## Introduction

Methane constitutes the majority of various gas resources, such as natural gas, shale gas, and landfill gas. More than 500 million tons are generated each year, an amount which is gradually increasing1. Currently, only a small amount of methane is used as industrial feedstock relative to the total amount produced, and most is burnt as an energy source2. This is because methane is highly stable and requires additional oxidants such as H2O or O2 to be used as a chemical feedstock, which necessitates additional equipment and operating costs3. If a catalyst were to be found, which could convert methane directly into chemical raw materials without the use of oxidants, the subsequent economically viable methane utilisation process could be widely commercialised4.

Recently, a non-oxidative catalytic methane-converting technology was reported and received great attention5. This innovative technology succeeded in producing ethylene and benzene in high yields without coke formation at temperatures above 1200 K, where dehydrogenation is thermodynamically favourable. The previous study focused on the role of Fe-single atom catalyst that was capable of producing CH3 radicals but did not participate in further C–C coupling of dehydrogenation5. Furthermore, a recent study on a new type of Fe single-atom catalyst for non-oxidative methane coupling reported that the Fe single sites were only active in the initial reaction period, then rapidly lost their activity and resulted in varying hydrocarbon distributions6. Since the C–H bonds of ethylene and benzene are more readily activated than that of methane, it is thermodynamically challenging to inhibit coke deposition at such high temperatures. Two understandings are possible: engineering the reactor such that the produced hydrocarbons rapidly exit the reaction zone before further dehydrogenation proceeds7, or reinterpreting the role of the Fe single-atom catalyst to play a role in C–C coupling in addition to the generation of CH3 radicals. The latter effect can increase the partial pressure of C2-hydrocarbons and the successive trimerisation rate in the reaction stream under mild reaction conditions, where coke formation is hindered. The proposed Fe single-atom catalyst was composed of FeC2 clusters embedded in a SiO2 substrate5. Subsequent studies focusing on the reactor7,8 and catalyst9 were also carried out using this type of catalyst. However, experimental observations at such high reaction temperatures originate from a mixture of gas-phase radical reactions and catalysis, which are difficult to decouple.

In this study, we investigate the energetics of methane coupling by focusing on the reactions occurring on the Fe single-atom catalyst using density functional theory (DFT) calculations. Although there has not been much research on how methane reacts on the catalyst surface under non-oxidative conditions, we establish a reaction network based on the Eley–Rideal mechanism, which has been suggested as a dominant for the oxidative methane coupling reaction10,11. The absence of adjacent adsorption at the single-atom active site supports this assumption as well. Based on the reaction network and energetics obtained using DFT, a microkinetic model is built to analyse the activity and selectivity of the Fe single-atom catalyst under various reaction conditions. Furthermore, we design and perform a series of reactions to demonstrate the role of the Fe-single atom catalyst computationally identified.

## Results

### Reaction network and energetics

The energetics of the C–H bond activation of methane and subsequent C–C bond formation on the FeC-confined SiO2 catalyst surface were analysed. According to previous experimental and theoretical investigations5,9, we built an initial surface geometry by substituting a SiO2 unit in the (001) surface of β-tridymite with an FeC2 unit, as shown in Fig. 1a, b. Then, all possible structures that could be formed during methane decomposition and C2-hydrocarbon production were generated (see Supplementary Data 1). We note that the lattice-confined C atoms can also participate in the formation of C2-hydrocarbons in this model network. To exclude duplication of geometries and simplify the reaction network, we focused on the local energetics of catalysis taking place on one of the two surface –Fe–C–Si– locations. The structural notation is based on the number of C and H atoms in the local structure of interest. For instance, structures ‘1.2’ and ‘3.2’ indicate –Fe–CH2–Si– and –Fe–C(C2H2)–Si–, respectively. Variations of structures comprising the same number of atoms are distinguished with a letter, such as ‘2.4a’ and ‘2.4b’. The obtained ab-initio energies of each structure were recalculated in terms of formation energies on the basis of CH4 and H2 and are summarised in Supplementary Data 1.

The possible reaction interconnecting structures were classified into five different categories. All the reaction categories include both forward and reverse reactions, but for concision, we denote them simply as H2 abstraction, transformation, dehydrogenation, CH4 insertion, and adsorption. The gaseous molecules involved in each reaction category are shown at the bottom of Fig. 1c.

Among the 108 reactions presented here, 42 transition states were randomly selected, and their formation energies were scaled with initial-state and final-state formation energies in each category, as shown in Fig. 1d and Supplementary Fig. 1a, respectively. The activation barriers for the adsorption (or desorption) of surface hydrocarbon species were calculated to be negligible and thus are not shown here. The initial-state scaling appeared to be slightly more suitable than the final-state scaling for all reaction categories. The coefficient of determination (R2) values of the initial-state scaling were 0.89, 0.64, 0.97, and 0.99 for H2 abstraction, transformation, dehydrogenation, and CH4 insertion, respectively, whereas those of the final-state scaling were 0.72, 0.05, 0.86, and 0.79, respectively. The Brønsted–Evans–Polanyi (BEP) relation was also applied to the scaling, but this approach was found to be much weaker than both transition-state scalings, showing R2 values of 0.19, 0.01, 0.87, and 0.88, respectively. Although the transition-state scaling was not necessarily stronger than the BEP relation12, previous studies have shown that the transition-state energies of particular surface reactions, such as hydrocarbon combustion and formation at high temperatures, are more suitably estimated by transition-state scaling13,14. According to Hammond’s postulate15, we can speculate that the transition states are structurally more similar to the initial states than to the final states. We note that reactions in the transformation category exhibited weaker proportionality than other categories regardless of the scaling method because various types of bond cleavage and formation are combined differently for each individual transformation between surface hydrocarbon species. The initial-state scaling derived as described above for each reaction category was used to predict the remaining transition-state energies for microkinetic modelling.

To investigate the role of the catalyst in C–C coupling, a typical pathway to produce ethylene on the surface was chosen in the reaction network, and its energetics were compared with that of the gas-phase reaction pathway. We note that CH3 radicals were assumed to be generated by the dehydrogenation of CH4 on the catalyst surface, as illustrated in the reaction network (Fig. 1); this applies to both the catalytic and gas-phase C–C coupling pathways. In the case of catalytic C–C formation (blue pathway in Fig. 2a), methane insertion to the surface (1.0 → 2.4b), dehydrogenation to produce surface radical species (2.4b → 2.3f), CH3 addition to form surface C2 species (2.3f → 3.6e), and remaining dehydrogenation followed by ethylene desorption (3.6e → … → 1.0) were assumed to occur in turn. In the case of gas-phase C–C coupling (green pathway in Fig. 2a), two CH3 radicals come together without surface mediation in the initial reaction stage (2CH3 → C2H6), followed by ethane insertion (1.0 → 3.6e). After hydrogen transfer from the adsorbate to surface carbon (3.6e → 3.6b), the surface ethylene desorbs, and surface hydrogen is removed by ·CH3 (1.2 → 1.1 → 1.0). We note that the essential difference between the two pathways is the C–C coupling manner to form surface C2 species (3.6e); additionally, the ethylene desorption routes (from 3.6e to 1.0) were set differently for auxiliary comparisons.

Figure 2b shows potential energy profiles, where only electronic energies obtained using DFT calculations are plotted. As expected from the highly unstable nature of CH3 radicals, gas-phase C–C coupling appeared to be significantly preferred over catalytic C–C coupling when comparing up to the formation of surface C2 species (3.6e). When the same pathways were plotted in terms of free energy calculated assuming typical reaction conditions (temperature of 1300 K and partial pressures of ·CH3 and CH4 of 0.01 and 0.99 bar, respectively), the gas-phase C–C coupling was also favoured (Fig. 1c). In this case, however, the insertion of ethane for subsequent dehydrogenation to form ethylene required a significant amount of energy comparable to that required for catalytic ethylene formation.

The above results indicate that the methane consumption rate and C–C coupling path can vary depending on conditions such as the reaction temperature and partial pressures of the gas components. That is, when the reaction proceeds at a low temperature, where ·CH3 radicals are barely present, gas-phase C–C coupling followed by catalytic dehydrogenation is more favourable than catalytic C–C coupling and subsequent dehydrogenation. In contrast, at a high temperature with a relatively high partial pressure of ·CH3, both reactions take place simultaneously at a similar rate.

The complexity of the overall reaction, in which the relative rates of individual reactions depend on the reaction conditions, is shown in Fig. 3. Changes in the free energy of activation for typical dehydrogenation and CH4 insertion are visualised as a function of CH4 conversion and reaction temperature (Supplementary Fig. 2). Here, we assumed that CH4 is converted to only·CH3 and H2, and so the partial pressures of CH4 and CH3 varied linearly depending on CH4 conversion. The increases in free-energy barriers with reaction temperature arise from negative entropies of activation, suggesting that the transition state is achieved by two reaction components in an associative manner16. The free energy barriers of dehydrogenation and CH4 insertion cross each other at various points for both variables. This complicates analysing the mechanism by comparing pathway energetics or descriptors.

### Microkinetic analysis

It is helpful to understand this complex reaction through plotting the overall reaction rate by considering individual reactions simultaneously. Figure 3 shows how the CH4 consumption rate and production rates change with reaction temperature and relative partial pressures of ·CH3 and CH4. We note that all the reactions presented in Fig. 1c were considered in this microkinetic analysis, except for gas-phase C–C coupling, to focus on the role of the Fe single-atom catalyst. The CH4 consumption rate monotonically increases with reaction temperature at a fixed ·CH3 partial pressure. On the other hand, the maximum rate occurs when the ·CH3 concentration varies at a fixed reaction temperature. The large quantity of ·CH3, which is the main reaction product, in the reactant phase is expected to be detrimental in terms of thermodynamics. However, the ·CH3 radicals also serve as an activator which removes the intermediates attached to the monoatomic active site.

As shown in Fig. 3b, the main products from the catalytic non-oxidative coupling of methane are·CH3 and H2, whose production rate profiles are almost identical to that of the CH4 consumption rate. The production rates of C2-hydrocarbons were an order of magnitude less than those of ·CH3 and H2. In particular, the ethylene formation rate is negligible compared with the acetylene formation rate, indicating that acetylene is the major product from the catalytic C–C coupling reaction. The C2 compounds are mainly produced at a high ·CH3 concentration (log($${P}_{{\mathrm{CH}}_{3}}$$) > −2), which suggests that the CH3 radical plays a major role in C–C coupling on the catalyst. However, when the reaction temperature is above 1100 K, the production rates of C2 compounds decrease, since they are easily decomposed, and the reaction again produces ·CH3 and H2.

The product selectivity is shown in Fig. 4a according to changes in the reaction temperature and ·CH3 concentration. As expected from the production rate (Fig. 3), CH4 decomposes mainly into ·CH3 and H2 at temperatures above 1300 K, whereas catalytic C–C coupling to produce acetylene is dominant at and below 1200 K. We note that the ·CH3 radicals also produce CH4 molecules by taking surface H species in the acetylene-production region, which subsequently lowers the CH4 consumption rate.

Since the major C2 product was identified as acetylene, the predominant pathway for acetylene formation was traced based on individual reaction rates. Two competitive reaction pathways were found as shown in Fig. 4b. Both pathways begin with a clean surface (1.0), but one proceeds by forming intermediate 2.4a, where CH3 is attached to surface carbon (2.4a, pink lines in Fig. 4b and c), whereas the other proceeds by forming intermediate 2.4b, where CH3 is attached to surface Fe (2.4b, yellow lines in Fig. 4b and c). The free energy diagram (Fig. 4c) shows that both pathways have comparable reaction barriers. However, microkinetic modelling of only these pathways shows that the pathway forming 2.4b (TOF = 1.3 × 10−4) is slightly faster than the other path (TOF = 5.9 × 10−6). This is presumably because the former proceeds via fewer reaction steps than the latter.

The degree of rate control (DRC) was calculated as an indicator to determine how the net production rate for each product is affected by the individual reaction (Eq. 1). The higher the absolute value of sensitivity is, the greater the effect on the overall reaction rate. Figure 5 shows the DRCs of selected states for ·CH3 and acetylene production with absolute values higher than 0.1. The DRCs for all the states considered in the present study are shown in Supplementary Fig. 3. If the value of a state is negative, the state must be destabilised to increase the overall reaction rate. On the contrary, if the value is positive, the overall reaction rate increases as the state stabilises. Therefore, in general, negative-valued states are strongly adsorbed species, which slow the subsequent reaction, and positive-valued states are often transition states at the rate-determining step with a large kinetic barrier.

The production rates of ·CH3 and acetylene were determined to be largely influenced by several common steps. In particular, the structure where the surface carbon is terminated with three hydrogens (1.3a, structure given in Supplementary Data 1) appears to be too stable for all reaction conditions. This is because once this structure forms, further reactions hardly proceed on the catalyst surface. On the other hand, in order to accelerate the overall reaction for both products, transition states 1.2–1.3a and 1.0–1.2 (see Supplementary Data 1 for structures) must be stabilised at high and low ·CH3 partial pressures, respectively.

In the case of ·CH3 production, the activation barriers of CH4 insertion (1.0–2.4a and 1.0–2.4b) must be stabilised under abundant ·CH3 conditions (log($${P}_{{\mathrm{CH}}_{3}}$$) > −3). However, acetylene production requires higher hydrogen transfer rates on carbon-attached structures (2.2a, 2.2b, etc.) under CH3-deficient conditions (log($${P}_{{\mathrm{CH}}_{3}}$$) < −3). Discrepancies in the optimal conditions for both reactions suggest that increasing the CH4 conversion and C–C coupling rate simultaneously is difficult over this catalyst.

### Reaction test

To experimentally confirm the role of catalysis in non-oxidative methane coupling, a SiO2-confined Fe catalyst (Fe@CRS, see experimental section within the Methods for the detailed preparation procedure) was prepared according to a previous report9, and a series of experiments was carried out. In order to elucidate the structure of the Fe@CRS catalyst, an Fe K-edge X-ray absorption study was performed (Fig. 6). In the X-ray absorption near-edge spectra, the spent Fe@CRS catalyst exhibited a white line at a position similar to those of the Fe foil and Fe3C spectra, whereas the fresh catalyst showed an oxide-like spectrum. This indicates that the initial FeOx clusters in the catalyst decomposed during the reaction and transformed into FeCx. The extended X-ray absorption fine structure of the spent Fe@CRS catalyst demonstrates that a SiO2-confined Fe structure with Fe–C and Fe–Si bonds was dominantly formed5, which validates our simulated model of the Fe single-atom surface structure.

Figure 7 shows the experimental results obtained using the Fe@CRS catalyst. Since the reaction occurs throughout both free space and on the catalyst surface in the heating zone of the reactor, we decoupled the variables into gas-phase residence time and catalyst contact time instead of using space velocity as a variable. Catalyst contact time had a great influence on increasing the CH4 conversion (Fig. 7a) than gas-phase residence time, indicating that C–H activation was mainly catalysed by Fe@CRS even though gas-phase activation also contributed to some extent. Catalytic CH4 conversion appears to be more active at high gas-phase residence times. This is because the radicals generated in the gas-phase reaction expose the active sites by removing adsorbed species from the catalyst surface, as predicted by the simulation.

Both the ethane and ethylene selectivities (Fig. 7b) increased with decreasing catalyst contact time, indicating that the gas-phase reaction is the major route to produce ethylene from ethane cracking. However, acetylene production appears to proceed mainly by catalysis, as the selectivity increased with catalyst contact time. These results agree with the above-discussed calculations showing that acetylene is the main C2 product afforded by catalysis. Benzene selectivity increased with catalyst contact time as well as gas-phase residence time, which supports that the trimerization of acetylene occurs in the gas phase. Other hydrocarbons, including C3, C4, and C5, were also produced during the reaction, but their total amount was as low as 20–30% compared with that of C2-hydrocarbons (see Supplementary Table 1). Coke deposition was also found to largely occur in some cases. It is noteworthy that the formation of C3–C5-hydrocarbons and coke increased with gas-phase residence time, indicating that the gas-phase radical reaction is considerably involved in these chain-growth reactions. Thus, further studies on the gas-phase reactions are necessary to aid in the design of catalysts and reactors.

## Discussion

In non-oxidative methane coupling, where a gas-phase radical reaction and surface catalysis are interconnected, the role of an Fe single-atom catalyst was investigated using DFT and microkinetic analyses. Based on these theoretical studies, we estimated that catalytic C–H activation occurs actively when the reaction temperature is above 1300 K, and most CH4 is converted into CH3 and H2. At lower reaction temperatures, the catalyst also appears to mediate the C–C coupling reaction, thereby producing mainly acetylene rather than ethylene. Both the C–H activation and C–C coupling reactions on the catalyst surface were calculated to be accelerated in the presence of CH3 radicals, which serve as scavengers to regenerate the catalytic active sites. However, a sensitivity analysis revealed that acetylene and ·CH3 productions cannot be simultaneously promoted by changing the reaction conditions. The product distribution estimated by the computational analyses was demonstrated by a series of experiments. Experimental non-oxidative methane coupling conducted using the Fe©CRS catalyst at 1293 K showed that methane conversion and acetylene selectivity increased with increasing catalyst contact time. The results of this study are valuable towards distinguishing the role of the catalyst in this complex reaction. However, it is necessary to consider additional gas-phase reaction kinetics to reveal the coke formation mechanism and design efficient reactors to further increase the yield of C2-hydrocarbons.

## Methods

### Computational

Electronic structure calculations were performed with DFT using the Vienna Ab-Initio Simulation Package17,18 implementing a plane-wave basis set. The vdW-DF2 exchange-correlation functional19,20 was used to take dispersion interactions into account within a generalised gradient approximation. The cut-off energy of the plane-wave basis was set to 450 eV, and the convergence criterion for the total energy between electronic steps was set to 1 × 10−4 eV.

The (001) surface of β-tridymite (SiO2, P63/mmc) embedded with a single Fe atom was used to simulate the catalyst according to a previous study5. Prior to generating the substrate structure, the lattice parameters of tridymite were optimised using the bulk structure to a = 5.28 Å and c = 8.65 Å. The substrate slab was built to contain a 4 × 4 surface SiO2 unit repeated periodically. Three SiO2 vertical layers were separated by a vacuum size of 10 Å. One surface Si atom and its two adjacent surface O atoms were replaced with an Fe and two C atoms, respectively. The top two layers of the slab were allowed to fully relax during geometry optimisations, whereas the bottom layer was kept fixed at the theoretical bulk-terminated geometry of tridymite.

The free energies of the gas and adsorbed species were calculated based on the ideal gas limit and harmonic limit, respectively, using the thermochemistry module implemented in the Atomic Simulation Environment package21. Vibrational frequencies were obtained using Hessian matrix calculations and normal mode analysis within a finite difference approximation. Transition-state geometries for the reactions were searched using the climbing-image nudged elastic band method and confirmed by frequency calculations to give one imaginary frequency.

Microkinetic modelling on the catalyst was performed under the mean-field approximation using the CatMAP software package22. The elemental reaction equations used for the microkinetic modelling are provided in the Supplementary Data 2. The contribution of each species, including transition-state species, to the overall reaction rate was estimated by the DRC23,

$${\mathrm{DRC}} = \frac{{\partial \,\ln \,r}}{{\partial \left( {\frac{{ - G_i}}{{k_{\mathrm{B}}T}}} \right)}},$$
(1)

where r is the overall reaction rate, Gi is the Gibbs free energy for species i, kB is Boltzmann’s constant, and T is temperature.

### Experimental

A cristobalite silica lattice-confined 0.4 wt% Fe catalyst (Fe@CRS) was synthesised according to the previous report9. Synthesised nanocrystalline fayalite (Fe2SiO4) was mixed with commercial quartz (Kanto) at a ratio of 1:107. The solid mixture was then milled with three millimeter of zirconia balls at 250 rpm for 15 h without exposing O2 for 15 h using a Pulverisette 7 premium line (Fritsch, Germany) apparatus. The resulting solid underwent the melt-fusing process at 1973 K for 6 h in air at a heating rate of 10 K min−1 leading to a nonporous material (SBET = 0.22 m2/g) with a high Fe3O4 dispersion.

Reaction measurements for non-oxidative methane coupling were carried out using a fixed-bed reactor system equipped with a quartz tube with an inner diameter of 7 mm at 1293 K. The tubular reactor showed a uniform profile by bringing R-type thermocouples in direct contact with the outer surface. The reaction proceeded by feeding 90% CH4/10% Ar mixed gas at the reaction temperature. The gas phase residence time (s) or catalyst contact time (s) was calculated by dividing the void space (cm3) or solid volume (cm3), respectively, by the feed flow rate (cm3/s). The void space throughout which the gas-phase reaction proceeds was calculated by excluding the solid volume derived from the skeletal density of the catalyst from the reactor volume. The skeletal density and particle size of the Fe@CRS catalyst were 2.40 g/cm3 and 425–850 μm, respectively9. The catalyst contact time and gas-phase residence time were varied by changing the flow rate and catalyst loading. The gas-hourly space velocity was varied from 0.1 to 0.6 s−1, and the catalyst loading was varied from 0 to 12 g. We analysed gaseous reactants and products using an online GC (YL 6500GC, Younglin) equipped with one TCD connected with a ShinCarbon ST column (Restek, Catalogue No. 80486-800) and two flame ionisation detectors (FIDs) connected with an RT-Alumina BOND column (Restek, Catalogue No. 19756) and an Rtx-VMS column (Restek, Catalogue No. 49915). The methane conversion and product selectivity were determined according to the carbon balance based on the GC analysis.

X-ray absorption spectroscopy (XAS) measurements was conducted at beamline 8C (Nano XAFS) of the Pohang Accelerator Laboratory, operated at an energy of 3.0 GeV and a ring current of 70–100 mA. The obtained XAS results were analysed with the software package Athena24.