Computational Study of H₂ Catalytic Combustion on Pd₃₈ Cluster Model and Pd(111) Slab Model

Abstract

Hydrogen is one of the exhaust gases produced by nuclear power stations. Due to the potential danger of incomplete combustion and the emission of hydrogen, hydrogen catalytic combustion is introduced to ensure the safety of nuclear power stations. Palladium is a widely used catalyst for hydrogen catalytic combustion. H₂ catalytic combustion on a Pd(111) slab model and Pd₃₈ cluster model were simulated using density functional theory (DFT), in order to analyze the H₂ oxidation mechanism on the catalyst surface.

1. Introduction

Hydrogen, H₂, is one of the exhaust gases produced by nuclear power stations. In nuclear power stations, large amounts of hydrogen are released into the containment vessel after a coolant loss. H₂ is a flammable and explosive gas. An H₂ concentration at 4–75.6% (volume concentration) will trigger an explosion when encountering a fire source, which would result in a serious accident. Therefore, the hydrogen produced by a nuclear power station needs to be treated in a timely manner. Many methods of hydrogen elimination have been applied in nuclear power station. H₂ catalytic combustion can reduce the H₂ concentration, to improve the safety performance of nuclear power stations. One advantage of H₂ catalytic combustion is that it can eliminate hydrogen at low concentrations. Another is that H₂ catalytic combustion can function at room temperature.

H₂ catalytic combustion is a flameless combustion. As early as 1970, Pfefferle studied catalytic combustion [1]. Compared to conventional combustion, catalytic combustion has a higher efficiency, less pollutants, and it is easier to control the combustion temperature. H₂ catalytic combustion was first studied by Sharer [2]. Subsequently, Kramer et al. studied low-temperature H₂ catalytic combustion on a Pd catalyst [3]. Zhang et al. reported the activity of a Pt catalyst for H₂ low-temperature catalytic combustion [4]. From then on, many works on H₂ catalytic combustion have been studied by researchers [4,5,6]. So far, Pt [7,8,9,10,11,12,13,14] and Pd [15,16,17] catalysts have been commonly applied for H₂ catalytic combustion. The catalytic reaction of H₂ and O₂ is a basic catalytic oxidation reaction. It is of great significance to select an appropriate catalyst for a heterogeneous catalytic combustion, by studying the adsorption mechanism of reactant molecules on the catalyst surface, especially the reaction rate of reactants on the catalytic sites and the adsorption/desorption effect of the catalytic surface [18].

With the development of modern computers, computational chemistry has made a great deal of progress. Sui et al. studied the H₂ catalytic oxidation reaction mechanism on a PdO surface using density functional theory (DFT) [19]. H₂ firstly dissociates into adsorbed H on the PdO surface, with an activation energy of 58.6 kJ/mol. Then, the H atom reacts with the O atom in PdO to form OH. Finally, two OH react to form H₂O and O, with an activation energy of 58.9 kJ/mol. Qi et al. investigated H₂ catalytic combustion on Pt(111) surfaces [18]. At first, H₂ and O₂ dissociate into adsorbed H and O on the Pt surface, respectively. Then, H and O react to form OH. Finally, OH reacts with H to form H₂O. At present, few literature works have systematically studied the mechanism of H₂ catalytic combustion using DFT. It is necessary to study the reaction mechanism of H₂ catalytic combustion on a Pd metal surface, which is helpful to guide the selection of a catalyst.

In many previous works, the researchers found that the size of particles played an important role in the catalytic system. Tsamis’ group studied the efficiency of Pd-doped porous silicon as a catalyst for hydrogen dissociation [20]. It was found that the catalytic activity of Pd-doped porous silicon was significantly higher than that of a planar surface covered with Pd. Abate [21] studied the effect of the size of Pd particles in H₂O₂ synthesis. They found that smaller particles of Pd showed a higher rate of combustion. In Huang’s work [22], they studied the influence of the size of Pd particles on Heck reactions. Their work showed that the size of the supported Pd particles were smaller, and the catalytic activity was higher. In this work, we wanted to investigate the effect of particle size on H₂ catalytic combustion using DFT calculation. A Pd₃₈ cluster model simulated the small Pd particles and a Pd(111) slab model simulated the large particles in the catalytic system. Comparing the reaction energy barrier of H₂ catalytic combustion on the surface of the two models, the influence of model morphology on the catalytic activity was studied. This method can guide the design of a catalyst rationally, through theoretical calculation, so that catalysts can be designed and prepared more efficiently.

2. Computing Method

The DMol3 module of Materials Studio software was applied for DFT calculation of H₂ catalytic combustion [23]. In this work, the exchange-correlation energy was calculated using the Perdew–Burke–Ernzerf (PBE) function with generalized gradient approximation (GGA). The wave functions were calculated using a localized double-numerical quality basis set with a polarization d-function (DNP). The effective core potential (ECP) was applied to represent the core electrons of the Pd metal atoms. The k-point mesh was set to 3 × 3 × 1 using Monkhorst–Pack with a thermal smearing of 0.005 hartree. In the configuration optimization, the convergences of the energy, maximum force, and maximum displacement were set as 2.0 × 10⁻⁵ Ha, 0.004 Ha/Å, and 0.005 Å, respectively. The van der Waals (vdW) interactions were performed using the empirical correction scheme of Grimme (DFT-D).

The transition states (TS) of the elementary reactions used a complete linear synchronous transit and quadratic synchronous transit (LST/QST) approach. The LST method applied a single interpolation to the maximum energy. The QST method searched for an energy maximum with constrained minimizations, to refine the transition states involved in the reaction. In addition, the transition states were optimized by imaginary frequency calculations, in order to confirm that every transition state led to the desired reactants and products [24,25].

The adsorption energy (E_ads) for adsorbate on Pd surfaces was calculated as follows:

$$E_{ads}=E_{A/Pd}-E_{Pd}-E_{A(g)}$$

where E_A/Pd is the total energy of Pd model with an adsorbate; E_Pd is the energy of the Pd model; E_A(g) is the energy of the adsorbate in gas-phase.

The activation energy (E_a) and reaction energy (ΔE):

$$\begin{gathered} E_a=E_{TS}-E_{IS} \\ \Delta E=E_{FS}-E_{IS} \end{gathered}$$

E_TS is the free energy of the transition state, E_IS is the free energy of the initial state, and E_FS is the free energy of final state.

3. Results and Discussion

3.1. Pd Catalyst Model

Based on our previous work [26], two models of Pd catalyst, a Pd₃₈ cluster model and a Pd(111) slab model, were built. The calculated lattice constant of bulk Pd was 3.90 Å, which is in good agreement with the previous experimental value of 3.62 Å [27]. In Figure 1a, a Pd(111) slab model was constructed as a four-layer symmetric periodic slab. The bottom two layers of the slab were fixed at their original atomic positions in the bulk, while the uppermost two layers and the adsorbates were allowed to relax during structural optimization. The crystal face of the Pd slab surface was a Pd(111) surface, and the vacuum thickness of 12 Å between clean slabs was established, to separate the interactions between the periodically repeating slabs using a 3 × 3 surface unit cell. In Figure 1b, the Pd₃₈ cluster was optimized to model Pd particles because the cluster model is more similar to Pd particles than the Pd slab. The Pd₃₈ cluster with 14 surfaces was constituted by 38 Pd atoms. The potential adsorption sites are marked in the two models.

3.2. Reactants and Intermediates on the Pd₃₈ Cluster Model and Pd(111) Slab Model

The adsorption behaviors of H₂ and its reaction intermediates in the two Pd models were investigated, to study the H₂ catalytic combustion mechanism. The adsorption geometry of reactants, intermediates, and products in the Pd₃₈ cluster model and Pd(111) slab model are shown in Figure 2 and Figure 3, respectively. Table 1 and Table 2 list the corresponding adsorption energies of the reactants, intermediates, and products in the Pd₃₈ cluster model and Pd(111) slab model, respectively.

The H₂ adsorption in the Pd(111) slab model exhibits four structures in Figure 2a(1)–a(4). In Figure 2a(1), the H₂ adsorbed on the top site of the Pd(111) slab model, marked as H₂-top, with an adsorption energy of −1.31 eV. The lengths of H-Pd bonds were 1.762 Å and 1.760 Å, respectively. In Figure 2a(2), H₂ adsorbs on the bridge site of the Pd(111) slab model, marked as the H₂-top, with an adsorption energy of −1.43 eV. One H atom adsorbs on a Pd atom and anther H atom adsorbs on the adjacent Pd atom. The lengths of H-Pd bonds were 1.788 Å and 1.792 Å, respectively. In Figure 2a(3), H₂ adsorbs on the fcc site of the Pd(111) slab model, marked as H₂-fcc. The corresponding adsorption energy was −1.66 eV. The lengths of the H-Pd bonds were 1.798 Å, 1.799 Å, and 1.799 Å, respectively. In Figure 2a(4), the adsorption structure is marked as H₂-hcp, where H₂ adsorbs on the fcc site of the Pd(111) slab model. The corresponding adsorption energy was −1.62 eV, which is close to that of H₂-fcc. The lengths of the H-Pd bonds were 1.802 Å, 1.801 Å, and 1.801 Å, respectively. Compared to the four adsorption structures, the adsorption energies were from 1.31 eV to 1.66 eV. This indicates that the four adsorption structures were stable. The adsorption energy of H₂-fcc was lowest of the four adsorption structures, which indicates that H₂ prefers to adsorb on the fcc site of the Pd(111) surface. Figure 2b(1)–b(4) shows the adsorption structures of O₂ on the Pd(111) surface. Similarly to the H₂ adsorption on the Pd(111) surface, there were also four adsorption structures of O₂ on the Pd(111) surface. O₂ can also adsorb on the top, bridge, fcc, and hcp sites; marked as O₂-top, O₂-bridge, O₂-fcc, and O₂-hcp, respectively. The corresponding adsorption energies were −0.68 eV, −0.77 eV, −0.98 eV, and −0.95 eV, respectively. It can be seen that O₂-fcc was the most stable adsorption structure, with the lowest adsorption energy. Figure 2c(1)–c(4) shows the H adsorption structures on the Pd(111) surface. H adsorbs on the top, bridge, fcc, and hcp sites, with adsorption energies of −2.08 eV, −2.49 eV, −2.57 eV, and −2.54 eV, respectively. H-fcc is the most stable adsorption structure, with the lowest adsorption energy of −2.57 eV. The O adsorption structures on the top, bridge, fcc, and hcp site are shown in Figure 2d(1)–d(4). The corresponding adsorption energies of the four structures were −2.37 eV, −3.06 eV, −3.25 eV, and −2.96 eV, respectively. The OH adsorption structures on the top, bridge, fcc, and hcp site are shown in Figure 2e(1)–e(4). The corresponding adsorption energies of the four structures were −1.03 eV, −1.39 eV, −1.40 eV, and −1.33 eV, respectively. In Figure 2f(1),f(2), H₂O adsorbs on the top and bridge site of the Pd(111) surface, with adsorption energies of −0.36 eV and −0.59 eV, respectively. According to the calculation results, the fcc site on the Pd(111) surface is beneficial for adsorption.

Figure 2. Adsorption configurations of all intermediates involved in the H₂ combustion reaction on the Pd(111) surface. (a(1)) H₂-top adsorption; (a(2)) H₂-bridge adsorption; (a(3)) H₂-fcc adsorption; (a(4)) H₂-hcp adsorption; (b(1)) O₂-top adsorption; (b(2)) O₂-bridge adsorption; (b(3)) O₂-fcc adsorption; (b(4)) O₂-hcp adsorption; (c(1)) H-top adsorption; (c(2)) H-bridge adsorption; (c(3)) H-fcc adsorption; (c(4)) H-hcp adsorption; (d(1)) O-top adsorption; (d(2)) O-bridge adsorption; (d(3)) O-fcc adsorption; (d(4)) O-hcp adsorption; (e(1)) OH-top adsorption; (e(2)) OH-bridge adsorption; (e(3)) OH-fcc adsorption; (e(4)) OH-hcp adsorption; (f(1)) H₂O-top adsorption; (f(2)) H₂O-bridge adsorption; Here, O, H, and Pd atoms are shown as red, white and blue balls, respectively, which are the same in Figure 2, Figure 3, Figure 4 and Figure 5.

Figure 2. Adsorption configurations of all intermediates involved in the H₂ combustion reaction on the Pd(111) surface. (a(1)) H₂-top adsorption; (a(2)) H₂-bridge adsorption; (a(3)) H₂-fcc adsorption; (a(4)) H₂-hcp adsorption; (b(1)) O₂-top adsorption; (b(2)) O₂-bridge adsorption; (b(3)) O₂-fcc adsorption; (b(4)) O₂-hcp adsorption; (c(1)) H-top adsorption; (c(2)) H-bridge adsorption; (c(3)) H-fcc adsorption; (c(4)) H-hcp adsorption; (d(1)) O-top adsorption; (d(2)) O-bridge adsorption; (d(3)) O-fcc adsorption; (d(4)) O-hcp adsorption; (e(1)) OH-top adsorption; (e(2)) OH-bridge adsorption; (e(3)) OH-fcc adsorption; (e(4)) OH-hcp adsorption; (f(1)) H₂O-top adsorption; (f(2)) H₂O-bridge adsorption; Here, O, H, and Pd atoms are shown as red, white and blue balls, respectively, which are the same in Figure 2, Figure 3, Figure 4 and Figure 5.

Figure 3 shows the adsorption structures of adsorbates in the Pd₃₈ cluster model. H₂ adsorbs on the Pd₃₈ cluster model shown in Figure 3a(1)–a(5). In Figure 3a(1)–a(4), H₂ adsorption on the (111) surface of the Pd₃₈ cluster is similar to that on the Pd(111) slab model. H₂ adsorbs on the top, bridge, fcc, and hcp site, with adsorption energies of −1.34 eV, −1.55 eV, −1.76 eV, and −1.73 eV, respectively. Compared to that on the Pd(111) slab model, the adsorption energies of H₂ on the Pd₃₈ cluster model are lower than that of H₂ on the Pd(111) slab model. This indicates that H₂ prefers to adsorb on the Pd₃₈ cluster model instead of the Pd(111) surface. In Figure 3a(5), H₂ adsorbs on the edge site of the Pd₃₈ cluster model, with an adsorption energy of −0.92 eV. The adsorption structures of O₂ on the Pd₃₈ cluster are shown in Figure 3b(1)–b(5). The adsorption energies of O₂ on the top, bridge, fcc, hcp, and edge site of the Pd₃₈ cluster were −0.83 eV, −0.95 eV, −1.06 eV, −1.02 eV, and −0.89 eV, respectively. Figure 3c(1)–c(5) shows the adsorption structures of the H atom on the top, bridge, fcc, hcp, and edge site of the Pd₃₈ cluster, with adsorption energies of −1.14 eV, −1.38 eV, −1.57 eV, −1.54 eV, and −1.11 eV, respectively. Figure 3d(1)–d(5) shows the adsorption structures of the O atom on the top, bridge, fcc, hcp, and edge site of the Pd₃₈ cluster, with adsorption energies of −2.45 eV, −2.59 eV, −3.62 eV, −3.56 eV, and −2.45 eV, respectively. The adsorption structures of OH on the top, bridge, fcc, hcp, and edge site of the Pd₃₈ cluster are shown in Figure 3e(1)–e(5). The corresponding adsorption energies were −1.57 eV, −2.51 eV, −2.46 eV, −2.41 eV, and −0.61 eV, respectively. There are three adsorption structures of H₂O on the top, bridge, and edge site of the Pd₃₈ cluster. The corresponding adsorption energies were −0.36 eV, −0.59 eV, and −0.44 eV. According to the adsorption energies, adsorbates prefer to adsorb on the Pd₃₈ cluster model than on the Pd(111) slab model. As the Pd₃₈ cluster model is a better electron donor, the adsorption of the adsorbate is more stable [26].

Figure 3. Adsorption configurations of all adsorbates involved in the H₂ combustion reaction on the Pd₃₈ cluster surface. (a(1)) H₂-top adsorption; (a(2)) H₂-bridge adsorption; (a(3)) H₂-fcc adsorption; (a(4)) H₂-hcp adsorption; (a(5)) H₂-edge adsorption; (b(1)) O₂-top adsorption; (b(2)) O₂-bridge adsorption; (b(3)) O₂-fcc adsorption; (b(4)) O₂-hcp adsorption; (b(5)) O₂-edge adsorption; (c(1)) H-top adsorption; (c(2)) H-bridge adsorption; (c(3)) H-fcc adsorption; (c(4)) H-hcp adsorption; (c(5)) H-edge adsorption; (d(1)) O-top adsorption; (d(2)) O-bridge adsorption; (d(3)) O-fcc adsorption; (d(4)) O-hcp adsorption; (d(5)) O-edge adsorption; (e(1)) OH-top adsorption; (e(2)) OH-bridge adsorption; (e(3)) OH-fcc adsorption; (e(4)) OH-hcp adsorption; (e(5)) OH-edge adsorption; (f(1)) H₂O-top adsorption; (f(2)) H₂O-bridge adsorption; (f(3)) H₂O-edge adsorption.

Figure 3. Adsorption configurations of all adsorbates involved in the H₂ combustion reaction on the Pd₃₈ cluster surface. (a(1)) H₂-top adsorption; (a(2)) H₂-bridge adsorption; (a(3)) H₂-fcc adsorption; (a(4)) H₂-hcp adsorption; (a(5)) H₂-edge adsorption; (b(1)) O₂-top adsorption; (b(2)) O₂-bridge adsorption; (b(3)) O₂-fcc adsorption; (b(4)) O₂-hcp adsorption; (b(5)) O₂-edge adsorption; (c(1)) H-top adsorption; (c(2)) H-bridge adsorption; (c(3)) H-fcc adsorption; (c(4)) H-hcp adsorption; (c(5)) H-edge adsorption; (d(1)) O-top adsorption; (d(2)) O-bridge adsorption; (d(3)) O-fcc adsorption; (d(4)) O-hcp adsorption; (d(5)) O-edge adsorption; (e(1)) OH-top adsorption; (e(2)) OH-bridge adsorption; (e(3)) OH-fcc adsorption; (e(4)) OH-hcp adsorption; (e(5)) OH-edge adsorption; (f(1)) H₂O-top adsorption; (f(2)) H₂O-bridge adsorption; (f(3)) H₂O-edge adsorption.

Figure 4. Elementary steps of H₂ oxidation on the Pd(111) surface: (a(1),a(2)) H₂ dissociation; (b(1),b(2)) O₂ dissociation; (c(1),c(2)) OH formation; (d(1)) H₂O formation.

Figure 4. Elementary steps of H₂ oxidation on the Pd(111) surface: (a(1),a(2)) H₂ dissociation; (b(1),b(2)) O₂ dissociation; (c(1),c(2)) OH formation; (d(1)) H₂O formation.

Figure 5. Elementary steps of H₂ oxidation on the Pd₃₈ surface: (a(3),a(4),a(5)) H₂ dissociation; (b(3),b(4),b(5)) O₂ dissociation; (c(3),c(4),c(5)) OH formation; (d(3),d(4)) H₂O formation.

Figure 5. Elementary steps of H₂ oxidation on the Pd₃₈ surface: (a(3),a(4),a(5)) H₂ dissociation; (b(3),b(4),b(5)) O₂ dissociation; (c(3),c(4),c(5)) OH formation; (d(3),d(4)) H₂O formation.

Table 1. Geometric parameters and adsorption energies of all adsorbates involved in H₂ combustion on the Pd(111) slab surface.

Figures	Name	Adsorption Energy/eV	Geometric Parameter/Å
Figure 2a(1)	H2-top	−1.31	Pd1-H1:1.762; Pd1-H2:1.760
Figure 2a(2)	H2-bridge	−1.43	Pd1-H1:1.788; Pd2-H2:1.792
Figure 2a(3)	H2-fcc	−1.66	Pd1-H1:1.798; Pd2-H1:1.799; Pd3-H1:1.799
Figure 2a(4)	H2-hcp	−1.62	Pd1-H1:1.802; Pd2-H1:1.801; Pd3-H1:1.801
Figure 2b(1)	O2-top	−0.68	Pd1-O1:1.964; Pd1-O 2:1.965
Figure 2b(2)	O2-bridge	−0.77	Pd1-O1:2.019; Pd2-O 2:2.017
Figure 2b(3)	O2-fcc	−0.98	Pd1-O 1:1.972; Pd2-O 1:1.970; Pd3-O 1:1.970
Figure 2b(4)	O2-hcp	−0.95	Pd1-O 1:1.993; Pd2-O1:1.990; Pd3-O1:1.991
Figure 2c(1)	H-top	−2.08	Pd1-H:1.646
Figure 2c(2)	H-bridge	−2.49	Pd1-H:1.736; Pd2-H:1.734
Figure 2c(3)	H-fcc	−2.57	Pd1-H:1.759; Pd2-H:1.760; Pd3-H:1.760
Figure 2c(4)	H-hcp	−2.54	Pd1-H:1.768; Pd2-H:1.768; Pd3-H:1.769
Figure 2d(1)	O-top	−2.37	Pd1-O:1.938
Figure 2d(2)	O-bridge	−3.06	Pd1-O:1.998; Pd2-O:1.998
Figure 2d(3)	O-fcc	−3.25	Pd1-O:2.034; Pd2-H:2.035; Pd3-O:2.035
Figure 2d(4)	O-hcp	−2.96	Pd1-O:2.079; Pd2-O:2.079; Pd3-O:2.078
Figure 2e(1)	OH-top	−1.03	Pd1-O:1.957
Figure 2e(2)	OH-bridge	−1.39	Pd1-O:2.016; Pd2-O:2.016
Figure 2e(3)	OH-fcc	−1.40	Pd1-O:2.055; Pd2-H:2.056; Pd3-O:2.055
Figure 2e(4)	OH-hcp	−1.33	Pd1-O:2.092; Pd2-O:2.093; Pd3-O:2.093
Figure 2f(1)	H2O-top	−0.36	Pd1-O:2.105
Figure 2f(2)	H2O-bridge	−0.59	Pd1-H1:1.887; Pd2-H2:1.886

Table 1. Geometric parameters and adsorption energies of all adsorbates involved in H₂ combustion on the Pd(111) slab surface.

Figures	Name	Adsorption Energy/eV	Geometric Parameter/Å
Figure 2a(1)	H2-top	−1.31	Pd1-H1:1.762; Pd1-H2:1.760
Figure 2a(2)	H2-bridge	−1.43	Pd1-H1:1.788; Pd2-H2:1.792
Figure 2a(3)	H2-fcc	−1.66	Pd1-H1:1.798; Pd2-H1:1.799; Pd3-H1:1.799
Figure 2a(4)	H2-hcp	−1.62	Pd1-H1:1.802; Pd2-H1:1.801; Pd3-H1:1.801
Figure 2b(1)	O2-top	−0.68	Pd1-O1:1.964; Pd1-O 2:1.965
Figure 2b(2)	O2-bridge	−0.77	Pd1-O1:2.019; Pd2-O 2:2.017
Figure 2b(3)	O2-fcc	−0.98	Pd1-O 1:1.972; Pd2-O 1:1.970; Pd3-O 1:1.970
Figure 2b(4)	O2-hcp	−0.95	Pd1-O 1:1.993; Pd2-O1:1.990; Pd3-O1:1.991
Figure 2c(1)	H-top	−2.08	Pd1-H:1.646
Figure 2c(2)	H-bridge	−2.49	Pd1-H:1.736; Pd2-H:1.734
Figure 2c(3)	H-fcc	−2.57	Pd1-H:1.759; Pd2-H:1.760; Pd3-H:1.760
Figure 2c(4)	H-hcp	−2.54	Pd1-H:1.768; Pd2-H:1.768; Pd3-H:1.769
Figure 2d(1)	O-top	−2.37	Pd1-O:1.938
Figure 2d(2)	O-bridge	−3.06	Pd1-O:1.998; Pd2-O:1.998
Figure 2d(3)	O-fcc	−3.25	Pd1-O:2.034; Pd2-H:2.035; Pd3-O:2.035
Figure 2d(4)	O-hcp	−2.96	Pd1-O:2.079; Pd2-O:2.079; Pd3-O:2.078
Figure 2e(1)	OH-top	−1.03	Pd1-O:1.957
Figure 2e(2)	OH-bridge	−1.39	Pd1-O:2.016; Pd2-O:2.016
Figure 2e(3)	OH-fcc	−1.40	Pd1-O:2.055; Pd2-H:2.056; Pd3-O:2.055
Figure 2e(4)	OH-hcp	−1.33	Pd1-O:2.092; Pd2-O:2.093; Pd3-O:2.093
Figure 2f(1)	H2O-top	−0.36	Pd1-O:2.105
Figure 2f(2)	H2O-bridge	−0.59	Pd1-H1:1.887; Pd2-H2:1.886

Table 2. Geometric parameters and adsorption energies of all adsorbates involved in H₂ combustion on the Pd₃₈ cluster surface.

Figures	Name	Adsorption Energy/eV	Geometric Parameter/Å
Figure 3a(1)	H2-top	−1.34	Pd1-H1:1.733; Pd1-H2:1.732
Figure 3a(2)	H2-bridge	−1.55	Pd1-H1:1.745; Pd2-H2:1.752
Figure 3a(3)	H2-fcc	−1.76	Pd1-H1:1.756; Pd2-H1:1.768; Pd3-H1:1.769
Figure 3a(4)	H2-hcp	−1.73	Pd1-H1:1.781; Pd2-H1:1.794; Pd3-H1:1.795
Figure 3a(5)	H2-edge	−0.92	Pd1-H1:1.881; Pd2-H2:1.882
Figure 3b(1)	O2-top	−0.83	Pd1-O1:1.968; Pd1-O2:1.968
Figure 3b(2)	O2-bridge	−0.95	Pd1-O1:2.114; Pd2-O2:2.121
Figure 3b(3)	O2-fcc	−1.06	Pd1-O1:2.070; Pd2-O1:2.077; Pd3-O1:2.077
Figure 3b(4)	O2-hcp	−1.02	Pd1-O1:2.059; Pd2-O1:2.066; Pd3-O1:2.067
Figure 3b(5)	O2-edge	−0.89	Pd1-O1:1.993; Pd2-O2:1.994
Figure 3c(1)	H-top	−1.14	Pd1-H:1.630
Figure 3c(2)	H-bridge	−1.38	Pd1-H:1.724; Pd2-H:1.729
Figure 3c(3)	H-fcc	−1.57	Pd1-H:1.738; Pd2-H:1.756; Pd3-H:1.754
Figure 3c(4)	H-hcp	−1.54	Pd1-H:1.749; Pd2-H:1.752; Pd3-H:1.750
Figure 3c(5)	H-edge	−1.11	Pd1-H:1.688
Figure 3d(1)	O-top	−2.45	Pd1-O:1.920
Figure 3d(2)	O-bridge	−2.59	Pd1-O:1.974; Pd2-O:1.979
Figure 3d(3)	O-fcc	−3.62	Pd1-O:2.011; Pd2-H:2.017; Pd3-O:2.018
Figure 3d(4)	O-hcp	−3.56	Pd1-O:2.062; Pd2-O:2.074; Pd3-O:2.073
Figure 3d(5)	O-edge	−1.73	Pd1-O:2.122
Figure 3e(1)	OH-top	−1.57	Pd1-O:1.924
Figure 3e(2)	OH-bridge	−2.51	Pd1-O:2.001; Pd2-O:2.009
Figure 3e(3)	OH-fcc	−2.46	Pd1-O:2.022; Pd2-H:2.028; Pd3-O:2.029
Figure 3e(4)	OH-hcp	−2.41	Pd1-O:2.069; Pd2-O:2.077; Pd3-O:2.078
Figure 3e(5)	OH-edge	−0.61	Pd1-O:2.143
Figure 3f(1)	H2O-top	−0.36	Pd1-O:2.075
Figure 3f(2)	H2O-bridge	−0.59	Pd1-H1:1.869; Pd2-H2:1.869
Figure 3f(3)	H2O-edge	−0.44	Pd1-H1:1.869; Pd2-H2:1.869

Table 2. Geometric parameters and adsorption energies of all adsorbates involved in H₂ combustion on the Pd₃₈ cluster surface.

Figures	Name	Adsorption Energy/eV	Geometric Parameter/Å
Figure 3a(1)	H2-top	−1.34	Pd1-H1:1.733; Pd1-H2:1.732
Figure 3a(2)	H2-bridge	−1.55	Pd1-H1:1.745; Pd2-H2:1.752
Figure 3a(3)	H2-fcc	−1.76	Pd1-H1:1.756; Pd2-H1:1.768; Pd3-H1:1.769
Figure 3a(4)	H2-hcp	−1.73	Pd1-H1:1.781; Pd2-H1:1.794; Pd3-H1:1.795
Figure 3a(5)	H2-edge	−0.92	Pd1-H1:1.881; Pd2-H2:1.882
Figure 3b(1)	O2-top	−0.83	Pd1-O1:1.968; Pd1-O2:1.968
Figure 3b(2)	O2-bridge	−0.95	Pd1-O1:2.114; Pd2-O2:2.121
Figure 3b(3)	O2-fcc	−1.06	Pd1-O1:2.070; Pd2-O1:2.077; Pd3-O1:2.077
Figure 3b(4)	O2-hcp	−1.02	Pd1-O1:2.059; Pd2-O1:2.066; Pd3-O1:2.067
Figure 3b(5)	O2-edge	−0.89	Pd1-O1:1.993; Pd2-O2:1.994
Figure 3c(1)	H-top	−1.14	Pd1-H:1.630
Figure 3c(2)	H-bridge	−1.38	Pd1-H:1.724; Pd2-H:1.729
Figure 3c(3)	H-fcc	−1.57	Pd1-H:1.738; Pd2-H:1.756; Pd3-H:1.754
Figure 3c(4)	H-hcp	−1.54	Pd1-H:1.749; Pd2-H:1.752; Pd3-H:1.750
Figure 3c(5)	H-edge	−1.11	Pd1-H:1.688
Figure 3d(1)	O-top	−2.45	Pd1-O:1.920
Figure 3d(2)	O-bridge	−2.59	Pd1-O:1.974; Pd2-O:1.979
Figure 3d(3)	O-fcc	−3.62	Pd1-O:2.011; Pd2-H:2.017; Pd3-O:2.018
Figure 3d(4)	O-hcp	−3.56	Pd1-O:2.062; Pd2-O:2.074; Pd3-O:2.073
Figure 3d(5)	O-edge	−1.73	Pd1-O:2.122
Figure 3e(1)	OH-top	−1.57	Pd1-O:1.924
Figure 3e(2)	OH-bridge	−2.51	Pd1-O:2.001; Pd2-O:2.009
Figure 3e(3)	OH-fcc	−2.46	Pd1-O:2.022; Pd2-H:2.028; Pd3-O:2.029
Figure 3e(4)	OH-hcp	−2.41	Pd1-O:2.069; Pd2-O:2.077; Pd3-O:2.078
Figure 3e(5)	OH-edge	−0.61	Pd1-O:2.143
Figure 3f(1)	H2O-top	−0.36	Pd1-O:2.075
Figure 3f(2)	H2O-bridge	−0.59	Pd1-H1:1.869; Pd2-H2:1.869
Figure 3f(3)	H2O-edge	−0.44	Pd1-H1:1.869; Pd2-H2:1.869

3.3. Reaction Mechanism of H₂ Catalytic Combustion on Pd₃₈ Cluster Model and Pd(111) Slab Model

The adsorption structure of H₂ catalytic combustion on the Pd(111) slab model and Pd₃₈ cluster model was optimized, in order to analyze the H₂ catalytic combustion mechanism on these two surface models. Figure 4 and Figure 5 show the H₂ catalytic combustion reaction on the Pd(111) slab model and Pd₃₈ cluster model, respectively. The corresponding activation energies are shown in Figure 6.

Table 3. Activation energies and reaction energies of the H₂ catalytic combustion on the Pd₃₈ cluster and Pd(111) slab.

Elementary Step	No.	Pd(111)/(eV)		Pd38/(eV)
		1	2	3	4	5
H2 → 2H	a	Ea: 0.45 Erxn: −0.12	Ea: 0.63 Erxn: −0.22	Ea: 0.34 Erxn: −0.23	Ea: 0.51 Erxn: −0.29	Ea: 0.21 Erxn: −0.32
O2 → 2O	b	Ea: 0.83 Erxn: −0.47	Ea: 0.94 Erxn: −0.32	Ea: 0.79 Erxn: −0.42	Ea: 0.82 Erxn: −0.35	Ea: 0.54 Erxn: −0.50
H + O → OH	c	Ea: 0.57 Erxn: −0.31	Ea: 0.62 Erxn: −0.19	Ea: 0.62 Erxn: −0.33	Ea: 0.51 Erxn: −0.24	Ea: 0.44 Erxn: −0.32
OH + H → H2O	d	Ea: 0.42 Erxn: −0.44		Ea: 0.62 Erxn: −0.33	Ea: 0.51 Erxn: −0.24

shows the activation energies and reaction energies of the elementary reaction steps.

The H₂ dissociation process on the Pd(111) surface is shown in Figure 4a(1). The H₂ is first adsorbed at the top site of the Pd(111) surface. As the reaction progresses, the length of the H-H bond is gradually elongated. Finally, the two H atoms gradually move to the hcp site on the Pd(111) surface. This elementary step is an exothermic reaction, with a reaction energy of −0.12 eV. The corresponding activation energy was 0.45 eV, which indicates that the elemental reaction proceeded relatively easily in both kinetics and thermodynamics. Another H₂ dissociation process on the Pd(111) surface is shown in Figure 4a(2). H₂ first adsorbs at the bridge site on the Pd(111) surface. The H-H bond length is gradually elongated as the reaction progresses. Then, two H atoms are gradually adsorbed to the top site of the two adjacent Pd atoms, respectively. The corresponding activation energy and reaction energy were 0.83 eV and −0.22 eV, respectively. The activation energy of this step was slightly higher than that in Figure 4a(1), which indicates that the H₂ dissociation process on the Pd(111) surface prefers to proceed in Figure 4a(2). Similar to the two steps of H₂ dissociation on the Pd(111) surface, two O₂ dissociation steps on the Pd(111) surface are shown in Figure 4 (b(1),b(2). The activation energies of the two O₂ dissociation steps were 0.63 eV and 0.94 eV, respectively. The corresponding reaction energies were −0.47 eV and −0.32 eV, respectively. The OH formation steps are shown in Figure 4c(1),c(2). In Figure 4c(1), H and O are adsorbed on the top site of the Pd(111) surface. As the reaction continues, O and H atoms become closer and bond to form OH. After the reaction, the OH adsorbs on the top site of the Pd(111) surface. The activation energy and reaction energy of the elemental step were 0.57 eV and −0.31 eV, respectively. In Figure 4c(2), the H atom adsorbs at the top site of the Pd(111) surface and the O atom adsorbs at the fcc site of Pd(111) surface. As the reaction progresses, the H atom gradually approaches the O atom to form OH, which adsorbs on the fcc site of the Pd(111) surface. The activation energy and reaction energy were 0.62 eV and −0.19 eV, respectively. Figure 4d(1) shows the elementary step of H₂O formation on the Pd(111) surface. OH adsorbs at the top site of Pd surface and H atom adsorbs at the adjacent fcc site. H atom bonds approaches the O atom in OH, to form H₂O, which adsorbs at the top site of the Pd(111) surface. The corresponding reaction activation energy and reaction energy were 0.42 eV and −0.44 eV.

The elementary steps of H₂ dissociation on the Pd₃₈ cluster surface are shown in Figure 5a(3)–a(5). The H₂ dissociation process in Figure 5a(3) is similar to that in Figure 4a(1). H₂ first adsorbs at the top site on the surface of the Pd₃₈ cluster. Then, two H atoms adsorb at the hcp site on the Pd₃₈ cluster surface after dissociation. The corresponding activation energy and reaction energy of the elemental step were 0.34 eV and −0.23 eV, respectively. The H₂ dissociation step in Figure 5a(4) is similar to that in Figure 4a(2), with activation and reaction energies of 0.51 eV and −0.29 eV, respectively. In Figure 5a(5), H₂ first adsorbs on the edge site of the Pd₃₈ cluster surface. Then, H-H is gradually elongated until breakage with the ongoing reaction. Finally, two H atoms dissociate and adsorb on the edge site of Pd₃₈ cluster. The activation energy and reaction energy of the elemental step were 0.21 eV and −0.32 eV, respectively. This elementary step in Figure 5a(5) has the lowest activation energy in the five elementary steps (a(1) to a(5) of H₂ dissociation. This indicates that H₂ dissociates more easily on the edge site of the Pd₃₈ cluster. The O₂ dissociation steps on the Pd₃₈ cluster surface are shown in Figure 5b(3)–b(5). The O₂ dissociation steps on the Pd₃₈ cluster are shown in Figure 5b(3),b(4) and are similar to those shown in Figure 4b(1),b(2). O₂ begins to dissociate from the top site on the Pd₃₈ cluster surface in Figure 5b(3), while O₂ dissociates from the bridge site on Pd₃₈ cluster surface in Figure 5b(4). The activation energy and reaction energy of the O₂ dissociation process in Figure 5b(3) are 0.79 eV and −0.42 eV, respectively. In Figure 5b(4), the activation energy and reaction energy of the O₂ dissociation process are 0.82 eV and −0.35 eV, respectively. Figure 5b(5) shows the O₂ dissociation step on the edge site of the Pd₃₈ cluster, in which two dissociated O atoms adsorb on the edge of the Pd₃₈ cluster. The corresponding activation energy and reaction energy were 0.54 eV and −0.50 eV, respectively. Figure 5c(3)–c(5) shows the OH formation process on the Pd₃₈ cluster. The OH formation process shown in Figure 5c(3),c(4) is similar to that shown in Figure 4c(1),c(2). As the reaction progresses, the O atom and H atom move closer to each other, to form OH, which vertically adsorbs on the fcc site and hcp site of the Pd₃₈ cluster in Figure 5c(3),c(4), respectively. The corresponding activation energy of OH formation on the Pd₃₈ cluster is lower than that on Pd(111). The OH formation step on the edge site of the Pd₃₈ cluster is shown in Figure 5c(5). The activation energy and reaction energy were 0.44 eV and −0.32 eV, respectively. Figure 5d(3),d(4) shows the H₂O formation step on the Pd₃₈ cluster. The formation process of H₂O in Figure 5d(3) is similar to that in Figure 4d(1). The activation energy and reaction energy were 0.62 eV and −0.33 eV, respectively. Figure 5d(4) shows the H₂O formation step on the edge site of Pd₃₈ cluster. OH first adsorbs on the edge site and the H atom adsorbs on the fcc site. As the reaction progresses, the H atom approaches the O atom of OH, to form H₂O, which adsorbs on the edge of the Pd₃₈ cluster.

The activation energies of each elementary reaction step are shown in Figure 6. Compared to the activation energies of each step, the activation energy of H₂ catalytic combustion on the Pd₃₈ cluster is lower than that on the Pd(111) surface. As seen in Figure 6, the activation energy of the O₂ dissociation step is the highest in the four steps, which is the rate-determining step in the H₂ catalytic combustion. The lowest activation energy of O₂ dissociation was 0.83 eV on the Pd(111) surface and 0.54 eV on Pd₃₈ cluster. These results indicate that the Pd₃₈ cluster model had a better catalytic activity than the Pd(111) slab model. The results show that H₂ catalytic oxidation is easier to carry out on Pd nanoparticles. Pd small cluster particles play an important role in improving the catalytic activity. This result is consistent with Singh’s work [6], where a low Pd loading catalyst showed a low activation energy for H₂ catalytic combustion. In addition, the activation energy of H₂ combustion on the edge site of the Pd₃₈ cluster model was lower than that on the other site, which indicates that the defect sites of the catalytic particles are more conducive to the catalytic reaction. The results show that the catalytic activity is related to the dispersion of the catalytic center in the nanoparticles.

4. Conclusions

The reaction mechanisms of H₂ catalytic combustion on the Pd₃₈ cluster model and Pd(111) slab model were investigated using density functional theory. The activation energies of the H₂ catalytic combustion in the Pd₃₈ cluster model were lower than those in the Pd(111) slab model. Compared to the activation energies, H₂ prefers to react on the Pd₃₈ cluster model, especially at the edge sites of the Pd₃₈ cluster model. The activation energy of the rate-determining step was only 0.54 eV, which was much lower than that in the Pd(111) slab model. This indicates that catalytic activity is highly related to the dispersion of the catalytic center in nanoparticles. The smaller nanoparticles with more marginal defect sites are beneficial for the H₂ catalytic combustion. These results provide theoretical guidance for catalyst design for H₂ oxidation combustion, which can reduce experimental costs and improve catalytic design efficiency.