Introduction

The direct oxidation methane-to-methanol (DOMM) under mild conditions is promising for producing basic chemicals and reducing greenhouse gas emissions1,2,3. However, DOMM still faces two enormous challenges: low reactivity and poor CH3OH product selectivity, which result from the high bond energy (434 kJ mol−1) of CH4 and the facile overoxidation of CH3OH to CO24,5. In recent years, Pd catalysts have been widely used for DOMM because the electron-deficient d orbitals of Pd atoms can coordinate the C-H bonds through ds hybridization to activate CH46,7,8,9,10. Moreover, the H2O2 generated in situ from O2 and H2 on the Pd surface provides abundant active oxygen species to accelerate the cleavage of C-H bonds7,11. Although numerous strategies, including alloying1,6, doping12,13 or loading14,15,16, have been used to modify Pd catalysts, further DOMM performance enhancements are urgently required.

Unveiling the origin of highly active sites by establishing a definitive structure–performance relationship is key to developing high-performance catalysts. Generally, the catalytic performance of metal nanocrystals strongly depends on particle shape and size, especially facet or surface structures. Single-crystal faceted nanocrystals as model systems can reveal the facet–activity relationship resulting from differences in surface atomic strain17,18, orientation19,20, and ligancy21,22, etc., which cause significant variations in the adsorption, activation, and conversion behavior of reactants23,24,25. For example, during the semi-hydrogenation of acetylene to ethylene, compared with the Pd {100} facet, which shows tensile strain, the Pd {111} facet, which is under compressive strain, has lower ethylene adsorption energies as a result of the downshift in the d-bond center of Pd, preventing the over-hydrogenation of acetylene17. For O2 activation on Pd {100} and {111} facets, the chemisorption of O2 is more favorable on the Pd {100} facet, reducing the magnetic moment to nearly zero and resulting in the generation of highly reactive singlet O2 from inert triplet O226,27.

The size–activity relationship of metal nanocrystals is related to the proportion of surface atoms (e.g., face, edge, and corner) and the electronic structure of the entire particle28. For example, in the aerobic oxidation of benzyl alcohol, the higher selectivity of larger Pd nanoparticles is a result of the larger proportion of high-coordination sites29, whereas for electrocatalytic CO2 reduction, smaller Pd nanoparticles provide more edge and corner atoms with low-coordination sites28, which is beneficial for CO2 adsorption, COOH* formation, and CO* removal, thus enhancing catalytic performance. Therefore, the underlying structure−property relationships of Pd nanocrystals for DOMM involved the facet- and size-dependent behavior is an intriguing issue; however, it has not yet been investigated.

In this study, two types of well-defined Pd nanocrystals, denoted {111}-dominated octahedral and {100}-dominated hexahedral nanocrystals, respectively, were synthesized using a one-step seed-mediated growth method. The {111}-faceted Pd nanocrystals exhibit significantly higher activity compared to the {100}-faceted counterparts, as well as previously reported Pd-based catalysts, achieving a remarkable methanol yield of 201.8 mmol·gPd−1·h−1 6,7. Experimental analysis, combined with density functional theory (DFT) calculations, demonstrate that regardless of the size of the octahedral or hexahedral Pd nanocrystals, the activity is determined by the exposed facets, an effect dominated by the d-orbital electronic structure of the surface Pd atoms. Crucially, the Pd {111} facet with a lower d-band center exhibits the weaker adsorption strength on key intermediates, such as *O2, *OH, and *CH3, enhancing the generation of H2O2 and reducing the energy barrier for CH4 activation and CH3OH formation, respectively, and resulting in DOMM activity three-times higher that of the Pd {100} facet.

Results

Synthesis and characterizations of Pd nanocatalysts

Figure 1a illustrates a schematic of the preparation of the Pd nanocrystals with controlled facets and sizes. First, Pd nano-cubes (denoted c-Pd) were prepared using a well-known wet-chemical protocol30,31. Subsequently, Pd nano-octahedra (denoted o-Pd) were obtained using the one-step seed-mediated growth method with c-Pd as seeds. The size of the c-Pd can be regulated by adding surface passivating ions (Br⁻ and Cl⁻), thereby indirectly controlling the size of the o-Pd nanoparticles. X-ray diffractometry (XRD) measurements revealed that the c-Pd and o-Pd nanocrystals having different sizes were the typical face-centered cubic phase (Fig. 1b, c), consistent with metallic Pd (JCPDS No. 46-1063)32. The transmission electron microscopy (TEM) images demonstrate that the synthesized Pd nanocrystals have two morphologies, hexahedral and octahedral, and four distinct sizes, having average edge lengths of approximately 5, 15, 30, and 48 nm (Fig. 1d–k). The high-resolution TEM images (HRTEM) clearly show the {100} and {111} facets on hexahedral and octahedral Pd nanocrystals (Fig. 1d1–k1). The integrated pixel intensities of the Pd (100) and (111) lattices indicate average Pd lattice spacings of 0.20 and 0.22 nm, respectively (Fig. 1d2–k2). X-ray photoelectron spectroscopy measurements indicate metallic Pd0 for both c-Pd and o-Pd nanocrystals, and the small quantity of Pd2+ species was attributed to spontaneous oxidation in air (Supplementary Fig. 1)33.

Fig. 1: Synthesis and characterization of Pd nanocatalysts.
Fig. 1: Synthesis and characterization of Pd nanocatalysts.The alternative text for this image may have been generated using AI.
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a Preparation schematic of c-Pd nanocrystals enclosed by the {100} facet and o-Pd nanocrystals enclosed by the {111} facet. XRD patterns of b c-Pd and c o-Pd at different sizes. TEM images of c-Pd with average edge lengths of d 5, e 15, f 30, and g 48 nm. TEM images of o-Pd with average edge lengths of h 5, i 15, j 30, and k 48 nm. HRTEM images and size distributions in insets.

Performance of direct CH4 conversion

The catalytic DOMM performance of the c-Pd enclosed by the {100} facet and o-Pd nanocrystals enclosed by the {111} facet under mild conditions was evaluated in a stainless-steel high-pressure reactor (Supplementary Fig. 2). As illustrated in Fig. 2a, all o-Pd nanocrystals exhibited higher CH3OH yields of 2.5–3.2 times than those of the c-Pd nanocrystals. With the increase in the average particle size, the CH3OH yield gradually decreased for both nanocrystals; however, the o-Pd nanocrystals (5 nm) demonstrated the maximum CH3OH yield of 201.8 mmol gPd−1 h−1 (cf. 64.6 mmol gPd−1 h−1 for c-Pd nanocrystals (5 nm)). In addition, the CH3OH selectivity of the o-Pd nanocrystals was also slightly higher than that of the c-Pd nanocrystals. However, only surface atoms act as active sites, determining the adsorption/desorption of reactants and the subsequent catalytic route; that is, internal (bulk) atoms are not involved in the reaction34. Therefore, to reveal the effect of facet-dependent catalytic performance accurately, the DOMM performance was evaluated assuming an equal quantity of surface Pd atoms on the four different sizes of o-Pd and c-Pd nanocrystals (Supplementary Fig. 3 and Supplementary Tables 1, 2). Regardless of average size, from 5 to 48 nm, the c-Pd or o-Pd nanocrystals produced the same CH3OH yield with respect to normalized surface atoms, indicating a zero-order correlation between catalytic activity and different sizes (Fig. 2b). However, taking the 5-nm Pd nanocrystals as an example, the CH3OH yield of the surface Pd atoms enclosed by {111} facet was about 543.1 mmol gSA−1 h−1, approximately 2.14-times than that of the Pd atoms on the {100} facet (Fig. 2b and Supplementary Table 3).

Fig. 2: DOMM performance.
Fig. 2: DOMM performance.The alternative text for this image may have been generated using AI.
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a Catalytic performance of DOMM of c-Pd nanocrystals enclosed by the {100} facet and o-Pd nanocrystals enclosed by the {111} facet. b Catalytic performance normalized by surface atoms for both Pd nanocrystals with average sizes from 5 to 48 nm. c Recycling tests for o-Pd nanocrystals. d Catalytic performance with different reaction times for o-Pd nanocrystals (Color consistency is maintained between the top-left panel of Fig. 2d and its bar chart for DOMM product representation.). e Comparison of catalytic performance and CH3OH selectivity for CH4 direct conversion with various catalysts. All reaction conditions: 10 mL of water, 20.5 mg of catalyst, 70 °C, feed gas at 3.0 MPa with 1.1% H2/2.2% O2/67.2% CH4/20.57% Ar/8.93% He. Each reaction was tested two times to obtain the error bars.

The turnover frequencies (TOFs) represent the intrinsic activity of each atom, and the surface Pd atoms on the {111} facet demonstrated higher TOFs than those of the {100} facet, for example, 76.6 h−1 vs. 33.1 h−1 for Pd nanocrystals (5 nm) (Supplementary Table 4). Thus, the o-Pd nanocrystals enclosed by {111} facet promoted DOMM, yielding more CH3OH than c-Pd nanocrystals enclosed by the {100} facet, mostly because of the higher intrinsic activity of the former for the DOMM (Supplementary Table 4). In addition, durability tests of o-Pd nanocrystals demonstrated that the CH3OH yield and selectivity decreased negligibly after 12 reuse cycles, indicating a robust structure (Fig. 2c and Supplementary Fig. 4). In addition, the relationship between reaction time and DOMM performance showed that in the initial stage (0.05 h), the liquid products were the CH3OOH and some CH3OH (Fig. 2d). Subsequently, the amount of CH3OOH rapidly decreased, gradually disappearing with the increase in reaction time. Meanwhile, the CH3OH yield gradually increased from 74.3 mmol gPd−1 h−1 at 0.05 h to 201.8 mmol gPd−1  h−1 at 0.5 h, and the CH3OH yield was constant beyond 0.5 h (Supplementary Figs. 5, 6 and Supplementary Table 5). Additionally, compared with those of reported advanced catalysts, the DOMM performance of the o-Pd nanocrystals enclosed by the {111} facet, i.e., 201.8 mmol gPd−1 h−1 with 99.7% selectivity, is the highest (Fig. 2e and Supplementary Table 6).

In situ characterization toward mechanism

To understand the origin of the facet-dependent catalytic behaviors for the DOMM, CH4 temperature-programmed desorption (TPD) measurements were used to assess the chemisorption capacity of CH4 molecules on the Pd {100} and {111} facets (Fig. 3a)35. Overall, the o-Pd nanocrystals had a lower desorption temperature (approximately 377 °C) than the c-Pd nanocrystals, which had a desorption temperature of 401 °C, suggesting that the surface Pd atoms on the {111} facet chemisorbed CH4 more weakly than those on the {100} facet. In addition, the active oxygen species were also key to accelerating the cleavage of the C-H bonds of CH4 molecules11, and their effects were tested via comparative experiments (Supplementary Table 7). Product analysis demonstrated that the in situ generation of H2O2 derived from the H2/O2 mixture promoted the DOMM process by approximately 75.3 and 31.1 times compared to reactions with (H2O2 + O2) or H2O2 as active oxygen species. As illustrated in Fig. 3b, the o-Pd nanocrystals had a higher H2O2 yield (32.9 mmol gPd−1 h−1) than the c-Pd nanocrystals (15.0 mmol gPd−1 h−1), implying that the {111} facet Pd atoms enhanced the in situ generation of H2O2, which is related to DOMM performance. In addition, the desorption of chemisorbed H2 and O2 on the surface Pd atoms was related to the stable formation of the *OOH intermediate, which determined the formation of H2O2. In addition, the desorption temperatures of H2 and O2 chemisorbed on {111} surface Pd atoms were lower than those on the {100} facet (Supplementary Fig. 7), demonstrating that the Pd {111} facet forms the *OOH intermediate more favorably as a result of the weak affinity of Pd atoms and O atoms, thus preventing O-O bond cleavage36. Moreover, electron paramagnetic resonance (EPR) spectroscopy was used to explore the key radical species37. As shown in Fig. 3c, DOMM over o-Pd nanocrystals yielded strong *OH and *CH3 radical signals, suggesting that the in situ generation H2O2 provided abundant *OH radicals, which could break the C-H bonds to form *CH3 radicals38. To investigate the changes in the species adsorbed on the surface of the o-Pd nanocrystals during the DOMM process, in situ diffuse reflectance infrared Fourier transform spectroscopy (DRIFTS) measurements were conducted (Fig. 3d). When the CH4/O2/H2 mixed gas flowed over the surface of the o-Pd nanocrystals, distinct adsorption bands at 3010 and 1300 cm−1 were observed, attributed to adsorbed *CH4 species39. The band at 1653 cm−1 corresponds to adsorbed *OH species, implying the decomposition of the in situ generated H2O2 to active *OH species. The band at 1344 cm−1 represents the deformation vibrations of adsorbed *CH3 species, indicating C-H bond activation of CH4 by active *OH species40,41,42. In addition, the bands at 1059 and 825 cm−1 correspond to the stretching vibrations of HOCH3* and *OOH species. The band related to the stretching of the C-O bond was observed at 2822 cm−1 because of the presence of *OCH3 species, and the band corresponding to peroxy-species (*OO) species adsorbed on Pd atoms was observed at 933 cm−143. As the reaction time was extended to 0.5 h, the band intensity of the C-O stretching vibration derived from the *OCH3 and HOCH3* species increased. Concurrently, there was a gradual decrease in the band intensities of *CH3, surface peroxide, and *OH species. Collectively, these observations reveal the reaction route of the DOMM process on the o-Pd surface44. In addition, almost-negligible bands at 2378 and 2310 cm−1 were assigned to the antisymmetric stretching vibrations of CO2 caused by the peroxidation of CH4 (Supplementary Fig. 8)42.

Fig. 3: DOMM mechanism.
Fig. 3: DOMM mechanism.The alternative text for this image may have been generated using AI.
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For c-Pd nanocrystals enclosed by the {100} facet and o-Pd nanocrystals enclosed by the {111} facet, a CH4-TPD and b in situ generated H2O2 productivity at 70 °C in O2 and H2 atmosphere. For o-Pd nanocrystals enclosed by the {111} facet, c EPR spectra of radical species (*CH3 and *OH), and d in situ DRIFTS spectra of adsorbed CH4, O2, and H2 at 70 °C.

Theoretical calculations

Next, DFT calculations were conducted to clarify the underlying mechanism of the facet-dependent catalytic DOMM behavior on the Pd {100} and {111} facets. The whole DOMM process was tightly related to the reaction triggering and conversion steps, where the former is the decomposition of H2O2 into *OH radicals, and the latter involves two elemental reactions, including OH-induced C-H bond activation and CH3OH formation11. The in situ generated H2O2 from O2/H2 was related to the formation of *OOH intermediates, which were affected by the adsorption energy of *O2 (ΔGads_OO) (Supplementary Fig. 9)45. Compared to the Pd {100} facet, the weaker ΔGads_OO on the Pd {111} facet enhanced H2O2 production (Supplementary Figs. 10a and Supplementary Table 8), aligning with experimental observations (Fig. 3b)11,46. Moreover, both Pd {100} and Pd {111} were capable of generating *OH rapidly via H2O2 decomposition with tiny barrier energy (ETS_OH, both ETS_OH < 0.1 eV) (Fig. 4a). Therefore, the initial reaction triggering step was mainly controlled by the adsorption energy of *O2, and a relative weaker *O2 adsorption energy on the Pd {111} facet resulted in higher *OH radical concentrations (Supplementary Fig. 10b).

Fig. 4: DFT calculations.
Fig. 4: DFT calculations.The alternative text for this image may have been generated using AI.
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DFT calculations on the Pd {100} and Pd {111} facets. a Reaction barrier diagram for the dissociation of H2O2 to *OH. b Reaction potential energy diagram of DOMM process. c DOS patterns. d ICOHP of *CH3 and *OH adsorbed on both facets. e Reaction rate indicator χ’ of on both facets. Thermodynamic and kinetic data on the Pd {111} facet were taken from our previously published work11,48.

Additionally, during the subsequent reaction conversion step, the OH-induced C-H bond activation to yield *CH3 radicals, resulted in the formation of Pd-C bonds and cleavage of Pd-O bonds, and CH3OH formation was related to the cleavage of Pd-C/O bonds (Supplementary Fig. 11). The DFT calculations indicate that the Pd {100} and {111} facets have similar energies for the adsorption of *CH3 species, −2.09 and −2.11 eV, respectively (Supplementary Figs. 10c). However, the adsorption energy of *OH species on the Pd {111} facet (ΔGads_OH = −2·70 eV) was obviously weaker than that on the Pd {100} facet (ΔGads_OH = −3·01 eV) (Supplementary Fig. 10d). These lead to a reduction in the energy consumption required to break the Pd-O bond during both the OH-induced C-H bond activation and CH3OH formation process on the Pd {111} facets. The reaction energy barrier of the OH-induced C-H bond activation (ETS1) on the Pd {100} and {111} facets were 1.21 and 0.90 eV, respectively, demonstrating that the C-H bond was more easily activated and cleaved on the Pd {111} facet (Fig. 4b). Further, the reaction energy barrier for CH3OH formation (ETS2) on the Pd {100} facet (ETS2 = 1.51 eV) was higher than that on the Pd {111} facet (ETS2 = 1.42 eV), indicating that the Pd {111} facet promoted CH3OH formation to a greater extent. Based on the aforementioned analysis, the differential adsorption intensity of *OH species on both Pd facets, rather than *CH3 species, determined the OH-induced C-H bond activation and CH3OH formation, and the weaker *OH adsorption on the Pd {111} facet promoted the subsequent reaction conversion step.

To reveal the facet–adsorption behavior of both Pd facets and the reaction intermediates, the d-band center energies (εd) of both Pd facets were calculated47. The Pd {111} facet exhibited a lower d-band center (spin-up εd = −6.85 eV and spin-down εd = −6.69) compared to the Pd{100} facet (spin-up εd = −6.66 eV and spin-down εd = −6.53) (Fig. 4c). The downshift in the εd of the Pd {111} facet would result in the weaker adsorption of reaction intermediates, such as *O2, *OH, and *CH3 species, consistent with the calculation results (Supplementary Fig. 9). Notably, the adsorption energy of *CH3 species on the Pd {111} facet was only marginally reduced by 0.02 eV compared to that of the Pd {100} facet. In contrast, the *OH adsorption energy was significantly reduced by 0.31 eV, which can be attributed to the stronger Pd-O bonds than Pd-C bonds, as confirmed by integrated crystal orbital Hamilton population (ICOHP) analysis (Fig. 4d). The significant Pd-O orbital overlap in the stronger Pd-O bond is reflected in the more negative ICOHP value. Thus, the Pd-O bond strength was more sensitive to the shift in the d-band center. Collectively, these findings demonstrate that the catalytic activity of the Pd {111} facet superior to that of the Pd {100} facet originates from the weakening of the adsorption of the key intermediate *OO and *OH species induced by the d-band centers of the different facets, which not only leads to the formation of a high concentration of *OH radical on the Pd {111} facet for participation in the DOMM, but also effectively reduces the reaction barrier in the DOMM, which ultimately yields the superior catalytic activity (Fig. 4e).

Discussion

In this study, we used well-defined, {100}-dominated hexahedral and {111}-dominated octahedral Pd nanocrystals as model systems to elucidate the structure-dependent catalytic behavior for the DOMM. Comprehensive experimental investigations and DFT calculations reveal that DOMM activity is primarily governed by the d-band electronic structure of the exposed crystal facet rather than the particle size. Specifically, the Pd {111} facet, which is characterized by a lower d-band center, facilitates the weaker adsorption of key intermediates (*O2 and *OH), enhances H2O2 generation, and lowers the energy barriers for both C–H bond activation and CH3OH formation. As a result, {111}-dominated octahedral Pd nanocrystals achieve an exceptional CH3OH yield of 201.8 mmol gPd⁻¹ h⁻¹, three times higher DOMM activity than that of the Pd {100} facet. These findings deepen our understanding of the structure−property relationships governing Pd-based catalysts for DOMM and provide a clear pathway for designing advanced catalysts with enhanced performance.

Methods

Chemicals

Sodium tetrachloropalladate(II) (Na₂PdCl₄, 98%), potassium tetrachloropalladate(II) (K₂PdCl₄, 98%), citric acid (CA, 99.5%), and carbon black (Vulcan XC-72R) were obtained from Macklin Reagent. Poly(vinylpyrrolidone) (PVP, ~55,000) and L-ascorbic acid (AA, 99.99%) were sourced from Sigma-Aldrich. Potassium bromide (KBr, 99.99%), potassium chloride (KCl, 99.99%), and formaldehyde (HCHO, 37%) were acquired from Aladdin Reagent and XILONG Scientific, respectively. Deionized (DI) water was used throughout. All chemicals were utilized as received.

Synthesis of cubic Pd nanocrystals (5–48 nm edge lengths)

All cubic Pd nanocrystals were prepared via aqueous polyol reduction at 80 °C using PVP (105 mg) and L-ascorbic acid (60 mg) in 8 mL DI water. For 5 nm cubes, KCl (185 mg) and KBr (5 mg) were dissolved before injecting K2PdCl4 (63 mg in 3 mL DI water). 15 nm cubes used KBr (600 mg) and K2PdCl4 solution (21 mg/mL, 3 mL). 30 nm cubes employed KBr (600 mg) and Na2PdCl4 (57 mg in 3 mL DI water). 48 nm cubes required Pd seed crystals (30 nm, 5 mg) with KBr (600 mg) and Na2PdCl4 (57 mg in 3 mL DI water). Each reaction proceeded for 3 h in capped vials. Products were isolated by centrifugation (7155 × g), washed 3 times with DI water, and redispersed (1.8 mg/mL).

Synthesis of octahedral Pd Nanocrystals (5–48 nm edge lengths)

5 nm octahedra were prepared by heating a mixture of PVP (105 mg), citric acid (180 mg), ethanol (3 mL), and water (8 mL) to 80 °C, followed by K2PdCl4 injection (21 mg/mL, 3 mL). Larger octahedra (15, 30, and 48 nm) were grown using corresponding 5, 15, and 30 nm cubic seeds with HCHO (0.1 mL), PVP (105 mg), H2O (8 mL) and K2PdCl4 solution (21 mg/mL, 3 mL). After 3 h at 60 °C, particles were centrifuged (7155 × g), washed, and redispersed (0.8 mg/mL).

Catalyst preparation (c-Pd/C and o-Pd/C)

Pd nanocrystals (5 mg) were deposited onto Vulcan XC-72R carbon (95 mg) in DI water (50 mL) via ultrasonication and stirring. The mixture was dried at 80 °C. c-Pd/C and o-Pd/C was annealed at 180 °C and 300 °C respectively for 3 h under Ar.

Characterization

TEM/HAADF-STEM was performed on Thermo Talos F200X G2, FEI Titan G2, or JEOL ARM200F instruments (200 kV). XRD used a Rigaku Smart-Lab (Cu Kα, λ = 1.5406 Å). XPS analysis employed a Shimadzu Axis Supra (Al Kα). Liquid products (CH3OH, CH3OOH) were quantified by ¹H NMR (D2O solvent, DMSO internal standard) or GC after CH3OOH reduction with hydrazine hydrate. Gas products (CO2) were analyzed by GC-FID with a methanizer. Productivity and selectivity were calculated as:

$${{\rm{Productivity}}}\left({{\rm{mmol}}}{{{\rm{g}}}}^{-1}{{{\rm{h}}}}^{-1}\right)=\frac{{{CH}}_{3}{OH}({{\rm{mmol}}})}{{{\rm{weigh}}}\; {{\rm{tof}}}\; {{\rm{Pd}}}\left({{\rm{g}}}\right)\times {{\rm{reaction}}}\; {{\rm{time}}}({{\rm{h}}})}$$
(1)
$${{CH}}_{3}{OH}\,{{\rm{selectivity}}}\left(\%\right)=\,\frac{{{CH}}_{3}{OH}({{\rm{mmol}}})}{{{\rm{All\; products}}}({{\rm{mmol}}})}\times 100\%$$
(2)

Catalytic performance test

The catalytic performance test was performed in a 50-mL stainless-steel autoclave containing H2O (10 mL), catalyst (20.5 mg), and feed gas (3.3% H2, 6.6% O2, 1.6% CH4, balance Ar/He) at 3.0 MPa and 70 °C for 0.5 h. After reaction, the stainless-steel autoclave was cooled in ice. Reusability tests involved catalyst recovery by centrifugation (7155 x g), drying (80 °C, vacuum), and reuse.

Turnover frequency (TOF)

$${TOF}=\frac{{Turnover}\; {number}\; {for}\,C{H}_{3}{OH}\; {formation}\,({TON})}{{Number}\; {of}\; {active}\; {site}\,(N)};{{{\rm{units}}}\; {{\rm{is}}}\; {{\rm{h}}}}^{-1}.$$
$${TON}=n(C{H}_{3}{OH})\times {N}_{A};$$

n is the amount of substance of the CH3OH; NA is 6.02 × 1023.

$$N=\frac{m\times W}{M}\times {N}_{A}$$

\({N}\) is the Number of active sites, m is the amount of substance of Pd, M is the relative atomic mass of Pd, W is the mass fraction of Pd in the catalyst.

EPR spectroscopy

Radicals were trapped using DMPO (100 mmol L−¹) in the reaction mixture. Spectra were recorded (Bruker A320 EPR). Control experiments included DMPO with H2O2/Fe2⁺ and H2O2/Fe2⁺/CH3OH.

CH4-TPD test

Samples (100 mg) were pretreated in Ar (400 °C), exposed to 5% CH4/Ar (50 °C, 0.5 h), purged with Ar, then heated to 700 °C (5 °C min⁻¹) under Ar flow (50 mL min⁻¹). Desorbed gases were detected by TCD.

In situ DRIFTS measurements

Pretreated samples (Ar, 100 °C, 1 h) were exposed to reactant gas (1.1% H2, 2.2% O2, 67.2% CH4, balance Ar/He) at 70 °C. Spectra (4000–650 cm⁻¹, 4 cm⁻¹ resolution) were collected over 64 scans.

DFT calculations

Spin-polarized DFT used VASP with PAW pseudopotentials, PBE functional, a 500 eV cutoff, and DFT-D3 van der Waals correction. Solvent effects were modeled via VASPsol. Pd(100) and Pd(111) slabs (4 × 4 × 1 supercells, 4 layers, 15 Å vacuum) were optimized. Transition states were located using CI-NEB. Adsorption energy: Eads = E(X/slab) − E(slab) − E(X). COHP analysis used LOBSTER.