On the behavior of solutions to the Neumann problem in unbounded domains

The paper deals with the uniformly elliptic equation (a^ij(x)u_xi)_xj=f(x) in an unbounded domain Ω⊂ℝⁿ and its solution u(x) that satisfies the homogeneous Neumann condition. The function f has a compact support. The domain Ω has the following structure: assume that {r_m} is an increasing sequence of positive numbers, h_m=r_m+1−r_m, and the ratio h_m+1/h_m lies between positive constants C₁ and C₂. The intersection of Ω with the spherical layer between the spheres of radius r_m and r_m+1 with center at the origin satisfies a certain inequality of isoperimetric type. It is shown in this paper that the set of solutions splits up into three classes: (i) the solutions for which and ; moreover, it is shown that these limits are attained with nearly the same speed (if C₁/C₂=1, then the speed is not less than the exponential one); (ii) the solutions for each of which a constant C exists such that and u(x)−C changes its sign for large |x|; here, the convergence to C is rapid (for C₁/C₂=1 this convergence is not slower than the exponential one); (iii) the solutions that do not change their sign for large |x| and increase or decrease to +∞ or −∞, respectively, with low speed (for C₁/C₂=1 with a linear speed) (one exception is possible here: a slow convergence to a constant). Bibliography: 7 titles. Dedicated to O.A. Oleinik This work was supported by the Soros International Science Foundation. Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 19. pp. 000-000, 0000.     

Thermotropic phase transition in soluble nanoscale lipid bilayers

The role of lipid domain size and protein-lipid interfaces in the thermotropic phase transition of dipalmitoylphosphatidylcholine (DPPC) and dimyristoylphosphatidylcholine (DMPC) bilayers in Nanodiscs was studied using small-angle X-ray scattering (SAXS), differential scanning calorimetry (DSC), and generalized polarization (GP) of the lipophilic probe Laurdan. Nanodiscs are water-soluble, monodisperse, self-assembled lipid bilayers encompassed by a helical membrane scaffold protein (MSP). MSPs of different lengths were used to define the diameter of the Nanodisc lipid bilayer from 76 to 108 A and the number of DPPC molecules from 164 to 335 per discoidal structure. In Nanodiscs of all sizes, the phase transitions were broader and shifted to higher temperatures relative to those observed in vesicle preparations. The size dependences of the transition enthalpies and structural parameters of Nanodiscs reveal the presence of a boundary lipid layer in contact with the scaffold protein encircling the perimeter of the disc. The thickness of this annular layer was estimated to be approximately 15 A, or two lipid molecules. SAXS was used to measure the lateral thermal expansion of Nanodiscs, and a steep decrease of bilayer thickness during the main lipid phase transition was observed. These results provide the basis for the quantitative understanding of cooperative phase transitions in membrane bilayers in confined geometries at the nanoscale.     

Concomitant length and diameter separation of single-walled carbon nanotubes

Gel electrophoresis and column chromatography conducted on individually dispersed, ultrasonicated single-walled carbon nanotubes yield simultaneous separation by tube length and diameter. Electroelution after electrophoresis is shown to produce highly resolved fractions of nanotubes with average lengths between 92 and 435 nm. Separation by diameter is concomitant with length fractionation, and nanotubes that have been cut shortest also possess the greatest relative enrichments of large-diameter species. Longer sonication time causes increased electrophoretic mobility in the gels; thus, ultrasonic processing determines the degree of both length and diameter separation of the nanotubes. The relative quantum yield decreases nonlinearly as the nanotube length becomes shorter. These techniques constitute a preparative, scalable method for separating nanotubes by two important attributes required for electronic and sensor applications.     

Cationic rhodium hydrogenation catalysts containing chelating diphosphine ligands: effect of chelate ring size

Earlier studies on the [(1,2-bis(diphenylphosphino)ethane)rhodium]^p+-catalyzed hydrogenation of 1-hexene and methyl-(Z)-α-acetamidocinnamate have been extended to catalysts containing larger chelating diphosphine ligands, i.e., Ph₂P(CH₂)_nPPh₂, where n = 3, 4 and 5. Comparisons include measurements of equilibrium constants for the binding of the olefinic substrates to the catalysts and of the catalytic hydrogenenation rates. Some related measurements also are reported for the corresponding catalyst systems containing the chiral ligand, 4R,5R-bis(diphenylphosphinomethyl)-2,2,-dimethyldioxalane (DIOP) and non-chelating PPh₃ ligands.     

Tellurium-mediated cycloaromatization of acyclic enediynes under mild conditions

The cycloaromatization of acyclic enediynes typically requires very high temperatures (>160 degrees C) and dilute conditions to proceed in a synthetically useful yield. These conditions hinder reaction throughput, inhibiting the use of this reaction for the large-scale production of materials. The reaction of sodium telluride with acyclic arenediynes yields the corresponding tellurepine, which under gentle heating extrudes Te degrees to yield the cycloaromatization product. We have developed conditions that form sodium telluride from inexpensive tellurium metal in situ, and that also perform the desilylation of silylated arenediynes in the same process. Under our conditions, we are able to perform desilylation and cycloaromatization at temperatures as low as 40 degrees C and on a scale as large as 5 g in standard laboratory glassware.     

Opposite trends in thermodynamic compatibility between copolyamide and chitosan in their binary blend

Gibbs energy, enthalpy, and entropy of mixing in binary blends of chitosan with ter‐copolyamide 6/66/610 at ambient conditions have been determined over the entire concentration range using thermodynamic cycle based on dissolution of individual polymers and their blends of different composition in a common solvent – formic acid. Experimental procedure included stepwise equilibrium vapor sorption of glacial formic acid on the cast films and isothermal microcalorimetry of dissolution of these films in liquid glacial formic acid at 25 °C. Formic acid appeared to be a very good solvent for individual polymers and their blends. Flory‐Huggins interaction parameter determined from sorption isotherms was negative and varied from ?2.56 to ?1.79 depending upon blend composition. The enthalpies of dissolution of individual polymers and their blends were strongly exothermic and varied from ?200 to ?40 Joule/g. Independent thermodynamic cycles for Gibbs free energy and enthalpy remarkably revealed similar trends in concentration dependence of different thermodynamic functions of mixing between chitosan and copolyamide. At high chitosan content, the binary blend is characterized by large and negative values of Gibbs free energy, enthalpy, and entropy of mixing that provide high polymer compatibility. On the contrary, at high copolyamide content the blends are incompatible and are characterized by positive values of enthalpy, entropy, and Gibbs free energy of mixing. Such complicated thermodynamic behavior is the result of the superposition of strong molecular interactions (H‐bonds) between polymers in the blend and isothermal fusion of copolyamide crystallites. Thermodynamic analysis correlates well with the data obtained by polarized microscopy. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 2603–2613, 2007     

Modeling viscoelastic properties of triblock copolymers: A DPD simulation study

Gel systems based on self‐assembled, amphiphilic ABA triblock copolymers in midblock‐selective solvent form stable, spatially extended networks with controllable morphology and tunable viscoelastic behavior. In this work, we systematically evaluate the mechanical properties of these gels using morphology calculations, and a nonequilibrium oscillatory shear technique based on the dissipative particle dynamics (DPD) method. Our simulations demonstrate that low molecular weight triblock copolymers with incompatible blocks self‐assemble into micelles connected with bridges and loop‐like chains comprised of the solvent‐selective polymer midblocks. The fraction of bridges, ?_b, generally increases with increasing relative volume of the midblock, x, defined as the ratio of midblock and endblock volumes ( ). For our model, ?_b reaches a plateau at approximately x > 9 for a strongly selective solvent. At this limit, the value of ?_b increases from 0.40 to about 0.66 as the copolymer concentration, c, increases from 0.2 to 0.5; however, this increase is less significant at higher concentrations. The elastic response of the gel studied here is comparable with the Rouse modulus. The elastic modulus increases with polymer concentration, and it exhibits a broad peak within 6 < x < 12. Finally, we present an approximate method to predict the elastic modulus of unentangled ABA triblock copolymers based solely on the morphology of the micellar gel, which can be gleaned from equilibrium DPD simulations. We demonstrate that our simulation results are in good qualitative agreement with other theoretical predictions and experimental data. © 2009 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 48: 15–25, 2010     

Critical phenomena in discrete-time interconnection networks

We present theoretical and numerical results for the performance of a multiprocessor network modeled as a ring and as a toroidal square lattice of nodes with local processors that generate messages for output ports/buffers. The output buffers are assumed to have infinite capacity, and the service time is deterministic. Two models are considered. One assumes that every processor generates messages with rate λ per time slot and per output port/buffer. The other model considers that the generation rate of a node depends on the intensity of the flow of arriving messages. Explicit expressions for the distribution of queue lengths, the average number of messages in the buffers, the average latency, and the critical network load depending on the distance between the source and the destination are obtained. Simulation results show excellent agreement with theoretical predictions based on the assumption of independent queues.