Model Study of CNT-Based PEMFCs’ Electrocatalytic Layers

Abstract

One of the most important problems in the development of proton-exchange membrane fuel cells (PEMFCs) is the selection of an efficient support material to serve as the electrocatalyst, which can ensure PEMFCs’ durability at low active metal loading, with minimal changes in the electrochemical surface and conductivity during long-term operations. Carbon nanostructures are now widely used in PEMFCs as such support materials, including carbon nanotubes (CNTs). In order to estimate the effect of the geometric parameters of a CNT-based support on the resulting size distribution of platinum nanoparticles for given synthesis conditions, in this work, we propose a semi-empirical model that assumes a random uniform distribution of platinum particles over the CNT surface. Based on the obtained distribution, the electrochemically active surface area (EASA) of the electrocatalyst is calculated and further used to evaluate the performance of the catalytic layer (CL) in the PEMFC. The applicability of the proposed model for calculating the parameters of CNT-based CLs and the output electrochemical characteristics of PEMFCs is shown.

1. Introduction

The widespread application of power supplies based on proton-exchange membrane fuel cells (PEMFCs) is still hampered by their low cost-effectiveness and relatively short service life. The above problems are for the most part associated with catalysts based on amorphous carbon black with platinum nanoparticles as an active sites (Pt/C), which are used in membrane–electrode assemblies (MEAs) of PEMFCs [1,2]. Catalysts of this composition are expensive and are deactivated under operating conditions due to the dissolution/agglomeration of platinum particles, corrosion of the support, and delamination of the catalytic layer (CL) [1,2,3,4]. Therefore, one of the most important issues in the development of PEMFCs is the selection of an efficient electrocatalyst support material capable of providing low metal loading and high values of electrochemically active surface area (EASA) and conductivity [5]. Such materials could be found among the carbon nanostructures that are currently widely used in PEMFCs [3,4,6,7]. The variety of shapes and structures of carbon supports allows for creating efficient electrocatalysts. Nowadays, there is a wide variety of carbon nanomaterials: nanofibers, various configurations of tubes, fullerenes and carbon black, reduced graphene oxide, and carbon nanotubes. Carbon nanotubes (CNTs) are two-dimensional nanostructures formed by rolled sheets of carbon atoms. They can be single-walled and multi-walled as well as form arrays, including vertically aligned carbon nanotubes (VACNTs).

A number of studies [3,4,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21] aimed at using CNTs both as components of gas diffusion layers (GDLs) and as microporous and catalytic layers (MPLs and CLs). In the design of PEMFCs, carbon nanotubes can completely replace the GDL and eliminate the need to create an MPL [8]. Thus, instead of ordinary carbon paper, a new nanotube-based paper was created and studied [8]. Carbon paper is modified with nanotubes by incorporating them into a carbon fiber matrix [9]. The inclusion of multi-walled CNTs (MWNTs) more than doubles the electrical conductivity. A doubling of the power density is also observed.

In addition to using CNTs as GDLs, CNTs were added to the compositions of some studied CLs with a view to improving the transport of reagents [10]. The authors of this work attribute the results obtained to the high adsorption capacity of CNTs, which keeps the reagents close to the electrocatalyst.

CNTs can be grown directly on a GDL [11], which has a positive effect on mass transfer in the cathode region. The optimal ratio of pores ensuring good transportation of reagent gases was achieved in the cited work. GDLs with suitable hydrophobicity and an appropriate pore structure can reduce water flooding of the cathode, especially at high current densities. A similar result was obtained in [12], where an improvement in mass transfer and electrical conductivity was noted when using CNTs compared to Vulcan XC-72R carbon black. The properties of CNTs can be used to increase the PEMFC performance efficiency at low humidity. For GDLs with an MPL and addition of hydrophilic CNTs, this problem becomes solvable [13].

CNTs have a developed surface, which makes it possible to effectively use them as a support for electrocatalysts [14]. A CNT-based platinum catalyst characterized by high EASA, low charge transfer resistance, low overvoltage losses during activation, and higher peak power density was synthesized [15]. A number of works [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34] have been devoted to the study of the morphological, structural, and electrochemical characteristics of CNT-based electrocatalysts.

During the synthesis of platinum nanoparticles on the CNT surface, a number of processes occur simultaneously, which are accompanied by both the growth and dissolution of nanoparticles. In particular, processes such as the coalescence of adjacent nanoparticles, autocatalytic growth, and Ostwald ripening can take place [16].

According to classical theory, the initial stage in the formation of nanoparticles is nucleation. The size of the nuclei depends on many factors, such as the temperature and composition of the reaction mixture, concentration of the reagents, pH of the solution, etc. The study of this stage is complicated due to the short lifetime of the nuclei and the existence of several ways of particle reduction [17].

The parameters of the resulting CL depend on the synthesis method used, and the type of support can be estimated using model calculations. The Ostwald ripening model [18] is often used as an analytical model of particle growth on a support to determine the particle size distribution. According to this model, the particle size changes as follows:

$$\frac{\mathrm{dr}}{\mathrm{dt}}=\frac{{\mathrm{A}}_{\mathrm{s}}}{{\mathrm{r}}^3}\left(1-\mathrm{r}/{\mathrm{r}}_{\mathrm{c}}\right)$$

(1)

where A_s is the Ostwald ripening rate constant, and r_c is the critical size of the nucleus. However, this model often does not allow for describing the process of nanoparticle growth with sufficient accuracy, and it is also not always possible to determine the necessary calculation parameters. Monte Carlo methods using a random variable generator can be utilized in this case. Models based on this technique have a sufficiently high accuracy in estimating the geometric parameters of synthesized nanoparticles [19,20].

In this work, a semi-empirical model using a random uniform distribution of particles over the CNT surface is proposed for the estimation of the effect of geometric parameters of the CNT-based support on the resulting size distribution of platinum nanoparticles under given synthesis conditions. Based on the obtained distribution, the EASA of the electrocatalyst is estimated and further used to model and evaluate the performance efficiency of the CL as part of the PEMFC.

2. Results and Discussion

The proposed model was tested for the CNT-based platinum catalyst parameters presented in [25,28,30,31]. The initial parameters that were used for the numerical simulation are listed in

Table 1. Parameters of the CNT-based catalyst [25].

Parameters	Values
CNT length (LCNT)	60 μm
CNT diameter (dCNT)	20 nm
CNT array density (NCNT)	3 × 109 tubes cm−2
Mass of platinum	0.1 mg cm−2
Average particle size	2.0–2.5 nm
EASA	77 m2·g−1
Minimum particle spacing (Rc)	0.5 nm
Minimum particle diameter (dw)	0.5 nm

[25].

As a result of the modeling with the parameter values presented in Table 1, the particle size distribution (Figure 1a) for the CNT-based catalytic layer was obtained, which is similar to the distribution obtained experimentally in [25] (Figure 1b).

Additionally, according to the simulation results, the average value of the particle size was obtained (2.1 nm), which corresponds to the experimental data (Table 1). The calculated EASA value for the hemispherical particles is 54 m²/g, which is lower than the value of 77 m²/g obtained using cyclic voltammetry. The discrepancy in the EASA values can be explained by the accepted assumption of the hemispherical surface of the particles. In the case of spherical particles, the calculated EASA value will be 108 m²/g. Thus, the real surface area of the particle turns out to be less than the area of the sphere corresponding in mass, and greater than the area of the corresponding hemisphere. The average particle surface shape was taken to be three-quarters of the sphere. In this case, the calculated value of the EASA was 81 m²/g, which practically coincides with the experimentally obtained result (Table 1).

Simulation was also carried out for CNT-based electrocatalysts, the characteristics of which are presented in [28,30,31]. The calculated CL parameters are shown in Figure 2 in comparison with the experimentally obtained data.

Calculations were also carried out using characteristics of a Pt/CNT electrocatalyst sample experimentally obtained in the laboratory. The CCL parameters include the tube diameter (d_CNT = 50 nm), CNT length (L_CNT = 100 μm), CNT array density (N_CNT = 2 × 10¹⁰ tubes cm⁻²), platinum loading (M_Pt = 0.3 mg/cm², or 20 wt.%), minimum distance between particles (R_c = 20 nm), and minimum particle size (d_n = 4 nm). The simulation gave the EASA value of 25 m²/g and average particle size of 5.2 nm. The resulting particle size distribution is shown in Figure 3a.

Figure 3b shows the TEM image and particle size distribution for the Pt/CNT sample. According to the TEM image, the average particle size for the Pt/CNTs was found to be 5.2 nm, which coincides with the simulation results. The experimentally obtained EASA values for the electrocatalyst were 27 m²/g, which also coincides with the simulation. However, the distribution of particles (Figure 3b) obtained from the simulation has a larger size spread since the distribution obtained with the TEM image includes only a small part of the CNT surface.

Table 2. Structural and electrochemical characteristics of electrocatalysts, where EG represents a reduction in ethylene glycol, and H₂ represents an impregnation of CL with the precursor and subsequent reduction of platinum with hydrogen.

No.	Synthesis Method	Pt Loading, % (mg/cm2)	Average Particle Size of Pt, nm	Average Particle Size of Pt, nm (Model)	EASA, m2/g	EASA (Model), m2/g
1	EG [25]	20 (0.1)	2.0–2.5	2.1	77	81.0
2	EG [28]	20 (0.3)	3.2	3.1	48.6	41.8
3	EG [30]	20 (0.55)	2.5–4	3.3	40.2	39.4
4	Imp. H2 [31]	15	1–3	2.3	46–70	56.3
5	Imp. H2	20 (0.3)	5.2	5.2	27	25.0

summarizes the literature data on the structural and electrochemical characteristics of Pt/CNT electrocatalysts in comparison with the results of the model calculations. To determine the experimental particle size distribution, the TEM images from [25,28,31] and the average size of the platinum particles obtained in [30] were used.

Thus, we can conclude that the developed model is applicable for predicting the characteristics of CNT-based electrocatalysts regardless of the synthesis method used. The error in determining the EASA is mostly associated with the complex shape of the particles, which affects the final result of the simulation.

The presented model allows predicting the CL characteristics depending on the parameters of the CNTs used (length and diameter), as well as on the Pt loading. As a result of the simulation, predicted values of the EASA and average particle size were obtained for various values of the platinum loading of the CL (Figure 4).

It is shown that as the platinum loading increases (Figure 4), the average particle size increases significantly, and as a consequence, the EASA decreases. Regarding the increase in the platinum loading from 0.05 mg/cm² to 0.3 mg/cm², the EASA decreased almost fourfold. According to the literature data (Table 2), the size of platinum particles increases with increasing platinum loading of the electrocatalyst. This effect is observed regardless of the electrocatalyst synthesis method, which is confirmed by the model calculations. Thus, electrocatalysts with a lower platinum content will be characterized by higher EASA values when used as a CNT support.

The PEMFC I–V curves calculated from Equations (5)–(8) based on the model CCL are shown in Figure 5 in comparison with the experimental data presented in [25,30,31] and with our experimental sample. The presented model accurately describes the PEMFC performance. The observed discrepancy between the experimental data and the simulation results is due to the lack of information provided in [25,30,31] to help future researchers determine the required parameters that should be used in the model.

The use of a larger number of parameters allows for achieving a complete correlation between the experimental data and the model curves in the entire potential range. Thus, the presented model has been successfully used to describe the I–V curves of PEMFCs with CNT-based CCLs.

The presented model can also be used to predict the performance of a PEMFC with a Pt/CNT-based CCL under various conditions. Figure 6 shows the I–V curves for the model PEMFC at various values of relative humidity of inlet gases.

A sharp drop in the PEMFC performance is observed under conditions of high humidity (Figure 6). This effect is possibly due to the flooding of the CLs, which is explained by the high saturation of the cathode with water because of its hydrophilicity. At lower gas humidity values (20%), current densities are shown to be quite high over a wide voltage range. The model calculations confirm that CNTs can be used to ensure the efficient operation of PEMFCs under low humidity conditions, as noted in [13].

3. Model Description

3.1. Calculation of Catalytic Layer Parameters

A random particle distribution model was developed for the calculation of the EASA of CNT-based platinum catalysts. The initial parameters in this model are the density of the CNT array (N_CNT), the CNT diameter and length (d_CNT, L_CNT), and the mass of platinum per unit area of the CL (M_Pt). A number of assumptions were made in developing the model:

• The CNTs are evenly distributed over the surface with a given density.
• The CNTs have a fixed diameter and length.
• Platinum particles are uniformly and randomly distributed over the CNT surface.
• The probabilities of adsorption of platinum particles at any of the CNT sites are equal.
• Platinum particles located at a distance of less than R_c agglomerate and form a larger particle.
• Nanoparticles have the smallest possible size, i.e., the critical size of the nucleus d_n.

To simplify the calculation of the particle distribution, the cylindrical surface of the CNT was broken down into smaller sections of length Δ_LCNT = L_CNT/N_c (N_c >> 1). Then, the cylindrical surface was unrolled and the obtained rectangular surface was arranged in correspondence with the former (Figure 7). To consider the closedness of the cylindrical surface and take into account the entire length of the CNT, the distance between the opposite sides of the rectangular region was taken as zero when calculating the distance between the particles. The model also assumes that the minimum distance between the particles (R_c) is set.

The calculation of the particle distribution included several steps. The first step consisted of the calculation of the number of nanoparticles of the specified minimum size d_n based on the CNT geometric parameters, platinum loading, and CNT array density according to the Equation:

$$N=\frac{{\mathrm{M}}_{\mathrm{Pt}}}{{\mathrm{N}}_{\mathrm{CNT}}\times 4/3\mathsf{\pi } \times {\left({\mathrm{d}}_{\mathrm{n}}/2\right)}^{3 }\times { \mathsf{\rho }}_{\mathrm{Pt}}},$$

(2)

where ρ_Pt is the density of platinum (ρ_Pt = 21.45 g/cm³). Then, random uniformly distributed values of particle coordinates were generated in the range corresponding to the height (Δ_LCNT) and width (length of the cylinder base, πd_CNT) of the region under consideration (Figure 7). The random numbers were generated using the Python 3.7 Random module. The next step was to calculate the distances between the centers of all pairs of particles taking into account the boundary conditions. If the distance to the particle from the given one was found to be less than R_c, the particle was removed from the surface (Figure 8).

The next step was to increase the mass of the randomly chosen particles. The number of additions of a fixed mass equal to the mass of the nucleus equaled the number of particles screened out at the previous step. As a result, the distribution of particles over the surface and their sizes were obtained. Then, the distances between the particles were recalculated, considering the increase in their size, and particles with a distance of less than R_c from the current particle were removed in a similar way. The rejection condition has the following general form:

$$\sqrt{{\left({\mathrm{x}}_1 - {\mathrm{x}}_2\right)}^2+{\left({\mathrm{y}}_1-{\mathrm{y}}_2\right)}^2}\le \left({\mathrm{d}}_1+{\mathrm{d}}_2\right)/2+{\mathrm{R}}_{\mathrm{c}},$$

(3)

where x₁, y₁, d₁, x₂, y₂, and d₂ are the coordinates and sizes of the first and second particles, respectively. With further increases in the sizes of the particles, the added mass was not fixed and was equal to the mass of the removed particle. The process of increasing the mass, calculating the distance between the particles, and rejecting the particles was continued until all the particles were at a distance greater than R_c. The result of the calculation is the distribution of particles over the cylindrical surface and their sizes (Figure 9).

The EASA was calculated on the basis of the obtained distribution of particles according to the equation:

$$\mathrm{EASA}=\frac{3\mathsf{\pi } \times {\mathrm{N}}_{\mathrm{CNT}} \times {{\displaystyle \sum }}_{\mathrm{i}=1}^{{\mathrm{N}}_0}{{\mathrm{d}}_{\mathrm{i}}}^2}{4\times { \mathrm{M}}_{\mathrm{Pt}}},$$

(4)

where ¾ is the particle shape parameter, N₀ is the final number of particles, and d_i is the diameter of the ith particle. The calculation of the particle size distribution was performed multiple times to obtain negligible error values of the EASA.

The model under consideration assumes an empirical determination of the minimum particle size d_n and minimum distance R_c.

3.2. PEMFC Performance Calculation

For the simulation of the I–V curves of the PEMFC with the CLs under consideration, a one-dimensional model was used, considering the processes of transfer of mass (liquid water), charge, and gas phase components (steam, oxygen, hydrogen) in the PEMFC layers. The model was implemented in the Python programming language (version 3.7, Python Software Foundation, 2018).

According to the model, the PEMFC includes four layers: cathode and anode gas diffusion layers (CGDL and AGDL), PEM, and cathode CL (CCL). The contribution of the anode CL (ACL) to the operation of the PEMFC is not considered in this model due to the high rate of hydrogen oxidation reactions and the thinness of the layer. CCLs are considered hydrophilic since they are based on hydrophilic CNTs. Hydrophobic carbon paper is considered an AGDL and CGDL. The initial model parameters of the PEMFC layers and their values are shown in

Table 3. Initial parameters of the PEMFC model with CNT-based CCL.

Parameters	Values
Cell temperature	30 °C
Gas pressure on the anode and cathode sides	1 bar
AGDL thickness	420 μm
PEM thickness	50 μm
CCL thickness	10 μm
GDL porosity	0.8
CCL porosity	0.8
GDL electrical resistivity	1.6 μΩ m2
CCL platinum loading	0.3 mg·cm−2

.

Model assumptions:

• The coefficient of water diffusion in the membrane does not depend on the water concentration (when the water content of the membrane is greater than five water molecules per SO₃⁻ group);
• The model is not applicable to low humidity conditions (when the water content of the membrane is less than five water molecules per SO₃⁻ group);
• There is no liquid water in the anode CL;
• The mass and charge transfer processes in the ACL do not significantly affect the PEMFC performance and are not considered in the model;
• The water evaporation and condensation rates are infinite;
• The processes of transfer of momentum, mass, and gas mixture components take place under isothermal conditions;
• The electrical resistivity of the CLs is negligible compared to the resistivity of the GDL;
• The rate of electrochemical reactions in the CCL is proportional to its water saturation (because water saturation determines the efficiency of the proton conductivity of the membrane);
• The cell temperature is constant and the same for all layers.

The transport of the gaseous medium components is described by the Stefan–Maxwell equations:

$$\{\begin{array}{c}{\mathrm{C}}_{\mathrm{g}}\frac{{\mathrm{dC}}_{1\mathrm{g}}}{\mathrm{dx}}=\frac{{\mathrm{C}}_{1\mathrm{g}}{\mathrm{N}}_{2\mathrm{g}}-{\mathrm{C}}_{2\mathrm{g}}{\mathrm{N}}_{1\mathrm{g}}}{{\mathrm{D}}_{12}}-\frac{{\mathrm{C}}_{\mathrm{g}}-{\mathrm{C}}_{1\mathrm{g}}-{\mathrm{C}}_{2\mathrm{g}}}{{\mathrm{D}}_{13}}{\mathrm{N}}_{1\mathrm{g}} \\ {\mathrm{C}}_{\mathrm{g}}\frac{{\mathrm{dC}}_{21\mathrm{g}}}{\mathrm{dx}}=\frac{{\mathrm{C}}_{21\mathrm{g}}{\mathrm{N}}_{1\mathrm{g}}-{\mathrm{C}}_{1\mathrm{g}}{\mathrm{N}}_{2\mathrm{g}}}{{\mathrm{D}}_{12}}-\frac{{\mathrm{C}}_{\mathrm{g}}-{\mathrm{C}}_{1\mathrm{g}}-{\mathrm{C}}_{2\mathrm{g}}}{{\mathrm{D}}_{23}}{\mathrm{N}}_{2\mathrm{g}}\end{array},$$

(5)

where C_ig are the concentrations of individual components of the gas phase, C_g is the total density of the gas phase, N_ig are the fluxes of individual components of the gas phase, and D_ij are the coefficients of mutual diffusion of the ‘i’ and ’j’ components, which depend on the porosity, saturation, and temperature [35].

The stationary distribution of water concentration in the membrane is described by the diffusion equation, which in this case is converted to a first-order differential equation:

$$-{\mathrm{D}}_{\mathrm{m}}\frac{{\mathsf{\rho }}_{\mathrm{mdry}}}{{\mathrm{M}}_{\mathrm{mdry}}}\frac{\mathrm{d}\mathsf{\lambda }}{\mathrm{dx}}+{\mathrm{n}}_{\mathrm{p}}\frac{{\mathrm{I}}_0}{\mathrm{F}}\frac{\mathsf{\lambda }}{{\mathsf{\lambda }}_{\max }}=\mathrm{const}=\frac{{\mathrm{I}}_0}{\mathrm{F}}{\mathrm{q}}_{\mathrm{wm}},$$

(6)

where λ is the water content of the membrane (the number of water molecules per ion-exchange center), D_m is the diffusion coefficient of water in the membrane, which depends on the temperature being represented by ${\mathrm{e}}^{-1/\mathrm{T}}$, ρ_mdry is the dry membrane density, M_mdry is the dry membrane equivalent mass, q_wmI₀/F is the water flux, and n_p is the electro-osmotic drag coefficient [36].

The distribution of potentials in the model PEMFC is subject to the equation:

$${\mathsf{\epsilon }}_{\mathrm{eq}}={\mathrm{U}}_{\mathrm{out}}+{\mathrm{I}}_0\left({\mathrm{R}}_{\mathrm{mem}}+{\mathrm{R}}_{\mathrm{ga}}+{\mathrm{R}}_{\mathrm{gc}}\right)+{\mathsf{\eta }}_{\mathrm{c}}\left(\mathrm{x}={\mathrm{h}}_{\mathrm{cc}}\right),$$

(7)

where ε_eq is the sum of the cathode and anode equilibrium potentials, U_out is the resulting output voltage of the model PEMFC, R_mem is the ohmic resistance of the membrane, R_ga and R_gc are the resistance of the anode and cathode GDL, respectively, η_c are the activation losses of the cathode electrochemical reaction, and h_cc is the CCL thickness.

The formation of water as a result of the cathodic reaction is described by the equation:

$$\frac{1}{\mathrm{F}}{\mathrm{i}}_{\mathrm{ex}0}\sin \mathrm{h}\left(\frac{{\mathsf{\eta }}_{\mathrm{c}}}{{\varphi }_{\mathrm{actc}}}\right){\mathrm{C}}_{\mathrm{Og}}\left(1-{\mathrm{s}}_{\mathrm{C}}\right),$$

(8)

where i_ex0 is the effective value of the exchange current of the oxygen reduction reaction, ϕ_actc is the Tafel constant for the reaction, which is directly proportional to the cell temperature, s_c is the CCL saturation, and C_Og is the oxygen concentration. The effective value of the exchange current also takes into account the catalyst surface area equal to the product of EASA by the specific platinum content.

4. Conclusions

To estimate the effect of geometric parameters of a CNT-based support under given synthesis conditions on the resulting characteristics of the CL (platinum particle size, EASA), a semi-empirical model was proposed using a random uniform distribution of particles over the CNT surface. Based on the obtained distribution, the EASA of the electrocatalyst was estimated and further used to evaluate the performance of the CL as part of the PEMFC.

The applicability of the proposed model for calculating the CL parameters and I–V curves of the model PEMFC was shown. The accuracy of the proposed simulation of the CNT-based CLs did not depend on the electrocatalyst synthesis method used. An increase in the EASA with decreasing platinum loading of the CL was shown for the CNT-based electrocatalysts. When calculating the I–V curves of the model PEMFC, it was found that the properties of the CNT-based CCLs could be successfully used to improve the PEMFC performance efficiency under conditions of low humidity of inlet gases. The results of the proposed model simulations highly correlate with previously published experimental data.