Towards understanding the lower CH₄ selectivity of HCP-Co than FCC-Co in Fischer–Tropsch synthesis

Abstract

In Fischer–Tropsch synthesis (FTS), the cobalt catalyst has higher C₅₊ and lower CH₄ selectivity in the hcp phase than in the fcc phase. However, a detailed explanation of the intrinsic mechanism is still missing. The underlying reason was explored combining density functional theory, Wulff construction, and a particle-level descriptor based on the slab model of surfaces that are prevalent in the Wulff shape to provide single-particle level understanding. Using a particle-level indicator of the reaction rates, we have shown that it is more difficult to form CH₄ on hcp-Co than on fcc-Co, due to the larger effective barrier difference of CH₄ formation and C–C coupling on hcp-Co particles, which leads to the lower CH₄ selectivity of hcp-Co in FTS. Among the exposed facets of fcc-Co, the (311) surface plays a pivotal role in promoting CH₄ formation. The reduction of CH₄ selectivity in cobalt-based FTS is achievable through phase engineering of Co from fcc to hcp or by tuning the temperature and size of the particles.

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1. Introduction

The ultimate goal for catalysis study is to design catalysts with high stability, activity, and selectivity. Although the design principles for catalysts of high stability and activity have been well-established,^1,2 it remains a substantial challenge to understand the relationship between the structure of a catalyst and its selectivity, especially for reactions with complex product distributions. A typical example for which selectivity is intricate is Fischer–Tropsch synthesis (FTS), which converts the simple mixture of CO and H₂ (syngas) to a broad spectrum of chemicals (C₁–C₇₀₊), including methane, paraffins, olefins, and different oxygenated species such as aldehydes, ketones, acids, and alcohols.³ Co-based catalysts are widely used in the low-temperature FTS process, due to their high activity, relatively high selectivity to long-chain paraffins, and low water–gas shift side reactions.^4–7 Still, in the FTS process, methanation is the main competing pathway for chain growth.^8–10 To date, a great deal of effort has been devoted to controlling the product distribution of FTS by changing the catalyst formulations.^11–15

It is generally accepted that the catalytic performances of inorganic solids can be tuned by their composition,^11,16 crystallographic structure (phase),^17,18 size,^19,20 shape,²¹ chelating molecular ligands,^22,23 and support.²⁴ Among these factors, the crystalline phase lies at the heart of the structural parameters for solids of given composition. It determines how the constituents (atoms or ions) are arranged in a highly ordered microscopic structure, and governs the electronic structures and surface atomic geometry of catalysts.^18,25 Besides the most stable phase, the various metastable phases of the catalysts can usually be stabilized by the reactive gas/liquid and often high-temperature environments,^26–29 which makes it possible to modulate the catalytic performances of materials by phase engineering.

Chen et al.³⁰ found that the 4H Cu and 4H/fcc Cu shells exhibit greater CO₂ reduction reaction activity and better C₂H₄ selectivity compared to fcc Cu, demonstrating the crystal phase-dependent C₂H₄ selectivity of Cu. Alumina with different crystal phases shows that CoMo/θ-Al₂O₃ has a much higher hydrodesulfurization selectivity than CoMo/γ-Al₂O₃ and CoMo/δ-Al₂O₃.³¹ The sorption and catalytic behavior observed between the alkaline ferrites studied are associated with the crystal structures and alkali composition.³² In FTS, Gnanamani et al.³³ used a different catalyst reduction process to obtain distinct phases of cobalt, and found that the hcp phase shows higher CO conversion and lower CH₄ selectivity than the fcc phase. Du et al.³⁴ found that the multistep activation process derived catalyst (CoZr/AC-RSCR) with more hcp phase has higher CO conversion and lower CH₄ selectivity than the H₂ reduction derived catalyst (CoZr/AC-R) with more fcc phase.

However, despite the extensive experimental observations of phase-dependent product selectivity of catalysts, the connection between the crystalline phase of a catalyst and its product selectivity remains obscure. A major obstacle in experiments is that the catalyst particles with known crystal phases are composed of multiple facets, due to which their identity and functioning mechanism may be different. It has been proposed by Zhong et al. that different crystal phases have different surface reactivities because of the distinct intrinsic surface strains.³⁵ It has been demonstrated that the isolated Pd sites on the surfaces of the particles determine the high selectivity in acetylene semi-hydrogenation reaction for intermetallic Pd-X (X = Ga or In) materials with multiple phases.³⁶ Several theoretical studies on the CH₄ selectivity of Fe₅C₂ surfaces also showed that the selectivity is highly facet dependent.^37,38

In FTS reactions, the selectivity of products is mainly determined by the competition of the coupling between C + H and C + C.^39,40 Due to the interrelated nature of all individual selectivities indicated by the linear relationship between the individual C₁–C₄ hydrocarbon selectivities and the C₅₊ selectivity,^8–10 CH₄ selectivity can be used as an indicator of the product selectivity in FTS.^41,42 Hydrogenation of carbon species, which is produced through CO dissociation, is the major pathway to CH₄ formation, and it has been well accepted that CH₃ + H → CH₄ is the rate-determining step.^38,41,43 Previously, the CH₄ selectivity of a few intuitively selected Co surfaces has been studied, such as the (100), (110), (111) and (311) of fcc-Co by Yu et al.⁴⁴ and the (0001) of hcp-Co by Cheng et al.⁴⁵ However, as the overall CH₄ selectivity of a Co particle is contributed by all its exposed facets, the fundamental cause for the lower CH₄ selectivity of hcp-Co than fcc-Co remains elusive.

In this work, we propose to use a theoretical approach to investigate the correlation between the crystalline phases of Co (hcp and fcc) and their CH₄ selectivity at the single particle level using the slab model of surfaces that are prevalent in the Wulff shape. CH₄ selectivity of hcp-Co and fcc-Co was evaluated by considering the competition between CH₄ formation and C₁ + C₁ coupling on all the prominent surfaces that covers the exterior of particles based on the Wulff theorem. The facet dependent effective barriers for CH₄ selectivity were then integrated into a particle-level energy descriptor⁴⁶ and compared between hcp and fcc Co. Finally, the conditions for phase transition between hcp and fcc were explored with respect to temperature and nanoparticle size to obtain catalysts with higher performances. We envision that our proposed strategy can effectively scoop out the selectivity information of probe reactions occurring on each facet and thereby reasonably add-up these results to represent those of the single particle level. As such, this approach allows us to understand the difference in the intrinsic selectivity of particles with different crystalline phases. We anticipate that this simplified strategy not only can lessen the gap between theoretical methodology and experimental understanding in cobalt-catalyzed FTS study, but may also be applicable to other catalysis systems to understand the correlation between the crystal phase and product selectivity.

2. Methods and models

2.1 Computational details

All DFT calculations were performed using the Vienna Ab Initio simulation package (VASP).^47,48 The electron–ion interaction was described with the projector augmented wave (PAW) method.^49,50 The electron exchange and correlation energies were treated within the generalized gradient approximation in the Perdew–Burke–Ernzerhof formalism (GGA-PBE).⁵¹ The different cutoff energies are given in Tables S1 and S2 (ESI†); according to the energy and lattice constant of bulk cobalt, the cutoff of plane wave basis set was set up to 450 eV. The Monkhorst–Pack k-point sampling was used. Electron smearing was employed according to the Methfessel–Paxton⁵² technique, with a smearing width σ = 0.2 eV. Due to the large influence of magnetic properties on the adsorption energies, spin polarization was taken into consideration. To locate transition states, we used the Nudged Elastic Band (NEB) method.⁵³ A vibrational frequency analysis was performed to verify whether a transition state is associated with a single imaginary frequency.

2.2 Model

Slab models of surfaces that are prevalent in the Wulff shapes were utilized as proxies of the cobalt particles. For the hcp-Co and fcc-Co surfaces, the optimized lattice constants of hcp-Co (a = b = 2.49 Å and c = 4.03 Å) with a 13 × 13 × 7 k-point agree well with the experimental (a = b = 2.51 Å, c = 4.06 Å)⁵⁴ and calculated (a = b = 2.49 Å, c = 4.03 Å)⁵⁵ values, which are given in Table S1 (ESI†). The optimized lattice constants of fcc-Co (a = b = c = 3.52 Å) with a 13 × 13 × 13 k-point also agree with the experimental (a = b = c = 3.54 Å)⁵⁶ and the calculated (a = b = c = 3.52 Å)⁵⁷ values, which are given in Table S2 (ESI†). We adopted the equilibrium shape of the hcp-Co and fcc-Co particles using Wulff construction at ab initio accuracy reported in the previous work (Fig. 1 and Table S3, ESI†).¹⁸ The hcp-Co particles are predicted to be covered by surfaces of (10−11), (10−10), (0001), (10−12), (11−20), and (11−21), while the fcc-Co particles are covered by the surfaces of (111), (100), (311), and (110). We choose unit cells p(4 × 4) for hcp-Co (0001), (11−21), fcc-Co (100), and (111), and p(4 × 2) unit cells for hcp-Co (10−10), (10−11), (10−12), (11−20) and fcc-Co (110), (311), as shown in Fig. 1 and Fig. S1 (ESI†). A 3 × 3 × 1 k-point sampling was used for all surfaces. The atoms in the top two layers of the slab models for hcp-Co and fcc-Co are fully allowed to relax while the bottom two layers atoms are fixed.

Fig. 1 Wulff construction and the utilized slab models of the prevalent surfaces of hcp-Co and fcc-Co in the Wulff shapes.

2.3 The adsorption energy and reaction barrier

The adsorption energy of C-containing species on cobalt surfaces was calculated according to the equation

E_ads = E_slab/X − E_slab − E_X, (1)

where E_slab/X is the total energy of the slab with the adsorbates. X is equilibrium geometry, E_slab is the energy of the clean surface, and E_X is the energy of the free adsorbate X in the gas phase.

The reaction barrier E_a and the reaction energy E_r were calculated according to the equations:

E_a = E_TS − E_IS (2)

E_r = E_FS − E_IS (3)

where E_IS, E_TS and E_FS are energies of the corresponding initial (IS), transition (TS), and final (FS) states, respectively. We used the zero point energy (ZPE) to correct all energies.

To estimate the relative activity of reactions on the particles of different Co phases, we use the concept of particle effective barrier which is a particle-level indicator of the reaction activity defined previously in the literature.⁴⁵ The particle effective barriers for CH₄ formation (E_{a_particle,CH4}) and the most feasible C₁ + C₁ coupling (E_{a_particle,C1+C1}) and the difference of them (ΔE_{a_particle}) were calculated in terms of rates as follows. First, we assume that the overall reaction rate (r) is the sum of the contributions from all the exposed surfaces. This assumption is valid as it has been recently proven unambiguously that in Co-catalyzed FTS reactions, the edges and vertices of the particles are responsible for the CO dissociation, while the C–C and C–H coupling reactions happen mainly on the flat surfaces of the catalyst particles.^58,59

Based on the Arrhenius formula, the reaction rate r on a particle with i surfaces can be calculated according to the equation^46,60

where r_i, k_i, S_i, A_i and E_a,i are the reaction rate, rate constant, surface area, number of sites per unit area, pre-exponential factor and reaction barrier on the i_th surface. A, S, m are the pre-exponential factor, surface area, and the number of sites per unit area, respectively, on hcp-Co and fcc-Co surfaces. We assume that m_i and A_i on different surfaces are the same and equal to the values on hcp-Co and fcc-Co surfaces and defined as the exposed relative ratio of surface i, which is equal to S_i/S. Thus, the effective barrier of the reaction on the particle E_{a_particle} can be calculated according to the equation

2.4 Phase transition calculation

To obtain the conditions of phase transition between hcp and fcc, the influence of temperature and size on the Gibbs free energy (G) of bulk Co phases was considered. The Gibbs free energy for the solid phase particle is defined as the summation of the Gibbs free energy in the bulk and the surface.^61,62 The Gibbs free energy of bulk hcp-Co and fcc-Co was calculated according to the equation in the literature^63,64

G = E^DFT + E^vib+ E^surf (6)

where E^DFT is the energy per cobalt atom of hcp-Co or fcc-Co. E^vib is the phonon energy per cobalt atom of hcp-Co or fcc-Co at different temperatures. The phonon energies are calculated by using the PHONON software.⁶⁵E^surf is the surface energy per cobalt atom of hcp-Co or fcc-Co at different sizes of cobalt nanoparticles. For simplicity, we assume the cobalt nanoparticles to be spherical of different sizes, and the surface energy⁶⁶ is calculated according to

E^surf = γ_mA_m (7)

where γ_m is the surface energy per unit surface area of hcp-Co or fcc-Co; A_m is the molar surface area:Here, M is the molar mass of hcp-Co or fcc-Co, A is the spherical surface area, ρ is the density of hcp-Co or fcc-Co, and V is the spherical volume.

3. Results and discussion

3.1 CH₄ formation

We first study the elementary steps of CH₄ formation on cobalt surfaces: C + H → CH, CH + H → CH₂, CH₂ + H → CH₃, and CH₃ + H → CH₄. The methanation potential energy surfaces are given in Fig. 2. The structures of the initial state (IS), transition state (TS), and final state (FS) for the elementary steps are provided in Fig. S1 and S2 (ESI†) and the reaction barriers and reaction energies of hcp-Co and fcc-Co are listed in Tables S4–S6 (ESI†). As shown in Fig. 2, the total energy of the TS has an increasing tendency along the hydrogenation reaction coordinate on most hcp-Co and fcc-Co surfaces, and the addition of the fourth hydrogen is usually the rate-limiting step, which is consistent with previous studies.^42,55,67 The increase of the reaction barrier illustrates that C hydrogenation becomes more and more difficult on cobalt surfaces to form CH₄. But on (10−12) and (311), the step with the highest barrier is CH₂ hydrogenation, which is different from the result of Qin et al.⁶⁸ and Yu et al.⁴⁴ This is likely due to the absence of zero point energy corrections in these studies.

Fig. 2 Methanation potential energy surface for C(g) + 2H₂(g) + O → CH₄(g) + O on cobalt surfaces (a) hcp-Co; (b) fcc-Co (relative energy (RE) is the adsorption energy of 4H atoms on cobalt surfaces). Schematic illustration of the effective barrier to CH₄ formation (E_{a_eff,CH4}) (c) hcp-Co (0001), (10−10), (10−11), (11−20), (11−21), fcc-Co (100), (110), (111); (d) hcp-Co (10−12) and fcc-Co (311).

Following the approach of Cheng,^42,69 the effective barriers of CH₄ formation (E_{a_eff,CH4}) were used to evaluate the catalytic activity of the Co surfaces to produce CH₄. As CH₄ formation happens by stepwise C hydrogenation, the rate-determining step is (CH₃ + H → CH₄) on most of the Co surfaces. Before the rate-determining step, the preceding hydrogenation steps may reach quasi-equilibrium. Thus, the rate of CH₄ formation can be calculated as the rate of the rate-liming step:

r_CH4 = A exp[−E_a,4/RT]θ_CH3θ_H = A exp[−(E_a,4 + E_r,3)/RT](θ_Cθ_*)³θ_Cθ_H (9)

Here, A is the pre-exponential factor. θ_CH3, θ_C, θ_H, and θ_* are the coverage of CH₃, C, H, and the free site on the surface, respectively. R is the ideal gas constant and T is the reaction temperature. E_a,4 is the reaction barrier of CH₃ hydrogenation, and E_r,3 is the reaction energy for C + 3H → CH₃. From eqn (10), it is evident that the effective barrier of CH₄ formation (E_{a_eff,CH4}) can be calculated as (E_a,4 + E_r,3), which is essentially the difference between the highest TS energy of CH₄ formation (E_TS4) and the initial state with one atomic carbon, one atomic oxygen and four atomic hydrogens adsorbed on cobalt surfaces (E_C+4H+O)^38,41,42 (Fig. 2c).

E_{a_eff,CH4} = E_a,4 + E_r,3 = E_TS4 − E_C+4H+O (10)

However, the difference between the barrier from CH₃ + H + O toward TS₃ is higher than that toward TS₄ by 0.53 eV on hcp-Co (10−12), and 0.21 eV on fcc-Co (311), respectively. Since the ratio of hydrogen to the free site coverage θ_H/θ_* is about 1 to 10 under typical reaction conditions (assuming equivalent coverages of H and free site *), the higher barrier of CH₃ + * → CH₂ + H than CH₃ + H → CH₄ causes r₋₃ = k₋₃θ_CH3θ_* ≪ r₄ = k₄θ_CH3θ_H. Thus, the reaction of CH₂ + H → CH₃ is irreversible and becomes the rate-determining step on (10−12) and (311), and the rate of CH₄ formation on these two surfaces should be calculated as:

R_CH4 = A exp[−E_a,3/RT]θ_CH2θ_H = A exp[−(E_a,3 + E_r,2)/RT](θ_H/θ_*)²θ_Cθ_H (11)

The effective barrier of CH₄ formation (E_{a_eff,CH4}) on (10−12) and (311) surfaces therefore is determined as the difference between the highest TS energy of CH₄ formation (E_TS3) and the initial state (E_C+4H+O) (Fig. 2d).

E_{a_eff,CH4} = E_a,3 + E_r,2 = E_TS3 − E_C+4H+O (12)

To assess the activity of the Co surfaces for CH₄ formation, the E_{a_eff,CH4} values for all the studied surfaces of hcp and fcc Co are obtained and are plotted in Fig. 3(a). Surprisingly, the CH₄ formation activity on hcp is not always lower than that on fcc. The E_{a_eff,CH4} of fcc-Co (110), which is the lowest one on fcc-Co, is slightly smaller than that of hcp-Co (0001) and hcp-Co (11−20). The rank of effective barriers is hcp-Co (10−11) > fcc-Co (100) > hcp-Co (10−10) > hcp-Co (11−21) > hcp-Co (10−12) > fcc-Co (311) > hcp-Co (11−20) ≈ fcc-Co (111) ≈ hcp-Co (0001) ≈ fcc-Co (110). It shows that hcp-(0001), hcp-(11−20), fcc-(111) and fcc(110) are all active and have similar catalytic activities in CH₄ formation. The rank of the effective barrier for CH₄ formation is consistent with previous ab initio studies.^44,66

Fig. 3 (a) The effective barriers of CH₄ formation (E_{a_eff,CH4}) on hcp-Co and fcc-Co surfaces and the average effective barrier of CH₄ formation (E_{a_particle,CH4}) for Co particles (dashed lines). (b) E_{a_eff,CH4} as a function of the sum of adsorption energies of C + 4H atoms (E_{ads_C+4H}) of each cobalt surface.

To assess the overall catalytic activity of hcp-Co and fcc-Co phases for CH₄ formation, we further calculated the particle effective barriers (E_{a_particle,CH4}) of CH₄ formation on hcp-Co and fcc-Co Wulff particles, assuming that the overall rate of reaction on the catalyst particle is the sum of the rates from all its exposed surfaces (see the Methods section). On hcp-Co, E_{a_particle,CH4} is 1.11 eV and on fcc-Co it is 1.02 eV (Fig. 3(a) and Table S7, ESI†). The results indicate that hcp-Co and fcc-Co have similar intrinsic catalytic activity to form CH₄.

A weak linear relationship was discovered between E_{a_eff,CH4} of the Co surface and the sum of adsorption energies of a C atom and 4 H atoms on it (E_{ads_C+4H}) with R² of 0.73 on ten surfaces, irrespective of the crystal phase (Fig. 3(b) and Table S8, ESI†). With the adsorption of C and 4 H atoms becoming stronger, the acquired E_{a_eff,CH4} follows a downhill trend on the cobalt surfaces, and the higher values of E_{ads_C+4H} are associated with lower E_{a_eff,CH4}. This linear correlation allows the fast estimation of the effective barrier of CH₄ formation on cobalt surfaces, as long as the adsorption energies of C and H are given (E_{a_eff,CH4} = −1.10E_{ad_C+4H} − 8.56).

3.2 C₁ + C₁ coupling

The carbon chain elongation process in FTS was investigated through the carbide mechanism⁷⁰ (CH_i + CH_j → CH_iCH_j, i, j = 0–3) and CO-insertion mechanism⁷¹ (CH_i + CO → CH_iCO, i = 0–3). The structures of the C₁ + C₁ coupling for IS, TS, FS are shown in Fig. S3–S12 (ESI†), and the reaction barriers and energies are given in Tables S5 and S6 (ESI†).

In the carbide mechanism, the CH_i + CH_j coupling rate is expressed as:

r_CHi+CHj = A exp[−E_a,i+j/RT]θ_CHiθ_CHj = A exp[−(E_a,i+j + E_i + E_j)/RT]θ_C²(θ_H/θ_*)^i+j (13)

where A, E_a,i+j, θ_CHi, θ_CHj, θ_C, θ_H, and θ_* are the pre-exponential factor, the reaction barrier for Ch_i + CH_j coupling, the coverage of CH_i, CH_j, C, H and the free site, respectively. The effective barrier of the coupling of CH_i + CH_j (E_{a_eff,i+j}) is equal to E_a,i+j + E_i + E_j, where E_i and E_j are the reaction energies for C + iH → CH_i and C + jH → CH_j, respectively.

In the CO-insertion mechanism, the CO + CH_j coupling rate is expressed as

r_COi+CHj = A exp[−E_a,CO+j/RT]P_COθ_CHj = A exp[−(E_a,CO+j + E_j)/RT]P_COθ_C(θ_H/θ_*)^j (14)

where P_CO is the pressure of CO. The effective barrier of the coupling of CO + CH_j (E_{a_eff,CO+j}) is E_a,CO+Ej, where E_j is the reaction energy for the C + jH → CH_j.^38,41,44,72

It turns out that the coupling of CH_i + CH_j, and CH_i + CO on ten cobalt surfaces resembles each other, and we hereby use the hcp-Co (0001) surface as an example to demonstrate the most feasible C₁ + C₁ coupling reaction. The energy profiles of the C₁ + C₁ coupling reaction on hcp-Co (0001) are shown in Fig. 4. The E_{a_eff,i+j} of C + CH (0.41 eV), C + CH₂ (0.86 eV), and C + CH₃ (1.28 eV) gradually increase with the hydrogenation of the carbon atom, indicating that the coupling ability of carbon species will be reduced, due to the increase of the coordination number of the C atom. A similar increasing trend of E_{a_eff,+j} is observed for CH + CH (0.10 eV), CH + CH₂ (0.52 eV), and CH + CH₃ (1.38 eV).

Fig. 4 Energy profiles of the carbide mechanism, the CO-inserted mechanism and the CH₄ formation on hcp-Co (0001) model surfaces. The energy of E_i + E_j is chosen as the zero point. (The blue dotted line is the effective barrier of CH₄ formation. The green rectangle is the most feasible pathway of the C₁ + C₁ coupling reaction and the related effective barrier.).

We take the hcp-Co (0001) as an example to discuss the C₁ + C₁ coupling reaction. Since CH₄ formation competes with C₁ + C₁ coupling, if the barrier of a particular C₁ + C₁ coupling reaction is higher than CH₄ formation, then the C1 species will be directly hydrogenated and this C₁ + C₁ coupling reaction will not be likely to happen. We found that the carbide mechanisms of C + C, C + CH, C + CH₂, CH + CH, CH + CH₂ reactions are all feasible on hcp-Co (0001), due to their lower E_{a_eff,i+j} than E_{a_eff,CH4}. However, the CO-insertion mechanism of C + CO, CH + CO, CH₂ + CO, and CH₃ + CO coupling reaction all has higher E_{a_eff,CO+j} than E_{a_eff,CH4}, indicating that the CO-insertion mechanisms are not favorable on hcp-Co (0001). Among the feasible carbide mechanisms, the CH + CH coupling has the lowest E_{a_eff,i+j}, and is probably the most favorable pathway among all the C₁ + C₁ coupling reactions on the hcp-Co (0001) surface.⁴⁵

For the other hcp-Co and fcc-Co surfaces, the most feasible C₁ + C₁ coupling pathways are divergent. It is CH + CH on hcp-Co (10−10), (10−12), (11−21), fcc-Co (110), (111), C + CH₃ on fcc-Co (311), C + CO on hcp-Co (10−11), fcc-Co (100), and CH + CO on hcp-Co (11−20), as shown in Table 1 and Fig. S13 (ESI†). For most of the surfaces, the reaction of CH + CH coupling is the most favorable pathway among all of the C₁ + C₁ coupling reactions, which is in line with the findings in ref. 44 and 73.

Table 1 The most feasible pathway of the C₁ + C₁ coupling reaction and the related effective barrier on hcp-Co (0001), (10−10), (10−11), (10−12), (11−20), (11−21) and fcc-Co (100), (110), (111), (311) model surfaces

	(10−10)	(10−11)	(10−12)	(11−20)	(11−21)	(100)	(110)	(111)	(311)
C1 + C1	CH + CH	C + CO	CH + CH	CH + CO	CH + CH	C + CO	CH + CH	CH + CH	C + CH3
E a_eff,i+j/Ea_eff,CO+j	0.05	1.42	1.21	0.18	0.67	1.05	0.20	−0.01	0.77

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3.3 CH₄ selectivity

As Cheng^42,72 and Yin, et al.⁴¹ have proven, the CH₄ selectivity in FTS can be approximately evaluated based on the effective barrier difference of CH₄ formation and C₁ + C₁ coupling.: ΔE_{a_eff}= E_{a_eff,CH4} − E_{a_eff,C1+C1}, where E_{a_eff,CH4} is the effective barrier of CH₄ formation, and E_{a_eff,C1+C1} is the effective barrier of the most feasible C₁ + C₁ coupling. The CH₄ selectivity is thus negatively associated with the value of ΔE_{a_eff} (Fig. 5).

Fig. 5 The effective barrier difference (ΔE_{a_eff} = E_{a_eff,CH4} − E_{a_eff,C1+C1}, the columns) and the particle effective barrier difference (ΔE_{a_particle}, the dashed lines) between CH₄ formation and the most feasible C₁ + C₁ coupling on hcp-Co (0001), (10−10), (10−11), (10−12), (11−20), (11−21), fcc-Co (100), (110), (111), (311), and Fe₅C₂ (510), (4̄11), (010), (111̄), (111). The ΔE_{a_eff} of Fe₅C₂ (510), (4̄11), (010), (111̄), (111) is from ref. 36.

It is worth noting that the CH₄ selectivity in hcp is not always lower than in fcc. Among the ten Co surfaces, hcp-Co (10−10) has the largest ΔE_{a_eff} (2.13 eV), indicating the lowest CH₄ selectivity, whereas fcc-Co (311) has the smallest ΔE_{a_eff} (0.55 eV) and the highest CH₄ selectivity. However, the ΔE_{a_eff} values of hcp-Co (0001), (10−12), and (11−20) are even lower than those of fcc-Co (100) and fcc-Co (111), implying higher CH₄ selectivity of these surfaces on hcp-Co. The ranks of the CH₄ selectivity of all the Co surfaces are shown in Fig. 5. To confirm the validity of our prediction, comparisons are made with the available experiments. Qin et al. have synthesized three types of hcp-Co particles exposing only one type of facet each. Comparison of their CH₄ selectivity shows that (10−11) has the lowest CH₄ selectivity, followed by (0001) and (11−20). Our calculated ΔE_{a_eff} values on (10−11), (0001), and (11−20) are 1.92 eV, 0.92 eV, and 0.87 eV, respectively, with the rank of CH₄ selectivity to be (10−11) < (0001) < (11−20), which is in excellent agreement with the experiments.⁶⁸

The overall CH₄ selectivities of hcp-Co and fcc-Co phases are defined by the particle effective barrier difference (ΔE_{a_particle}) between CH₄ formation (E_{a_eff,CH4}) and the most feasible C₁ + C₁ coupling (E_{a_eff,C1+C1}) on hcp-Co and fcc-Co Wulff particles, assuming that the overall rate of reaction on the catalyst particle is the sum of the rates from all its exposed surfaces. (See the Methods section). The ΔE_{a_particle} on hcp-Co and fcc-Co surfaces is 0.86 eV and 0.60 eV, respectively. Therefore, the CH₄ selectivity in FTS is higher on fcc-Co than on hcp-Co, which is consistent with the experimental observations.^33,74,75 Notably, the (311) surface of fcc-Co plays a pivotal role in determining the overall CH₄ selectivity of the fcc phase. Moreover, our results suggest that the CH₄ selectivity can be suppressed by decreasing the exposure ratio of the facets that are prone to form CH₄: (0001), (10−12), and (11−20) on hcp-Co and (110) and (311) on fcc-Co.

In FTS, Co-based FTS generally has higher CH₄ selectivity than Fe-based FTS.^42,76 Herein, we compare the ΔE_{a_eff} of exposed facets of hcp and fcc Co with those of χ-Fe₅C₂ particles,³⁶ the most widely accepted active phase in iron-catalyzed FTS.^11,23,37 Most of the χ-Fe₅C₂ surfaces have large ΔE_{a_eff}, with the exception of Fe₅C₂(111), which has lower ΔE_{a_eff} than the cobalt surfaces of (10−10), (10−11), (11−21), and (100). The overall CH₄ selectivity of χ-Fe₅C₂, hcp-Co, and fcc-CO can be compared using ΔE_{a_particle} (Fig. 5, dashed lines and Table S9, ESI†). The ΔE_{a_particle} (1.20 eV) of Fe₅C₂ particles is larger than that of hcp-Co and fcc-Co, indicating that the Co particle of both phases has higher CH₄ selectivity than the χ-Fe₅C₂ particle. This is probably the reason why Co-based FTS generally shows higher CH₄ selectivity than Fe-based FTS.⁴¹

3.4 Tuning of the phase transition between hcp and fcc Co

On the basis of the understanding of the activity and selectivity for CH₄ and C₂₊ production on the cobalt phases, one can further use the knowledge for catalyst design through phase engineering. Operando characterization shows that in FTS, the cobalt catalyst can exist in both hcp and fcc phases.^77,78 In order to get more hcp phase which has lower CH₄ selectivity and higher C₅₊ selectivity, the thermodynamic conditions for the modulation of the ratio of hcp and fcc Co phases by changing the temperature and particle size are explored.

The thermodynamic stability of the Co particles of different phases can be evaluated using the Gibbs free energy per cobalt atom, which involves the energy, phonon energies at certain temperature, and surfaces energies at a given particle size.^55–58 The details of the theory and calculation methods are shown in Section 2.4. The temperature and particle sizes in which the Gibbs free energy difference is zero represent the condition of phase transition between hcp-Co and fcc-Co, which is given in Fig. 6 and Table S10 (ESI†). The black solid curve corresponds to the conditions of phase transition of bulk cobalt. At 718 K, a phase transition from hcp-Co to fcc-Co happens, which is very close to the experimental transition temperature of 695 K.^79,80 This shows that bulk hcp-Co is more stable than bulk fcc-Co below 718 K, and keeping a low operating temperature (<718 K) is beneficial for stabilizing the hcp-Co phase. On the other hand, the particle size also plays an important role in the phase transition as shown in Fig. 6. The transition temperatures of different particle sizes (10 nm, 15 nm, 20 nm, 25 nm, and 30 nm) are 1059 K, 946 K, 890 K, 855 K, and 833 K, respectively. With the increase in the size of cobalt particles, the phase transition temperature gradually decreases until it approaches the bulk transition temperature. When the particle size is larger than 30 nm, hcp-Co is stable only at a relatively narrow temperature window between 718 K and 833 K. Therefore, the ratio of hcp phase particles could be maximized by controlling the operating temperature below 718 K and size of the Co particles as small as possible within the range of 10–30 nm, which will likely lower CH₄ selectivity and improve C₅₊ selectivity of the FTS process as observed experimentally.⁸¹

Fig. 6 Calculated Gibbs free energy difference for hcp-Co and fcc-Co phases as a function of temperature at different particle sizes.

4. Conclusion

In this work, we proposed a strategy combining DFT, Wulff construction, and a particle-level energy descriptor to understand the relationship between the crystal phase and their catalytic selectivity, using the CH₄ formation on hcp-Co and fcc-Co as an example. Hcp-Co has lower intrinsic catalytic activity than fcc-Co to form CH₄, due to the higher E_{a_particle,CH4}. A linear relationship was discovered between E_{a_eff,CH4} of a Co surface and the sum of adsorption energies of a C atom and 4H atoms on it (E_{ads_C+4H}), which provides an efficient way of predicting the intrinsic catalytic activity of Co surfaces for CH₄ formation. The most feasible C₁ + C₁ coupling pathway is CH + CH on hcp-Co (0001), (10−10), (10−12), (11−21), fcc-Co (110), (111), C + CH₃ on fcc-Co (311), C + CO on hcp-Co (10−11), fcc-Co (100), and CH + CO on hcp-Co (11−20). The CH₄ selectivity, represented by the effective barrier difference between CH₄ formation and the most feasible C₁ + C₁ coupling, decreases in the sequence of surfaces (311) > (110) > (10−12) > (11−20) > (0001) > (111) > (11−21) > (100) > (10−11) > (10−10). The lower CH₄ selectivity of hcp-Co than fcc-Co can be well explained by its higher ΔE_{a_particle}. Tuning the CH₄ selectivity of the Co catalyst can be achieved not only by controlling the morphology of the particles to avoid the active facets for CH₄ formation such as (0001), (10−12), (11−20) on hcp-Co and (110), (311) on fcc-Co, but is also achievable through engineering the crystal phases by controlling the temperature and particle size. Our research provides a pathway to understand the phase-selectivity relationship for complex reactions.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

The authors are grateful for the financial support from the National Science Fund for Distinguished Young Scholars of China (Grant no. 22225206), the National Key R&D Program of China (2022YFA1604103), the National Natural Science Foundation of China (No. 21972160, 21972157, 22372187), CAS Project for Young Scientists in Basic Research (YSBR-005), Key Research Program of Frontier Sciences CAS (ZDBS-LY-7007), Major Research plan of the National Natural Science Foundation of China (92045303), Informatization Plan of Chinese Academy of Sciences, (Grant no. CAS-WX2021SF0110), Youth Innovation Promotion Association CAS (2020179), the Doctoral Research Funds in Shanxi (QZX-2023012), the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (2023L304), the Doctoral Research Funds of Yuncheng University (YQ2023025), and the Autonomous Research Project of SKLCC (Grant no. 2023BWZ005).

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Footnote

† Electronic supplementary information (ESI) available: Detailed data about the CH₄ formation and C₁ + C₁ coupling on hcp-Co and fcc-Co surfaces. See DOI: https://doi.org/10.1039/d3cp06041a

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