Fixed-point results for multi-valued operators in quasi-ordered metric spaces

Nadler’s contraction principle (Nadler, 1969 [17]) has led to the fixed-point theory of set-valued contraction in nonlinear analysis. The purpose of this paper is to present some new fixed-point theorems involving multi-valued operators in the setting of quasi-ordered metric spaces.     

Hirshfeld surface analysis of two new phosphorothioic triamide structures

Hirshfeld surfaces and two‐dimensional fingerprint plots are used to analyse the intermolecular interactions in two new phosphorothioic triamide structures, namely N,N′,N′′‐tris(3,4‐dimethylphenyl)phosphorothioic triamide acetonitrile hemisolvate, P(S)[NHC₆H₃‐3,4‐(CH₃)₂]₃·0.5CH₃CN or C₂₄H₃₀N₃PS·0.5CH₃CN, (I), and N,N′,N′′‐tris(4‐methylphenyl)phosphorothioic triamide–3‐methylpiperidinium chloride (1/1), P(S)[NHC₆H₄(4‐CH₃)]₃·[3‐CH₃‐C₅H₉NH₂]⁺·Cl⁻ or C₂₁H₂₄N₃PS·C₆H₁₄N⁺·Cl⁻, (II). The asymmetric unit of (I) consists of two independent phosphorothioic triamide molecules and one acetonitrile solvent molecule, whereas for (II), the asymmetric unit is composed of three components (molecule, cation and anion). In the structure of (I), the different components are organized into a six‐molecule aggregate through N—H...S and N—H...N hydrogen bonds. The components of (II) are aggregated into a two‐dimensional array through N—H...S and N—H...Cl hydrogen bonds. Moreover, interesting features of packing arise in this structure due to the presence of a double hydrogen‐bond acceptor (the S atom of the phosphorothioic triamide molecule) and of a double hydrogen‐bond donor (the N—H unit of the cation). For both (I) and (II), the full fingerprint plot of each component is asymmetric as a consequence of the presence of three fragments. These analyses reveal that H...H interactions [67.7 and 64.3% for the two symmetry‐independent phosphorothioic triamide molecules of (I), 30.7% for the acetonitrile solvent of (I), 63.8% in the phosphorothioic triamide molecule of (II) and 62.9% in the 3‐methylpiperidinium cation of (II)] outnumber the other contacts for all the components in both structures, except for the chloride anion of (II), which only receives the Cl...H contact. The phosphorothioic triamide molecules of both structures include unsaturated C atoms, thus presenting C...H/H...C interactions: 17.6 and 21% for the two symmetry‐independent phosphorothioic triamide molecules in (I), and 22.7% for the phosphorothioic triamide molecule of (II). Furthermore, the N—H...S hydrogen bonds in both (I) and (II), and the N—H...Cl hydrogen bonds in (II), are the most prominent interactions, appearing as large red spots on the Hirshfeld surface maps. The N...H/H...N contacts in structure (I) are considerable, whereas for (II), they give a negligible contribution to the total interactions in the system.     

Polyethersulfone/polyacrylonitrile blend ultrafiltration membranes with different molecular weight of polyethylene glycol: preparation,morphology and antifouling properties

Asymmetric ultrafiltration (UF) membranes were prepared from blends of polyethersulfone (PES)/polyacrylonitrile (PAN) via phase inversion method induced by immersion precipitation. Polyethylene glycol (PEG) with four different molecular weights was used as pore former and hydrophilic polymeric additive. N,N‐dimethylformamide (DMF) and water were used as solvent and coagulant (nonsolvent), respectively. The effects of different proportion of PES/PAN and molecular weight of PEG on morphology and performance of the prepared membranes were investigated. Performance of the membranes was evaluated using UF experiments of pure water and buffered bovine serum albumin (BSA) solution as feed. The contact angle measurements indicated that the hydrophilicities of PES/PAN membrane increase by increasing the PAN concentration in the casting solution. However, performance of the membranes improves by increasing the PAN concentration in the casting solution up to 20% and then decreases with further addition of PAN. It was found out that the rejection of BSA decreases with increasing the PAN concentration in the casting solution. Furthermore, it was found that the performance of the membranes increases by increasing the molecular weight of PEG up to 1500 Da and then decreases with the higher molecular weights. The morphology of the prepared membranes was studied by scanning electron microscopy. Copyright © 2011 John Wiley & Sons, Ltd.     

Fourier spectral embedded boundary solution of the Poisson’s and Laplace equations with Dirichlet boundary conditions

A Fourier spectral embedded boundary method, for solution of the Poisson’s equation with Dirichlet boundary conditions and arbitrary forcing functions (including zero forcing function), is presented in this paper. This iterative method begins by transformation of the Dirichlet boundary conditions from the physical boundaries to some corresponding regular grid points (which are called the numerical boundaries), using a second order interpolation method. Then the transformed boundary conditions and the forcing function are extended to a square, smoothly and periodically, via multiplying them by some suitable error functions. Instead of direct solution of the resulting extended Poisson’s problem, it is suggested to define and solve an equivalent transient diffusion problem on the regular domain, until achievement of the steady solution (which is considered as the solution of the original problem). Without need of any numerical time integration method, time advancement of the solution is obtained directly, from the exact solution of the transient problem in the Fourier space. Consequently, timestep sizes can be chosen without stability limitations, which it means higher rates of convergence in comparison with the classical relaxation methods. The method is presented in details for one- and two-dimensional problems, and a new emerged phenomenon (which is called the saturation state) is illustrated both in the physical and spectral spaces. The numerical experiments have been performed on the one- and two-dimensional irregular domains to show the accuracy of the method and its superiority (from the rate of convergence viewpoint) to the other classical relaxation methods. Capability of the method, in dealing with complex geometries, and in presence of discontinuity at the boundaries, has been shown via some numerical experiments on a four-leaf shape geometry.     

Different molecular assemblies in two new phosphoric triamides with the same C(O)NHP(O)(NH)<Subscript>2</Subscript> skeleton: crystallographic study and Hirshfeld surface analysis

Different molecular assemblies were compared in two new structures [4-CH₃-C₆H₄C(O)NH]P(O)[NH]₂(CH₂)₃, 1, and [4-CH₃-C₆H₄C(O)NH]P(O)[NHC₆H₃(3,4-CH₃)₂]₂, 2, belonging to the families of “cyclic phosphoric triamide” and “phosphoric triamide”, respectively. The differences in the hydrogen bond motifs were discussed (by single crystal X-ray diffraction) as a result of three factors: (1) action of two N atoms with a non-planar environment in 1 as an H-bond acceptor, (2) different orientations of three N–H bond vectors in two molecules and (3) different conformations of C=O and P=O groups. These differences lead to more complicated hydrogen bond pattern of 1, with respect to that of 2, as structure 1 may be considered as a model of four-acceptor–three-donor versus a two-acceptor–three-donor system in 2. The main discrepancies of 1 and 2, monitored by the Hirshfeld surface analysis, are related to the contribution portions of O···H/H···O contacts, in which compound 1 not only involves the greater existence of classical hydrogen bonds but also contains the further C–H···O weak interactions in its crystal packing with respect to compound 2. Instead, in 2, the shortage of O···H/H···O contacts has been partially compensated by the C···H/H···C interactions, due to the presence of more unsaturated carbon acceptors. The differences in assemblies are also reflected in the solid-state IR spectra, especially for the N–H vibration frequencies. The new compounds were further studied by 1D NMR experiments (¹H, ¹³C, ³¹P), 2D NMR techniques [HMQC and HMBC (H–C correlation), HSQC (N–H correlation)], high-resolution ESI–MS, EI–MS spectrometry and IR spectroscopy.     

Fe-NMR spectroscopy revisited: ferrocene derivatives containing heterosubstituents

⁵⁷Fe-NMR spectra (16.1 MHz; natural abundance) of mono- and 1,1′-disubstituted ferrocene derivatives (1–15), including two ferrocenophanes (14, 15), could be measured in a much shorter time than expected according to previous reports (500 MHz spectrometer, standard equipment; solutions in 5-mm tubes). Substituents were ^tBu, alkynyl, silyl, germyl, stannyl and boryl groups, and the bridges in the ferrocenophanes were SiMe₂ (14) and Me₂SiSiMe₂ units (15). The δ⁵⁷Fe values cover a fairly large range of >450 ppm. There is a large difference (365 ppm) in ⁵⁷Fe nuclear shielding for the two ferrocenophanes, with the higher ⁵⁷Fe shielding in the [1]ferrocenophane 14. Parallel trends were observed between δ⁵⁷Fe of ferrocene derivatives and δ⁵⁵Mn of comparable sandwich complexes derived from [(η⁵-C₅H₅)Mn(η⁶-C₆H₆)].     

The synergistic cooperation of NH⋯O and CH⋯O hydrogen bonds in the structures of three new phosphoric triamides

The supramolecular assemblies of three new phosphoric triamides, {(C₆H₅CH₂)(CH₃)N}₂(4-CH₃-C₆H₄C(O)NH)P(O) (1), {(C₆H₁₁)(CH₃)N}₂(4-CH₃-C₆H₄C(O)NH)P(O) (2) and {(C₂H₅)₂N}₂(4-CH₃-C₆H₄C(O)NH)P(O) (3) were studied by single crystal X-ray diffraction as well as by Hirshfeld surface analysis. It was found that a synergistic cooperation of NH?O and CH?O hydrogen bonds occurs in all three structures, but forming unique supramolecular architectures individually. Along with the presence of centrosymmetric dimers in 1, 2 and 3, based on a classical NH?O hydrogen bond, the presence of weak CH?O interactions play an additional and vital role in crystal architecture and construction of the final assemblies, collectively identified as a centrosymmetric dimer (0D), a 1-D array and a 3-D network, respectively. These differences in superstructures are related to the effect of aromatic, bulk and flexible groups used in the molecules designed, with a similar C(O)NHP(O) backbone. The NH?O contacts in 1, 2 and 3 are of the “resonance-assisted hydrogen bond” types and also the anti-cooperativity effect can be considered in the multi-acceptor sites P═O in 1 and 2 and C═O in 3. All three compounds were further studied by 1D NMR experiments, 2D NMR techniques (HMQC and HMBC (H–C correlation)), high resolution ESI–MS, EI–MS spectrometry and IR spectroscopy methods.     

The first phosphoramide–mercury(II) complex with a Cl2Hg–OP[N(C)(C)]3 segment

Mercury(II) exhibits a strong preference for linear coordination which has been attributed to relativistic effects splitting the 6p orbitals and promoting sp hybridization. If the two ligands attached to the mercury(II) ion are weak donors, the metal ion can act as a good Lewis acid and expand its coordination number. Moreover, mercury has a special affinity for softer bases, such as S and N atoms, and has much less affinity for hard bases, such as those including an O atom. The asymmetric unit of dichlorido[tris(piperidin‐1‐yl)phosphane oxide‐κO]mercury(II)–dichloridomercury(II) (2/1), [HgCl₂{(C₅H₁₀N)₃PO}]₂·[HgCl₂], is composed of one HgCl₂{(C₅H₁₀N)₃PO} complex and one half of a discrete HgCl₂ entity located on an inversion centre. The coordination environment around the Hg^II centre in the complex component is a distorted T‐shape. Bond‐valence‐sum calculations confirm the three‐coordination mode of the Hg^II atom of the complex molecule. The noncovalent nature of the Hg…Cl and Hg…O interactions in the structure are discussed.     

Dwell fatigue crack nucleation model based on crystal plasticity finite element simulations of polycrystalline titanium alloys

In this paper a crystal plasticity-based crack nucleation model is developed for polycrystalline microstructures undergoing cyclic dwell loading. The fatigue crack nucleation model is developed for dual-phase titanium alloys admitting room temperature creep phenomenon. It is a non-local model that accounts for the cumulative effect of slip on multiple slip systems, and involves evolving mixed-mode stresses in the grain along with dislocation pileups in contiguous grains. Rate dependent, highly anisotropic behavior causes significant localized stress concentration that increases with loading cycles. The crystal plasticity finite element (CPFE) model uses rate and size-dependent anisotropic elasto-crystal plasticity constitutive model to account for these effects. Stress rise in the hard grain is a consequence of time-dependent load shedding in adjacent soft grains, and is the main cause of crack nucleation in the polycrystalline titanium microstructure. CPFE simulation results are post-processed to provide inputs to the crack nucleation model. The nucleation model is calibrated and satisfactorily validated using data available from acoustic microscopy experiments for monitoring crack evolution in dwell fatigue experiments.