The Rellich-Kondrachov theorem for unbounded domains

Summary The full Kondrachov compactness theorem for Sobolev imbeddings of the type W ₀ ^m,p (G) W ₀ ^j,r (G) on bounded domains G in R ⁿ is extended to a large class of unbounded domains with reasonable n- 1 dimensional boundaries. A Poincaré inequality is obtained for such domains and a compactness theorem for traces of functions in W ₀ ^m,p (G) on lower dimensional hyperplanes is also proved.     

Direct simulation of breakdown to turbulence following oblique instability waves in a supersonic boundary layer

The late stages of transition to turbulence in a Mach two boundary layer are investigated by direct numerical simulation of the compressible Navier-Stokes equations. The primary instability at this Mach number consists of oblique waves, which are known to form a pattern of quasi-streamwise vortices. It is found that breakdown does not follow immediately from these vortices, which decay in intensity. The generation of new vortices is observed by following the evolution of the pressure and vorticity in the simulation, and analysed by consideration of vorticity stretching. It is found that the slight inclined and skewed nature of the quasi-streamwise vortices leads to a production of oppositely signed streamwise vorticity, which serves as a strong localised forcing of the shear layer alongside the original vortices, formed by convection and stretching of spanwise vorticity. The shear layer rolls up into many new vortices, and is followed by a sharp increase in the energy of higher frequencies and in the skin friction.     

Ionic effects on the behavior of thermoresponsive PEO–PNIPAAm block copolymers

The temperature‐dependent aggregation and recovery of the copolymer poly(ethylene oxide)₂₂‐b‐poly(N‐isopropylacrylamide)₂₉ with a C₁₂ end‐cap in aqueous solutions of salts and acids are investigated. Salt solutions affected the critical aggregation temperature of the copolymer in a manner predictable according to the Hofmeister series, with the kosmotropic adipic ion lowering the critical aggregation temperature and the chaotropic iodide raising it. Also, both salts and acids increased the size of copolymer aggregates formed with heating, due to the electrostatic shielding of aggregated structures provided by the electrolytes. Additionally, the presence of ionic additives caused a thermohysteretic increase in the size of copolymer aggregates with temperature cycling. The transitions of polymer structure with increasing temperature were surprisingly sharp with the C₁₂ end‐cap present, and particularly broad in samples in which the end cap had been cleaved. This observation suggested that the hydrophobic end group was responsible for imparting some degree of order to the polymer at low temperatures, which allowed for rapid reconfiguration with increasing temperature. Finally, in addition to the transitions expected from the least critical solution temperature behavior of the polymer blocks, we have observed an unexpected additional transition which we attribute to the contraction of the poly(ethylene oxide) chains of the copolymer aggregates at higher temperatures. This work illustrates the importance of considering the environment and composition of thermoresponsive block copolymers in certain applications, particularly in solutions with even modest electrolyte concentrations (1–10 mM), as it can have a profound effect on transition temperatures and morphology. © 2014 Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2014 , 52, 507–516     

Multiple C−H Bond Activations in Corannulene by a Dirhenium Complex

The reaction of Re₂(CO)₈(μ-C₆H₅)(μ-H), 1 with corannulene (C₂₀H₁₀) yielded the product Re₂(CO)₈(μ-H)(μ-η²-1,2-C₂₀H₉), 2 (65 % yield) containing a Re₂ metalated corannulene ligand formed by loss of benzene from 1 and the activation of one of the CH bonds of the nonplanar corannulene molecule by an oxidative-addition to 1 . The corannulenyl ligand has adopted a bridging η²-σ+π coordination to the Re₂(CO)₈ grouping. Compound 2 reacts with a second equivalent of 1 to yield three isomeric doubly metalated corannulene products: Re₂(CO)₈(μ-H)(μ-η²-1,2-μ-η²-10,11-C₂₀H₈)Re₂(CO)₈(μ-H), 3 (35 % yield), Re₂(CO)₈(μ-H)(μ-η²-2,1-μ-η²-10,11-C₂₀H₈)Re₂(CO)₈(μ-H), 4 (12 % yield), and Re₂(CO)₈(μ-H)(μ-η²-1,2-μ-η²-11,10-C₂₀H₈)Re₂(CO)₈(μ-H), 5 (12 % yield), by a second CH activation on a second rim double bond on the corannulene molecule. The isomers differ by the relative orientations of the coordinated Re₂(CO)₈(μ-H) groupings. All new products were characterized structurally by single crystal X-ray diffraction analysis.     

Role of radicals in UV‐initiated postplasma grafting of poly‐ε‐caprolactone: An electron paramagnetic resonance study

Electron paramagnetic resonance (EPR) measurements were performed on poly‐ε‐caprolactone (PCL) films at different stages of the postplasma‐grafting process. PCL films prepared by solvent casting (SC) or electrospinning (ESP) yield very similar EPR spectra after Ar‐plasma treatment and subsequent exposure to air, but the EPR signal is much stronger in the PCL‐ESP films. The free radicals appear to be mainly, and possibly exclusively, oxygen centered. The radicals generated by UV irradiation in PCL‐ESP films were studied in situ with EPR, using a UV‐LED (λ = (285 ± 5) nm). Their EPR spectrum is distinctly different from the plasma‐induced signal, indicative of carbon‐centered radicals, and appears to be independent of the plasma pretreatment. UV‐induced homolytic splitting of (hydro)peroxide bonds was not observed. Both the plasma‐ and UV‐induced radicals decay at room temperature (RT), even in an inert atmosphere. This study demonstrates the potential of electrospun films and UV‐LEDs for the study of plasma‐ and UV‐generated free radicals with EPR in polyesters, and raises questions with respect to the validity of some generally accepted molecular mechanisms underpinning the postplasma grafting technique for polyesters. © 2012 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2012     

The structure of poly-2,5-benzoxazole (ABPBO) and poly-2,6-benzothiazole (ABPBT) fibers by X-ray diffraction

The determination of the crystalline structure of oriented fibers of poly-2,5-benzoxazole (ABPBO) and poly-2,6-ben-zothiazole (ABPBT) is described. Both unit cells are metrically orthorhombic, with the parameters: a = 6.061 (17), b = 3.384 (13), c (fiber axis) = 11.575 (6) Å for ABPBO; and a = 6.044 (6), b = 3.417 (7), c (fiber axis) = 12.194 (18) Å for ABPBT. The fiber repeat consists in each structure of two fused ring groups arranged in a planar, zigzag conformation. The conformational torsion angle and orientation of chains within the unit cells are derived from a linked-atom least squares refinement technique. Polymer chains pack laterally through van der Waals interactions. A plausible disorder model which involves defects in chain direction is presented. Refinement of a static disorder model for ABPBO in which 50% of the chains have their chain directions reversed leads to a lower R residual and sum of constraints.