Fractal behavior in trapping and reaction: A random walk study

We investigate the trapping of a random walker in fractal structures (Sierpinski gaskets) with randomly distributed traps. The survival probability is determined from the number of distinct sites visited in the trap-free fractals. We show that the short-time behavior and the long-time tails of the survival probability are governed by the spectral dimensiond. We interpolate between these two limits by introducing a scaling law. An extension of the theory, which includes a continuous-time random walk on fractals, is discussed as well as the case of direct trapping. The latter case is shown to be governed by the fractal dimensiond.     

The mean spherical model in a random external field and the replica method

We provide a quick elementary solution of the mean spherical model in a random external field. This also allows an immediate proof of the self-averaging property of the free energy. We calculate the free energy by means of the replica method, i.e., for any (not necessarily integer) replica numbern, and show that when a phase transition occurs the limits andn 0 are not interchangeable.     

Quasi‐random integration in quantum chemistry: Efficiency,stability, and application to the study of small atoms and molecules constrained in spherical boxes

This project consists of two parts. In the first part, a series of test calculations is performed to verify that the integrals involved in the determination of atomic and molecular properties by standard self‐consistent field (SCF) methods can be obtained through Halton, Korobov, or Hammersley quasi‐random integration procedures. Through these calculations, we confirm that all three methods lead to results that meet the levels of precision required for their use in the calculation of properties of small atoms or molecules at least at a Hartree–Fock level. Moreover, we have ensured that the efficiency of quasi‐random integration methods that we have tested is Halton=Korobov>Hammersley?pseudo‐random. We also find that these results are comparable to those yielded by ordinary Monte Carlo (pseudo‐random) integration, with a calculation effort of two orders of smaller magnitude. The second part, which would not have been possible without the integration method previously analyzed, contains a first study of atoms constrained in spherical boxes through SCF calculations with basis functions adapted to the features of the problem: Slater‐type orbitals (STOs) trimmed by multiplying them by a function that yields 1 for 0 < r < (R‐δ), polynomial values for (R‐δ) < r < R and null for r > R, R being the radius of the box and δ a variationally determined interval. As a result, we obtain a equation of state for electrons of small systems, valid just in the limit of low temperatures, but fairly simple. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

On the statistical independence of various column contributions to band broadening. Part 1: Second moment contributions from statistically independent,longitudinal diffusion in both phases

The total length-based second moment contribution from longitudinal sample diffusion in both phases on a column, σD , is derived by adding individual partial differential contributions to a partial differential equation accounting for the longitudinal diffusion processes only. Although each diffusion-dispersed sample part is equilibrated between two phases, the resulting σ,D (= 2D _mt _m + 2D _st _s) can be interpreted as the sum of two independent contributions in accordance with the variance addition rule. (D _m and D _s are the mean diffusion coefficients and t _mand t _s the mean residence times of the sample in the mobile and stationary phases, respectively.) The same σD expression is derived from the random walk model of Giddings by treating the diffusional process in each phase as statistically independent of the other processes. Under these conditions the broadening contribution from longitudinal diffusion in the mobile phase is shown to be independent of the velocity profile.     

Derivation of the transport equation for electrons moving through random impurities

A single (nonrelativistic, spinless) electron subject to a constant external electric field interacts with impurities located on an infinitely extended lattice by a potential of random strength. The random strength is given by a field of Gaussian random variables. We show the existence of the averaged dynamics and prove that in the weak coupling limit, 0, ₂ t= fixed, one obtains the usual transport equation for the velocity distribution.Work supported by a Max Kade Foundation fellowship.On leave of absence of the Fachbereich Physik der Universität München.     

STATISTICS OF A NOVEL CLASS OF LIQUID CRYSTAL POLYMERS

A novel class of the liquid crystal polymers with mesogens laterally attached to flexible main chain backbones has been synthesized recently by us. While the flexible backbones tend to take configuration of maximum entropy the mesogenic side groups tend to orientate because of the anisotropie soft dispersion forces. A mean field theory is presented to describe the competition between these two trends which makes it possible for this class of polymers to show anisotropic (liquid crystal) phase within certain temperature range where the polymer main chains and the side chain mesogens will take approximately parallel arrangement, which is different from the normal side chain liquid crystal polymers. The temperature range of the liquid crystal phase, its ordering and phase transition all depend on the flexibility of the backbone, the strengthof the anisotropic forces of the side groups and the hinge elasticity. The results show that the liquid crystalisotropic phase transition is of the first order. The phase diagram, i. e. the dependence of the transition temperature on the structure of the polymers is also given.     

Effects of processing parameters on the mechanical properties of polypropylene random copolymer

The mechanical properties of polypropylene random copolymer (PP-R) with different processing parameters were studied. Special attention is devoted to the investigation of the influence of masterbatch addition on the variation in the mechanical properties of injection moulded PP-R. Tensile, instrumented Charpy impact, Shore D hardness, differential scanning calorimeter (DSC) and Vicat softening temperature (VST) tests were conducted on the test samples containing different colour masterbatches varying from 0.5 to 10 wt%. The observed changes in the mechanical behaviour are explained by the type and level of masterbatch content. The natural UV weathering performance of the PP-R material was studied from the masterbatch type point of view. The effect of processing parameters on material performance was studied on samples which were directly obtained from extruded pipes and on injection moulded samples.

Finally, the effects of storage time on the polymer properties were investigated.     

Cationic copolymerization of indene with styrene derivatives: Synthesis of random copolymers of indene with high molecular weight

Random copolymers with high molecular weights of indene and p‐methylstyrene (pMeSt) were synthesized by cationic polymerization with trichloroacetic acid/tin tetrachloride in CH₂Cl₂ at low temperatures. When indene and pMeSt (1:1 v/v), for example, were polymerized at ?40 °C, both monomers were consumed at very similar rates to give a copolymer with high molecular weight [number‐average molecular weight (M_n): 8–9 × 10⁴]. This is indeed quite unexpected behavior for the combination of these two monomers because pMeSt polymerized over 1000 times faster than indene in the homopolymerization under the reaction conditions previously described. The product copolymer of indene and pMeSt had a random monomer sequence in it that was confirmed by NMR analyses and thermal‐property measurements. In sharp contrast with pMeSt, styrene and p‐chlorostyrene, which have no electron‐donating groups on the phenyl ring, led to low molecular weight polymers (M_n < 10,000) in the copolymerization with indene (1:1 v/v). © 2002 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 40: 2449–2457, 2002     

Microstructure and ferroelectric properties of r.f. magnetron sputtering derived PZT thin films deposited on interlayer (PbO/TiO2)

Lead zirconate titanate (PZT) thin films were deposited on Pt/Ti/SiO₂/Si and interlayer/Pt/Ti/SiO₂/Si substrate by radio frequency (r.f.) magnetron sputtering with a Pb_1.1Zr_0.53Ti_0.47O₃ target. The crystallization of the PZT thin films was formed only by substrate temperature. When interlayer (PbO/TiO₂) was inserted between the PZT thin film and the Pt electrode, the grain growth and processing temperature of the PZT thin films were considerably improved. Compared to PZT/Pt structure, the dielectric constant and polarization properties of the PZT/interlayer/Pt structure were fairly improved. In particular, PZT/interlayer/Pt at the substrate temperature of 400 °C showed prevalent ferroelectric properties (_r=475.97, tanδ=0.0591, P_r=23 μC/cm²). As a result of an X-ray photoelectron spectroscopy (XPS) depth-profile analysis, it was found that PZT/interlayer/Pt deposited only by substrate temperature without the post-annealing process via r.f. magnetron sputtering method remained independent of each other regardless of substrate temperatures.     

Metallocene ethylene-1-octene copolymers: Influence of comonomer content on thermo-mechanical, rheological, and thermo-oxidative behaviours before and after melt processing in an internal mixer

The influence of the comonomer content in a series of metallocene-based ethylene-1-octene copolymers (m-LLDPE) on thermo-mechanical, rheological, and thermo-oxidative behaviours during melt processing were examined using a range of characterisation techniques. The amount of branching was calculated from ¹³C NMR and studies using differential scanning calorimetry (DSC) and dynamic mechanical analysis (DMA) were employed to determine the effect of short chain branching (SCB, comonomer content) on thermal and mechanical characteristics of the polymer. The effect of melt processing at different temperatures on the thermo-oxidative behaviour of the polymers was investigated by examining the changes in rheological properties, using both melt flow and capillary rheometry, and the evolution of oxidation products during processing using infrared spectroscopy.The results show that the comonomer content and catalyst type greatly affect thermal, mechanical and oxidative behaviour of the polymers. For the metallocene polymer series, it was shown from both DSC and DMA that (i) crystallinity and melting temperatures decreased linearly with comonomer content, (ii) the intensity of the β-transition increased, and (iii) the position of the tan δ_max peak corresponding to the α-transition shifted to lower temperatures, with higher comonomer content. In contrast, a corresponding Ziegler polymer containing the same level of SCB as in one of the m-LLDPE polymers, showed different characteristics due to its more heterogeneous nature: higher elongational viscosity, and a double melting peak with broader intensity that occurred at higher temperature (from DSC endotherm) indicating a much broader short chain branch distribution.The thermo-oxidative behaviour of the polymers after melt processing was similarly influenced by the comonomer content. Rheological characteristics and changes in concentrations of carbonyl and the different unsaturated groups, particularly vinyl, vinylidene and trans-vinylene, during processing of m-LLDPE polymers, showed that polymers with lower levels of SCB gave rise to predominantly crosslinking reactions at all processing temperatures. By contrast, chain scission reactions at higher processing temperatures became more favoured in the higher comonomer-containing polymers. Compared to its metallocene analogue, the Ziegler polymer showed a much higher degree of crosslinking at all temperatures because of the high levels of vinyl unsaturation initially present.