Nanoscale transport characteristics and catalyst utilization of vertically aligned carbon nanotube catalyst layers for fuel cell applications: Comprehensive stochastic modeling of composite morphological structures

Journal of Catalysis 377 (2019) 465–479

Contents lists available at ScienceDirect

Journal of Catalysis journal homepage: www.elsevier.com/locate/jcat

Nanoscale transport characteristics and catalyst utilization of vertically aligned carbon nanotube catalyst layers for fuel cell applications: Comprehensive stochastic modeling of composite morphological structures Seungho Shin, Jiawen Liu, Ali Akbar, Sukkee Um ⇑ Department of Mechanical Engineering, Hanyang University, 222 Wangsimni-ro, Seongdong-gu, Seoul 04763, South Korea

a r t i c l e

i n f o

Article history: Received 19 December 2018 Revised 20 July 2019 Accepted 29 July 2019 Available online 16 August 2019 Keywords: Polymer electrolyte fuel cell Vertically aligned carbon nanotube Nanoscale transport phenomenon Catalyst utilization Lattice Boltzmann method Statistical analysis

a b s t r a c t Nanoscale transport characteristics and catalyst utilization of vertically aligned carbon nanotube (VACNT) catalyst layers (CLs) are evaluated using a fully statistical modeling approach based on the inherent random nature of the catalyst layer structures for fuel cell applications. Composite morphological structures of the catalyst layers are stochastically modeled with a 95% confidence level, and transport phenomena inside the catalyst layers are simulated using the D3Q19 lattice Boltzmann method (LBM). The effective diffusion coefficient of VACNT catalyst layers is predicted to be higher than that of the conventional catalyst layer, despite a relatively small pore diameter and a low-Knudsen diffusion coefficient. Consequently, the VACNT catalyst layers exhibit improved catalyst utilization compared to the conventional catalyst layers. These statistical results obtained from a series of numerical experiments confirm that the PEFC catalyst layers containing the VACNT catalyst supports can provide more efficient reactant transport, resulting in enhanced catalyst utilization for electrochemical reactions. Ó 2019 Elsevier Inc. All rights reserved.

1. Introduction In polymer electrolyte fuel cell (PEFC)1 systems, mass transport based on composite catalyst layers remains one of the major performance-determining characteristics because effective transport phenomena of electrons, ions, and reactant gases, are prerequisites for the occurrence of complete electrochemical reactions [1–3]. The transport phenomena and characteristics of the catalyst layers depend on the morphological composite structures and compositions of the catalyst layers. For instance, on the cathode side of the membrane–electrode assembly (MEA),2 electrons are conducted through the carbon structures; protons (H+) are conducted through the proton-conductive polymer electrolytes (i.e., ionomers), such as Nafion; and the reactant gases (i.e., oxygen for cathode) are transported through the pore structures to facilitate the electrochemical reactions at the surface of the carbon-supported platinum catalysts (Pt/Cs),3 as depicted in Fig. 1. However, in conventional fuel cell

⇑ Corresponding author. 1 2 3

E-mail address: [email protected] (S. Um). PEFC: polymer electrolyte fuel cell. MEA: membrane electrode assembly. Pt/C: carbon-supported platinum catalyst.

https://doi.org/10.1016/j.jcat.2019.07.053 0021-9517/Ó 2019 Elsevier Inc. All rights reserved.

systems, the transport phenomena of the electrons, ions, and reactant gases are greatly associated with the heterogeneous morphological composite structures of the catalyst layers and the interconnections of carbons, ionomers, and pores, respectively [1,3,4]. In particular, the oxygen transport phenomena inside conventional catalyst layers are intrinsically limited by the complex pore structures, such as closed and dead-end pores. Vertically aligned carbon nanotube (VACNT)4-supported Pt catalysts (Pt/CNTs) are a promising alternative to the conventional Pt/Cs for improving the nanoscale catalyst layer structures and fuel cell performance [5–7]. By adopting the VACNTs as carbon supports, the catalyst layer structure can form fully interconnected electron transport paths and continuous pore structures, as depicted in Fig. 1. Recent experimental studies showed that the VACNT structures can provide enhanced electric conductivity, gas diffusion characteristics, and catalyst utilization for PEFC applications [8–12]. For example, Yasuda et al. developed an iron-nitrogen-doped carbon nanotube catalyst by employing VACNT, and showed that the VACNT-supported, nonprecious metal catalysts, exhibited higher oxygen reduction reaction (ORR)5 activity than conventional 4 5

VACNT: vertically aligned carbon nanotube. ORR: oxygen reduction reaction.

466

S. Shin et al. / Journal of Catalysis 377 (2019) 465–479

Fig. 1. Schematic diagrams of the conventional and VACNT catalyst layers for fuel cell applications.

carbon-supported catalysts owing to the mesoporous nature of the VACNTs, which resulted in efficient mass transport [10]. Tian et al. reported that the VACNTs can provide significant improvements for the fabrication of low-Pt loading MEAs when used as a carbon support because there are adequate electron, proton, and reactant transport paths in the VACNT catalyst layers to provide improved catalyst utilization [11]. VACNT-based catalyst layers exhibited comparable performances to conventional catalyst layers, even though extremely low amounts of Pt catalysts were deposited on the surfaces of the VACNTs. Similarly, Murata et al. demonstrated that the VACNTs used as a carbon support enabled the operation of PEFCs at high-current densities with low-Pt loading [12], mainly owing to the enhanced pore continuity and electrical contact. Significant progress has been made in fabricating highperformance PEFC catalyst layers. However, it is still challenging to evaluate substantive degrees of catalyst utilization, which primarily depend on several factors, such as the interconnection of electric/ionic conduction paths and the reactant diffusion. Nevertheless, several indices have been proposed to access and evaluate the catalyst utilization for PEFC catalyst layers [13–17]. The effectiveness factor, which is mathematically expressed as the ratio of catalytic activity to the maximum catalyst activity [13], has been commonly used as an important indicator of the catalyst utilization. For example, Wang and Eikerling introduced a theoretically modified effectiveness factor for PEFCs and estimated the catalyst utilization in terms of the effectiveness factor by considering a simplified spherical Pt/C agglomerate model [14]. Without considering the entire catalyst layer structures, they mathematically predicted the catalyst utilization in two types of single agglomerate models based on the calculated proton concentration and penetration depth. However, these effectiveness-factor-based approaches have limitations because the definition of the effectiveness factor cannot address the complex pore structures and mass transport phenomena inside the catalyst layers [15–17]. Therefore, the effectiveness factor may not be sufficient to evaluate the transport characteristics and the related catalyst utilization for all the PEFC catalyst layers, and a modified utilization concept needs to be defined in conjunction with the nanoscale transport phenomena. In a recent research study, Shin et al. proposed a transportbased effective catalyst utilization factor, which was closely associated with the tortuous reactant mass transport paths inside the PEFC catalyst layers [18]. In this study, the reactant mass transport phenomena inside conventional catalyst layers were simulated with the lattice Boltzmann method (LBM)6 and Pt catalysts that

6

LBM: lattice Boltzmann method.

are suitable for the ORR were classified along with the complex nanoscale transport paths. Numerous studies have reported that the LBM is useful for simulating the mass transport phenomena inside heterogeneous porous media. It is also easily applied to PEFC catalyst layers because the LBM enables the simulation of intrinsic particle interactions, which are ultimately related to mass transport [19–24] For instance, Kim et al. applied the LBM to stochastically modeled PEFC catalyst layers to evaluate the effective properties by assuming spherical Pt/C agglomerates with ionomers of uniform thicknesses, in which catalyst layers were reconstructed by the Gaussian random field method [22,23]. Gao used a LB simulation to calculate the effective transport characteristics inside the catalyst layers [24]. The catalyst layers were simplified into a bundle of cylindrical pore structures based on the estimated pore diameters that were measured from binary tomographic images obtained using focused ion beam/scanning electron microscopy (FIB/SEM) without separating the catalysts and ionomers. Most of these pioneering approaches modeled the internal structures of the catalyst layers based on a limited number of computational specimens or computational tomographic images of a specific catalyst layer [21–24]. However, as previously mentioned, the PEFC catalyst layers are highly random and heterogeneous. Consequently, it is extremely difficult to exactly reproduce the morphological composite structures with the same compositions. In addition, the internal structures of the catalyst layers yield difficulties in (a) reflecting the random nature of the PEFC catalyst layers and (b) predicting reliable transport characteristics and fuel cell performance based on simplified catalyst and pore structures even though the agglomerated carbon structures are modeled by the molecular dynamics simulation method considering the physical forces such as Van der Waals force [25]. Additionally, dispersions of the carbon agglomerates and ionomer within the catalyst ink are inherently non-uniform and the ionomer thickness on the surface of catalyst particles cannot be characterized by a single value [21,26–28]. Direct three-dimensional imaging of the ionomer networks inside the catalyst layers is still a challenge, and thereby, numerous studies point out that a distribution of film thickness is spatially random [26–28] and cannot be accurately predicted. With regard to the random nature of catalyst structures, a stochastic modeling approach can improve the accuracy of the previous modeling efforts [21,23,26–34]. In this respect, Shin et al. proposed a quasi-random nanostructural modeling (QRNM)7 method. In their model, heterogeneous nanostructures of the composite catalyst layers were stochastically

7

QRNM: quasi-random nanostructural modeling.

467

S. Shin et al. / Journal of Catalysis 377 (2019) 465–479

modeled with a 95% confidence level to account for the random nature of the catalyst layers [31–32]. The nanoscale transport phenomena were simulated using the single-relaxation time (SRT) LBM [18,19,21], to predict the mass transport characteristics of conventional PEFC catalyst layer and validated against published experimental data. Their statistical research shows that a threedimensional LBM model based on this QRNM method can successfully predict the micro/nano-scale pore structures and corresponding mass transport characteristics of catalyst layers which show a good agreement with published experimental data [18,35]. Subsequently, the effective transport characteristics and catalyst utilization of the catalyst layers were statistically evaluated and the maximum catalyst utilization of conventional catalyst layers was predicted to be 9% [18]. The objective of this study was to build a fully statistical nanoscale morphological and transport model for advanced fuel cell composite catalyst layers using the LBM. A stochastic QRNM method was employed to address the inherent random nature of the composite catalyst layer structures, and a series of numerical experimental data at a 95% confidence level was randomly generated for both the conventional and VACNT catalyst layers for reliable statistical analyses. Nanomorphological, structure-based catalyst utilization was defined and evaluated for two different catalyst layers. Subsequently, nanoscale transport phenomena inside the catalyst layers were simulated using the LBM to predict the effective transport characteristics. The structure–transport characteristics, such as the tortuosity, Knudsen diffusion coefficient, and the effective diffusion coefficient of the composite catalyst layers, were statistically estimated from a series of numerical data. Finally, the transport-coupled catalyst utilization was compared to elucidate the effects of the morphological catalyst layer structures on the effective transport characteristics of the conventional and VACNT catalyst layers.

2. Methods 2.1. Quasi-random nanostructural modeling methods Nanoscale internal structures of VACNT catalyst layers were statistically modeled by a quasi-random nanostructural modeling method [31,32] following an experimental fabrication sequence [11,12], which included carbon nanotube (CNT) growth on the sprayed catalyst precursors, deposition of Pt catalyst particles, and the impregnation of ionomers onto the CNTs. The QRNM procedures for the VACNT catalyst layer modeling are briefly summarized as follows: (1) Vertically aligned Pt/CNT catalyst nodes with a CNT diameter of 20 nm [12] were randomly generated by the Monte Carlo method, (2) the Pt catalysts were assumed to be uniformly deposited on the surface of CNTs, (3) the dispersed ionomers were nonuniformly generated on the surface of the pre-generated Pt/CNTs or other ionomers, and (4) the remaining nodes were classified as pore nodes. The conventional carbon-supported catalyst layers were also stochastically modeled in the same manner as the VACNT catalyst layers for comparative statistical analyses. Note that some previous researches have been based on an assumption that the conventional catalyst agglomerated models cover the entire catalyst particle with a thin ionomer layer. However, as reported in many previous studies, three-dimensional images of the ionomer layer obtained by a series of 2D electron microscopy images evidently shows that the carbon supports are partially covered by the ionomer [36–38]. Based on these previous researches, in this study, the ionomers are modeled to reflect the inhomogeneous dispersion of ionomers on the surface of Pt/Cs or Pt/VACNTs. The volume-based modeling parameters for the QRNM method were mathematically calculated from the mass-based

experimental fabrication parameters of the VACNT catalyst layers as follows:

/Pt ¼ mPt

1

qPt

¼

/CNT ¼ mCNT

1

cPt 1 ; l qPt

qCNT

¼

cCNT 1 ; l qCNT

/Pt=CNT ¼ /Pt þ /CNT ¼ /io ¼ mio

1

qio

ð1Þ

¼ mCNT k

ð2Þ


 1 cPt cCNT ; þ l qPt qCNT 1

qio

¼

cCNT l

k

1

qio

ð3Þ

;

ð4Þ


    1 cPt 1 k ; /po ¼ 1  /Pt=CNT þ /io ¼ 1  þ cCNT þ l qPt qCNT qio

ð5Þ

where /Pt , /CNT , and /io , are the volume fractions of the Pt catalyst, CNT support, and ionomer, respectively, and /po is the volume fraction of the pore space (i.e., porosity). mk and ck are the mass per unit volume and mass loading per unit area of the kth component of the catalyst layer, respectively. Additionally, qk is the density of the kth component and k is the ionomer-to-carbon weight ratio (i.e., I/C ratio). The selection of actual size of a representative elementary volume (REV)8 is still unclear although numerous deterministic approaches have been proposed. In this study, the relative gradient error criterion of Li et al. [39] was employed to determine the REV size and the relative gradient error criterion was set to be 0.01. The resolution of REV was set to 20 nm in accordance with the diameter of CNTs and carbon particles [12,15–16,26] because the Pt catalyst particles were assumed to be uniformly deposited on the surface of CNTs and carbon particles. Consequently, A REV of 0:8 lm  0:8 lm  0:8 lm was adopted as the computational domain of a catalyst layer specimen with a granularity of 20 nm (i.e., 40  40  40 nodes). The number of computational specimens was set to 25 to ensure a 95% confidence level. A detailed description and validation of the QRNM method and the REV selection can be found elsewhere [31,32,39]. 2.2. Lattice Boltzmann simulation The reactant mass transport phenomena inside the fuel cell catalyst layers were simulated using the single-relaxation time (SRT) D3Q19 Bhatnagar–Gross–Krook lattice Boltzmann method (BGK– LBM) [40–42]. In the LB simulations, the density and velocity of the fluid were estimated in the context of statistical mechanics by tracking the evolution of the particle distribution function, f a ðx; tÞ, which represents the probability of finding a fluid particle around a certain position x at time t with velocity ca . The discretized form of the evolution of the distribution function is as follows [40–42]:

f a ðx þ ca Dt; t þ DtÞ  f a ðx; tÞ ¼ 

Dt  eq f ðx; tÞ  f a ðx; tÞ ; Tr a

ð6Þ

where Dt is the unit time scale and T r is the relaxation time, which is associated with kinematic viscosity, v , as follows:

Tr ¼

3Dt 1 vþ ; 2 Dx2

ð7Þ eq

where Dx is the unit length scale and f a ðx; tÞ is the equilibrium distribution function, which is expressed as

8

REV: representative elementary volume.

468

S. Shin et al. / Journal of Catalysis 377 (2019) 465–479

eq f a ðx; tÞ

! ca  u 1 ðca  uÞ2 1 u2 ; ¼ xa qðx; tÞ 1 þ 2 þ  cs 2 2 c2s c4s

ð8Þ

Table 1 Physical properties and equivalent lattice properties.

where xa is a weight factor, q is the density of fluid around position x at time t, u is the velocity, and cs is the speed of sound, expressed as [40–42]

c cs ¼ pffiffiffi ; 3

ð9Þ

q¼

f a;

Value

Temperature Pressure

353 K 1 atm

Physical properties Catalyst layer domain length Density of oxygen Kinematic viscosity of oxygen

0.8 lm 1.089 kg/m3

Inlet velocity of oxygen

where c is the unit discrete velocity c ¼ Dx=Dt. The detailed information about the D3Q19 lattice and the weight factor xa were described in a previous publication [18]. The fluid density q and velocity u at each position x can be calculated from the mass and momentum conservation as follows: 18 X

Quantity

Lattice properties (in lattice unit) Relaxation time Number of lattices in catalyst layer domain length Density of oxygen Kinematic viscosity of oxygen Inlet velocity of oxygen

2:157  105 m2 =s 0.001 m/s 1 120 1.089 0.1667 5:077  108

ð10Þ

a¼0

u¼

18 1X

q

f a ca :

ð11Þ

a¼0

The LBM modeling properties in the lattice unit can be obtained by introducing nondimensional numbers, such as the Reynolds number. For the LBM simulations in this study, the physical domain size of REV (i.e., 0:8 lm  0:8 lm  0:8 lm) was discretized by a 120  120  120 lattice unit. The inlet velocity of the reactant gas at the gas diffusion media (GDM)/CL interface (i.e., x = 0) was set to 0.001 m/s based on the multidisciplinary macroscale GDM model of Kim et al. [43], and the outlet pressure was set to 0.1 MPa (i.e., 1 atm). The kinematic viscosity of oxygen is 2:157  105 m2 =s at 80 °C. The detailed information about the physical and equivalent lattice properties is shown in Table 1. The slip velocity at the gas–solid interface, uw can be calculated as follows [44–46]:

pffiffiffiffiffiffiffi uw  0:4Kn Rex ; u

ð12Þ

where u is a bulk velocity, Kn is the Knudsen number, and Rex is the Reynolds number Rex ¼ qux=l at the leading edge of x. Eq. (12) implies that the slip at the gas–solid interface can be significant for high Knudsen and Reynolds numbers. However, it should be noted that the inlet velocity at the GDM/CL interface in this study was set to be 0.001 m/s [43], which corresponds to extremely low-Reynolds numbers Rex with x = 20 nm (i.e., carbon particle size),

Rex ¼

ux

m

¼ 9:57  107 :

ð13Þ

Consequently, even in the extreme case where the reactant flow channel is tens of nanometers (i.e., Kn  10), the slip velocity of Eq. (12) becomes

pffiffiffiffiffiffiffi uw  0:4Kn Rex ¼ 3:91  103 ; u

ð14Þ

which indicates that the slip of the reactant particles at the gas– solid interface within the catalyst layers is negligible. Therefore, similar to previous studies [21–23,46], a no-slip boundary condition at the fluid-solid interface was adopted to elucidate the reactant transport phenomena inside the catalyst layers. 3. Results and discussion

in Fig. 2(a) and (b). A representative elementary volume of 0:8 lm  0:8 lm  0:8 lm was extracted from the computational catalyst layer specimens, which had thicknesses of 8 lm [12,15–16]. For the conventional catalyst layers with a reference composition, a carbon support loading of 0.4 mg/cm2 with a Pt catalyst loading of 0.3 mg/cm2 (i.e., Pt/C volume fraction of 0.2675) was considered for the cathode catalyst layers. The ionomer-tocarbon weight ratio (i.e., I/C ratio, k) was set to 1.4 (i.e., ionomer volume fraction of 0.35). The detailed morphological properties are listed in Table 2. As shown in Fig. 2, the conventional and VACNT catalyst layers formed mostly heterogeneous nanostructures, thereby producing complex interconnections/disconnections for each transport component. The interconnected networks of each transport components were morphologically classified by the modified Hoshen-Kopelman algorithm based on the percolation theory [47–49]. For the conventional catalyst layers in Fig. 2(a), 70.5% and 94.7% of Pt/Cs (black color) and ionomers (red color) generated successively interconnected conduction networks for electrons (e) and protons (H+), respectively. Additionally, 90.2% of pore spaces branching from the GDM side created continually interconnected reactant transport paths (O2 for the cathode catalyst layers). In contrast, for the VACNT catalyst layer with the same volumetric compositions as those listed above, all Pt/CNTs are interconnected from the GDM side to the membrane side, as shown in Fig. 2(b), indicating that perfect conduction networks for electrons formed in the VACNT catalyst layers. Additionally, 83.8% and 88.6% of ionomers and pore spaces in the VACNT catalyst layers formed successfully interconnected transport paths for protons and reactants, respectively, which were relatively lower than those of conventional catalyst layers. It should be note that the liquid water is known to contribute significantly to the transport of proton. However, the formation of liquid water by saturation is highly undesirable phenomenon for catalyst utilization it hinders reactant mass transport even though the liquid water can provide additional proton transport paths. Therefore, the formation and existence of liquid water as a consequence of the ORRs are not considering in this study, and the ionomers are assumed to exclusively conduct protons. The complete catalyst layer model including electrochemical reactions and consequent formation of liquid water will be addressed in a future work. In previous research studies, Shin et al. introduced a morphology-based effective catalyst utilization factor (M–ECUF)9 as an indicator of the maximum possible number of catalysts that can be utilized for electrochemical reactions as follows [31,32]:

3.1. Morphological analysis of catalyst layer The morphological nanostructures of conventional and VACNT catalyst layers were modeled using the QRNM method, as shown

9

M-ECUF: morphology-based catalyst utilization factor.

S. Shin et al. / Journal of Catalysis 377 (2019) 465–479

469

Based on this definition, the catalyst sites that are simultaneously in contact with interconnected Pt/Cs (or Pt/CNTs), ionomers, and pores were classified as the morphology-based electrochemically active catalyst sites to which the electrons, ions, and reactants can be accessible (i.e., the closed pores are excluded). For the conventional catalyst layer, only 23.2% of the catalyst sites were classified as morphologically utilized Pt/C catalyst sites (i.e., M–ECUF = 0.232), as shown in Fig. 2(a). In contrast, the morphology-based ECUF of the VACNT catalyst layer was predicted to be 0.54, which is more than twice the value of the conventional catalyst layer. In the present study, the catalyst sites fully covered with ionomers were excluded from the calculation of utilized catalyst sites because the gas concentration in the ionomers across the interface between the pore space and ionomers significantly decreased. The equilibrium concentration of oxygen in the Nafion ionomer that dissolved through the interface can be determined by Henry’s law as follows [50–54]:

cO2 ;i ¼

Fig. 2. Three-dimensional views of stochastically generated nano-morphologies of (a) conventional and (b) VACNT catalyst layer.

Table 2 Morphological properties for the reference case composition. Quantity

Value Conventional catalyst layers

VACNT catalyst layers

Catalyst layer thickness Pt catalyst loading Carbon support loading I/C weight ratio

8 lm 0.3 mg/cm2 0.4 mg/cm2 1.4

8 lm 0.3 mg/cm2 0.28 mg/cm2 2

Pt catalyst density Carbon support density Ionomer density

21.45 g/cm3 2 gcarbon/cm3 (Carbon) 2 g/cm3

21.45 g/cm3 1.4 gCNT/cm3 2 g/cm3

Pt catalyst volume fraction Carbon support volume fraction Ionomer volume fraction Porosity

0.0175 0.25

0.0175 0.25

0.35 0.3825

0.35 0.3825

CM ¼

Morphology - based electrochemically active catalyst sites : Total number of catalyst sites ð15Þ

p cO2 RT ¼ ; H H

ð16Þ

where cO2 ;i is the concentration of oxygen dissolved in the ionomer, H is the Henry’s law constant with units of Pa  m3 =mol, p is the partial pressure of oxygen, R is the universal gas constant, and T is the temperature. Fig. 3(a) shows the correlation between the oxygen dissolution ratio and Henry’s law constant for oxygen in the commercially available Nafion ionomer [50–54]. As shown in Fig. 3(a), only 6–15% of the oxygen in the gas phase can dissolve in the ionomer for Henry’s law constants in the range of 2.0–5.0  104 Pam3/mol even though the oxygen concentration profile was calculated with an assumption of complete electrochemical reaction at the surface of catalyst (i.e., oxygen concentration at the surface of the catalyst = 0). Consequently, the catalyst utilization may not possibly occur when the catalysts are fully covered by ionomers because of the significant mass transport limitations imposed by the ionomer layers. As shown in Fig. 3(b), only the 2 mol/m3 of oxygen was expected to dissolved in ionomer with the Henry’s constant of 2.0  104 Pam3/mol, when considering the molar fraction of oxygen in the air of 2 bar. Therefore, in this study, the catalyst sites completely covered by ionomers were excluded from the evaluation of effective catalysts. Fig. 4(a) shows a statistical comparison of the morphologybased ECUFs between the conventional and VACNT catalyst layers as a function of carbon support volume fraction. The corresponding mass-based carbon support loadings are marked in Fig. 4(a). The ionomer volume fractions of both conventional and VACNT catalyst layers were fixed at 0.35. For reliable statistical analyses, a total of 25 computational catalyst layer specimens were selected to achieve a 95% of confidence level [18]. As shown in Fig. 4(a), the M–ECUFs of both conventional and VACNT catalyst layers decreased in a sinusoidal manner as the carbon support volume fraction increased from 0.25 to 0.31. However, the VACNT catalyst layers showed improved catalyst utilization compared to the conventional catalyst layers, mainly owing to the perfect conduction networks of the CNTs, as shown in Fig. 2(b). For the VACNT catalyst layer with a Pt catalyst loading of 0.3 mg/cm2 and an ionomer volume fraction of 0.25, 10–45% of the Pt catalysts were estimated to be effectively utilized over the entire carbon support volume fraction, which is approximately twice as high as the catalyst utilization of conventional catalyst layers. This result implies that the VACNT catalyst layers can provide improved catalyst utilization for the fuel cell catalyst layers. Fig. 4(b) shows the variation of the interconnection of ionomers and pore spaces and the corresponding morphology-based ECUFs as a function of the ionomerto-carbon ratio for VACNT catalyst layers with a loading of

470

S. Shin et al. / Journal of Catalysis 377 (2019) 465–479

an optimum value at an ionomer-to-carbon ratio of 2.0. The contour plot of M–ECUF as functions of the CNT loading and I/C ratios is shown in Fig. 4(c) and clearly indicates that the catalyst utilization for VACNT catalyst layers was highly susceptible to the composition of the catalyst layers. Accordingly, the VACNT catalyst

Fig. 3. Variation of (a) oxygen dissolution ratio as a function of the Henry’s law constant for oxygen in ionomers at 80 °C and 2 atm and (b) oxygen concentration profiles in the vicinity of the ionomer/pore interface.

0.28mgCNT =cm2 . The interconnected Pt/CNT ratio is not shown in Fig. 4(b) because of the perfect conduction networks for electrons in the VACNT catalyst layers (i.e., 100% of interconnected Pt/CNT ratios for all I/C ratio ranges) [11,12]. In the low range of I/C ratios, the interconnection of ionomers inside the VACNT catalyst layer was significantly improved from 10% to 85% by increasing the I/C ratio from 1.4 to 2.0, while the interconnection of pore spaces decreased slightly (from 98% to 85%) in the same range of ionomer contents. Consequently, the morphology-based ECUFs improved proportionally to the interconnected ionomer ratio in this range of I/C ratios, and 45% of the catalyst sites were predicted to be utilized for the ORR. In contrast, at the higher range of I/C ratios, the M–ECUF showed a strong dependency on the interconnection of pore space. As the ionomer contents increased for I/C ratios of 2.0–2.8, the interconnection of the pore spaces drastically decreased from 0.85 to 0.16 owing to the excessive ionomers in the VACNT catalyst layer, while the interconnection of ionomers was slightly improved, reaching saturation. Therefore, for highionomer contents, the morphology-based ECUFs decreased significantly from 0.45 to 0.07 with a strong dependence on interconnected pore ratios, indicating that the excessive ionomers in the VACNT catalyst layer led to losses of the utilized catalyst sites by blocking the pore spaces and the reactant mass transport. Consequently, the morphology-based ECUFs of the VACNT catalyst layers varied as a function of the ionomer contents, and exhibited

Fig. 4. (a) Statistical comparison of the morphology-based effective catalyst utilization factor between conventional (red, dashed line) and VACNT (black, solid line) catalyst layers as a function of carbon support volume fraction with a fixed ionomer volume fraction of 0.35. (b) Variations of the interconnected ionomer ratio (black, solid line), pore ratio (red, dashed line), and corresponding M–ECUF (blue, dotted line) as a function of the I/C ratio for the VACNT catalyst layers with CNT loading of 0.28 mg/cm2. (c) Contour of M–ECUF as functions of CNT loading and I/C ratio.

S. Shin et al. / Journal of Catalysis 377 (2019) 465–479

layers should be designed using the optimum CNT loading of 0.28 mg/cm2 and an I/C ratio of 2.0 for the VACNT catalyst layers with a catalyst loading of 0.3 mgPt/cm2. The morphological structures of the VACNT catalyst layers are further analyzed in terms of the inter-nanotube distance. In this study, the inter-nanotube distance is defined as the distance from the center of each VACNT to the center of the nearest VACNT, and mathematically calculated from the length of the common perpendicular of two centerlines of VACNTs.

! ! ! P1 P2  ð u1  u2 Þ ! ! d¼ : ð u1  u2 Þ

ð17Þ

! ! where P 1 and P 2 are the center points of each CNT, and u1 and u2 are the direction vectors of the centerline of each CNT. As presented in Fig. 5(a), the average inter-nanotube distance continuously decreases from 24.0 nm to 22.4 nm as the VACNT volume fraction is increased from 0.25 to 0.31. Despite the I/C ratio of the VACNT catalyst layers is fixed to be 2.0, the interconnected ionomer ratio increases from 0.8 to 0.94 in the VACNT volume fraction range. This result clearly demonstrates that the formation of interconnected ion transport path is directly affected by the inter-nanotube distance even though the I/C ratio is exactly the same. Therefore, the interconnected ionomer ratio increases in proportion to the VACNT loading due to the decrease in the inter-nanotube distance as shown in Fig. (b). However, in this range of the average inter-

nanotube distance, the interconnected pore ratio is predicted to drastically decrease from 0.9 to 0.16, which signifies that the decrease in the inter-nanotube distance can directly result in the lower pore interconnection and corresponding poor reactant mass transport. Consequently, as the average inter-nanotube distance decreases from 24 nm to 22.4 nm, the morphology-based ECUF continuously decreases in accordance with the decrement of the average inter-nanotube distance. The maximum M–ECUF is predicted to be 0.46 at the optimal inter-nanotube distance of 23.8 nm with a fixed I/C ratio of 2.0. 3.2. Transport characteristics of catalyst layer As previously mentioned, the catalyst layers form mostly heterogeneous pore structures and the reactant gases are primarily transferred through the pore spaces of the interconnected pore structures, whereas the reactants in closed pores and dead-end pores have limited contribution to the electrochemical reactions owing to the morphological structures of pore. Therefore, the morphological structure of pores and consequential reactant transport phenomena affect the catalyst utilization inside the catalyst layer, and hence must be taken into account. The reactant transport phenomena inside the fuel cell catalyst layers can be physically described by the advection-diffusion process. The total mass flux of transported species in advectiondiffusion process is defined as follow:

nA ¼ jA þ qA v

Fig. 5. (a) Variation of inter-nanotube distance and interconnected ionomer ratio as a function of VACNT volume fraction, and (b) effects of inter-nanotube distance on interconnected ionomer, pore ratio, and corresponding morphology-base effective catalyst utilization.

471

ð18Þ

where nA is total mass flux, jA is diffusion flux, and qA v is mass flux of A due to bulk motion. In the lattice Boltzmann method of this study, the total mass flux in advection-diffusion process is simulated by tracking the evolution of the particle distribution function with streaming and collision processes to account for both bulk flux and diffusion flux. The mass transport phenomena in the fuel cell catalyst layers were simulated using the D3Q19 Bhatnagar–Gross–Krook lattice Boltzmann method (BGK–LBM). The number of lattices (i.e., grid points) in each length of catalyst layer domain are set to be 120 as given in Table 2, and thereby, each of computational catalyst layer specimen consisted of 1,728,000 lattice points. For each computational catalyst layer specimen, approximately 100,000– 150,000 time steps were needed to reach a steady state, and with a Visual Studio C++ code it took 62–92 h on a single CPU (4.0 GHz). A total of 20 CPUs were used simultaneously for this study, and it took 150 days to generate all statistical data sets of total 1050 catalyst layer. The feed stream reactant gases transfer through the complex pore structures from the channel to catalyst sites within the catalyst layers can be varied depending on multiple factors [21–24]. Therefore, the velocity distribution at the GDM/CL interface is inherently depending on the complex pore structures. Consequently, the local velocity value at the specific location of the GDM/CL interface cannot be accurately predicted due to the spatial randomness of pore structures and should be dealt with statistically valid value based on the ensemble average [55]. In this study, the average inlet velocity at the GDM/CL interface was derived from the multidisciplinary macroscale GDM model of Kim et al. [43] and was set to 0.001 m/s in order to model and analyze the mass transport phenomena inside the catalyst layers. Fig. 6 shows the computed mass transport paths for the conventional and VACNT catalyst layers. The streaklines of the reactants from the GDM/CL interface nodes were directly visualized using the calculated velocity vectors as follows [56]:

dx dy dz ¼ ¼ : ux ðx; y; zÞ uy ðx; y; zÞ uz ðx; y; zÞ

ð19Þ

472

S. Shin et al. / Journal of Catalysis 377 (2019) 465–479

tuosity and porosity for porous media is commonly described using the classical Bruggeman equation [57–58,60]

s¼

Fig. 6. Streaklines of fluid released from the GDM/CL interface of (a) conventional and (b) VACNT catalyst layers.

The tortuosity of the catalyst layer was predicted from the streaklines of the individual reactant transport paths, defined as [57,58]

s¼

2  2  P 2  Le uðsÞ hui ¼ Ps ¼ ; L hux i s ux ðsÞ

 2 Le a 0:5  ð/po Þ ¼ ð/po Þ : L

ð22Þ

Fig. 7 shows the variation of the statistically estimated tortuosities of the conventional and VACNT catalyst layers as a function of porosity. The tortuosity data were compared with the Bruggeman equation for various exponents a [61]. For conventional catalyst layers, the estimated tortuosity-porosity correlation was approximated by the Bruggeman equation with a = 0.746 (R2 = 0.996), indicating that the Bruggeman approximation of a = 0.5 underestimates the tortuosity of the composite catalyst layers. The average tortuosity of the conventional catalyst layers decreased from 2.46 to 1.68 as the porosity increased. The tortuosities of the conventional catalyst layers were verified by Sabharwal and coworkers against multiple numerical/experimental data of previous researches [35] and showed a good agreement. In contrast, the tortuosity of the VACNT catalyst layers was approximately 30% lower than that of the conventional catalyst layers in the range of 1.79– 1.25, which indicates that the VACNT catalyst layers produced more favorable pore structures for reactant mass transfer. Correspondingly, this resulted in a significant discrepancy with the classical Bruggeman equation. The reactant transport inside the catalyst layers significantly depends on the complex pore structures. In order to identify the precise pore size distribution (PSD) of both conventional and VACNT catalyst layers, a 4 nm voxel resolution is adopted to identify the precise pore size distributions (PSDs) of both conventional and VACNT catalyst layers and compared with the 20 nm resolution PSDs. The PSD data are extracted from a total of 25 catalyst layer specimens for a reference composition listed in Table 2. Fig. 8(a) and (b) show the PSD data of the conventional catalyst layers with 4 nm and 20 nm voxel resolutions, respectively. Detailed validation of the computational PSDs of the conventional catalyst layers can be found in a previous research [32]. As shown in Fig. 8(a), pore sizes smaller than 20 nm appear in the 4 nm voxel resolution PSDs of the conventional catalyst layer. The average pore diameter that are calculated from the 4 nm resolution PSD data is estimated to be 26.6 nm, which is 2.2% lower than the pre-estimated average pore diameter (i.e., 27.2 nm) extracted from the 20 nm resolution data. Similarly, in the PSD data of VACNT catalyst layers with a 4 nm resolution, pores smaller than 20 nm are

ð20Þ

where Le is the length of a streamline and L is the length of the computational catalyst layer specimen. In addition, hui and hux i represent the magnitudes of the tangential velocities u and ux along a streamline s, respectively. For the reference composition given in Table 2, the tortuosities of the conventional and VACNT catalyst layers were estimated to be 2.062 and 1.458, respectively, implying that the tortuosity of the VACNT catalyst layer was lower than that of the conventional catalyst layer. Note that some research groups defined the tortuosity differently as [59]

s¼

P Le uðsÞ ¼ Ps : L s ux ðsÞ

ð21Þ

The average tortuosity of the catalyst layers for various porosity conditions was statistically analyzed from a set of 25 numerical experimental data for every case. The correlation between the tor-

Fig. 7. Comparison of tortuosity between conventional (black, solid line) and VACNT (red, dashed line) catalyst layers for various catalyst layer compositions.

S. Shin et al. / Journal of Catalysis 377 (2019) 465–479

473

Fig. 8. Pore size distribution histograms of conventional catalyst layers with (a) 4 nm voxel resolution and (b) 20 nm voxel resolution, and pore size distribution histograms of VACNT catalyst layers with (c) 4 nm voxel resolution and (d) 20 nm voxel resolution.

observed and the average pore diameter is predicted to be 23.0 nm, which is 2.5% lower than the average pore diameter (i.e., 23.6 nm) extracted from the 20 nm resolution data, as shown in Fig. 8(c). For statistical comparison, the PSD histograms of both conventional and VACNT catalyst layers with a 4 nm resolution are redimensionalized and compared with the 20 nm voxel resolution data. As displayed in Fig. 8(d) and (b), no significant difference between the 4 nm and 20 nm voxel resolution PSDs is observed and the constructed PSD histograms with a 20 nm voxel resolution predict the general trends of the PSD with a 4 nm resolution. Additionally, the 4 nm resolution PSD data clearly reveal that approximately 90% of pore volume is classified as pores with 20 to 120 nm in diameter which have a dominant influence on mass transport phenomena. Consequently, a 20 nm voxel resolution is adopted to investigate reactant transport characteristics. Fig. 9(a) and (b) shows the computational tomographic images of the conventional and VACNT catalyst layers depicted in Fig. 6(a) and (b), respectively. Pore structures that were favorable to reactant mass transport were separately visualized and contoured based on the pore diameters. Pore diameter data were extracted from a total of 1050 catalyst layer models consisting of 25 specimens per each composition for comprehensive statistical analysis, and the pore diameters of each catalyst layers were predicted to be in range of 20–120 nm, as reported in previous research [62]. As shown in Fig. 9(a), the conventional catalyst layers were predicted to form heterogeneous pore structures ranging in diameter from 20 (i.e., voxel size) to 80 nm, with an average pore diameter of 27 nm.

A pore size distribution (PSD) data of the conventional catalyst layers had been given and compared in our previous research [32] and showed good agreement with FIB-SEM experimental data [62]. Contrary to the conventional catalyst layers, the VACNT catalyst layers showed an improved organization of the pore structures from the GDM side to the membrane side, as shown in Fig. 9 (b). This indicates that the VACNT catalyst layers generate structures that are more favorable composite to the reactant mass transfer. However, the average pore diameters of the VACNT catalyst layers were estimated to be in the range of 20–70 nm, with an average pore diameter of 24 nm, which is relatively lower than that of the conventional catalyst layers. The effects of the carbon support volume fraction and ionomer volume fraction on the average pore diameters are statistically analyzed in Fig. 9(c) and (d), respectively. The width of the pore size distribution was also displayed in terms of statistical deviation. The correlations between the average pore diameter and carbon support volume fraction and between the average pore diameter and ionomer volume fraction were estimated as second-order polynomials and showed negligible deviation from the numerical data. As shown in Fig. 9(c) and (d), the VACNT catalyst layers exhibited relatively small pore diameters compared to those of the conventional catalyst layers for all volumetric compositions. As the carbon support volume fraction increased from 0.25 to 0.32 (i.e., porosity range of 0.40–0.33) the average pore diameter of the VACNT catalyst layer gradually changed from 23.6 to 20.3 nm, which is approximately 1–3 nm smaller than that of

474

S. Shin et al. / Journal of Catalysis 377 (2019) 465–479

Fig. 9. Computational tomography of (a) conventional and (b) VACNT catalyst layers contoured by the pore diameter. Variations of average pore diameter with the (c) carbon support volume fraction and (d) ionomer-to-carbon ratio.

conventional catalyst layer, as shown in Fig. 9(c). Similarly, in the 0.25–0.5 I/C ratio range, the VACNT catalyst layer produced 1.5– 5 nm smaller average pore diameters than those of the conventional catalyst layer and were predicted to have an average pore diameter of 27.9–20.5 nm, as shown in Fig. 9(d). Therefore, it is reasonable to deduce that the VACNT catalyst layers could produce more suitable composite pore structures for the mass transport of oxygen, owing to the narrow pore spaces, than those of the conventional catalyst layers. Knudsen diffusion is generally considered to be a predominant reactant diffusion mechanism within the catalyst layer of fuel cell systems because the pore diameter of the catalyst layer is smaller than the mean free path of the reactant particles (i.e., approximately 100 nm for a particle diameter of 3 Å at 298 K and 1 atm) [45], as shown in Fig. 9. Fig. 10(a) shows the statistical comparison of the Knudsen diffusion coefficients between the conventional and VACNT catalyst layers as a functions of porosity. The Knudsen diffusion coefficient was calculated from the average pore diameter data in Fig. 9 as follows: po

DKn ¼

d 3

 1 8RT 2 pM

ð23Þ po

where DKn is the Knudsen diffusion coefficient, d is the average pore diameter, and M is the molecular mass. As specified in Eq.

(23), the Knudsen diffusion coefficient DKn is proportional to the po

average pore diameter, d . Therefore, as shown in Fig. 10(a), the Knudsen diffusion coefficient of oxygen inside the VACNT catalyst layers was predicted to be 10–15% lower than those of the conventional catalyst layers in all porosity ranges owing to the relatively narrow pore spaces inside the VACNT catalyst layers, as previously discussed (Fig. 9). The Knudsen diffusion coefficient of oxygen inside the VACNT catalyst layers continuously increased from 3.54  106 to 4.67  106 m2/s, corresponding to an increase in the porosity and the average pore diameter, whereas the Knudsen diffusion coefficient inside the conventional catalyst layers increased from 3.98  106 to 5.60  106 m2/s. The predicted pore diameters and the associated Knudsen diffusion coefficients are shown in Figs. 9 and 10(a), respectively, demonstrating that the nanotube forest of the VACNT catalyst layers can act as a diffusion barrier due to the small pore diameters [63]. The complex pore structures inside the catalyst layers lead to tortuous reactant mass transport paths, and therefore, the mass diffusion phenomena inside the catalyst layers are strongly dependent on the morphological structures of catalyst layers. Therefore, substantive mass diffusion phenomena need to be evaluated in terms of the effective diffusion coefficient considering the porosity and tortuosity to account for the pore structures and the extended mass transport paths. The effective diffusion coefficient, Deff was evaluated from the statistically pre-determined tortuosity and

S. Shin et al. / Journal of Catalysis 377 (2019) 465–479

Fig. 10. Statistical comparison of the (a) Knudsen diffusion coefficient and (b) effective diffusion coefficient between conventional (black, solid line) and VACNT (red, dashed line) catalyst layers.

the Knudsen diffusion coefficient data shown in Figs. 7 and 10 (a), respectively, as follows [49,58]:

Deff ¼

/po

s

DKn :

ð24Þ

Note that some research groups define the effective diffusion coefficient instead as follows [59]:

Deff ¼

/po

s2

DKn ;

ð25Þ

with the tortuosity defined by Eq. (21). Although the mathematical expressions of the tortuosity and effective diffusion coefficient in Eq. (25) are different from those in Eq. (24), these definitions are equivalent [64]. Fig. 10 (b) shows the statistically estimated effective diffusion coefficient in the porosity range of 0.3–0.5 for both the conventional and VACNT catalyst layers. As previously discussed (Figs. 9 and 10(a)), the VACNT catalyst layers produced relatively small pore diameters and, consequently, low-Knudsen diffusion coefficients compared to those of the conventional catalyst layers, which implies that the reactant transport inside the VACNT catalyst layer could be limited by the vertically aligned CNT structures. However,

475

despite the small pore diameters and the low-Knudsen diffusion coefficient of the VACNT catalyst layers, the effective diffusion coefficient of the VACNT catalyst layers was predicted to be higher than that of the conventional catalyst layer by 15–25%, as shown in Fig. 10(b), because the tortuosity of the VACNT catalyst layer was significantly lower compared to that of the conventional catalyst layer in all porosity ranges, as predicted in Fig. 7 This statistical analysis evidently indicates that the morphological structure of the VACNT catalyst layers and consequential organized pore structures can improve the oxygen mass transport phenomena inside the PEFC catalyst layers, as reported in previous studies [10–12]. The effective diffusion coefficient of PEFC catalyst layers can also be affected by the inlet velocity at the GDM/CL interface. In order to analyze the effects of the inlet velocity on the mass transport phenomena inside the VACNT catalyst layers, the average inlet velocity at the GDM/CL interface is increased from 0.001 m/s to 0.004 m/s. Fig. 11 shows the variation of the statistically predicted tortuoisties of the VACNT catalyst layers for various inlet velocity conditions. The composition of each catalyst layer is set as Table 2. As shown in Fig. 11(a), the tortuosity of the VACNT catalyst layer gradually increases from 1.45 to 1.56 as the inlet velocity is increased from 0.001 m/s to 0.004 m/s. This result implies that the increment of the inlet velocity causes more complex reactant transport phenomena and low-effective diffusion coefficients inside the porous catalyst layers. Consequently, as the inlet velocity is increased from 0.001 m/s to 0.004 m/s, the effective diffusion coefficient is estimated to decrease by 6.8%, as shown in Fig. 11 (b). However, despite the intensified transport limitations due to the tortuous transport paths, the effective diffusion coefficient of the VACNT catalyst layers iss predicted to be higher than that of conventional catalyst layers. The increase of tortuosity is associated with morphological structures of porous media and increased flow inertia. Fig. 12 displays computational tomographic images of the VACNT catalyst layer for inlet velocities of 0.001 m/s and 0.004 m/s, respectively. Pore spaces are contoured by local u-velocity of reactant flow, and the reactant transport paths are visualized based on the calculated velocity fields. As previously mentioned, the morphological structures of PEFC catalyst layers are highly random and heterogeneous. Therefore, complex pore structures lead to tortuous reactant transport paths and even local back-flows [65–66], as shown in Fig. 12. For instance, at a low inlet velocity condition (i.e., u = 0.001 m/s) in Fig. 12(a), only a small local back-flow region (BFR) is observed (i.e, white border area) and almost every uvelocities within the entire velocity fields are positive. However, as the flow inertia is increased (i.e, u = 0.004 m/s) in Fig. 12(b), the local back-flow is enhanced due to the complex pore structures and comparatively strong negative u-velocities are observed within the velocity fields. Consequently, the local back-flow regions induced by the increased flow inertia lead to more tortuous reactant transport paths. 3.3. Transport-based catalyst utilization The morphology-based ECUF in Eq. (15) can serve as a useful indicator for estimating the theoretical maximum catalyst utilization [31,32]. However, the catalyst utilization can be greatly affected by the complex pore structures of the composite catalyst layers, as discussed in Figs. 6 and 9, because the catalysts located on the dead-end pores (i.e., the reactant mass transfer is limited by the stagnant reactant flow) have a limited contribution to the active catalytic reactions. Consequently, these morphologically effective catalysts may not participate in the fuel cell reactions, and therefore, the M–ECUFs may overestimate the catalyst utilization [18]. In this regard, the M–ECUF should be modified by considering reactant mass transport phenomena. Shin et al. introduced

476

S. Shin et al. / Journal of Catalysis 377 (2019) 465–479

Fig. 11. (a) Variation of the statistically predicted tortuoisties of the VACNT and conventional catalyst layers for various inlet velocity conditions, and (b) corresponding effective diffusion coefficient.

the transport-based effective catalyst utilization factor (i.e., T–ECUF)10 to reflect the effects of the practical reactant mass transfer on the catalyst utilization as follows:

CT ¼

Transport - based electrochemically active catalyst sites : Total number of catalyst sites ð26Þ

Fig. 13 shows the three-dimensional reactant mass transport paths (i.e., streaklines) and the corresponding distribution of transport-based effective catalysts (orange) for the conventional and VACNT catalyst layers. Among the morphologically classified effective catalysts (yellow), the catalysts adjacent to the preestimated reactant transport paths were classified as transportbased utilized catalysts according to the definition of T–ECUF. The distribution of the morphology-based utilized catalysts shown in Fig. 2 is also presented in Fig. 13. Compared to the conventional catalyst layers, a significant improvement of the catalyst utilization was observed in the VACNT catalyst layer. For the conventional catalyst layer shown in Fig. 13(a), the T–ECUF was predicted to be 0.071. By contrast, the T–ECUF of the VACNT catalyst layer 10

T-ECUF: transport-based catalyst utilization factor.

Fig. 12. Computational tomographic images contoured by u-velocity with reactant flow paths inside the VACNT catalyst layer for inlet velocity of (a) 0.001 m/s and (b) 0.004 m/s.

shown in Fig. 13(b) was determined to be 0.137, which is approximately equal to two times the value of the convention catalyst layer. The morphology- and transport-based ECUFs of the VACNT catalyst layers were statistically estimated and compared with those of the conventional catalyst layers, as shown in Fig. 14. For a carbon support volume fraction of 25% with a 0.3 mg/cm2 Pt catalyst, the M–ECUFs of the conventional and VACNT catalyst layers yielded an asymmetric bell-shaped curve for the ionomer volume fraction values, which were influenced by the interconnection of ionomers and pores, as shown in Fig. 4(b). However, the M–ECUFs of the VACNT catalyst layers were predicted to be higher than those of the conventional catalyst layers for most ionomer volume fraction conditions except for the ionomer volume fractions that were lower than 0.275, as shown in Fig. 14(a). In particular, the maximum M–ECUF of the VACNT catalyst layers was computed to be more than two times greater than that of the conventional catalyst layers owing to the perfect electron conduction networks of the VACNTs and the favorable composite structures for reactant

S. Shin et al. / Journal of Catalysis 377 (2019) 465–479

477

Fig. 14. Statistical comparison of the (a) M–ECUFs and (b) T–ECUFs between conventional (red, dashed line) and VACNT (black, solid line) catalyst layers.

Fig. 13. Three-dimensional distribution of morphology-based (yellow) and transport-based (orange) effective catalysts in (a) conventional and (b) VACNT catalyst layer.

transport. The maximum M–ECUF of the VACNT catalyst layers was predicted to be 0.473 (i.e., 0.071 mg/cm2), whereas that of the conventional catalyst layers is 0.235 (i.e., 0.142 mg/cm2). This result implies that the VACNT catalyst layers produce enhanced composite morphological structures for the ORR reaction, and may be a promising alternative for improving the catalyst utilization of fuel cell catalyst layers. Note that for ionomer volume fractions less than 0.275, the M–ECUF of the VACNT catalyst layers was evaluated to be slightly lower than that of the conventional catalyst layer because there was not sufficient ionic connectivity inside the VACNT catalyst layers in this lower ionomer volume fraction range, as reported in a previous research [32]. The T–ECUFs of the VACNT catalyst layer were also predicted and compared with those of the conventional catalyst layers, as shown in Fig. 14 (b). Similar to the M–ECUF, the T–ECUF curves of the conventional and VACNT catalyst layers exhibited asymmetric bell-shapes for an ionomer volume fractions in the range of 0.250–0.450, and showed optimum ionomer volume fractions at 0.325 and 0.350, respectively. However, as previously mentioned, the T–ECUFs of conventional and VACNT composite catalyst layers were predicted to be lower than the M–ECUFs. The maximum

T–ECUF of the conventional catalyst layers was estimated to be 0.082, whereas that of the VACNT catalyst layers was 0.152 (i.e., 85% higher than that of the conventional catalyst layers). The statistical results elicited based on the numerical experimental data shown in Fig. 14 demonstrated that effective catalyst utilization is directly dependent on the composite structures of the catalyst layer, and identifying the VACNT catalyst layers yielded much higher effective catalyst utilization by improving the substantive transport phenomena of electron, ions, and reactants. This comprehensive statistical analysis conclusively indicated that the PEFC catalyst layers that used VACNTs as carbon supporters can provide more efficient reactant transport, thus resulting in enhanced catalyst utilization for active electrochemical reactions.

4. Conclusions The advanced PEFC composite catalyst layers containing VACNTs as carbon supporters were modeled using a statistical approach based on the inherent random nature of the structures of the catalyst layers. For a reliable statistical analysis, each of the data points was extracted from a total of 25 computational catalyst layer specimens to achieve a 95% confidence level. The structure–transport characteristics of the VACNT catalyst layers, such as the tortuosity and Knudsen and effective diffusion coefficients, were statistically estimated from a series of numerical experimental data derived by the LBM. The catalyst utilization and reactant transport paths of VACNT catalyst layers were compared with

478

S. Shin et al. / Journal of Catalysis 377 (2019) 465–479

those of conventional catalyst layers in pursuit of PEFC performance improvements. The tomographic images of the composite catalyst layers revealed that the VACNT catalyst layers could provide pore structures that were more favorable to reactant mass transfer with significantly lower tortuosity. However, the VACNT catalyst layers were predicted to have relatively lower average pore diameters than those of the conventional catalyst layers in the porosity range of 0.25–0.50, thereby resulting in lower Knudsen diffusion coefficients owing to the nanotube forest effect of the VACNT catalyst layers on the mass diffusion. Despite the relatively lower Knudsen diffusion coefficient, the effective diffusion coefficient of the VACNT catalyst layers was computed to be higher mainly owing to the improved pore structures and low tortuosity. Additionally, the effects of the inlet velocity on the effective diffusion coefficient are also investigated and those results clearly indicates that the formation of excessively strong velocity fields inside the VACNT catalyst layers can rather have a negative effect on the reactant mass transport phenomena through the pore structures. However, despite the transport limitations due to the tortuous transport paths, the effective diffusion coefficient of the VACNT catalyst layers is predicted to be higher than that of the conventional catalyst layers. Consequently, the VACNT catalyst layers yielded improved catalyst utilization compared to that of the conventional catalyst layers according to the morphology- and transport-based effective catalyst utilization factors (M–ECUF and T–ECUF). Declaration of Competing Interest There are no competing interest to declare. Acknowledgements This work was supported by the Korea Evaluation Institute of Industrial Technology [grant number 201800000000249] and the National Research Foundation of Korea [grant numbers 201800000002799 and 201800000002384]. References [1] M. Mench, Fuel Cell Engines, John Wiley & Sons, New Jersey, 2008. [2] R. O’hayre, S.W. Cha, F.B. Prinz, W. Colella, Fuel Cell Fundamentals, John Wiley & Sons, New York, 2016. [3] M. Eikerling, K. Malek, Q. Wang, in: PEM Fuel Cell Electrocatalysts and Catalyst Layers, Springer, London, 2008, p. 381. [4] M. Uchida, Y. Aoyama, N. Eda, A. Ohta, Investigation of the microstructure in the catalyst layer and effects of both perfluorosulfonate ionomer and PTFEloaded carbon on the catalyst layer of polymer electrolyte fuel cells, J. Electrochem. Soc. 142 (12) (1995) 4143–4149. [5] M. Chhowalla, K.B.K. Teo, C. Ducati, N.L. Rupesinghe, G.A.J. Amaratunga, A.C. Ferrari, D. Roy, J. Robertson, W.I. Milne, Growth process conditions of vertically aligned carbon nanotubes using plasma enhanced chemical vapor deposition, J. Appl. Phys. 90 (10) (2001) 5308–5317. [6] M.B. Jakubinek, M.A. White, G. Li, C. Jayasinghe, W. Cho, M.J. Schulz, V. Shanov, Thermal and electrical conductivity of tall, vertically aligned carbon nanotube arrays, Carbon 48 (13) (2010) 3947–3952. [7] M. Yu, H.H. Funke, J.L. Falconer, R.D. Noble, High density, vertically-aligned carbon nanotube membranes, NANO Lett. 9 (1) (2008) 225–229. [8] W. Zhang, P. Sherrell, A.I. Minett, J.M. Razal, J. Chen, Carbon nanotube architectures as catalyst supports for proton exchange membrane fuel cells, Energy Environ. Sci. 3 (9) (2010) 1286–1293. [9] A. Bonnefont, P. Ruvinskiy, M. Rouhet, A. Orfanidi, S. Neophytides, E. Savinova, Advanced catalytic layer architectures for polymer electrolyte membrane fuel cells, Wiley Interdisc. Rev.: Energy Environ. 3 (5) (2014) 505–521. [10] S. Yasuda, A. Furuya, Y. Uchibori, J. Kim, K. Murakoshi, Iron–nitrogen–doped vertically aligned carbon nanotube electrocatalyst for the oxygen reduction reaction, Adv. Funct. Mater. 26 (5) (2016) 738–744. [11] Z.Q. Tian, S.H. Lim, C.K. Poh, Z. Tang, Z. Xia, Z. Luo, P.K. Shen, D. Chua, Y.P. Feng, Z. Shen, J. Lin, A highly order–structured membrane electrode assembly with vertically aligned carbon nanotubes for ultra–low Pt loading PEM fuel cells, Adv. Energy Mater. 1 (6) (2011) 1205–1214. [12] S. Murata, M. Imanishi, S. Hasegawa, R. Namba, Vertically aligned carbon nanotube electrodes for high current density operating proton exchange membrane fuel cells, J. Power Sources 253 (2014) 104–113.

[13] E.W. Thiele, Relation between catalytic activity and size of particle, Ind. Eng. Chem. 31 (7) (1939) 916–920. [14] Q. Wang, M. Eikerling, D. Song, Z. Liu, Structure and performance of different types of agglomerates in cathode catalyst layers of PEM fuel cells, J. Electroanal. Chem. 573 (1) (2004) 61–69. [15] M. Lee, M. Uchida, H. Yano, D.A. Tryk, H. Uchida, M. Watanabe, New evaluation method for the effectiveness of platinum/carbon electrocatalysts under operating conditions, Electrochim. Acta 55 (28) (2010) 8504–8512. [16] M. Lee, M. Uchida, D.A. Tryk, H. Uchida, M. Watanabe, The effectiveness of platinum/carbon electrocatalysts: dependence on catalyst layer thickness and Pt alloy catalytic effects, Electrochim. Acta 56 (13) (2011) 4783–4790. [17] W.K. Epting, S. Litster, Effects of an agglomerate size distribution on the PEFC agglomerate model, Int. J. Hydrogen Energy 37 (10) (2012) 8505–8511. [18] S. Shin, A.R. Kim, S. Um, Computational prediction of nanoscale transport characteristics and catalyst utilization in fuel cell catalyst layers by the lattice Boltzmann method, Electrochim. Acta 275 (2018) 87–99. [19] S. Succi, E. Foti, F. Higuera, Three-dimensional flows in complex geometries with the lattice Boltzmann method, EPL (Europhys. Lett.) 10 (5) (1989) 433. [20] C.K. Aidun, J.R. Clausen, Lattice-Boltzmann method for complex flows, Annu. Rev. Fluid Mech. 42 (2010) 439–472. [21] L. Chen, G. Wu, E.F. Holby, P. Zelenay, W.Q. Tao, Q. Kang, Lattice Boltzmann pore-scale investigation of coupled physical-electrochemical processes in C/Pt and non-precious metal cathode catalyst layers in proton exchange membrane fuel cells, Electrochim. Acta 158 (2015) 175–186. [22] S.H. Kim, H. Pitsch, I.D. Boyd, Accuracy of higher-order lattice Boltzmann methods for microscale flows with finite Knudsen numbers, J. Comput. Phys. 227 (19) (2008) 8655–8671. [23] S.H. Kim, H. Pitsch, Reconstruction and effective transport properties of the catalyst layer in PEM fuel cells, J. Electrochem. Soc. 156 (6) (2009) B673–B681. [24] Y. Gao, Using MRT lattice Boltzmann method to simulate gas flow in simplified catalyst layer for different inlet–outlet pressure ratio, Int. J. Heat Mass Transf. 88 (2015) 122–132. [25] K. Malek, M. Eikerling, Q. Wang, T. Navessin, Z. Liu, Self-organization in catalyst layers of polymer electrolyte fuel cells, J. Phys. Chem. C 111 (36) (2007) 13627–13634. [26] F.C. Cetinbas, R.K. Ahluwalia, N. Kariuki, V. De Andrade, D. Fongalland, L. Smith, J. Sharman, P. Ferrira, S. Rasouli, D.J. Myers, Hybrid approach combining multiple characterization techniques and simulations for microstructural analysis of proton exchange membrane fuel cell electrodes, J. Power Sources 344 (2017) 62–73. [27] F.C. Cetinbas, R.K. Ahluwalia, N.N. Kariuki, D.J. Myers, Agglomerates in polymer electrolyte fuel cell electrodes: Part I. Structural characterization, J. Electrochem. Soc. 165 (13) (2018) F1051–F1058. [28] F.C. Cetinbas, R.K. Ahluwalia, Agglomerates in polymer electrolyte fuel cell electrodes: Part II. Transport characterization, J. Electrochem. Soc. 165 (13) (2018) F1059–F1066. [29] P.P. Mukherjee, C.Y. Wang, Stochastic microstructure reconstruction and direct numerical simulation of the PEFC catalyst layer, J. Electrochem. Soc. 153 (5) (2006) A840–A849. [30] P.P. Mukherjee, Q. Kang, C.Y. Wang, Pore-scale modeling of two-phase transport in polymer electrolyte fuel cells—progress and perspective, Energy Environ. Sci. 4 (2) (2011) 346–369. [31] S. Shin, A.R. Kim, S. Um, Statistical prediction of fuel cell catalyst effectiveness: quasi-random nano-structural analysis of carbon sphere-supported platinum catalysts, Int. J. Hydrogen Energy 41 (22) (2016) 9507–9520. [32] S. Shin, A.R. Kim, S. Um, Integrated statistical and nano-morphological study of effective catalyst utilization in vertically aligned carbon nanotube catalyst layers for advanced fuel cell applications, Electrochim. Acta 207 (2016) 187– 197. [33] M. Wang, J. Wang, N. Pan, S. Chen, Mesoscopic predictions of the effective thermal conductivity for microscale random porous media, Phys. Rev. E 75 (3) (2007) 036702. [34] G. Inoue, M. Kawase, Effect of porous structure of catalyst layer on effective oxygen diffusion coefficient in polymer electrolyte fuel cell, J. Power Sources 327 (2016) 1–10. [35] M. Sabharwal, L.M. Pant, N. Patel, M. Secanell, Computational analysis of gas transport in fuel cell catalyst layer under dry and partially saturated conditions, J. Electrochem. Soc. 166 (7) (2019) F3065–F3080. [36] M. Lopez-Haro, L. Guétaz, T. Printemps, A. Morin, S. Escribano, P.H. Jouneau, P. Bayle-Guillemaud, F. Chandezon, G. Gebel, Three-dimensional analysis of Nafion layers in fuel cell electrodes, Nat. Commun. 5 (2014) 5229. [37] L. Guetaz, M. Lopez-Haro, S. Escribano, A. Morin, G. Gebel, D.A. Cullen, K.L. More, R.L. Borup, Catalyst-layer ionomer imaging of fuel cells, ECS Trans. 69 (17) (2015) 455–464. [38] V. Berejnov, D. Susac, J. Stumper, A.P. Hitchcock, 3D chemical mapping of PEM fuel cell cathodes by scanning transmission soft X-ray spectrotomography, ECS Trans. 50 (2) (2013) 361–368. [39] J.H. Li, L.M. Zhang, Y. Wang, D.G. Fredlund, Permeability tensor and representative elementary volume of saturated cracked soil, Can. Geotech. J. 46 (8) (2009) 928–942. [40] A. Mohamad, Lattice Boltzmann Method: Fundamentals and Engineering Applications with Computer Codes, Springer Science & Business Media, London, 2011. [41] S. Succi, The lattice Boltzmann Equation: For Fluid Dynamics and Beyond, Oxford University Press, New York, 2001.

S. Shin et al. / Journal of Catalysis 377 (2019) 465–479 [42] S. Succi, The Lattice Boltzmann Equation: For Complex States of Flowing Matter, Oxford University Press, New York, 2018. [43] A.R. Kim, S. Shin, S. Um, Multidisciplinary approaches to metallic bipolar plate design with bypass flow fields through deformable gas diffusion media of polymer electrolyte fuel cells, Energy 106 (2016) 378–389. [44] F. White, I. Corfield, Viscous Fluid Flow, McGraw-Hill, New York, 2006. [45] J. Welty, C. Wicks, G. Rorrer, R. Wilson, Fundamentals of Momentum, Heat, and Mass Transfer, John Wiley & Sons, New Jersey, 2009. [46] L. Chen, L. Zhang, Q. Kang, H.S. Viswanathan, J. Yao, W. Tao, Nanoscale simulation of shale transport properties using the lattice Boltzmann method: permeability and diffusivity, Sci. Rep. 5 (2015) 8089. [47] J. Hoshen, R. Kopelman, Percolation and cluster distribution. I. Cluster multiple labeling technique and critical concentration algorithm, Phys. Rev. B 14 (8) (1976) 3438. [48] J.M. Hammersley, Percolation processes II. The connective constant, Math. Proc. Cambridge Philos. Soc. 53 (1957) 642–645. [49] J.M. Hammersley, Monte Carlo methods for solving multivariable problems, Ann. N. Y. Acad. Sci. 86 (3) (1960) 844–874. [50] D.M. Bernardi, M.W. Verbrugge, Mathematical model of a gas diffusion electrode bonded to a polymer electrolyte, AIChE J. 37 (8) (1991) 1151–1163. [51] S. Um, C.Y. Wang, K.S. Chen, Computational fluid dynamics modeling of proton exchange membrane fuel cells, J. Electrochem. Soc. 147 (12) (2000) 4485– 4493. [52] S. Gottesfeld, T.A. Zawodzinski, Polymer electrolyte fuel cells, Adv. Electrochem. Sci. Eng. 5 (1997) 195–301. [53] T. Suzuki, K. Kudo, Y. Morimoto, Model for investigation of oxygen transport limitation in a polymer electrolyte fuel cell, J. Power Sources 222 (2013) 379– 389. [54] L. Xing, M. Mamlouk, R. Kumar, K. Scott, Numerical investigation of the optimal NafionÒ ionomer content in cathode catalyst layer: an agglomerate two-phase flow modelling, Int. J. Hydrogen Energy 39 (17) (2014) 9087–9104.

479

[55] K. Mitra, Y. Kuo, Digital Signal Processing: A Computer-based Approach, vol. 2, McGraw-Hill, New York, 2006. [56] W. Janna, Introduction to Fluid Mechanics, CRC Press, New York, 2015. [57] S. Litster, W.K. Epting, E.A. Wargo, S.R. Kalidindi, E.C. Kumbur, Morphological analyses of polymer electrolyte fuel cell electrodes with nano–scale computed tomography imaging, Fuel Cells 13 (5) (2013) 935–945. [58] Y. Wang, S. Wang, Evaluation and modeling of PEM fuel cells with the Bruggeman correlation under various tortuosity factors, Int. J. Heat Mass Transf. 105 (2017) 18–23. [59] Z. Sun, X. Tang, G. Cheng, Numerical simulation for tortuosity of porous media, Microporous Mesoporous Mater. 173 (2013) 37–42. [60] D.W. Chung, M. Ebner, D.R. Ely, V. Wood, R.E. García, Validity of the Bruggeman relation for porous electrodes, Modell. Simul. Mater. Sci. Eng. 21 (7) (2013) 074009. [61] J.A. Currie, Gaseous diffusion in porous media. Part 2. Dry granular materials, Br. J. Appl. Phys. 11 (8) (1960) 318. [62] S. Thiele, R. Zengerle, C. Ziegler, Nano-morphology of a polymer electrolyte fuel cell catalyst layer—imaging, reconstruction and analysis, NANO Res. 4 (9) (2011) 849–860. [63] S.H. Ko, C. Grigoropoulos (Eds.), Hierarchical Nanostructures for Energy Devices, Royal Society of Chemistry, Cambridge, 2014. [64] L. Shen, Z. Chen, Critical review of the impact of tortuosity on diffusion, Chem. Eng. Sci. 62 (14) (2007) 3748–3755. [65] S. Das, N.G. Deen, J.A.M. Kuipers, A DNS study of flow and heat transfer through slender fixed-bed reactors randomly packed with spherical particles, Chem. Eng. Sci. 160 (2017) 1–19. [66] D. Hlushkou, S. Bruns, U. Tallarek, High-performance computing of flow and transport in physically reconstructed silica monoliths, J. Chromatogr. A 1217 (23) (2010) 3674–3682.