Development of a Novel Water Jet Polisher Using Soft Abrasives for Small Complex-Structure Heat Pipes of Aluminum Alloy Produced Using Additive Manufacturing.

It is a challenge to polish the interior surface of a small bent pipe with complex structures and sizes less than 0.5 mm. This is because of the fact that traditional polishing methods could destroy, block, or break the small complex structures. For a small bent pipe made of aluminum alloy produced using additive manufacturing, the defects, such as adhered powders and spatters, are easy to jam the pipe without polishing, possibly resulting in catastrophic failure for aerospace applications. To overcome this challenge, a novel water jet polisher was developed using soft polymethyl methacrylate (PMMA) abrasives. After polishing a specific area, the adhered powders on the interior surface were reduced from over 140 to 2, 3, and 6 by the soft abrasives with mesh sizes of 200, 400, and 600, respectively. The surface roughness Sa was decreased from 3.41 to 0.92 µm after polishing using PMMA abrasives with a mesh size of 200. In comparison, silica abrasives were also employed to polish the small bent pipes, leading to the bent part of pipes breaking. However, this kind of failure was absent when using soft abrasives. Computational fluid dynamics calculations elucidate that a peak erosion rate of silica abrasives for a bent pipe with a turn angle of 30° is 2.18 kg/(m2·s), which is 17 times that of soft abrasives. This is why the small bent pipe was broken using silica abrasives, whereas it remained intact when polished with soft abrasives. In addition, water jet polishing has a lower erosion rate, a relatively smooth erosion curve, and less erosion energy, leaving the bent parts intact. The developed soft abrasive water jet polisher and the findings of this study suggest new possibilities for cleaning the adhered powders and spatters and polishing the interior surface of small bent pipes with complex structures.

Atomic surface of quartz glass induced by photocatalytic green chemical mechanical polishing using the developed SiO2@TiO2 core-shell slurry.

High-performance devices of quartz glass demand an atomic surface, which induces a challenge for chemical mechanical polishing (CMP) with a high material removal rate (MRR). Moreover, traditional CMP usually employs toxic and corrosive slurries, leading to the pollution of the environment. To overcome these challenges, a novel green photocatalytic CMP is proposed. In the CMP, SiO2@TiO2 core-shell abrasives were developed, and the CMP slurry included the developed abrasives, sodium carbonate, hydrogen peroxide and sorbitol. After photocatalytic CMP, the surface roughness Sa of quartz glass is 0.185 nm, with a scanning area of 50 × 50 µm2, and the MRR is 8.64 µm h-1. To the best of our knowledge, the MRR is the highest on such a big area of atomic surface for quartz glass. X-ray photoelectron spectroscopy reveals that SiO2@TiO2 core-shell abrasives were used as photocatalysts motivated by simulated solar light, generating electrons and holes and producing hydroxyl radicals through hydrogen peroxide. As a result, OH- could combine with Si atoms on the surface of quartz glass, forming Si-OH-Si bonds. Then the formed bonds were removed based on the balance between chemical and mechanical functions. The proposed CMP, developed SiO2@TiO2 abrasives and slurry provide new insights to achieve an atomic surface of quartz glass with a high MRR.

Atomic surface achieved through a novel cross-scale model from macroscale to nanoscale.

Chemical mechanical polishing (CMP) is widely used to achieve an atomic surface globally, yet its cross-scale polishing mechanisms are elusive. Moreover, traditional CMP normally employs toxic and corrosive slurries, resulting in potential pollution to the environment. To overcome these challenges, a novel cross-scale model from the millimeter to nanometer scale is proposed, which was confirmed by a newly developed green CMP process. The developed CMP slurry consisted of hydrogen peroxide, sodium carbonate, sodium hydroxycellulose, and silica. Prior to CMP, fused silica was polished by a ceria slurry. After CMP, the surface roughness (Sa) was 0.126 nm, the material-removal rate was 88.3 nm min-1, and the thickness of the damaged layer was 8.8 nm. The proposed model was built by fibers, through integrating Eulerian and Lagrangian models and reactive force field-molecular dynamics. The results predicted by the model were in good agreement with those of CMP experimentally. A model for large-sized fibers revealed that a direct contact area of 11.12% was obtained for a non-woven polishing pad during the CMP experiments. Another model constructed via combining Eulerian and Lagrangian functions showed that the stress at the intersections of the fibers varied mainly from 0.1 to 0.01 MPa and was higher than the stress at other parts. An increase in viscosity led to a decrease in the areas with low stress, demonstrating that viscosity enhanced the stress and facilitated the removal of material. At the microscale and nanoscale, the stress of the abrasive surface exposed to the workpiece changed from 2.21 to 6.43 GPa. Stress at the interface contributed to the formation of bridging bonds, further promoting the removal of material. With increasing the compressive stress, the material-removal form was transformed from a single atom to molecular chains. The proposed model and developed green CMP offer new insights to understand the cross-scale polishing mechanism, as well as for designing and manufacturing novel polishing slurries, pads, and setups.

Unprecedented atomic surface of silicon induced by environmentally friendly chemical mechanical polishing.

Silicon (Si) dominates the integrated circuit (IC), semiconductor, and microelectronic industries. However, it is a challenge to achieve a sub-angstrom surface of Si. Chemical mechanical polishing (CMP) is widely used in the manufacturing of Si, while toxic and polluted slurries are usually employed in CMP, resulting in pollution to the environment. In this study, a novel environmentally friendly CMP was developed, in which a slurry is composed of ceria, hydrogen peroxide, sodium pyrophosphate, sodium carboxymethyl cellulose, sodium carbonate, and deionized water. After CMP, the surface roughness Sa was 0.067 nm with a measurement area of 50 × 50 µm2, and a sub-angstrom surface is achieved. To the best of our knowledge, it is the lowest surface roughness in such a large area. Transmission electron microscopy shows that the thickness of the damaged layer after CMP is 2.8 nm. X-ray photoelectron spectroscopy and infrared Fourier transformation reveal that during CMP, a redox reaction firstly took place between Ce3+ and Ce4+. Si and ceria are hydroxylated, forming Si-OH and Ce-OH, then dehydration and condensation occur, generating Si-O-Ce. These findings propose new insights to fabricate a sub-angstrom surface of Si for use in IC, semiconductor, and microelectronic industries.

Experimental Investigation on the Surface Formation Mechanism of NdFeB during Diamond Wire Sawing.

Diamond wire sawing is widely used in processing NdFeB rare earth permanent magnets. However, it induces periodic saw marks and fracture chipping pits, which severely affect the flatness and surface quality of the products. In this study, the lateral motion of the diamond wire was monitored to determine the surface formation mechanism. Then, a white light interferometer and an SEM were used to observe the sawed surface profile. Finally, the surface quality was quantitatively studied by identifying the area rate of fracture chipping pits with an image recognition MATLAB script. According to the observation results, the calculation formula of PV which is related to the process parameters was deduced. Additionally, by combining the fracture rate and wire vibration, a novel method was proposed to investigate the optimal process parameters. It can be found that the surface quality sawed at P = 0.21 MPa, vf = 0.2 mm/min, and vs = 1.8 m/s remains better than when sawed at P = 0.15 MPa, vf = 0.1 mm/min, and vs = 1.8 m/s, which means the sawing efficiency can be doubled under such circumstances, i.e., when the surface quality remains the same.

Dispersion and Polishing Mechanism of a Novel CeO2-LaOF-Based Chemical Mechanical Polishing Slurry for Quartz Glass.

Quartz glass shows superior physicochemical properties and is used in modern high technology. Due to its hard and brittle characteristics, traditional polishing slurry mostly uses strong acid, strong alkali, and potent corrosive additives, which cause environmental pollution. Furthermore, the degree of damage reduces service performance of the parts due to the excessive corrosion. Therefore, a novel quartz glass green and efficient non-damaging chemical mechanical polishing slurry was developed, consisting of cerium oxide (CeO2), Lanthanum oxyfluoride (LaOF), potassium pyrophosphate (K4P2O7), sodium N-lauroyl sarcosinate (SNLS), and sodium polyacrylate (PAAS). Among them, LaOF abrasive showed hexahedral morphology, which increased the cutting sites and uniformed the load. The polishing slurry was maintained by two anionic dispersants, namely SNLS and PAAS, to maintain the suspension stability of the slurry, which makes the abrasive in the slurry have a more uniform particle size and a smoother sample surface after polishing. After the orthogonal test, a surface roughness (Sa) of 0.23 nm was obtained in the range of 50 × 50 µm2, which was lower than the current industry rating of 0.9 nm, and obtained a material removal rate (MRR) of 530.52 nm/min.

Microstructure and Mechanical Properties of Nanoparticulate Y2O3 Modified AlSi10Mg Alloys Manufactured by Selective Laser Melting.

AlSi10Mg has a good forming ability and has been widely accepted as an optimal material for selective laser melting (SLM). However, the strength and elongation of unmodified AlSi10Mg are insufficient, which limits its application in the space industry. In this paper, yttrium oxide (Y2O3) nanoparticles modified AlSi10Mg composites that were manufactured using SLM. The effects of Y2O3 nanoparticles (0~2 wt.% addition) on the microstructure and mechanical properties of AlSi10Mg alloys were investigated. An ultimate tensile strength of 500.3 MPa, a yield strength of 322.3 MPa, an elongation of 9.7%, a good friction coefficient of 0.43, and a wear rate of (3.40 ± 0.09) ×10-4 mm3·N-1·m-1 were obtained with the addition of 0.5 wt.% Y2O3 nanoparticles, and all these parameters were higher than those of the SLMed AlSi10Mg alloy. The microhardness of the composite with 1.0 wt.% Y2O3 reached 145.6 HV0.1, which is an increase of approximately 22% compared to the unreinforced AlSi10Mg. The improvement of tensile properties can mainly be attributed to Orowan strengthening, fine grain strengthening, and load-bearing strengthening. The results show that adding an appropriate amount of Y2O3 nanoparticles can significantly improve the properties of the SLMed AlSi10Mg alloy.

Atomic-level flatness on oxygen-free copper surface in lapping and chemical mechanical polishing.

Oxygen-free copper (OFC) serves as a core component of high-end manufacturing, and requires high surface quality. It is always a significant challenge to manufacture high-quality atomic-level surfaces. In this study, SiO2 nanospheres with good dispersibility were prepared and a late-model environmentally friendly chemical mechanical polishing (CMP) slurry was developed. The CMP slurry consists of SiO2 nanospheres, CeO2 nanospheres, H2O2, NaHCO3, polyaspartic acid and deionized water. After CMP, the average roughness (Sa) of the OFC wafer reached 0.092 nm with an area of 50 × 50 µm2. Atomic-level flatness on the oxygen-free copper surface was acquired, which has never been reported before. Moreover, the mechanical removal mechanism of abrasive particles and the chemical reactions during lapping and CMP are proposed in detail. The thickness and composition of the damaged layer after lapping and polishing were analyzed. The lapping-damaged layer consists of a lattice distortion region, moiré fringes, grain boundary, superlattice and edge dislocations, and the polishing-damaged layer contains a handful of stacking faults with single-layer or multi-layer atoms. The chemical action involves three reactions: oxidation, corrosion and chelation. The processing method and its mechanistic explanation pave the way for the fabrication of high-performance OFC surfaces for use in vacuum, aerospace, military and electronic industries.

Thiourea-Assisted Facile Fabrication of High-Quality CsPbBr3 Perovskite Films for High-Performance Solar Cells.

Inorganic CsPbBr3 perovskite solar cells have attracted widespread attention recently because of their decent efficiency and good ambient stability. Nevertheless, the fabrication of high-quality CsPbBr3 perovskite via the conventional solution-processing strategy still faces great challenges because the solubility of CsBr in the conventional solvent is poor. Here, we develop a facile thiourea-assisted two-step spin-coating process to fabricate a CsPbBr3 perovskite film with high phase purity and crystallinity and enlarged crystal grains. Thiourea is introduced into the PbBr2 layer during the first-step spin-coating process, which promotes the wettability of the PbBr2 layer and produces the space for growing large perovskite grains. The green high-concentration CsBr/H2O solution is adopted at the second-step spin-coating process, enabling enough CsBr to be deposited by a facile one-step process. By optimizing the content of thiourea, a compact CsPbBr3 perovskite film with a smooth surface, large grains, and high phase purity and crystallinity is formed. Consequently, the fabricated perovskite solar cell with the architecture of FTO/TiO2/CsPbBr3 film/carbon exhibits a superior performance with a high efficiency of 9.11%. In addition, the unencapsulated device preserves over 90% of its initial efficiency after storage at ambient conditions for 45 days.

Smart MSN-Drug-Delivery System for Tumor Cell Targeting and Tumor Microenvironment Release.

Tumor-targeted delivery and controlled release of antitumor drugs are promising strategies for increasing chemotherapeutic efficacy and reducing adverse effects. Although mesoporous silica nanoparticles (MSNs) have been known as a potential delivery system for doxorubicin (DOX), they have restricted applications due to their uncontrolled leakage and burst release from their large open pores. Herein, we engineered a smart drug-delivery system (smart MSN-drug) based on MSN-drug loading, cell membrane mimetic coating, on-demand pore blocking/opening, and tumor cell targeting strategies. The pore size of DOX-loaded MSNs was narrowed by polydopamine coating, and the pores/channels were blocked with tumor-targeting ligands anchored by tumor environment-rupturable -SS- chains. Furthermore, a cell membrane mimetic surface was constructed to enhance biocompatibility of the smart MSN-drug. Confocal microscopy results demonstrate highly selective uptake (12-fold in comparison with L929 cell) of the smart MSN-drug by HeLa cells and delivery into the HeLa cellular nuclei. Further in vitro IC50 studies showed that the toxicity of the smart MSN-drug to HeLa cells was 4000-fold higher than to the normal fibroblast cells. These exciting results demonstrate the utility of the smart MSN-drug capable of selectively killing tumor cells and saving the normal cells.

Modular Construction of Prussian Blue Analog and TiO2 Dual-Compartment Janus Nanoreactor for Efficient Photocatalytic Water Splitting.

Janus structures that include different functional compartments have attracted significant attention due to their specific properties in a diverse range of applications. However, it remains challenge to develop an effective strategy for achieving strong interfacial interaction. Herein, a Janus nanoreactor consisting of TiO2 2D nanocrystals integrated with Prussian blue analog (PBA) single crystals is proposed and synthesized by mimicking the planting process. In situ etching of PBA particles induces nucleation and growth of TiO2 nanoflakes onto the concave surface of PBA particles, and thus enhances the interlayer interaction. The anisotropic PBA-TiO2 Janus nanoreactor demonstrates enhanced photocatalytic activities for both water reduction and oxidation reactions compared with TiO2 and PBA alone. As far as it is known, this is the first PBA-based composite that serves as a bifunctional photocatalyst for solar water splitting. The interfacial structure between two materials is vital for charge separation and transfer based on the spectroscopic studies. These results shed light on the elaborate construction of Janus nanoreactor, highlighting the important role of interfacial design at the microscale level.

Comparison of interfacial bridging carbon materials for effective carbon-based perovskite solar cells.

As a hole transport materials (HTMs) and noble-metal free perovskite solar cells (PSCs), carbon-based PSCs (C-PSCs) have drawn much attention due to its low cost and excellent stability. The interfacial engineering, between perovskite and counter electrodes (CEs), played a crucial role in charge collection and affected the performance of C-PSCs. Herein, the systematic investigation of interfacial bridging carbon materials has been carried out to improve the interfacial contact. Results demonstrated that the morphology of interfacial bridging carbon materials played more important role than their energy band and conductivity, where carbon nanotube (CN) showed much better interfacial bridging effect and energy level alignment than other carbon materials. We achieved both the high power conversion efficiency (PCE) (15.09%) and stability in C-PSCs without HTMs due to the optimal interfacial bridging carbon material. It could retain 67% of their initial PCE after storing for 340 h under the rigorous environmental condition without any encapsulation (air atmosphere, 85 °C, 65% RH).

Intermediate-Controlled Interfacial Engineering for Stable and Highly Efficient Carbon-Based PSCs.

A major bottleneck hindering the performance and commercial application of cost-effective carbon-based perovskite solar cells (C-PSCs) is the contact issue at the interface of the perovskite layer and the carbon counter electrode. Herein, a new approach of intermediate-controlled interfacial engineering (IIE) utilizing an ultra-low-cost acetylene black material is developed for the first time that can improve the interfacial contact of C-PSCs. We achieved both high efficiency (16.41%) without hole-transport materials and good stability as a result of the optimal heterogeneous interfacial contact. Devices without any encapsulation consistently exhibit excellent environmental stability, retaining 93% of their original efficiency by storing in an ambient atmosphere (30 °C, 30% RH) for 2000 h and achieving 81% of their original efficiency by storing in a terrible air environment (85 °C, 65% RH) for 312 h. In addition, to acquire a deep understanding of carrier transport, a comparison of heterogeneous interfaces fabricated using different methods has been undertaken. In C-PSCs fabricated by the IIE method, the lower radioactive recombination and faster carrier transfer result in a shorter carrier lifetime. We present a promising future for the industrialization of C-PSCs by reducing the costs and improving the performance.

Current Progress of Electrocatalysts for Ammonia Synthesis Through Electrochemical Nitrogen Reduction Under Ambient Conditions.

Ammonia, one of the most important chemicals and carbon-free energy carriers, is mainly produced by the traditional Haber-Bosch process operated at high pressure and temperature, which results in massive energy consumption and CO2 emissions. Alternatively, the electrocatalytic nitrogen reduction reaction to synthesize NH3 under ambient conditions using renewable energy has recently attracted significant attention. However, the competing hydrogen evolution reaction (HER) significantly reduces the faradaic efficiency and NH3 production rate. The design of high-performance electrocatalysts with the suppression of the HER for N2 reduction to NH3 under ambient conditions is a crucial consideration for the development of electrocatalytic NH3 synthesis with high FE and NH3 production rate. Five kinds of recently developed electrocatalysts classified by their chemical compositions are summarized, with particular emphasis on the relationship between their optimal electrocatalytic conditions and NH3 production performance. Conclusions and perspectives are provided for the future design of high-performance electrocatalysts for electrocatalytic NH3 production. The Review can give practical guidance for the design of effective electrocatalysts with high FE and NH3 production rates.

A two-dimensional Ru@MXene catalyst for highly selective ambient electrocatalytic nitrogen reduction.

Compared with the traditional Haber-Bosch process, electrochemical ammonia synthesis has attracted much attention owing to its low energy consumption, low pollution potential, and sustainability. However, owing to the influence of high overpotential and low selectivity, the nitrogen reduction reaction (NRR) process was of limited applicability in industry. Here, we report a high-performance Ru@Ti3C2 MXene catalyst for an ambient electrocatalytic NRR. In a 0.1 M KOH electrolyte, the NH3 yield of the Ru@MXene catalyst reached 2.3 µmol h-1 cm-2, furthermore, at -0.4 V (vs. RHE) the Faraday efficiency was 13.13%.

Macroscale Superlubricity Enabled by Graphene-Coated Surfaces.

Friction and wear remain the primary modes for energy dissipation in moving mechanical components. Superlubricity is highly desirable for energy saving and environmental benefits. Macroscale superlubricity was previously performed under special environments or on curved nanoscale surfaces. Nevertheless, macroscale superlubricity has not yet been demonstrated under ambient conditions on macroscale surfaces, except in humid air produced by purging water vapor into a tribometer chamber. In this study, a tribological system is fabricated using a graphene-coated plate (GCP), graphene-coated microsphere (GCS), and graphene-coated ball (GCB). The friction coefficient of 0.006 is achieved in air under 35 mN at a sliding speed of 0.2 mm s-1 for 1200 s in the developed GCB/GCS/GCP system. To the best of the knowledge, for the first time, macroscale superlubricity on macroscale surfaces under ambient conditions is reported. The mechanism of macroscale superlubricity is due to the combination of exfoliated graphene flakes and the swinging and sliding of the GCS, which is demonstrated by the experimental measurements, ab initio, and molecular dynamics simulations. These findings help to bridge macroscale superlubricity to real world applications, potentially dramatically contributing to energy savings and reducing the emission of carbon dioxide to the environment.

Alternative electrodes for HTMs and noble-metal-free perovskite solar cells: 2D MXenes electrodes.

The high cost of hole transporting materials (HTMs) and noble metal electrodes limits the application of perovskite solar cells (PSCs). Carbon materials have been commonly utilized for HTMs and noble-metal-free PSCs. In this paper, a more conductive 2D MXene material (Ti3C2), showing a similar energy level to carbon materials, has been used as a back electrode in HTMs and noble-metal-free PSCs for the first time. Seamless interfacial contact between the perovskite layer and Ti3C2 material was obtained using a simple hot-pressing method. After the adjustment of key parameters, the PSCs based on the Ti3C2 electrode show more stability and higher power conversion efficiencies (PCE) (13.83%, 27% higher than that (10.87%) of the PSCs based on carbon electrodes) due to the higher conductivity and seamless interfacial contact of the MXene electrode. Our work proposes a promising future application for MXene and also a good electrode candidate for HTM and the noble-metal-free PSCs.

Interfacial Bonding Energy on the Interface between ZChSnSb/Sn Alloy Layer and Steel Body at Microscale.

To investigate the performance of bonding on the interface between ZChSnSb/Sn and steel body, the interfacial bonding energy on the interface of a ZChSnSb/Sn alloy layer and the steel body with or without Sn as an intermediate layer was calculated under the same loadcase using the molecular dynamics simulation software Materials Studio by ACCELRYS, and the interfacial bonding energy under different Babbitt thicknesses was compared. The results show that the bonding energy of the interface with Sn as an intermediate layer is 10% larger than that of the interface without a Sn layer. The interfacial bonding performances of Babbitt and the steel body with Sn as an intermediate layer are better than those of an interface without a Sn layer. When the thickness of the Babbitt layer of bushing is 17.143 Å, the interfacial bonding energy reaches the maximum, and the interfacial bonding performance is optimum. These findings illustrate the bonding mechanism of the interfacial structure from the molecular level so as to ensure the good bonding properties of the interface, which provides a reference for the improvement of the bush manufacturing process from the microscopic point of view.

Quantitative fabrication, performance optimization and comparison of PEG and zwitterionic polymer antifouling coatings.

A versatile fabrication and performance optimization strategy of PEG and zwitterionic polymer coatings is developed on the sensor chip of surface plasma resonance (SPR) instrument. A random copolymer bearing phosphorylcholine zwitterion and active ester side chains (PMEN) and carboxylic PEG coatings with comparable thicknesses were deposited on SPR sensor chips via amidation coupling on the precoated polydopamine (PDA) intermediate layer. The PMEN coating showed much stronger resistance to bovine serum albumin (BSA) adsorption than PEG coating at very thin thickness (∼1nm). However, the BSA resistant efficacy of PEG coating could exceed that of PMEN due to stronger steric repelling effect when the thickness increased to 1.5∼3.3nm. Interestingly, both the PEG and PMEN thick coatings (≈3.6nm) showed ultralow fouling by BSA and bovine plasma fibrinogen (Fg). Moreover, changes in the PEG end group from -OH to -COOH, protein adsorption amount could increase by 10-fold. Importantly, the optimized PMEN and PEG-OH coatings were easily duplicated on other substrates due to universal adhesion of the PDA layer, showed excellent resistance to platelet, bacteria and proteins, and no significant difference in the antifouling performances was observed. These detailed results can explain the reported discrepancy in performances between PEG and zwitterionic polymer coatings by thickness. This facile and substrate-independent coating strategy may benefit the design and manufacture of advanced antifouling biomedical devices and long circulating nanocarriers. STATEMENT OF SIGNIFICANCE: Prevention of biofouling is one of the biggest challenges for all biomedical applications. However, it is very difficult to fabricate a highly hydrophilic antifouling coating on inert materials or large devices. In this study, PEG and zwitterion polymers, the most widely investigated polymers with best antifouling performance, are conveniently immobilized on different kinds of substrates from their aqueous solutions by precoating a polydopamine intermediate layer as the universal adhesive and readily re-modifiable surface. Importantly, the coating fabrication and antifouling performance can be monitored and optimized quantitatively by a surface plasma resonance (SPR) system. More significantly, the SPR on-line optimized coatings were successfully duplicated off-line on other substrates, and supported by their excellent antifouling properties.

Dynamics of the functions [Formula: see text] with the real parameter.

In this paper, the dynamics of the functions [Formula: see text] with the real parameter is studied. We say that a real parameter [Formula: see text] belongs to the set [Formula: see text] for a positive integer n if [Formula: see text] has an attracting cycle of n-order. We prove that the Fatou set [Formula: see text] is a completely invariant attracting basin for every parameter [Formula: see text]. Further, regarding the set [Formula: see text] for [Formula: see text], we prove the following results: (1) There exists [Formula: see text] such that [Formula: see text]. (2) For every positive integer [Formula: see text], the set [Formula: see text] is non-empty. (3) For every prime number [Formula: see text], the set [Formula: see text] has at least two components.