The finite element approximation of a coupled reaction-diffusion problem with non-Lipschitz nonlinearities

Summary. A coupled semilinear elliptic problem modelling an irreversible, isothermal chemical reaction is introduced, and discretised using the usual piecewise linear Galerkin finite element approximation. An interesting feature of the problem is that a reaction order of less than one gives rise to a "dead core" region. Initially, one reactant is assumed to be acting as a catalyst and is kept constant. It is shown that error bounds previously obtained for a scheme involving numerical integration can be improved upon by considering a quadratic regularisation of the nonlinear term. This technique is then applied to the full coupled problem, and optimal and error bounds are proved in the absence of quadrature. For a scheme involving numerical integration, bounds similar to those obtained for the catalyst problem are shown to hold. Received May 25, 1993 / Revised version received July 5, 1994     

Development of Molecular Mechanics Torsion Parameters for alpha,beta-Cyclopropyl Ketones and Conformational Analysis of Bicyclo[m.1.0]alkan-2-ones

Conformations of cyclopropyl methyl ketone have been studied using ab initio methods in an effort to quantify the effects of conjugative overlap between the cyclopropane ring and an adjacent ketone carbonyl. Results were comparable with previous experimental and theoretical studies. Cyclopropyl methyl ketone exhibits a global energy minimum in the s-cis conformer and a local energy minimum near the s-trans conformer. The potential energy curve obtained was used to derive torsion parameters which were employed in molecular mechanics studies of the conformations of the set of bicyclo[m.1.0]alkan-2-ones having larger ring sizes from five- to 16-membered. Similar conformations for the cyclopropyl ketone substructure are observed for all the medium and large ring systems examined. Possible synthetic ramifications of local conformational anchoring by this functional group array are discussed.     

Shape-tunable polymer nodules grown from liposomes via ring-opening metathesis polymerization

A new inisurf (acting as surfactant and initiator) molecule for ring-opening metathesis polymerization (ROMP) was synthesized and used in aqueous solution in order to control the size and shape of polymer nodules grown from liposomes. Nodules were observed to grow in size with conversion of monomer, and depending on the monomer used, they adopted either a spherical or comet-like shape. Here, we investigate polymer production from a liposome surface. We use a hydrophobic derivative of the Grubbs catalyst positioned at the liposome surface to allow for ROMP of monomers dissolved in the aqueous outer phase. We obtain nodules of polymer that can grow up to tens of micrometers, unveiling new efficient possibilities of polymerization from a membrane in an aqueous solution.     

Matrix-assisted laser desorption/ionisation mass spectrometry for the direct analysis of enzymatically digested kappa- iota- and hybrid iota/nu-carrageenans

Enzymatically digested oligosaccharides of kappa-, iota- and hybrid iota/nu-carrageenans were analysed using matrix-assisted laser desorption/ionisation time-of-flight mass spectrometry in the negative-ion mode. nor-Harmane was used as matrix. Depending on the stock concentration and the laser intensity applied, the oligosaccharides exhibited losses of sulphate units (neutralised by the Na+ ion, and thus non-stable), leaving the primary backbone structure in most cases with only the deprotonated sulphate groups (carrying the negative charge, stable). This meant that kappa- and iota-oligosaccharides could not be easily distinguished from one another since they share the same primary backbone structure. However, for the hybrid iota/nu-oligosaccharides the primary backbone structure could be identified since the nu-carrageenan repeating unit differs from that of the kappa/iota-carrageenan unit. For all types of oligosaccharides, the results indicated cleavage of an anhydrogalactose unit from the non-reducing end. Specifically, for the hybrid oligosaccharides of iota/nu-carrageenans, this type of fragmentation means that the nu-carrageenan unit is not positioned on the non-reducing end of the hybrid oligosaccharides. Dehydration reactions, and exchange reactions of Na+ with K+ and Ca2+, were also observed.     

Interdependence of redox state,hydrogen bonding,anion recognition and charge partition in crystals of (EDT-TTF-CONHMe)6 [Re6Se8(CN)6] (CH3CN)2(CH2Cl2)2

Neutral pi-conjugated molecules and their radical cations co-exist in [(EDT-TTF-CONHMe+*)4(EDT-TTF-CONHMe0)2] [Re6Se8(CN)6]4- (CH3CN)2(CH2Cl2)2 whose crystal structure reveals that, upon one-electron oxidation, an activation of the N-H and C-H hydrogen bond donor ability is coupled to a deactivation of the hydrogen bond acceptor character of the carbonyl oxygen atom: this is expressed in the supramolecular hydrogen bond pattern and, ultimately, into charge localisation and partition in the solid state.     

Mechanism of the aromatic hydroxylation of thiophene by acid-catalyzed peracid oxidation

The oxidation of thiophene (1) with peracids in a strongly acidic environment yielded thiophen-2-one (4) as the product of an apparent direct hydroxylation of the thiophene aromatic ring together with the anticipated thiophene-S-oxide dimers, 2a,b, as the main products. Formation of the latter dimers can be rationalized in a straightforward manner by initial oxidation at the sulfur atom of thiophene (1) to yield thiophene-S-oxide followed by subsequent dimerization in a Diels-Alder type reaction. Trapping experiments in the presence of a competing dienophile indicated that thiophen-2-one (4) did not originate from the monomeric thiophene-S-oxide but was the product of an independent reaction pathway. The extent of thiophen-2-one (4) formation correlated with the acidity of the reaction medium and was suppressed in the presence of water, the latter presumably acting as a competing base. As evidenced by the use of 2,5-dideuterated thiophene (1-D), its mechanism of formation involved a 1,2-hydride shift, a feature commonly described in the peracid-mediated epoxidation of aromatic hydrocarbons and indicative for the occurrence of cationic intermediates. In agreement with all these observations we propose a mechanism involving initial protonation of thiophene followed by nucleophilic attack of the peracid in position 2 of the thiophene ring. Intramolecular epoxidation may lead to the formation of thiophene 2,3-epoxide as a highly reactive intermediate that then undergoes heterolytic ring opening and a 1,2-hydride shift to yield thiophen-2-one (4) after a final, acid-catalyzed, isomerization of the double bond.     

Two-step upflow anaerobic sludge bed system for sewage treatment under subtropical conditions with posttreatment in waste stabilization ponds

A pilot-scale sewage treatment system consisting of two upflow anaerobic sludge bed (UASB) reactors followed by five waste stabilization ponds (WSPs) in series was studied under subtropical conditions. The first UASB reactor started up in only 1 mo (stable operation, high chemical oxygen demand [COD] removal efficiency, low volatile fatty acids concentration in the effluent, alkalinity ratio above 0.7, biogas production above 0.1 Nm3/kg of CODremoved). Removal efficiencies up to 90% were obtained in the anaerobic steps at a hydraulic retention time of 6 + 4 h (80% removal in the first step). Fecal coliform removal in the whole system was 99.9999% (99.94% in anaerobic steps and 99.98% in WSPs). COD balances over UASB reactors are provided. A minimum set of data necessary to build COD balances is proposed. Intermittent sludge washout was detected in the reactors with the COD balances. Sludge washout from single-step UASB reactors should be monitored and minimized in order to ensure constant compliance with discharge standards, especially when no posttreatment is provided. The system combined high COD and fecal coliform removal efficiency with an extremely low effluent concentration, complying with discharge standards, and making it an attractive option for sewage treatment in subtropical regions.     

Continuous flow in open microfluidics using controlled evaporation

This paper presents a method for programming the flow rate of liquids inside open microfluidic networks (MFNs). A MFN comprises a number of independent flow paths, each of which starts with an open filling port, has a sealed microchannel in which assays can be performed, and an open capillary pump (CP). The MFN is placed over Peltier elements and its flow paths initially fill owing to capillary forces when liquids are added to the filling ports. A cooling Peltier element underneath the filling ports dynamically prevents evaporation in all filling ports using the ambient temperature and relative humidity as inputs. Another Peltier element underneath the CPs heats the pumps thereby inducing evaporation in the CPs and setting the flow rate in the microchannels. This method achieves flow rates in the microchannels ranging from approximately 1.2 nL s(-1) to approximately 30 pL s(-1), and is able to keep 90% of a 0.6 microL solution placed in an open filling port for 60 min. This simple and efficient method should be applicable to numerous assays or chemical reactions that require small and precise flow of liquids and reagents inside microfluidics.     

Palladium(II)- and mercury(II)-catalyzed rearrangements of propargyl acetates

The scope and utility of the metal-catalyzed rearrangement of propargyl acetates first reported by Rautenstrauch were expanded. Treatment of a series of appropriate acetate substrates with Pd(II)- and Hg(II)-catalysts afforded synthetically useful fused 5,6-bicyclic-1,4-cyclopentadienyl acetates and 2-cyclopentenones. It was found that the substituents at the terminal alkynyl and alkenyl positions of the acetate substrate had a significant impact on the outcome of the reaction.