Chemical and Thermal Sintering of Supported Metals with Emphasis on Cobalt Catalysts During Fischer–Tropsch Synthesis

Understanding and Tuning the Effects of H2O on Catalytic CO and CO2 Hydrogenation

Catalytic COx (CO and CO2) hydrogenation to valued chemicals is one of the promising approaches to address challenges in energy, environment, and climate change. H2O is an inevitable side product in these reactions, where its existence and effect are often ignored. In fact, H2O significantly influences the catalytic active centers, reaction mechanism, and catalytic performance, preventing us from a definitive and deep understanding on the structure-performance relationship of the authentic catalysts. It is necessary, although challenging, to clarify its effect and provide practical strategies to tune the concentration and distribution of H2O to optimize its influence. In this review, we focus on how H2O in COx hydrogenation induces the structural evolution of catalysts and assists in the catalytic processes, as well as efforts to understand the underlying mechanism. We summarize and discuss some representative tuning strategies for realizing the rapid removal or local enrichment of H2O around the catalysts, along with brief techno-economic analysis and life cycle assessment. These fundamental understandings and strategies are further extended to the reactions of CO and CO2 reduction under an external field (light, electricity, and plasma). We also present suggestions and prospects for deciphering and controlling the effect of H2O in practical applications.

Issues and challenges of Fischer-Tropsch synthesis catalysts

Depletion of oil and gas resources is a major concern for researchers and the global community. Researchers are trying to develop a way to overcome these issues using the Fischer-Tropsch synthesis (FTS) process. The FTS reaction converts a mixture of hydrogen and carbon monoxide gases into a liquid fuel. The reactions are performed in the reactor and in the presence of a catalyst. A series of catalysts, such as iron, cobalt, nickel, and ruthenium, have been used for the FTS process. In iron-based catalysts, the Fe5C phase is the active phase that produces C5+ hydrocarbons. At higher conversion rates, the presence of water in the products is a problem for cobalt catalysts because it can trigger catalyst deactivation mechanisms. Ni-based catalysts play key roles as base catalysts, promoters, and photothermal catalysts in FTS reactions to produce different useful hydrocarbons. Ruthenium catalysts offer not only high activity but also selectivity toward long-chain hydrocarbons. Moreover, depending on the Ru particle size and interaction with the oxide support, the catalyst properties can be tuned to enhance the catalytic activity during FTS. The detailed reaction pathways based on catalyst properties are explained in this article. This review article describes the issues and challenges associated with catalysts used for the FTS process.

Surface topology of MXene flakes induces the selection of the sintering mechanism for supported Pt nanoparticles

Sintering of metal nanocatalysts leading to particle growth and subsequent performance deactivation is a primary issue that hinders their practical applications. While metal-support interaction (MSI) is considered as the critical factor which influences the sintering behavior, the underlying microscopic mechanism and kinetics remain incompletely understood. Here, by using in situ scanning transmission electron microscopy (STEM) and theoretical analysis, we reveal the selection rule of the sintering mechanism for Pt nanoparticles on a two-dimensional (2D) MXene (Ti3C2T x ) support, which relies on the surface topology of MXene flakes. It is demonstrated that the sintering of Pt nanoparticles proceeds via Ostwald ripening (OR) in the surface defect (such as steps and pore edges) regions of MXene flakes due to strong MSI on the Pt/MXene interface; conversely, weak MSI between Pt nanoparticles and the planar surface of MXene leads to prevalent particle migration and coalescence (PMC) for sintering. Furthermore, our quantitative analysis shows a significant divergence in sintering rates for PMC and OR processes. These microscopic observations suggest a clear "sintering mechanism-MSI" relationship for Pt/MXene nanocatalysts and may shed light on the design of novel nanocatalysts.

An In Situ TEM Study of the Influence of Water Vapor on Reduction of Nickel Phyllosilicate - Retarded Growth of Metal Nanoparticles at Higher Rates

Unavoidable water formation during the reduction of solid catalyst precursors has long been known to influence the nanoparticle size and dispersion in the active catalyst. This in situ transmission electron microscopy study provides insight into the influence of water vapor at the nanoscale on the nucleation and growth of the nanoparticles (2-16 nm) during the reduction of a nickel phyllosilicate catalyst precursor under H2/Ar gas at 700 °C. Water suppresses and delays nucleation, but counterintuitively increases the rate of particle growth. After full reduction is achieved, water vapor significantly enhances Ostwald ripening which in turn increases the likelihood of particle coalescence. This study proposes that water leads to formation of mobile nickel hydroxide species, leading to faster rates of particle growth during and after reduction.

Tuning the Size of TiO2-Supported Co Nanoparticle Fischer-Tropsch Catalysts Using Mn Additions

Modifying traditional Co/TiO2-based Fischer-Tropsch (FT) catalysts with Mn promoters induces a selectivity shift from long-chain paraffins toward commercially desirable alcohols and olefins. In this work, we use in situ gas cell scanning transmission electron microscopy (STEM) with energy-dispersive X-ray spectroscopy (EDS) elemental mapping, and near-ambient pressure X-ray photoelectron spectroscopy (NAP-XPS) to demonstrate how the elemental dispersion and chemical structure of the as-calcined materials evolve during the H2 activation heat treatment required for industrial CoMn/TiO2 FT catalysts. We find that Mn additions reduce both the mean Co particle diameter and the size distribution but that the Mn remains dispersed on the support after the activation step. Density functional theory calculations show that the slower surface diffusion of Mn is likely due to the lower number of energetically accessible sites for the Mn on the titania support and that favorable Co-Mn interactions likely cause greater dispersion and slower sintering of Co in the Mn-promoted catalyst. These mechanistic insights into how the introduction of Mn tunes the Co nanoparticle size can be applied to inform the design of future-supported nanoparticle catalysts for FT and other heterogeneous catalytic processes.

Size-dependent diffusion of supported metal nanoclusters: mean-field-type treatments and beyond for faceted clusters

Nanostructured systems are intrinsically metastable and subject to coarsening. For supported 3D metal nanoclusters (NCs), coarsening can involve NC diffusion across the support and subsequent coalescence (as an alternative to Ostwald ripening). When used as catalysts, this leads to deactivation. The dependence of diffusivity, DN, on NC size, N (in atoms), controls coarsening kinetics. Traditional mean-field (MF) theory for DNversus N assumes that NC diffusion is mediated by independent random hopping of surface adatoms with low coordination, and predicts that DN ∼ hN-4/3neq. Here, h = ν exp[-Ed/(kBT)] denotes the hop rate, and neq = exp[-Eform/(kBT)] the density of those adatoms. The adatom formation energy, Eform, approaches a finite large-N limit, as does the effective barrier, Eeff = Ed + Eform, for NC diffusion. This MF theory is critically assessed for a realistic stochastic atomistic model for diffusion of faceted fcc metal NCs with a {100} facet epitaxially attached to a (100) support surface. First, the MF formulation is refined to account for distinct densities and hop rates for surface adatoms on different facets and along the base contact line, and to incorporate the exact values of Eform and neqversus N for our model. MF theory then captures the occurrence of local minima in DNversus N at closed-shell sizes, as shown by KMC simulation. However, the MF treatment also displays fundamental shortcomings due to the feature that diffusion of faceted NCs is actually dominated by a cooperative multi-step process involving disassembling and reforming of outer layers on side facets. This mechanism leads to an Eeff which is well above MF values, and which increases with N, features captured by a beyond-MF treatment.

Direct Observation of Ni Nanoparticle Growth in Carbon-Supported Nickel under Carbon Dioxide Hydrogenation Atmosphere

Understanding nanoparticle growth is crucial to increase the lifetime of supported metal catalysts. In this study, we employ in situ gas-phase transmission electron microscopy to visualize the movement and growth of ensembles of tens of nickel nanoparticles supported on carbon for CO2 hydrogenation at atmospheric pressure (H2:CO2 = 4:1) and relevant temperature (450 °C) in real time. We observe two modes of particle movement with an order of magnitude difference in velocity: fast, intermittent movement (vmax = 0.7 nm s-1) and slow, gradual movement (vaverage = 0.05 nm s-1). We visualize the two distinct particle growth mechanisms: diffusion and coalescence, and Ostwald ripening. The diffusion and coalescence mechanism dominates at small interparticle distances, whereas Ostwald ripening is driven by differences in particle size. Strikingly, we demonstrate an interplay between the two mechanisms, where first coalescence takes place, followed by fast Ostwald ripening due to the increased difference in particle size. Our direct visualization of the complex nanoparticle growth mechanisms highlights the relevance of studying nanoparticle growth in supported nanoparticle ensembles under reaction conditions and contributes to the fundamental understanding of the stability in supported metal catalysts.

A hollow void catalyst of Co@C(Z-d)@void@CeO2 for enhancing the performance and stability of the Fischer-Tropsch synthesis

To enhance the catalyst performance of Fischer-Tropsch synthesis (FTS), removing the mass-transfer restriction in the catalysis synthesis is essential. Although the core-shell nanostructures can improve the activity and stability of the catalyst, they can restrict the reactants' diffusion towards the active sites and the transfer of the products from these sites in FTS. Creating an adequate porosity between the core and the outer shell of the catalyst structure can tackle this issue. In this work, the synthesized cobalt-based nano-catalyst is encapsulated with two shells and a middle porous shell. The first shell is a carbon shell at the core of the catalyst derived from ZIF-67, the second one is the outer shell of ceria, and the middle porous shell is formed by removing the sacrificial silica shell through the etching technique. The characterization and performance tests represent significant evidence of the etching treatment's impact on the FTS catalyst performance. Besides, molecular dynamics simulation is also utilized to clarify its effect. The FTS catalytic performance is enhanced more than 2 times with the etched catalyst versus the catalyst without it at 17.5 bar and a (H2/CO) ratio of 1.2. In addition, not only does the etched catalyst with high porosity play the role of a nanoreactor and intensify its catalytic performance, but it also has higher stability.

Graphene-confined ultrafast radiant heating for high-loading subnanometer metal cluster catalysts

Thermally activated ultrafast diffusion, collision and combination of metal atoms comprise the fundamental processes of synthesizing burgeoning subnanometer metal clusters for diverse applications. However, so far, no method has allowed the kinetically controllable synthesis of subnanometer metal clusters without compromising metal loading. Herein, we have developed, for the first time, a graphene-confined ultrafast radiant heating (GCURH) method for the synthesis of high-loading metal cluster catalysts in microseconds, where the impermeable and flexible graphene acts as a diffusion-constrained nanoreactor for high-temperature reactions. Originating from graphene-mediated ultrafast and efficient laser-to-thermal conversion, the GCURH method is capable of providing a record-high heating and cooling rate of ∼10⁹°C/s and a peak temperature above 2000°C, and the diffusion of thermally activated atoms is spatially limited within the confinement of the graphene nanoreactor. As a result, due to the kinetics-dominant and diffusion-constrained condition provided by GCURH, subnanometer Co cluster catalysts with high metal loading up to 27.1 wt% have been synthesized by pyrolyzing a Co-based metal-organic framework (MOF) in microseconds, representing one of the highest size-loading combinations and the quickest rate for MOF pyrolysis in the reported literature. The obtained Co cluster catalyst not only exhibits an extraordinary activity similar to that of most modern multicomponent noble metal counterparts in the electrocatalytic oxygen evolution reaction, but is also highly convenient for catalyst recycling and refining due to its single metal component. Such a novel GCURH technique paves the way for the kinetically regulated, limited diffusion distance of thermally activated atoms, which in turn provides enormous opportunities for the development of sophisticated and environmentally sustainable metal cluster catalysts.

Metal Nanocatalyst Sintering Interrogated at Complementary Length Scales

Metal nanoparticle (NP) sintering is a prime cause of catalyst degradation, limiting its economic lifetime and viability. To date, sintering phenomena are interrogated either at the bulk scale to probe averaged NP properties or at the level of individual NPs to visualize atomic motion. Yet, "mesoscale" strategies which bridge these worlds can chart NP populations at intermediate length scales but remain elusive due to characterization challenges. Here, a multi-pronged approach is developed to provide complementary information on Pt NP sintering covering multiple length scales. High-resolution scanning electron microscopy (HRSEM) and Monte Carlo simulation show that the size evolution of individual NPs depends on the number of coalescence events they undergo during their lifetime. In its turn, the probability of coalescence is strongly dependent on the NP's mesoscale environment, where local population heterogeneities generate NP-rich "hotspots" and NP-free zones during sintering. Surprisingly, advanced in situ synchrotron X-ray diffraction shows that not all NPs within the small NP sub-population are equally prone to sintering, depending on their crystallographic orientation on the support surface. The demonstrated approach shows that mesoscale heterogeneities in the NP population drive sintering and mitigation strategies demand their maximal elimination via advanced catalyst synthesis strategies.

Facile CO bond cleavage on polynuclear vanadium nitride clusters V4N5. Phys Chem Chem Phys 2022;24:29765-29771. [PMID: 36458914 DOI: 10.1039/d2cp04304a] [Citation(s) in RCA: 0] [Impact Index Per Article: 0]

Identifying the structural configurations of precursors for CO dissociation is fundamentally interesting and industrially important in the fields of, e.g., Fischer-Tropsch synthesis. Herein, we demonstrated that CO could be dissociated on polynuclear vanadium nitride V₄N₅^- clusters at room temperature, and a key intermediate, with CO in a N-assisted tilted bridge coordination where the C-O bond ruptures easily, was discovered. The reaction was characterized by mass spectrometry, photoelectron spectroscopy, and quantum-chemistry calculations, and the nature of the adsorbed CO on product V₄N₅CO^- was further characterized by a collision-induced dissociation experiment. Theoretical analysis evidences that CO dissociation is predominantly governed by the low-coordinated V and N atoms on the (V₃N₄)VN^- cluster and the V₃N₄ moiety resembles a support. This finding strongly suggests that a novel mode for facile CO dissociation was identified in a gas-phase cluster study.

Comment on "Real-time atomistic simulation of the Ostwald ripening of TiO2 supported Au nanoparticles" by B. Zhu, R. Qi, L. Yuan and Y. Gao, Nanoscale, 2020, , 19142

Using Monte Carlo simulations (MCS) in combination with an analytical model for the metal-metal interaction with the parameters based on density functional theory (DFT), Zhu, Qi, Yuan, and Gao predicted that the Ostwald ripening of Au nanoparticles on TiO₂ occurs primarily via the detachment and attachment of Au dimers. I show that this and some other predictions are not properly validated because the parameters employed in the analytical model in order to describe the Au-Au interaction are in fact inconsistent both with DFT and experimental thermodynamical data.

Solar Energy Catalysis

When it comes to using solar energy to promote catalytic reactions, photocatalysis technology is the first choice. However, sunlight can not only be directly converted into chemical energy through a photocatalytic process, it can also be converted through different energy-transfer pathways. Using sunlight as the energy source, photocatalytic reactions can proceed independently, and can also be coupled with other catalytic technologies to enhance the overall catalytic efficiency. Therefore, sunlight-driven catalytic reactions are diverse, and need to be given a specific definition. We propose a timely perspective for catalytic reactions driven by sunlight and give them a specific definition, namely "solar energy catalysis". The concept of different types of solar energy catalysis, such as photocatalysis, photothermal catalysis, solar cell powered electrocatalysis, and pyroelectric catalysis, are highlighted. Finally, their limitations and future research directions are discussed.

Recent Progress on Sulfated Nanozirconia as a Solid Acid Catalyst in the Hydrocracking Reaction

Zirconia has advantageous thermal stability and acid–base properties. The acidity character of ZrO2 can be enhanced through the sulfation process forming sulfated zirconia (ZrO2-SO4). An acidity test of the catalyst produced proved that the sulfate loading succeeded in increasing the acidity of ZrO2 as confirmed by the presence of characteristic absorptions of the sulfate group from the FTIR spectra of the catalyst. The ZrO2-SO4 catalyst can be further modified with transition metals, such as Platinum (Pt), Chromium (Cr), and Nickel (Ni) to increase catalytic activity and catalyst stability. It was observed that variations in the concentrations of Pt, Cr, and Ni produced a strong influence on the catalytic activity as the acidity and porosity of the catalyst increased with their addition. The activity, selectivity, and catalytic stability tests of Pt/ZrO2-SO4, Cr/ZrO2-SO4 and Ni/ZrO2-SO4 were carried out with their application in the hydrocracking reaction to produce liquid fuel. The percentage of liquid fractions produced using these catalysts were higher than the fraction produced using pure ZrO2 and ZrO2-SO4 catalyst.

Ceria–tungsten–tin oxide catalysts with superior regeneration capacity after sulfur poisoning for NH3-SCR process

A combined study on the anti-sintering ability, SO2-poisoning mechanism and thermal regeneration property of CeWSnOx catalysts for NH3-SCR reaction.

Photo-assisted thermal catalytic Fischer-Tropsch Synthesis over Co-Cu/CeO2

Generally, the increase in temperature in the Fischer-Tropsch synthesis accelerates the conversion of CO but reduces the selectivity of high value-added products due to the increase in the percentage of...

Sabatier principle of metal-support interaction for design of ultrastable metal nanocatalysts

[Figure: see text].

Sintering Rate and Mechanism of Supported Pt Nanoparticles by Multiscale Simulation

Thermal stability is the key issue in the industrial application of supported metal nanocatalysts. A combination method of density functional theory calculations, machine learning, and molecular dynamics simulation is adopted to study the sintering behavior of supported platinum (Pt) nanoparticles on graphene or TiO₂ nanosheet, and analyze sintering mechanisms under different temperatures, particle sizes, and metal support interactions (MSIs). The results show that the agglomeration of supported nanoparticles is mainly based on the mechanism of small particle migration and growth. Small-sized particles with high surface energy determine the sintering rate. In addition, the increase of temperature is conducive to the agglomeration of particles, especially for systems with strong MSI. Based on the analysis of the sintering process, a sintering kinetic model of supported Pt nanoparticles related to particle size, temperature, and MSI is established, which provides theoretical guidance for the design of supported metal catalysts with high thermal stability.

Bubble-templated synthesis of nanocatalyst Co/C as NADH oxidase mimic

Designing highly active nanozymes for various enzymatic reactions remains a challenge in practical applications and fundamental research. In this work, by studying the catalytic functions of natural NADH oxidase (NOX), we devised and synthesized a porous carbon-supported cobalt catalyst (Co/C) to mimic NOX. The Co/C can catalyze dehydrogenation of NADH and transfers electrons to O₂ to produce H₂O₂. Density functional theory calculations reveal that the Co/C can catalyze O₂ reduction to H₂O₂ or H₂O considerably. The Co/C can also mediate electron transfer from NADH to heme protein cytochrome c, thereby exhibiting cytochrome c reductase-like activity. The Co/C nanoparticles can deplete NADH in cancer cells, induce increase of the reactive oxygen species, lead to impairment of oxidative phosphorylation and decrease in mitochondrial membrane potential, and cause ATP production to be damaged. This ‘domino effect’ facilitates the cell to approach apoptosis.

Si-doped Al2O3 nanosheet supported Pd for catalytic combustion of propane: effects of Si doping on morphology, thermal stability, and water resistance

Catalytic combustion of propane as typical light alkanes was important for the purification of industrial VOCs and automobile hydrocarbon emissions. Si-doped Al₂O₃ nanosheet was synthesized by a hydrothermal method, and effects of Si content on the morphology and thermal stability of Al₂O₃ were investigated. The doping of SiO₂ could tune the thickness of Al₂O₃ nanosheets and significantly improve its thermal stability, the θ phase was still maintained, and the specific surface area was as high as 56.3 m² g^-1 after calcination at 1200 °C. And then the Si-doped Al₂O₃ nanosheets were used as support of Pd catalysts (Pd/Si-Al₂O₃ nanosheets) for catalytic combustion of propane, especially Pd/3.6Si-Al₂O₃ nanosheets, which presented high activity, stability, and resistance to sintering and H₂O due to the promotion of Si on the thermal stability of Al₂O₃ and the stabilization (dispersion, isolation, and strong interaction) of PdO_x species.

Correlation between the TiO2 encapsulation layer on Pt and its electrochemical behavior

Supported metal catalysts with partial encapsulation resulting from strong metal-support interactions show distinctive structural features which strongly affect their functionalities. Yet, challenges in systematic synthesis and in-depth characterization for such systems limit the present understanding of structure-property relationships. Herein, the synthesis and characterization of two Pt/TiO2 models are conducted by a simple change of the synthesis order, while keeping all other parameters constant. They differ in containing either bare or encapsulated Pt nanoparticles. The presence of an extremely thin and inhomogeneous TiO2 layer is clearly demonstrated on 2-3 nm sized Pt nanoparticles by combination of imaging, energy dispersive X-ray spectroscopy and electron energy loss spectroscopy performed in a transmission electron microscope. The two Pt/TiO2 systems exhibit differences in morphology and local structure which can be correlated with their electrochemical activity and stability using cyclic voltammetry experiments. Beyond enhanced particle stability, we report an increase in H+ intercalation on titania and reduced Pt activity due to partial encapsulation by TiO2. Finally, the growth of an encapsulation layer as a result of cyclic voltammetry measurements is discussed. These results shed light on the in-depth structure-property relationship of catalysts with strong metal-support interactions which leads to enhanced functional materials for electrochromic devices and energy applications.

Passivation of Co/Al2O3 Catalyst by Atomic Layer Deposition to Reduce Deactivation in the Fischer–Tropsch Synthesis

The present work explores the technical feasibility of passivating a Co/γ-Al2O3 catalyst by atomic layer deposition (ALD) to reduce deactivation rate during Fischer–Tropsch synthesis (FTS). Three samples of the reference catalyst were passivated using different numbers of ALD cycles (3, 6 and 10). Characterization results revealed that a shell of the passivating agent (Al2O3) grew around catalyst particles. This shell did not affect the properties of passivated samples below 10 cycles, in which catalyst reduction was hindered. Catalytic tests at 50% CO conversion evidenced that 3 and 6 ALD cycles increased catalyst stability without significantly affecting the catalytic performance, whereas 10 cycles caused blockage of the active phase that led to a strong decrease of catalytic activity. Catalyst deactivation modelling and tests at 60% CO conversion served to conclude that 3 to 6 ALD cycles reduced Co/γ-Al2O3 deactivation, so that the technical feasibility of this technique was proven in FTS.

Rare-Earth-Based Metal-Organic Frameworks as Multifunctional Platforms for Catalytic Conversion

The development of catalytic conversion is very important for human society. In the catalytic process, metal-organic frameworks (MOFs) can be utilized to obtain effective catalysts for their porous structures and adjustable properties. In addition, the introduction of rare-earth (RE) elements with unique properties for catalysts can realize good catalytic performances. Thus, the RE-MOF related catalysts for catalytic conversion are summarized. Due to the cooperation of RE elements and porous MOF structures, the RE-based MOFs can be used as promising catalysts or precursors/supports for other catalysts in the areas of energy conversion, environmental governance, and organic synthesis. These aggregated studies highlight the RE-MOFs as promising candidates for catalytic conversion.

Antisolvent Route to Ultrathin Hollow Spheres of Cerium Oxide for Enhanced CO Oxidation

Cerium(IV) oxide (CeO₂), or ceria, is one of the most abundant rare-earth materials that has been extensively investigated for its catalytic properties over the past two decades. However, due to the global scarcity and increasing cost of rare-earth materials, efficient utilization of this class of materials poses a challenging issue for the materials research community. Thus, this work is directed toward an exploration of making ultrathin hollow ceria or other rare-earth metal oxides and mixed rare-earth oxides in general. Such a hollow morphology appears to be attractive, especially when the thickness is trimmed to its limit, so that it can be viewed as a two-dimensional sheet of organized nanoscale crystallites, while remaining three-dimensional spatially. This ensures that both inner and outer shell surfaces can be better utilized in catalytic reactions if the polycrystalline sphere is further endowed with mesoporosity. Herein, we have devised our novel synthetic protocol for making ultrathin mesoporous hollow spheres of ceria or other desired rare-earth oxides with a tunable shell thickness in the region of 10 to 40 nm. Our ceria ultrathin hollow spheres are catalytically active and outperform other reported similar nanostructured ceria for the oxidation reaction of carbon monoxide in terms of fuller utilization of cerium. The versatility of this approach has also been extended to fabricating singular or multicomponent rare-earth metal oxides with the same ultrathin hollow morphology and structural uniformity. Therefore, this approach holds good promise for better utilization of rare-earth metal elements across their various technological applications, not ignoring nano-safety considerations.

Nanocrystallites, adsorption, surface tension, and Wulff rule

Chemisorption on the surface of metal nanocrystallites (NCs) sometimes induces their reshaping. This interesting phenomenon was observed experimentally in various systems. Related theoretical studies imply that it can be described using the Wulff rule with the surface tension dependent on the coverage of the NC facets by adsorbate. There is, however, no agreement as to how the surface tension should be calculated in this case. Relying on the laws of statistical physics, I clarify the situation in this area in general and also in the framework of the mean-field approximation in three situations: (i) with adsorption-desorption equilibrium, (ii) with a fixed amount of adsorbate at a NC, and (iii) with a fixed amount of adsorbate at facets of a NC. Under these conditions, the surface tension is shown to be described by the same expressions.