles such as carbon black or CPt are blended into a mixture of dispersion agents and ionomer solution, the particles may form agglomerates after stirring. The characteristic length scale of these agglomerates is a result of the forces acting on the agglomerate and interparticle forces acting among the particles. Microscopically the forces that have significant impacts on the dispersion and agglomerate formation are the van der Waals force (or the London force), the Coulomb force, surface tension when a liquid-vapor interface is present (such as bubbles), shear stress, and so on. When the particle size decreases down to the nano-meter range, the effects of certain forces become more pronounced. Research of the fluid with particles blended within falls in the category of colloid science. The hydrodynamic and transport behavior of colloid fluids deviates from ideal solution of simple fluid and particles. This is primarily due to the predominant interparticle forces that become significant as the length scale of particles decreases. The book by Cosgrove [259] gives a good introduction on the phenomena involved in common colloids. The book by Witten and Pincus [260] that focuses on structured fluids provides a comprehensive account on the physics and phenomena of agglomeration and dispersion of particles in different fluids.

In the literature, a great deal of research works can be found in pharmaceutical applications, e.g. granulation [261], which involves dispersion of particles in colloids and drying of the mixture to form granules, lubricant with carbon black particulates [262], or soot formed by combustion [263], etc. Modeling work based on classical theories and molecular simulations has been reported.

Large scale molecular dynamics simulations on the PEMFC CL were pioneered by [77] where CGMD simulations were performed to investigate the agglomerate structure. Similar approach was employed to study the interaction between ionomers, Pt and water molecules on a carbon surface [264]. More recently, simulations and reconstruction of the CL by using the CGMD method have been reported [257, 258]. In their CGMD investigation, effects of sonication during CL fabrication were taken into consideration. Only global data were used for validation in these reports because it is premature for true, local validation as no experimental characterization for such small domains are yet available. To this end, simulations performed by using the discrete element method [265] over a physical domain size much larger than those used in the CGMD simulations proved to be quite effective and reliable.

6.1.1 Ultrasound Sonication

The sonication process that employs ultrasound waves to slurry, such as the catalyst ink of solid aggregates with dispersion solvent, has many applications in chemical engineering and manufacturing. At low sonication energy intensities, e.g. the ultrasonic bath of about 1 W cm−2, physical effects such as enhanced mass transport and breakage of large agglomerates can take place. For higher sonication intensities, chemical reactions may occur due to high energy released from collapse of bubbles, or the so-called sonication cavitation. When a bubble in the liquid collapses, shock waves form and propagate in the fluid, which cause interparticle collisions and result into breakage of agglomerates. Chemical reaction induced by sonication cavitation falls in the category of sonochemistry–a review on the applications of sonication cavitation to material chemistry can be found in [266] and the references cited therein.

For the preparation of catalyst ink, sonication serves at least two purposes, i.e. to enhance mixing of the dispersion agent and the CPt particles at early stage of the application, and to maintain small and uniform agglomerate size by inter-agglomerate collisions caused by the fluctuating flow field induced by the ultrasound. The sonication process plays an important role in MEA fabrication; however, models developed thus far to describe the process are scarce.

6.2 Sprays and Spray Drying

The catalyst ink can be applied to either on the GDL or on the membrane by screen printing or spray coating. When the ink is screen-printed onto a surface, the thickness and composition of the catalyst layer cannot be easily varied. Fabricating the catalyst layer by the spray route has the advantages of flexibility in control of catalyst thickness, composition and porosity. One major disadvantage of CL fabrication by using a spray is the utilization of the ink due to loss of catalyst ink during application. Nevertheless, sprays and spray drying provide vast flexibility in engineering the catalyst layer.

6.2.1 Atomization, Mixing and Drying of Colloids

During the process when a fluid is injected from a nozzle and sprayed into atmosphere, the fluid is subjected to forces including inertial force of the fluid, surface tension, viscous shear on the droplet interface and within the droplet. The breakup regimes of a spray can be characterized by the Weber number and the Ohnesorge number, e.g. the review by [267]. The droplet size of typical sprays for hydrocarbon fuels is in the micron and sub-micron range. This length scale is still approximately two orders of magnitude larger than the agglomerate size observed in the common PEMFC CL; therefore, it is expected that spraying may not have significant impact on the agglomeration, unless perhaps only have extremely high flow speed out of the spray nozzle, e.g., [250]. In view of the length scale of the droplets, it is likely that formation of macro-pores in a CL is closely affected by the droplet size distribution and the rate of deposition of the droplets on the membrane or the GDL surface.

The above discussion is based on a non-evaporating spray in which the saturation vapor pressure of the spray fluid is rather low in the atmosphere environment where the fluid is injected into. When the saturation pressure of the inject fluid becomes high, e.g. for dispersion agents of high volatility or the injected fluid is at high temperature, vaporization of the fluid may have important impacts on the final droplet size distribution. When the vapor pressure of the fluid is much higher than its corresponding pressure at the ambient temperature, the inject fluid may become a super-heated jet, in which the jet is abruptly disintegrated during the spraying process due to the bubbles formed within the fluid, e.g. [268]. As the thermodynamic state of the injected fluid approaches or exceeds its critical point, surface tension of the mixture vanishes, and the atomization of the fluid becomes mixing of dense fluids, e.g. [269]. The final form of the CL is, hypothetically, dominated by the mixing of the fluids and the rate of deposition of the fluid onto the target surface, in which the transport resembles that in a chemical vapor deposition method. For a vaporizing spray of a colloid with particles blended in the mixture, the final form of the droplets strongly depends on the ratio of the evaporation rate and the diffusion of the particles within the droplet. This ratio can be manipulated to control the microstructure of the CL. An example of this can be seen in nanoparticles via the spray route. Furthermore, if chemical reactions take place during the spray process, such as pyrolysis, one can have additional means to control the microstructure of the CL. Discussion on this area is out of the scope of this review. Further information in this regard can be referred to [270].

6.3 Sintering or Drying by Convection

The last stage of the CL fabrication is sintering at elevated temperature (yet below the glass point of the electrolyte) or drying by natural or forced convection. During this process, excessive dispersion agents inside the CL diffuses through the gas pore pathway and is carried away by the gas flow on the CL surface by evaporation. Depending on the liquid content of the droplets when they are deposited on the membrane or the GDL surface, drying of the CL may have certain effects on the pore formation and microstructure.

6.4 Engineered Microstructure

The literature reviewed thus far focuses on fabrication of catalyst layer that is largely random for the microstructure. A new trend of engineering the catalyst layer in an organized way has emerged since those documented in [8, 9, 271], e.g., using the nanostructured thin film (NSTF) catalyst [12], porous carbon [272], ordered mesoporous carbon (OMC, see Fig. 14) [273], carbon gel [274], glassy carbon (GC) [275, 276], etched nanorods [277], etc. Some modeling works on ordered CL structure have also been reported independently, e.g. [278, 279, 280, 281]. The idea of engineered catalyst layers, or engineered catalytic reactors in general, is to control and ultimately optimize the microstructure to achieve maximum performance and possibly durability. For the case of a PEMFC catalyst layer, the goal is to improve water and heat management and to prolong the lifespan of the catalyst layer. In order to achieve similar performance and power density, the length scale of the catalyst support is likely falling into nano-meter range, where the aforementioned methodology and strategies are applicable. In fact, with known geometry, the engineered catalyst layer can be modeled easier, in a way similar to the research done in the semiconductor industry.
Fig. 14

SEM images of a OMS and b OMC, and TEM images of c OMS and d OMC. Reprinted with permission from Ref. [273]. Copyright © 2008, Elsevier

7 Modeling Water Transport in the Catalyst Layer

Water transport in the PEMFC’s catalyst layers is of paramount importance to all other transport processes [7, 282, 283], yet it remains one of the weakest link among all transport models developed for the catalyst layer to date. Some analytical models or empirical models on PEM fuel cells are useful in predicting the polarization curves of a fuel cell at conditions when water flooding is present, e.g. those by Kulikovsky [284, 285, 286], Boyer et al. [287], and Xia et al. [288]. However, these models did not consider the thermodynamics behavior of water in a heterogeneous environment. Water in the ionomer phase of the CL affects proton transport, which has an impact on the sites of electrochemical reactions. Liquid water may exist in the CL structure and hinders the transport of reactant gases. The complexity of water transport in the catalyst layer is at least three-fold: (1) transport of heat, species from electrochemical reactions and liquid water are intimately coupled; (2) water exists in three different phases (vapor, liquid, absorbed in ionomer) and phase change occurs among these phases; (3) the mechanisms of water transport in confined, nano-scale space are not well understood. Many well-established multiple phase flow models, e.g., [289, 290, 291, 292], which were developed for porous media under ideal conditions (isothermal, uniform surface properties, no source of water from the porous media, etc.,) are not strictly applicable for the environment of PEMFC catalyst layers.

When water is formed near a Pt surface covered by the ionomer, it exists as absorbed water in the ionomer phase. Water molecules are transferred through the ionomer to the pore phase by diffusion and electro-osmotic drag eventually. The thermodynamic state of water in the pore phase depends on several factors, e.g. local relative humidity and rates of heat and mass transfer. If local humidity is low, water vapor may diffuse through the pores and no liquid water will be present in the CL. Similarly, if the heat generated from the exothermic reactions is sufficient to match or exceeds the latent heat required for water vaporization, water vapor may form and expand from the pore, causing a convective flow. In this scenario, the gaseous flow is driven by pressure, instead of diffusion, which is the underlying assumption of most of the aforementioned models. If local humidity is high and reaction heat is low, liquid water may form on the ionomer surface and move across the CL pores, driven by the capillary pressure that originates from the sites of water production. Whether liquid water forms in the CL or not, one can examine the heat production rate versus the latent heat required for vaporization as a function of current density, see Fig. 15. At low current density, say i < 16,000 A m−2, the amount of heat release due to reaction is less than the heat required for water vaporization, therefore, condensation of water is expected unless local humidity is low enough to transfer water vapor out of the pores by diffusion. When at high current density conditions, the reaction heat should be sufficient to vaporize liquid water. It is noted, however, this simple analysis assumes that all the reaction heat is used for phase change, where in reality some heat will be conducted through the solid material of the MEA to the coolant.
Fig. 15

Estimated heat generation and latent heat required for vaporization

8 Outlook

In this paper we have presented a review on the literature related to modeling and computational methods for studying the transport in PEMFC catalyst layers. The review started by revisiting the mathematical framework based on the classical volume averaging method, followed by reviewing several popular macroscopic models, including the well-known macrohomogeneous models, different versions of agglomerate models, and the pore diffusion limited used in commercial software. The complete set of volume averaged equations is then used as the yardstick to examine these macroscopic models. It is concluded these macroscopic models are the simplified subset of equations from the VAM equation after certain assumptions are made. Although these macroscopic models have had some successes in matching global experimental data, the underlying assumptions remain unverified. Advances in microscopy and availability of experimental observations with increasingly higher resolution provide the opportunity to validate these models and revisit presumed morphological catalyst layer structures such as agglomerates. Future macroscopic modeling work should start from the classical VAM formulation and attempt to close the model with new evidence from advanced microscopy.

For the other class of modeling, i.e., the mesoscopic and microscopic models, we have looked at powerful and well-established computational methods, including QM, MD, LBM, and KMC (in the order of length scale each method is suitable for), and the emerging technique of the pore scale model (PSM). It is clear that each of these methods has its advantages and disadvantages, and the upper limit of length scale for every model is set by computer resources available. These microscopic models are, more or less, based on first principles, thus the need for model-closure is less demanding than the macroscopic models. However, the length scales of these models are still far smaller than the REV used in the macroscopic models, which necessitates some novel multi-scale simulation methodology to bridge the gap. Multi-scale modeling and simulations are a natural trend for investigation of the transport in the CL. Through our review, it is found that the porescale model that solves the couple transport within an REV (of a typical macroscopic model) is a promising method to bridge the macroscopic models and microscopic models. The PSM model is on one hand relying on a holistic, theoretical framework to uplink to the macroscopic platform, and on the other it provides places to incorporate microscopic simulations. Similar to the case of macroscopic models, the issue of validation of these mesoscopic and microscopic models is of paramount importance. In recent years, advanced microscopy techniques such as high-resolution SEM, scanning transmission electronic holography microscopy (STEHM) [293], and nano-CT [294] etc., have become available and affordable. New techniques to resolve in situ phenomena at the nano-meter scale are also under development. It is recommended that these new tools should be employed to investigate the transport phenomena taking place at the nano-meter scale, especially in situ, and validate the mesoscopic and the PSM models.

The development of ultra-low catalyst loading catalyst has become a major area of focus in the last decade. Earlier agglomerate based models provided some insight in transport, performance and design trade-offs of such CLs for the anode [295, 297]. For the cathode, a major issue is the experimentally observed increase in resistance to oxygen transport when platinum loading is reduced. Recent experiments point to a key role played by water in this phenomenon [298, 299]. Darling [300] recently presented a comparative analysis of various models with a focus on the proper prediction of the increased resistance and found that, in the context of agglomerate models, the best predictions are obtained by accounting for transport limitations at the nanoscale associated with a combination of localized diffusion and slow adsorption. Further research in modeling and understanding transport limitations and the role of water in ultra-low Pt cathodes is of critical importance to achieving cost reduction targets.

From the literature reported on processes related to CL fabrication, i.e. mixing and dispersion of the catalyst colloid, application of the catalyst ink, and drying of the ink on membrane or GDLs, it is found that there are potentially several routes to model the aggregation process and the final microstructure of the CL at its beginning of life. Techniques to model the dispersion processes can be found in the area of colloid science; however, there are few existing models that are readily available to be applied to the case of the CL. New simulation techniques are needed to investigate the particle-ionomer and particle-particle interactions during the mixing and sonication processes. In-situ observation of these phenomena by using advanced microscopy will certainly shed some light on understanding the problem and the results can validate the model and numerical simulations for particle aggregation in catalyst colloids.



The authors would like to acknowledge the financial support in part from Canada Research Chair, National Research Council Canada, Natural Science and Engineering Research Council of Canada, MITACS Centre of Excellences, and the CaRPE-FC research network. This work resulted in part from a Fellowship at the Hanse-Wissenschaftskolleg Institute for Advanced Study, Delmenhorst, Germany. PCS also acknowledges the support of the National Natural Science Foundation of China (NSFC 21776226), National Key Research and Development Program of China (No. 2017YFB0102702),and the Hubei-100 Plan of China. The authors also benefited from many insightful discussions with Drs. Kyle Lange and Marc Secanell.


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Copyright information

© Shanghai University and Periodicals Agency of Shanghai University 2019

Authors and Affiliations

  1. 1.School of Automotive EngineeringWuhan University of TechnologyHubeiChina
  2. 2.Institute for Integrated Energy SystemsUniversity of VictoriaVictoriaCanada
  3. 3.School of Energy and Power EngineeringChongqing UniversityChongqingChina

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