Modeling of micro/meso-scale reactive transport ... - Oxford Journals

Modeling of micro/meso-scale reactive transport phenomena in catalyst layers of proton exchange membrane fuel cells

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Yu Xiao, Jinliang Yuan* and Bengt Sunde´n Department of Energy Sciences, Faculty of Engineering, Lund University, PO Box 118, 22100 Lund, Sweden

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Abstract In this article, we outline and review the methods that are currently used to simulate the micro/mesoscale flow and reactive transport processes in the porous catalyst layers (CLs) of a fuel cell. The approaches beyond the atomic scale (molecular dynamics) and below the conventional continuum scale (Navier– Stokes solvers) use coarse-grained pseudo-particles which can either move on a fixed lattice or continuously in space. The focus is mainly put on the development of the off-lattice pseudo-particle models, such as coarse-grained molecular dynamics (CG-MD), dissipative particle dynamics (DPD) and smoothed particle hydrodynamics (SPH) methods. As an example, a CG-MD method is performed as a microscopic structure reconstruction technique to reflect the self-organized phenomena during the formation steps of a CL. In addition, we also highlight the combined nano-scale elementary kinetic processes and the issues on the coupling of DPD and SPH to finite element (FE) modeling techniques. This article also highlights the critical aspects and addresses the future trends and challenges for these models.

Keywords: CG-MD, microscopic transport phenomena; reaction; catalyst layer; fuel cell *Corresponding author: [email protected]

Received 21 February 2012; revised 10 April 2012; accepted 19 April 2012

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1 INTRODUCTION Proton exchange membrane fuel cell (PEMFC) systems show great promise for providing efficient environmentally acceptable power sources for transportation and related applications. In a typical PEMFC, a proton exchange membrane (Nafion type, for example) is in contact with the anode and cathode catalyst layers (CLs), and further with two gas diffusion layers (GDLs). Especially, the CLs play an important role on the performance of a fuel cell and have received tremendous attention in both scientific and industrial communities [1]. In spite of the significant achievements in research and development, some fundamental processes in the CLs still need to be investigated more, such as the transport of gases in the complex pore networks, the migration of water and protons near the membrane and the catalytic reactions at active sites of platinum (Pt) in both the anode and the cathode (Figure 1 [2]). Such studies require the theory and the simulation method at a hierarchy of scales, because the main structural effects occur at well-separated scales, viz., at catalyst nanoparticles (a few nanometers), at agglomerates of carbon/Pt and the secondary pores in between (100 nm) and at the

macroscopic component level (10 mm). The approaches beyond the atomic scale and below the conventional continuum scale often employ coarse-grained pseudo-particles which can either move on a fixed lattice (lattice-based pseudoparticle models) or continuously in space (off-lattice pseudoparticle models) [3]. The modeling and simulation at the micro/meso-scales aim to bridge the atomistic and the continuum methods, and avoid their shortcomings. Therefore, various simulation methods have been evaluated and extended to study the microscopic structures and the phase separation of these polymer nano-composites, including coarse-grained molecular dynamics (CG-MD) [4, 5], dissipative particle dynamics (DPD) [6, 7] and lattice Boltzmann (LB) [8, 9]. In these methods, a polymer system is usually treated with a field description or the microscopic particles that incorporate molecular details implicitly [10]. Therefore, they are able to simulate the phenomena on the length and time scales currently inaccessible by the molecular dynamics (MD) methods [11, 12]. In this article, we mainly discuss the off-lattice pseudoparticle models, such as CG-MD, DPD and smoothed particle hydrodynamics (SPH) [13, 14], all based on the understanding

International Journal of Low-Carbon Technologies 2012, 7, 280– 287 # The Author 2012. Published by Oxford University Press. All rights reserved. For Permissions, please email: [email protected] doi:10.1093/ijlct/cts046 Advance Access Publication 19 June 2012

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Modeling of micro/meso-scale reactive transport phenomena

Figure 1. Schematic illustration of a cathode CL [2].

of MD models. The CG-MD method is developed as a systematic technique to provide an insight into the self-organized phenomena and the microscopic structures of the CLs, based on the experimental fabrication procedure [15]. The combined nano-scale elementary kinetic processes in the three-phase boundary region in the CLs are then discussed and the seamless coupling schemes between atomistic and continuum scales are also introduced.

2 CL MICROSTRUCTURE FORMATION To improve the structure –dynamics relationships in the CLs, the meso-scale approaches have been employed to understand the self-organization phenomena and transport properties that have to be considered in the fabrication and operation of the CLs for PEMFCs [1]. Because of computational limitations, full automatic models are not able to probe the random morphology of the system. For meso-scale simulations of CL formation, a CG-MD technique [16] was employed as an example. As shown in Figure 2a, the whole CG model of Nafion copolymer includes a part of the atomistic level configuration of a Nafion chain, i.e. one side chain and a four-monomer unit (-(CF2CF2)4-) are coarsegrained as spherical beads with the volume of 0.315 nm3 (r ¼ 0.43 nm). The Nafion oligomer consists of 20 repeated monomers [4]. In Figure 2b, the carbon slab is modeled by eight layers of the zigzag graphene structure, in which 36 carbon atoms are coarse grained as one bead with the radius of 0.43 nm, as shown by the shaded rings. Thus, the typical carbon slab with the size of 5 5 2.5 nm3 is represented by 224 non-polar beads (type of N0 [17, 18]). Similar to carbon slabs, the 10 10 6 fcc-Pt (111) cluster is represented by 30 non-polar beads (type of Na) in Figure 2c. In this case, each bead consists of 20 Pt atoms with the radius of 0.43 nm, as shown by the shaded rings. Following the fabrication processes during ink preparation of the catalyst-coated membranes (CCMs), the CG-MD model can consolidate the main features of microstructure formation

Figure 2. (a) Schematic drawing of the complete Nafion copolymer, (b) Coarse-grain representation of the carbon slab (c) Atomistic (in the shaded rings) and coarse-grain representation of one fcc-Pt(111) cluster.

in each step (Figure 3). The self-aggregation of Pt/C nanoparticles is easily observed in step I. After removing all the solvents to mimic sintering process in step III, the polytetrafluoro-ethylene chains (added in step II) embed themselves in the carbon aggregates to generate more hydrophobic property. In step IV, Nafion ionomers are added and three different solvents, viz., ethylene glycol, isopropanol and hexanol, are examined, respectively, to exploit the effect of the solvents on the evolution of the CL microstructure. After the heating and drying processes in step V, the CL microstructure segregates into the hydrophobic and hydrophilic regions. One of the important findings is that the hydrophilic beads form a threedimensional network of irregular water-filled channels, which evolve the primary pore structure in the agglomerate. The evolving structural characteristics of the CLs are particularly important to further analyze the transport of protons, electrons, reactant molecules (O2) and water, as well as the distribution of electrocatalytic activity at Pt/water interfaces. In principle, such micro-scale simulation studies allow relating of these properties to a selection of solvents, carbon ( particle sizes and wettability), catalyst loading and level of membrane hydration in the CL [19].

3 COUPLING ELECTROCHEMICAL REACTIONS IN CLS A Tafel – Heyrovsky – Volmer three-step reaction model [20] for the hydrogen oxidation is considered at the anode, and a Damjanovic three-step reaction model [21] for the oxygen reduction mechanism is assumed at the cathode. Assuming noninteraction between adsorbed intermediate species and between intermediates and water, the rates of these elementary steps are described in Table 1. The value of the surface covering fraction (ui) can be obtained by a solution of the conservation equations in Ref. [19], specially, the ionomer coverages on Pt and C can be obtained from the CG-MD simulations. International Journal of Low-Carbon Technologies 2012, 7, 280– 287 281

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Figure 3. Step formation of a CL based on the CCMs preparation procedure.

The kinetic parameters (associated with each elementary step) are given by [19] ki ¼

hv kB T 1 exp ½DG0 þ f1 ðhðtÞÞ þ f2 ðui ðtÞÞ 1 exp kB T h RT

ð1Þ where n is the vibrational frequency of the atomic bond on the Pt or C surface. DG 0 represents the activation barrier, as determined from ab initio calculated data. According to the superposition principle [22], the potential drop h(t) is the sum of the ones related to the thickness of the adsorbed water layer (Df1) and the drop related to the dipolar surface density (Df2).

hðtÞ ¼ Df1 þ Df2 ðsÞ 282 International Journal of Low-Carbon Technologies 2012, 7, 280– 287

ð2Þ

In Equation (2), the last term is a function of the electronic density s, which can be expressed as

@s ¼ JðtÞ JFar @t

ð3Þ

where JFar is the Faradaic current density [23]. In the case of a volumetric electrode with a total electronic current I(t), the current density J(t) can be calculated by [22] JðtÞ ¼

IðtÞ

1 Selectrode eZA g

ð4Þ

where g is the specific contact area between metal parts (Pt/C in a PEMFC electrode) and electrolyte fractions (Nafion); Selectrode and eZA are the geometrical surface and thickness of the volumetric electrode, respectively. These physicochemical


Table 1. Competitive reactions considered for HOR and ORR on the Pt surface [19]. Electrochemical mechanisms HOR

ORR O2ðgasÞ þ Hþ þ e þ s , OOHðadsÞ

H2ðgasÞ þ 2s , 2HðadsÞ

½v ¼ kORR1 CO2 CH þ us expðaORR1 ðF=RTÞhF Þ

½v ¼ kHOR1 CH2 u2s kHOR1 u2H

kORR2 uOOH expðð1 aORR1 ÞðF=RTÞhF Þ

H2ðgasÞ þ s , HðadsÞ þ Hþ þ e

OOHðadsÞ þ H2 OðgasÞ þ 2s , 3 OHðadsÞ

½v ¼ kHOR2 CH2 us expðð1 aHOR2 ÞðF=RTÞhF Þ

½v ¼ kORR2 QH2 O u2s kORR2 u3OH

kHOR2 CH þuH expð aHOR2 ðF=RTÞhF Þ HðadsÞ , s þ Hþ þe

OHðadsÞ þHþ þe , H2 OðgasÞ þ s

½v ¼ kHOR3 uH exp ðð1 aHOR3 ÞðF=RTÞhF Þ

½v ¼ kORR3 CH þ uOHS expðaORR3 ðF=RTÞhF Þ

kHOR3 CH þ us expðaHOR3 ðF=RTÞhF Þ

kORR3 us QH2 O expðð1 aORR3 ÞðF=RTÞhF Þ

Figure 4. General depiction of the transition region used in coupling methods [25].

The continuity of atomistic and continuum regions in CLs necessitates a seamless coupling between these two regions. This is carried out using a transition region [25], as depicted in Figure 4. In view of the large discrepancy between the length and time scales in nano- and macro-domains, the development of the transition region has been the main concern of the researchers [26, 27].

region. The transition region consists of an overlap of the two domains where both atoms and FE nodes are present. The atoms and nodes share the same position. Four-noded quadrilateral elements are used in the immediate vicinity of the interface. When one moves away from the interface to the continuum direction, eight-noded quadrilateral elements are used, and the FEs become larger and the nodes become increasingly sparse. The overlap atoms in the transition region serve the purpose of providing the MD atoms along the interface with a complete set of neighbors for them to interact with according to the non-local formulation of the MD region. They are not true atoms and do not contribute their energies to the system but only have the effects on the MD atoms. Their positions are determined through interpolating the displacements from the nodes of the elements in which they reside.

4.1 Coupling the MD and FE equations

4.2 Coupling the DPD and FE equations

In Figure 4 [25], one side of the transition region is neighbored by the MD region and the other by finite element (FE)

There are many numerical methods that can be used for solving the continuum partial differential equations (PDEs),

parameters can be obtained from Refs [22, 24], or deduced from the micro-scale models, such as CG-MD simulations [19]. The boundary conditions, such as the local concentrations of species, can be given as assumptions or provided from the coupling models at a larger scale [22].

4 COUPLING CONTINUUM EQUATIONS

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and the most popular one is the FE method [6]. The FE proceeds by dividing the continuum domain into a number of elements, each connected to the next by the nodes. The discretization process converts the PDE’s into a set of coupled ordinary equations that are solved at the nodes of the FE mesh and interpolated throughout the interior of the elements using shape functions. By a standard Galerkin procedure [28], the PDEs are transformed into an FE incremental-iterative form (for iteration ‘i’) 2

3( ) ) ( 1 tþDt ˆ ði1Þ ðiÞ tþDt extði1Þ M þ K K F vp 4 Dt 5 DV ¼ DPðiÞ 0 KTvp 0 2 3( ) 8 1 t 9 1 < M V= tþDt ði1Þ ˆ ði1Þ Kvp M þtþDt K V 5 tþDt ði1Þ þ Dt 4 Dt : ; P KTvp 0 0 ð5Þ where DV (i) and DP (i) are the vectors of nodal velocity and ˆ ði1Þ is the matrix pressure increments, and tþDt K tþDt ^ ði1Þ

K

¼ tþDt K ði1Þ þ Km

ð6Þ

where tþDt K ði1Þ and Km are the convective and viscous matrices, respectively [6]. In particular, the issues on the coupling of DPD to FE equations are addressed in the following. In Figure 5, the entire fluid domain is divided into a local one and a global one [6]. Fluid flow in the local domain is modeled with both DPD and FE methods, while fluid flow in the global domain is only modeled by the FE method. The boundary between the local and global domains is shown as the line ABCD. It is assumed that the boundary ABCD is fit along the lines of the FE mesh. When coupling the DPD and FE Navier– Stokes equations, the incremental-iterative FE

Figure 5. Two domains within a flow field. The discrete particle and FE models of the same local domain [6].

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equations in the local domain can be obtained as 2 3( ) 1 ðiÞ ˆ ði1Þ Kvp M þ tþDt K 4 Dt 5 DV DPðiÞ KTvp 0 tþDt extði1Þ tþDt intði1Þ F F þ ¼ 0 0 2 3 1 tþDt ði1Þ Mþ K Kvp 5 tþDt Vði1Þ 4 Dt tþDt ði1Þ P KTvp 0 ( 1 ) Mt V þ Dt 0 where the internal nodal force vector is ð ði1Þ tþDt intði1Þ FKi ¼ NK;j tþDt tij dV

ð7Þ

ð8Þ

V

to replace the viscous nodal force, which equals to Kmt þDt V ði1Þ on the right-hand side of Equation (5); in ði1Þ Equation (8), tþDt tij are viscous stresses at the end of time step. Equation (7) is suitable for coupling with the DPD model because the shear stresses can be evaluated by using the interintði1Þ from the DPD solution. It is further action forces tþDt fi assumed that Equation (5) written for one FE is evaluated for all FEs within the local domains (where both FE and DPD models are employed). The boundary conditions are generally used at the common FE – DPD boundary [29, 30]: (1) Velocities of DPD particles at the common boundary are equal to the velocities in the meso-scale, at all meso-scale time steps, t

FE ; VDPD ¼ t V

tþDt

FE ; . . . VDPD ¼ tþDt V

ð9Þ

(2) The number of DPD particles remains constant within the local domain by imposing periodic boundary conditions. Practical implementation of these conditions is specific for each problem and must be appropriately formulated. The driven cavity flow is presented as an example to illustrate the applicability of the DPD þ FE method [6]. As shown in Figure 6a, the flow in the cavity is induced by a horizontal motion of the upper wall with a constant velocity of V. A local DPD domain (a small square region in Figure 6b) is selected at the right upper corner of the cavity and the flow is solved both by the FE method alone and by the FE – DPD multiscale bridging method. In Figure 6c, the red vertical line goes through the middle of the local (FE þ DPD) domain and extends through the global (FE) domain to the bottom of the cavity. Along the line, the velocity vector profiles calculated by the FE and by the FE þ DPD multiscale method are compared with good agreement.


Figure 7. SPH particles attachment to FEs [29].

Figure 6. Driven cavity problem: (a) the flow domain is divided into the global FE domain and the local FE þ DPD domain in (b). (c) The velocity component vy along the vertical middle line is calculated both by the FE method alone and by the FE þ DPD multiscale method [6].

4.3 Coupling the SPH and FE equations The SPH equations are derived using an approach that parallels the derivation of the classical displacement-based FEM. From this perspective, the SPH method can be viewed as a special case of the FEM, where the connectivity of the elements is constructed by a search for the nearest neighbors [29, 30]. The SPH method can be easily embedded within existing FE code architecture, if the particles are viewed as the elements whose connectivity must be determined for each time step. Figure 7 shows how SPH particles can be attached to standard FEs in the interface [29]. The smaller real circle represents the SPH particle; the larger broken circle around the particle i represents the support domain of the SPH particle; the smaller broken circle around the FE node represents the background particle. The background particle has SPH particle property, and the particle variables are consistent with those of the corresponding FE nodes, such as the particle mass, position, velocity and stress. The SPH – FEM attachment algorithm adds FE nodes to SPH neighbor list in the mode of background parties. The approximation of particle density, momentum and energy for the particle i is done with contributions from particles n1, n2, . . ., n5 and FE nodes n6, n7, n8. Figure 8 shows the solution procedures [31]. At the start of each time step, all the relative data to SPH approximations are transferred from FE nodes to the corresponding background

Figure 8. Solution procedure for SPH– FEM attachment algorithm [31].

particles. The background particles only increase the particle number in the support domain, and they are only passively searched by other SPH particles. At the end of each time step, the relative data are transferred from SPH particles to the corresponding FE nodes in the interface, updating the FE data. For the FE part, the transfer of the data from the interface particles to the corresponding FE nodes is just like loading boundary conditions, and this is feasible because the shape function of FE has the property of the Kronecker d tensor [32]. For the SPH part, FE nodes are added to the neighbor list of the particles, and the boundary effects are avoided. As a result, the continuity between SPH particles and FEs is guaranteed. International Journal of Low-Carbon Technologies 2012, 7, 280– 287 285

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4.4 Challenges for the coupling methods Both of the coupling methods described above adopt the transition region between the atomistic and continuum domains. It results in a much easier computation by allowing the interface atoms to occupy the same location as a node, thereby defining the nodal positions by the MD atom positions. However, it is difficult to establish at exactly what point of mesh refinement the continuum region loses its validity and the underlying assumptions should be kept in mind. At the same time, for the MD region, it will be also checked if the molecular details of the system can be ignored [25]. In addition, the respective local and non-local mismatch between the two regions introduces spurious effects across the interfaces of the transition region. It attributes to the introduction of the overlap atoms as a means of maintaining the proper neighboring for the non-local MD atoms. The overlap atom energies are omitted from the total energy functions because they introduce redundant degrees of freedom. The basic idea behind these ‘ghost-forces’ is that some atoms in the MD region have missing force contributions because the overlap atom energies are not included in the calculations [31]. One of the main goals of multiscale modeling is to drastically reduce the number of degrees of freedom of engineering problems while maintaining accuracy in the regions of interest. So far, the developed methods have not achieved the goal of attaining large realistic system sizes. Even with the use of parallel computing techniques, these methods still require a long time to simulate the problems of interest. All single-scale modeling methods provide the results that lead to an understanding of the properties under the specific conditions. This information is then passed on to the designers to learn from and incorporate into future designs. Thus, the multiscale modeling should be developed to demonstrate the usefulness for engineering design. Clearly, the results directing to real engineering problems can be used to assist in the CLs design.

5 CONCLUSIONS This article highlights the significant efforts made by the research community in modeling micro/meso-scale reactive transport phenomena in the porous CLs of a fuel cell. The modeling of the nanocomposites in the CLs is a multiscale problem. In order to develop a seamless coupling scheme between atomistic and continuum scales, the development of the transition region has been the main concern of the research community. In particular, the issues on the coupling of MD, DPD and SPH with FE modeling techniques are addressed, respectively. However, there still exist major challenges and limitations for these methods. In practice, it is difficult to accurately relate the parameters used in a simulation to continuum variables such as the fluid viscosity. In addition, the exact point of mesh refinement and ‘ghost-forces’ are needed to be kept in mind for engineers. In the future, these 286 International Journal of Low-Carbon Technologies 2012, 7, 280– 287

deficiencies may be mitigated by the simulations based on hybrid multiscale particle-based models.

ACKNOWLEDGEMENTS The European Research Council (ERC, 226238-MMFCS) supports the current research.

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