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Topics in Catalysis

, Volume 61, Issue 12–13, pp 1290–1299 | Cite as

How Au Outperforms Pt in the Catalytic Reduction of Methane Towards Ethane and Molecular Hydrogen

  • José I. Martínez
  • Federico Calle-Vallejo
  • Pedro L. de Andrés
Original Paper
  • 194 Downloads

Abstract

Within the context of a “hydrogen economy”, it is paramount to guarantee a stable supply of molecular hydrogen to devices such as fuel cells. Besides, catalytic conversion of the environmentally harmful methane into ethane, which has a significantly lower Global Warming Potential, is an important endeavour. Herein we propose a novel proof-of-concept mechanism to accomplish both tasks simultaneously. We provide transition-state barriers and reaction Helmholtz free energies obtained from first-principles Density Functional Theory by taking account vibrations for \(2\hbox {CH}_4(\hbox {g}) \rightarrow \hbox {C}_2\hbox {H}_6(\hbox {g}) + \hbox {H}_2(\hbox {g})\) to show that \(\hbox {H}_2\) can be produced by subnanometer \(\hbox {Pt}_{38}\) and \(\hbox {Au}_{38}\) nanoparticles. The active sites for the reaction are located on different planes on the two nanoparticles, thus differentiating the working principle of the two metals. The complete cycle to reduce \(\hbox {CH}_4\) can be performed on Au and Pt with similar efficiencies, but Au requires only half the working temperature of Pt. This sizable decrease of temperature can be traced back to several intermediate steps, in excellent agreement with previous experiments, but most crucially to the final one where the catalyst must be cleaned from H(\(\star\)) to be able to restart the catalytic cycle. This highlights the importance of including in catalytic models the final cleaning steps. In addition, this case study provides guidelines to capitalize on finite-size effects for the design of new and more efficient nanoparticle catalysts.

Keywords

Nanoparticle catalyst Methane reduction Ethane evolution Hydrogen production Density functional theory Phonons Thermodynamics Transition-state 

1 Introduction

The steam reforming process, in which methane and water react over a Ni catalyst, is the most widely used commercial source for the production of molecular hydrogen [1]. The rate-limiting step in this reaction is the dissociative chemisorption of methane: a single C–H bond breaks as the molecule collides with the metal surface, leaving chemisorbed \(\hbox {H}(\star\)) and \(\hbox {CH}_3(\star\)) fragments. Several experimental and theoretical groups have studied this reaction, mostly on Ni and Pt surfaces, and molecular beams have been used to measure how the dissociative sticking probability varies with the translational and vibrational energy of methane [2, 3, 4, 5, 6, 7]. Typically, the barriers for dissociation on these surfaces are relatively large [8] and the dissociative sticking probabilities are small. Nevertheless, it has been found that reactivity increases dramatically with increasing collision energy and vibrational excitation of the molecule, as well as substrate temperature.

Nanostructured versions of the most common catalysts are appealing candidates to open new routes that enhance the efficiency of this energy-intensive reaction. Some of their advantages are the following: (i) Nanoparticles possess large effective surface areas in the nanoparticle. Hence more active sites will be available to react with the gas-phase molecules participating in the reaction, substantially increasing the multidirectional sticking probability (and thereby the catalytic activity); and (ii) the high morphological versatility of nanoparticles provides a wide multiplicity of available sites. If the reaction is structure-sensitive, those multiple sites contribute differently to the overall catalytic activity, and geometry optimization is paramount to enhance the catalyst’s performance [9]. In addition, modern theoretical chemistry computational approaches can already provide realistic descriptions of catalytic nanoparticles. Instead of describing complex nanostructures using approximations such as Wülff constructions, it is preferable to simulate entire nanoparticles where terraces, edges, and corners coexist and finite-size effects are observed [10, 11, 12]. Understanding the interplay between those size and geometry factors can ultimately provide realistic models of catalytic processes [13].

Methane is the primary component of natural gas and shale gas. It is extremely flammable and may form explosive mixtures with air. It is also violently reactive with oxidizers, halogens, and some halogen-containing compounds, and an efficient asphyxiator, able to displace oxygen in a closed space. These properties, together with the facts that it is one of the primary greenhouse gases with a Global Warming Potential of 25, compared to 5.5 for ethane and 1 for \(\hbox {CO}_{2}\) (taken as reference) over a 100-year period [14, 15], and that its activation often requires high temperatures, make the development of efficient catalysts for the selective transformation of methane a task of paramount importance. Additionally, there is a continuous natural supply of methane via the bacteria (methanotrophs), unlike ethane that is mainly produced from other secondary non-natural processes, is oxidized rapidly in the atmosphere by the hydroxyl radical, and it is quite short-lived (about 2 months) [16].

Multiple natural and industrial ways of producing methane exist: from the biological methanogenesis [17] and fermentation of organic matter to serpentinization [18] and the well-known centenary Sabatier and Fischer-Tropsch processes, among others; all of them efficient sources of methane. In striking contrast, apart from the combustion resulting in CO and \(\hbox {CO}_2\) as reaction products [19, 20], not so many ways to reduce methane are available. One of the few clean processes to reduce methane is found naturally in the Earth’s atmosphere, where it is photo-chemically transformed into ethane and molecular hydrogen. Indeed, atmospheric ethane results from the photochemical action of sunlight on methane gas: ultraviolet photons with wavelengths shorter than 160 nm can photo-dissociate the methane molecule into a methyl radical and a hydrogen atom, \(\hbox {CH}_4(\hbox {g}) \rightarrow \hbox {CH}_3(\hbox {g})+\hbox {H}(\hbox {g})\). When two methyl radicals recombine the result is ethane \((2\hbox {CH}_3(\hbox {g}) \rightarrow \hbox {C}_2\hbox {H}_{6}(\hbox {g}))\), which can be accompanied by the generation of molecular hydrogen. Although ethane is still a greenhouse gas, it is much less abundant than methane (ethane: < 0.01% vs. methane: \(\sim\) 0.44% on earth). This photo-dissociation reaction has been analyzed under collision-free conditions for the room-temperature gas-phase dissociation of methane for primary H atom formation, resulting in a photolysis frequency of 121.6 nm, which translates into a photon energy of \(\approx 10\) eV [21].

Therefore, the search for catalysts able to carry out the on-surface methane reduction into ethane and molecular hydrogen may prove attractive to transform methane in a clean way with the additional benefit of storing energy in the H–H bond. Apparently, the viability of such an on-surface reaction will be directly linked to a reaction pathway involving kinetic energy barriers that can be overcome with moderate molecular collision energies, vibrational excitation of the molecule, and substrate temperature. Such conditions are needed to make the steam reforming catalyst industrially appealing.

In this communication we describe, from a first-principles modeling standpoint, a viable mechanism for the complete catalytic cycle of the reaction 2\(\hbox {CH}_4\)(g) \(\rightarrow\) \(\hbox {C}_2\hbox {H}_{6}\)(g) + \(\hbox {H}_2\)(g) on the truncated octahedra subnanometer \(\hbox {Pt}_{38}\) and \(\hbox {Au}{38}\) nanoparticles. This particular shape displays a high surface-to-volume ratio for a 38-atoms cluster, is very stable, and contains short (111) and (100) terraces where active undercoordinated atoms are abundant.

The reaction starts with two gas-phase methane molecules being reduced on one terrace of the nanoparticles. The elementary steps are: (i) sequential adsorption/dissociative chemisorption of two methane molecules, \(2\times (\hbox {CH}_4(g)\rightarrow \hbox {CH}_3(\star )+\hbox {H}(\star ))\), where \(\star\) denotes a free adsorption site; (ii) associative desorption of ethane, \(2\hbox {CH}_3(\star ) \rightarrow \hbox {C}_2\hbox {H}_{6}\)(g); and, finally, (iii) associative desorption of molecular hydrogen, \(2\hbox {H}(\star ) \rightarrow \hbox {H}_2\)(g). The facets where the cycle is completed are different for Pt and Au: four-atom (100) terraces, and seven-atom (111) terraces, respectively. Although the catalytic cycle is energetically viable on both \(\hbox {Pt}_{38}\) and \(\hbox {Au}_{38}\) nanoparticles, our calculations predict that the limiting step of the reaction, which is the cleaning of the nanoparticle from poisoning \(\hbox {H}(\star )\), can be achieved on \(\hbox {Au}_{38}\) at about half the temperature required for \(\hbox {Pt}_{38}\). All stationary configurations (initial, transition, and final states) have been obtained from first-principles Density Functional Theory (DFT) calculations. The energies for these states have been used to determine the canonical probabilities for a thermal fluctuation to overcome kinetic barriers along the reaction path, yielding the time scale for the reaction. Our mechanism offers a viable and clean route for: (i) the catalytic reduction of methane towards ethane and molecular hydrogen; and (ii) within the context of a “hydrogen economy”, the stable supply of molecular hydrogen to technologically and industrially attractive devices such as fuel cells.

2 Computational Methods and Models

2.1 Ab Initio Electronic Energies

The total energies and forces were minimized using DFT as implemented in the plane-wave package Quantum Espresso [22]. Within the DFT+D formalism, a semi-empirical van der Waals (vdW) \(R^{-6}\) correction has been used to add long-range dispersive forces [23, 24, 25]. This vdW contribution to the total energy is almost negligible around the transition-states. It only becomes significant (> 15%) when the distances between the adsorbates and the nanoparticles increase to values > 4 Å, which only matters for a good description of energies at asymptotic distances. In order to capture the influence of the electronic spin in the total energy of the systems involving “subsystems” with unpaired electrons, all the calculations have been carried out within a spin-polarized framework. The Perdew–Burke–Ernzerhof (PBE) parametrization has been used for the exchange and correlation potential [26]. The ionic cores were described by Projector Augmented Wave (PAW) method [27]. All atoms in the nanoparticles were free to move in all directions. The atomic relaxations were carried out with conjugate gradient minimization scheme until the maximum force on any atom was below 0.01 eVÅ\(^{-1}\). Geometrical optimization was performed with a plane-wave cutoff of 500 eV. The nanoparticles were simulated in a cubic box of \(30\times 30\times 30\) Å\(^{3}\); the smaller distance between images was larger than 20 Å. The Brillouin zone was sampled at the Gamma point only. The Fermi level was smeared with the Methfessel–Paxton approach with a Gaussian width of 0.01 eV, and all energies were extrapolated to T = 0 K [28]. All these parameters yield total electronic energies with accuracies of \(\varDelta E \approx \pm \, 0.01\) eV (converged to a precision better than 10\(^{-6}\) eV). DFT adsorption energies for \(\hbox {H}_2\), \(\hbox {CH}_4\) and \(\hbox {C}_2\hbox {H}_{6}\) were calculated with respect to the clean optimized nanoparticles and the gas-phase species in the following way: \(\varDelta E_{ADS}=E_{A,\star }-E_{\star }-E_{A}\), where \(\star\) represents the clean nanoparticle, and A and (\(A,\star\)) are the gas-phase and adsorbed states for a given species. The gas-phase references were calculated in the same cubic box of \(30\times 30\times 30\) Å\(^{3}\) using the Gamma point only. These values have been corrected with Helmholtz’s free energies derived from vibrational modes of the different configurations. Apart from the enthalpies of adsorption, dissociation and desorption (H), the important feature to ascertain the feasibility of a particular state is the height of the barrier (\(\varDelta E\)) at the transition-state (TS). The TSs have been investigated within the climbing-image nudged elastic band (CI-NEB) method [29, 30, 31], where the initial, final, and all the intermediate states (12 image states for all the barriers have been analyzed) were completely free to relax. Out of necessity, we use the approximation of a rigid barrier model, which should be acceptable up to temperatures of about 30% the melting temperatures of the nanoparticles, i.e. up to around 400 K for Au and 600 K for Pt.

2.2 Thermochemistry

The influence of temperature on the energetics of the different steps has been analyzed by introducing the effect of vibrations on the system. For each stationary quasi-equilibrium state reported in Figs. 1, 2 and 3 we have computed the frequencies of the vibrational modes. A model of harmonic independent oscillators has been used to compute the partition function in the canonical ensemble, where the system is in contact with a thermostat at temperature T: [32]
$$\begin{aligned} Z(N,V,T)= \varPi _{i} \frac{e^{-\frac{\hbar w_i}{2 k_B T}}}{1-e^{-\frac{\hbar w_i}{k_B T}}}. \end{aligned}$$
(1)
The number of particles N is fixed, and the volume V corresponds to a zero stress condition. The origin for energies is taken at the bottom of the harmonic well. To ensure convergence of the series we include modes \(w>1\) cm\(^{-1}\) only; a reasonable approximation at the temperatures of interest.
Fig. 1

Pictorial sketch of the different intermediate stages along the whole catalytic cycle 2\(\hbox {CH}_4\)(g) \(\rightarrow\) \(\hbox {C}_2\hbox {H}_{6}\)(g)+\(\hbox {H}_2\)(g) on \(\hbox {Pt}_{38}\) (a) and \(\hbox {Au}_{38}\) (b) nanoparticles. The top views of the four-atom (100) and seven-atom (111) facets, where the catalytic cycle is completed on Pt and Au, respectively, are also given as insets in both panels

Fig. 2

Computed reaction pathway (including enthalpies and transition-state barriers) for the whole catalytic 2\(\hbox {CH}_4\)(g) \(\rightarrow\) \(\hbox {C}_2\hbox {H}_{6}\)(g)+\(\hbox {H}_2\)(g) cycle completed on the \(\hbox {Pt}_{38}\) nanoparticle. Labels 1–8 correspond to the geometries depicted in Fig. 1a. DFT energies (black), Helmholtz free-energies at T = 0 K (blue) and Helmholtz free-energies at T = 400 K (red)

Fig. 3

Computed reaction pathway (including enthalpies and transition-state barriers) for the whole catalytic 2\(\hbox {CH}_4\)(g) \(\rightarrow\) \(\hbox {C}_2\hbox {H}_{6}\)(g)+\(\hbox {H}_2\)(g) cycle completed on the \(\hbox {Au}_{38}\) nanoparticle. Labels 1–8 correspond to the geometries depicted in Fig. 1b. DFT energies (black), Helmholtz free-energies at T = 0 K (blue) and Helmholtz free-energies at T = 400 K (red)

All the thermodynamic magnitudes can be obtained from the partition function, in particular the contribution of vibrational modes to the Helmholtz’s free energy: [33]
$$\begin{aligned} F = - k_B T \ln {(Z)} \end{aligned}$$
(2)
This free energy is used to include the effect of temperature in the energies of the different steps and on the largest barrier (limiting step). To further interpret these results it is also possible to break up Helmholtz’s function in the associated vibrational entropies and internal energies as follows:
$$\begin{aligned} {\left\{ \begin{array}{ll} S = - \left( \frac{\partial F}{\partial T} \right) _{V,N} \\ U = F + T S \end{array}\right. } \end{aligned}$$
(3)

2.3 \(\hbox {Au}_{38}\) and \(\hbox {Pt}_{38}\) Geometrical Models

The shape of the \(\hbox {Pt}_{38}\) and \(\hbox {Au}_{38}\) nanoparticles corresponds to a truncated octahedron with minimum and maximum atom-to-atom nanoparticle diameters of 7.3 and 8.5 Å for \(\hbox {Pt}_{38}\), and 7.6 and 8.9 Å for \(\hbox {Au}_{38}\) (see Fig. 4). There are eight (111) and six (100) terraces that are denoted 111T and 100T, respectively. At the intersection of neighboring facets there are 12 edges between neighboring (111) facets, and 24 edges between neighboring (100) facets. These are denoted 111E and 100E, respectively. Finally, there are twenty-four corners at the intersection between two neighboring (111) and (100) terraces; these are called kinks. These atoms, together with the eight central atoms of (111) terraces (called centers), are the only two non-equivalent atoms in the outer shell of the nanoparticle. The outer shell has 32 atoms while the inner core of the nanoparticle consists of a regular 6-atom octahedron.
Fig. 4

Perpendicular view to a (100) terrace (left panel) and to a (111) edge (right panel) of the \(\hbox {Pt}_{38}\) and \(\hbox {Au}_{38}\) nanoparticles (truncated octahedron) included in this study, where T stands for terrace and E for edge

Pt and Au nanoparticles have been chosen because they show high catalytic efficiencies in numerous technological and industrial processes. Particularly, in the field of heterogeneous catalysis, platinum and its alloys have been successfully used in many catalytic applications, including the water–gas shift [34, 35, 36], fuel cells and light harvesting [13, 37, 38, 39]. Moreover, gold nanoparticles have also been used as efficient catalysts in a number of chemical reactions [40]. Interestingly enough, gold surfaces can be used for selective oxidation reactions [41, 42], but can also be reductive, as in the case of nitrate reduction [43]. Additionally, \(\hbox {Pt}@\hbox {Au}\) composite nanoparticles have recently been shown to enhance the kinetics of the oxygen reduction reaction (ORR), and oxygen evolution reaction (OER) in rechargeable \(\hbox {Li}-\hbox {O}_{2}\) cells [44].

3 Results and Discussion

3.1 Reaction Path Description: Energetics and Transition-State Barriers

In gas phase, the reaction 2\(\hbox {CH}_4\)(g) \(\rightarrow\) \(\hbox {C}_2\hbox {H}_{6}\)(g) + \(\hbox {H}_2\)(g) is endothermic, requiring approximately 0.7 eV to proceed (17 kcal/mol). Moreover, it has a large kinetic barrier of about 6.2 eV at the initial step of \(\hbox {CH}_4\) dehydrogenation.

In the presence of \(\hbox {Au}_{38}\) and \(\hbox {Pt}_{38}\) the reaction proceeds following some elemental steps shown schematically in Figs. 1, 2 and 3. While the overall reaction enthalpy is obviously not modified by the presence of the catalyst, the barrier of the rate-limiting step is reduced to \(\approx 1\) eV for Au and 2 eV for Pt (see below). In black, we show the DFT internal electronic energies for each of the equilibrium configurations. In blue (\(T=0\) K) and red (\(T=400\) K) we give the corrections to these values obtained from the vibrational Helmholtz’s free energy. The origin is always set out in the first step to facilitate comparisons. Although the zero-point energy correction for each phase in the reaction can be substantial due to the fair number of modes involved, differences between them amount to fractional corrections of 15% or less for T = 0 K. At non-zero temperatures, however, these differences increase due to the disparate nature of vibrations on the two metals and the different relevant adsorption sites. In particular, the whole pathway for Au transforms from endothermic to quasi-exothermic, except for the last steps where the enthalpy of reaction is recovered. It is interesting to notice that at 400 K the population of the intermediate steps in Au is significantly increased because of thermodynamical equilibrium. As a whole, Au is “softer” than Pt and the contribution to Helmholtz’s free energy is larger. On the other hand, Pt is more active chemically and makes the reaction exothermic even for very low temperatures, but the final steps become more costly.

To facilitate its analysis the whole process has been divided in the following eight steps:
  • 1. Methane adsorption. The reaction starts with the adsorption of a \(\hbox {CH}_4\) molecule on a favorable site on the nanoparticle. For \(\hbox {Pt}_{38}\) this is the top of a kink atom (black dots in the inset of Fig. 1a). The molecule is physisorbed by one of the H atoms on the nanoparticle with an equilibrium distance of 2.12 Å from the Pt atom. The adsorption energy is − 0.12 eV. For \(\hbox {Au}_{38}\), in contrast, the same process takes place on a kink atom on a (111) facet (black dots in the inset of Fig. 1b). The equilibrium distance is 3.13 Å, and the adsorption energy is − 0.06 eV. We observe that the interaction with Au is weaker compared with Pt because the different chemical activity of both species.

  • 2. Methane dehydrogenation. We consider the elementary dissociative reaction \(\hbox {CH}_4\)(\(\star\)) \(\rightarrow\) \(\hbox {CH}_3\)(\(\star\))+H(\(\star\)). On \(\hbox {Pt}_{38}\), one hydrogen atom from the adsorbed methane molecule moves to the closest 100E bridge edge (blue dots in the inset of Fig. 1a). The total energy is reduced by − 0.24 eV (exothermic) and the kinetic barrier is 0.70 eV. On \(\hbox {Au}_{38}\), this is an endothermic process that requires about 0.69 eV of extra energy and has a barrier of 0.86 eV at T = 0 K, which reduces substantially the required energy by 0.46 eV at T = 400 K. H is adsorbed on-bridge at the closest 111E edge (blue dots in the inset of Fig. 1b). The larger barrier is related to the weaker interaction between Au and methane compared to that for Pt and methane. This apparent disadvantage will turn out to be beneficial at the end of the process when the nanoparticle needs to be restored to its original state. Indeed, it is known that a single Pt atom can dehydrogenate methane without a barrier, while on a surface a barrier appears due to the increased coordination of Pt to other Pt atoms [6].

  • 3–4. Adsorption and dehydrogentation of a second methane molecule. At this point \(\hbox {CH}_3\)(\(\star\)) is adsorbed atop a kink (2.07 Å) on the platinum 100T terrace, and H(\(\star\)) atom is adsorbed on-bridge near a 100E edge (1.77 Å). The previous steps 1 and 2 are repeated at a neighboring site with a second methane molecule adsorbed atop a vicinal empty kink (black dots in the insets of Fig. 1). The second adsorbed \(\hbox {CH}_4\) molecule loses again one H atom that is adsorbed on-bridge in the remaining symmetrical empty 100E edge of the same (100) facet for \(\hbox {Pt}_{38}\) (blue dots in the inset of Fig. 1a), and in the remaining symmetrical empty 100E edge of the same (111) facet for \(\hbox {Au}_{38}\) (blue dots in the inset of Fig.  1b). Steps 3 and 4 are similar to 1 and 2 in enthalpy, barriers and respective optimum sites for both Pt and Au, cf. Table 1.

  • 5–6. Associative desorption of ethane: \(\mathbf{2CH}_\mathbf{3}(\star )\rightarrow \mathbf{C}_{\mathbf{2}}{} \mathbf{H}_{\mathbf{6}}(\mathbf{g})\). After the two consecutive processes of dissociative chemisorption of methane, the two \(\hbox {CH}_3\)(\(\star\)) fragments adsorbed on neighboring sites combine to produce ethane. Up to now, the barriers on both Pt and Au are quite similar, being the main difference the exothermic versus endothermic reaction steps. The process of associative desorption of ethane raises the first important difference in terms of kinetic barriers: 1.24 eV for Pt compared to 0.87 eV for Au. The reaction on both substrates is exothermic, but is more stable on gold, namely − 0.32 eV compared to − 0.07 eV for platinum, accompanied by a similar reduction in the barrier. These barriers are plotted together in Fig. 5 (red for Pt, and blue for Au). The significance in time, or in working temperature, is shown in Fig. 6c; the difference in barriers implies approximately a 100 K higher temperature on Pt to get the process completed in the same time.

  • 7–8. Associative desorption of \(\mathbf{H}_\mathbf{2}\). The most substantial difference between the two catalysts is found in the associative desorption of \(\hbox {H}_2\). In the presence of chemisorbed H(\(\star\)), a known poison for the reaction [45, 46], the first dehydrogenation step of the next cycle would become significantly more difficult, and the barriers would grow accordingly. Therefore, it is imperative to remove atomic hydrogen from the nanoparticle. A convenient way of desorbing it is by forming molecular hydrogen.

    On \(\hbox {Pt}_{38}\) two H(\(\star\)) remain adsorbed on-bridge at 100E opposite edges of the same (100) facet (step 7 of Fig.  1a). The H-cleaning process will be completed in this nanoparticle by the migration of one of the H(\(\star\)) atoms (denoted \(H_{A}\star\)) to be adsorbed on-bridge in the closest 100E edge (denoted \(H_{B}\star\)), shown in step 7’ of Fig.  1a. This migration allows for the formation, and subsequent desorption, of an \(\hbox {H}_2\) molecule from the two adsorbed H(\(\star\)) atoms, leaving \(\hbox {Pt}_{38}\) clean and ready to start over a new catalytic cycle (step 8 of Fig. 1a).

    On the other hand, on \(\hbox {Au}_{38}\) the two H(\(\star\)) remain adsorbed on-bridge at the 111E mirror edges of the same (111) facet (step 7 of Fig. 1b). Atomic H diffusion on Au nanostructures is fast and efficient [47], which facilitates the recombination and subsequent desorption of an \(\hbox {H}_2\) molecule, leaving \(\hbox {Au}_{38}\) clean in its initial configuration (step 8 of Fig. 1b).

Table 1

All values in eV

i

\(\varDelta U_e\)

\(F_0\)

\(\varDelta F_0\)

\(\varDelta F_{400}\)

\(\hbox {Pt}_{38}\)

 0

0

2.45

0

0

 1

− 0.13

2.41

− 0.04

− 0.40

 2

− 0.37

2.32

− 0.13

− 0.35

 3

− 0.50

2.29

− 0.16

− 0.41

 4

− 0.68

2.19

− 0.26

− 0.40

 5

− 0.75

2.30

− 0.15

− 0.28

 6

− 0.63

2.31

− 0.14

− 0.18

 7

− 0.56

2.29

− 0.16

− 0.20

 8

+ 0.77

2.26

− 0.19

− 0.21

\(\hbox {Au}_{38}\)

 0

0

2.45

0

0

 1

− 0.06

2.44

− 0.01

− 0.36

 2

+ 0.63

2.30

− 0.15

− 0.46

 3

+ 0.54

2.30

− 0.15

− 0.64

 4

+ 0.87

2.14

− 0.31

− 0.86

 5

+ 0.55

2.33

− 0.12

− 0.57

 6

+ 0.59

2.33

− 0.12

− 0.25

 7

+ 0.77

2.26

− 0.19

− 0.21

For each elemental step i (labels 0–8) we give the increment in the internal electronic energy, \(\varDelta U_e\), w.r.t the step 0, zero-point vibration Helmholtz’s free energy at T = 0 K, \(F_0\), increments with respect to the first step, \(\varDelta F_0\), and Helmholtz’s free energy increment at T = 400 K, \(\varDelta F_{400}\)

 
Fig. 5

Schematics of the initial (IS), transition (TS) and final (FS) states for the elementary step \(2\hbox {CH}_3\)(\(\star\)) \(\rightarrow\) \(\hbox {C}_2\hbox {H}_{6}\)(\(\star\))+2H(\(\star\)) on the \(\hbox {Pt}_{38}\) (top panel) and \(\hbox {Au}_{38}\) (bottom panel) nanoparticles. The middle panel shows the CI-NEB energy barriers for \(\hbox {Pt}_{38}\) (red) and \(\hbox {Au}_{38}\) (blue) nanoparticles

At T = 0 K the barrier for this final step in \(\hbox {Pt}\) is 2.06 eV for Pt and 0.62 eV for Au. Temperature effects suggest that these values could typically end up around 2 and 1 eV (see Fig. 6b). The difference in estimated times to complete this step amounts to several orders of magnitude (see Fig. 6c). Regarding thermodynamics, the reaction enthalpies for \(\hbox {Pt}\) and \(\hbox {Au}\) are 1.33 and 0.18 eV, respectively, cf. Figs. 2 and 3. These values are robust estimations since they are related to global energy conservation on the whole cycle. Therefore, this last step reinforces the fact that catalysis on \(\hbox {Au}\) should be more attractive than on \(\hbox {Pt}\) for this particular reaction, implying a higher working temperature on Pt of at least 100 K.

At this point it is important to remark that all different non-equivalent adsorption sites for the intermediates participating in the catalytic cycle have been checked for both \(\hbox {Pt}_{38}\) and \(\hbox {Au}_{38}\) nanoparticles. Here we only report the most stable ones (which agree well with those reported in previous literature for these nanoparticles, e.g. Refs. [9, 12]) to construct the most viable and realistic reaction path as given above. Table 1 summarizes the energies used to draw Figs. 2 and 3. Besides, we have also analyzed several possible secondary reaction pathways deviating from the catalytic cycle proposed here. As an example, we have computed the barriers for subsequent dehydrogenations of \(\hbox {CH}_3\)(\(\star\)) (in steps 3–4) towards \(\hbox {CH}_{2}(\star )+\hbox {H}(\star ), \hbox {CH}(\star )+2\hbox {H}(\star )\) and \(\hbox {C}(\star )+3\hbox {H}(\star )\). The computed transition-state barrier for \(\hbox {CH}_3(\star ) \rightarrow \hbox {CH}_{2}(\star )+\hbox {H}(\star )\) are 3.6 and 4.1 eV on \(\hbox {Pt}_{38}\) and \(\hbox {Au}_{38}\), respectively, which renders this step, and the ones derived from it, energetically unlikely. Moreover, we have checked minimum-energy pathways (MEPs) and transition-state (TS) barriers on other adsorption sites for all the participating adsorbates, where the intensity of the adsorption was weaker, also resulting in larger kinetic barriers that hinder the advance of the whole catalytic cycle. Additional plausible competing channels for the proposed reactions, such as \(\hbox {H}_2\) molecular associative desorption from adsorbed \(\hbox {CH}_4\)(\(\star\)) and \(\hbox {CH}_3\)(\(\star\)), or dehydrogenation of the adsorbed final \(\hbox {C}_2\hbox {H}_{6}\)(\(\star\)) have also been carefully checked, yielding very large barriers that prevent the evolution of those secondary reaction channels. All these attempts reinforce our confidence on the reaction path proposed in this study.
Fig. 6

a (Top) Difference of Helmholtz’s free energies (in eV) as a function of temperature (in K) between steps 5 and 4 (associative desorption of ethane) in Au (yellow) and Pt (grey). (Bottom-left) Internal vibrational energy (in meV) and entropic contribution (in meV) to the free energy plotted in the top panel (bottom-right). b Helmholtz’s free energy barrier (in eV) as a function of temperature (in K) at the transition-state for associative desorption of molecular hydrogen; steps 6 and 8 in Au (yellow) and Pt (grey). c Typical times to overcome the transition-state barrier for the associative desorption of ethane (Au in solid yellow and Pt in solid gray), and the associative desorption of molecular hydrogen (Au dotted yellow and Pt dotted gray)

The proposed catalytic cycle can be understood fairly well by focusing on the highest barriers along the path; this is a good approximation since the exponential Boltzmann factors select the few most relevant steps. We identify two such crucial steps on the transition-states for the associative desorption of ethane \((\hbox {TS}_{\mathrm{C}_2\mathrm{H}_{6}}\), between steps 4 and 5), and the associative desorption of molecular hydrogen \((\hbox {TS}_{\mathrm{H}_2}\), last step).

Regarding the comparison with related available experimental information, we have to point out that several works in the literature have inspired the present study. In particular, those on several faces of Pt, Pd and Ni [6, 48, 49, 50] and clusters of Pt [51]. These experiments agree in general terms with the adsorption energies and barriers that we obtain, but none of them focuses on the final effort to clean the catalyst and start a new cycle again. The combined experimental and theoretical study of Vajda et al. on very small clusters of Pt indeed shows the dehydrogenated propane molecule in a bound state by around 1 eV, which is fully consistent with our predictions (see Fig. 2). Finally, the nanoparticles we study have an important feature when compared with surface science experiments: the presence of several orientations on the same system allows for processes to take place in different coordination environments that are close together. In that direction work by Anghel et al. on Pt(110)-(1\(\,\times\,\)2), and by Franke et al. on Pt(321), also display different orientations close together, and the results of their calculations are fully compatible with our values.

Finally, to analyze the effect of temperature we computed vibrational Helmholtz’s free energies for each local equilibrium step and for the transition-state associated to the last one, which we find effectively limits the rate for the whole cycle.

3.2 Associative Desorption of Ethane

First, we analyze the electronic energy in the region around the transition-state for the associative desorption of ethane: 2\(\hbox {CH}_3\)(\(\star\)) \(\rightarrow\) \(\hbox {C}_2\hbox {H}_{6}\)(g). In Fig. 5 Pt (red) shows a larger barrier than Au (blue) by around 0.37 eV; if the height of the barrier was independent of temperature, this would determine the reaction to be a factor \(e^{-\frac{0.37}{k_B T}}\) times slower on Pt than on Au, i.e. \(\approx 10^{-7}\) at \(T=300\) K. Moreover, the final state (desorbed ethane) is \(\approx -0.25\) eV more stable for Au than for Pt, determining in equilibrium at 300 K a ratio of populations between the initial and final states of \(\approx 1:10^{5}\) for Au compared to 1 : 1 for Pt. Therefore, all these considerations based on chemical arguments favor Au over Pt on this particular step.
Fig. 7

(Left panel) Relevant normal modes with amplitudes directed towards the transition-state for the formation (and subsequent desorption) of \(\hbox {C}_2\hbox {H}_{6}\) on \(\hbox {Au}_{38}\) (ranging between 130 and 360 cm\(^{-1}\)). (Right panel) Relevant normal modes with amplitudes directed towards the transition-state for the formation (and subsequent desorption) of \(\hbox {H}_2\) on \(\hbox {Pt}_{38}\) (ranging between 550 and 1650 cm\(^{-1}\))

Next, we analyzed the effect of temperature on the quasi-equilibrium initial (4) and final (5) states. In Fig. 6a we plot the difference between Helmholtz’s free energies versus T. The different role of vibrations for \(\hbox {CH}_3\)(\(\star\)) and \(\hbox {C}_2\hbox {H}_{6}\)(g) is already apparent from the zero-point energies quoted in Table 1, but its evolution with temperature is even more interesting: the molecule becomes less and less stable at hight temperatures both for Au and Pt. The reason can be traced back to the entropy S (bottom-right panel of Fig. 6a): while the vibrational internal energies U of both Au and Pt (bottom-left panel of Fig. 6a) are stabilized by about 100 meV from 4 to 5, the entropic term \(-T \times S\) is destabilized by about 200 meV for Pt, and it nearly doubles for Au. Therefore, the softer material (Au) is affected more significantly by temperature and the reaction “changes the trend” (regarding the two considered materials: Pt and Au) due to the mere contribution of zero-point vibrations.

To compute the effect of temperature on this transition-state \({\text{TS}}_{{{\text{C}}_{2} {\text{H}}_{6} }}\) is difficult in view of the large amount of modes needed to be described with enough accuracy. A rigid model for the barrier where the effect of temperature on the initial and final points is interpolated to the transition-state predicts that at 400 K both barriers would be comparable. Although this procedure is not warranted down to the finest details, it outlines at least the tendency on the barrier as the temperature increases. The rates for this step have been computed by looking at normal modes possessing amplitudes that make the methyl groups approach the barrier simultaneously from both sides. The rate for the transition is then obtained by multiplying the Boltzmann factor (giving the probability to pick up a thermal fluctuation at temperature T to overcome the barrier after “fluctuation”, \(\varDelta E\)) times the number of attempts to pass the barrier given by the frequency of the relevant normal mode, \(\varGamma e^{-\frac{\varDelta E}{k_BT}}\).

Figure 7 shows esquematically the most relevant vibrational modes, i.e. the modes related to overcoming the highest barriers. For Au (see the left panel of Fig. 7) this is the step where the two adsorbed methane molecules couple to form \(\hbox {C}_2\hbox {H}_{6}\). Because these vibrations also involve substrate atoms, the relevant frequencies for this process are in a band between 130 and 360 cm\(^{-1}\). On the contrary, for Pt (see the right panel of Fig. 7) the highest barrier, i.e. the limiting step, corresponds to the associative desorption of 2H(\(\star\)) to form \(\hbox {H}_2\). The relevant modes associated to this process are in the range between 550 and 1650 cm\(^{-1}\) (again, these modes also involve vibrations of the substrate atoms).

3.3 Associative Desorption of Molecular Hydrogen

The most significant difference arises at the moment of cleaning the substrate to restart the catalytic cycle. This corresponds to the associative desorption of molecular hydrogen. The barrier to form and desorb \(\hbox {H}_2\) on \(\hbox {Pt}_{38}\) is \(\approx 2\) eV, doubling the corresponding one on \(\hbox {Au}_{38}\).

In this particular case it is simpler to include the effect of vibrations on the transition-state due to the significant reduction in the number of normal modes involved w.r.t. the previous steps. On the initial configurations (steps 7’ and 7 for \(\hbox {Pt}_{38}\) and \(\hbox {Au}_{38}\), respectively, in Fig. 1) we have six modes associated with two chemisorbed H atoms. On the final configuration only the internal stretch of the hydrogen molecule is left and we can follow the softening of all the modes except one in a quasi-adiabatic picture. Furthermore, hindered rotations here for \(\hbox {TS}_{\hbox {H}_2}\) are not an issue due to the simplicity of the geometry (contrasting with the previous \(\hbox {TS}_{\hbox {C}_2\hbox {H}_{6}}\) case). The result of this approach for the relative variation of the barrier with respect to the initial state is given in Fig. 6c. In both cases the barrier is lowered by about 0.1–0.2 eV, being again Au more affected than Pt by temperature. In this step, mainly two modes contribute to the process: one around atop sites and the other around the hollow sites, with frequencies of 550 and \(1650\,\hbox {cm}^{-1}\). Again, beside the details for the vibrations behind the transformation, Boltzmann’s factors dominate in this range of temperatures and fully determine the time taken for the process.

Therefore, we conclude that temperature-dependent corrections on the initial and final states around the barrier (and on the barrier itself) of this most crucial step do not significantly alter our main conclusions.

4 Conclusions

We have proposed an energetically-viable mechanism that explains the complete catalytic cycle of 2\(\hbox {CH}_4\)(g) \(\rightarrow\) \(\hbox {C}_2\hbox {H}_{6}\)(g)+\(\hbox {H}_2\)(g) on platinum and gold nanoparticles made of 38 atoms with truncated-octahedron shapes. DFT has been used to obtain transition-state barriers, kinetic rates, and reaction enthalpies. We have also computed the relevant vibrational modes to obtain Helmholtz’s free energies.

Using these values we have evaluated the performance of both catalysts. The proposed catalytic reaction is a temperature-activated mechanism that requires to surmount barriers of \(\approx 2\) eV on \(\hbox {Pt}_{38}\) and \(\approx 1\) eV on \(\hbox {Au}_{38}\). Remarkably, Au nanoparticles outperform those of Pt in several intermediate steps, but most crucially in the final one where H(\(\star\)) must be removed from the nanoparticles to be able to restart the catalytic reaction. The reason for the improved activity of Au can be rationalized in terms of the equilibrated compromise between the adsorption strength of the adsorbed intermediates. Pt nanoparticles bind adsorbates in a way such that it is more difficult to clean the surface at the end of the cycle. Furthermore, Au is more affected by the operating temperature since its vibrational modes are typically softer than on Pt.

Therefore, the complete catalytic cycle to decompose \(\hbox {CH}_4\) can be performed on Au with similar efficiency than Pt, but using only half the temperature. Such a substantial reduction in the working temperature of the catalyst is appealing from an industrial standpoint and evinces the benefits of using nanostructured catalytic materials in which a compromise exists between the relative strength of the reaction intermediates in complex pathways.

Finally, in conclusion, the results of this work provide: (i) on one side, an excellent starting point to elucidate the optimal particle size for the reduction of methane towards ethylene and hydrogen evolution, capitalizing on geometric and finite-size effects; and (ii) on the other hand, within the context of a “hydrogen economy”, a viable route towards the stable supply of molecular hydrogen to technologically and industrially attractive devices such as fuel cells.

Notes

Acknowledgements

This work has been supported by the Spanish MINECO (Grants MAT2014-54231-C4-1-P and MAT2017-85089-C2-1-R), and the EU via the ERC-Synergy Program (Grant ERC-2013-SYG-610256 Nanocosmos) and the EU Graphene Flagship (Grant agreements 696656 Graphene Flagship-core 1 and 785219 Graphene Flagship-core 2). JIM acknowledges funding from Nanocosmos and “Ramón y Cajal” MINECO Program through Grant RYC-2015-17730, and thanks CTI-CSIC for use of computing resources. FC-V thanks “Ramón y Cajal” MINECO Program through Grant RYC-2015-18996.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • José I. Martínez
    • 1
  • Federico Calle-Vallejo
    • 2
  • Pedro L. de Andrés
    • 1
  1. 1.Materials Science FactoryInstitute of Material Science of Madrid (ICMM-CSIC)MadridSpain
  2. 2.Departament de Ciència de Materials i Química Fisica & Institut de Química Teòrica i Computacional (IQTCUB)Universitat de BarcelonaBarcelonaSpain

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