A computational investigation of the finite-time blow-up of the 3D incompressible Euler equations based on the Voigt regularization

We report the results of a computational investigation of two blow-up criteria for the 3D incompressible Euler equations. One criterion was proven in a previous work, and a related criterion is proved here. These criteria are based on an inviscid regularization of the Euler equations known as the 3D Euler–Voigt equations, which are known to be globally well-posed. Moreover, simulations of the 3D Euler–Voigt equations also require less resolution than simulations of the 3D Euler equations for fixed values of the regularization parameter \(\alpha >0\). Therefore, the new blow-up criteria allow one to gain information about possible singularity formation in the 3D Euler equations indirectly, namely by simulating the better-behaved 3D Euler–Voigt equations. The new criteria are only known to be sufficient criterion for blow-up. Therefore, to test the robustness of the inviscid-regularization approach, we also investigate analogous criteria for blow-up of the 1D Burgers equation, where blow-up is well known to occur.     

The effect of two distinct fast time scales in the rotating,stratified Boussinesq equations: variations from quasi-geostrophy

Inspired by the use of fast singular limits in time-parallel numerical methods for a single fast frequency, we consider the limiting, nonlinear dynamics for a system of partial differential equations when two fast, distinct time scales are present. First-order slow equations are derived via the method of multiple time scales when the two small parameters are related by a rational power. We find that the resultant system depends only on the relationship of the two fast time scales, i.e. which fast time is fastest? Using the theory of cancellation of fast oscillations, we show that with the appropriate assumptions on the nonlinear operator of the full system, this reduced slow system is exactly that which the solution will converge to if each asymptotic limit is considered sequentially. The same result is also obtained via the method of renormalization. The specific example of the rotating, stratified Boussinesq equations is explored in detail, indicating that the most common distinguished limit of this system—quasi-geostrophy, is not the only limiting asymptotic system.     

Effect of crystallographic orientation on the anodic formation of nanoscale pores/tubes in TiO2 films

Self-organized nanopores and nanotubes have been produced in thin films of titanium (Ti) prepared using filtered cathodic vacuum arc (FCVA), DC- and RF-sputter deposition systems. The anodization process was performed using a neutral electrolyte containing fluoride ions with an applied potential between 2 and 20 V (for clarity the results are only presented for 5 V). Scanning electron microscopy (SEM), atomic force microscopy (AFM) and X-ray diffraction (XRD) techniques were used to characterise the films. It was found that the crystallographic orientation of the Ti films played a significant role in determining whether pores or tubes were formed during the anodic etching process.     

Photochemical functionalization of gallium nitride thin films with molecular and biomolecular layers

We demonstrate that photochemical functionalization can be used to functionalize and photopattern the surface of gallium nitride crystalline thin films with well-defined molecular and biomolecular layers. GaN(0001) surfaces exposed to a hydrogen plasma will react with organic molecules bearing an alkene (C=C) group when illuminated with 254 nm light. Using a bifunctional molecule with an alkene group at one end and a protected amine group at the other, this process can be used to link the alkene group to the surface, leaving the protected amine exposed. Using a simple contact mask, we demonstrate the ability to directly pattern the spatial distribution of these protected amine groups on the surface with a lateral resolution of <12 mum. After deprotection of the amines, single-stranded DNA oligonucleotides were linked to the surface using a bifunctional cross-linker. Measurements using fluorescently labeled complementary and noncomplementary sequences show that the DNA-modified GaN surfaces exhibit excellent selectivity, while repeated cycles of hybridization and denaturation in urea show good stability. These results demonstrate that photochemical functionalization can be used as an attractive starting point for interfacing molecular and biomolecular systems with GaN and other compound semiconductors.     

Formation of silver nanoparticles at the air-water interface mediated by a monolayer of functionalized hyperbranched molecules

Nanofibrillar micellar structures formed by the amphiphilic hyperbranched molecules within a Langmuir monolayer were utilized as matter for silver nanoparticle formation from the ion-containing water subphase. We observed that silver nanoparticles were formed within the multifunctional amphiphilic hyperbranched molecules. The diameter of nanoparticles varied from 2-4 nm and was controlled by the core dimensions and the interfibrillar free surface area. Furthermore, upon addition of potassium nitrate to the subphase, the Langmuir monolayer templated the nanoparticles' formation along the nanofibrillar structures. The suggested mechanism of nanoparticle formation involves the oxidation of primary amino groups by silver catalysis facilitated by "caging" of silver ions within surface areas dominated by multibranched cores. This system provides an example of a one-step process in which hyperbranched molecules with outer alkyl tails and compressed amine-hydroxyl cores mediated the formation of stable nanoparticles placed along/among/beneath the nanofibrillar micelles.     

1H, 13C, and 73Ge NMR spectral analysis of substituted aryltrimethylgermanes

NMR chemical shifts of 1H, 13C, and 73Ge are reported for a series of monosubstituted aromatic trimethylgermanes of the type XC6H4Ge(CH3)3; X = p-N(CH3)2, p-OCH3, p-OC2H5, p-C(CH3)3, p-Si(CH3)3, p-Ge(CH3)3, p-Sn(CH3)3, p-CH3, m-CH3, -H, m-OCH3, p-Cl, p-Br, m-F, m-CF3, p-CF3, o-OCH3, and o-CH3. The relatively narrow 73Ge resonances show a strong correlation with Hammett sigma constants, with a correlation coefficient of 0.976 and 0.876 for 73Ge chemical shifts in meta- and para-substituted derivatives, respectively. The 13C chemical shifts of the methyl carbons bonded to germanium also display a relationship, with correlation coefficients of 0.904, 0.993, and 0.911 for para-, meta- and all derivatives, respectively. Comparisons of the Hammett plots for the homologous series XC6H4M(CH3)3; M = C, Si, Ge, Sn, show that, in general, correlation coefficients decrease while slopes increase significantly down the group, presumably reflecting the corresponding increase in chemical shift range of the group 14 atom. The Hammett constant derived for the p-Ge(CH3)3 group of +0.13 compares with the NMR-derived constants of -0.12 for p-C(CH3)3, +0.14 for p-Si(CH3)3, and -0.14 for p-Sn(CH3)3. The indication of electron release by carbon and tin can be rationalized through traditional hyperconjugative arguments for carbon and by the low electronegativity and consequent inductive effect of tin. The small electron attraction suggested by the positive constants for silicon and germanium can be simply, and perhaps naively, attributed to pi-acceptor interactions with the benzene ring.     

The near edge structure of cubic boron nitride

We compare the near edge structure (NES) of cubic boron nitride (cBN) measured using both electron energy loss spectroscopy (EELS) and X-ray absorption spectroscopy (XAS) with that calculated using three commonly used theoretical approaches. The boron and nitrogen K-edges collected using EELS and XAS from cBN powder were found to be nearly identical. These experimental edges were compared to calculations obtained using an all-electron density functional theory code (WIEN2k), a pseudopotential density functional theory code (CASTEP) and a multiple scattering code (FEFF). All three codes were found to reproduce the major features in the NES for both ionisation edges when a core-hole was included in the calculations. A partial core hole (1/2 of a 1s electron) was found to be essential for correctly reproducing features near the edge threshold in the nitrogen K-edge and to correctly obtain the positions of all main peaks. CASTEP and WIEN2k were found to give almost identical results. These codes were also found to produce NES which most closely matched experiment based on χ² calculations used to qualitatively compare theory and experiment. This work demonstrated that a combined experimental and theoretical approach to the study of NES is a powerful way of investigating bonding and electronic structure in boron nitride and related materials.     

Random benzotrithiophene-based donor-acceptor copolymers for efficient organic photovoltaic devices

A series of benzotrithiophene-containing random terpolymers for polymer solar cells is reported. Through variations of the two other components in the terpolymers, the absorption profile and the frontier energy levels are optimized and maximum power conversion efficiencies are nearly doubled (5.14%) relative to the parent alternating copolymer.     

Synthesis of novel thieno[3,2-b]thienobis(silolothiophene) based low bandgap polymers for organic photovoltaics

Thieno[3,2-b]thienobis(silolothiophene), a new electron rich hexacyclic monomer has been synthesized and incorporated into three novel donor-acceptor low-bandgap polymers. By carefully choosing the acceptor co-monomer, the energy levels of the polymers could be modulated and high power conversion efficiencies of 5.52% were reached in OPV devices.