An in situ time-dependent study of the photodimerisation of chloro-derivatives of trans-cinnamic acid using infrared microspectroscopy with a synchrotron radiation source

The photodimerisation of single crystals of substituted cinnamic acid has been monitored continuously by infrared microscopy using a synchrotron source. The beta-form of 2,4-dichloro-trans-cinnamic acid dimerises under ultraviolet irradiation to form the corresponding beta-truxinic acid derivative in a reaction which follows strictly first order kinetics. By contrast the corresponding reactions in single crystals of beta-2-chloro-trans-cinnamic acid and beta-4-chloro-trans-cinnamic acid deviate somewhat from first order kinetics as a result of solid-state effects. In all three cases the reactions proceed smoothly from monomer to dimer with no hint of any reaction intermediate.

On the time-dependent analysis of Gamow decay

Gamow's explanation of the exponential decay law uses complex 'eigenvalues' and exponentially growing 'eigenfunctions'. This raises the question, how Gamow's description fits into the quantum mechanical description of nature, which is based on real eigenvalues and square integrable wavefunctions. Observing that the time evolution of any wavefunction is given by its expansion in generalized eigenfunctions, we shall answer this question in the most straightforward manner, which at the same time is accessible to graduate students and specialists. Moreover, the presentation can well be used in physics lectures to students.

The cyclical behaviour of sector and regional diversification benefits: 1987-2002

This paper investigates the time series behaviour of the relative benefits of sector and regional diversification strategies, using the notion of cross-sectional dispersion introduced by Solnik and Roulet (2000). Using monthly data over the period 1987:1 to 2002:12, four sector and four regional classifications are examined in the UK. The results indicate that sector and regional dispersion indices are highly time varying and so dwarf any lower frequency cyclical components that may be present. Nonetheless, periods of high dispersion are closely followed by periods of low dispersion, suggestive of cyclical behaviour of sector and regional diversification benefits. Then, using the HP-filter we isolated the cyclical component of the various dispersion indices and found that the sector dispersion indices are generally above the regional dispersion indices. This implies that a sector diversification strategy is likely to offer greater risk reduction benefits than a regional diversification approach. Nonetheless, we find that in some periods, certain regional diversification strategies are of equal or greater benefit than certain sector approaches. The results also appear to be quite sensitive to the classifications of sectors and regions. Hence, the appropriate definition of sectors and regions can have important implications for sector and regional diversification strategies.

Small- and waiting-time behaviour of the thin-film-equation

We consider the small-time behavior of interfaces of zero contact angle solutions to the thin-film equation. For a certain class of initial data, through asymptotic analyses, we deduce a wide variety of behavior for the free boundary point. These are supported by extensive numerical simulations. © 2007 Society for Industrial and Applied Mathematics

The effect of vessel diameter on time dependent gas hold-up variations in highly viscous impeller agitated liquids

Time dependent gas hold-up generated in the 0.3 and 0.6 m diameter vessels using high viscosity castor oil and carboxy methyl cellulose (CMC) solution was compared on the basis of impeller speed (N) and gas velocity (V-G). Two types of hold-up were distinguished-the hold-up due to tiny bubbles (epsilon(ft)) and total hold-up (epsilon(f)), which included large and tiny bubbles. It was noted that vessel diameter (i.e. the scale of operation) significantly influences (i) the trends and the values of epsilon(f) and epsilon(ft), and (ii) the values of tau (a constant reflecting the time dependency of hold-up). The results showed that a scale independent correlation for gas hold-up of the form epsilon(f) or epsilon(ft) = A(N or P-G/V)(a) (V-G)(b), where "a" and "b" are positive constants is not appropriate for viscous liquids. This warrants further investigations into the effect of vessel diameter on gas hold-up in impeller agitated high viscosity liquids (mu or mu(a) > 0.4 Pa s). (C) 2003 Elsevier B.V. All rights reserved.

Solvent-dependent modulation of metal–metal electronic interactions in a dinuclear cyanoruthenate complex: a detailed electrochemical, spectroscopic and computational study

The dinuclear complex [{Ru(CN)4}2(μ-bppz)]4− shows a strongly solvent-dependent metal–metal electronic interaction which allows the mixed-valence state to be switched from class 2 to class 3 by changing solvent from water to CH2Cl2. In CH2Cl2 the separation between the successive Ru(II)/Ru(III) redox couples is 350 mVand the IVCT band (from the UV/Vis/NIR spectroelectrochemistry) is characteristic of a borderline class II/III or class III mixed valence state. In water, the redox separation is only 110 mVand the much broader IVCT transition is characteristic of a class II mixed-valence state. This is consistent with the observation that raising and lowering the energy of the d(π) orbitals in CH2Cl2 or water, respectively, will decrease or increase the energy gap to the LUMO of the bppz bridging ligand, which provides the delocalisation pathway via electron-transfer. IR spectroelectrochemistry could only be carried out successfully in CH2Cl2 and revealed class III mixed-valence behaviour on the fast IR timescale. In contrast to this, time-resolved IR spectroscopy showed that the MLCTexcited state, which is formulated as RuIII(bppz˙−)RuII and can therefore be considered as a mixed-valence Ru(II)/Ru(III) complex with an intermediate bridging radical anion ligand, is localised on the IR timescale with spectroscopically distinct Ru(II) and Ru(III) termini. This is because the necessary electron-transfer via the bppz ligand is more difficult because of the additional electron on bppz˙− which raises the orbital through which electron exchange occurs in energy. DFT calculations reproduce the electronic spectra of the complex in all three Ru(II)/Ru(II), Ru(II)/Ru(III) and Ru(III)/Ru(III) calculations in both water and CH2Cl2 well as long as an explicit allowance is made for the presence of water molecules hydrogen-bonded to the cyanides in the model used. They also reproduce the excited-state IR spectra of both [Ru(CN)4(μ-bppz)]2– and [{Ru(CN)4}2(μ-bppz)]4− very well in both solvents. The reorganization of the water solvent shell indicates a possible dynamical reason for the longer life time of the triplet state in water compared to CH2Cl2.

Time-dependent gel to gel transformation of a peptide based supramolecular gelator

A dipeptide with a long fatty acid chain at its N-terminus gives hydrogels in phosphate buffer in the pH range 7.0–8.5. The hydrogel with a gelator concentration of 0.45% (w/v) at pH 7.46 (physiological pH) provides a very good platform to study dynamic changes within a supramolecular framework as it exhibits remarkable change in its appearance with time. Interestingly, the first formed transparent hydrogel gradually transforms into a turbid gel within 2 days. These two forms of the hydrogel have been thoroughly investigated by using small angle X-ray scattering (SAXS), powder X-ray diffraction (PXRD), field emission scanning electron microscopic (FE-SEM) and high-resolution transmission electron microscopic (HR-TEM) imaging, FT-IR and rheometric analyses. The SAXS and low angle PXRD studies substantiate different packing arrangements for the gelator molecules for these two different gel states (the freshly prepared and the aged hydrogel). Moreover, rheological studies of these two gels reveal that the aged gel is stiffer than the freshly prepared gel.

Boundary value problems for third-order linear PDEs in time-dependent domains

We present the extension of a methodology to solve moving boundary value problems from the second-order case to the case of the third-order linear evolution PDE qt + qxxx = 0. This extension is the crucial step needed to generalize this methodology to PDEs of arbitrary order. The methodology is based on the derivation of inversion formulae for a class of integral transforms that generalize the Fourier transform and on the analysis of the global relation associated with the PDE. The study of this relation and its inversion using the appropriate generalized transform are the main elements of the proof of our results.

The Klein-Gordon equation in a domain with time-dependent boundary

We solve a Dirichlet boundary value problem for the Klein–Gordon equation posed in a time-dependent domain. Our approach is based on a general transform method for solving boundary value problems for linear and integrable nonlinear PDE in two variables. Our results consist of the inversion formula for a generalized Fourier transform, and of the application of this generalized transform to the solution of the boundary value problem.