Key factors for designing single-atom metal-nitrogen-carbon catalysts for electrochemical CO2 reduction

The electrochemical CO₂ reduction (CO₂RR) is a sustainable approach to mitigate the increased CO₂ emissions and simultaneously produce value-added chemicals and fuels. Metal-nitrogen-carbon (M-N-C) based single-atom catalysts (SACs) have emerged as promising electrocatalysts for CO₂RR with high activity, selectivity, and stability. To design efficient SACs for CO₂RR, the key influence factors need to be understood. Here, we summarize recent achievements on M-N-C SACs for CO₂RR and highlight the significance of the key constituting factors, metal sites, the coordination environment, and the substrates, for achieving high CO₂RR performance. The perspective views and guidelines are provided for the future direction of developing M-N-C SACs as CO₂RR catalysts.     

Stable monovalent aluminum(i) in a reduced phosphomolybdate cluster as an active acid catalyst

Low-valent aluminum Al(i) chemistry has attracted extensive research interest due to its unique chemical and catalytic properties but is limited by its low stability. Herein, a hourglass phosphomolybdate cluster with a metal-center sandwiched by two benzene-like planar subunits and large steric-hindrance is used as a scaffold to stabilize low-valent Al(i) species. Two hybrid structures, (H₃O)₂(H₂bpe)₁₁[Al^III(H₂O)₂]₃{[Al^I(P₄Mo^V₆O₃₁H₆)₂]₃·7H₂O (abbr. Al₆{P₄Mo₆}₆) and (H₃O)₃(H₂bpe)₃[Al^I(P₄Mo^V₆O₃₁H₇)₂]·3.5H₂O (abbr. Al{P₄Mo₆}₂) (bpe = trans-1,2-di-(4-pyridyl)-ethylene) were successfully synthesized with Al(i)-sandwiched polyoxoanionic clusters as the first inorganic-ferrocene analogues of a monovalent group 13 element with dual Lewis and Brønsted acid sites. As dual-acid catalysts, these hourglass structures efficiently catalyze a solvent-free four-component domino reaction to synthesize 1,5-benzodiazepines. This work provides a new strategy to stabilize low-valent Al(i) species using a polyoxometalate scaffold.

Monovalent aluminum(i) species was successfully stabilized using a reduced phosphomolybdate scaffold as a dual-acid catalyst for a four-component domino reaction.

Low- or sub-valence aluminum compounds are increasingly growing into a significant frontier subject in coordination and modern organic synthetic chemistry owing to their unique singlet carbene character, Lewis acid/base properties and catalytic reactivity.¹ However, low-valence aluminum(i) compounds have inherent electron deficiency and exhibit thermodynamic instability, making them prone to self-polymerization with metal–metal bonds² or disproportionation³ to metallic Al and Al(iii) species. Inspired by the special stabilizing effect of metallocene compounds, a ligand stabilization strategy has recently been undertaken to stabilize the low-valence aluminum center.^4,5 In this regard, the utilized ligand should satisfy two key criteria: (i) sufficient steric hindrance is required to inhibit monomer polymerization; and (ii) a suitable electronic effect is needed to stabilize the aluminum(i) center. A few organometallic Al(i) compounds protected by bulky organic groups have been prepared such as [(Cp*Al)₄] (Cp* = C₅Me₅),⁶ and [(CMe₃)₃SiAl₄].⁷ However, despite having the ligand effect, most of these Al(i) compounds still decompose in aqueous solutions or heating conditions. In contrast to organometallic Al(i) compounds, inorganic Al(i) structures, i.e. monomeric monohalides, only exist in gaseous form at high temperature⁸ and to the best of our knowledge, no stable inorganic Al(i) compound is known at room temperature due to thermodynamic instability. Therefore, exploring efficient strategies to synthesize stable inorganic Al(i) compounds remains highly desired but a great challenge.Polyoxometalates (POMs), a diverse family of inorganic molecular clusters based on early-transition metals (W, Mo, V, Nb, and Ta), have extensively attracted attention in research in various fields of materials science, coordination chemistry, medicinal chemistry and catalysis science.^9–11 Owing to their adjustable constituent elements and well-defined structures, POMs have been considered as promising inorganic ligands to stabilize high- and low-valent metal ions. For instance, Rompel et al.¹² reported one Keggin-type [α-CrW₁₂O₄₀]⁵⁻ anion in which a labile {Cr^IIIO₄} tetrahedral unit was assembled at the center of the cluster. Li and co-workers employed a monolacunary Keggin-type inorganic ligand to stabilize a high-valent Cu³⁺ ion.¹³ As a unique member of the POM family, the hourglass-type phosphomolybdate cluster {M[P₄Mo^V₆O₃₁]₂}ⁿ⁻ (abbr. M{P₄Mo₆}₂), consisting of two [P₄Mo^V₆O₃₁]¹²⁻ (abbr. {P₄Mo₆}₂) subunits bridged by one metal (M) center, represents a fully reduced metal-oxo cluster. With all Mo atoms in the oxidation state of (+5), a more negative charge is endowed to the cluster surface.^14,15 Such high electron density of {M[P₄Mo^V₆O₃₁]₂}ⁿ⁻ polyoxoanions provides an electron-rich local environment for the possible stabilization of unusual-valence metals. It is worth noting that the [P₄Mo^V₆O₃₁]¹²⁻ subunit presents near-planar triangular structures with the side sizes ranging from 7.50–7.92 Å (Fig. S1^†). The structural feature can supply sufficient steric hindrance to restrain the polymerization of low-valence metal species. Moreover, the six Mo atoms in each [P₄Mo^V₆O₃₁]¹²⁻ subunit arrange in a planar hexagonal-ring structure like a benzene ring, implying that such {M[P₄Mo^V₆O₃₁]₂}ⁿ⁻ clusters may have a similar delocalized electron structure to conjugated benzene or cyclopentadiene. These features make [P₄Mo^V₆O₃₁]¹²⁻ a promising candidate with respect to organic protecting groups to construct an inorganic ‘ferrocene’ analogue of Al(i) (Scheme 1). Therefore, we hypothesize that hourglass-type polyoxoanion clusters are promising to stabilize the labile Al(i) center and isolate inorganic Al(i) species.Open in a separate windowScheme 1Similar ferrocene-like sandwich structure features of an inorganic hourglass-type [Al^I(P₄Mo^V₆O₃₁)₂]²³⁻ polyanion to an organometallic [(η⁵-Cp*)₂Al^I]⁺ cation.Herein, we show a [P₄Mo^V₆O₃₁]¹²⁻ cluster as an inorganic scaffold to stabilize the Al(i) center in two hybrid compounds, (H₃O)₂(H₂bpe)₁₁[Al^III(H₂O)₂]₃{[Al^I(P₄Mo^V₆O₃₁H₆)₂]₃·7H₂O (abbr. Al₆{P₄Mo₆}₆) and (H₃O)₃(H₂bpe)₃[Al^I(P₄Mo^V₆O₃₁H₇)₂]·3.5H₂O (abbr. Al{P₄Mo₆}₂) (bpe = trans-1,2-di-(4-pyridyl)-ethylene), in which the labile Al(i) center is sandwiched by two [P₄Mo^V₆O₃₁]¹²⁻ sides, forming an inorganic moiety of a ‘ferrocene’ analogue. Both Al₆{P₄Mo₆}₆ and Al{P₄Mo₆}₂ are experimentally determined at room temperature for the first time, and prepared by hydrothermal reactions of Na₂MoO₄·2H₂O, H₃PO₄, AlCl₃·6H₂O, ethanol and N-containing bpe at 160 °C with slightly different pH values. Notably, the combination of ethanol, N-containing bpe and high hydrothermal temperature is a prerequisite to the isolation of Al(i) species. First, both ethanol and N-containing bpe were used to provide a reducing environment under hydrothermal conditions. By combining high temperature and pressure, sufficient energy is supplied to reduce Mo⁶⁺ and Al³⁺ ions to Mo⁵⁺ and Al⁺ species, respectively. Then, Mo⁵⁺ species and phosphoric acid molecules are assembled to form [P₄Mo₆O₃₁]¹²⁻ subunits, which are subsequently combined with Al⁺ ions to form hourglass-type [Al(P₄Mo₆O₃₁)₂]²³⁻, hence effectively stabilizing Al(i) species (Fig. 1). From the perspective of stereochemistry, two highly negative [P₄Mo₆O₃₁]¹²⁻ fragments, resembling the methyl cyclopentadiene organic group, sandwich one low-valent metal Al(i) center. Hence, the construction of a strong reducing hourglass-like skeleton makes it possible to stabilize the existing Al⁺ species.Open in a separate windowFig. 1Ball-and-stick diagram showing the assembly of the hourglass-type cluster {Al(P₄Mo₆)₂}.Single crystal X-ray diffraction revealed the hourglass-type {Al(P₄Mo₆)₂} cluster in Al₆{P₄Mo₆}₆ and Al{P₄Mo₆}₂ (Table S1^†), in which the [P₄Mo₆O₃₁]¹²⁻ subunits have a C₃ symmetry and display a near-planar structure formed by six edge-sharing {MoO₆} octahedra with alternating short Mo–Mo single bonds and long non-bonding Mo⋯Mo contacts. The side sizes of the {P₄Mo₆} subunit range from 7.50–7.92 Å, which supplies sufficient steric hindrance to restrain the polymerization or disproportionation of low-valence Al(i) species. All Mo atoms are in a reduced oxidation state of +5 and the central Al atoms are in the +1 oxidation state, as confirmed by bond valence calculations (Table S2^†). Thus, the synthesized Al{P₄Mo₆}₂ represents a fully reduced metal–oxygen cluster. Moreover, the six Mo atoms in each {P₄Mo₆} subunit present a benzene-like planar hexagonal-ring structure with a similar π-type delocalization electron interaction with Al(i) instead of organic bulky groups. Such π-type delocalization electron interaction constructs an inorganic ‘ferrocene’ analogue of Al(i) and produces sufficient delocalization energy to stabilize Al(i) species. Considering the formation mechanism of traditional metallocenes, {P₄Mo₆} subunits with a similar strong electron-donating ability and suitable steric-hindrance effect on Cp rings, augment the stability of Al(i) species. Al₆{P₄Mo₆}₆ and Al{P₄Mo₆}₂ compounds also present the first isolation of aluminum-sandwiched hourglass-type clusters in POM chemistry. Importantly, regarding the inherent and strong hydrolysis of aluminum species in water, these low-valent Al(i)-containing clusters represent the first example of stable solid-state inorganic sub-valent Al(i) compounds at room temperature.The asymmetric structure of Al₆{P₄Mo₆}₆ consists of two crystallographically independent {Al(P₄Mo₆)₂} clusters sandwiched by central Al(1) and Al(4) atoms, two bridging [Al(H₂O)₂]³⁺ (Al(2) and Al(3)) cations and six protonated bpe cations (Fig. S2^†). Aluminum centers involve two kinds of oxidation states: the central Al(1) and Al(4) are in the +1 state, while the bridging Al(2) and Al(3) are in the +3 state. Both Al(1) and Al(4) display the six-coordinated octahedral configuration and bridge two {P₄Mo₆} subunits to form two {Al^I(P₄Mo₆)₂} clusters. The average lengths of Al–O bonds are 2.318–2.324 Å for Al(1) and Al(4) (Table S3^†), which are slightly longer than those of classic Al–O bonds (1.90 Å) for Al(2) and Al(3), but close to that of the Al–O bond in silica-supported alkylaluminum(i) composites.^16–20 The long Al–O lengths for Al(1) and Al(4) centers may be ascribed to the lower electron cloud density located at the surface of the Al(i) cation, resulting in slightly longer bonds with the surrounding oxygen donors.^5,21 Moreover, the small distorted extents (sum((d_ij − d_ave)/d_ave)²/coordination number) of {Al(1)O₆} (3.86 × 10⁻⁴) and {Al(4)O₆} (1.89 × 10⁻³) indicate that they are in regular octahedral geometry. Moreover, another structural feature of Al₆{P₄Mo₆}₆ is that {Al^I(P₄Mo₆)₂} clusters are connected by bridging [Al(H₂O)₂]³⁺ cationic fragments (Al(2) and Al(3)), forming an unusual chain-like arrangement (Fig. 2a). It is worth noting that the 1-D chain contains a large repeating monomer with the maximum spacing of 81.69 Å, consisting of twelve Al-containing fragments ({–Al2–Al1–Al3–Al4–Al3–Al1–Al2–Al1–Al3–Al4–Al3–Al1–}). Such a long repeating monomer is rare. Each repeating monomer has two types of symmetric systems: Al(2) in the middle of the monomer plays a center of mirror symmetry and divides the whole repeating monomer into two equidistant half-units of {–Al1–Al3–Al4–Al3–Al1–}; Al(4) in each half-unit further acts as the reverse symmetric center of two {–Al3–Al1–Al2–} subunits. The two types of symmetrical systems form the infinitely extending chain-like structure in Al₆{P₄Mo₆}₆. Since bpe is a rigid and conjugated molecular structure, an effective π⋯π stacking interaction emerges and results in a honeycomb-like supramolecular organic moiety, which accommodates these 1-D inorganic chains and stabilizes the whole Al₆{P₄Mo₆}₆ framework (Fig. S3 and S4^†).Open in a separate windowFig. 2(a) One-dimensional (1D) inorganic structure in Al₆{P₄Mo₆}₆ with a length of repeating units of 81.69 Å, consisting of twelve Al-containing fragments ({–Al2–Al1–Al3–Al4–Al3–Al1–Al2–Al1–Al3–Al4–Al3–Al1–}). (b) Four kinds of coordination environments of {AlO₆} octahedra, respectively (i = 1 − x, y, 0.5 − z; ii = 0.5 − x, 1.5 − y, 1 − z).Al{P₄Mo₆}₂ has a similar structure to Al₆{P₄Mo₆}₆ (Table S4^†), wherein the most obvious difference is that {Al^I[P₄Mo₆]₂} clusters exist in isolated form and interact with the surrounding protonated bpe cations via hydrogen bonding to form into a 3-D supramolecular framework (Fig. S5 and S6^†). The different peripheral environment around the {Al^I[P₄Mo₆]₂} cluster can affect its acidity and catalytic activity.The solid-state ²⁷Al NMR spectrum of Al₆{P₄Mo₆}₆ depicts two distinct resonances at δ = −22.34 and 27.33 ppm due to the octahedrally coordinated Al^III and Al^I sites, respectively (Fig. 3a), indicating two types of Al local environments in Al₆{P₄Mo₆}₆. In contrast, Al{P₄Mo₆}₂ displays only one sharp signal at δ = 7.20 ppm due to the octahedrally coordinated Al^I sites (Fig. 3b). The observed narrow peak-width corresponds to the highly symmetric charge distribution at the aluminum nucleus, similar to the ferrocene analogue [(η⁵-Cp*)₂Al^I]⁺.⁵ Noticeably, Al^I resonance in Al₆{P₄Mo₆}₆ appears at a lower magnetic field compared to Al{P₄Mo₆}₂, due to the different peripheral environment around the hourglass {Al(P₄Mo₆)₂} cluster. XPS spectra of Al₆{P₄Mo₆}₆ and Al{P₄Mo₆}₂ further affirm the valence states of Al and Mo elements (Fig. S7 and Table S5^†). The Al 2p XPS profile of Al₆{P₄Mo₆}₆ reveals two peaks at 74.39 and 73.75 eV ascribed to Al^III and Al^I, respectively (Fig. 3c). The area ratio of the two peaks is close to 1 : 1, in consistence with the chemical structure of Al₆{P₄Mo₆}₆. The high-resolution Al 2p XPS spectrum of Al{P₄Mo₆}₂ displays a weaker broad peak attributed to the low amount of Al⁺ (Fig. 3d). Moreover, the structural stabilities of Al₆{P₄Mo₆}₆ and Al{P₄Mo₆}₂ were investigated by soaking them in water for 24 hours. Fig. S9–S11^† show the comparison of XRD, IR and XPS spectra of Al₆{P₄Mo₆}₆ and Al{P₄Mo₆}₂ before and after soaking in water. It can be found that the characteristic diffraction peaks in XRD after soaking for 24 hours still show good agreement with the simulated data (Fig. S9^†). The characterized absorption bands in IR spectra also exhibit good match with the original Al₆{P₄Mo₆}₆ and Al{P₄Mo₆}₂ (Fig. S10^†). The XPS spectra of Al₆{P₄Mo₆}₆ after soaking in water were also obtained. There is basically no change in the high-resolution spectra of Al 2p with the Al^I/Al^III atomic ratios of ca. 1 : 1 (Fig. S11^†). The spectroscopic and theoretical observations verify that the low valence Al(i) species can stably exist in the reduced phosphomolybdates in the solid state (Fig. S12 and Table S6^†). Moreover, the acidities of Al₆{P₄Mo₆}₆ and Al{P₄Mo₆}₂ were measured to be 0.27 and 0.442 mmol g⁻¹, respectively, demonstrating the promising potential of Al₆{P₄Mo₆}₆ and Al{P₄Mo₆}₂ as dual-acid catalysts.Open in a separate windowFig. 3(a and b) ²⁷Al NMR spectra of solid Al₆{P₄Mo₆}₆ and Al{P₄Mo₆}₂; (c and d) XPS spectra of Al in Al₆{P₄Mo₆}₆ and Al{P₄Mo₆}₂.The catalytic performance of Al₆{P₄Mo₆}₆ and Al{P₄Mo₆}₂ was evaluated via a solvent-free four-component domino reaction for the synthesis of pharmaceutical intermediate 1,5-benzodiazepine (Table 1). With Al₆{P₄Mo₆}₆ and Al{P₄Mo₆}₂ as catalysts, the yields of the final product 8aaa reach 83% and 75%, respectively (Table 1, entries 1 and 2). Almost no 8aaa is observed without the acid catalysts, even when the reaction is set for a long time (Table 1, entry 3). This clarifies the excellent catalytic performance of Al₆{P₄Mo₆}₆ and Al{P₄Mo₆}₂. Typical Brønsted acid p-TsOH and Lewis acid AlCl₃ as control samples yield only 43% and 29% 8aaa, respectively (Table 1, entries 4 and 5), much lower than those attained by Al₆{P₄Mo₆}₆ and Al{P₄Mo₆}₂ catalysts. Moreover, (H₂en)₁₂[{Na_0.8K_0.2(H₂O)}₂{Na[Mo₆O₁₂(OH)₃(HPO₄)₂(PO₄)₂]₂}₂]·7H₂O^22,23 (abbr. {Na[P₄Mo₆]₂}) in contrast achieved 72% yield of 8aaa in 30 min, slower than that of Al₆{P₄Mo₆}₆ and Al{P₄Mo₆}₂. This indicates the advantage of the unique dual-acid features of Al(i)-stabilized reduced phosphomolybdate clusters with multiple Lewis and Brønsted acid active centers, in which the synergistic effect between the Al species and reduced phosphomolybdate cluster contributes to the catalytic activity.Comparison tests of one-pot synthesis of 1,5-benzodiazepine 8aaavia a four-component domino reaction^a

Preparation, nano purification, quality control and labeling optimization of [64Cu]-5,10,15,20-tetrakis (penta fluoro phenyl) porphyrin complex as a possible imaging agent

Cu-64 was produced via the ⁶⁸Zn (p,αn)⁶⁴Cu nuclear reaction (≈200 mCi, >95 % chemical yield at 180 μA for 1.1 h irradiation, (radionuclidic purity >96 %, copper-67 as impurity) followed by purification with amino functionalized nano magnetic oxide, Fe₃O₄ aiming to remove trace amount of heavy metal ions from aqueous media due to achieve ultra pure [⁶⁴Cu] CuCl₂ for labeling step. [⁶⁴Cu] labeled 5,10,15,20-tetrakis(penta fluoro phenyl) porphyrin ([⁶⁴Cu]-TFPP) was prepared using freshly prepared [⁶⁴Cu] CuCl₂ (Cu-64; T _1/2 = 12.7 h) and 5,10,15,20-tetrakis(penta fluoro phenyl)porphyrin (H₂TFPP) for 60 min at 100 °C under reflux condition (radiochemical purity: >97 % ITLC, >98 % HPLC, specific activity: 14–16 GBq/mmol). Stability of the complex was checked in final formulation and human serum for 24 h. The partition coefficient was calculated for the compound (log P = 0.73). The biodistribution of the labeled compound in vital organs of wild-type rats was studied using scarification studies and PET imaging up in 2 and 4 h after injection. A detailed comparative pharmacokinetic study performed for ⁶⁴Cu cation and [⁶⁴Cu]-TFPP. The complex is mostly washed out from the circulation through kidneys and liver and can be an interesting tumor imaging/targeting agent due to high specific uptake and rapid excretion through the urinary tract.     

Preparation of conductive,flexible and transparent films by in situ deposition of polypyrrole nanoparticles on polyethylene terephthalate

In this study polypyrrole (PPy) nanoparticles were deposited as a thin film on the modified surface of polyethyleneterephthalate (PET) by in situ chemical polymerization in the presence of sodium dodecylsulfate (SDS), sodium dodecylbenzenesulfonate (DBSNa) and mixture of them as the surfactant. The surface of PET was modified by KOH before deposition and was investigated for conductivity and adhesion of PPy nanoparticles to PET. Resulting conductive flexible films were characterized by UV–Vis spectroscopy, fieldemission scanning electron microscopy, contact angle measurements and four-point-probe technique for conductivity. Direct morphological observation (FESEM) and electrical measurements indicated that the morphology, conductivity and the nature of deposited PPy films depend on surfactant, surface modification of PET and monomer concentration. In optimized process condition, uniform conductive films of PPy were obtained with good adhesion to PET.     

Preparation,characterization and heterogeneous catalytic applications of GO/Fe3O4/HPW nanocomposite in chemoselective and green oxidation of alcohols with aqueous H2O2

Graphene oxide ‐ Fe₃O₄ ‐ NH₃⁺H₂PW₁₂O₄₀ ^‐ magnetic nanocomposite (GO/Fe₃O₄/HPW) was prepared by linking amino ‐ functionalized Fe₃O₄ nanoparticles (Fe₃O₄ ‐ NH₂) on the graphene oxide (GO), and then grafting 12 ‐ tungstophosphoric acid (H₃PW₁₂O₄₀) on the graphene oxide ‐ magnetite hybrid (GO ‐ Fe₃O₄ ‐ NH₂). The obtained GO/Fe₃O₄/HPW nanocomposite was well characterized with different techniques such as FT ‐ IR, TEM, SEM, XRD, EDX, TGA ‐ DTA, AGFM, ICP and BET measurements. The used techniques showed that the graphene oxide layers were well prepared and the various stages of preparation of the GO/Fe₃O₄/HPW nanocomposites successfully completed. This new nanocomposite displayed excellent performance as a heterogeneous catalyst in the oxidation of alcohols with H₂O₂. The as ‐ prepared GO/Fe₃O₄/HPW catalyst was more stable and recyclable at least five times without significantly reducing its catalytic activity.     

Synthesis of diaryl sulfides via nickel ferrite‐catalysed C─S bond formation in green media

NiFe₂O₄ magnetic nanoparticles (MNPs) were synthesized, characterized and applied as an air‐stable, inexpensive and magnetically separable nanocatalyst for the synthesis of structurally diverse sulfides. Efficient methodologies were developed for the synthesis of unsymmetric diaryl sulfides via odourless and one‐pot reactions of triphenyltin chloride/S₈ or arylboronic acid/S₈ as thiolating agents with aryl halides or nitroarenes as starting materials in the presence of base (K₂CO₃ or NaOH) and NiFe₂O₄ MNPs as a catalyst in water or poly (ethylene glycol) as solvent at 80–110 °C. Free from ligand and the unpleasant smell of thiols and with the use of magnetically reusable nanocatalyst, green solvents and commercially available and cheap sulfur source and starting materials, these methods are more eco‐friendly and practical than available protocols for the synthesis of sulfides.     

A combined treatment of hydration and dynamical effects for the modeling of host–guest binding thermodynamics: the SAMPL5 blinded challenge

As part of the SAMPL5 blinded experiment, we computed the absolute binding free energies of 22 host–guest complexes employing a novel approach based on the BEDAM single-decoupling alchemical free energy protocol with parallel replica exchange conformational sampling and the AGBNP2 implicit solvation model specifically customized to treat the effect of water displacement as modeled by the Hydration Site Analysis method with explicit solvation. Initial predictions were affected by the lack of treatment of ionic charge screening, which is very significant for these highly charged hosts, and resulted in poor relative ranking of negatively versus positively charged guests. Binding free energies obtained with Debye–Hückel treatment of salt effects were in good agreement with experimental measurements. Water displacement effects contributed favorably and very significantly to the observed binding affinities; without it, the modeling predictions would have grossly underestimated binding. The work validates the implicit/explicit solvation approach employed here and it shows that comprehensive physical models can be effective at predicting binding affinities of molecular complexes requiring accurate treatment of conformational dynamics and hydration.     

Super face d-antimagic labeling for disjoint union of toroidal fullerenes

The discovery of the fullerene molecules and related forms of carbon such as nanotubes has generated an explosion of activity in chemistry, physics, and materials science. Classical fullerene is an all-carbon molecule in which the atoms are arranged on a pseudospherical framework made up entirely of pentagons and hexagons. A toroidal fullerene (toroidal polyhex) is a cubic bipartite graph embedded on the torus such that each face is a hexagon. In this paper we examine the existence of entire labeling, where face-weights of all 6-sided faces of disjoint union of toroidal fullerenes form an arithmetic progression with common difference \(\hbox {d}\in \{1,2,3\}\).     

Associated Prime Submodules of Finitely Generated Modules

As generalizations of annihilators and associated primes, we introduce the notions of weak annihilators and weak associated primes, respectively. We first study the properties of the weak annihilator of a subset X in a ring R. We next investigate how the weak associated primes of a ring R behave under passage to the skew monoid ring R*M. Let R be a semicommutative ring, and M an ordered monoid and φ: M → Aut(R) a compatible monoid homomorphism. Then we can describe all weak associated primes of the skew monoid ring R*M in terms of the weak associated primes of R in a very straightforward way.